History of Virtual Work Laws: A History of Mechanics Prospective

Birkhauser

Science Networks. Historical Studies Founded by Erwin Hiebert and Hans Wußing Volume 42 Edited by Eberhard Knobloch, Helge Kragh and Erhard Scholz

Editorial Board: K. Andersen, Aarhus D. Buchwald, Pasadena H.J.M. Bos, Utrecht U. Bottazzini, Roma J.Z. Buchwald, Cambridge, Mass. K. Chemla, Paris S.S. Demidov, Moskva E.A. Fellmann, Basel M. Folkerts, München P. Galison, Cambridge, Mass. I. Grattan-Guinness, London

J. Gray, Milton Keynes R. Halleux, Liège S. Hildenbrandt, Bonn Ch. Meinel, Regensburg J. Peiffer, Paris W. Purkert, Bonn D. Rowe, Mainz A.I. Sabra, Cambridge, Mass. Ch. Sasaki, Tokyo R.H. Stuewer, Minneapolis V.P. Vizgin, Moskva

Danilo Capecchi

History of Virtual Work Laws A History of Mechanics Prospective

Birkhauser

Danilo Capecchi Università La Sapienza, Rome (Italy)

ISBN 978-88-470-2055-9 DOI 10.1007/978-88-470-2056-6

ISBN 978-88-470-2056-6 (eBook)

Library of Congress Control Number: 2011941587 Springer Milan Heidelberg New York Dordrecht London © Springer-Verlag Italia 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover design: deblik, Berlin Typesetting with LATEX: PTP-Berlin, Protago TEX-Production GmbH, Germany (www.ptp-berlin.eu) Printing and Binding: Grafiche Porpora, Segrate (Mi) Printed in Italy Springer-Verlag Italia S.r.l., Via Decembrio 28, I-20137 Milano Springer is a part of Springer Science+Business Media (www.springer.com)

Preface

Lagrange, in the Méchanique analitique of 1788, identified three programs of research, or paradigms, in the history of statics: the lever, the composition of forces, and the principle of virtual work. The paradigm of the lever would have been in force from antiquity until the early XVIII century, when Varignon was asserting the parallelogram law for composition and decomposition of forces. The principle of virtual work would have become dominant since the XIX century. This picture is in my opinion quite realistic, although the final predicted by Lagrange was never fully realized because the principle of virtual work has never replaced the rule of the composition of forces, but at most has outflanked it. Also the picture is too schematic. In fact, some form of law of virtual work has always existed in mechanics, always however with limited applications. The law of virtual work, as usually presented in modern textbooks of mechanics, says that there is equilibrium for one or more bodies subjected to a system of forces if and only if the total virtual work is zero for any virtual displacement. In Chapter 2 of this book the meaning of the terms work and virtual is described in some detail; here I will only to mention that, since Lagrange in the second half of the XVIII century, the law of virtual work had no appreciable changes in its formulation. The view on its role in mechanics is instead still varying, passing from the enthusiasm of the XIX century to a modest presence in modern rational mechanics as well as, all considered, in the engineering field, albeit with some important exceptions. The present book starts from the first documented formulations of laws of virtual work. They usually have only a vague analogy to the modern ones and only mathematically. Attention is paid to Arabic and Latin mechanics of the Middle Ages. With the Renaissance there began to appear slightly different wordings of the law, which were often proposed as unique principles of statics. With Bernoulli and Lagrange the process reached its apex. The book ends with some chapters dealing with the discussions that took place in the French school on the role of the Lagrangian law of virtual work and its applications to continuum mechanics. Even though the book takes a particular point of view, it presents an important slice of history of mechanics. Essential reference is made to primary sources; secondary literature is mainly used to frame the contributions of the scientists consid-

vi

Preface

ered in their times. To allow a better understanding of the ideas of the authors studied, English translations are always accompanied by original quotations (Appendix). No pre-conceived historical hypotheses have been explicitly assumed though. The mere existence of the book suggests that I have in mind a continuous chain connecting concepts from antiquity up to now. However the nature of the chain is complex and I leave it to the reader to unveil it. The book is the result of a twenty year study of mechanics and its history and should be of interest to historians of mathematics and physics. It should also arouse interest among engineers who are now perhaps the most important witnesses of classical mechanics, and with it, of the law of virtual work. I want to acknowledge Giuseppe Ruta, Romano Gatto, Antonino Drago for contributing comments and suggestions to specific parts. Cesare Tocci for suggestions regarding the whole book, and finally I want to acknowledge Raffaele Pisano for his reading and the debates we have had. Editorial considerations Figures related to quotations are nearly all redrawn to allow a better comprehension. Symbols of formulas are always those of the authors, except in easily identifiable cases. Translations of text from French, Latin, German and Italian are as much as possible close to the original. Rome, September 2011

Danilo Capecchi

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Virtual velocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Virtual displacement laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Virtual work laws as principles of mechanics . . . . . . . . . . . . . . . . . 1.4 Virtual work laws as theorems of mechanics . . . . . . . . . . . . . . . . . 1.5 Contemporary tendencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Final remarks. The rational justification of virtual work laws . . . .

1 2 4 5 9 10 12

2

Logic status of virtual work laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The theorem of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Proofs of the virtual work theorems in the literature . . . . 2.1.1.1 Physics and rational mechanics treatises . . . . . . 2.1.1.2 Statics handbooks . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.3 Poinsot’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Force as a primitive concept . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Equilibrium case . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Motion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Work as a primitive concept . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Equilibrium case . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 Motion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 17 23 23 24 26 27 28 28 29 31 31 32

3

Greek origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Different approaches to the law of the lever . . . . . . . . . . . . . . . . . . 3.1.1 Aristotelian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Physica and De caelo . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Mechanica problemata . . . . . . . . . . . . . . . . . . . . 3.1.1.3 A law of virtual work . . . . . . . . . . . . . . . . . . . . . 3.1.2 Archimedean mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 Proof of the law of the lever . . . . . . . . . . . . . . . . 3.2 The mechanics of Hero of Alexandria . . . . . . . . . . . . . . . . . . . . . . .

33 34 34 35 38 43 45 48 51

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3.2.1

The principles of Hero’s mechanics . . . . . . . . . . . . . . . . . 3.2.1.1 A law of virtual work . . . . . . . . . . . . . . . . . . . . . 3.2.1.2 Hero’s inclined plane law . . . . . . . . . . . . . . . . . . The mechanics of Pappus of Alexandria . . . . . . . . . . . . . . . . . . . . . 3.3.1 Pappus’ inclined plane law . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 58 59 60

4

Arabic and Latin science of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Arabic mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Weight as an active factor in Arabic mechanics . . . . . . . 4.1.1.1 Liber karastonis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1.2 Kitab al-Qarastun . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Comments on the Arabic virtual work law . . . . . . . . . . . . 4.2 Latin mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Weight as a passive factor in the Latin mechanics . . . . . . 4.2.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.1 Proposition I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.2 Proposition II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.3 Proposition VI. The law of the Lever . . . . . . . . 4.2.2.4 Proposition VIII . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.5 Proposition X. The law of the inclined plane . . 4.2.3 Comments on the Latin virtual work law . . . . . . . . . . . . .

63 66 68 69 73 74 75 80 80 81 84 86 86 88 89

5

Italian Renaissance statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Renaissance engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Daniele Barbaro and Buonaiuto Lorini . . . . . . . . . . . . . . . 5.2 Nicolò Tartaglia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Definitions and petitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.1 Proof of propositions I–IV . . . . . . . . . . . . . . . . . 5.2.2.2 The law of the lever . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.3 The law of the inclined plane . . . . . . . . . . . . . . . 5.3 Girolamo Cardano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 De subtitilate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 De opus novum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Guidobaldo dal Monte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The centre of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 The virtual work law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Giovanni Battista Benedetti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Effect of the position of a weight on its heaviness . . . . . . 5.5.2 Errors of Tartaglia and Jordanus . . . . . . . . . . . . . . . . . . . . 5.6 Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The concept of moment. A law of virtual velocities . . . . 5.6.2 A law of virtual displacements . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Proof of the law of the inclined plane . . . . . . . . . . . . . . . .

91 95 96 97 98 98 100 101 103 104 105 107 108 109 109 115 116 116 118 120 121 127 131

3.3

Contents

ix

6

Torricelli’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The centrobaric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Galileo’s centrobaric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Torricelli’s joined heavy bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Torricelli’s fundamental concepts on the centre of gravity 6.4 Torricelli’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Analysis of the aggregate of two bodies . . . . . . . . . . . . . . 6.4.2 Torricelli’s principle as a criterion of equilibrium . . . . . . 6.5 Evolution of Torricelli’s principle. Its role in virtual work laws . . 6.5.1 A restricted form of Torricelli’s principle . . . . . . . . . . . . .

135 135 138 140 141 144 146 148 153 154

7

European statics during the XVI and XVII centuries . . . . . . . . . . . . . 7.1 French statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Gille Personne de Roberval . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1.1 The inclined plane law . . . . . . . . . . . . . . . . . . . . 7.1.1.2 The rule of the parallelogram . . . . . . . . . . . . . . . 7.1.2 René Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2.1 The concept of force . . . . . . . . . . . . . . . . . . . . . . 7.1.2.2 The application to simple machines . . . . . . . . . . 7.1.2.3 The refusal of virtual velocities . . . . . . . . . . . . . 7.1.2.4 Displacements at the very beginning of motion 7.1.2.5 A possible precursor . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Blaise Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Post Cartesians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Nederland statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Simon Stevin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.1 The rule of the parallelogram of forces . . . . . . . 7.2.1.2 The law of virtual work . . . . . . . . . . . . . . . . . . . . 7.2.2 Christiaan Huygens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 British statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 John Wallis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Isaac Nevton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 160 160 161 164 164 167 170 171 173 175 176 177 178 180 184 187 189 190 193

8

The principle of virtual velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The concept of force in the XVIII century . . . . . . . . . . . . . . . . . . . . 8.1.1 Newtonian concept of force . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Leibnizian concept of force . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Johann Bernoulli mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Dead and living forces according to Bernoulli . . . . . . . . . 8.2.2 The rule of energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Varignon: the rule of energies and the law of composition of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Elements of Varignon’s mechanics . . . . . . . . . . . . . . . . . . 8.3.2 The rule of the parallelogram versus the rule of energies

195 195 195 197 199 199 201 210 210 213

x

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Contents

The Jesuit school of the XVIII century . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Vincenzo Angiulli and Vincenzo Riccati . . . . . . . . . . . . . . . . . . . . . 9.1.1 The principle of actions of Vincenzo Angiulli . . . . . . . . . 9.1.1.1 The action of a force . . . . . . . . . . . . . . . . . . . . . . 9.1.1.2 The principle of actions . . . . . . . . . . . . . . . . . . . 9.1.1.3 The measure of actions . . . . . . . . . . . . . . . . . . . 9.1.1.4 The principle of action and the principles of statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1.5 The applications to simple machines . . . . . . . . . 9.1.2 The principle of actions of Vincenzo Riccati . . . . . . . . . . 9.2 Ruggiero Giuseppe Boscovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 A virtual work law for Saint Peter’s dome . . . . . . . . . . . . 9.2.1.1 The mechanism of failure and the forces . . . . .

217 218 218 219 221 223

10 Lagrange’s contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 First introduction of the virtual velocity principle . . . . . . . . . . . . . 10.1.1 The first ideas about a new principle of mechanics . . . . . 10.1.2 Recherches sur la libration de la Lune . . . . . . . . . . . . . . . 10.1.2.1 Setting of the astronomical problem . . . . . . . . . 10.1.2.2 The symbolic equation of dynamics . . . . . . . . . 10.1.2.3 The virtual velocity principle . . . . . . . . . . . . . . . 10.1.3 The Théorie de la libration de la Lune . . . . . . . . . . . . . . . 10.2 Méchanique analitique and Mécanique analytique . . . . . . . . . . . . 10.2.1 Méchanique analitique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1.1 Constraint reactions . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Mécanique analytique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2.1 Criticisms of Lagrange’s proof . . . . . . . . . . . . . 10.3 The Théorie des fonctions analytiques . . . . . . . . . . . . . . . . . . . . . . . 10.4 Generalizations of the virtual velocity principle to dynamics . . . . 10.4.1 The calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Elements of D’Alembert’s mechanics . . . . . . . . . . . . . . . . 10.4.2.1 D’Alembert principle . . . . . . . . . . . . . . . . . . . . .

237 240 240 242 245 247 250 251 252 253 258 259 263 264 268 273 274 277

11 Lazare Carnot’s mechanics of collision . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Carnot’s laws of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The first fundamental equation of mechanics . . . . . . . . . . 11.1.2 Geometric motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 The second fundamental equation of mechanics . . . . . . . 11.2 Gradual changing of motion. A law of virtual work . . . . . . . . . . . . 11.3 The moment of activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281 285 287 289 291 293 295

12 The debate in Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The criticisms on the evidence of the principle . . . . . . . . . . . . . . . . 12.1.1 Vittorio Fossombroni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1.1 Invariable distance systems . . . . . . . . . . . . . . . .

299 300 300 301

225 228 230 233 234 235

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12.1.1.2 The equation of forces . . . . . . . . . . . . . . . . . . . . . 12.1.1.3 The equation of moments . . . . . . . . . . . . . . . . . . 12.1.2 Girolamo Saladini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 François Joseph Servois . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The criticisms on the use of infinitesimals . . . . . . . . . . . . . . . . . . . . 12.2.1 Gabrio Piola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1.1 Piola’s principles of material point mechanics . 12.2.1.2 System of free material points . . . . . . . . . . . . . . 12.2.1.3 System of constrained material points . . . . . . . .

302 304 306 308 311 312 312 314 315

13 The debate at the École polytechnique . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 One of the first professor of mechanics, Gaspard de Prony . . . . . . 13.1.1 Proof from the composition of forces rule . . . . . . . . . . . . 13.2 Joseph Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 First proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Third proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 André Marie Ampère . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Pierre Simon Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 319 320 321 323 325 326 328 332

14 Poinsot’s criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Considérations sur le principe des vitesses virtuelles . . . . . . . . . . 14.2 Théorie générale de l’équilibre et du mouvement des systèmes . . 14.2.1 Poinsot’s principles of mechanics . . . . . . . . . . . . . . . . . . . 14.2.1.1 System of material points constrained by a unique equation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1.2 System of material points constrained by more equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Demonstration of the virtual velocity principle . . . . . . . . . . . . . . . .

335 336 339 342

15

Complementary virtual work laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Augustin Cauchy formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Kinematics of plane rigid bodies . . . . . . . . . . . . . . . . . . . .

353 354 356

16 The treatises of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Siméon Denis Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Jean Marie Duhamel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Gaspard Gustave Coriolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 362 365 367

17 Virtual work laws and continuum mechanics . . . . . . . . . . . . . . . . . . . . 17.1 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Joseph Louis Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1.1 Mono-dimensional continuum . . . . . . . . . . . . . . 17.1.1.2 Three-dimensional continuum . . . . . . . . . . . . . . 17.1.2 Navier’s equations of motion . . . . . . . . . . . . . . . . . . . . . . . 17.2 Applications in the theory of elasticity . . . . . . . . . . . . . . . . . . . . . .

375 375 375 376 377 381 383

344 346 348

xii

Contents

17.2.1 Alfred Clebsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 The Italian school . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Gabrio Piola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Eugenio Beltrami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 Enrico Betti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 387 388 390 392

18 Thermodynamical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Pierre Duhem’s concept of oeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Virtual transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Activity, energy and work . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3 Rational mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.1 Free systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3.2 Constrained systems . . . . . . . . . . . . . . . . . . . . . .

395 396 397 398 401 401 402

Appendix. Quotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.14 Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.15 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.16 Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.17 Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.18 Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405 405 406 407 409 412 423 426 433 437 441 448 452 454 457 463 464 467 471

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

473

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

489

1 Introduction

Hereinafter, the law of virtual work is not given its contemporary meaning. To do so would be misleading in that it would attempt to write a history based on unavoidable recourse to categories of thinking that did not exist in the past. I use instead the broader meaning of law of equilibrium, where forces appear together with the motion of their points of application independently of the logical status assumed, be it a theorem, a principle or an empirical law. In this sense the laws of virtual work represent a particular historical point of view on mechanics. Since the Greek origins of mechanics, there have been two alternative formulations of laws of virtual work (hereinafter VWL). The first, which dates back to Aristotle’s school, today goes under the name of laws of virtual velocities, in which the effects of forces are assumed depending on the virtual velocities of their points of application. The second, which has been known at least since the Hellenistic period, today goes under the name of laws of virtual displacements, in which instead the effects of forces are assumed depending on virtual displacement of their points of application. The two approaches, though conceptually different, are mathematically equivalent. In the early days of VWLs, virtual motions were considered primarily as possible motions, those which one would have imagined the body, or system of bodies, to assume within the respect of constraints, for example, following a disturbance induced by a small force that alters the equilibrium. If one imagines that a balance rotates around the fulcrum, at the same time one would imagine that the weights of which it is burdened move. But with this type of ‘natural’ conception there coexists another, though not fully conscious at the beginning, in which the virtual motion is seen as purely geometric. On the one hand one sees the balance in equilibrium under assigned weights; on the other hand one imagines the unloaded balance moving with a motion that takes place with a time flowing in a super-celestial world. This way of viewing virtual motions began to emerge from the ‘subconscious’ to become the ‘natural’ one only in the XIX century with Poinsot and Ampère.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_1, © Springer-Verlag Italia 2012

2

1 Introduction

Below is a brief history of VWLs, almost a summary of the book, from which it is clear that the various formulations that have occurred during approximately two thousand years, from Aristotle to Galileo, showed no appreciable progress. After Galileo, there was instead an abrupt change of direction and in a few generations very sophisticated formulations were reached.

1.1 Virtual velocity laws The reconstruction of the historical development of the laws of virtual velocities is currently very incomplete. It goes back to Aristotelian Mechanica problemata of the fourth century BC [12], with the law: “heavy bodies located at the end of a lever are equilibrated when, in their possible motion, velocities are in inverse ratio to weights”. Its explicit formulation, however, is documented only by Galileo who introduced it especially in Le mecaniche [119] and in the Discorsi sulle cose che stanno in sù l’acqua [115]; in the latter memoir he associated explicitly the law of virtual velocities to Aristotle. The law of ‘virtual velocities’ of Aristotle’s school took the functioning of the lever as the main reference. Velocity did not have its current quantitative meaning, but was rather the concept of the common man for which there was no well-defined measurement, and at most a formulation of a criterion of more or less. Moreover, even force – regardless of its metaphysical uncertainty – was a somewhat indefinite quantity. It could be measured by weight, and then introduced into the calculations, but uncertainties still remained. Its direction was not well defined, or rather was defined tacitly: the force applied, for example, to the end of a lever was implicitly considered orthogonal to it. The law of virtual velocities, although formulated on the basis of magnitudes not well quantified, led to correct results already in the Aristotelian school. The ‘velocity’ of the points of a lever that rotates around its fulcrum can be said to vary in proportion to the distance from it and this was enough to determine a quantitative relationship between the forces and the distances from the fulcrum. The idea of a VWL arose from the motion of points on a circle which rotates around its centre: Remarkable things occur in accordance with nature, the cause of which is unknown, and others occur contrary to nature, which are produced by skill, for the benefit of mankind. [...] It is strange that a heavy weight can be moved by a small force, and that, too, when a greater weight is involved. For the very same weight, which a man cannot move without a lever, he quickly moves by applying the weight of the lever. Now the original cause of all such phenomena is the circle; and this is natural, for it is in no way strange that something remarkable should result from some thing more remarkable [12].1

In their analysis of the motion of the circle, Aristotle’s followers concluded that the points which tend to move more easily require less force than those which tend to move less easily. If to ‘more or less easily’ is given the meaning of ‘more or less force’, then one obtains a trivial tautology, but if it is given the meaning of ‘more or less quickly’, then a form of VWL is obtained. 1

pp. 133, 135.

1.1 Virtual velocity laws

3

Nowhere in the Mechanica problemata did Aristotle use the word or the concept of equilibrium. At most equilibrium can be seen in dynamical key as the result of the cancellation of effects of opposing forces. Effect measured on the basis of virtual motion. The higher the virtual velocity of the point of application of a force the greater the effect. The study of equilibrium on the basis of possible motions seemed a contradiction in terms for those who could not conceive, with Aristotle, rest as motion in power. And there were many who did not share the ideas of Aristotle. But even if this metaphysical difficulty is ignored, the Aristotelian law of virtual work was too ‘complex’ from a logical point of view to be assumed as a principle, i.e. it had to be demonstrated. According to the mathematicians of the time a demonstration had to be based on the existing model of geometry and had to consist of a derivation from evident propositions. The intuitive Aristotelian considerations had no probative value. For this reason in ancient Greece, the law of the lever among scientists but also among technicians, followed a different approach, based on the concept of centre of gravity. Unfortunately we have few documents relating to the mechanical studies of Greek mathematicians posterior to Aristotle. There are essentially the basic texts of Archimedes on hydrostatics and centres of gravity, and some studies on the balance atributed to Euclid. The most complete witness of Greek mechanics is contained in the Mechanica of Hero of Alexandria, which had an applicative character. However it can be said that Greek mechanicians assumed as their main conceptual model the lever and the law which regulates its behaviour was proved with considerations ‘beyond any doubts’ from principles, fixed by Archimedes (see Chapter 3) which are also ‘beyond any doubts’. Here equilibrium is the key concept, while motion is not considered, except to deny it. In the modern era Galileo was the first to assume a VWL with a dynamical connotation where equilibrium resulted from cancellation of opposing trends. The name he gave to these trends was ‘momento’ (moment), a term which remained long in the history of mechanics: Moment is the propension of descending, caused not so much by the Gravity of the moveable, as by the disposure which divers Grave Bodies have in relation to one another; by means of which Moment, we oft see a Body less Grave counterpoise another of greater Gravity Moment is the propension to go downward, caused not so much on by severity of the gravity of a mobile, but by the mutual disposition of the different heavy bodies, by the moment of which you will see many times a less heavy body counterbalance another more heavy [119].2 (A.1.1)

Galileo was not able to combine disparate magnitudes, such as weight and velocity, and the idea of momento was expressed in the language of proportions that remains at a somewhat imprecise level. In the study of the lever, shown in Le mecaniche, Galileo saw virtual motion as that motion generated by altering the equilibrium with a small weight. He then retained for it a certain degree of reality.

2

p. 159. Translation in [121].

4

1 Introduction

1.2 Virtual displacement laws The idea of virtual displacement is in principle simpler than that of virtual velocity, because a displacement could be detected unambiguously even in antiquity. So it was natural that in addition to the law of virtual velocities, also the law of virtual displacements had emerged. It had developed along two completely different paths. The first, which is generally the most emphasised, took the functioning of devices for lifting and shifting – the machines – as the main reference. The beginning is found in the writings of Hero of Alexandria, but it is present more clearly as a general law of mechanics in Thabit’s Liber karastonis in the IX century and in Jordanus de Nemore’s De ratione ponderis in the XIII century. Jordanus assumed the law that moving a weight p at height h is equivalent to moving a weight q = p/k up to hk, whatever k. The logical status of Jordanus’ virtual displacement law is still disputed: is it a principle or a theorem derived from the Aristotelian laws of motion? It had however a general character and was used in various demonstrations. Important is that of the inclined plane, which for the first time was referred to correctly. Note that Jordanus’ is a law of equivalence, or conservation, but not of equilibrium. To obtain equilibrium it is necessary to present an ad absurdum argument. The examination of the proof of the law of the lever, reported in the De ratione ponderis, shows the way (see Chapter 4). Consider a lever with two weights P and Q placed at distances p and q in inverse proportion to P and Q respectively. For the law of equivalence, weight P can be replaced by a weight equal to Q placed at a distance q from the fulcrum of the lever, on the same side of P, since by hypothesis the relation Pp = Qq holds true. What is obtained in this way is a lever with equal arms and equal weights and, as such, in equilibrium, thus satisfying the principle of sufficient reason. This means that the balance was in equilibrium even before the change of weight P with the weight Q. The ideas of Jordanus found their natural successor in René Descartes, who focused on the concept of what we now call work, which he called ‘force’: The same force that can lift a weight, for example of 100 pounds to a height of two feet, can also lift 200 pounds to a height of one foot, or 400 pounds to a height of 1/2 foot, and other [96].3 (A.1.2)

But there was a second source of the law of virtual displacement that put equilibrium in the spotlight. This is Torricelli’s principle, according to which the centre of gravity of a system of bodies in equilibrium cannot sink for any virtual displacement compatible with constraints. It is a generalisation of the ancient empirical principle that the center of gravity of a heavy body moves necessarily down when there are no obstacles that prohibit it. Torricelli’s principle was already formulated by Galileo: Because, as it is impossible for a heavy body or a mixture of them to move naturally upward, moving away from the common centre towards which all heavy things converge, so it is impossible that it spontaneously moves, if with this motion its own centre of gravity does not approach the common centre [emphasis added] [118].4 (A.1.3) 3 4

vol. 2, p. 435. p. 215.

1.3 Virtual work laws as principles of mechanics

5

And it can be traced back to medieval times, but only with Evangelista Torricelli could it take the form of a physical law expressed in the language of mathematics. Torricelli’s principle was originally formulated for only two bodies: “Two joined bodies cannot move by themselves, if their common centre of gravity does not sink”, but its extension to more bodies is straightforward. It had two great advantages over the other formulations of VWLs: it was ‘convincing’ for it appealed to everyday experience – and therefore no particular objection can be taken to assume it as the basis of statics – and could be easily generalised to a system of bodies. Starting from Torricelli’s ideas, John Wallis reworded the principle of Torricelli, saying that the sum of the products of forces times displacements of their points of application in the direction of forces must be equal to zero. According to Varignon Wallis was the man “who went farther than any other authors [before Bernoulli]” [238].5

1.3 Virtual work laws as principles of mechanics However Torricelli’s principle was not received enthusiastically and was basically ignored by nearly all other mechanicians. A good number of scholars (including Pardies, Lamy, Rouhalt and Borelli), acknowledged the truth of the fact of the annulment of the virtual work of forces, but no one considered it possible to take this as a principle of statics because it was not self-evident, as the epistemology of time required for a principle. Moreover the principle, although very general, in many cases failed. It was successful for simple machines (lever, inclined plane, wedge, etc.), in which the directions of force and motion remain constant during virtual motion. It failed where this condition did not occur, such as the motion of a body on a curved profile. René Descartes was the first to realise that, for the validity of any VWL, it was necessary to consider not the actual motion of bodies but that it would progress along straight lines or planes tangent to the constraints that limit the motion, i.e. the motion at the very beginning. This observation generalised the approach already used in statics by Galileo and Roberval, which replaced the existing constraints with equivalent ones. For example (Galileo), the inclined plane with a lever perpendicular to it. Besides this important technical improvement, Descartes claimed clearly the role of a principle of mechanics for a VWL in the formulation he gave it, that moving a weight p at height h is equivalent to moving a weight q = p/k up to hk. For him it was a sufficiently clear and distinct proposition and was also enough to solve all problems of statics. Descartes’ idea of virtual motion was generalized further by Christiaan Huygens who introduced the concept of infinitesimal displacements in Torricelli’s principle. His early works on the subject date back to 1667 (see Chapter 7) and concern the equilibrium of three or more ropes at the ends of which forces are applied. The memoirs of Huygens, related only to special cases, however were not published while he 5

Preface.

6

1 Introduction

was alive and it is unclear whether they came, indirectly, to the notice of Johann Bernoulli. It was Johann Bernoulli who refined the wording of the VWL in a systematic way by introducing explicitly the concept of infinitesimal displacements. The law he formulated is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary formulations of VWLs. The new edition of Johann Bernoulli’s works, which also includes unpublished letters, provides a starting point for an interpretation of the origins of VWL in Bernoulli, slightly lowering the aura of mystery that until now it has been wrapped. In February 1714 Johann Bernoulli published Manoeuvre des vaisseaux (see Chapter 8), a book dedicated to the theory of sailing. The preparation of this book was stimulated by the publication of another book by Renau of Elissgaray, a marine engineer, on the same topic and the discussion that followed on the composition of forces. Bernoulli had recently embraced Leibniz’s ideas of dead and living forces. He distinguished between the impulsive forces (living forces) and the forces that act continuously (dead forces), like the wind that pushes on the sails of ships. And the forces that act continuously are characterized by their energies, i.e. the product of the force by the component of the virtual infinitesimal displacement in the direction of the force, named by Bernoulli virtual velocities. In the end Bernoulli, as indeed did Huygens, tended to consider virtual velocities and virtual displacements essentially the same, and finally to consider the term ‘virtual velocity’ as a synonym for ‘infinitesimal displacement’. This fact created a never-ending controversy because velocities and infinitesimal displacements are not perfectly matched, and while velocity was a well established and accepted concept in the XVIII century, the infinitesimal displacement remained shrouded in an aura of mystery. The formulation of Bernoulli’s VWL is commonly associated with a letter of Bernoulli to Pierre Varignon in 1715. Paradoxically, this letter appeared in the Nouvelle mécanique ou statique of 1725, a book which presented Varignon’s rule of composition of forces as the fundamental principle of statics alternative to any VWLs. Bernoulli’s statement affirmed that for a system of forces that maintains a point, a surface, on a body in equilibrium, the sum of positive energies equals that of negative energies, considered with their absolute value. Bernoulli was well aware of the importance of his principle. In his letter to Varignon he wrote that the composition of forces is not but a small corollary of his principle. Varignon of course did not share this enthusiasm and did not record in his book this part of Bernoulli’s letter. For Varignon, Bernoulli’s law is at most a theorem, to be proved case by case. Though he did not give a general proof, he devoted a large part of his Nouvelle mécanique [238] to prove it in ‘all cases’ where, using the rule of the composition of forces, it is known there is equilibrium. Bernoulli’s VWL was not immediately accepted as a possible principle of mechanics. Bernoulli himself seemed to have changed his attitude and, in his writings, referred to it only once, in 1728 in the Discourse sur le lois de la communication du mouvement [35]. Here he introduced again the virtual velocity, but as the velocity that each element of a body gains or loses, over the velocity already acquired, in an infinitely small time, according to its direction (see Chapter 8). The above defini-

1.3 Virtual work laws as principles of mechanics

7

tion is not equivalent to that contained in the letter to Varignon, because it explicitly named a velocity rather than a displacement, which is in general the variation dv of a given motion. This new point of view is due to the fact that now Bernoulli’s interest is motion of bodies and not just their equilibrium. No reference or comment is made to his earlier definition of the virtual velocity, as if he had never written anything about it. After Bernoulli, probably the most significant contribution to the development of VWLs was due to Vincenzo Riccati who introduced the principle of action in the Dialogo di Vincenzo Riccati della compagnia di Gesù of 1749 and the De’ principi della meccanica of 1772. Vincenzo Angiulli moved in the wake of Riccati with his Discorso sugli equilibri of 1770. Although the idea of the principle of action was essentially Riccati’s, the less original Angiulli was closer to the foundational aspects of mechanics. Angiulli tried to prove his VWL not from other mechanical principles but from ‘indubitable’ metaphysical principles, including the equivalence of cause and effect. He began with the Leibnizian concept of dead force, which is presented as an infinitesimal pulse, such as f ds (where f is the intensity of the pulse, identified with the ordinary force, and ds the infinitesimal displacement of the point where the force is applied) continually renewed by gravity or some other cause and continuously destroyed by constraints. In the absence of constraints, the pulses can be accumulated and the action of the dead force is that of cumulative pulses; the action of the dead force generates then the living force and therefore the motion. With the introduction of infinitesimals Angiulli could enunciate his principle of actions, which he qualifies as a theorem because it is demonstrated with his metaphysical considerations: The equilibrium comes from the fact that the actions of the forces which must be equilibrated, if born, would be equal and opposite, and therefore the equality, and opposition of the actions of the forces is the actual cause of equilibrium. […] The equilibrium is nothing but the impediment of the motions, that is of the effects of the forces, to which it is not surprising if the prevention of the causes, i.e. of the actions themselves is reached [4].6 (A.1.4)

The principle of action implies the relation ∑ f ds = 0, where f ds are the elementary actions emerging in the infinitesimal displacements ds, compatible with constraints. It is therefore a possible formulation of VWL. For Angiulli, the ontological status of the constraints was that of ‘hard bodies’, i.e. idealised bodies that absorb all the pulses and the living force. Constraints obey an economy criterion, acting only as much as it is needed. In practice Angiulli made the assumption of smooth constraints without being aware of the problematic nature of the fact. Note that the constraints have only the effect of destroying the motions and do not exert any reactive force, as this is a foreign concept to Leibniz’s mechanics. Half a century after the letter of Bernoulli to Varignon, Lagrange gave the VWL a more efficient form. Officially, he referred to Bernoulli, but its role was actually very different. When in 1764, for the first time, Lagrange exposed Bernoulli’s principle 6

pp. 16–17.

8

1 Introduction

of virtual velocities, he recast it by talking about the equilibrium of bodies and not of forces and applied it considering all the motions compatible with constraints and not only rigid motions. He did not conceive the law of virtual work as a theorem, derived for example in the context of Newtonian mechanics; it was rather an alternative principle. This position should be clear from the introductory part of the Mécanique analytique, published more than twenty years later, where he presented the various ways of addressing the problems of equilibrium of bodies: the lever, the rule of the parallelogram and the principle of virtual velocities. The first edition of the Mécanique analytique [145] of Lagrange, with the prominence it gave to his VWL, was the genesis of a wide debate on its logic status and was also the occasion for a critical analysis of the principles of statics. It was an occasion that, in the history of classical mechanics, has an equivalent only in the debate at the beginning of the XVIII century on the principles of dynamics, and of which today one no longer understands the significance. The list of scientists who became interested in the problem should make us reflect on the extent of the effort that was made and the opportunity to learn a lot by following their teachings: Lazare Carnot, Lagrange, Laplace, Poinsot, Fourier, Prony, Ampère and then also Cauchy, Gauss, Poisson and Ostrogradsky. Lobachewsky too was involved, but the content of his contribution has been lost. To have again such heated discussion of scientists on the fundaments of mechanics it will be necessary await up to the introduction of relativistic mechanics, one century later. To understand the reasons of the debate one needs to reflect that, though the logical status of dynamics was undoubtedly controversial, there was generally agreement that one could give statics a shared formulation. But Lagrangian VWL seemed to many to not meet the assumptions of epistemology of the times. Although one could say – but not everyone agreed even on that – the VWL was prior to all the laws of mechanics in the sense that these laws could be derived from it, one could not admit it was evident; in particular it seemed less simple and evident for example of the law of the lever. Lagrange also agreed and, in the second edition of the Mécanique [148], wrote: And in general I can say that all the general principles that can be discovered in the science of equilibrium, will not be but the same as the principle of virtual velocities considered differently, and from which it differs only in form. But this principle is not only itself very simple and general, it has, in addition, the precious and unique advantage of being translated into a general formula that includes all the problems that can be posed on the equilibrium of bodies. […] As to the nature of the principle of virtual velocities, it is not so obvious that it can be claimed as a primitive principle [emphasis added] [148].7 (A.1.5)

Young Lagrange was attracted by the “precious and unique advantage of being translated into a general formula that includes all the problems that can be put on the equilibrium of the body” and did not hesitate to take an instrumental position. There is no doubt that he was essentially a mathematician and, in line with the times, strongly attracted by the formal aspects of Calculus. Although this position is subject to criticism, credit must be given to Lagrange for an originality that allowed him to go against the perhaps too rigid epistemology of the times. His attitude certainly con7

pp. 22–23.

1.4 Virtual work laws as theorems of mechanics

9

tributed to the development of the more liberal epistemology of the XX century, born to a large extent with the advent of non-Euclidean geometry. Lazare Carnot reached a VWL for colliding bodies according to his laws of impact in a mechanical theory, in principle, without forces. A fundamental concept developed by Carnot which influenced the subsequent debate, Poinsot’s included, is that of geometric motion, that is motion considered in itself independently of any force (see Chapter 11).

1.4 Virtual work laws as theorems of mechanics Concluding, one could not accept the statement of the Lagrangian VWL as a principle and had to prove it by reduction to a theorem of another approach to mechanics. This question promoted, as already mentioned, a heated debate, especially in France where the main contributions were those of Lazare Carnot, Fourier, Ampère and Poinsot. For Italy it is worth noting the contribution of Vittorio Fossombroni. The reasons for the attention paid to Lagrange’s VWL were not only scientific, however. It was no coincidence that the interest was polarised in France. Here the Cartesian tradition was still alive and national pride was still an obstacle to a full acceptance of Newtonian physics and metaphysics. VWLs seemed to offer the opportunity to develop a completely ‘continental’ mechanics, freed from the concept of force of a Newtonian matrix. Joseph Fourier tried several demonstrations. In the probably most successful one, Lagrange’s VWL is reduced to the law of the lever, replacing weights with active forces that exert their action by threads, rings and levers. André Marie Ampère, following Carnot, introduced the concept of virtual velocity as a vector tangent to the trajectories compatible with the constraints, where time “has nothing to do with”. Vittorio Fossombroni in a memoir of 1794, demonstrated Lagrange’s VWL in the case of a free rigid body starting from the cardinal equations of statics. Of some interest is Fossombroni’s attempt to replace infinitesimal virtual displacements, which created some embarrassment, with finite displacements of arbitrary value. He showed that if the forces are parallel to each other and if their points of application are arranged along a line, the virtual work of these forces is zero for any finite rigid motion. This idea was generalised, to the case of forces in the space with application points lying on a plane, by Poinsot who felt the same embarrassment in the use of infinitesimal quantities. Louis Poinsot gave in my opinion the most successful proof of a VWL. Since his mechanical theory was based on the rule of composition of forces and reduced to mathematical formulas, his proof was and is still considered by mathematicians and physicists, more interesting than the more geometric Fourier’s type, based on the law of the lever, and has become a model for almost all textbooks of statics. Poinsot accepted the principle that a material point subject to a certain active force is equilibrated on a surface if and only if the force is orthogonal to it. In addition he considered other principles, among which the principle of composition of forces and the principle of solidification, according to which if one adds constraints – both

10

1 Introduction

internal and external – to a system of bodies in equilibrium, the equilibrium is not altered. On the basis of his principles Poinsot was able to fully characterise statics and write the equations of equilibrium in which only the constraint equations and the components of the active forces applied to various points of this system appear. To demonstrate his VWL, Poinsot gave up the virtual displacement concept, to adopt that of virtual velocity – contemporary meaning. According to Poinsot real time and virtual time run on different universes: It must be noted further that the system is supposed to move in any way, without reference to forces that tend to move it: the motion that you give is a simple change of position where the time has nothing to do at all [197].8 (A.1.6)

By replacing virtual displacements with virtual velocities, it is then easy to prove a VWL in the form ∑ f dv = 0, where v are the virtual velocities of the points of application of the active forces f . One cannot stress enough the fact that Poinsot’s virtual velocity is purely geometric and his virtual work is only a mathematical definition. Poinsot thus closed the circle that had opened with Aristotle. The laws of virtual work were initially manifested as laws of virtual velocities, then the laws had split into virtual displacement and virtual velocity laws. With Bernoulli there was a partial but ambiguous reunification; Poinsot brought everything back to the baseline by eliminating the laws of virtual displacements.

1.5 Contemporary tendencies But not everyone followed Poinsot in dealing with VWLs as theorems of mechanics and considering the virtual work as a purely mathematical concept. It is possible to identify a line of thought that instead of diminishing the mechanical meaning of the virtual work tended to enhance it. This line of thinking had its precursors in Descartes and Leibniz. Then it became precise with Lazare Carnot, who introduced the concept of work of a ‘power’ along an arbitrary path, named by him moment of activity, giving it the meaning of a physical magnitude and a key place in mechanics. A few years after Lazare Carnot’s contribution, Gaspard Coriolis established definitely in 1829 the term ‘work’ to indicate the Bernoullian energies. This change of terminology also implied a change of the ontological status. Also referring back to the ideas of Lazare Carnot, virtual work began to take on the role of well-defined mechanical magnitudes. Coriolis adopted the molecular model of matter, where everything is reduced to material points treated as centres of force. In this mechanics, there are no constraints in the classical geometrical sense: there are ‘material’ constraints composed of material points carrying out repulsion actions against the particles that wish to penetrate them. Since there are no constraints, the infinitesimal displacements are not subject to any limitation and can be identified with – and indeed they were – real motions. So next to virtual work, there was room for ‘real work’. Coriolis addressed for the first time the thorny problem of friction. While in a traditional formulation of VWL it was difficult to consider the reactions, without which it is impossible 8

p. 13, part II.

1.5 Contemporary tendencies

11

to introduce friction, for Coriolis there was no difficulty. Friction is represented by the tangential components of the interaction between two bodies resulting from the superposition of forces exerted by the material points constituting the bodies. The following passage serves to illustrate the idea: We are led to realize that the principle of virtual velocities in the equilibrium of a machine, composed of more bodies, cannot take place without considering first the sliding friction, where the virtual displacements cause the slipping of the bodies, one on others, and finally that the rolling when bodies cannot take that virtual motion without deformation near the contact points. Frictions are recognized always, for experience, able to maintain equilibrium in a certain degree of inequality between the sum of the positive work and the sum of the negative work, here taking as negative the elements belonging to the smaller sum. It follows that the sum of the elements to which they give rise has precisely the value that can cancel the total sum and is equal to the small difference between the sum of the positive and negative elements [79].9 (A.1.7)

Parallel to the discussion on the concept of virtual or real work, a new science, thermodynamics, was developing, where real work had a physical meaning in every respect. Sadi Carnot put the work that he indicated with the term ‘engine power’ at the centre of his Reflexions sur la puissance du feu of 1824 [62]. Work moved from thermodynamics to mechanics with Rankine, Helmholtz and Duhem in the XIX century. In Duhem’s mechanics, a VWL came from the principle of conservation of energy, basically in its variational version. The connection of Duhem’s VWL to ‘real’ work or energy marked in some way a reconciliation with the principle of the impossibility of perpetual motion. Until then the two principles were kept strictly separate. Lagrange in particular, in his writings, never referred to the perpetual motion. The role of VWLs in contemporary classical mechanics is not well defined, it is however not essential. In theoretical treatises on rational mechanics, which takes a strong axiomatic point of view, VWLs are often not even mentioned, even though the axioms upon which mechanics is erected, such as Lagrange or Hamilton equations, could be derived from them. In the applied mechanics of rigid bodies, VWLs are present but not important. They are used to solve some particular problems, in which for the presence of constraints it would be difficult to use other methods. But when considering the mechanical theory as a whole, it is generally preferred to start from the cardinal equations. The constraints are taken into account by introducing auxiliary unknowns such as the reactive forces which are then removed in the solution of the single problems. There are no conceptual difficulties in dealing with constraints friction; it is enough to provide the appropriate ‘constitutive’ relationships. In continuum mechanics the role of VWL is instead rather important. But this does not depend on its ability to address the various conditions of constraints, but rather on the mathematical expressions the virtual work law takes, that makes it easier for approximate solutions in many cases, for example with the finite element method. 9

p. 117.

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1 Introduction

Table 1.1. Various versions of virtual work laws Scholar

Century

Aristotle Hero Thabit Jordanus Galileo Stevin Dal Monte Torricelli Descartes Wallis Bernoulli Riccati Angiulli Lagrange Fossombroni Carnot L Fourier Ampère Poinsot Piola Servois

IV BC I X XIII XVI XVI XVI XVII XVII XVII XVIII XVIII XVIII XVIII XVIII XVIII XIX XIX XIX XIX XIX

Real mot. d d d v d d d d d d d d d

Geom. mot.

Logic status

Real work

v

p t p t p t t p p p p p p p t ? t t t ? p

• ? • • • ?

d

d v d v v d d

? •

In the table above the main characteristics of the various VWL formulations are reported. It is distinguished if the virtual motions are real or fully geometric, in the sense that if they run as the time of the forces or not, if displacement (d) or velocity (v) is concerned. The logic status is distinguished, i.e. the law is considered a principle (p) or a theorem (t) and if the virtual work is a physical magnitude (bullet) or instead a pure mathematical expression.

1.6 Final remarks. The rational justification of virtual work laws The history of the various forms that VWLs have taken also focused on attempts that have been made to give them a rational justification. The degree of satisfaction achieved was different from period to period. Together with a certain agreement there was however always a tension towards overcoming the law, searching for a more powerful expression. Aristotle seemed at first sight convincing enough to justify the law according to which the efficacy of weights placed on the arms of a balance depends on their distance from the fulcrum. He was considered persuasive by many mathematicians, but not by his contemporaries, accustomed to the high standards of rigor exemplified by Euclidean geometry. The justification of Jordanus de Nemore’s proposition that what can lift p to h can also lift p/n to nh was criticized by his immediate successors. It was perfected by Tartaglia, but Tartaglia’s arguments were the sub-

1.6 Final remarks. The rational justification of virtual work laws

13

ject of severe criticism by Archimedean mathematicians of 1500, in particular by Guidobaldo dal Monte. Galileo did not attempt any proof, but justified his law of moments on an intuitive level; the same approach was followed by Johann Bernoulli with his rule of energy. Descartes considered self evident the simple virtual work law as expounded by Jordanus de Nemore. Riccati and Angiulli tried an external justification of Bernoulli’s rule of energy, from ‘certain’ metaphysical principles, such as the equality between causes and effects; principles not accepted by most mathematicians. Lagrange at the beginning considered as evident a virtual work law substantially coincident with Bernoulli’s rule of energy, which he called the principle of virtual velocities. Then he presented a very simple and elegant justification, which however did not meet completely the standards of rigor of the times according to which any recourse to geometric intuition was not allowed, that instead Lagrange had introduced, albeit without the use of figures. Fourier, Ampère, Laplace, Poinsot and many other scientists attempted to reduce their laws of virtual work, substantially coincident with the principle of virtual velocities of Bernoulli and Lagrange, to the elementary principles of statics, essentially the law of the lever and the rule of the parallelogram of forces. These attempts were followed by others who felt them as not entirely satisfactory. From the above, must it be concluded that the laws of virtual work have never been rationally justified? Or with a term that has a more restricted, but stronger meaning, have they never been proved? The answer is not simple. To understand why just recall that the other fundamental laws of mechanics such as the law of the lever and the rule of the parallelogram of forces followed the same fate. Many explanations were proposed but always something was found to complain about. Even contemporary scholars have dedicated themselves to attempt to justify VWLs, albeit with less passion and strength [283]. The problem has a bit shifted and transformed itself into the question: Is the mechanics L – statics and dynamics – resulting from the adoption as a founding principle the most advanced law of virtual work equivalent to the mechanics N resulting from the adoption of the most advanced version of Newtonian mechanics? The problem is a little bit easier than that to justify a VWL because the acceptance of various principles assumed for the advanced version of Newtonian mechanics can be object of a less strict scrutiny. However the problem absorbs more philosophers of science than specialists of classical mechanics. To the latter the problem seems not difficult to solve and in a positive way, following the reasoning of Chapter 2 of the present text. On the other hand, Poinsot in his Mémoire sur la théorie générale de l’équilibre et du mouvements des systèmes of 1806 had given the problem a fully satisfactory response according to the modern standards of evidence, by proving the equivalence between a VWL and a Newtonian mechanics enlarged with a series of principles to take into account the presence of constraints. Concluding from the historical path and also from a modern logical analysis it can be concluded that the laws of virtual work have been considered justified in a fairly satisfactory way in the past and today, no less than many other fundamental laws of mathematical-physics.

2 Logic status of virtual work laws

I use the term law of virtual work to mean any rule of equilibrium that includes both forces and possible displacements of their points of application. In this chapter I will give a more restricted meaning by referring only to the most (modern) advanced formulations and I will consider their logical status, i.e. whether they are principles of an autonomous mechanics or theorems of ‘another’ mechanics. To demonstrate a law, a proposition, means in the broadest sense, to bring it to laws ‘assumed as known’, with a sense that varies with the epistemological frame of reference. Until the development of modern axiomatic theories and their application to physical science by the neopositivists of the XX century, a principle was considered as acceptable if it had an intuitive nature of evidence, possibly established a priori. Currently there is a more liberal view and one does not require evidence of the principles, only that they must have sufficient strength and do not lead to logical contradictions. The problem of provability of a VWL clashes immediately with the fact that even today there does not exist a reference theory of mechanics that is fully defined and universally accepted. This is true even for systems of material points, although there do exist some axiomatizations [382, 360, 390]. A major difficulty encountered in various formulations of VWL and mechanics concerns the ontological status attributed to constraints and reactive forces. Before the XVIII century, constraints had been treated only as passive elements not able to act. Only after studying elasticity and accepting models of matter based on particles considered as centres of forces, have researchers begun to think of constraints as capable of administering active forces. In dynamics, according to Lagrange, the first scholars to assimilate constraint reaction to active forces were the Bernoullis, Clairaut and Euler, in the period 1736 to 1742. “The use of these forces dispensed from taking account of the constraints and allows one to make use of the laws of motion of free bodies” [145].1 In statics, constraint reactions are less problematic; they can be considered as the forces necessary to maintain the constraint. The first to introduce them in calculations was probably Varignon in his Nouvelle mécanique ou statique of 1725 [238]. 1

p. 178–179.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_2, © Springer-Verlag Italia 2012

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It was simply the difficulty of incorporating the reactive forces in a consistent mechanical theory that led Johann Bernoulli to the formulation of an effective law of virtual work, known after Lagrange as the principle of virtual velocities, which provides a criterion of equilibrium without intervention of these undesirable ‘beings’. But his statement that "the sum of power each multiplied by the distance traveled from the point where they are applied, in the direction of this power will always be zero", always questioned its nature; in this regard the following comment by Fossombroni is of interest: That common faculty of primitive intuition, so everyone is easily convinced by a simple axiom of geometry, as for example, that the whole is greater than the part, certainly does not need to agree on the aforementioned mechanical truth, which is much more complicated than that one of the common axioms, as the genius of the great Men who have admitted the axiom, exceeds the ordinary measure of human intelligence, and it is therefore necessary for those who are not satisfied to obtain a proof resting on foreign theories […] or to rest on the faith of chief men despising the usual reluctance to introduce the weight of authority in Mathematics [109].2 (A.2.1)

The success of Lagrange’s Mécanique analytique [145] which assumed Bernoulli’s principle of virtual velocities, suitably reformulated, as the source of all mechanics, opened a heated debate on its plausibility. In this chapter I do not refer to these efforts, nor to the preceding others, but try mainly to clarify in what sense one can prove a law of virtual work, be it Lagrange’s or otherwise. Attempts to demonstrate can be divided into two categories. In the first, which I refer to as foundational, one tries to deduce a VWL without reference to existing criteria of equilibrium; in the second category, which I refer to as reductionist, one tries to deduce a VWL by a pre-existing criterion of equilibrium of a pre-existing mechanics. It should be said however that there is a certain arbitrariness in this dichotomy, because any reductionistic attempt can be reformulated as a foundational one, as will be clear in the following. Attempts were made in the first direction by Vincenzo Riccati and Vincenzo Angiulli, Johann Bernoulli, Lazare Carnot and Lagrange himself. The first two thought they could demonstrate the law of virtual work with metaphysical considerations, using a reference mechanics of Leibnizian type but without a pre-existing criterion of equilibrium. Carnot tried to reach a VWL starting from the law of impact using a mechanics of reference in principle without force; Lagrange made use of the law of the pulley. Attempts in the second direction were made by French scientists of the École polytechnique. Summing up and using the categories of Lagrange’s mechanics, they assumed as reference mechanics those derived from the law of the lever and the rule of the parallelogram. The demonstration of Poinsot was the one that most influenced subsequent treatises of statics. In the recent scientific literature, the problem of the logic status of VWL is only addressed in the manuals of statics, where the author gives his idea in a few pages on the subject, usually referring to a limited number of ‘basic versions’ [283]. Given the predominantly teaching character of the manuals, problematic aspects tend to be hidden to provide greater certainty. To my knowledge, there are no recent theoretical 2

pp. 13–14.

2.1 The theorem of virtual work

17

works on basic aspects of VWLs, as there are no recent theoretical works on the foundation of the mechanics of a material point, even if the argument is far from exhausted. It seems further that there are only a few recent studies on the history of VWL [197], and that it is treated marginally in the numerous monographs on Lagrange, Laplace, etc. In this chapter I will try to highlight the logical status of VWLs focusing mainly on the reductionist approach that is more widespread. Just because there is no generally accepted formulation of classical mechanics I will not consider the situation in its generality and assume only what is more consolidated, in particular I will assume a mechanics of material points and forces applied to them – i.e. corpuscular mechanics. The wording of the VWL will emerge in a natural way along the line of least resistance, avoiding inessential complications. I will simplify the constraint conditions, limiting myself to dealing with holonomic, bilateral and independentof-time constraints, that can be represented mathematically by an algebraic equation only of the position variables, because I think they equally capture the essence of the problem and an extension to more general constraints is possible involving only technical complications. The reductionist approach to VWLs examined, assuming an already given mechanics, presupposes the concept of force. However, it is possible to tackle the problem from a different point of view, in which it is not necessary to posit the concept of force, giving as a primitive the concept of work. This view will be discussed briefly at the end of the chapter. According to this approach also, the VWL will be stated as a principle or deduced from the most fundamental laws, always related to the concept of work, which will now be virtual in a different way.

2.1 The theorem of virtual work A constrained system S of a finite number n of material points is a system the configuration of which is defined by a number m of degrees of freedom less than the 3n that would be needed to describe the configuration of the system as supposedly free. I will indicate with M the space, or better the manifold, of the possible configurations of dimension m for S and with N the space of configurations of dimension 3n in the absence of constraints. Each space of configuration M and N is associated with a space of tangent vectors indicated below with MT and with NT respectively. For example, for a material point P constrained to move on a surface, N has dimension 3 and M has dimension 2. The vector space NT is the space associated with the ordinary three-dimensional vector space; the space MT is the set of vectors that lie on the tangent plane to the surface in the position occupied by P. In the case of two material points, constrained to keep a constant distance, the space of all configurations N has dimension 6, corresponding to the 6 degrees of freedom of two free material points in three dimensional space. The space M of compatible configurations has instead dimension 5, because the degrees of freedom are reduced by a constraint equation which expresses the invariance of the distance between the two points. The vector space NT consists of pairs of vectors representing the displacements of the two material points which can be any, and the space MT is represented by the pairs of vectors

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2 Logic status of virtual work laws

virtual displacement

MT

C M possible displacement

Fig. 2.1. Tangent manifold

that have the same component in the line joining the two material points. The space MT of vectors u tangent to M is called the tangent manifold of M and vectors u are called virtual displacements. It is clear from the definition that the virtual displacements in general are not possible displacements, which are motions taking place on M. The difference between possible and virtual displacements is shown in Fig. 2.1. The virtual displacements coincide with the possible motion only for infinitesimal values. If it is considered that the virtual displacements occur in the direction tangent to the constraints and that the possible velocities are tangent to the constraints, it is instead possible to identify the virtual displacements with possible velocities, considering time as an arbitrary parameter. There are essentially two ways to study the equilibrium of a constrained system. In the first way it is assumed that there are known external forces, named active forces, and forces due to the constraints, named reactive forces or constraint reactions, the presence of which should be inferred indirectly from the empirical evidence that motions of the material points of a constrained systems are different from those registered without constraints. The value of constraint reactions is not given, depending on the geometry of constraints and the active forces. In the second mode there are only active forces while the constraints are characterized exclusively by their geometry; in this paragraph I will examine the first situation. On the system S of material points there are active forces fi , with a given law of variation in time and space and reactive forces ri , associated to the constraints, a priori unknowns. Collecting the active forces in the vector f and the reactive forces in the vector r, assume the following principle of equilibrium: P1 . A system of material points constrained to a manifold M starting with zero velocity, is in equilibrium in a given configuration C if and only if the following relation is satisfied at any time: f + r = 0.

(2.1)

Notice that the sufficient part of the principle (i.e. if f + r = 0 then there is equilibrium) calls for this other principle: P∗1 . If constraints can furnish reactive forces r such that f + r = 0 then they actually furnish them.

2.1 The theorem of virtual work

19

MH MT f ΠH ( f ) C

ΠT ( f ) M

Fig. 2.2. Orthogonal projections

In the following, in all considerations relating to equilibrium, I will assume implicitly a certain configuration C and any instant of time while the initial condition is rest. Denote by ΠT the projection operator from NT to MT, and by ΠH the projection operator from NT to MH, the complementary vector space of NT orthogonal to MT . Then equation (2.1) is equivalent to the two relations: ΠT ( f + r) = ΠT ( f ) + ΠT (r) = 0 ΠH ( f + r) = ΠH ( f ) + ΠH (r) = 0.

(2.2)

Fig. 2.2 clarifies the meaning of (2.2) on a two-dimensional space. The space of admissible configurations is defined by the curve M, the tangent space MT is the line tangent to M in C – the position occupied by the material point P – the orthogonal space is the line MH orthogonal in C to MT. Define now the virtual work of forces acting on S as the linear form on NT : L(u) = ( f + r) · u, where u is a vector of NT and dot denotes the inner, or scalar, product. Then consider the two other linear forms La (u) = f · u and Lr (u) = r · u, respectively called virtual work of active forces and reactive forces. Note that the virtual work coincides with the classical definition of work but it refers to a virtual displacement and not to a possible displacement. If the virtual displacements are identified with velocities, then the virtual work has the mechanical significance of power. The following theorem of virtual work can easily be proved: T1 . A system of material points on a manifold M is equilibrated if and only if L f (u) + Lr (u) = 0 for any u in NT . Indeed L f (u) + Lr (u) = ( f + r) · u = 0 ∀u ∈ N T ↔ f + r = 0, for the same definition of scalar product. To check the balance, with theorem T1 , it is necessary to specify the manner in which the reactive forces vary on the manifold M. A traditional way to characterize the reactive forces is to introduce the concept of smooth constraint, which can be expressed as: D1 *. A system of constraints associated to a manifold M and a system of material points S is smooth if and only if it is able to furnish reactive forces r such that ΠT (r) = 0.

20

2 Logic status of virtual work laws

That is in a system of smooth constraints the reactive forces belong to the space MH orthogonal to the tangent space MT in C. This mean that if ΠT ( f ) > 0, i.e. if there is at least a force fi that has a non-zero component in the direction of the displacement ui allowed by constraints, the equilibrium is not possible and the system moves, however small is the force fi . This corresponds to the intuitive concept of smooth constraints as constraints without friction. The characterization of smooth constraint can also be given, equivalently, referring to the linear form Lr (u), reaching the definition: D1 . A system of constraints associated to a manifold M and a system of material points S is smooth if and only if Lr (u) = 0 for any u in MT . Or, alternatively, in a less formal way, using the definitions of virtual displacement and work: D1 . A system of constraints associated to a manifold M and a system of material points S is smooth if and only if the virtual work of reactive forces is zero for any virtual displacement. Usually the characterization of smooth constraints assumes only the condition that r belongs to MH . But it is equally important to stress that constraints are able to exercise all the forces belonging to MH regardless of their intensity. So if constraints are smooth their reactions could be any values in a known direction, and it is possible to apply the criteria of balance P1 or T1 , to state the two theorems: T∗2 . If the constraints are smooth, a system of material points on the manifold M is equilibrated if and only if ΠT ( f ) = 0. T2 . If the constraints are smooth, a system of material points on the manifold M is equilibrated if and only if L f (u) = 0 for any u in MT . Proof of T∗2 is simple and is implicitly contained in equations (2.2). Necessary part: if a system of material points is in equilibrium for P1 it is f + r = 0, then equations (2.2) hold, and from the first of them, because constraints are smooth and ΠT (r) = 0, it is ΠT ( f ) = 0. Sufficient part: assume ΠT ( f ) = 0, because for smooth constraints ΠT (r) = 0, the first relation of (2.2) is satisfied. The second relation ΠH ( f ) + ΠH (r) = ΠH ( f ) + r = 0, is also satisfied because the constraints (smooth), by definition, can provide all the reactions orthogonal to MT , and therefore also r = −ΠH ( f ). It follows that f + r = 0, and then the system of material points is in equilibrium. The demonstration of T2 immediately follows from T∗2 for the properties of scalar product. It is worth noting that in the case of constraints that are not smooth, to check the equilibrium may not be easy. As an example consider the material point of Fig. 2.3 in which there is also a tangential component of the constraint reaction, due to friction. If the point is in equilibrium it is certainly f = −r, but for an arbitrary value of f it is not said that there will be equilibrium because for example the friction is not enough and the constraint is not able to provide r = − f . T2 is a theorem of virtual work as is T1 ; although commonly only T2 is called theorem of virtual work. It would thus appear to have solved the problem of the logic

2.1 The theorem of virtual work

21

r ΠH ( r ) ΠT ( r )

f Fig. 2.3. Not smooth constraint

status of the law of virtual work, at least as formulated above: if properly formulated, it is a theorem of statics. Unfortunately, such a conviction is no longer anything but an illusion, disguised in the words with which the concept of smooth constraints has been introduced. In fact, it is given only a definition but it does not provide any ‘decisions’ criterion and the definition leads to circularity: if the constraint is smooth, reactive forces are orthogonal to virtual motion and if the reactive forces are orthogonal to virtual motion then the constraint is smooth. To justify the usefulness of the theorem of virtual work, and then the opportunity of referring to T2 as a VWL, an operating criterion is necessary to determine in advance whether a constraint is smooth or not, and this criterion cannot exist because the constraints are usually defined analytically only by the variety M and are not observable, i.e. they are not entities on which to have a priori reasoning. The only way to use T2 (and T∗2 ) it seems is to assume the following principle: P2 . All constraints are smooth. Then from P2 , by applying modus ponens to T∗2 and T2 , two theorems are obtained: T∗3 . A system of material points on the manifold M is equilibrated if and only if ΠT ( f ) = 0. T3 . A system of material points on the manifold M is equilibrated if and only if L f (u) = 0 for any u in MT . Today theorem T3 is usually called principle of virtual work; for historical reasons even here it is a theorem. It derives from a principle of the mechanics of material points (P1 ) and a principle (P2 ) that seems external to it. Given the critical role of P2 in the proof of T3 it itself is often called the principle of virtual work. In the following I will not accept this use and with the term virtual work principle I always refer to T3 . T3 may be a theorem of the reference mechanics only if P2 holds good. It is then clear that the problem of provability of the virtual work principle is closely related to the problem of the characterization of constraints and, ultimately, of the reference mechanics, so that it be complete. If in the reference mechanics there are

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no assumptions about the constraints it does not make sense to think seriously about the provability of the virtual work principle. Is it possible to say anything more about the constraints within a reference particle mechanics in which ‘solids’ are assumed to consist of material points which act as centres of forces that underlie the cohesion? Rather than the locus of points expressed by an algebraic equation a constraint can be associated, and it normally is, with a body sufficiently ‘hard’ to be considered impenetrable. When a particle approaches the body that acts as a constraint, forces awake – the reactive forces – which are opposed to opening up the parts of that body. Knowing the laws of forces as functions of distance of the centres, the laws of interaction between the body and the particle could be determined, at least in principle. In practice this is not possible and recourse to an approximate description is necessary with an empirical character, in the broadest sense, which will provide the necessary characterization of constraints. In this way there would be no problem to determine whether a particular constraint system is smooth or not on the basis of its constitutive relationships and definitions D1 and D∗1 and the theorem of virtual work T2 then would make sense because there is an operational criterion to be applied case by case to decide on the basis of empirical observations if T2 can be applied or not. Note however that in mechanics, in fact, one tends to apply the principle of virtual work T3 ; generally, the assumption of smooth constraints is not object to scrutiny because it is not practically possible to do so. Assuming that the constraints are formed of bodies, in the past it was thought, and sometimes it is still thought, to prove the principle P2 and the theorem T3 showing first a seemingly weaker assumption, namely that: P3 . All the surfaces of the material bodies are smooth constraints for material points. It is clear that this principle expresses ideals; in practice constraints are never smooth and there are horizontal forces, or friction. Ignoring this fact and accepting the ideal nature of P3 , is it possible to accept it? It seems doubtful that criteria of symmetry and sufficient reason – in the sense that the reaction forces must be ‘always’ orthogonal because there is no reason that they are not – can be applied in the particle model in which the very concept of the surface of a body presents difficulties. But even by accepting P3 , P2 cannot be proved without any other assumption. In fact, P3 does not say anything about the internal constraints between material points, where the attribute smooth appears unintuitive. So, if the reference mechanics is not changed, even assuming P3 , P2 is not certified and therefore the virtual work principle is not a theorem. Before concluding this section I would like to briefly refer to the extension of VWLs to dynamics, extension that was made for the first time by Lagrange in 1763 (see Chapter 10) [142]. Though VWLs can be extended quite easily to dynamics, of course they will no longer provide a criterion for equilibrium, but a criterion of balance that leads to the equations of motion. Including among forces also the forces of inertia equal to −ma, the theorem takes the form T4 :

2.1 The theorem of virtual work

23

T4 . For smooth constraints a motion of a system of material points moving on the manifold M is such that the virtual work of all forces, the inertial ones included, is zero for any virtual displacement. The proof is immediate, because the equation of motion can be written as −ma + f + r = 0 and by assuming I = −ma in the form (I + f ) + r = 0, the same assumed in the proof of theorem T2 .

2.1.1 Proofs of the virtual work theorems in the literature In the technical and teaching literature the provability of the virtual work theorems, T1 , T2 or T3 is addressed some ways differently than the one reported above. The difference in presentation depends on the audience to which it refers. There are the ‘proofs’ described in treatises of physics and mechanics, those of specific texts of statics and those of texts of continuum mechanics. For the last area, the virtual work principle is usually presented for systems that are either unconstrained or subject to constraints that require simply the vanishing of the displacements of certain points, so it becomes a theorem which can be easily derived from the principles of continuum mechanics, generally described by partial differential equations. More difficult and interesting is the approach in the other two types of texts. 2.1.1.1 Physics and rational mechanics treatises As regards the presentation of the classic texts on mechanics the work by Capriglione and Drago [283, 301] which I sum up briefly, seems conclusive. The goal is usually to demonstrate the virtual work principle, the hypothesis of smooth constraints is implicit and there is always the consciousness of the author of the manual that he is proving the theorem of virtual work in the form T2 but not the virtual work principle T3 . It can be said that although there is no complete agreement on how to define the virtual displacement, in most cases the infinitesimal displacement du instead of velocity are adopted, but the ‘degree of virtuality’ of this displacement is not always clear. This problem did not appear at the beginning of the present chapter where it was assumed that the motions have only a virtual geometric characterization, in which time does not intervene. The concept of virtual displacement has, however, developed historically with reference to a magnitude that evolves over time; the virtual velocities are obtained by the derivatives of displacements with respect to time. Traces of this historical development have remained in the demonstrations in physics textbooks. Sometimes there is distinction between the time with which the forces change – referred to as the ‘real time’ – and the time with which the motion varies – referred to as the ‘virtual time’ – flowing independently of each other. In this case it may be that the real time is frozen and only the virtual time flows; virtual displacements have in this case only a purely geometric characterization, consistent with the presentation in § 2.1. The proof of the necessary part of the virtual work principle, that is if a system is in equilibrium then the virtual work of active forces is zero, takes place essentially as presented in the previous pages where the criterion of equilibrium was provided

24

2 Logic status of virtual work laws

by the annulment of the forces acting on individual material points. If the system is in equilibrium, then the balance of forces f + r = 0 subsists. Multiplying both sides by the virtual motion du, it is: L = ( f + r) · du = 0.

(2.3)

If the constraints are smooth r · du = 0 = 0, so (2.3) provides the necessary part of the virtual work principle, L = L f = f · du = 0. The proof of the sufficient part, i.e. the virtual work of active forces is always zero, then the system is in equilibrium, is usually treated in a manner substantially different from what has been done in previous paragraphs: The proof is by reduction to the absurd. It is assumed that despite being valid (*) [L f = 0], the system is put in motion, namely that at least one of its points, say the i-th, is affected, in the time dt subsequent to t, by a displacement dri , compatible with constraints. Since the material point under consideration starts from rest, it is necessarily: Fi dri > 0, then the sum of all partial work relating to other parts of the system that actually moves, it is also:

∑ Fi dri > 0,

(1)

since the sum is made up entirely of non-negative terms and at least one of them, by assumption, is not null. (a) But Fi = Fi + Ri , for which we rewrite (1) as: (a)

∑(Fi

+ Ri ) · dri > 0.

(2)

At this point one makes the assumption of smooth constraints and the absurd is obtained (a) ∑ Fi · dri > 0 [L f > 0], because against the hypothesis [283].3 (A.2.2)

The demonstration is taking place assuming that true motions exist, then considering these as virtual motions and assuming that the constraints are smooth. Rather than to show that L f = 0 for all virtual displacement is equivalent to f + r = 0, and then an existing equilibrium criterion is fulfilled, it is shown that to admit the motion is in contradiction with L f = 0 for all virtual displacements, using an argument of dynamic type. The asymmetry between the demonstration of the necessary and sufficient condition is not convincing for me. Besides, the use of virtual displacements taking place in real time does not permit the direct extension of the proof to the case of timedependent constraints and the case of an impulsive force. This form of proof of the sufficient part of the virtual work principle was probably introduced for the first time by Poisson in his Traité de mécanique of 1833 [200]. The central point of the proof lies in taking the dynamic assumption that the resultant forces Fi and displacements dri , which are generated by the absurd, necessarily have the same sense, and this assumption is not at all obvious, as will be explained further in Chapter 16. 2.1.1.2 Statics handbooks Generally theorem T1 or at most the necessary part of T2 , is proved, which is sufficient for applications. Instead of a system of particles, reference is made to a system 3

pp. 331–348.

2.1 The theorem of virtual work

25

of rigid bodies connected to each other and with the outside by a system of rigid hinged rods, thus tacitly admitting that any kind of constraint can be reproduced by an appropriate system of rods. Normally the following additional assumptions are made: 1) constraints (connecting rods) exert forces Rh (reactive forces) that have the same ontological status of the active forces Fk ; 2) the direction of the reactive forces Rh is that of the rods; 3) for each rigid body the equilibrium is defined by the satisfaction of the cardinal equations of statics between Rh and Fk ; 4) an infinitesimal displacement field is assumed, i.e. virtual displacements are coincident with virtual velocities (unless an inessential constant); 5) all forces and displacements are independent of time. Assumption 2 is a principle analogous to P2 because the rod imposes the point in touch with a body to move on a sphere and then the reactive force, being collinear to the rod, is orthogonal to the surface. The proof below differs a little from those normally presented in statics handbooks [285], because it avoids any recourse to matrix calculus which, making the proof automatic, the proof hides the nature of assumptions, implicit and explicit. As first, at least in many handbooks, it is shown that the work of a system of forces applied to a rigid body can be estimated from the resultants F and static moment MO – about a point O fixed – of forces, in the form: L = FuO + MO θ,

(2.4)

where uO is the virtual displacement of a reference point O and θ the rigid body rotation, according to Fig. 2.4. Having proved (2.4), indicating respectively with F a and MOa the resultant of forces and moments of active forces and with F r and MOr the corresponding quantities of reactive forces, it is easily shown that there is equilibrium for the body if and only if the relation holds true: L = (F a + F r )uO + (MOa + MOr )θ = 0,

y

(2.5)

y

MO

θ F

uO

x

O

Fig. 2.4. Force and virtual displacement for a rigid body

O

xx

26

2 Logic status of virtual work laws

for any virtual displacement uO , θ, i.e. that a generalization of theorem T1 is true, as can be proved into two steps: a) the condition L = 0 is necessary for the equilibrium. Indeed if there is equilibrium the cardinal equations of statics are satisfied: Fa + Fr = 0 MOa + MOr = 0

(2.6)

and then L = 0; b) the condition L = 0 is sufficient for the equilibrium. Indeed if L = 0 for any values of uO , θ, from (2.5) it is easy to see that the cardinal equation of statics (2.6) are satisfied. In the case of smooth constraints and inextensible rods, the virtual work of the reactive forces is zero and from (2.5) and (2.6), verified at equilibrium, one can derive the necessary part of the theorem T2 : L = F a uO + MOa θ = 0.

(2.7)

Usually nothing is said on the demonstration of the sufficient part of the theorem T2 , which is a bit more complicated. 2.1.1.3 Poinsot’s proof The examination of Poinsot’s proof, which will be discussed in more detail in Chapter 14, has presently a considerable interest because it influenced much of the demonstrations of statics handbooks and is considered as the best one ever given. The reasons for this success are essentially two: (a) Poinsot considers a reference mechanics where the equilibrium is determined by the balance of forces which can be expressed by simple analytical relations – equations of equilibrium of a particle – and (b), primarily, he uses virtual velocities, obtained considering a non-physical time, instead of virtual infinitesimal displacements. Poinsot takes for granted the assumption P3 , or more precisely its modified version, that does not require the concept of constraint reaction, whereby a particle is in equilibrium on a surface if and only if the applied force is perpendicular to it. In the words of Poinsot: Indeed, it is shown that if a point has no freedom in space other than to move on a fixed surface or line, there may not be equilibrium unless the resultant of forces which press it is perpendicular to the surface or the curved line [197].4 (A.2.3)

In fact he does not prove what he says and does not even have the opportunity to do so because his constraints have the ontological status of relations between the positions of points and are not bodies. Poinsot is not the only one to think it logically necessary that a constraint cannot resist tangential forces. Laplace is also convinced of this: 4

p. 467.

2.2 The principle of virtual work

27

The force of pressure of a point on a surface perpendicular to it, could be divided into two, one perpendicular to the surface, which would be destroyed by it, the other parallel to the surface and under which the point would have no action on this surface, which is against the supposition [156].5 (A.2.4)

The reasoning is not conclusive, in fact it reduces to the trivial tautology that the constraint is not acting in a tangential direction because it does not act in the tangent direction. Lagrange also expressed similar ideas: Now if one ignores the force P, and assuming that the body is forced to move on this surface, it is clear that the action or rather the resistance that the surface opposes to the body cannot act but in a direction perpendicular to the surface [149]. 6 (A.2.5)

But he seems to realize the problematic nature of the concept of smooth constraint because, often, in his writings he associates the constraints, expressed by mathematical relations, to constraints made of inextensible wires deprived of bending stiffness. In this case the orthogonality of the reaction to the surface – for example the spherical surface described by a material point with a wire connected to a fixed point – is more convincing, even if this evidence has in fact an empirical rather than logic basis, making reference to our everyday experiences. In addition to P3 , Poinsot considers other principles. The first principle is that of solidification, for which if one adds constraints – both internal and external – to a system of bodies in equilibrium, the equilibrium is not altered. The second principle, presented by Poinsot as the fundamental property of the equilibrium, states that necessary condition for the equilibrium of an isolated system of particles is that all the forces applied at various points can be reduced to any number of pairs of equal and opposite forces. The third principle is required by the second, even if not explicitly, and concerns the possibility to decompose a force into other forces using the rule of the parallelogram. A fourth principle concerns the mechanical superposition for constraints, for which if on a system of points there act more constraints, they are able to absorb the sum of the forces that each constraint is able to absorb separately. Based on these principles, he quite convincingly proves the principle P2 , with reference to the definition D∗1 ; more precisely he proves that if L = 0, M = 0, etc. are the relations among the coordinates x, y, z, x , y , z , etc., which define constraints, the reactive forces are orthogonal to the resulting constraints. The demonstration of the virtual work principle follows the same reasoning of the first part of § 2.1 and in my opinion is free from any criticism.

2.2 The principle of virtual work The term principle has a meaning not entirely unique, as is the case with all important concepts. It is the foundation of a science, which in turn may allow more than one set of principles. Among the principles, even at the time of Lagrange, there 5 6

vol. 1, p. 9. vol. 9, p. 378.

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2 Logic status of virtual work laws

were axioms, principles and theorems otherwise proved. Today this distinction no longer holds and there is the tendency to group all the principles under the term axioms. Compared to the views of the XIX century, essentially Aristotelian, now the premises, i.e. the explicans of the theories, are neither required to be certain nor the premises of the conclusions, i.e. the explicandum, to be better known. The first requirement is ignored because it would exclude almost all scientific laws, for which instead of a certain knowledge plausible conjectures are considered. The second requirement is usually ignored, and there are theories – such as atomic and quantum, for example – that explain relatively well-known phenomena with beings of indecipherable nature. Moreover, the boundary between what seems known and what does not, is largely the result of metaphysical and epistemological conceptions of the times and does not reside only in the object. With the use, things that were not known or obvious, become of public domain, an example of this are the concepts of force, atom, energy that sparked diffuse controversy when they were introduced. According to the modern epistemology therefore the principle of virtual work may also be accepted as a principle of mechanics if it proves to have sufficient logic strength to describe all the mechanics – of course combined with other axioms – and to produce results in agreement with the experimental evidence. In the following I will try to analyze in more detail the consequences of taking a VWL as a principle without any reference to another mechanical theory. There can be considered essentially three formulations, which gradually move away from what was presented as a theorem in the previous pages. It is posssible: a) to assume forces as the primitive quantities and virtual displacements that take place in a virtual time; b) to assume forces as the primitive quantities and virtual displacements that take place in the real time (in the same time with which the forces vary); c) to assume work as the primitive quantity that takes place in real time. In the following I will refer only to cases a) and c), being easy to extrapolate to case b) the considerations valid for the first two. With the usual ambiguity, which that should not bother us, the virtual displacements are treated as virtual velocities and virtual work as virtual power.

2.2.1 Force as a primitive concept 2.2.1.1 Equilibrium case Assuming some concept of force, even a pre-Newtonian one, that in principle can always be replaced by a weight attached to a rope. I introduce only the active forces, while the reactive forces do not appear explicitly, in the limit the concept can also be missed, which avoids many problems of both logical and ontological type. In the case of a system of n material points, with the symbols used in the preceding paragraphs, if f is the vector of the active forces and u the vector of the virtual velocities/virtual displacement – the virtual work is defined as the product f · u, and the principle of virtual work could be enunciated in the form of T3 :

2.2 The principle of virtual work

29

PP1 . A system of particles constrained to move on a manifold M is in equilibrium if and only if the virtual work of the active forces is zero for any virtual displacement. Note however that now, because there is no reference mechanics and a priori criterion of equilibrium, equilibrium is not intended as a balance of forces, but simply as rest. The principle can also be seen with Poincaré, as a methodological principle, a stipulation. If the applications to an empirical case do not work, it is always possible to say: it is because there are hidden forces, for example frictions. One might ask whether a mechanical theory based on PP1 is acceptable as follows: does it provide satisfactory results from an empirical point of view? Has it something different from Newtonian mechanics? The first thing that catches the eye is the extraordinary simplicity of the principle in the case of constrained systems. The idea of constraint reaction that creates difficulties should not be formulated. All breaks down in the examination of only the active forces. All the rules of simple machines and the rule of the parallelogram, which can be used for alternative formulations of mechanics, become simple theorems. The necessary part of the principle is falsified by every experience, however. That is, if a system is in equilibrium under a system of active forces f , it is not true that the virtual work is zero for every possible virtual displacement; it could be both positive and negative. An example of this is obtained by considering the equilibrium of a heavy object on an inclined plane. It is found empirically that, for a very rough surface there is equilibrium for a material point even when the plane is tilted several degrees, but the weight force can make a positive virtual work – for downward virtual displacements – and a negative virtual work – for upward virtual displacements. The sufficient part of the virtual work principle PP1 , i.e. if the virtual work is zero for each value of virtual velocities, then the system of particles is in equilibrium, seems instead to be always empirically verified. This is somehow a consequence of the principle P∗1 , in the sense that if the reactive forces r associated with smooth constraints are sufficient to maintain in equilibrium a system of points (L = 0), then the actual non-smooth constraints furnish effectively the reaction r and the system is in equilibrium. To treat the case with friction it is necessary to reformulate the virtual work principle by involving the forces of friction, that is the forces in the direction of the virtual displacement: PP2 . A system of particles constrained to move on a manifold M is in equilibrium if and only if the virtual work of the active and reactive forces is zero for any virtual displacement. 2.2.1.2 Motion case The question of applicability of the virtual work principle to motion arose also when it was considered as a theorem of a mechanics of reference, but in that case, just because there is a mechanics of reference, one could think that motion would be otherwise faced within this mechanics. Adopting the virtual work principle as a proper principle one should instead investigate whether it is possible to study the motion of

30

2 Logic status of virtual work laws

bodies and how. To use the virtual work principle in the case of motion it is necessary to operate in a mechanics where it is at least possible to define the mass and also to talk about inertial reference frames; in fact it seems necessary to assume most of the concepts of Newtonian mechanics. Supposing that these difficulties have been overcome, a possible statement of the virtual work principle could be the following: PP3 . The virtual work of the active and inertia forces for a system of material points constrained on a manifold M is always zero in any inertial reference system. (Here the term inertia forces must be considered merely a nominal definition of −ma.) Statement PP3 may seem asymmetrical with that represented by PP1 . In fact it can be divided into two parts, one necessary and the other sufficient. To calculate the virtual work in a factual situation, with a and f ‘real’, one gets L = 0; vice versa if one requires L = 0 for any virtual displacements, one obtains the real value of a. The falsification of the virtual work principle in the presence of constraints that are not smooth was not taken very seriously by scientists from the XVI to the XIX century. The problem of the correspondence of scientific theories to physical reality is probably as old as science, but it was put into evidence by Galileo Galilei. Guidobaldo dal Monte’s polemic on the law of isochronism of the pendulum proposed by Galileo is well known. Dal Monte, along with his contemporaries, argued that the law of isochronism was not verified in practice; Galilei argued instead that the ideal pendulum would obey the law. Then Dal Monte replied that physics must relate to the real world and not an imaginary ideal world, a ‘paper world’ [86].7 Today the position of Galileo is generally accepted, but it is also clear that it is not possible to abstract from accidents the essence of phenomena, mainly because one cannot always tell in advance which attributes are accidental and which ones are substantial. If in many situations, the friction is presented as an accident, which masks the substantial reality of the problem, in other situations this is no longer true. And the justification that the cases in which the presence of friction are important only to technology and not to science is senseless. Without entering the merits of theories for the study of the motion of constrained body systems, think of how strange would appear a world in which the virtual work principle holds true strictly; in this world without friction, life itself would be impossible. And though in some calculations friction can be neglected, in describing the substance of the world it must be considered. A theory that does not have the conceptual tools to address important issues must be considered unsatisfactory and its basic principles as incomplete. Then a mechanics with a rigid axiomatic structure with PP1 incorporated, cannot be the ‘mechanics’. The only solution to solve the aporia that appears – PP1 should be taken as an axiom but it cannot be taken as an axiom – is to adopt a liberal epistemology, not rigidly axiomatic, which may allow acceptance of PP1 in some cases and non-PP1 in others, without being able to decide which option to choose. The choice whether to apply PP1 or non-PP1 is so delegated to ‘moral’ considerations, in charge of the i, though not explicitly verbally. 7

Preface.

2.2 The principle of virtual work

31

2.2.2 Work as a primitive concept The interpretation of work as a primitive concept is perhaps the most interesting interpretation of the virtual work principle that has been revived in recent years in the international literature [328]. Taking, perhaps without a clear understanding of the historical reference, considerations of the late XIX century, it involves a complete revision of classical mechanics where the concept of work, which is taken as a scalar quantity susceptible of measurement, is accepted as primitive while the force is simply a definition. So there is a reversal from what happens in Newtonian mechanics, in which it is the force that is primitive and the work is a defined magnitude. This possibility is not completely counterintuitive. The idea of work and fatigue as physical magnitudes surely already appeared in Galileo [119] though it was only the energetic movement of the late XIX century that captured the attention of physicists. The mathematical formulation of this idea could take the following line: consider a system of material points S and a vector space V that contains the virtual motions (velocities or displacements) of S that are considered to be in real time. The imposition of an element of V to S gives the virtual work L, to be treated as a physical quantity that is in principle measurable. The assumption, empirical in nature, is that L is a linear form defined on V. The vector space F dual of V is the space of the forces. In other words, the components of the forces acting on a system of material points, whose motions are defined by the virtual components uk with respect to a fixed coordinate system, are those numerical values fk that determine the virtual work, according to the relation: L = ∑ fk uk . This formulation is valid for both discrete and continuous systems. In the case of a discrete system, in which V is a finite dimensional space, the foregoing considerations are substantially equivalent to those developed for the first time in Lagrange’s Mécanique analytique, when the generalized forces are introduced, with the important difference that there the generalized forces are defined in terms of other forces, regarded as primitive and known quantities. 2.2.2.1 Equilibrium case In this new formulation of mechanics the virtual work principle can be expressed either directly, without reference to forces, or indirectly, considering the forces that now are defined quantities. Moreover it can be a principle or can be obtained from a more fundamental law and therefore to appear as a theorem. In the following I treat only the first point of view, postponing the second to Chapter 18, dedicated to energetism. Considering the work of friction, it is possible to formulate the principle: PP4 . The equilibrium is possible if and only if the virtual work of all forces applied to the system is never positive, whichever the virtual displacement assumed.

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Here, as for PP1 , there is no reference to an a priori equilibrium criterion, and equilibrium is simply rest. In this formulation of the virtual work principle, the possibility of also considering the virtual work done by the reactive forces, i.e. the frictions, is not ruled out, even though in practical application it has to rely on the assumption of smooth constraints. The same difficulties in studying the problem of constrained bodies in the context of Newtonian mechanics are found but with an important difference: the virtual work is a primitive magnitude and therefore it seems more natural to characterize smooth and non-smooth constraints: PP5 . The virtual work made by the reaction forces cannot be positive. This characterization is implicit in the commonly accepted principle of the impossibility of perpetual motion. Consider, for example, a heavy material point that is bilaterally constrained to move on a rough plane, slightly sloped, and which is in equilibrium. The work Lr made by the reactive forces (frictions) is always negative, the work La of the weight can be both positive and negative. For a virtual downward motion it is Lr < 0 and La > 0, so it is possible to have L = Lr + La = 0; for a virtual upward motion it is Lr < 0 and La < 0 so that L = La + Lr < 0. 2.2.2.2 Motion case It is not entirely clear how to generalize principle PP4 to dynamics. In [328] it is suggested to consider the forces of inertia as ordinary forces. This suggestion, however, is controversial because, if work is the primitive concept, the concept of force, albeit of inertia, should be eliminated. Probably the only satisfactory way, from a logical and epistemological point of view, is to define kinetic energy and to add kinetic power (the time derivative of kinetic energy) to static power. This is what Duhem did, as in discussed in Chapter 18, somehow solving also the difficulty of introducing the concept of mass. With this addition it is possible to enunciate the following principle (only valid for smooth constraints): PP6 . In each moment and for any system of material points the virtual work, measured as the sum of that of all the efforts applied to the system and the kinetic power, is zero whatever is the considered virtual displacement.

3 Greek origins

Abstract. This chapter explains the meaning of the partition of Greek mechanics into Aristotelian and Archimedean. In the first part the Aristotelian mechanics is considered that identifies as a principle the following VWL based on virtual velocity: The effectiveness of a weight on a scale or a lever is the greater the greater its virtual velocity. In the central part the Archimedean mechanics is considered where there is no reference to any VWL. In the final part devoted to the late Hellenistic mechanics, the VWL of Hero of Alexandria is considered for which the possibility of raising a weight is determined by the ratio of its virtual displacement and that of the power. The law is presented not as a principle but as a simple corollary of equilibrium. In ancient Greece, mechanics was the science that dealt primarily with the study of equipment or machines (in Greek mhqan†), to transport and lifting weights, also as a response to other technological problems of the times. The search for equilibrium was not of practical interest – excluding the case of weighing by means of a balance – and mechanics, at least at the beginning, did not take care of it. From this point of view mechanics was very different from modern statics which is instead seen as the science of equilibrium. Pappus wrote that “the mechanician Heron and his followers distinguished between the rational part of mechanics (involving knowledge of geometry, arithmetic, astronomy, and physics) and its manual part (involving mastery of crafts such as bronze-working, building, carpentry, and painting)” [181].1 In the following, mechanics is only used in the sense of rational mechanics with Pappus’ meaning; as such its status was neither well-defined for its social appreciation nor for its epistemology. Regarding the social appreciation there were contrasting valuations. Mechanics concerned problems of everyday life, connected to manual work and as such considered negligible by the intellectual aristocracy. But its applications gave rise to wide interest. To cite the famous attribution to Archytas of Tarentum of a dove “which flew according to the rules of mechanics. Obviously it was kept suspended 1

p. 447.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_3, © Springer-Verlag Italia 2012

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by weights and filled with compressed air” [201].2 Aristotle appreciated the activity of mechanicians, as is clear from the prologue to his Mechanica problemata. To the contrary Plutarch (46–127 AD) in his Vitae parallelae, wrote that Archimedes felt ashamed for his interest in mechanics and wanted to be remembered only for his mathematical works. Even though Plutarch attributes his own conception to Archimedes, his opinion indicates the low consideration mechanics held in some circles [192].3 As far as the epistemological status is concerned, mechanics was considered by Aristotle as a mixed science: These are not altogether identical with physical problems, nor are they entirely separate from them, but they have a share in both mathematical and physical speculations, for the method is demonstrated by mathematics, but the practical applications belong to physics [12]. 4

This classification was used through all the Middle Ages. Archimedes probably did not share this opinion and considered mechanics as a branch of pure mathematics. The mechanics of Greek, as has happened in many areas of Western knowledge, is the basis of modern conceptions. Available sources are not numerous, but they are important. The earliest references are to the pythagorean Archytas of Tarentum (c 428–350 BC). For sufficiently precise written documentation, however, reference to Aristotle (384–322 BC), Euclid (fl 365–300 BC), Archimedes (287–312 BC), Hero (I century AD) and Pappus of Alexandria (fl 320 AD) is needed. In the following I first present the ideas of Aristotle, who is usually credited for a mechanics based on a law of virtual work. Then I introduce the principles of Archimedes’s mechanics, alternative to the Aristotelian and where there is no use of any virtual work laws. Finally, a hint of the mature Hellenistic mechanics, that although influenced more by Archimedes is also influenced by Aristotle.

3.1 Different approaches to the law of the lever 3.1.1 Aristotelian mechanics The principal Aristotelian treatises on mechanical arguments are the Physica, De caelo and the Mechanica problemata. They were largely studied and commented upon, both for their philological aspects and for their content. In the following I will give a very concise summary. Firstly I will consider the Physica and De caelo which describe motion according to nature (free motion) and motion against nature (forced motion) with both qualitative (causes) and quantitative (mathematical laws of motion) considerations. We could say that the context is quite general, it concerns all kinds of forces and bodies and can be defined as ‘theoretical physics’. I shall then consider the Mechanica problemata. In this treatise, which can be defined as ‘engineering based’, the approach is less systematic than the Physica and De caelo. 2 3 4

vol. I, p. 483. Marcellus. p. 331.

3.1 Different approaches to the law of the lever

35

3.1.1.1 Physica and De caelo The first Aristotelian thesis developed in the Physica and De caelo refers to the motion according to nature of a heavy body: it is downward along the line connecting the body with the centre of the world. The space traversed in the fall, in a given time, is directly proportional to the weight and inversely proportional to the resistance of the medium: Further, the truth of what we assert is plain from the following considerations. We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness [14].5 A, then, will move through B in time Γ, and through Δ, which is thinner, in time E (if the length of B is equal to Δ), in proportion to the density of the hindering body. For let B be water and Δ air; then by so much as air is thinner and more incorporeal than water, A will move through Δ faster than through B. Let the speed have the same ratio to the speed, then, that air has to water [14].6 A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement [13].7

The Aristotelian law on motion according to nature is assumed by most scholars with the mathematical relation v = p/r, where v is the velocity, p the weight and r the resistance of the medium. There are however some objections to this view [287],8 [349].9 Perhaps the main objection is that of regarding velocity as a definite kinematic quantity, summarizing space and time with their ratio, which to a modern is just velocity. This assumption is clearly anachronistic. Not only because in Greek mathematics there was no sense in the ratio between two heterogeneous quantities, like space and time, but also because there was no use for the quantification of velocity, which was, in fact introduced only as an intuitive concept, something that allowed one to say something is greater or lesser [14]10 [287].11 To the community of mechanics scholars, the first known writings on the quantification of velocity, which were presented within the mathematics of proportions, and on the systematic use of this quantification is commonly considered that by Gerardus de Brussel, in the first half of 1200, moreover expressed in a form not completely explicit as ‘petitiones’ of his famous book Liber de motu, where it is said the velocity (motus) is measured by the space traversed in a given time [127]. After Gerardus de Brussel many medieval and all Renaissance scholars read Aristotle as most modern scholars do. It must however be said that Aristotle in some places con5

IV, 8, 215a. IV, 8, 215b. 7 I, 6, 274a. 8 Chapter 7. 9 Chapter 9. 10 VI, 2; VII, 4. 11 Chapter 3. 6

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ceives of velocity as a well-defined kinematical quantity. This occurs, for example, in the preceding quotation of Physica, where velocity and resistance are considered as inversely proportional to each other, and in the following passage: For since the distinction of quicker and slower may apply to motions occupying any period of time and in an equal time the quicker passes over a greater length, it may happen that it will pass over a length twice, or one and a half times, as great as that passed over by the slower: for their respective velocities may stand to one another in this proportion [emphasis added] [14].12

where it is said quite clearly that velocity can be measured with space covered in a given time. But Aristotle does not develop this reasoning and any time he refers to mathematical laws he speaks about space and time separately and not about velocity. The second Aristotelian thesis on motion, clearly stated in the Physica, concerns the motion against nature of a heavy body: it occurs along a straight line and the space covered, in a given time, is directly proportional to the ‘force’ applied to the body and inversely proportional to its weight: Then, A the movent have moved B a distance Γ in a time Δ, then in the same time the same force A will move 1/2B twice the distance Γ, and in 1/2 Δ it will move 1/2B the whole distance for Γ: thus the rules of proportion will be observed. Again if a given force move a given weight a certain distance in a certain time and half the distance in half the time, half the motive power will move half the weight the same distance in the same time. Let E represents half the motive power A and Z half the weight B: then the ratio between the motive power and the weight in the one case is similar and proportionate to the ratio in the other, so that each force will cause the same distance to be traversed in the same time [14].13

The difficulty for the modern reader in the interpretation of Aristotle’s writing lies mainly in giving sense to ‘force’. Aristotle, at the beginning of book VII of the Physica explains which are precisely the kinds of forces to consider, but this is not enough to achieve a complete understanding: The motion of things that are moved by something else must proceed in one of four ways: for there are four kinds of locomotion caused by something other than that which is in motion, viz. pulling, pushing, carrying, and twirling [14].14 […] Thus pushing on is a form of pushing in which that which is causing motion away from itself follows up that which it pushes and continues to push it; pushing off occurs when the movent does not follow up the thing that it has moved: throwing when the movent causes a motion away from itself more violent than the natural locomotion of the thing moved, which continues its course so long as it is controlled by the motion imparted to it [14].15

Most scholars maintain that ‘force’ had the same meaning as it has today, though not in the Newtonian sense of cause of motion variation, but in the less demanding sense of muscular activity [287, 171, 305]. Expressed in modern terms, and synthesizing space and time into velocity, this position assumes the direct proportionality between 12 13 14 15

VI, 2, 233b. VII, 5, 249b. VII, 5, 243a. VII, 5, 243b.

3.1 Different approaches to the law of the lever

37

force ( f ), velocity (v) and weight (p); with a formula f = pv. Other scholars consider the modern concept of work as being the closest to ‘force’ [370, 295, 136]. This interpretation is strongly advised in the case of a thrown object, where it is difficult to see a force in the preceding sense. Even the fact that a force must act together against a resistance seems to confirm this position. Finally, there are those who think that to interpret Aristotle’s writings it is enough to make references to the common sense of an uneducated person. Some scholars indeed think that the learning process of a person resumes the historical process (the ontogenesis resumes the phylogenesis) and the scientific conceptions of classical Greece could be understood by assuming the identity of a youth who has not yet studied Newtonian mechanics [373, 359, 369]. This last position has the merit of averaging the two preceding, because sometimes it is more straightforward to translate ‘force’ with force, sometimes with work, and sometimes more with static moment. Before taking a position it must be said that the precise differentiation between force and work will occur only in the XVIII century, and as late as the XIX century ‘force’ will ambiguously be used to mean both force and work [129]. Moreover, it must be noticed that the various interpretations cannot be decided upon empirically. The experimental context needed to verify the Aristotelian laws of motion is different from that foreseen by the modern paradigm, the Newtonian for example. With Newton one has to observe the motion of a material point in a void space under a force with assigned direction and intensity. With Aristotle one has to study the motion of an extended body, which moves against resistances of the external medium which tends to oppose the applied force and maintains the body in a status of constant velocity. For Aristotle it is implicit that a resistance opposes a force, otherwise there would be no motion, or an infinity velocity motion would occur, which is impossible. The causes of resistance to a body’s motion are not made explicit by Aristotle; in a passage of De caelo, he attributes them to weight: Again, a body which is in motion but has neither weight nor lightness, must be moved by constraint, and must continue its constrained movement infinitely. For there will be a force which moves it, and the smaller and lighter a body is the further will a given force move it. Now let A, the weightless body, be moved the distance CE, and B, which has weight, be moved in the same time the distance CD. Dividing the heavy body in the proportion CE : CD, we subtract from the heavy body a part which will in the same time move the distance CE, since the whole moved CD: for the relative speeds of the two bodies will be in inverse ratio to their respective sizes. Thus the weightless body will move the same distance as the heavy in the same time. But this is impossible. Hence, since the motion of the weightless body will cover a greater distance than any that is suggested, it will continue infinitely [13].16

In other passages resistances are attributed even to the medium [14]17 and one would not be mistaken by assuming friction too as a resistance. When results furnished by Aristotelian laws of motion are compared with those of modern mechanics (Newtonian or Lagrangian), one sees that the Aristotelian laws are ‘true’ whatever the interpretation of ‘force’, when the parameter time is not considered. They are in general ‘false’ when this parameter is considered. For example, 16 17

III, 2, 301b. IV, 8, 215b.

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by interpreting ‘force’ as force one gets agreement only for equal intervals of time; by interpreting ‘force’ as work one gets agreement only if the parameter time is excluded completely. In what follows, for both motions, according to nature and against nature (natural and violent motions), I will refrain as much as possible from adopting a preconceived position on the ontological status of the various mechanical concepts, because often there is no need to do this and different interpretations do not necessarily conflict. When I have to choose I will opt for the (pre-Newtonian) common sense. 3.1.1.2 Mechanica problemata In the Mechanica problemata (known also as Mechanical problems, Mechanica) there is no explicit affirmation of the general theoretical principles contained in the Physica and De caelo, in particular no reference to the laws of natural and violent motion. Also for this reason and for its practical contents, the attribution of this treatise to Aristotle is still debated.18 In the following I will not enter into the merit of this attribution and, for the sake of simplicity, I will talk about Mechanica problemata as an Aristotelian work, instead of, as frequently seen, a pseudo-Aristotelian one. The writing is largely dedicated to the solution of problems, some mechanical in nature; they are referred to in Table 3.1 [296].19 The object of the mechanical problems is mainly the study of the shifting of heavy bodies. Nowhere in the text do the concept and the word equilibrium (isorropein) occur. The functioning of machines or devices, among them the wedge, pulley and winch, is reconnected to the lever. The validation of the law of the lever is suggested and may be the first in the history of mechanics. In the following I will comment on this validation. At the beginning Aristotle refers all the mechanical effects to the properties of the circle: Remarkable things occur in accordance with nature, the cause of which is unknown, and others occur contrary to nature, which are produced by skill for the benefit of mankind. Among the problems included in this class are included those concerned with the lever. For it is strange that a great weight can be moved by a small force, and that, too, when a greater weight is involved. For the very same weight, which a man cannot move without a lever, he quickly moves by applying the weight of the lever. Now the original cause of all such phenomena is the the circle; and this is natural, for it is in no way strange that something remarkable should result from something more remarkable, and the most remarkable fact is the combination of opposites with each other [12].20

The cause of the farthest points of a circle moving more easily than the closest under a given force is identified by Aristotle in the fact that in circular motion the compo18 During the Middle Ages and Renaissance the attribution of the Mechanica problemata to Aristotle was substantially undisputed. For the attributions in more recent periods see [15]. It is worth noticing that Fritz Kraft considers the Mechanica problemata to be an early work by Aristotle, when he had not yet fully developed his physical concepts [348, 378]. A recent paper by Winter considers Archytas of Tarentum as the author of Mechanica problemata [398]. 19 pp. 136–137. 20 pp. 331–333.

3.1 Different approaches to the law of the lever

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Table 3.1. Problems of Mechanica problemata 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Why larger balances are more accurate than smaller ones. Why the balance seeks the level position when supported from above, but not when supported from below. Why small forces can move great weights by means of the lever, despite the added weight of the lever. Why rowers in the middle of ships contribute most to their movement. Why the rudder, though small, moves the huge mass of the ship. Why ships go faster when the sail yard is raised higher. Why in unfavourable winds the sail is reefed aft and slackened afore. Why round and circular bodies are most easy to move. Why things are drawn more easily and quickly by means of greater circles. Why a balance is more easily moved when without weights than when weighted. Why heavy weights are more easily carried on rollers than in carts. Why a missile thrown from a sling moves farther than one thrown by the hand. Why larger handles move windlasses more easily. Why a stick is more easily broken over the knee when the hands are equally spaced, and farther apart. Why seashore pebbles are rounded. Why timbers are weaker the longer they are, and bend more when raised. Why a wedge exerts great force and splits great bodies. Why two pulleys in blocks reduce effort in raising or dragging. Why a resting axe does not cut wood, and a striking axe splits it. How a steelyard can weigh heavy objects with a small weight. Why dentists use forceps rather than the hand for extraction. Why nutcrackers operate without a blow. Why the lines traced by points of a rhombus are not of equal length. Why concentric circles trace paths of different length when rolled jointly on this or that circumference. Why beds are made in length double the width. Why long timbers are most easily carried by their centres. Why longer timbers are harder to raise to the shoulder. Why swing beams at wells are counterweighted. Why of two men carrying a beam, the man nearer the centre of the beam feels more of the weight. Why men move feet back and shoulders forward to rise from sitting. Why objects in motion are more easily moved than those at rest. Why objects thrown ever stop moving. Why objects move at all when not accompanied by the moving power. Why bodies thrown cannot move far, but related to the thrower. Why objects in a vortex finish at the centre.

nent against nature of the farthest points is proportionally less than that according to nature. Indeed the farthest points describe a larger circle and the motion in this circle is closest to the linear one, which is considered to be natural (for the meaning of this term in the Mechanica problemata see below). Let ABΓ be a circle, and from the point B above the centre let a line be drawn to Δ; it is joined to the point Γ if it travelled with velocities in the ratio of BΔ to ΔΓ it would move along the diagonal BΓ. But, as it is, seeing that it is in no such proportion it travels along the arc BEΓ. Now if of two objects moving under the influence of the same force one suffers

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3 Greek origins more interference, and the other less; it is reasonable to suppose that the one suffering the greater interference should move more slowly than that suffering less, which seems to take place in the case of the greater and the less of those radii which describe circles from the centre. For because the extremity of the less is nearer the fixed point than the extremity of the greater, being attracted towards the centre in the opposite direction, the extremity of the lesser radius moves more slowly. This happens with any radius which describes a circle; it moves along a curve naturally in the direction of the tangent, but is attracted to the centre contrary to nature. The lesser radius always moves in its unnatural direction for because it is nearer the centre which attracts it, it is the more influenced. That the lesser radius moves more than the greater in the unnatural direction in the case of radii describing the circles from a fixed centre is obvious from the following considerations [12]21 .

Shortly after Aristotle ‘proves’ his assertion by simple geometric considerations that refer to Fig. 3.1. In reality it must be said that Aristotle’s arguments make difficult reading. For instance the interpretation of the locution “according to nature” presents some problems. For Bottecchia [15]22 the motion according to nature is what happens along the circumference, while the motions along the tangent and the radius would be both against nature. But this makes the text unintelligible. The most common interpretation, which avoids this problem, assumes the motion according to nature along the tangent and the motion against nature towards the centre of the circle. Moreover to follow the Aristotelian classification of natural motion, Duhem suggests considering the natural downward motion from the horizontal radius, as shown in Fig. 3.2.23 For more considerations about natural and violent motions see the work by Christiane Vilain [394]. In the Aristotelian text there are also some other ambiguities, as for example the role played by the force, for which there is no precise indication on the direction of application and nature (muscular force or weight). For this purpose it is interesting to note the comment by Giuseppe Vailati, for whom any translation of the Aristotelian text is problematic because of the ambiguities of the Greek language: B

Δ E

Γ

A Fig. 3.1. Motion against nature in the circle

21 22 23

pp. 341–343. p. 67. Rotated ninety degrees clockwise, [15] p. 67.

3.1 Different approaches to the law of the lever

Δ

E

M

A

41

Ψ

Z X K Y B

Θ

Ω H

Γ

P

Fig. 3.2. Motion of points at different distances from the centre in the circle

‘Ambiguities’ that could be considered as a linguistic document of the belief in a primordial connection between the different compatible velocities of the various points the positions of which depend on each other, and the ease with which different points can be moved, other things being equal [391].24 (A.3.1)

For this reason I prefer not to comment on the wording of Aristotle because mine would be only one of the possible interpretations. In any case, perhaps, more interesting of the real intentions of Aristotle are the possible interpretations of a reader of the Mechanica problemata subsequent to Aristotle. I will return to this point in the following paragraphs and in the chapter on Medieval mechanics. Here I merely state the interpretation of Aristotle’s proof shared by most commentators. For them Aristotle proves two things: a) for a given amount of natural (vertical) motion, the motion against nature is greater for points closer to the centre of the circle (XZ against BP); b) for a given time, with the most distant points that describe longer arcs, the motion according to nature is greater for points further away from the centre (HK against ΘZ). This is considered enough to attribute to Aristotle the idea that a force applied (tangentially-vertically) to the points of the circle farthest from the centre has a greater effect because these points move faster then the closest, or, which is the same, that the effect of a force depends on the velocity of its point of application in a possible motion. But this is a law of virtual work, according to the meaning assumed in the book. It is however a qualitative law and as such useless to obtain mathematical expressions for equilibrium or motion. Aristotle transformed this qualitative law into a quantitative one in the attempt to solve the problem related to the law of the lever: “Why is it that small forces can move great weights by means of a lever?”. 24

p. 10.

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Table 3.2. Two versions of the lever law (problem 3) proof in Mechanica problemata A. Quoniam autem ab aequali pondere celerius movetur maior earum, quae a centro sunt; duo vero pondera quod movet et quod movetur. Quod igitur motum pondus ad movens, longitudo patitur ad longitudinem; semper autem quanto ab hypomoclio distabit magis, tanto facilius movebit. Causa autem est, quae retro commemorata est: quoniam quae plus a centro distat, maiorem describit circulum. Quare ab eadem potentia plus separabitur movens illud, quod plus ab hypomoclio distabit [162].25 B. But since under the impulse of the same weight the greater radius from the centre moves the more rapidly, and there are three elements in the lever, the fulcrum, that is the cord or centre, and the two weights, the one which causes the movement, and the one that is moved; now the ratio of the weight moved to the weight moving is the inverse ratio of the distances from the centre [emphasis added]. Now the greater the distance from the fulcrum, the more easily it will move. The reason has been given before that the point further from the centre describes the greater circle, so that by the use of the same force, when the motive force is farther from the lever [sic! correct: fulcrum], it will cause a greater movement. Γ

Α Η

Ε

Δ

Γ

Κ

Β

Δ

Fig. 3.3. The equilibrium in the lever Let AB be the bar, Γ the weight, and Δ the moving force, E the fulcrum; and let H be the point to which the moving force travels and K the point to which the weight moved travels [12].26

The main part of the solution Aristotle suggested for this problem is reported in Table 3.2 in two different translations: the first translation from Greek into Latin (A); a modern translation into English (B): A noteworthy aspect clearly stated in the Aristotelian text – “and there are three elements in the lever, the fulcrum, that is the cord or centre, and the two weights, the one which causes the movement, and the one that is moved” – is the assimilation of weight to a motive power, at least as far as the effects are concerned. And this is independent of the fact that the tendency of a heavy body to direct itself towards the centre of the world is associated with a force, interior or exterior, or it simply depends on the nature of the body. It is true that in theoretical treatises like the Physica and De caelo, Aristotle is reluctant to consider a weight as a motive power, but many of Aristotle’s successors, Latin and Arabic, perceived weight as a special kind of force. And it is certain that although weight was not put completely on the same footing as force, it was often treated as such [321, 186, 246, 343]. In the proof of the law of the lever, according to which “the ratio of the weight moved to the weight moving is the inverse ratio of the distances from the centre”, Aristotle assumes that the effect of a force or a weight vary linearly with its virtual velocity, i.e. linearly with its distance from the fulcrum, transforming so its qualitative 25 26

p. 4. pp. 353, 355.

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law into a quantitative one. This seemed quite natural to many later commentators, simply because it was usually enough to turn any laws into a linear proportion. It was not the same view of Bernardino Baldi (and Guido Ubaldo dal Monte), who while appreciating a lot of the Mechanica problemata denounces the lack of evidence of the quantitative law: Thus when Aristotle discloses the reason for which the lever moves a weight more easily, he says that this happens because of the greater length on the side of the power that moves; and this [accords] quite well with his first principle, in which he assumes that things at the greater distance from the centre are moved more easily and with greater force, from which he finds the principal cause in the velocity with which the greater circle overpowers the lesser. So the cause is correct, but it is indeterminate; for I still do not know, given a weight and a lever and a force, how I must divide the lever at the fulcrum so that the given force may balance the given weight. Therefore Archimedes, assuming the principle of Aristotle, went on beyond him; nor was he content that the force be on the longer side of the lever, but he determined how much [longer] it must be, that is, with what proportion it must answer the shorter side so that the given force should balance the given weight [19].27 (A.3.2)

Fig. 3.4. Simple machines of the Mechanica problemata [18]29 (reproduced with permission of Biblioteca Alessandrina, Rome)

3.1.1.3 A law of virtual work Some scholars, historians and scientists believe that the Aristotelian mechanics contains in a nutshell the modern principle of virtual work, at least as regards a formulation restricted to weights. In my opinion they attributed to much to Aristotle. Galileo was among the first to associate a law of virtual work, in the form of virtual velocities, to the Mechanica problemata: The second principle is that the moment and the force of gravity is increased by the speed of motion, so that absolutely equal weights, but combined with unequal speed, have strength, moment and virtue unequal, the faster and more powerful, according to the proportion of its speed to the speed of the other. […] Such equivalence between gravity and speed is found in all the mechanical instruments, and was considered to be, by Aristotle, a principle in his Mechanica problemata; so we can still take as a quite true assumption that absolutely

27 29

pp. 54–55. Translation by [298], p. 14. p. 6.

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3 Greek origins unequal weights alternately be counterbalanced and to make equal moment, every time their gravitas are in inverse proportion with the speed of motions, that is one is less heavy than the other, as much it is able to move more quickly than the other [115].30 (A.3.3)

He was however preceded by Giuseppe Moletti (1531–1588), professor of mathematics at Padua, just before Galileo. Moletti during his stay in Padua lectured on mechanics and used the Mechanica problemata as an authoritative text. Below some quotations from his lectures edited by Laird [351] are reported, related to the marvellous properties of the circle on the footsteps of Aristotle: But in order not to confuse us I shall first discuss the principle that Aristotle states for both the machines we have given as examples, that is, for the pulley, and the lever. [...] All the operations of machines, then, consist in their movement, and consequently the same machine will have a greater or a lesser effect to the extent that the movement that it makes is nearer to its own, proper movement. [...] Thus it is clear from the preceding demonstrations that the less a weight is constrained to move in a circle, or the farther a force is from the centre, with so much more speed it will move and so much more effect the force will have [351].31 (A.3.4)

Pierre Duhem goes further and sees the principle of virtual work also in the Aristotelian law of violent motion, for which force, speed, and weight are linked by mathematical relationships expressed by means of proportion. For example, Duhem believes he can derive the law of conservation of the product of weight for the height by the law of forced motion (see § 4.2). Like almost all interpretations of historical texts Duhem’s is subject to criticism, not because it is incompatible with the Aristotelian mechanics, but because the Mechanica problemata has such ambiguities that many interpretations are possible and it seems unlikely that the one based on modern categories, like those used by Duhem, may be the right one. For example, it is doubtful that Aristotle could have considered homogeneous the speed of a free body with that of a body hanging from the end of a scale that moves partially of natural motion and partially of violent motion. Giovanni Vailati shares, with some cautions, Duhem’s view limited to Mechanica problemata [391]. Ernst Mach is once again more cautious and says verbatim: Let us now consider some details. The author of the Mechanical problems mentioned on p. 511 remarks about the lever that the weights which are in equilibrium are inversely proportional to the arms of the lever or to the arcs described by the endpoints of the arms when a motion is imparted to them. With great freedom of interpretation we can regard this remark as the incomplete expression of the principle of virtual displacements [355].32

Note that the main feature of the laws of virtual work, including that of Aristotle, is that the effect of a force is somehow considered a posteriori in the sense that it depends not only on the force in itself, but also on the motion which is determined by the presence of constraints. It is possible that this idea may have come to Aristotle, or who for him, for his metaphysical conceptions of motion and rest, which identify the rest as motion in power and therefore somehow ‘aware’ of its future. But it is also possible, and it seems more likely, that the dependence on the effect of a force 30 31 32

pp. 72–73. pp. 86–87; 122–123. pp. 6–7.

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by its motion was obtained without any ideological mediation as an empirical intuition. The same goes for the laws of natural and violent motions. They are classified in the general Aristotelian metaphysics, but do not necessarily flow from it, rather it is more likely to be the result of empirical observations which were then absorbed in some way by Aristotle.

3.1.2 Archimedean mechanics Developments in Greek mechanics after Aristotle are poorly documented. There remain a few of the writings of Archimedes (287–212 BC) and Euclid (fl 300 BC). Then the writings of Hero (fl I century AD) and Pappus of Alexandria (fl 320 AD), which have educational purpose. In their treatises, Archimedes and Euclid mainly focused on theoretical foundations of mechanics, with particular emphasis on the demonstration of the law of the lever. The approach of the Mechanica problemata could not satisfy the mathematicians of Hellenistic periods, used to standards of no less rigor than the modern ones. They could not accept such a leap of logic that led from Aristotle’s empirical evidence, not problematic, of the increased effectiveness of the applied forces farther from the fulcrum of a lever, to the mathematical formulation of the law that this different effect is proportional to the virtual velocity and then the distance. Though equilibrium was not a central problem for applications of mechanics it became a central theoretical problem for Archimedes. He realized that once the problem of equilibrium was solved, the problem of rising was also solved. Indeed if a weight p equilibrates a weight q in a lever, a weight only slightly heavier than p will lift q. But there is an advantage of this shifting of the theory from transport to equilibrium, because the equilibrium is much easier to study in a rigorous way. In the following I will refer to only the approach of Archimedes, reserving the right to return to Hero and Pappus in the following paragraphs and to Euclid in Chapter 4, devoted to Arabic mechanics. Archimedes set his mechanical theory on a few suppositiones (suppositions, principles), partly empirical in nature, which certainly appear more convincing than the Aristotelian. His goal was not only to formulate the law of the lever, but also to address the equilibrium of extended bodies, which are found in his theory of hydrostatics. The equilibrium of a body, or a set of bodies, was reduced to the determination of its centre of gravity and to make sure it is well constrained. Today there remains only a text where Archimedes treats the problem of equilibrium, the Aequiponderanti (in English, On the equilibrium of plane figures). However, there are indications that he wrote some other texts, or that the Aequiponderanti is only a part of a larger treatise. See for example the writing by Hero referred to in the following. Archimedes was the first scientist to set rational criteria for determining centres of gravity and his theory was the first known physical theory formalised on a mathematical basis. For my purpose, I shall mainly examine Book I of the Aequiponderanti [10, 11, 291, 287] where Archimedes, besides studying the rule governing the law of the lever, also evaluates the centres of gravity of various geometrical plane figures. He gave the basic elements of the theory of the centre of gravity, establishing seven

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suppositiones, shown in Table 3.3 [11].33 Using them Archimedes is able to prove thirteen propositiones (propositions, theorems), shown in Table 3.4 which allows us to calculate the centre of gravity of composed bodies, starting by knowing the centres of gravity of the simple bodies from which they are formed. Tables 3.3 and 3.4 show that the organisation of Archimedes’ theory differed from that used by Euclid. For example, Archimedes does not intend to develop a complete theory of mechanical science and write a detailed treatise. Instead he concentrates on facing a problem, important but unique: the determination of the centres of gravity for bodies of any shape. Archimedes draws on Euclid’s Elements, which Table 3.3. The suppositiones of Archimedes’ centre of gravity theory S1 S2

S3 S4 S5

S6 S7

Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken. When equal and similar plane figures coincide if applied to one another, their centres of gravity similarly coincide. In figures which are unequal but similar the centres of gravity will be similarly situated. By points similarly situated in relation to similar figures I mean points such that, if straight lines be drawn from them to the equal angles, they make equal angles with the corresponding sides. If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be in equilibrium at the same distances. In any figure whose perimeter is concave in (one and) the same direction the centre of gravity must be within the figure.

Table 3.4. The first seven propositiones of Archimedes’ centre of gravity theory P1 Weights which balance at equal distances are equal. P2 Unequal weights at equal distances will not balance but will incline towards the greater weight. P3 Unequal weights will balance at unequal distances, the greater weight being at the lesser distance. P4 If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity. P5 If three equal magnitudes have their centres of gravity on a straight line at equal distances, the centre of gravity of the system will coincide with that of the middle magnitude. Cor. 1. The same is true of any odd number of magnitudes if those which are at equal distances from the middle one are equal, while the distances between their centres of gravity are equal. Cor. 2. If there be an even number of magnitudes with their centres of gravity situated at equal distances on one straight line, and if the two middle ones be equal, while those which are equidistant from them (on each side) are equal respectively, the centre of gravity of the system is the middle point of the line joining the centre of gravity of the two middle ones. P6,7 Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balance at distances reciprocally proportional to the magnitudes.

33

pp. 189–192.

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were widely accepted at the time. The order of the propositions follows a basic logic, similar, although not identical, to those of Euclid. It advances from the simpler cases to those more complex. In Archimedes’ work however, the axiomatic structure and the concatenation of the propositions are less evident. The first four propositions for example are demonstrated directly from the suppositions and are not consequent to one or the other; the second after the first, the third after the second, etc. The main difference between the organisation of Euclidean and Archimedean theories can be found in the epistemological status of the principles. Indeed not all of Archimedean principles are ‘necessary’ as Euclid’s ones; some of them are indeed simply true or empirically evident [276, 374]. A fundamental supposition is the empirical assertion according to which if equal weights are positioned on a lever, at different distances from a fulcrum, there is no equilibrium but instead a tendency for the more distant weight to move downward. At first glance such an assertion could be interpreted in a dynamical key, holding that the greater weight has the higher tendency to move downward, thus breaking the horizontal equilibrium. But it could also be seen as a matter of fact which does not necessarily need to be explained. Some Archimedes’ suppositions seem to have a level of evidence more or less analogous to that of certain postulates of Euclid’s Elements. For example, the first part of S1 and S4 could derive from the principle of sufficient reason; the epistemological state of S5 and S6 is instead difficult to determine. Note that the suppositions S4 , S5 , S6 and S7 refer to the centre of gravity concept which was not introduced explicitly by Archimedes in the Aequiponderanti. When one considers the order of suppositions and propositions, a degree of organisational coherence is evident between the first four suppositions and propositions, as they complete each other. It seems that Archimedes wanted to reduce as much as possible the content of the suppositions, declaring only the parts impossible to demonstrate – either because self-evident or because empirically evident – leaving to the propositions the role of making precise the whole concept. In particular, the first part of supposition 1 and the whole of proposition 1 refers to the two sides of the implication: equal weight ↔ equilibrium. And that Archimedes considers supposition 1 as evidently known (equal weights → equilibrium), which provides the sufficient condition for equilibrium and not proposition 1 (equal weights ← equilibrium), which provides the necessary condition is a questionable choice. From a purely logical point of view, Archimedes could have chosen proposition 1 as the first part of supposition 1. But Archimedes’ actual choice is more convincing because it can be considered as self evident, or when this is not the case, supposition 1 is easier to be verified experimentally than proposition 1. The nature of the suppositions being not completely self evident, it seems more natural to deny their opposites than to affirm them, leading naturally to the use of the reduction ad absurdum, rarely practiced by Euclid, as the preferred kind of proof. In what follows I report the demonstration of the first proposition to show the typical way of Archimedean argument by reduction to the absurd, then I will take back the sixth proposition on the law of the lever.

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Table 3.5. Formal expressions of proof for proposition 1 Assumptions and suppositions (S)

Propositional logic

Content

1 – Negation of proposition 1 2 – 1 + S3 3 – S1 4 – Absurdum from 2 and 3

¬(E → U) or ¬U ∧ E U ∧ ¬E or ¬(U → E) U →E E →U

Weights are not equal Equal weights are not in equilibrium Equal weights are in equilibrium Weights are equal

Weights which balance at equal distances are equal. For if they are unequal, take away from the greater the difference between the two. The remainder will not then balance [supposition 3], which is absurd [supposition 1]. Therefore the weight cannot be unequal [11].34

In order, I shall use the formalism of classic propositional logic which enhances the understanding of Archimedes’ assumptions. Proof of proposition 1 can be synthesised using the following Table 3.5 [276]:35 In the table the following symbols are used, U: equal weights, E: equilibrium, while ¬(E → U) ↔ ¬U ∧ E and U ∧ ¬E ↔ ¬(U → E) are trivial theorems of propositional logic. 3.1.2.1 Proof of the law of the lever Here the proof of proposition 6 is reported, according to which two heavy bodies with commensurable weights (the ratio is a rational number) balance when they are suspended at distances inversely proportional to weights. The proof of proposition 7, the case of not commensurable weights, which makes use of the exhaustion method, is not presented, because both less interesting and more problematic. The main reason to concentrate on the original Archimedean proof is that in many texts in the history of science, the demonstrations of the lever attributed to Archimedes, are actually often only a rough paraphrase. This is also due to the interpretation of Ernst Mach, which I will report shortly thereafter. Propositions 6, 7. Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balance at distances reciprocally proportional to the magnitudes. I. Suppose the magnitudes A, B to be commensurable, and the points A, B to be their centres of gravity. Let DE be a straight line so divided at C that: A : B = DC : CE. We have then to prove that, if A be placed at E and B at D, C is the centre of gravity of the two taken together. Since A, B are commensurable, so are DC, CE. Let N be a common measure of DC, CE. Make DH, DK each equal to CE, and EL (on CE produced) equal to CD. Then EH = CD, since DH = CE. Therefore LH is bisected at E, as HK is bisected at D. Thus LH, HK must each contain N an even number of times. Take a magnitude O such that O is contained as

34 35

p. 190. p. 89.

3.1 Different approaches to the law of the lever

A

49

B O

E

C

H

L

D K

Fig. 3.5. The lever many times in A as N is contained in LH, whence A : O = LH : N. But B : A = CE : DC = HK : LH. Hence, ex aequali, B : O = HK : N, or O is contained in B as many times as N is contained in HK. Thus O is a common measure of A, B. Divide LH, HK into parts each equal to N, and A, B into parts each equal to O. The parts of A will therefore be equal in number to those of LH, and the parts of B equal in number to those of HK. Place one of the parts of A at the middle point of each of the parts N of LH, and one of the parts of B at the middle point of each of the parts N of HK. Then the centre of gravity of the parts of A placed at equal distances on LH will be at E, the middle point of LH (Prop. 5, Cor. 2), and the centre of gravity of the parts of B placed at equal distances along HK will be at D, the middle point of HK. Thus we may suppose A itself applied at E, and B itself applied at B. But the system formed by the parts O of A and B together is a system of equal magnitudes even in number and placed at equal distances along LK. And, since LE = CD, and EC = DK, LC = CK, so that C is the middle point of LK. Therefore C is the centre of gravity of the system ranged along LK. Therefore A acting at E and B acting at B balance about the point C [11].36

The demonstration opens with the assertion that to prove proposition 6 it is enough to show that the centre of gravity of the two weights A and B coincides with the fulcrum of the lever. The first part of the proof is purely geometric. Given the extremes E and D of the lever with fulcrum C, extends to the right of DK = EC and to the left of EL = CD, so that C is the midpoint between L and K. Then choose the point H that satisfies the ratio CD : CE = LH : KH, involving HD = DK, HE = EL. Since CD and CE are commensurable, LH and KH are also commensurable. Let N be the measure in common between CD and CE, and let n = CE / N and m = CD / N, it will be also LH / N = 2n, HK / N = 2m. Divide then HL into 2n parts and HK in 2m and at the midpoint of each part put a weight O = A/2n = B/2m. We have so many equal weights evenly distributed on LK: 2n of them centred on E and 2m centred on D, as clear from Fig. 3.6 for the case of n = 4, m = 3. There are now two situations, a lever 1 (DE), with weights A and B hanging from D and E and a lever 2 (LK), with 2(n + m) equidistant weights O. At this point Archimedes can apply his suppositions, showing first that the lever 2 is balanced, then that lever 1 is equivalent to lever 2 and thus balanced too. Lever 2 is balanced 36

pp. 192–193.

50

3 Greek origins

C L

H

N

E

O

A

B

2n

2m

D

K

Fig. 3.6. The lever

because for corollary 2 of proposition 5, 2(n+m) weights have their centre of gravity in C. As far as lever 1 one can say that the 2n weights O on the left (with 2nO = A) of the lever 1, have E as centre of gravity, they are so ‘equal’ to A, and the 2m weights O on the right, have D as their centre of gravity and are so equal to B. They can thus be replaced by the two weights A and B to obtain a lever that is in equilibrium for supposition 6. The demonstration of Archimedes has been criticised since ancient times. Lagrange enumerates the scientists who, in modern time, tried to improve the demonstration: Stevin, Galileo, Huygens, Daniel Bernoulli. Lagrange’s position is almost clear; he believes that the law of the lever cannot be entirely deduced a priori but it is also based on empirical principles. Moreover he believes that the demonstration by Archimedes, is, all considered, to be preferred to those proposed by other theorists, because “It should be said that by altering the simplicity of this proof it is added quite nothing of exactitude”[148].37 In modern times the criticism that had the greatest success is that of Ernst Mach, who accused the proof of Archimedes of circularity (that is the proof implicitly assumes what needs to be proved) [355].38 I do not quote here Mach’s considerations, I simply point out that only recently it has been shown clearly that they are inconsistent by Toeplitz, Dijksterhuis, Stein, Goe, Suppes [317].39 For Galletto, who has conducted an in-depth study of its logical status, the demonstration of Archimedes is correct in the sense that it follows deductively by his suppositions without any logic error. If there should be be any criticism, then it might concern the plausibility of the suppositions, but this is another story. Galletto, however, acknowledges some gaps in the Archimedean text, including the failure to introduce the concept of centre of gravity. Together with other authors, including Vailati, Galletto believes that the concept of gravity was defined by Archimedes in some other work, now lost. He believes that the concept used is that reported by Pappus of Alexandria, for which the centre of gravity is the point of suspension of neutral equilibrium.

37 38 39

p. 3. pp. 9–11; 512–513. pp. 470–472.

3.2 The mechanics of Hero of Alexandria

51

3.2 The mechanics of Hero of Alexandria About the life of Hero of Alexandria, mathematician and engineer, author of fundamental treatises of mathematics, mechanics, optics, etc., almost nothing is known, there is only widespread consensus that he lived in the first century AD [330].40 Hero was among the first researchers to combine theoretical knowledge with the technical; to make a modern parallel one can compare him to the engineers of the École polytechnique in the early XIX century. He is the heir and successor of a major scientific revolution that occurred in the Hellenistic period, the representatives of which in mechanics are Ctesibus (fl 285–222 BC) and Philo of Byzantium (c. 280 - c. 220 BC), which sees its climax with Archimedes (c. 287 - c. 212 BC) and the sunset with Pappus of Alexandria (c. 290 - c. 350 AD) [379]. Although Hero’s originality is not comparable to that of Archimedes, in him there is a more complete summary of theoretical and practical knowledge, in particular between mathematics and mechanical practice. Hero’s writing that contains a theoretical study of mechanics is the Mechanica [330]. The work has been received in its entirety only in the Arabic version of Qusta ibn Luka. According to Carra de Vaux the manuscript was carried in Europe by Jacob Glolius (1596–1667) at the beginning of the XVII century and translated by him into Latin; unfortunately this translation was lost [131].41 A summary of Golius work was published by Anton Brugmans in 1785 [52]. Of Qusta ibn Luka’s manuscript there are today two Western translations,42 one into French by Carra de Vaux [131], to which I shall refer, and another into German with a few fragments in Greek added [132], the examination of which gives the impression that the Arabic text is not faithful to the original Greek. The Mechanica is generally regarded as a compilation and dissemination text, but it is not unlikely that Hero has made his way; in any case the text is a testimonial of an impressive accumulation of knowledge of which no other documents remains. It is divided into three books, the first two of a theoretical nature, the third more applicative, that brings considerably complex war machines and lifting of weights. Some of them are shown in Fig. 3.7. In the first book, problems of general mechanics are dealt with, also of kinematics, including the problem of the wheel of Aristotle, issue 24 of the Mechanica problemata. The book ends with an analysis (incorrect) of the inclined plane (see subsequent sections), the weight distribution of a beam on supports and some considerations on the centres of gravity: He said: the centre of gravity or point of inclination is such a point that, when a load is suspended at it, it is divided into two equal parts. Therefore Archimedes and his followers in mechanics have particularized this theorem and distinguished between the point of suspension and the centre of gravity [131].43 (A.3.5) 40

p. 25. p. 8. 42 There is also an English version by Jutta Miller [133] on the web page of the Max Plank Institute, Berlin. 43 p. 73. Translation in [133]. 41

52

3 Greek origins

Fig. 3.7. Hero’s machines for the lifting of large weights and a press [131]44 (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

The second book is dedicated to the so-called Hero’s five machines, shaft with the wheel, lever, block and tackle, wedge and screw (the screw was not named in Aristotle’s Mechanica problemata and probably it is an invention of the Hellenistic period). Missing from the list is the inclined plane which is treated separately. Hero refers to them as the powers (d‘namic). The fact that from all the devices of the technology of the times Hero concentrated only on five of them, very different in form from each other, could be explained because with their combination all the machines used in practice can be obtained. The introduction of the wedge, generally not used in combination with other machines, could be explained by the fact that the screw is reducible to it. The exclusion of the inclined plane is not so easy to explain. In both the first two books the influence of Aristotle and Archimedes is evident. Aristotle and the Mechanica problemata were never mentioned in the text, but there is the reference of the operation of all machines to the circle, at least in principle,

44

a) p. 167; b) p. 169; c) p. 172; d) p. 182.

3.2 The mechanics of Hero of Alexandria

53

and the idea of increasing the effectiveness of forces with the increasing of speed. Towards the end of the book there is also a list of seventeen problems, which have a formulation very close to those of the Mechanica problemata. Archimedes was named many times as the author of contributions not present in his extant writings. For examples to Archimedes, as well as the demonstration of the law of the lever, is also attributed that of the angular lever: Some have thought that inverse proportionality is not present in irregular scales. Let us therefore also imagine a differently heavy and dense scale beam of any material that is in equilibrium when it is suspended at point γ. Here, we understand with equilibrium the rest and standstill of the scale beam, even if it is inclined to any side [emphases added]. Then we suspend weights at random points, namely δ and , and we let the beam again be in balance after their suspension. Now Archimedes has proven that also in this case weight to weight is inverse to distance to distance [131].45 (A.3.6)

γ

ε

δ

η

ξ

θ

Fig. 3.8. The angular lever

Hero attributes to Archimedes also a book on the supports: It is now urgently needed to give some explanations concerning pressure, transport and support with regard to quantity, as are suitable for an introduction. For Archimedes has already adopted a reliable procedure on this part in his book with the title Book of Supports [131].46 (A.3.7)

Of course, given the large interval of time between Hero and Archimedes, Hero’s quotations are not first-hand and therefore must be considered with caution.

3.2.1 The principles of Hero’s mechanics Hero declares continuously that all the wondrous and paradoxical effects of all the simple machines can be explained by means of the lever, the law and reason of which are attributed exclusively to Archimedes. The circle is also assumed as an explanatory model, but it is submitted to the lever.

45 46

pp. 88–89. Translation in [133]. p. 77. Translation in [133].

54

3 Greek origins

δ

β

ζ

α

ε

θ

γ

η

Fig. 3.9. Two circles with he same centre Let us assume two circles around the same centre [Fig. 3.9], namely point α, whose two diameters are the lines βγ and δ. Let the two circles be mobile around the point α, their centre, and let them be perpendicular to the horizon. If we now suspend at the two points β and γ equal weights, namely η and ζ, then it is clear that the circles do not incline to any side, since the weights ζ and η are equal and the spaces βα and αγ are also equal, so βγ is a scale beam that can be set in motion around the point of suspension, namely point α. If we now shift the weight at γ and suspend it at , then the load ζ will sink and set the circles in rotation. If we however increase weight θ, then it will again keep the balance of weight ζ and load θ then relates to load ζ like the distance βα to the distance α and we thus imagine the line β as a balance that can be set in motion around the point of suspension, namely point α. Archimedes has proven this in his work on the balancing of inclination. From that it is evident that it is possible to move an immense bulk with a small force [131].47 (A.3.8) That the five powers that move a load are similar to circles around one centre is proven by the figures that we have designed in the preceding; but it appears to me that they look more similar to the balance than to the circles [emphasis added], because in the preceding the bases of the proof for the circles resulted from the balance. For it was proven that the load suspended from the smaller side relates to the one suspended from the larger side like the larger scale beam to the smaller one [131].48 (A.3.9)

But Hero’s claims to reduce all the machines to the lever seems to me in some cases only a rhetorical artifice. In fact, besides the law of the lever Hero uses other principles to explain the functioning of the block and tackle and the wedge, at least this is my opinion and I am not convinced of the frequent attempt of historians to justify them somehow with the lever. The block and tackle is explained simply by assuming the additivity of equal parallel forces due to different pieces of ropes. Consider for example the block of pulleys of Fig. 3.10a [131].49 Hero affirms that the ratio between the power and resistance (weight) is 1 : 6 because each of the six pieces of the rope sustaining the weight is tied equally by the power and there are six pieces of rope. To explain why the rope has a constant tension it would be possible to see each pulley as a balance with fulcrum in its centre and equal arms which is equilibrated 47 48 49

pp.106–107. Translation in [133]. p. 127. Translation in [133]. p. 99.

3.2 The mechanics of Hero of Alexandria

55

κ η ζ

Fig. 3.10. System of pulleys (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

only if the tension of the rope at its extremities are equal, but it seems to me more natural to assume as a matter of fact that the tension of the rope is constant. In the following quotation Hero explains the functioning of a simple block of two pulleys, one fixed and one mobile, as shown in Fig. 3.10b. Let us now imagine a different weight at ζ and fasten to it the pulley η, pull over this pulley a rope and tie its two ends to a firm crossbeam, so that the weight ζ floats, then each of the two tightened parts of the rope lifts the weight of half the load. If somebody now unties the one end of the rope tied at k and stands there himself and holds the rope, then he carries half the load and the whole load is twice the force that holds it [131].50 (A.3.10)

For the wedge Hero’s reasoning is not completely clear to me and thus I prefer not to report it. It seems however that he assumed as an explanatory principle a law of virtual work according to which the efficacy of the wedge depends on the ratio of transversal and longitudinal displacements or velocities. Hero’s failure to reduce all the principles to the lever can be considered as a defect for those who assumed that science should have an axiomatic structure. But it can also be seen as the effect of a pragmatic epistemology, that allows a prolific approach to mechanics. 3.2.1.1 A law of virtual work While there is no doubt that Hero uses a quite advanced form of virtual displacement law, it is not easy to decide on its logic status. There is no doubt that Hero, to explain the operation of machines (with some reservations for the block and tackle, wedge and the screw), refers to the law of the lever. But to this explanatory principle Hero joins a kinematical analysis, which suggests a law of virtual work. Notice that the kinematical analysis is not a simple geometric exercise; it is required by the nature 50

pp. 115–116. Translation in [133].

56

3 Greek origins

itself of mechanics which is the science of lifting and shifting of heavy bodies, and the relative displacement of the weight to be lifted and the applied power has a technological relevance. This remains true even when, mainly for the sake of simplicity, the search of the moving power is replaced by the search for the equilibrating power. And what was said above about Hero’s approach is valid also for his followers in the Renaissance, as for example Guidobaldo dal Monte and Simon Stevin. To compare Hero’s approach with that of Aristotle, one should refer to the law of the lever. In the Aristotelian text the law of the lever is a theorem derived from the principle of virtual velocities. From the inverse relationship between weight and speed, with a simple geometrical reasoning, the inverse relationship between weights and distances follows. In Hero’s text, instead the law of the lever is a principle. From the inverse relationship between weights and distances it is possible with simple geometric reasoning to obtain the law of virtual velocities or, which is the same, of virtual displacements as a theorem. The same reasoning applies to all machines. On this matter it is of some interest to confront points of view different from mine, i.e. the comments of Clagett who sees Hero’s virtual work as a principle [287],51 Duhem who consider Hero as substantially Aristotelian [305]52 and Giusti [333] who considers Hero as essentially Archimedean. In the following quotation, which deals with how to raise 1000 talents with only 5 talents, Hero sets out clearly what is stated above: A delay occurs however with this tool and those similar to it of great power, because the smaller the moving force is in relation to the load to be moved, the more time we need [emphasis added], so that force to force and time to time are in the same (inverse) ratio. An example for this is the following: Since the force in wheel β was two hundred talents and it moved the load, one requires one rotation for the rope wound around α to wind up, so that the load through the motion of wheel β moves the amount of the circumference of α.

Fig. 3.11. A series of shafts with the wheel connected in series (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome) 51 52

Chapter 1. vol. 1, p. 186.

3.2 The mechanics of Hero of Alexandria

57

If it is moved, however, through the motion of cogwheel δ, the wheel on γ has to move five times for the axle a to move once, because the diameter of β is five times the diameter of the axle γ. Thus five times the amount of γ is equal to a single β, if we make the respective axles and wheels equal to one another. But if not, then we find a proportionality similar to this one. The cogwheel δ moves at β and the five revolutions of δ take fivefold the time of 40 talents. Thus the ratio of the moving force to the time is inverse. The same shows with multiple axles and multiple wheels and is proven in the same way [131].53 (A.3.11)

Notice that Hero in the previous quotation makes reference to time, but Clagett contends that it is quite clear from the text that one can identify time with covered space [287]. So one can read a virtual work law according to which the ratio between the moving force and the moved weight is inverse to the ratio of the corresponding covered spaces. In the following quotations Hero refers with some emphasis to ‘ralentissement de la vitesse’ which occurs in all the machine where with a small force a heavy body is raised. A delay occurs however with this tool and those similar to it of great power, because the smaller the moving force is in relation to the load to be moved, the more time we need, so that force to force and time to time are in the same (inverse) ratio. […] That the delay also occurs with this tool [131].54 (A.3.12)

The interpretation of time with space sustained by Clagett is contrasted by Mark Schiefsky [380], who sustains – really with some oscillations – that Hero does not make reference to a single machine and compares the space covered by power and resistance in a given time, as Clagett thinks. He should instead make reference to two distinct machines with different mechanical performance for which it is a matter of fact, easily verifiable in practice, that the machine with a lower performance will take less time than a machine with a greater performance to lift the same weight at the same height. If the interpretation of Schiefsky were correct it would be more difficult to see in Hero some form of virtual work law. It remains the fact that a reader subsequent to Hero can read him as Schiefsky, but also as Clagett – and me – and derive from Hero a virtual law. The law of virtual work is as a matter of fact contained in Hero’s work, independently of his intentions. In any case there is at least a circumstance where Hero uses clearly the virtual work displacement law and as a principle: to explain why a lesser force is needed to displace a weight hung by a wire when one presses farther from the hinge. This can lead to the conclusion that somehow Hero could consider the virtual work law as an explanatory principle, where the use of the circle or the lever becomes difficult to handle. It should be stressed however that he did not apply, unfortunately, any virtual work law to the inclined plane. Let us, for instance, imagine the firm support that the load is suspended from at point α and let the rope be the line αβ. Let us now draw the line αγ perpendicular to αβ and let us assume on line αβ two randomly positioned points, δ and . […] Thus if we pull the load from , it comes to κ; if we pull it, however, from point δ, then it reaches η, so that the load 53 54

p. 132. Translation in [133]. pp. 131–132, 134. Translation in [133].

58

3 Greek origins is lifted higher from point δ than from point . The load, however, that is lifted to a higher point, strains the force more than the one lifted to a lower point, because the one lifted to a higher point takes more time [131].55 (A.3.13)

3.2.1.2 Hero’s inclined plane law In the following I report the treatment of the inclined plane by Hero, that even though not associated in any way to virtual work laws is important because the explanation of its operation affects the entire history of mechanics until at least to Galileo. Hero speaks briefly of the inclined plane apart from other machines in the first book, without illustrations and without any explicit quantification. The reasoning is quite simple. Consider Fig. 3.12 [219]56 in which a cylinder is placed on the inclined plane BC. The vertical plane FD divides the cylinder into two parts. Hero argues that the right side of FED is balanced by an equivalent part FHD of the left side, and therefore the only part that must be supported by an external force is the lunula highlighted in gray in Fig 3.13. The explanation of Hero is clearly wrong according to modern conceptions of statics, because the body DHFE although symmetric with respect to the vertical is not balanced. However, it is clever, interesting for its high rhetoric value and provides results that are not easily contestable by experience. In particular, it provides a zero force if the plane is horizontal and a force equal to the weight of the entire cylinder if the plane is vertical. However it should be said that the text contains some ambiguity, as Hero does not speak explicitly of a force parallel to the inclined plane, but rather a force needed to pull the cylinder up: We will make recourse to some power or weight applied to the other side, to equilibrate the given weight, so that an excess of power prevails on the weight and pulls it upward [131].57 (A.3.14)

α

γ ζ θ

δ η ε

β Fig. 3.12. Force applied transversely to a pendulum 55 56 57

pp. 149–150. Translation in [133]. p. 41. p. 71.

κ

3.3 The mechanics of Pappus of Alexandria

59

F

E

I

C

H G

D K

A

B

Fig. 3.13. Hero’s proof of the inclined plane law

3.3 The mechanics of Pappus of Alexandria The last contribution of Greek mechanics is that of Pappus of Alexandria (IV century AD). Of the great work of Pappus, in eight books and entitled Synagoge or Mathematical collections, we possess only an incomplete part, the first book being lost, and the rest having suffered considerably. The last part, Book VIII, treats principally of mechanics, the properties of the centre of gravity, and some mechanical powers. Interspersed are some questions of pure geometry. Of Book VIII with the Greek version, translated into Latin by Commandino and Hultsch [181, 182] there exists also an Arabic version translated into English by David Jackson [183].58 The Arabic version is more complete than the Greek one and probably closer to the original; on the other hand the Greek version contains fragments of Hero’s Mechanics. Pappus’ book 8 describes the five Hero’s simple machines and explains how they work, referring for the theory to Archimedes and Hero. The sum of what concerns the knowledge of the centre of gravity is then for the most part as we have given it. You can learn the basic principles through which this science is acquired if you look at Archimedes’ book ‘On Aequilibria’, and at Heron’s work ‘On Mechanics’, while here we shall set out in order those points connected with this that most people find unclear, among which is the following [183].59 In this way we learn how to move a given weight with a given force. It is said that this section of Mechanics is one of Archimedes’ discoveries and that when he discovered it he said, “Give me a place to stand that I may move the world for you!” Heron of Alexandria gave a most clear exposition of this operation in his book called “The Drawing of Weight” in which he makes use of a lemma proven in his books on Mechanics where he also mentions the five powers which are: wedge, lever, screw, compound pulley, and shaft with wheels. These are the things by which, in general, a given weight is moved by a given power, I mean each of these powers [183].60

The importance of the text of Pappus is not so much in its content, which is essentially an epitome of the mechanics of Hero: 58 59 60

To point out an edition by John Wallis in 1688. p. 9. p. 23.

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3 Greek origins

Fig. 3.14. A complex mechanical device. The baroulkos or weight hauler (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome) That is something which has been explained in the book On Drawing Weight. As regards the five powers which we mentioned and which we said had been mentioned by Hero, we shall treat of them briefly as an, aide-memoire for lovers of knowledge [183].61

as the fact that in the Renaissance, along with the De architectura of Vitruvius, it was the only witness to the Greek-based mechanical technology. Important aspects of the mechanical theory in the text of Pappus are the famous definition of the centre of gravity as the suspension point of equilibrium (see Chapter 6), the law of the inclined plane, less convincing than Hero’s and still wrong. Finally, perhaps most importantly, the implicit reference to the law of virtual work, reported in the case of machines where the transmission is by wheels, as shown in the following two quotations: In his book “Barulcus”, however, he explains how a given weight is moved by a given power derived from positioning toothed wheels, when the ratio of the wheel’s diameter to that of the axle is five to one, and when the weight to be moved is 1000 talents and the motive power 5 talents [183].62 I say that the ratio of the speed of movement of wheel A to the speed of movement of wheel B is as the ratio of the number of teeth on wheel B to the number of teeth on wheel A [183].63

3.3.1 Pappus’ inclined plane law The treatment of the inclined plane of Pappus of Alexandria has even a greater interest than that of Hero, not so much for its quality, which is not excellent, but because it was well known in the Renaissance, it was the object of praise before and heavy criticism after. Before any comments I refer to Pappus’ analysis, which in reality is also not easy to decipher.

61 62 63

p. 62. p. 23. p. 56.

3.3 The mechanics of Pappus of Alexandria

K

D

61

C A

G F H

E

X

L B

M

N

Fig. 3.15. Pappus’ proof of the inclined plane law

With reference to Fig. 3.15: a) a weight A requires a non-zero force C to be carried on a horizontal plane; b) the weight A is balanced on the inclined plane by a weight B determined considering the angled lever with fulcrum L, with A supposed concentrated in E and B in G; c) to carry this weight B on the inclined plane, a force D proportional to C given by the proportion C : D = A : B is necessary; d) the force necessary to move A on the inclined plane is obtained by adding to D the force C. Some mathematics gives the relationship: D : C = GE : FG, that for the horizontal plane (GE/FG = 1) gives as it should be D = C, but for the vertical plane (FG = 0) furnishes an infinitive value for D, which is absurd. This fact probably could not have impressed Pappus who could have said that in practice there is never a vertical plane. Note that Pappus’ formula is deprived of any practical value because C is not generally known. Moreover as in Hero the direction of the force to move the weight along the inclined plane is not defined. It is not easy to justify Pappus’ assumptions, especially c) and d) that to our sensibility have little sense. In my opinion the only way in which Pappus and his followers accepted this proof was that it was at the moment the only way to link the inclined plane with the lever and so to succeed in the reductionist purpose to reduce the whole of mechanics to the lever.

4 Arabic and Latin science of weights

Abstract. In this chapter Latin and Arabic Middle Ages mechanics are compared, both based on virtual displacements VWLs. In the first part Arabs are considered who, with Thabit ibn Qurra, use as a principle of equilibrium a VWL for which the effectiveness of a weight on a scale is proportional to its virtual displacement measured along the arc of the circle described by the arm from which the weight is suspended. In the second part Latin scholars are considered who, with Jordanus de Nemore, assume as principles two distinct VWLs. A VWL is associated with the concept of gravity of position for which the efficacy of a weight on a scale is the greater the more its virtual displacement is next to the vertical. Another VWL is associated with the resistance of a weight to be lifted, which depends on the lifting entities in a given time. In formulas: What can raise a weight p of a height h can raise p/n of nh. It was and still is an axiom of historiography that, since its origins, mechanics has followed two main routes classified as Aristotelian and Archimedean [287, 125]. The Aristotelian route is associated with Mechanica problemata by Aristotle. Its laws are proved ‘dynamically’, as balance of tendencies of weights going downward. The level of formalization and rigor varies from author to author but it is usually not excellent. The Archimedean route is associated with the Aequiponderanti (and to a lesser extent with Euclid’s book on the balance) and dynamics is, instead, reduced to a minimum. Weights are considered as plane figures (geometrical instead of physical entities) with the main concern being evaluation of the center of gravity; moreover there is more attention given to rigorous proof than to physical aspects. The level of formalization and rigor is usually excellent. This dichotomy is a bit simplistic, I think. It forgets for example, that in ancient mechanics – and even up to the XVIII century – the basic law was in any case that of the lever. The Aristotelian and Archimedean approach differed only in the manner of its proof. One could say they differed in the ‘meta-mechanics’, in which different principles were used – such as a VWL or the theory of centres of gravity – to demonstrate as a theorem the law of the lever, then taking it as the principle in dealing with it later in ‘mechanics’. Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_4, © Springer-Verlag Italia 2012

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Recent studies, as for example those by Jaouiche [136] and Knorr [345] suggest a different interpretation. According to these authors, while it is certain that there are two different approaches in mechanics, the assumption that one is derived from Aristotle and the other from Archimedes, is yet to be proved. For example Knorr observes that the principles of the so-called Aristotelian mechanics express nothing but a diffuse knowledge which is not unique to Aristotle [345]. This holds true also for the violent law of motion according to which a displacement of a body is proportional to the force applied. This law was formulated by Aristotle in his De caelo and Physica, as shown in Chapter 3, but it is not difficult to accept that he simply formalized what was but a diffuse kind of knowledge. According to Knorr most of the so-called Aristotelian mechanics could be found in Archimedes’ work and also, may be in lost treatises [345]. Archimedean mechanics would represent simply the more formalized approach adopted by a mature Archimedes, avoiding, in the proofs, physical concepts like force for example, whose meaning is difficult to grasp with certainty. Indeed there is a general change in attitude among the historians of ancient science, especially those educated in mathematics and physics. They are becoming convinced that the development of mathematics and mechanics was quite independent of that of philosophy, and that at the most they influenced each other. Therefore, the labels ‘Aristotelian’, ‘Platonic,’ etc. for ancient scientists are probably less important than usually supposed [327]. The theses by Jaouiche and Knorr would explain why, in nearly all the medieval technical writings on mechanics, both Latin and Arabic, there is no explicit reference to the name or to the works of Aristotle. Not even in the historical periods when diffusion of the theoretical works, Physica and De caelo, was at large. This holds true also for those medieval mathematicians who were familiar with Aristotle’s philosophical works, among them Thomas Bradwardwine. They seem to be moving on two levels. On the one hand the mathematician, working on a technical text and organizing it more geometrico; on the other hand the philosopher working on a philosophical text, involving more or less the same arguments. In the chapter, for the sake of simplicity and according to tradition, I will continue to use the labels ‘Aristotelian’ and ‘Archimedean’, the former where weight is treated as a motive power (active factor) or a resisting effect (passive factor), kinematics concepts are introduced (i.e. virtual motion) and the concept of centre of gravity is not relevant, the latter when the centre of gravity and the rules of their evaluation are dominant concepts and kinematics has no role. The Greek concept of mechanics is revived in the Renaissance, with the synthesis of Archimedean and Aristotelian routes. This is best represented by Mechanicorum liber by Guidobaldo dal Monte [86] who reconsiders the mechanics by Pappus Alexandrinus, maintaining the original purpose that was to reduce all simple machines to the lever. During this period mechanics was considered a theoretical science and it was mathematical, although its object had a physical nature and had social utility [350]. Texts in the Arabic Middle Ages diverged from the Greek and Renaissance ones mainly because they divide mechanics into two parts. In particular al-Farabi (c 870–

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950) established the epistemological status of mechanics by differentiating it in the science of weights and in the science of devices, both considered part of mathematics, divided into seven disciplines: arithmetics, geometry, perspective, music, science of weight and science of devices [248].1 The science of devices referred to practical use and construction of machines. The science of weights, probably also because centered on the balance, was a science of equilibrium and not of transport as was the Greek mechanics. Besides translations of Aristotle’s theoretical works, Physica2 and De caelo,3 available since the IX century, the scholars of Islamic lands had surely access to mechanical writings by Pappus and Hero written in Greek. Also circulating were two treatises on the balance attributed to Euclid (known as the Euclid book on the balance4 and De ponderoso et levi). It seems instead that of Archimedes’ mechanical work, only that on floating bodies was known. Regarding the Aristotelian Mechanica problemata “it can now definitively be established that this text was known to Arabic authors, thanks to the rediscovery of a significant passage of a partial Arabic epitome found in the Fifth Book of the Kitab mizan al-hikma [by al-Khazini]” [247, 249]. In the Latin world a process similar to that registered in the Arabic world occurred. Even here a science of weights was constituted, named Scientia de ponderibus.5 Besides this there was a branch of learning called mechanics, sometimes considered an activity of craftsmen, sometimes of engineers (Scientia de ingeniis). Texts on mechanics available in the Latin Middle Ages were: Liber de canonio, translated anonymously from Greek into Latin in an unknown period; Liber karastonis, translated into Latin by Gerardo da Cremona from a Thabit ibn Qurra’s Arabic treatise; De ponderoso et levi, from Arabic, attributed to Euclid; Aequiponderanti by Archimedes, translated from Greek by Moerbeke in 1269; De insidentibus aquae, by Archimedes, translated from Greek in 1269, also by Moerbeke; Liber Archimedis de insidentibus in humidum or Liber Archimedis de ponderibus, from Arabic and Latin sources, uncorrectly attributed to Archimedes; and Excertum de libro Thabit de ponderibus, an epitome of the Liber karastonis. There are also indications that in some way Mechanica problemata by Aristotle circulated in some form. Finally there are the various treatises attributed to Jordanus de Nemore [171, 50, 345], which I will discuss in the following sections. On the whole there are few works where a deep comprehension of Middle Ages mechanics has been attempted from logical, epistemological and ontological points of view. The most widely known are [305, 287, 171, 297, 50, 51], who studied the 1

p. 12. The Arabic translation of Physica has a long and complicated history. The first translation is attributed to Ibn-an-Nadima (786–803). The best, and only extant today, is by Isahaq-ibn-Hunayn, at the end of the nineth century [334]. 3 De caelo was translated by al-Kindi circle during the 9th century. 4 The attribution of this text to Euclid is controversial. It is known only recently in an Arabic version by Franz Woepcke [399]. 5 The expression Scientia de ponderibus comes from the translation from Arabic into Latin of al-Farabi’s work (Science of devices was instead translated as Scientia de ingenii) by Dominicus Gundissalinus [248], p. 17. 2

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ancient treatises in their native language. The present chapter follows these attempts, particularly those of Duhem. The objective is not to discuss new sources but to put known sources under a different light. The study, carried out in some detail, on the way the principles of mechanics evolved, leads to the examination of the evolution of the various proofs of the law of the lever. Among these proofs, I only consider those in the Aristotelian route traced in the scientia de ponderibus of the Arabic and Latin Middle Ages, because only they are related to virtual work laws, which is my historical point of view. In the first part of the chapter I refer to the most relevant analyses of the principles of scientia de ponderibus developed in the Islamic lands, by referring mainly to Ibn Qurra Thabit’s treatises because they are the most ancient available texts. I also mention the contributions by al-Muzaffar al-Isfizari, who partially followed Thabit, and to al-Kazini. In the second part, devoted to the Latin Middle Ages, reference is mainly made to Jordanus de Nemore’s (XIII century) treatises because they are the first comprehensive texts left to us.

4.1 Arabic mechanics Both the science of weights and science of devices (machines) were relevant for Arabic technology. An important reason for the attention paid to the science of devices was the need to solve problems of water lifting in the Iranian plateau, where there were numerous underground aqueducts [309]. The science of weights was instead motivated by a more diffuse need, though apparently less demanding, that of precise balances.The interest in the balance in Islamic scientific learning was culturally nurtured by its role as a symbol of good morals and justice. Considered the tongue of justice and a direct gift of God, the balance was made a pillar of society and a tool of good governance. But probably the most important reasons should be sought in the eminent importance of balances for commercial purposes. In a vast empire with lively commerce between culturally and economically fairly autonomous regions, more and more sophisticated balances were, in the absence of standardization, key instruments governing the exchange of currencies and goods, such as precious metals and stones. Abd ar-Rahman al-Khazini, around 1120, wrote the Kitab mizan alhikma, dedicated to the description of an ideal balance conceived as a universal tool of a science at the service of commerce, the so-called balance of wisdom. This was capable of measuring absolute and specific weights of solids and liquids, calculating exchange rates of currencies, and determining time [248].6 As mentioned in the introduction besides Aristotle’s theoretical and technical works, Arabic scholars had no doubt access to two treatises on the balance attributed to Euclid (The Euclid book on the balance and De ponderoso et levi), a book on floating bodies by Archimedes. These Greek texts were joined by Arabic texts, by various authors as for example Thabit ibn Qurra (or Qorrah) (836–901), Al Muzaffar al-Isfizari (1048–1116) and Abd ar-Rahman al-Khazini (fl. 1115–1130). They also knew mechanical writings by Pappus and especially Hero of Alexandria. In partic6

pp. 3–4.

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Fig. 4.1. The balance of wisdom (modified from [343]7 )

ular Hero’s Mechanica was the object of many comments and translations [309]. Interesting for this purpose is the book Kitab al-Hiyal (The book of ingenious devices) of the three brothers Banu Musa, scholars from IX century Baghdad [259]. Their book is an outstanding contribution in the field of mechanical sciences. In the form of a catalogue of machines, it is a large illustrated work on mechanical devices including automata. The book described a total of a hundred devices and how to use them. It was based partly on the work of Hero of Alexandria and Philon of Byzantium and contained original work by the brothers. Some of these inventions include: valve, float valve, feedback controller, automatic flute player, a programmable machine, trick devices, and self-trimming lamp. Noteworthy is also the work by alKaraji (c. 953–c. 1029) about hydraulics [310]. But, as it does not seem that Arabs introduced new elements about virtual work laws in the Heronian texts, in the following I will concentrate on the science of weights, where instead there was an important Arabic contribution. Particularly I will discuss only a treatise on the balance attributed to Thabit, to which I will refer in the following with its Latin name Liber karastonis which is on the Aristotelian route; for works on the Archimedean route see [246]. Al-Sabi Thabit ibn Qurra al-Harrani (836–901) was a native of Harran and a member of the Sabian sect. In his youth Thabit was a money changer in Harran. The mathematician Muhammad ibn Musa ibn Shakir, one of three sons of Musa ibn Shakir, who was traveling through Harran, was impressed by his knowledge of languages – his native language was Syriac but he knew perfectly Greek and Arabic also – and invited him to Baghdad; there, under the guidance of the brothers, Thabit became a great scholar in mathematics and astronomy. Here he translated and revised many of the important Greek works; particularly all the works of Archimedes that have not been preserved in the original language, and Apolonius’ Conics. Manuscripts of Euclid’s Elements which were translated by Hunayn ibn Ishaq were revised by 7

p. 93.

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Thabit. Although he contributed to a number of areas, the most important of his work was in mathematics where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept from natural number to real numbers, ‘integral calculus’, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of the science of weights [290]. The Liber karastonis is one of the most important treatises in the science of weights of Arabic origin. The medieval science of weights owed an extraordinary debt to the production of a single work, a treatise on the balance, Kitab al Qarastun, by the ninth century mathematicianastronomer Thabit ibn Qurra. It retained a prominent place within the theoretical section on mechanics in the rich compendium compiled by al-Khazini, Kitab mizan al-hikma, two centuries later. Beginning from the 12th century, it exercised a major influence on mechanical studies in the Latin West, through the translation as the Liber karastonis made by Gerardo da Cremona. Four centuries later, writings on mechanics still clearly betrayed their provenience through elaborations and commentaries on this work [345].8

It was translated into Latin during the XII century by Gerardo da Cremona [171].9 A large number of manuscripts exist, all derived from a unique progenitor. In what follows I will refer to the text edited by Moody and Clagett, which is derived partially from a manuscript conserved in Paris and partially from a text edited by Bucher, based on a manuscript conserved in Milan [171],10 [264]. It will be enough for my purpose that is not to present a philological correct version of the text, but only to evidence some particular aspects that would have emerged in any way independently of the interpretation of particular words or phrases. Besides the Liber karastonis, at least three manuscripts in Arabic exist, with analogous subjects [246] [136],11 called Kitab al-Qarastun, controversially attributed to Thabit [345]12 [136],13 one conserved in London, one in Kraków and another in Beirut.14 The first manuscript was edited, translated into French and commented on by Khalil Jaouiche [136]. The second, while in Berlin, was edited and translated into German by Eilhard Wiedmann [397], and subsequently studied by Mohammed Abattouy [246]. The third one has been studied by Wilbur Richard Knorr [345].

4.1.1 Weight as an active factor in Arabic mechanics In Arabic mechanics the main dynamic concept is the motive power associated with weight. The fact that weight plays the role of an active factor, like a ‘force’ is stressed 8

p. 5. pp. 77–118. 10 pp. 84–85. 11 pp. 2–3. 12 p. 47. 13 p. 31. 14 Mohammed Abattouy registers a recent hitherto unknown copy in Florence [247], p. 17. 9

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over the fact that it opposes resistance to upward motion. From this idea it follows a virtual work law for which the efficacy of a weight on a balance depends on the virtual displacement of the point at which the weight is hung. This will be clear even with a superficial reading of Thabit’s book. 4.1.1.1 Liber karastonis The Liber karastonis is composed of a prologue followed by eight propositions and finally a comment. They all relate to the karaston, that is the steelyard or Roman balance, which is a straight-beam balance with arms of unequal length. It incorporates a counterweight which slides along the calibrated longer arm to counterbalance the load and indicates its weight. The exposition of the theory, though it is classified in the Aristotelian route, has a high standard of rigor, not far from the texts of the Archimedean route, with the exception of the first propositions, where the reader is asked to accept much more than in the Archimedean route. Immediately after the prologue the following propositio (proposition) I is presented: I. I say, therefore, in the case of two spaces which two moving bodies describe in the same time, that the proportion of the one space to the other is as the proportion of the power of the motion of that which cuts the one space to the power of the motion of that which cuts the other space.15 I posit the following example for this proposition. In the case of two walkers, one walks thirty miles and the second walks sixty miles in the same time. It is noted, therefore, that the power of the motion [emphasis added] of he who walks the sixty miles is double the power of the motion of he who walks the thirty miles, just as the space sixty miles is double the space thirty miles. This proposition is admitted per se and is immediately evident to the intellect [171].16 (A.4.1)

The term proposition in ancient texts usually means theorem; but what is written just after “This proposition is admitted per se and is immediately evident to the intellect”, qualifies it rather as a principle. The assertion of evidence, not completely shared by the modern reader, suggests that in the cultural climate of the period, the proportionality between force and displacement were part of common knowledge and Arabic natural philosophy, be it derived from the Aristotelian texts or not. Before attempting to comment on proposition I, let me clarify its content. In the proposition, the motion of two bodies, whose shapes are not specified, is discussed. The distances covered are introduced also (plane in the Arabic version), without specifying the kind of pattern. Because in the core of the Liber karastonis the arcs of a circle are considered, it can be presumed an affirmation of general character is present and then the paths can be any thing. By entering the merit of proposition I, two things should be stressed. First, there is no distinction between natural and violent motions here. This confirms the attitude of Arabic scholars to consider weight as an intrinsic mover and consequently to 15

A similar statement is found also in the Liber Euclidi de ponderosi et levi: “bodies are equal in strength whose motions through equal places, in the same air or the same water, are equal in times” and in some following propositions [171], p. 27. 16 p. 90.

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consider both natural and violent motions as a consequence of a force. For the association of a force to weight see for example the references listed in [246].17 Secondly, reference is made to “virtus motus” (power of motion), and not “virtus”. This creates interpretation problems, which can be overcome, as Jaouiche does, by considering the Latin version to be affected by an error of translation from Arabic into Latin by Gerardo da Cremona [136].18 According to Jaouiche the Arabic version of proposition I suggests “virtus of mobile (force du mobile)” instead of “virtus of motion (virtus motus)” [136].19 To confirm his thesis, Jaouiche considers the presence of the sentence which specifies the equality of times, simply to pay homage to the tradition and then not essential [136].20 I prefer an interpretation which consists in looking at proposition I as a reinterpretation of the ‘Aristotelian’ laws of motion. In these laws the measure of force was known a priori, independently of motion; Thabit instead suggests measuring force a posteriori by the effects it produces; more precisely, by the space covered in a given time: the greater the space covered the greater the acting force. It then seems correct to speak about “virtus motus” instead of “virtus”. The suggested interpretation of proposition I, which is so considered as a virtual work law formulation, makes it easier to understand the proof of the law of lever (proposition III) whose statement is given below: III. Since this is manifest now, then I propose [the following with respect to] every line which is divided into two different segments and imagined to be suspended by the dividing point and where there are suspended on the respective extremities of the two segments two weights, and the proportion of the one weight to the other, so far as being drawn downward is concerned, is inversely as the proportion of the lines. [I say that in these circumstances] the line is in horizontal equilibrium [171].21 (A.4.2)

The Arabic version is analogous. In what follows I refer to Abattouy’s translation: This being proved, I say that if the line AB is suspended from point G and there are set at its ends, at point A and B, two weights proportional to its two parts and inversely proportional to them, the line AB will be parallel to the horizon [246].22

The proof of proposition III, has to relay, besides proposition I, on the following proposition II: II. Then I say that in the case of every line which is divided into two parts and fixed at the division point and where the whole line is moved with a movement not directed to its natural place, then such a movement produces two similar sectors of two circles. The radius of one of these circles is the longer line and the radius of the second is the shorter line. And the proportion of the arc which the point of the extremity of one of the two lines describes to the arc which the point of the extremity of the other line describes is as the proportion of the line whose revolution produces the one arc to the line producing the other arc [171].23 (A.4.3) 17 18 19 20 21 22 23

pp. 33–35. p. 120. pp. 146–147. pp. 50–63. pp. 92, 94. Translation in [171]. pp. 37–38. p. 90. Translation in [171].

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D A

G

a

b E

B

T Fig. 4.2. The lever with different arms in the Latin manuscript

which in itself only expresses a theorem of plane geometry, according to which for an assigned angle arcs and radii of a circle are proportional, for an assigned angle. But mainly it needs the following comment Thabit added to proposition II after having proved it: We have already said that in the case of two spaces which two moving bodies describe in the same time, the proportion of the power of the motion of one of the bodies to the power of the motion of the other is as the proportion of the space which the first motion cuts to the other space. And point A with the motion of the line has already cut arc AT and point B with the motion of the line has already cut arc BD, and this in the same time. Therefore, the proportion of the power of the motion of point B to the power of the motion of point A is as the proportion, one to the other, of the two spaces which the two points describe in the same time, evidently the proportion of arc BD to arc AT. This proportion has already been shown to be the same as the proportion of line GB to line AG [171].24 (A.4.4)

From the previous passage it is convenient to distinguish between what Thabit says and why he says it. Thabit clearly affirms that the “power of motion” of the point B of the longest arm of the balance is greater than that of the point A, or more generally that the power of motion of a point of a balance is directly proportional to its distance from the fulcrum. Note that displacement is measured according to arcs of circles that the weights describe in their motion; this is not peculiar to Thabit, but can be found also in the works by al-Isfizari: Since the two weights A and B were supposed to be equal, the motion took place because the arc BO, along which the weight B moves with a natural motion is greater than the are AS, along which A moves with violent motion [246].25

and by Galileo himself [119].26 Thabit justifies his affirmation by saying “We have already said”, which can only make reference to proposition I. But this induces, at least for modern readers I think, a serious interpretation problem. Indeed proposition I when adapted to weight seems to make sense only for downward motions, but in the previous passage Thabit is considering both upward and downward motions. One (Thabit?) could overcome this difficulty by assuming that if a weight suspended from one side of a balance moves 24 25 26

p. 92. English translation, my accommodation. p. 44. p. 164.

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upward it could move downward too the same distance in the same amount of time, when the rotation of balance is imagined to revert and then one can always make reference to a possible downward motion. The same problem occurs in Galileo’s demonstrations about equilibrium with the use of the concept of ‘momento’ (see Chapter 5). A translation into modern concepts of Thabit’s reasoning, not consistent for modern mechanics, could be based on the following assumptions: a) there are two bodies A and B of equal weight; b) if the two bodies A and B are suspended from a balance with different arms, with B on the longest arm, the balance will sink on the side of B; c) because B describes a greater arc than A, in the same amount of time, in point B there is a greater power of motion (VWL); d) the power of motion of B is proportional to the distance of B from fulcrum. Here is Thabit’s proof of proposition III: The demonstration of this follows: I cut from BG the longer segment an amount equal to AG the shorter segment. This cut off line is GE. If then, two equal weights [a and b] are suspended at points A and E, the line AE will be in horizontal equilibrium, since the power of motion at the two points is equal as we have demonstrated. So that if I incline point A to point T, the weight [b in E] there suffices for its return to a position of horizontal equilibrium through arc AT. And when we change the weight from point E to point B, and if we wish the line to remain in horizontal equilibrium, it is necessary for us to add something extra to the weight at A, so that the proportion of its total to the weight which is at B is as the proportion of BG to AG. Since the power of the point B exceeds the power of point A by the amount that BG exceeds AG, as we have shown, hence the weight which is at the point of the stronger power is less than the weight which is at the point of weaker power according as is the proportion of arc to arc. Therefore, when there is a weight at point B and a second weight at point A and the proportion of weight a to weight b is as the proportion of GB to AG, the line is in horizontal equilibrium [171].27 (A.4.5)

The proof unfolds into two steps. In the first step a symmetry situation is considered, equal weights being located at the same distance from the fulcrum. The equilibrium is not considered as being self-evident, but is justified by means of proposition I. The two weights compensate each other because powers of motion of their points of suspension are equal as they pass equal arcs (for proposition II) in the same amount of time. This part of the proof is followed by the comment that if one weight is inclined downward the other will force it back and the horizontal position is recovered. That is, in modern terms, the horizontal position is stable (which today is known to be false). In the second step Thabit proposes to lengthen the arm by moving the weight from E to B so that GB>GE. Thabit says two things: a) the balance inclines toward the side B; b) to resume equilibrium the weight A must be increased until the inverse proportionality between weights and distances holds good. The first assumption is not justified by Thabit, perhaps because its justification is contained in the second. The second assumption is justified by the comment added to proposition II, previously referred to where it was concluded that the power of motion of a point on a 27

p. 94. Translation in [171].

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b

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a

g b

d

a

Fig. 4.3. The lever with different arms in the Arabic manuscript

balance arm is directly proportional to its distance from the fulcrum. So when the two weights are inversely proportional to the distances their powers are equal and the balance is in a status of equilibrium. 4.1.1.2 Kitab al-Qarastun Arabic manuscripts are quite different from those in Latin. In what follows I will use essentially the version edited by Jaouiche, with a few references to the other manuscripts.The order of propositions, indeed not numbered, in the Arabic versions is different from the Latin one. Jaouiche uses the name postulate for the proposition corresponding to proposition I (which, moreover, is not located at the beginning). He uses the name lemma for the proposition corresponding to proposition II, while proposition III is named theorem 1. The texts of propositions are virtually the same as those in the Liber karastonis, except for secondary aspects. The texts of explanations and/or proofs are instead very different; shorter and much less satisfactory than those of the Latin version. Postulate I is not followed by any comment; similarly the lemma is not followed by the dynamical comment where the proportionality between power and distance from fulcrum is affirmed. Thus the proof of theorem 1 (Jaouiche nomenclature) is incomplete, as clear from the following piece which refers to it in full: I say if ab is suspended [Fig. 4.3] at g and if at both ends, a and b, two weights proportional and equivalent to these two segments are applied, [ab] is parallel to the horizon. Indeed, taking on the longest side ag a segment gd equal to gb, if one applies to d a weight equal to the weight applied to b, [ab] is parallel to the horizon. If the weight which is in d is tilted down, the weight which is in b will rise and pass the arc dd equal to the arc bb because gd is equal to gb. If then the weight is moved from point d to point a, the latter being in the lower position and one wants to raise it up to the higher position a, one must increase the weight in b such that the ratio of total [weight] [in b] to the weight in a is equal to the ratio of the arc aa to the arc dd, which are passed in the same time though they are uneven. But this ratio is equal to the ratio of one of two segments of the straight line to the other [136].28 (A.4.6)

The version of Berlin’s manuscript is substantially equivalent: We cut from the longer AG [a segment] like GB and that is GD. If a weight equal to the weight at B is suspended from point D, AB will be parallel to the horizon, so that if it is inclined from the higher D to the lower D, the weight at B would move it and raise it up to the higher D, making it traverse the arc DD. But the arc DD is equal to the arc BB, for GD 28

p. 149.

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4 Arabic and Latin science of weights is as GB. Nevertheless, the arc DD and the arc AA are traversed in the same time. Hence if we move the weight from D to the lower A and we wish that it is raised up to the higher A, we will need to add to the weight at B an addition such that the ratio of the whole to the weight at A will be as the ratio of the arc AA to the arc DD, if these two arcs are traversed in the same time even though they are different. This ratio is the ratio of one of the two parts of the line to the other [246].29

A similar reasoning is developed by Al-Isfizari: Therefore, we have here two distinct notions each one of them requiring the sinking, namely the weight and the distance. The excess of one of them over the other in weight is as the excess of this latter over the former in distance. The equality between them required the counterbalance and that the beam is extended in parallelness to the horizon, so that the line AB remains parallel to the horizon. The ratio of the arc BO to the arc AS is as the ratio of the line GB to the line GA, as it was demonstrated by Euclid in his book [246].30

and the explanation of the reason for equilibrium of the balance with equal weight and arms is more detailed. As compensation for its shortness, the proof of theorem 1 is followed by a reference to a case where the balance arm is deprived of weight: If the axis is heavy and it is divided into two unequal segments, we increase the thickening of the shortest segment until the axis is parallel to the horizon. […] We are then reduced to the case already treated in the axis free of weight [136].31 (A.4.7)

The obvious conclusion is that Arabic manuscripts do not add anything of importance to the lever law interpretation, at least from the point of view of the role played by weight.

4.1.2 Comments on the Arabic virtual work law Thabit’s argument to prove the law of the lever is based on the assumption that equilibrium is determined by the equity of causes of motion, that is, of the motive powers of weights. This is an axiom of ancient philosophy, which, however, has little meaning when translated into the precise language of physics. Indeed Thabit’s reasoning is successful because there is a shift in the meaning of the term power (virtus). According to proposition I, power should be interpreted in the usual way, i.e. as a muscle force. But this position when carried out coherently leads to a paradoxical consequence. If two equal powers – equivalent to two equal muscle forces – are applied to the extremities of a balance with different arms, the balance cannot be in equilibrium whatever the ratio of the arms might be. To overcome this paradox, power should be given a different meaning, that of the efficacy of the power or the capacity to produce a rotation of the balance; in modern terms the meaning of static moment. This is a rhetorical artifice which has a relevant heuristic role but no value from a logical point of view. The same problem arises for many proofs of the lever law, that of Galileo included. 29 30 31

p. 38. p. 44. p. 15.

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Anyway the choice of Thabit to measure by the ease with where power its point of application moves is a form of virtual work law. By comparing this form with those of modern virtual work principles, some similarities and differences are found. One of the differences is that in the modern laws, displacements are evaluated along the vertical direction, and are consequently straight, while Thabit makes reference to curvilinear paths. Regarding the role of time, which is implicit in Thabit’s principle, it can be said that there is not a substantial difference. Actually, even in the modern principle, though it does not appear in the enunciation, time plays an essential role; indeed various virtual displacements occur in time; they are contemporary (in respect of the congruency of the system moved) and then occur in the same given time. Anyway, neither Thabit nor adherents of the modern principle have any interest in the effective measurement of time.

4.2 Latin mechanics According to many historians of science, the reasons for interest in the mechanics of Latin Middle Ages is different from that of Arabic Middle Ages, less oriented to the needs of society. The development of mechanics in Europe in the XIII and XIV centuries should be referred to the general revival of interest in the texts from the Greek and Arabic worlds, and then somehow separated from applications. It must be said though that if in the Latin Middle Ages there was no need to study the science of weights in order to design appropriate scales for trade, there was a stimulus to improve the general knowledge of statics required by construction of the Gothic cathedrals, very daring buildings that saw their heyday in the XIII century. So it is likely that it was not just a cultural interest to date from the XIII century the writings of Jordanus de Nemore, which I will discuss below, which represented a significant improvement compared to those of Thabit, especially because it covered a wider range of problems. In the Latin Middle Ages various treatises on the scientia de ponderibus circulated, as already noted. Some were Latin translations from Greek or Arabic, a few were written directly in Latin. An outstanding scholar was Jordanus of Nemore or Jordanus Nemorarius. Practically nothing is known about his life. He appeared at the beginning of the XIII century. Besides writings about mechanics he was the author of many mathematical works [344, 337, 305]. Treatises attributed to Jordanus are: Elementa Jordani super demonstratione ponderum (version E), Liber Jordani de ponderibus (cum commento) (version (P), Liber Jordani de Nemore de ratione ponderis (version R, discovered by Duhem). They used to be commented upon up to the XVI century; recently they have been studied by historians of science with various tendencies. For the medieval comments there are manuscripts of the XIII century, classified by Moody and Clagett as Corpus Christi [50]; the manuscript of the XIV century published by Petrus Apianus in 1533, referred to as Aliud commentum [50]; the manuscripts of the XIV century named Commentum Henrici Angligena [50]; the Questiones super tractatum de ponderibus, by Blasius of Parma, of the XIV-XV centuries [50]. For modern comments there are essentially those by Duhem, Clagett, Moody and Brown, in the already cited works.

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Fig. 4.4. A modern view of the Erfurt cathedral

Duhem’s studies are the only ones which exhibit a deep understanding of mechanical concepts, and notwithstanding some justified criticisms on the historical approach, they remain still fundamental. In what follows I will mostly refer back to Duhem, seeking to get an understanding of concepts rather than to attempt a historical and philological reconstruction of treatises by Jordanus, which may be found in the secondary literature32 giving only a few hints about this reconstruction, to show that its results are in evident contrast to those obtained by a scrutiny of the fundamental concepts of the various treatises. The law of the lever, or more in general a form of virtual work law, could be derived by a contemporary physicist with the Aristotelian violent motion law alone, for which if A move B to Γ then A moves 1/2 B to 2 Γ (in a given time), in a different and easier way than that carried out by the Arabic mathematicians. This has been clearly shown by Duhem: Consider a lever with power α and resistance β; the resistance is at a certain distance from the fulcrum. If the power α can move β so that it describes in a time δ the arc γ, it would move the weight β/2, located at a double distance from the fulcrum, in the same time δ and pass an arc 2γ. It needs so the same power to move a certain weight, located at a certain distance from the fulcrum, and a half weight to a double distance. From this we can easily justify the theory of the lever as given in the Mechanica problemata [305].33 (A.4.8) 32 There are various hypotheses about the roots of Jordanus’ mechanical works. Quite convincing is the hypothesis of the Arabic roots: [248], p. 17; [312], pp. 4, 12; [50, 287]. 33 vol. 2, p. 122.

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Table 4.1. The suppositions by Jordanus S1 S2 S3 S4 S5 S6 S7

The movement of every weight is toward the center (of the world), and its strength is a power of tending downward and of resisting movement in the contrary direction. That which is heavier descends more quickly. It is heavier in descending, to the degree that its movement toward the center (of the world) is more direct. It is heavier in position when in that position its path is of descent is less oblique. A more oblique descent is one which in the same distance, partakes less of the vertical. One weight is less heavy in position, than another, if it is caused to ascend by the descent of the other. The position of equality is that of equality of angles to the vertical, or such that these are right angles, or such that the beam is parallel to the plane of the horizon.

Duhem’s argument leads to a mathematical relation analogous to that found with the Arabic version of the virtual work law as referred to in the preceding section: the key factor for equilibrium is the product of weight times virtual displacement (in a given time). It must be said that Duhem keeps his argument at a superficial level, cavalierly confusing concepts of force and static moment or work, more or less the same that Thabit did. If he had taken seriously Aristotle’s law of violent motion which was formulated only for free bodies he would have never applied it to bodies suspended from a lever. In what follows I will try to understand whether Duhem’s argument is similar or not to that carried out by Latin medieval scholars, who since the XIII century wrote treatises on mechanics, or more precisely on the scientia de ponderibus. I will take as the basic treatise the Liber Jordani de Nemore de ratione ponderis (version R, in the following De ratione) as edited by Ernest Moody and Marshall Clagett [171]. They tried to get the most plausible version from a mixing of various manuscripts. A more philologically accurate reconstruction of the text can be found in Joseph Edward Brown [50].34 I will also refer to some comments and especially to the version of De ratione edited by Nicolò Tartaglia [224] and to Quesiti et inventioni diverse, also by Tartaglia [223], which largely represents a paraphrase of it. For more comments see [273]. The De ratione is quite a complex treatise, divided into four books. In the first book there are the principles and theorems of the science of weights. The second and third books are more technical and concern the solutions of some of the problems of the balance, with arms endowed or not with natural weight. The fourth book is about various issues, among which the fall and breaking of bodies. The first book, the one concerning the principles and theorems, starts with seven suppositions (suppositiones) – referred to in Table 4.1 (A.4.9). The suppositions have different logical status. Some look like principles in the modern sense (S2 , S3 ), some look like definitions (S5 ), some others are difficult to classify. Supposition S1 is the most complex one. It contains: 34

p. 75.

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a) a principle in the modern sense (“Omnis ponderosi motum esse ad medium”); b) a definition (that of “virtu”) (“virtutemque ipsius esse potentia ad inferiora tendendi virtutem ipsius et motui contrario resistendi”). Suppositions from S3 to S6 introduce the gravity of position concept. In supposition S3 Jordanus makes a generic assertion, for which a body weighs more the more directly it goes toward the centre of the world. It implies that ‘heaviness’ depends not only on the body, but also on its possible, or virtual, motion, assuming so a form of virtual work law. In supposition S4 the meaning of S3 is specified, with the introduction of the locution gravity according to position. A body is heavier than another, by position, when its descent is less oblique. It is then stated precisely when a motion is less or more oblique in supposition S5 : a direction is more oblique than another when it is closer to the horizon. Which is in clear contrast to the modern use of the term obliquity, but coherent with Jordanus’ ideas for which the reference direction is the vertical one. Supposition S6 on the one hand can be seen as a definition of ‘less heavy’, on the other hand it describes a factual situation, the rising of a less heavy body caused by a more heavy body. The same holds for supposition S7 , which on the one hand can be seen as a definition of equilibrium and on the other hand as a factual situation representing equilibrium. Suppositions S6 and S7 make sense only for two weights belonging to a balance. This holds true for S1 as well, because it refers to contrary or upward motions. Indeed considering contrary motions would require implicitly the assumption of a force causing them; but De ratione concerns only weights and then contrary motion cannot be due but by the weight placed on the side of the balance which is opposite to that to be raised. Supposition S6 makes it clear that Jordanus would consider a weight to be able to rise another weight and then to act as a motive power. However in Jordanus’ treatise it is never explicitly stated that both weights suspended from the end of a balance tend to go down. Rather it seems that as a body is pushed up it loses its heaviness. It is not clear if this corresponds to a Jordanus’ philosophical conception or if it is simply due to his difficulty in quantifying the tendency of bodies to move downward. Jordanus’ suppositions contain certain keywords which would deserve a comment because their meaning is not so easy to grasp: “gravis”, “ponderosus”, “velocitas”, “virtus”, “gravitas secundum situm”. In what follows, for the sake of space, I shall limit myself to commenting on the last two keywords which have a particular importance. The interpretation of “virtus” is quite a delicate question. One is tempted to associate it with the meaning of force. There are however reasons not to do this. The most important is that virtus, besides the tendency to go downward, represents the resistance to go upward. In the De ponderoso et levi, virtus was connected to velocity, at least for the motion according to nature: “Bodies are equal in virtus when their motions are equal in equal times and equal spaces in the same air or water” [171].35 Nothing is instead said for the motion against nature. Nicolò Tartaglia wrote in the Quesiti et inventioni diverse: “Definition four. Bodies are of the same virtus or power when in equal time 35

p. 26.

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[223]36

they pass equal spaces” and added to the supposition I of the De ratione he edited this interesting comment, not appearing in the text edited by Moody and Clagett: The motion of every body is toward the centre and its virtus is a power of tending downward, and we can understand the power from the arm length or from its velocity which is determined by the length of the balance arms and to resistance to the contrary motion [224].37 (A.4.10)

The final part of Tartaglia’s supposition P1 explicitly asserts that the weights are not free but are suspended from a balance and proposes a method to evaluate the virtus: virtus is measured by velocity. Jordanus does not explain what causes virtus, but his use of a unique term for both motion against and according to nature, should indicate he is thinking of a unique cause. A modern term to translate virtus could be ‘heaviness’, but this creates ambiguities. For this reason in what follows virtus will often not be translated, or in some cases it will be translated as strength. Concerning the concept of gravity of position, it can be said that there is widespread agreement among historians [287, 305] that it is partially derived from Mechanica problemata. This could be evident enough from suppositions, particularly from supposition S3 , and is suggested by the preface of version P which does not start directly with the suppositions, as the other treatise attributed to Jordanus does, but presents an ample discussion from which I refer to the outstanding points: It is therefore clear that there is more violence in the movement over the longer arc, than over the shorter one; otherwise the motion would not become more contrary (in direction) Since it is apparent that in the descent (along the arc) there is more impediment acquired, it is clear that the gravity is diminished on this account. But because this comes about by reason of the position of the heavy bodies, let it be called positional gravity in what follows. For in reasoning in this way about motion, as if the motion were the cause of heaviness or lightness, we conclude, from the fact that a motion is more contrary (in direction) that the cause of this contrariety is more contrary – that is, that it contains a greater element of violence. For if a heavy body descends, this occurs by nature; but that its descent is along a curved path, is contrary to its nature, and hence this descent is compounded of the natural and the violent. But since, in the ascent of a weight, there is nothing due to its nature, we have to argue as we do in the case of fire, because nothing ascends by nature. For we reason concerning the ascent of fire, as we do concerning the descent of a heavy body; from which it follows that the more a heavy body ascends, the less positional lightness it has, and therefore the more positional gravity [171].38 (A.4.11)

Beside the consideration of motion along an arc of a circle with different radii, one should make note of the explicit introduction of the locution “gravitas secundum situm”. Of course the preface of version P does not demonstrate the influence of the Mechanica problemata on the derivation of this concept. Perhaps the preface was added for the sake of ‘completeness’ by the editor. 36 37 38

p. 82r. p. 3. p. 150. Translation in [171].

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It is not easy to express with a unique term the gravity of position concept. For downward motion, with a little forcing, the gravity of position can be represented by the product of the weight (p), considered as an active force multiplied by the (virtual) velocity of sinking (v), mathematically pv (this is essentially what the Arabic mechanics did). It is difficult to say whether Jordanus would recognize himself in this representation. In effect he never gave a mathematical expression to gravity of position. For him it remains a qualitative concept, defined by the more or the less, which is useful to prove certain assertions but not to furnish numerical laws. Jordanus used the concept of gravity of position, for example, to show that a balance with different arms and equal weights is not in a horizontal equilibrium but sinks toward the longer arm, or to show that a balance with equal weights and arms is in a stable equilibrium configuration. When he needed a mathematical law he used a different approach, described in the following. Another issue is raised by historians of science on Jordanus’ concept of gravity of position. If gravity is supposed to be a quality, a form of the body, it is possible to think the gravity could change by varying the disposition of the body with respect to another body. If instead the gravity is conceived as a force (internal or external) independent of the position of the body, absolute gravity, it effectiveness can be greater or lower depending on the resistance of the constraints.

4.2.1 Weight as a passive factor in the Latin mechanics In the Latin mechanics, the two dynamical concepts of virtus and gravitas secundum situm associated with weight, make sense for both upward and downward motions. In the first case they appear as passive factors because the weight for its virtus or gravitas opposes a resistance when an applied force tends to raise it; in the second case they appear as active factors, because the weight for its virtus or gravitas is responsible for a motive power directed downward. Jordanus uses the concept of gravitas secundum situm mainly as an active factor and the concept of virtus (resistendi) as a passive factor. The passive factor is the only one used to formulate in mathematical terms a virtual work law, and from this point of view it is the key dynamical concept. This will be clear from the examination of a few theorems of the De ratione.

4.2.2 Propositions After the suppositions, the De ratione continues with forty three propositiones (theorems); Table 4.2 refers to the first ten (A.4.12). Among them the propositiones P1 , P2 , P6 , P8 and P10 , dealing expressly with the principles of mechanics, have a particular relevance. Though the reference treatise is the De ratione (R version), I will consider also the versions E and P and some comments, by Middle Ages and Renaissance scholars.

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Table 4.2. The first ten propositions by Jordanus P1 P2

P3 P4 P5 P6

P7

P8

P9 P10

Between any heavy bodies, the strengths are proportional to the weights. When the beam of a balance of equal arms is in the horizontal position, then, if equal weights are suspended from its extremities, it will not leave the horizontal position; and if it is moved from the horizontal position, it will revert to it. But if unequal weights are suspended, the balance will fall on the side of the heavier weight until it reaches the vertical position. In whichever direction a weight is displaced from the position of equality, it becomes lighter in position. When equal weights are suspended from a balance of equal arms, inequality of the pendants by which they are hung will not disturb their equilibrium. If the arms of the balance are unequal, then, if equal weights are suspended from their extremities, the balance will be depressed on the side of the longer arm. If the arms of a balance are proportional to the weights suspended, in such manner that the heavier weight is suspended from the shorter arm, the weights will have equal positional gravity. If two oblong bodies, wholly similar and equal in size and weight, are suspended on a balance in such manner that one is fixed horizontally onto one arm, and the other is hung vertically, and so that the distance from the axis of support to the point from which the vertically suspended body hangs, is the same as the distance from the axis to the mid point of the other body then they will be of equal positional gravity. If the arms of a balance are unequal, and form an angle at the axis of support, then, if their ends are equidistant from the vertical line passing through the axis of support, equal weights suspended from them will, as so placed, be of equal heaviness. Equality of the declination conserves the identity of the weight. If two weights descend along diversely inclined planes, then, if the inclinations are directly proportional to the weights, they will be of equal strength in descending.

4.2.2.1 Proposition I Table 4.3 refers to different accounts of the proposition I for version E (italic), P (small caps) and R (A.4.13). Notice that proposition I, at least for versions E and P, is somehow equivalent to proposition I of the Liber karastonis. Its logical status is however different; there it was a principle, here it is a theorem. In short Jordanus is more prudent than Thabit; instead of assuming the proportionality between weight and velocity – notice that now velocity is considered like a well-defined kinematical quantity (see § 3.1.1.1) – he assumes a weaker statement which asserts only the monotony between weight and velocity as expressed by the supposition S2 , according to which the greater the weight, the greater the velocity.

Table 4.3. Different accounts of Jordanus de Nemore’s proposition I The proportion of the velocity of descent, among heavy bodies, is the same as that of weight, taken in the same order, but the proportion of the descent to the contrary ascent is the inverse proportion. BETWEEN ANY TWO HEAVY BODIES, THE PROPER VELOCITY OF DESCENT IS DIRECTLY PROPORTIONAL TO THE WEIGHT, BUT THE PROPORTION OF DESCENT AND OF THE CONTRARY MOVEMENT OF ASCENT IS THE INVERSE. Between any heavy bodies, the strengths are proportional to the weights.

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The delicacy of proposition I is highlighted by the fact that it is given different accounts; the statements of versions E and P are substantially the same but differ from that of version R in two important aspects. They refer to the relation between weight and velocity rather than to weight and virtus and they consider explicitly both the downward and upward motions. It is possible that the substitution of the term “virtus” (strength) in version R to the term “velocitas” was made to allow a unitary treatment of upward and downward motions, because the concept of virtus is independent of the direction of motion. Here is the proof of proposition I of version R: Let there be two weights, ab and c, of which c is the lighter. And let ab descend to d, and let c descend to e. Again, let ab be raised to f , and c raised to h. I then say that the distance ad is to the distance ce, as the weight ab is to the weight c; for the velocity of descending is as great as the power of the weight. But the power of the combined weight consists of the powers of its components. Let the weight a then be equal to the weight e, so that a’s power is the same as that of e. If then the ratio of the weight ab to the weight c is less than the ratio of the power of ab to the power of e, the ratio of the weight ab to (its component weight) a will likewise be less than the ratio of the power of ab to the power of a. And therefore the ratio of the power of ab to that of b will likewise be less than the ratio of the weight ab to the weight b. Consequently the ratio of the same weights will be both greater and less than the ratio of their powers. Since this is absurd, the proportion must be the same in both cases. Hence the weight ab is to the weight c, as the distance ad is to the distance ce, and conversely as the distance ch is to the distance a f [171].39 (A.4.14)

h

f

a

b c

d

e

Fig. 4.5. Downward and upward motions

The first part of the above passage proves proposition I as given in version R; the second part proves what is added in versions E and P. The text makes quite a direct reference to suppositions S1 and S2 and an indirect reference to S3 , by assuming vertical paths of weights instead of circular, as Thabit did. According to suppositions S1 and S2 Jordanus can assume that virtus grows with weight; he goes ahead and assumes also the additivity with respect to weight. Additivity is assumed explicitly “But the power of the combined weight consists of the powers of its components”. It is assumed implicitly when Jordanus affirms that the strength of the portion of ab equal to c equals that of c; this means also that posit 39

pp. 174, 176. English translation adapted.

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c = a, the residual part of the virtus is that of ab − c = b. The final part of the quoted passage, “Hence the weight ab is to the weight c, as the distance AD is to the distance CE, and conversely as the distance CH is to the distance AF”, is a simple corollary and, by relating strength and velocity, states the proportionality between weight and velocity for the downward motion: “the weight ab is to the weight c, as the distance AD is to the distance CE”, and the inverse proportionality for upward motion: “as the distance CH is to the distance AF”. The proof of the first part of proposition I appears clearly circular to a modern reader and then inconsistent, because it assumes what is to prove. This has been noticed even by Brown [50].40 The fact that Jordanus did not consider additivity and proportionality as equivalent notions, as they would be by modern mathematicians, is probably due to his lack of familiarly with the algebraic calculus. The proof consists of a reductio ad absurdum. Suppose, says Jordanus, the proportionality between strength and weight is not direct but the ratio of weight to weight is less than the ratio of strength to strength. Then, with p(.) that means strength, it follows: (a + b)/a < p(a + b)/p(a) = [p(a) + p(b)]/p(a), but, Jordanus continues, then (a + b)/b > [p(a) + p(b)]/p(b) = p(a + b)/p(b). In short, at the same time that the ratio of weight to weight is both less and greater than the ratio of strength to strength, which is absurd; then the assumption that the ratio of weight to weight is less than the ratio of strength to strength should be denied. The proof is clearly too hasty; it is made explicit in the version of the De ratione edited by Tartaglia and in some writings of Middle Ages commentators, with the aid of proposition 30 of Euclid’s Elements book V.41 Even the conclusion, weight and velocity (space) are proportional, is too hasty, probably because Jordanus had modified the enunciation of proposition I in versions E and P to arrive quickly at R and he may have not finished his work, deferring the discussion of the ratio of strength to velocity to a subsequent (not yet existing) proposition. To note that in the final and initial parts of the proof of the R version, distances of descent are associated with velocities, with time implicit. It looks as if a metric for the velocity has been introduced by measuring it against the space covered in a given time. Concerning the upward motion, Jordanus’s text leaves one still more bewildered because of its terseness. Indeed, upward motion is only mentioned in the final sentence: “Hence the weight ab is to the weight c, as the distance ad is to the distance ce, and conversely as the distance chis to the distance a f ”, where ch and a f are upward motions. Though the proof of proposition I leaves one unsatisfied, its conclusion is clear. In the downward motion distance ad and ce covered in an assigned time, are proportional to weights ab and c respectively; in the upward motion, distance a f and ch 40

p. 208. This proposition states that given four quantities, A, B, H, K, if (A+B)/A > (H+K)/H, then (A+B)/B< (H+K)/K [221], p. 104, 105. So assumed A = a, B = b, H = p(a); K = p(b), c = a from (a + b)/c < p(a + b)/p(c) i.e. (a + b)/a < [p(a) + p(b)]/p(a) it follows (a + b)/b > [p(a) + p(b)]/p(b) = p(a + b)/p(b). 41

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covered in an assigned time are inversely proportional to weights ab and c respectively. I repeat that these conclusions, particularly the one concerning upward motion, make sense only when the weights are thought to be suspended from the arms of a balance, where the weight which sinks from one side raises the weight on the other side. Moreover, the sinking weight which acts as a motive power, must be taken to be unchanged, at the same distance and with constant velocity. In this way the result of proposition I can be formulated by asserting that what can raise p at the height h can raise p/n a at the height nh. This is exactly the formulation of the virtual work law Duhem considered at the beginning of Section 4.1; the argumentation is however much more articulated and convincing. 4.2.2.2 Proposition II Propositio II When the beam of a balance of equal arms is in the horizontal position, then, if equal weights are suspended from its extremities, it will not leave the horizontal position; and if it is moved from the horizontal position, it will revert to it. But if unequal weights are suspended, the balance will fall on the side of the heavier weight until it reaches the vertical position [171].42

f

l

r z m

s

n

x c

d

a

b g

k y

h t

e Fig. 4.6. The lever with equal arms

This proposition was carefully considered in the Renaissance, and its conclusion, in Thabit’s footsteps, that the balance returns to its horizontal position when removed (stable equilibrium) will be according, to the various authors, confirmed or denied. For instance Tartaglia agrees with Jordanus; Benedetti claims for unstable equilibrium (balance assumes the vertical position under perturbation of the horizontal one). Dal Monte is for indifferent equilibrium (balance stays where it is left). This last position is that accepted by modern mechanics. The problem could not be solved empirically in the Middle Ages and the Renaissance for various reasons: the use of systematic experiments to verify a theory was not accepted, the presence of

42

p. 176. Translation in [171].

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imperfection (inequality on masses, friction) made it difficult to read any conclusions, etc. The first part of the proposition, equal weights hanging from a balance with equal arms are equilibrated, rather than being taken as a postulate, is demonstrated in the same manner as Thabit did, arguing that the two weights are moving with the same obliquity, so they have the same gravity of position and equilibrate themselves. The second part is proved by showing that when assuming a position different from the horizon, the gravity of position of the weight that is lower (b in Fig. 4.6) is less than the weight that is higher (c in Fig. 4.6) because in a virtual rotation of the balance, the higher c is lowered more than the lower b, so its gravity of position is greater and the scale returns horizontally: Let it now be supposed that the balance is pushed down on the side of b, and elevated correspondingly on the side of c. I say that it will revert to the horizontal position. for the descent from c toward the horizontal position is less oblique than the descent from b toward e. For let there be taken equal arcs, as small as you please, which we will call dc and bg; and let the lines czl and dmn, and also bkh and gyt, be drawn parallel to the horizontal, and let fall, vertically, the diameter f rzmakye. Then zm will be greater than icy, because if an are cx, equal to cd, is taken in the direction of f , and if the line xrs is drawn horizontally, then rz will be smaller than zm; and since ri is equal to ky, zm will be greater than ky. Since therefore any arc you please, which is beneath c, has a greater component of the vertical than an arc equal to it which is taken beneath b, the descent from c is more direct: than the descent from b; and hence c will be heavier in its more elevated position, than b. Therefore it will revert to the horizontal position [171].43 (A.4.15)

Note that in the proof Jordanus assumes, rightly in my view, arbitrarily small arcs because the motion is to be considered at the very beginning. But he does not make the passage to the limit and consider them as infinitesimals – the times were not right – and then he fails to notice that in the limit, for infinitesimal arcs, vertical displacements of A and B are equal, then the gravity of their position are equal, then equilibrium is indifferent. However the reduction to infinitesimal motion would lead to an evaluation of the gravity of position different from that proposed by Jordanus. If the motion on a given circle with infinitesimal displacements is assumed, everything is going as for finite displacements; gravity of position is maximum at the horizontal position of the balance and is zero in the vertical position; in an intermediate position the gravities of the weights are equal and the balance is in equilibrium. But if circles of different radius are concerned, the consideration of infinitesimal displacements does not attribute the greater gravity, with the same inclination of the lever arm, to the weights that are on the larger circle. Considering finite displacements instead enables this attribution. The concept of gravity of position, although interesting and suggestive, seems to take more than a simple infinitesimal reinterpretation in order to be adopted by modern statics.

43

pp. 176, 178. Translation in [171].

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4.2.2.3 Proposition VI. The law of the Lever On the basis of the virtual work law implicit in proposition I, which if my interpretation is correct is a theorem of statics, it was not difficult for Jordanus to prove the law of the lever: Proposition VI If the arms of a balance are proportional to the weights suspended, in such manner that the heavier weight is suspended from the shorter arm, the weights will have equal positional gravity. Let the balance beam be ACB, as before, and the suspended weights a and b; and let the ratio of b to a be as the ratio of AC to BC. I say that the balance will not move in either direction. For let it be supposed that it descends on the side of B; and let the line DCE be drawn obliquely to the position of ACB. If then the weight d, equal to a, and the weight e equal to a are suspended, and if the line DG is drawn vertically downward and the line EH vertically upward, it is evident that the triangles DCG and ECH are similar, so that the proportion of DC to CE is the same as that of DG to EH. But DC is to CE as b is to a; therefore DG is to EH as b is to a. Then suppose CL to be equal to CB and to CE, and let l be equal in weight to b; and draw the perpendicular LM. Since then LM and EH are shown to be equal, DG will be to LM as b is to a, and as l is to a. But, as has been shown, a and l are inversely proportional to their contrary (upward), motions. Therefore, what suffices to lift a to D, will suffice to lift l through the distance LM. Since therefore l and b are equal, and LC is equal to CB, l is not lifted by b; and consequently a will not be lifted by b, which is what is to be proved [171].44 (A.4.16) D d

L l

a A G

C

H

M E

B b e

Fig. 4.7. The lever

The proof is clear enough, except for some prolixity when showing the similitude of triangles. Substantially Jordanus says: suppose, for argument’s sake, the balance is not equilibrated and rises on the left, but this is impossible (absurd) because, for proposition I, a weight d in D is equivalent to a weight l = b in L symmetric to B, and the balance should behave as a balance with equal arms and weight, which was proved in a preceding proposition (proposition P2 ) to be equilibrated. 4.2.2.4 Proposition VIII Propositio P8 If the arms of a balance are unequal, and form an angle at the axis of support, then, if their ends are equidistant from the vertical line passing through the axis of support, equal weights suspended from them will, as so placed, be of equal heaviness [171].45 44 45

pp. 182, 184. Translation in [171]. pp. 184, 186. Translation in [171].

4.2 Latin mechanics

87

c d

m p

z r

b

e

h f

l

ad

n

k k

g

t

o

x

y

Fig. 4.8. The angled lever with equal weights

The proof is laborious due to the need of Jordanus to take finite displacements. It is conducted by reduction ad absurdum, assuming that the balance is moving and by showing that in this case there is the negation of proposition P1 , whereby a weight in its descent cannot lift a weight equal to itself, i.e. the absurd. Note that on this occasion Jordanus compares directly the descent of a heavy body with its ascent. That is, he rewrites proposition P1 by asserting that if a weight p descends a distance of h, a weight np will rise a distance of h/n. This fact, not usually commented on by historians of science, should be kept in mind for a correct interpretation of the proofs, such as that of proposition P6 , in which one is reduced to the comparison of raising weights. Here Jordanus’ reasoning: with reference to Fig. 4.8, assume by absurdly that the angled lever with two equal weights on the ends, and placed at the same distance (measured from the vertical) from the fulcrum, is not in equilibrium, but rotate for example clockwise. The weight at b describes the arc bm, while the weight at a the arc ax. Then the ascent mp of b would be greater than the descent tx of a, which is impossible given the equality of a and b and the proposition P1 . The same applies if the lever rotates counterclockwise, in this case the ascent would be ln and the descent rh. The reasoning would be easier if infinitesimal displacements could be considered as made in modern formulations, in which case b would have the same vertical displacement of a, and this is precisely the condition required for equilibrium. However the arguments developed with the use of finite displacements enable recognition of both equilibrium and stability, although this was probably not completely clear to Jordanus. Let the axis be c, the longer arm ac, and the shorter arm bc and draw the vertical line ceg; and let the lines ac and be, perpendicular to this vertical, be equal. […] For let ag and be be extended by a distance equal to their own length, to k and to z; and on them let the arcs of circles, mbhz and kxal, be drawn; and let the arcs ax and al be equal to each other, and similar to the arcs mb and be and let the arcs ay and a f also be equal and similar. If then a is heavier in this position than b, let it be supposed that a descends to x and that b is raised to m. Then draw the lines zm, kxy, k f l; and let mp be erected perpendicularly on zbp, and xt and ed on kad. And because nt is equal to ed, while ed is greater than xt –

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4 Arabic and Latin science of weights on account of similar triangles – mp will also be greater than xt. hence b will be lifted vertically more than a will descend vertically, which is impossible since they are of equal weight [171].46 (A.4.17)

There are hints that Jordanus also knew the general law of the angled lever with unequal weights for which the weights should be in inverse proportion to their distance from the vertical, but he did not provide the demonstration [171]47 . 4.2.2.5 Proposition X. The law of the inclined plane Proposito P10 If two weights descend along diversely inclined planes, then, if the inclinations are directly proportional to the weights, they will be of equal strength in descending [171].48

d

m n g

z

h

e y

x t

k

a

r b

l c

Fig. 4.9. The inclined plane

The proof of the law of the inclined plane is preceded by proposition P9 (see Table 4.2), for which the gravity of position is constant along an inclined plane. This is not clear and also it is not clear to me the meaning of the proposition P9 . The proposition seems self-evident. Perhaps a reason could be the assertion that the gravity of position depends only on the ratio between the length of the plane and its vertical projection. The proof is very similar to the one given for the lever. It proceeds by reductio ad absurdum, replacing the situation of equilibrium of weights e and h placed on opposite inclined planes dc and dk, to the lifting of weights g = e and h located on the same side of inclined planes da and dk. Suppose by absurdity that h and e are not balanced and that, for example, e descends a distance of er and h ascends a distance of xm. For proposition P1 , g is equivalent to h because the two weights 46 47 48

p. 186. Translation in [171]. Proposition R 3.01, p. 204 p. 190. Translation in [171].

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are inversely proportional to the obliquity of the planes and then there is the inverse proportion between g and h and their ascents zn and xm for an assignment descent of e, so h on the side ad can be replaced by g on side kd. But e and g have the same gravity of position and then are balanced, therefore there cannot be motion. Hence the absurdum. Let there be a line abc parallel to the horizon, and let bd be erected vertically on it; and from d draw the lines da and dc, with dc of greater obliquity. I then mean by proportion of inclinations not the ratio of the angles, but of the lines taken to where a horizontal line cuts off an equal segment of the vertical. Let the weight e, then, be on dc, and the weight h on da; and let e be to h as dc is to da. I say that those weights are of the same virtus in this position. For let dk be a line of the same obliquity as dc, and let there be on it a weight g, equal to e. If then it is possible, suppose that e descends to l, and draws h up to m and let gn be equal to it, which in turn is equal to el. Then let a perpendicular on db be drawn from g to h, which will be ghy; and another from l, which will be tl. Then, on ghy, erect the perpendiculars nz and mx; and on lt, erect the perpendicular er. Since then the proportion of nz to ng is as that of dy to dg, and hence as that of db to dk, and since likewise mx is to mh as db is to da, mx will be to nz as dk is to da – that is, as the weight g is to the weight h, but because e does not suffice to lift g to n, it will not suffice to lift h to m. Therefore they will remain as they are [171].49 (A.4.18)

4.2.3 Comments on the Latin virtual work law Proposition I at first sight could seem to be derived in a straightforward way from Aristotle’s law of violent motion as expounded in the Physica or De caelo and one can assume that Jordanus used these treatises as reference. There are reasons however to doubt this thesis. Firstly, there is no mention of Aristotle in Jordanus’ writings, with the exception of the preface to version P of De ratione, which in any case is related to the Mechanica problemata only. Secondly, the setting of De ratione is different from that of Physica and De caelo, because the weights are not free in the space but suspended from a balance. In short, it is possible Jordanus followed a different line of thinking than that suggested by Duhem at the beginning of the section – a line of thinking which is not Aristotle’s. Proposition I when interpreted as suggested previously is a theorem of statics, stating that which can raise p to the height h can raise np to the height h/n. More precisely it is a form of the virtual work law and presents strong analogies with modern virtual work principles, at least in the versions considering them as a balance of work. The main difference is that in the modern laws the work of the two weights, rising and sinking, have a different algebraic sign. In Jordanus, works of different situations of rising are instead equated to a unique work of sinking. That makes Jordanus’ law useful only indirectly as an equilibrium criterion. The proof of the law of the lever, for example, is indeed obtained only by a reductio ad absurdum, while with the modern principle it is sufficient to write an algebraic equation. With his law, Jordanus was able to prove easily, and for the first time correctly, the inclined plane law by assuming direct proportionality between the raised weight and the vertical component of displacement along the inclined plane; but this will not be considered here. 49

p. 190. Translation in [171].

5 Italian Renaissance statics

Abstract. This chapter deals with Italian Renaissance mechanics in which the Aristotelian approach with WVLs is joined to the Archimedean without VWLs. The first part presents the mechanics of Nicolò Tartaglia, who takes as a principle the VWL associated with Jordanus de Nemore’s concept of gravity of position. The final part shows the mechanics of Galileo Galilei, who uses as a principle the VWL based on virtual velocities with the concept of moment for which the efficacy of a weight on a scale is the greater the greater its virtual speed. And shows as a corollary the VWL based on virtual displacements according to which anything that can lift a weight p of a height h can raise p/n of nh. In the central part the contributions of Girolamo Cardano, Guidobaldo dal Monte and Giovanni Battista Benedetti are presented, all of which somehow refer to a VWL. In the Middle Ages it was possible to identify in Europe two distinct traditions of mechanics; the science of weights, in particular that of Jordanus Nemorarius, and the philosophy of motion. Alongside these theoretical traditions there was the task of practical ‘mechanicians’ somehow continuing the tradition of the Roman period. In the XVI century there was a recovery of the ancient knowledge and Jordanus’ tradition is seconded by the Hellenistic, Aristotelian (Mechanica problemata) and Archimedean traditions [298, 296]. The postclassical tradition of the Mechanica problemata is a typical phenomenon of Renaissance, as it was practically unknown during the Middle Ages. However at least a Greek manuscript dating from the twelfth century survived, which testifies that the text was potentially accessible to Medieval philosophers. Few of the Hellenistic writings reached Europe. The Mechanica by Hero of Alexandria was known for sure in an Arabic translation only in the XVII century. Renaissance mathematicians had access to it through some epitome contained in Pappus’ Book 8 of the Mathematical collections (which remained in manuscript form until 1588) and Book 10 of Vitruvius’ De architectura. There is however at least a clue that some knowledge of Hero’s text should exist in the Renaissance; indeed Nicola Antonio (or Colantonio) Stigliola referred to in his book De gli elementi mechanici a treatment

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_5, © Springer-Verlag Italia 2012

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of the inclined plane substantially equivalent to that of Hero [219].1 Hero’s other two manuscripts were also known, the Pneumatica and the Automata, that were objects of translations and comments. Very important from a technological point of view, they were lacking theoretical arguments on mechanics. For the Archimedean tradition, considerations similar to that of the Mechanica problemata hold good, for some of his manuscripts were known in the Middle Ages, in Greek and Latin, but they had no impact in mechanics and mathematics. Archimedes’ ideas spread in the Renaissance thanks to Tartaglia’s editions of Moerbeke’s Latin translation of books on the centre of gravity and on floating bodies [222]. According to Drake [298, 296] in Italy, the leading nation of the period, there were two main schools in mechanics, sharing one or more of the traditions mentioned above. The North one, formed by Giovanni Battista Benedetti (Venice, 1530–1590), Nicolò Tartaglia (Brescia, 1499?-1557), Girolamo Cardano (Pavia, 1501–1576). The Centre one, formed by Federico Commandino (Urbino, 1509–1575), Guidobaldo dal Monte2 (Pesaro, 1545–1607), Bernardino Baldi (Urbino, 1553–1617). The school of the North would be more interested in practical issues, which require the study of motion and that is why they focused on Jordanus’s or Mechanica problemata instead of Archimedean mechanics devoid of any reference to kinematics. There are, however, differences and difficulties in this classification scheme. Tartaglia was critical to the setting of the Aristotelian Mechanica problemata, and appreciates Jordanus. Cardano followed the Mechanica problemata, and more generally the physics of Aristotle. Benedetti did not accept the approach of the Mechanica problemata, nor that of Jordanus, but tended to follow Archimedes, at least for what concerns the study of equilibrium. In the Centre, next to the strictly Archimedean of Commandino approach, one must register the approach of dal Monte that showed a certain appreciation to Mechanica problemata and that of Bernardino Baldi, who also was a follower of Mechanica problemata but perhaps less attentive to the Archimedean approach, though he made use of the theory of centres of gravity in statics. This partition in schools has to be mitigated and integrated considering the succession of generations and the dissemination of the various texts on mechanics and mathematics (see Tables 5.1, 5.2). It is undeniable that there was a close correlation between the professional or mathematical culture and the preference toward Archimedean mechanics. In the North, for various reasons, there was less attention paid to Archimedes, not only because of a greater emphasis devoted to practical aspects and motion, but also because of the lake of cultural tools (mathematical) to appreciate the mechanics of Archimedes. Tartaglia was certainly a talented mathematician, but more for his intelligence and originality than for education. The same applies to Cardano. Benedetti, a generation after, was able to read more 1

p. 41. According to Romano Gatto it is probable that during the Middle Ages at least a Greek copy of the Mechanica survived [326] because in Montfaucon’s Bibliotheca bibliothecarum manuscriptorum [172], p. 472, it is attested the presence – among Libri Greci – of the title Heronis Mechanica, & alia multi quae rare reperiuntur. According to Gatto however Stigliola had no direct access to Hero, but he read Leonardo da Vinci [325], p. 300. 2 For the spelling of dal Monte’s name see [305], vol. 2, p. 351.

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easily Archimedean texts, which were published in Italian too, and to appreciate Archimedes’ mechanical side. In central Italy the same was true of Bernardino Baldi, who certainly did not have the mathematical training of Commandino or dal Monte, although he was essentially a contemporary of Galileo. From the schema outlined above, scholars of southern Italy, Francesco Maurolico (Messina, 1494–1575), Nicola Antonio Stigliola (1546–1623) and Luca Valerio (Napoli, 1553–1618) remain on the outside. All were expert mathematicians, followers of Archimedes in mechanics and as such less interesting from my point of view. An in depth study of the southern Italy school is due to Romano Gatto [323, 324, 325, 125]; see also [367]. The Aristotelian text Mechanica problemata already presented in Chapter 3, was of considerable importance in the Renaissance. By its nature it was able to mobilize people of different backgrounds, humanists interested in the philosophical aspect and mathematicians and engineers interested in its theoretical and technological content. There is agreement that the Mechanica problemata as such remained without direct influence from the decline of Hellenistic science until the Greek revival of the Renaissance. Latin writers of the Middle Ages who encountered the Greek text were insufficiently impressed by it to continue the discussion.

Table 5.1. Hero, Jordanus, Archimedes’ texts Heronian texts 1501 De expetendis et fugientis rebus. Valla 1521 Di Lucio Vitruvio Pollione de architectura libri dece traducti de latino in vulgare affigurati. Cesariano 1550 De subtilitates. Cardano 1575 Spiritalium liber. Commandino 1588 Mathematica collectiones. Commandino 1589 Gli artificiosi et curiosi moti spirituali. Aleotti 1589 Automata. Baldi 1581 Pneumatica. Baldi 1592 Spiritali di Herone Alexandrino, ridotte in lingua volgare. Giorgi Jordanus’ texts 1533 Liber de ponderibus. Apianus 1546 Quesiti et inventioni diverse. Tartaglia 1565 Jordani opuscolorum de ponderositate. Tartaglia

1543 1551 1558 1570? 1565 1588

Archimedean texts Opera Archimedis. Tartaglia De insidentibus aquae. Tartaglia (in Italian) Archimedis opera non nulla. Commandino Momenta omnia mathematica. Maurolico (published 1685) Archimedis De iis quae vehuntur in aqua libri duo. Commandino In duos Archimedis aequeponderantium libros paraphrasis. Dal Monte

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Table 5.2. Editions of Mechanica problemata [15, 378] Year Author

Title

1517 Vittore Fausto

Aristotelis Mechanica. Parisiis

1525 Niccolò Leonico

Opuscola. Venetia

1565 Alessandro Piccolomini In mechanicas quaestiones Aristotelis paraphrasis. Romae 1570 Girolamo Cardano

Opus novum de proportionibus numerorum. Basileae

1573 Antonio Guarino

Le mechanice d’Aristotile trasportate di greco in volgar idioma, con le sue dimostrazioni nel fine. Modena

1582 Vannocci Biringucci

Parafrasi di monsignor Alessandro Piccolomini sopra le mecaniche d’Aristotile. Roma

1585 Giovanni B. Benedetti

De mechanicis in diversarum speculationum mathematicarum et physicarum liber. Taurini

1599 Henri de Monanthenil

Aristotelis mechanica, graeca, emendata, latina facta et commentariis illustrata. Parisiis

1613 Francesco Maurolico

Problemata mechanica cum appendice, et a magnetem, et a piroxidem nautica pertinentia. Messane

1581 Giuseppe Biancani

Aristotelis loca mathematica ex universis ipsius operibus collecta. Bononiae

1621 Bernardino Baldi

In mechanica Aristotelis problemata exercitationes. Moguntiae

1627 Giovanni de Guevara

In Aristotelis mechanicas commentarij. Rome

The XV century saw the rapid multiplication of Greek copies. The beginning of the XVI century saw two important Latin translations by two humanists. The first was due to Vittore Fausto (1480–1511), but the most largely circulating copy was the second translation by Niccolò Leonico Tomeo (1456–1531). Table 5.2 reports a quite exhaustive list of the translations and commentaries of the Mechanica problemata. In the course of later development, the Mechanica problemata gave way to more sophisticated mathematical treatment of the problems discussed qualitatively in it. There were also vernacular versions, a very important one being by Oreste Vannocci Biringucci (1558–1585), the nephew of the homonymous author of De la pirotechnica), encouraged in the translation enterprise by Alessandro Piccolomini (1508– 1579) who felt an Italian translation of Mechanica problemata to be necessary so that also engineers could profit from it. Highly original additions were offered lastly by Bernardino Baldi. The background of merging of humanist with practical concerns is traceable primarily to the emergence of architecture as a distinguished profession in the XV century, and particularly with the revival of interest in the text of Vitruvius, and the engineering treaties by (Taccola and) Francesco di Giorgio Martini.

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5.1 Renaissance engineering The XV and XVI centuries, especially in Italy, saw the emergence of a considerable number of skilled professional engineers. The contribution of these engineers, who have not yet even been fully counted, is known only very superficially [305]. A study, which will require the work of a large research group, is essential to reconstruct their role with some accuracy. A first examination of the sources consulted3 leads to the conclusion that the widespread opinion that the professional Italian engineers had no relevant direct influence in development of the mechanical theory should be shared [364, 293, 1, 368, 89, 45, 27, 92, 163, 108]. They however had an indirect influence in stimulating mathematicians and philosophers to develop theories which could help the solution of the technological problems connected to the enormous development of industry and architecture of the Renaissance [272, 331, 288, 251, 352]. A major limitation to the possibility that professional engineers could contribute to the development of mechanical theory, in addition to the characteristics of their necessarily practical business, is due to the fact that almost all scientific texts were written in Greek or Latin. There was however some important contamination especially with the traditions of the Problemata mechanica and Hero’s writings. Giuseppe Cereda and Vittorio Zonca made reference to Aristotle when they spoke of their machines. Giambattista Aleotti, an engineer at the court of Ferrara, quoted Archimedes, Aristotles and Hero [342]. Antonio Guarino, an engineer at the court of Modena, translated into Italian from the Greek the Mechanica problemata. Daniele Barbaro in his paraphrase of Vitruvius’s book showed a really noteworthy knowledge of mechanical theory [21]. A different story is about the most famous engineer of the Italian Renaissance, Leonardo da Vinci (1452–1519), who left hundreds of drawings and pages devoted to mechanics.4 It is difficult to give a full account of the opinions of historians on his role for science in general and mechanics in particular. One goes from an enthusiastic vision of the early XIX century, especially on the side of historians of science educated in literature, to a more mature appreciation of Duhem and finally to a fierce criticism of Truesdell, who minimises both the originality and contribution to the subsequent science development of the work of Leonardo. Eduard Dijksterhuis eventually considers studying Leonardo not for his contributions to science, but 3 Including the writings of Taccola (Siena, 1381–1458), Leon Battista Alberti (Genova, 1404– 1472), Francesco di Giorgio Martini (Siena, 1439–1501), Leonardo da Vinci (Vinci, 1452– 1519), Vannoccio Biringuccio (Siena, 1480–1539), Francesco de’ Marchi (Bologna, 1504–1576), Giovanni Battista Bellucci (San Marino, 1506, 1554), Daniele Barbaro (Venezia, 1513–1570), Bonaiuto Lorini (Firenze, 1540–1611), Domenico Fontana (Ticino, 1543–1607), Giuseppe Ceredi (Piacenza, f. 1560), Camillo Agrippa (Milano, fl 1570), Vittorio Zonca (Padova, 1568–1602), Giambattista Aleotti (near Ferrara 1546–1636), Antonio Guarino (1504–1590). 4 The many interests of Leonardo were previously considered in the early 1400, by the Sienese Mariano Taccola interested in the writings of mechanical and technical military of Pneumatica by Philo of Byzantium (280–220 BC). In recent times Giambattista Venturi published in 1797 a famous essay on the scientific work of Leonardo [393] and in the years 1880–1936 his notebooks and manuscripts were published in facsimile, and today all Leonardo’s works are printed with a diplomatic transcription. Leonardo was also studied in depth by Pierre Duhem [306], Clifford Truesdell and Roberto Marcolongo [357, 388].

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for the opportunity offered by his copious notes to follow the maturation of various scientific concepts [292]. The difficulty of analyzing the role of Leonardo is also due to the nonexistence of an organic edition of his works. In this situation I preferred to not refer to Leonardo’s contribution to the development of the law of virtual work, which is however probably also important.

5.1.1 Daniele Barbaro and Buonaiuto Lorini As examples of the level of knowledge of mechanical science and the laws of virtual work, I recommend excerpts from the Dieci libri di architettura di M. Vitruvio [21] by Daniele Barbaro [21] and the Fortificationi by Buonaiuto Lorini [163]. In his commentary on Book X of Vitruvius’s Dieci libri di architettura, Barbaro refers to Aristotle’s Mechanica problemata and tries to bring to the lever the various simple machines. The explanation of the operations is quite brief, reflecting the text of Vitruvius. However I feel it to be of a certain interest, considering the time of publication, the explanation of how the system of pulleys, i.e. the block and tackle, works: There is no doubt that if a weight is attached to a simple rope, let’s say a thousand pounds, all the work and force is supported by the rope, then if that rope will be doubled and to that a pulley is suspended where to hang the weight, the rope is to get half of fatigue, and a half force is enough to lift that weight. And if there are more pulleys? [...] If the first doubling takes away half of the weight, the second doubling to which a half remains, will take away half of that half and the whole weight will be taken away by the fourth part of the force which lifted the first weight [21].5 (A.5.1)

The explanation does not refer to the lever and is substantially the same as Hero’s, based on simple considerations of equilibrium. Barbaro however did not know Hero and probably not even the works of Pappus of Alexandria that had not yet been published by Commandino. Buonaiuto Lorini is at least a generation younger than Barbaro, and then he was able to read the latest developments in mechanics, of dal Monte certainly, but perhaps also of Cardano and Benedetti, of whom I shall speak below. In Book V of his Fortificationi Lorini shows both the mechanics of simple machines, with some theoretical considerations, and the complex construction equipment for lifting heavy weight, earth and water. He quotes Hero, Archimedes and Guidobaldo dal Monte. In particular, unlike Barbaro he refers the operation of the pulley to the lever, as dal Monte did. An interesting reference is to a law of virtual work for which, to a greater ratio of weight and power, there corresponds a greater ratio of the motion of the power with respect to that of the weight: The secret of all the inventors of mills and other machines is to look for, just like you said, to accompany force with speed, a really difficult thing, because since the same power has to multiply into many, which one after another may lift, or carry a load, it is necessary that the time it is multiplied likewise, as for example it would be if you were to carry a weight of one thousand pounds from one place to another, through the sheer force of one man, which shall take only a part, that will be fifty pounds [163].6 (A.5.2) 5 6

p. 446. p. 238.

5.2 Nicolò Tartaglia

97

Fig. 5.1. The simple machines of Daniele Barbaro (reproduced with permission of Biblioteca Centrale della Facoltà di Architettura of Università La Sapienza, Rome)

5.2 Nicolò Tartaglia Nicolò (or Niccolò) Tartaglia was born in Brescia probably in 1499 and died in Venice in 1557. He received no formal education, except for a period of fifteen days in a “scuola per scrivere” when he was fourteen. He learned to read Latin but, with a single exception, he wrote only in a not very elegant Italian [294]. In 1537 he published his first book, the Nova scientia, inspired by practical problems of gunnery. In 1543 his editions of Euclid (in Italian) and Archimedes (in Latin) were published, see Table 5.1. In 1551 he published in Italian the first book of Archimedes’ De insidentibus in aquae, in 1546 the Quesiti et inventioni diverse, where his version of the science of weights is reported. Although the book was largely a paraphrase of Jordanus’ De ratione ponderis first book, of which he possessed a copy published posthumously in 1565 [224], Tartaglia did not cite the fact, and this brought upon him the accusation of plagiarism. Of Tartaglia’s writings on mechanics I will refer only to Book VIII of Quesiti et inventioni diverse, because it is the only one related to the virtual work laws.

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5.2.1 Definitions and petitions Book VIII of Quesiti et inventioni diverse starts with some definitions (seventeen) and petitions (six). Table 5.3 reports the statement of the main definitions and petitions [298, 223]. (A.5.3) Examination of the table makes evident the more formal approach of Tartaglia with respect to Jordanus de Nemore. Definitions III and IV relate to virtue (see § 4.2). The third definition has a qualitative character and applies to both the upward and downward motions. The fourth definition, which is still about virtue, is more problematic. Meanwhile it identifies the measure of virtue – that below he often names power – with speed, it seems to apply only to the downward motion. Of some importance seems to me Definition XIV, which takes away any ambiguity to the introduction of weight. Table 5.3. Tartaglia’s definitions and petitions Definition III

By virtus of a heavy body is understood and assumed that power which it has to tend or go downward, as also to resist the contrary motion which would draw it upward.

Definition IV

Bodies are said to be of equal virtus or power when in equal times they run through equal spaces.

Definition XIII

A body is said to be positionally more or less heavy than another when the quality of the place where it rests and is located makes it heavier [or less heavy] than the other, even though both are simply equal in heaviness.

Definition XIV

The heaviness of a body is said to be known when one knows the number of pounds, or other named weight, that it weighs.

Definition XVII The descent of a heavy body is said to be more oblique when for a given quantity it contains less of the line of direction, or of straight descent toward the centre of the world. Petition II

Likewise we request that it be conceded that that body which is of greater power should also descend more swiftly; and in the contrary motion, that is, of ascent, it should descend more slowly - I mean in the balance.

Petition III

Also we request that it be conceded that a heavy body in descending is so much the heavier as the motion it makes is straighter toward the centre of the world.

Petition VI

Also we request that it be conceded that no body is heavy in itself.

After definitions, petitions follow, which to Tartaglia are those propositions that should be asked the opponent being accepted for the conduct of the demonstrations (they are then postulates). Notice that the second petition is linked to the fourth definition, comparing power with speed, both for downward and upward motions. Here he used the word power, which in the fourth definition was identified with virtue.

5.2.2 Propositions Tartaglia considers fifteen propositions (theorems), some of them are shown in Table 5.4 [298, 223]. (A.5.4)

5.2 Nicolò Tartaglia

99

Table 5.4. Tartaglia’s propositions I II

The ratio of size of bodies of the same kind is the same as the ratio of their power. The ratio of the power of heavy bodies of the same kind and that of their speeds (in descent) is concluded to be the same; also that of their contrary motions (that is, of their ascents) is concluded to be the same, but inversely. III If there are two bodies simply equal in heaviness, but unequal positionally, the ratio of their powers and that of their speeds will necessarily be the same. But in their contrary motions (that is, in ascent) the ratio of their powers and that of their speeds is affirmed to be inversely the same. IV The ratio of the power of bodies simply equal in heaviness, but unequal in positional force, proves to be equal to that of their distances from the support or centre of the scale. V When a scale of equal arms is in the position of equality, and at the end of each arm there are hung weights simply equal in heaviness, the scale does not leave the said position of equality; and if it happens that by some other weight [or the hand] imposed on one of the arms it departs from the said position of equality, then, that weight or hand removed, the scale necessarily returns to the position of equality. VI Whenever a scale of equal arms is in the position of equality, and at the end of each arm are hung weights simply unequal in heaviness, it will be forced downward to the line of direction on the side where the heavier weight shall be. VII If the arms of the scale are unequal, and at the ends thereof are hung bodies simply equal in heaviness, the scale will tilt on the side of the longer arm. VIII If the arms of the balance are proportional to the weights imposed on them, in such a way that the heavier weight is on the shorter arm, then those bodies or weights will be equally heavy positionally. XIV The equality of slant is an equality of [positional] weight. XV If two heavy bodies descend by paths of different obliquities, and if the proportions of inclinations of the two paths and of the weights of the two bodies be the same, taken in the same order, the power of both the said bodies in descending will also be the same.

Before going into the validity of the proof of the various propositions, I want to stress Tartaglia’s ideas. He found in Jordanus’s writings two possible principles of statics, one based on the concept of gravity of position, the other on the capability of a weight to lift another. According to my interpretation, Jordanus used both, the first for qualitative proof, the second to establish mathematical relations. Tartaglia makes instead a choice and decides to base his mechanics only on the gravity of position. This notwithstanding, he maintains a trace of Jordanus’ ideas and to state the equilibrium of a lever or an inclined plane considers the equivalence of weight disposed on the same side and not on the opposite. Tartaglia reconsiders Jordanus’ original proposition II by splitting it into three ‘propositions’ and modifying in part the conclusion. In the first proposition he ‘proves’ that greater weights have greater power. In the second that speed and power are in the same proportion in downward motion and in the inverse proportion in upward motion. In the third proposition that speed and weight are in direct proportion in downward motion and in inverse in upward motion. In the fourth proposition he proves that the power of weights is proportional to their distances from the fulcrum.

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A

B

C

E

F

D

Fig. 5.2. Ratio of sizes (A, B, C) and powers (D, E, F)

5.2.2.1 Proof of propositions I–IV Proposition I The ratio of the size of bodies of the same kind is the same as the ratio of their power. Let there be the two bodies AB and C of the same kind; let AB be the greater, and let the power of the body AB be [represented by the line] DE, and that of the body C [by the line] F. Now I say that that ratio which the body AB bears to the body C is that of the power DE to the power F. And if possible (for the adversary), let it be otherwise, so that the ratio of the body AB to the body C is less than the ratio of the power DE to the power F. Now let the greater body AB include a part equal to the lesser body C, and let this be the part A, and since the force or power of the whole is composed of the forces of the parts, the force or power of the part A will be D, and the force or power of the remainder B will necessarily be the remaining power E; and since the part A is taken equal to C, the power D (by the converse of Definition 7) will be equal to the power F, and the ratio of the whole body AB to its part A (by Euclid V.7, 2) will be as that of the same body AB to the body C (A being equal to C), and similarly the ratio of the power DE to the power F will be as that of the said power DE to its part D (D being equal to F). Therefore [by the adversary’s assumption] the ratio of the whole body AB to its part A will be less than that of the whole power DE to its part D. Therefore, when inverted (by Euclid V.30), the ratio of the body AB to the residual body B will be greater than that of the whole power DE to the remaining power E, which will be contradictory and against the opinion of the adversary, who wants the ratio of the greater body to the less to be smaller than that of its power to the power of the lesser body. Thus, the contrary destroyed, the proposition stands [223].7 (A.5.5)

In his proposition I, Tartaglia assumes bodies of the same material but different size, so there is no doubt on the meaning of the proposition. He takes for granted, even if not explicitly stated in his petitions, that a heavier body has more power than a lighter. Tartaglia reproduces the framework of proof of proposition II of Jordanus, making it more clear. But there are still some points not acceptable to a modern reader. Without specifying exactly what it is and how to measure the power of a body, Tartaglia accepts additivity: the power of a body is given by the sum of the power of its parts. Like Jordanus, he does not notice, however, that in this way he takes for granted what he wants to prove (see § 4.2.2.2). A modern reader is baffled by the almost miraculous demonstration such as Tartaglia’s, as would be that of Jordanus. There is the impression that with this way of reasoning one can prove anything, for example, that beauty is proportional to size.

7

p. 88r. Translation in [298].

5.2 Nicolò Tartaglia

101

Plate 1. Tartaglia’s books on Scientia de ponderibus (reproduced with permission, respectively, of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome, and of Max Planck Institute for the History of Science, Berlin)

Tartaglia’s proof of his proposition II is based on the same reasoning. This time things are slightly clearer because the definitions third and fourth and petition second, connect somehow power and speed, in particular they argue that it has a higher speed if there is a higher power. The first part of proposition II, that bodies fall down with speeds proportional to their size, is proved with arguments similar to that used in proposition I. Additivity of speed with power is assumed and proportionality demonstrated. To demonstrate the inverse relationship between power and speed Tartaglia assumes that the resistance to upward motion is proportional to the power of the body. So that power that will barely fit in the other arm to lift the body AB, will be sufficient to lift faster the body C and the relationship of speed of C to AB is that of ED to F (Fig. 5.2). From propositions I and II follows the proportionality (direct or inverse) between weight (size) and speed. The logical status of proposition III is not clear; to a modern reader it seems an immediate consequence of propositions I and II, however, a demonstration is proposed by following exactly the arguments of proposition I. In proposition IV Tartaglia aims to quantify the concept of gravity of position, at least for bodies connected to the arms of a balance. The proof again follows the same line of argument, with some more difficulty. Tartaglia seems to make the assumption that the sum of distances corresponds to the sum of weights. 5.2.2.2 The law of the lever With proposition IV the demonstration of the law of the lever should be immediate, it would suffice to argue that the two weights at each end of the lever are equal

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D

A

L

H F M

C

B

E

Fig. 5.3. Equilibrium of the lever with different arms

in gravity of position and therefore balanced. Tartaglia, however, prefers to repeat Jordanus’s approach, where instead of the equilibrium of opposing tendencies it is considered the equivalence of weights that tend to move in the same direction. In the passage below, Tartaglia refers the lever with weights E and D to the lever in which the weights are D and L = E, on the same side. Through his proposition IV he argues that they are equally heavy for position and D may be replaced by L arriving at a balance with equal arms (LC = EC) and equal weights, and as such, in equilibrium for the proposition V (not commented here). Proposition VIII If the arms of the balance are proportional to the weights imposed on them, in such a way that the heavier weight is on the shorter arm, then those bodies or weights will be equally heavy positionally. […] First let there be the bar or balance ACB and the weights A and B hung thereon, and let the ratio of B to A be as that of the arm AC to the arm BC. I say that this balance will not tilt to either side. And if (for the adversary) it is possible for it to tilt, let us assume it to tilt on the side of B and to descend obliquely as the line DCE in place of ACB, and [let us] take D as A and E as B; and the line DF falls perpendicularly, and the line EH rises similarly. […] and put L equal in heaviness to B and descending along the perpendicular LM, then, since it is manifest that LM and EH are equal, the proportion of DF to LM will be as the simple heaviness of the body B to that of the body, or as the simple heaviness of the body L to that of D […]. Whence if the said two heavy bodies, that is, D and L, were simply equal in heaviness, standing then in the same positions or places at which they are presently assumed to be, the body D would be positionally heavier than the body L (by the Fourth Proposition) in that ratio which holds between the whole arm DC and the arm LC. And since the body L is simply heavier than the body D (by our assumption) in the same ratio as that of the arm DC to the arm LC, then the said two bodies D and L in the level position would come to be equally heavy, because by as much as the body D is positionally heavier than the body L, by so much is the body L simply heavier than the body D; and therefore in the level position they come to be equally heavy. […] Therefore if the body B (for the adversary) is able to lift the body A from the level position to the point D, the same body B would also be able and sufficient to lift the body L from the same level position to the point where it is at present, which consequence is false and contrary to the Fifth Proposition […]. Thus, the adversary’s position destroyed, the thesis stands [223].8 (A.5.6) 8

pp. 92v–93r. Translation in [298].

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103

At the end of his proposition Tartaglia refers to the demonstration of Archimedes, stating that since the matter of his treatise is quite different from the Archimedean, he has considered to demonstrate the law of the lever with other principles as more appropriate. Note that Tartaglia did not study the angular lever (Jordanus’ proposition VIII). This is because the use of the concept of gravity of position does not work in such a case [305].9 5.2.2.3 The law of the inclined plane The proof of the law of the inclined plane is preceded by a lemma similar to that reported by Jordanus, according to which the gravity of position along an inclined plane is constant. Tartaglia does not make the step that it would seem natural to explicitly state that the gravity of position is inversely proportional to the obliquity (with the meaning he gave to this term). The lack of this step is critical because in the proof of the law of the inclined plane Tartaglia uses it effectively. From this point of view the demonstration of Tartaglia is less satisfactory than that of Jordanus. The proof is developed as in the case of the lever, bringing the equilibrium to an equivalence. But the reasoning is less strict, because it asserts without explanation that the two heavy bodies H and G are equally heavy for position as they have weights inversely proportional to their obliquities, which although intuitive, has not yet been demonstrated by Tartaglia. In his beautiful work, Storia del metodo sperimentale, Raffaello Caverni [284] considers Tartaglia’s demonstration as the first truly exemplary proof, of higher value than that of Jordanus, of whom Caverni seems however to not know the De ratione. Caverni reports and comments on the demonstration of Tartaglia, justifying it with the statement of the proposition XIV [284],10 which for me is a logical gap. D

N

G

M

E

H Z

X

Y T

K

A

Fig. 5.4. Equilibrium on the inclined plane 9

Vol. 1, p. 121. vol. IV, pp. 321–232.

10

R B

L C

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Proposition XV If two heavy bodies descend by paths of different obliquities, and if the proportions of inclinations of the two paths and of the weights of the two bodies be the same, taken in the same order, the power of both the said bodies in descending will also be the same. […] Then let the letter E represent a heavy body placed on the line DC, and the letter H another on the line DA, and let the ratio of the simple heaviness of the body E to that of the body H be the ratio of DC to DA I say that the two heavy bodies in those places are of the same power or force. And to demonstrate this, I draw DK of the same tilt as DC, and I imagine on that a heavy body, equal to the body E, which I letter G, in a straight line with EH, that is, parallel to CK. […] Also the ratio of MX to NZ will be as that of DK to DA; and (by hypothesis) that is the same as that of the weight of the body G to the weight of the body H, because G is supposed to be simply equal in heaviness with the body E. Therefore, by however much the body G is simply heavier than the body H, by so much does the body H become heavier by positional force than the said body G, and thus they come to be equal in force or power. And since that same force or power that will be able to make one of the two bodies ascend (that is, to draw it up) will be able or sufficient to make the other ascend also, [then], if (for the adversary) the body Eß is able and sufficient to make the body H ascend to M, the same body E would be sufficient to make ascend also the body G equal to it, and equal in inclination. Which is impossible by the preceding proposition. Therefore the body E will not be of greater force than the body H in such place or position; which is the proposition [223].11 (A.5.7)

5.3 Girolamo Cardano Girolamo Cardano was born in Pavia in 1501 and died in Rome in 1576. He was educated at the university of Pavia, and subsequently at that of Padua, where he graduated in medicine. He was, however, excluded from the College of physicians at Milan on account of his illegitimate birth, and it is not surprising that his first book should have been an exposure of the fallacies of the College. In 1547 he accepted a chair of medicine at Pavia university. The publication of his works on algebra and astrology had gained for him a European renown. In 1551 his reputation was crowned by the publication of his great work, De subtilitate rerum, here after De subtilitate, which embodied the soundest physical learning of his time and simultaneously represented its most advanced spirit of speculation [294]. Cardano’s writings on mechanics are only a small part of his interests, which were mainly medical and astrological, for he wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music. His role in mechanics is controversial. Duhem is convinced that he borrowed abundantly from Leonardo da Vinci [305], but Drake is doubtful on the purpose [298].12 Though Cardano cites Archimedes, Hero, and Ctesibius, he was strongly influenced by Aristotle. 11 12

pp. 97r, 97v. Translation in [298]. p. 26.

5.3 Girolamo Cardano

105

A F G H

P O N B

C

D

M

E S U

R Q

Fig. 5.5. The balance with equal arms

Statics is dealt with in the last part of the De subtilitate [54] first book, with some other considerations scattered elsewhere. Other considerations on statics are in De opus novum de proportionibus [56]. In the following I will comment mainly on Cardano’s considerations of the balance.

5.3.1 De subtitilate The text of the De subtilitate where Cardano discusses the effectiveness of the weights placed on the arms of a balance is difficult to read because it is not always entirely consistent. The aim is to comment on issue 2 of Mechanica problemata, in which Aristotle discusses the stability of the scale with fulcrum above or below the beam. Cardano speaks about a balance CD hanging in A, as shown in Fig. 5.5: Next we must consider weights that are placed upon a balance. Let there be a balance whose point of suspension is at A, let the point where the arms of the beam are joined be B, and let the beam be CD. It is clear that CD moves about B as a fixed centre, because CD cannot be separated from B. Let the angles ABC and ABD be right angles [54].13 (A.5.8)

But then he develops all his considerations as if the balance were hanging in B. He compares the effectiveness of the same weight p placed respectively in F and C to conclude that it is heavier in C. He argues the conclusion in two ways, for the first way he refers to qualitative considerations based on common experience, mainly on the evidence that weights more distant from the fulcrum are more effective: I say that a weight placed at C (the beam being in the horizontal position CBD) will be heavier than if the beam were put in any other position, as, for instance, with the end of the beam at F [...], therefore, I shall show by two arguments that this happens when the weight and beam are placed at C rather than at F […]. The first of these arguments may be explained in this way. It is clear that, in steelyards and in those instruments which raise weights, the farther the weight is from the point of suspension, the heavier it seems […] it is also clear that, the farther the balance-arm descends toward C from A, the heavier the weight becomes 13

p. 23.

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and, therefore, the more swiftly it moves; but, for the opposite reason, in the movement from C toward Q, the weight is lighter and the motion is slower – a fact which is proved by experience [54].14 (A.5.9)

The second way of argument is more complex, it refers to the principles of Aristotelian physics and is quantitative in nature. To bring coherence to the argument of Cardano one should refer instead of a balance in two different positions (FR and CD), to points C and F of a circle, describing equal arcs in equal times. With simple geometrical considerations Cardano shows that for a given rotation of the circle, the point F describes a path OP, measured on the vertical line, less than that BM of C, and then it moves with a less vertical velocity. The greater velocity of C with respect to F leads to the conclusion that the weight p is more effective (heavier) in C then in F. Cardano’s reasoning is different from that of all his predecessors, particularly of Tartaglia, although it is similar. He seems to refer to the law of virtual work of Thabit, reported in Chapter 4, for which the effectiveness, or force, of a weight is measured by (is proportional to) its virtual velocity. The difference is that Thabit considers motions along arcs, Cardano vertical motions. The second argument may be demonstrated as follows: Let the arc CH be laid off equal to the arc CE […] therefore, BN is greater than OP, and, because of this, BM is greater than OP. Now, while the end of the beam is moved from C to E, the weight descends through the distance BM and is thus brought closer to the centre than it was at C. While the beam is moved through the length of the arc FG, the weight descends through OP. And BM is greater than OP. Now, supposing that in equal times this weight passes from C to E and from F to G, it descends still more quickly from C than from F; therefore, it is heavier at C than at F [emphasis added] [54].15 (A.5.10)

Cardano closes his argument by asserting that from above it is not difficult to see how the balance is stable with the fulcrum over and unstable with the fulcrum below [54].16 In fact Cardano’s conclusion is not clear, although it can be inferred easily from the conclusions reached by him when the balance has the fulcrum in B. In fact, taking for example the fulcrum in A one sees that if the two weights are not at the same level, the higher will be more effective than the lower and the balance will return with weights at the same level. Cardano returns to problems of statics in other books of the De subtilitate, particularly interesting are the considerations on the block and tackle: The fourth example of subtilitates is the block and tackle. But because the ratio of times is as that of powers, [the boy] will pull four times more slowly with two pulleys, six times with three pulleys [...] so it will happen that the boy in a hour will pull just the same weight with the pulley that a man, six times more strong, being above, can pull on the spot with a single rope [54].17 (A.5.11)

Notice that also Cardano, like Hero, speaks about time instead of space. 14 15 16 17

Liber primus, pp. 23–24. Translation in [57]. Liber primus, p. 24. Translation in [57]. p. 24. Liber XVII, p. 467–468.

5.3 Girolamo Cardano

107

5.3.2 De opus novum In De opus novum Cardano addresses topics of statics on several occasions. In particular, he gives a personal demonstration (wrong) of the inclined plane, which is different from that of Pappus (wrong too) and Jordanus Nemorarius (correct), which were probably both known by him. In essence Cardano believes that the force required to move a weight on an inclined plane is proportional to the angle of the plane with the horizon, instead of the sinus as it should be [56].18 Of some interest is the demonstration of the law of the balance, in which Cardano measures the effectiveness of the weights based on the virtual displacements, as he did in De subtilitate, except that now the motion instead of being measured on the vertical seems to be measured along the arc, as Thabit did. a

e

f

b m

h

p o

n g i

d k

c

Fig. 5.6. The law of balance Proposition forty five Show the law of balance If the beam bd is put in e and f and if the ratio of eb to b f is as that of g to h, I say there will be equilibrium. Otherwise h would move to k, until it reaches the line ad. If h were not fixed [at the beam bd] it would move along [the vertical] eh; but because it is fixed it will move along the curve hk. Take a point [m] near [to b] in be and n at equal distance in b f . Because all eb is moved in any part with a same force, i.e. the weight h, and because the point in h moves along hk and the point in m along mp, the ratio of hk to mp equals that of the force in mp to the force in hk, and so the force will be nearly infinite in b [56].19 (A.5.12)

18 19

p. 63. p. 34.

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5.4 Guidobaldo dal Monte Guidobaldo dal Monte was born near Pesaro in 1545 and died in Pesaro in 1607. He entered the university of Padua in 1564 having as a companion Torquato Tasso, studied mathematics with Federico Commandino and was teacher of Bernardino Baldi. He was one of the greatest mathematicians and mechanician of the late XVI century. He was also a highly competent engineer as well as a director of the Venice arsenal and last but not least the brother of a prominent cardinal. He was the mentor of Galileo and secured for him his first academic position as a lecturer in mathematics at Pisa and Padua universities [294]. In 1577 he published the Mechanicorum liber [86, 377], translated into Italian by Filippo Pigafetta in 1581 as Le mechaniche [88]. The book had an enormous editorial success and was read for the whole XVII century. In 1588 he published the Archimedis aequeponderantium, a paraphrase of Archimedes’ Aequeponderanti [87], in 1600 an important book on perspective. Dal Monte was one of the major critics of the approach of Jordanus de Nemore. According to him those of Jordanus and his followers, among which he includes Tartaglia, are not valid demonstrations and goes so far as to say that Jordanus should not even be counted among the true mathematicians. Bernaldino Baldi went still further and considered as paralogisms the demonstrations of Jordanus [18].20 Criticisms of dal Monte must be placed in his time to be understood. Scholars of mathematics of the period, particularly those of central and southern Italy, could not fail to be charmed by the elegance and rigor of geometry as it was revealed by the recently published Greek translations of Euclid and Archimedes. Archimedes, moreover to his mathematical theory flanked a consistent mechanical theory and with the same standards of rigor. It was therefore natural to accept the argument of Archimedes in mechanics and reject those by Jordanus. Although to a modern observer the full refusal of Jordanus seems unjustified because the De ratione ponderis has a Euclidean approach based on definitions, axioms and theorems. It is certainly the ancient text in which the Euclidean approach is extended further outside geometry. It is all in all a very modern text. Dal Monte, however, could hardly accept to reason with concepts such as gravity of position which remained a bit undefined and made recourse to empirical intuition. Given that Jordanus’ theses were then quite common in Italy, dal Monte somehow felt the need to re-establish the ‘truth’, by writing the Mechanicorum liber and Archimedis aequeponderantium that can be seen as the natural completion of the work of spreading Archimedes’s mechanical thought. Commandino indeed had only previously published his text on floating bodies and his anxiety over the rigor led dal Monte to make criticisms that today seem ungenerous, such as those that consider wrong the demonstrations based on the parallelism of descent lines of heavy bodies. The hostility towards the approach of Jordanus also led dal Monte to refuse the 20

p. 32.

5.4 Guidobaldo dal Monte

109

correct proof of the inclined plane for the incorrect one by Pappus of Alexandria. Which brought upon the blame, among others, of Evangelista Torricelli [233].21 Although the Mechanicorum liber on the one hand had given up the fertility of Jordanus’s approach, based on the concept of gravity of position and a law of virtual work, playing in some way a conservative role, it expanded the scope of mechanics. The medieval science of weights, in which attention was focused on demonstrating the law of the lever, is led back to the Greek tradition of mechanics as a science of machines, influenced in this by the Mechanica problemata, but especially by Hero’s approach, then known only through the work of Pappus of Alexandria.

5.4.1 The centre of gravity In the following, instead of the Mechanicorum liber I will refer to its Italian translation Le mechaniche, which was more diffuse. The text begins with the definition of the centre of gravity, which is worthy to be reported because of the great weight this concept will have in formulating the principle of Torricelli. Dal Monte takes the definition of Pappus, with the addition of a definition due to Commandino: The centre of gravity of any body is a certain point within it, from which, if it is imagined to be suspended and carried, it remains stable and maintains the position which it had at the beginning, and is not set to rotating by that motion. This definition of the centre of gravity is taught by Pappus of Alexandria in the eighth book of his Collections. But Federico Commandino in his book On Centres of Gravity of Solid Bodies explains this centre as follows: The centre of gravity of any solid shape is that point within it around which are disposed on all sides parts of equal moments, so that if a plane be passed through this point cutting the said shape, it will always be divided into parts of equal weight [88].22 (A.5.13)

Is still unclear the role that dal Monte gives to the centre of gravity for a system of bodies. On the one hand the bodies are taken individually, subject to gravity converging toward the centre of the world, on the other hand, the gravity is considered to be concentrated in the centre of gravity of the whole, which is determined by the Archimedean rules. It should be noted that dal Monte, with many other mathematicians of the time, will definitely realize that, from a practical point of view, to consider the lines of action of gravity parallel to each other or to consider them converging to the centre of the world did not matter much, nevertheless he believed that, to establish the ‘reality’ of things, one could not accept this approximation.

5.4.2 The balance After studying the balance with equal weights and arms and with suspension points above and below the centre of gravity, correctly recognizing the stability in the first case and instability in the second, dal Monte then goes on to study the still controversial case, the quality of the equilibrium of a balance when it was suspended 21

vol. 3, p. 439. p. 1. Translation in [298]. Notice that the second definition is the same as that referred to in § 3 by Hero. 22

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Plate 2. The two editions of dal Monte’s Mechanics (reproduced with permission, respectively, of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome, and of Biblioteca Alessandrina, Rome)

for his centre of gravity C as shown in Fig. 5.7a, clearly stating that it is in neutral equilibrium: A balance parallel to the horizon, having its centre within the balance and with equal weights at its extremities, equally distant from the centre of the balance, will remain stable in any position to which it is moved. […] I say, first, that the balance DE will not move and will remain in that position Now since the weights A and B are equal, the centre of gravity of the combination of the two weights A and B will be at C. Hence the same point C will be the centre of gravity of the balance and of the whole weight. And since the centre of gravity of the balance, C, remains motionless while the balance AB together with the weights moves to DE, the centre of gravity is not moved [88].23 (A.5.14)

According to dal Monte, Jordanus, Cardano and Tartaglia, who assumed a stable state of equilibrium for the horizontal scale, were wrong and even went against Archimedes: Now since they say that the weight placed at D is heavier in that position than is the weight placed at E in its lower position, then, when the weights are at D and E, the point C will no longer be their centre of gravity, inasmuch as they would not be stable if suspended from C. But that centre will be on the line CD, by Archimedes, On Plane Equilibrium, 1.3. It will not be on CE, the weight D being heavier than the weight E; let it therefore be at H, from which, if they were suspended, the weights would remain stationary. And since the centre of gravity of the weights joined byAB is at the point C, but that of those placed at D and E 23

p. 10. Translation in [298].

5.4 Guidobaldo dal Monte

F

L

N

D

111

F D H C

A

B

M

C H

K

O E E

a)

G

G b)

T K

V

Fig. 5.7. Equilibrium in the balance with equal arms and weights

is the point H, when the weights A and B are moved to DE, the centre of gravity C would be moved toward D and would approach closer to D, which is impossible. For the weights remain the same distance apart, and the centre of gravity of any body stays always in the same place with respect to that body [88].24 (A.5.15)

His searches were picky, but not as rigorous as he claims, in an attempt to refute the views of Jordanus. With reference to Fig. 5.7a he starts by underlining the weakness of Jordanus’ claims that the weight is heavier in D than in E, recovering and improving the limit analysis of the gravities of position in D and E, when the spaces covered become very small. Things being taken as before, and from the points D and E the lines DH and EK being drawn perpendicular to the horizon, let there be taken another equal circle LDM, with centre N, which is tangent to the circle FDG at the point D.[…] But the ratio of angle MDH to HDG is smaller than any other ratio that exists between greater and smaller quantities; therefore the proportion of the weights at D and E will be the smallest of all possible ratios, or, rather, will not be a ratio at all. […] we shall find ratio diminishing ad infinitum, and it follows thus that the ratio of the weight placed at D to that at E is not so small that one infinitely less cannot be found. And since the angle MDG can be divided in infinitum, so also one may divide in infinitum the excess of weight which D has over E [88].25 (A.5.16)

He sets out very clearly the view of the school of Jordanus that the weight in D, in Fig. 5.7a, is heavier than that in E but of a very small amount. Indeed with reference to Fig. 5.7b where the portion of circle passing from E is redrawn in D as LDM, it is possible to see that the mixed angles HDG (the obliquity of D) and MDH (the obliquity of E) differ by an angle that is small as one likes, in particular, smaller than any angle bounded by straight segments. The challenge of this statement is showing that there are countless angles whose difference with the angle HDG tends to zero. So the gravity of the weight in D is equal to that of the weight in E, and the equilibrium is indifferent. 24 25

pp. 11–12. Translation in [298]. pp. 13–14. Translation in [298].

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Fig. 5.8. Difference in gravity of position

Then dal Monte shows, paradoxically, that if one followed the reasoning of Jordanus, assuming that the forces of weights converge toward the centre of the world (or earth), then the weight in E would be heavier than the weight in D, because the angle ODS (the obliquity of D) is greater than TES (the obliquity of E), and then the balance should assume a vertical direction rather than return to a horizontal position (Fig. 5.8a). The arguments of dal Monte against the concept of gravity of position continue; for him it is an ill defined concept and its quantification depends on the size and arrangement of the arcs that are considered, a hiatus with the preceding argument where infinitesimals were spoken of: For the weight placed at L would move freely toward the centre of the world along LS, and the weight at D along DS. But since the weight at L weighs wholly on LS, and that at D on DS, the weight at L will weigh more on the line CL than that at D on DC. Therefore the line CL will more sustain the weight than the line CD; and in the same way, the closer the weight is to F, it will be shown for this reason to be more sustained by the line CL, since the angle CLS is always less, which is obvious. For if the lines CL and LS should come together, which would happen at FCS, then the line CF would sustain the whole weight that is at F and would render it motionless, nor would it have any tendency to descend [gravezza] along the whole circumference of the circle [88].26 (A.5.17)

Dal Monte maintains with Jordanus, that the gravity position is greater where the descent path is closer to the line joining this point with the centre of the world. According to this logic and with reference to Fig. 5.8b the point where the gravity of position is greater is O and not A. Dal Monte justifies this greater propensity because the arm of the balance offers the least resistance. That is, in modern terms, 26

p. 19. Translation in [298].

5.4 Guidobaldo dal Monte

L

113

F

D C

A

G

M B

E

H S

K

Fig. 5.9. Indifferent equilibrium of the balance with equal arms and weights

instead of directly recognizing that there is a greater tendency to fall down because the component of weight is greater, he says there is a greater tendency because there is minor resistance. With an argument closely related to his idea of gravity, dal Monte proves once again that the equal weights placed in E and D at the ends of the balance of Fig. 5.9 are equally heavy and that the balance remains stationary (indifferent equilibrium) in the position DE. It is one thing, he says, to consider the weights in D and E separately, in which case they would move to S along DS or ES respectively; the other is to consider them together, so their centre of gravity would move to S along CS, while the weights in D and E along DH and EK, as shown in Fig. 5.9. But since C cannot sink, the weights remain at their place, D and E. If the weight placed at E is heavier than the weight placed at D, the balance DE will never remain in that position, as we have undertaken to maintain, but it will move to FG. To which we reply that it makes a great deal of difference whether we consider the weights separately, one at a time, or as joined together; for the theory of the weight placed at E when it is not connected with another weight placed at D is one thing, and it is quite another when the weights are joined in such a way that one cannot move without the other. For the straight and natural descent of the weight placed at E, when it is without connection to another weight, is made along the line ES; but when it is joined with the weight D, its natural descent will no longer be along the line ES, but along a line parallel to CS. For the combined magnitude of the weights E and D and the balance DE has its centre of gravity at C, and, if this were not supported at any place, it would move naturally downward along the straight line drawn from the centre of gravity C to the centre of the world S until C reached S. […] But if the weights E and D are joined together and we consider them with respect to their conjunction, the natural inclination of the weight placed at E will be along the line MEK, because the weighing down of the other weight at D has the effect that the weight placed at E must weigh down not along the line ES, but along EK. The same is true of the weight at E; that is, the weight at D does not weigh down along the straight line DS, but along DH, both of them

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being prevented from going to their proper places [...].Thus the descent of the weight at D will be equal to the rise of the weight at E, and the weight at D will not raise the weight at E. From which it follows that the weights at D and E, considered in conjunction, are equally heavy [88].27 (A.5.18)

The first chapter ends with a study of unequal arm balance treated according to the Archimedean approach. In the sense that the equilibrium of the balance is guaranteed if it is suspended from its centre of gravity, which as demonstrated by Archimedes divides the beam of the balance in parts inversely proportional to the weights. In the second chapter, Della leva, dal Monte examines the lever as if it were something different from the balance. Probably the reason for this apparent duplication of treatment, in addition to the somewhat pedantic writing, comes from the fact that, besides the weight that is to be raised, it is considered also a muscle force. This force in the limit might not be vertical, but simply perpendicular to the lever arm; an examination of the text does not completely dissolve the ambiguity, made even greater by the fact that in the drawings the points of the lever to which the force is applied are identified only by letters and the direction of the force is never highlighted.

Fig. 5.10. The simplest example of block of pulleys (reproduced with permission of Biblioteca Alessandrina, Rome)

A particularly interesting chapter is the third, dedicated to the block and tackle which were of great significance in applications for the lifting of heavy weights, especially in construction. The theory of a block of pulleys had never been treated with clarity in the modern era, in particular, their study was not addressed by Jordanus. Dal Monte, following the suggestion of the Mechanica problemata refers the operation block of pulleys to the lever. I will return to this point in Chapter 7, which shows Descartes criticisms of dal Monte’s demonstration. In the last two chapters, devoted respectively to the wedge and the screw, dal Monte refers to the famous Pappus’ demonstration of the inclined plane, by fully ac27

pp. 34–36. Translation in [298].

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115

cepting it. The reason for this acceptance must be sought in the fact that this demonstration fitted with his reductionist attempt to reduce all the machines to the lever. The proof of Jordanus would have been ignored by dal Monte even had he considered it as correct, but this was not the case, because he assumed a different principle.

5.4.3 The virtual work law Dal Monte complements his static analysis with a kinematic analysis also thereby arriving at a statement of a law of virtual displacements. For example, after studying the statics of the lever he switches to analyze the relationships between force, weight and motion. G F C

B

K

H

A

E

D Fig. 5.11. The shaft with the wheel Let there be the lever AB with its fulcrum C, and let the weight D be attached at the point B, and let the power at A move the weight D by means of a lever AB. Then the space of the power at A is to the space of the weight as CA is to CB. But let there be the lever at AB, whose fulcrum is B, and the moving power is at A and the weight at C; I say that the space of the moved power to the space of the weight carried is as BA to BC [88].28 (A.5.19)

But dal Monte does not give importance to this fact and he is ready to deny it is true and therefore not worthy of being engaged in laws of mechanics. Corollary From these things it is evident that the ratio of the space of the power which moves to the space of the weight moved is greater than that of the weight to the same power. For the space of the power has the same ratio to the space of the weight as that of the weight to the power which sustains the same weight. But the power that sustains is less than the power that moves; therefore the weight will have a lesser ratio to the power that moves it than to the power that sustains it. Therefore the ratio of the space of the power that moves to the space of the weight will be greater than that of the weight to the power [88].29 (A.5.20) 28 29

pp. 76–77. Translation in [298]. pp. 77–78. Translation in [298].

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Dal Monte basically says that although the kinematic analysis indicates that the distances traveled by the weight and power in equilibrium are inversely proportional to them, in practice one must take into account that when the weight is moving it is not in balance and the power must be a bit greater than that necessary for the equilibrium and thus the ratio of the distance traveled by the weight and that of power will be greater than the ratio between strength and weight.

5.5 Giovanni Battista Benedetti Giovanni Battista Benedetti was born in Venice in 1530 and died in Turin in 1590. He received his first and only systematic education in philosophy, music and mathematics from his father. Though never mentioned by Tartaglia he was nevertheless a his pupil for a short time. In 1558 Benedetti became court mathematician for Duke Ottavio Farnese in Parma. In 1567 he was invited by the Duke of Savoy, Emanule Filiberto, to the court in Turin, where until his death he remained an important adviser to the court [294]. In his first book, the Resolutio of 1553 [28] he exposed, in the letter of dedication, the theory of falling bodies, according to which bodies of same density fall with equal speed, independently of the weight. In mechanics his chief work was the Diversarum speculationum of 1585 [29]. The book deals largely with questions of dynamics; there were however fundamental contributions to statics, where a quite modern concept of static moment of a force is referred to. Though the Diversarum speculationum may be considered a commentary of the Mechanica problemata, Benedetti’s approach was essentially Archimedean. He criticises both Tartaglia and Jordanus de Nemore for their kinematic approaches. In the following I will comment shortly upon the law of the lever and the efficacy of a force applied to a arm of a balance.

5.5.1 Effect of the position of a weight on its heaviness In the Diversarum speculationum Benedetti is critical toward the exposition on the matter of both Mechanica problemata and De ratione ponderis. He begins his remarks by stating that the effect of a weight on the end of an arm of a balance depends on the inclination of the arm: The ratio of [the effect of] the weight at C to [the effect of] the same weight at F will be equal to the ratio of the whole arm BC to the part, BU [29].30 (A.5.21)

To justify this fact Benedetti refers not to the greater ease of motion and thus to greater virtual velocity, but rather to a greater or lesser resistance offered by the arm, considered as a constraint for the motion. The same argument used by dal Monte in his Mechanicorum liber. For Benedetti the more the slope, the greater the effect of the constraint:

30

Chap, 2, p. 142. Translation in [298].

5.5 Giovanni Battista Benedetti

A

F

u

C

O

e M

117

B

D

Q

Fig. 5.12. A weight pending from an arm To make this clearer let us imagine a string Fu perpendicular [to the horizontal], with the weight that had been at F now hanging at the extremity u of the string. It will be clear from this that the weight will produce the same effect as if it had been at F. […] And I would make the same assertion if the arm were in position eB. […] For the weight hanging by a string from u is the same [in its effect] as that which had hung freely from a string at point E of arm BE. And this would be due to the fact that in part it hung from the centre B. And if the arm were in position BQ, the whole weight would be suspended from the centre B, just as in position BA it would rest wholly upon that centre [29].31 (A.5.22)

Chapter 2 of the Diversarum speculationum ends with a comment of some interest, which states that if it is true that the heavy bodies tend toward the centre of the world according to straight lines with the direction dependent on their position on the balance, it may be assumed that these directions differ only slightly and then the lines, such as CO and BQ of Fig. 5.12, can be considered as parallel. Now I call side BC horizontal, supposing that it makes a right angle with CO, whence angle CBQ is less than a right angle by the size of an angle equal to that which the two lines CO and BQ make at the centre of the region of the elements.Yet this makes no difference, since that angle is too small to be measured [29].32 (A.5.23)

In Chapter 3 Benedetti changes to make considerations of quantitative character and clearly states that the effect of a force – led by a weight attached to a rope or a muscle – on an arm of a balance, however inclined, is proportional to the distance of the line of action of the force from the fulcrum. First this statement is referred to in a not problematic case of vertical forces: From what we have already shown it may easily be understood that the length of Bu [Fig. 5.12], which is virtually perpendicular from centre B to the line of inclination Fu, is the quantity that enables us to measure the force of F itself in a position of this kind, i.e., a position in which line FU constitutes with arm FB the acute angle BFu [29].33 (A.5.24)

then it is referred to also to forces or weights which act in inclined directions. Benedetti’s argument, even if unequivocal, in the substance does not appear entirely convincing. 31 32 33

Chapter 2, p. 142. Translation in [298]. Chapter 2, p. 143. Translation in [298]. Chapter 2, p. 143. Translation in [298].

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t b

o

i

a o

b

a i

e

t e c c

Fig. 5.13. The ‘static moment’ of weights and forces To understand this better, let us imagine [Fig. 5.13] a balance boa fixed at its centre o, and suppose that at its extremities two weights are attached, or two moving forces, e and c, in such a way that the line of inclination of e, that is be, makes a right angle with ob at point b, but the line of inclination of c, that is ac, makes an acute angle [Fig. 5.13a] or an obtuse angle [Fig. 5.13b] with oa at point a Let us imagine, then, a line ot perpendicular to the line of inclination ca […]. Imagine, then, that oa is cut at point i, so that oi is equal to ot, and that a weight is suspended at i, equal to c and with a line of inclination parallel to that of weight e. But we assume that the weight or force c is greater than e in proportion as bo is greater than ot. Obviously, then, according to Archimedes, De ponderibus, boi will not move from its position. Again, if in place of oi we imagine ot rigidly connected [in the same line] with ob and subjected to force c acting along line tc, the result will obviously be the same – bot will not move from its position [29].34 (A.5.25)

Curious, to say the least, is the way in which Benedetti explains in Chapter 4 of the Diversarum speculationum the reason of the different effectiveness of two forces which are at different distances from the fulcrum of a lever. Somehow repeating the reasoning above, and at the same time denying it. In essence he argues that one must consider the lever as a solid body and not as a rod, as shown in Fig. 5.14. The force applied in n shall weigh more on the fulcrum compared to the force applied in u, because the line ni is closer to the vertical than ui, it will therefore be less effective. e

o

n

x

i

t

u

s

Fig. 5.14. A three-dimensional lever

5.5.2 Errors of Tartaglia and Jordanus Benedetti in Chapter 7 of the Diversarum speculationum exposes those he believed were errors made by Tartaglia (and Jordanus de Nemore). Among his criticisms one 34

Chapter 3, p. 143. Translation in [298].

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of the most interesting is relative to the analysis of the balance with equal weights and arms. For Tartaglia the balance has the horizon as position of stable equilibrium. For Benedetti instead, the equilibrium was unstable. First he ‘proves’ that Tartaglia’s argumentation is fallacious given his premises, i.e. the parallelism of gravity actions. In such a case the gravity of position for the weights a and b of Fig. 5.1535 should be the same: k m

e

i b

x l

y

o

n

z c

a

b

w

q

p

j

s a v

d f

r t

Fig. 5.15. Tartaglia’s fallacious reasoning And in the second part of the fifth proposition he [Tartaglia] fails to see that no difference in weight is produced by virtue of position in the way in which he argues. For if body b must descend on arc il, body a must ascend on arc vs, equal and similar to arc il and placed in the same way. Therefore, just as it is easy for body a to ascend on arc vs it is easy for body b to descend on arc vs. And this fifth proposition is the second proposed by Jordanus [29].36 (A.5.26)

Benedetti believes that the reasoning of Tartaglia (and Jordanus) is not consistent because he regards the two weights a and b as if they were independent and both could move downward. They are in fact constrained by the beam ab and when one of them falls the other rises. So it should be compared the paths il and vs, which have the same vertical projections xy and d j respectively. But the reasoning of Tartaglia, according to Benedetti, is wrong in the merit too, because the lines of action of gravity are not parallel but converging toward the centre of the world u. With reference to Fig. 5.16, consider with Benedetti the balance aob displaced from the horizontal position. Assuming the two weights a and b have the same absolute gravity, their efficacy depends on the distances of their line of action, respectively ao and bo from the fulcrum o. For the weight a the distance is represented by ot for the weight b by oe, and it is simple to prove that ot > oe. Then the gravity of weight a is greater than that of weight b, and the balance is going to tilt up the vertical position, i.e. the horizon is an unstable equilibrium position. Benedetti closes the chapter criticizing the (correct) solution given by Tartaglia on the inclined plane, saying it is worthless without specifying the reason. 35 36

The figure in not in Benedetti’s text, it is derived from [223], p. 89r. Chapter 7, p. 148. Translation in [298].

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n

b

s o e t a

c

u Fig. 5.16. Unstable equilibrium for the balance with equal arms and weights

5.6 Galileo Galilei Galileo Galilei was born in Pisa in 1564 and died in Florence (Arcetri) in 1642. In Pisa he undertook the study of mathematics under the guidance of Ostilio Ricci, a pupil of Nicolò Tartaglia. Of this period are the first contact with Christopher Clavius (1538–1612) and Guidobaldo dal Monte. In 1589, Galileo obtained a professorship of mathematics at Pisa. In 1592 he moved to Padua as a professor of mathematics. In 1610 he published the Sidereus nuncius [114], a work that made him famous around the world; still in 1610 he was named Philosopher and Mathematician of the Grand Duke of Tuscany. In 1616 the Holy Office condemned the Copernican theory and Galileo was warned (is it true?) not to defend it. In 1632 he published his Dialogo sopra i due massimi sistemi [116] on Ptolemaic and Copernican systems. In June 1633, in the guise of penance and kneeling, in front of catholic cardinals, Galileo was forced to pronounce the solemn recantation and admission of guilt [294]. In 1638 he published, still incomplete, the Discorsi e dimostrazioni matematiche sopra due nuove scienze [118, 347, 297, 296]. The contribution that Galileo provided to statics is far less decisive than that to dynamics, nonetheless it is important. Though there may be doubts on the originality of some of his writings, it is certain that no one before him had formulated and solved his own problems with extraordinary clarity. The main works of Galileo, which specifically concern the equilibrium are: Le mecaniche (1593–1594 early manuscripts, first printed in a French version by Mersenne in 1634 and in Italian, in 1649, after the death of Galileo [119], the Discorso intorno alle cose che stanno in su l’acqua e scritture varie, printed in 1612 [115] and the already cited Discorsi e dimostrazioni matematiche sopra due nuove scienze [118].

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5.6.1 The concept of moment. A law of virtual velocities In Le mecaniche Galileo introduces a concept and a term, that of moment (“momento”), that will be of great fortune and adopted, at least in Italy, until the early XIX century: Moment is the propension of descending, caused not so much by the Gravity of the moveable, as by the disposure which divers Grave Bodies have in relation to one another; by means of which Moment, we oft see a Body less Grave counterpoise another of greater Gravity: as in the Steelyard, a great Weight is raised by a very small counterpoise, not through excess of Gravity, but through the remotenesse from the point whereby the Beam is up held, which conjoyned to the Gravity of the lesser weight adds thereunto Moment, and Impetus of descending, wherewith the Moment of the other greater Gravity may be exceeded. MOMENT then is that IMPETUS of descending, compounded of Gravity, Position, and the like, whereby that propension may be occasioned [119].37 (A.5.27)

The concept is taken up and elaborated in the Discorso intorno alle cose che stanno in su l’acqua: Moment for mechanics, means that virtue, that force, that effectiveness with which the motor moves and the mobile resists [emphasis added], a virtue which depends not only on the simple gravity, but on the speed of motion, from the different angles of the spaces over which the motion is made, because a heavy body makes more impetus in a very inclined space than in one less inclined. The second principle [the first was that equal weights with equal speed have equal forces and moments] is, that the moment and the force of gravity is increased by the speed of motion so that absolutely equal weights, but combined with unequal velocities, are of force, moment and virtue unequal, and the fastest is more powerful, according to the proportion of its speed to the speed of the other. Of this we have a very suitable example in the balance with unequal arms, where absolutely equal weights do not press and are not equally strong, but that which is at the greatest distance from the center, around which the balance moves, sinks and raises the other, and it is the motion of ascending fast, the other slow: and such is the force and virtue that the speed of motion gives to the mobile that receives, and it can compensate as much weight as added to the other mobile; so that if one arm of a balance were ten times longer than the other, in order to move the balance around its middle, the end of that passed ten times more space than the end of this, a weight placed at the greater distance can sustain and equilibrate another ten times heavier than itself, and this because, moving the balance, the lower weight will move ten times faster than the other [115].38 (A.5.28)

From the reading of passages quoted above it is clear as Galileo espouses the view that the downward velocity of a heavy body increases its efficacy or ‘force’ to go down while the upward velocity increases its resistance to be lifted. His conception is rather uncommon in statics and differs from dal Monte and Benedetti’s who instead believed that there was no increase of ‘force’ due to velocity, but only a greater velocity due to lower resistance of constraints. It also differs from Jordanus’s concept of gravity of position, measured by the rate of possible descent, i.e. a purely geometric motion therefore not increasing the ‘force’ or the resistance of weights. Galileo tried without success to provide a measure of the static force equivalent to the increase of the effectiveness of a weight with his speed in the last section 37 38

p. 159. Translation in [121]. pp. 68–69.

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of Le mecaniche and during the sixth day added to the Discorsi e dimostrazioni matematiche sopra due nuove scienze in the Florentine edition of 1715. Only in the XIX century, when the difference between force and energy was fully clarified, was it recognized that the force due to the speed (kinetic energy) is incommensurable with the static force. Paolo Galluzzi [321]39 attaches great importance to the use of the term moment and identifies in the Le mecaniche the place where Galileo first introduced the definition in a technical sense. He preferred it to the more generic medieval term gravitas, used in the De motu [113] a few years before, which gave rise to ambiguity because sometimes it pointed to the sheer weight. In the Le mecaniche, after having proved the law of the lever according to Archimedes and similarly to what he will do in the first day of the Discorsi (with a reasoning similar to that of Stevin, that probably he did not know) Galileo examines the equilibrium of the lever using the concept of moment: D A

C

B

E Fig. 5.17. The lever Now being that Weights unequall come to acquire equall Moment, by being alternately suspended at Distances that have the same proportion with them; I think it not fit to over passe with silence another congruicy and probability, which may confirm the same truth; for let the Ballance AB, be considered, as it is divided into unequal parts in the point C, and let the Weights be of the same proportion that is between the Distances BC, and CA, alternately suspended by the points A, and B: It is already manifest, that the one will counterpoise the other, and consequently, that were there added to one of them a very small Moment of Gravity, it would preponderate, raising the other, so that an insensible Weight put to the Grave B, the Ballance would move and descend from the point B towards E, and the other extream A would ascend into D, and in regard that to weigh down B, every small Gravity is sufficient, therefore not keeping any accompt of this insensible Moment, we will put no difference between one Weights sustaining, and one Weights moving another [emphasis added]. Now, let us consider the Motion which the Weight B makes, descending into E, and that which the other A makes in ascending into D, we shall without doubt find the Space BE to be so much greater the Space AD, as the Distance BC is greater than CA, forming in the Center C two angles DCA, and ECB, equall as being at the Cock, and consequently two Circumferences AD and BE alike; and to have the same proportion to one another, as have the Semidiameters BC, and CA, by which they are described: so that then the Velocity of the Motion of the descending Grave B cometh to be so much Superiour to the Velocity of the other ascending Moveable A, as the Gravity of this exceeds the Gravity of that; and it not being possible that the Weight A should be raised to D, although slowly, unless the other Weight B do move to E swiftly, it will not be strange, or inconsistent with the Order of 39

p. 199–221.

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Nature, that the Velocity of the Motion of the Grave B, do compensate the greater Resistance of the Weight A, so long as it moveth slowly to D, and the other descendeth swiftly to E, and so on the contrary, the Weight A being placed in the point D, and the other B in the point E, it will not be unreasonable that that falling leasurely to A, should be able to raise the other hastily to B, recovering by its Gravity what it had lost by it’s Tardity of Motion. And by this Discourse we may come to know how the Velocity of the Motion is able to increase Moment in the Moveable, according to that same proportion by which the said Velocity of the Motion is augmented [emphasis added] [119].40 (A.5.29)

Galileo’s reasoning is not very clear. He starts with a balance with two weights A and B inversely proportional to the arms that are assumed to be equilibrated. Then he imagines a motion led by a small weight added on one side, which because of its smallness does not alter the ratio of the weights of the two bodies. As a result of the motion the heavy bodies A and B acquire velocities proportional to the distances from the fulcrum, then velocities and weights are in inverse relationship, then the moments are the same and A and B are in equilibrium. But the reasoning is circular because the equilibrium is proved after it was assumed. Galileo’s reasoning would not be circular, and perhaps this could be his intention, if he had not specified at the beginning that the equilibrated weights were in inverse relationship to distance, as follows: consider a balance in equilibrium with the two weights A and B and upset the balance by adding a small weight on the side of B. In the following motion there will be determined two moments that will compensate only if the weights are in inverse proportion to distances. The principle Galileo invokes for the equilibrium is the equality of moments, i.e. a law of virtual work, expressed by means of velocity. He states that this principle can be deduced from the Mechanica problemata of Aristotle: This equality between gravity and speed is found in all the mechanical instruments, and was considered by Aristotle in his Mechanical questions; so we can still take for granted that absolutely unequal weights alternatively counterweight and make themselves of the same moment, once their gravities have contrary proportion with the speed of their motions [115].41 (A.5.30)

By considerations similar to those developed for the lever with straight arms, Galileo demonstrated that for the angular lever the magnitude that defines the equilibrium is the distance from the fulcrum of the vertical line through the weight. There is also another thing, before we proceed any farther, to be considered; and this is touching the Distances, whereat, or wherein Weights do hang: for it much imports how we are to understand Distances equall, and unequall; and, in sum, in what manner they ought to be measured. […] But if elevating the Line CB, moving it about the point C, it shall be transferred into CD, so that the Ballance stand according to the two Lines A C, and CD, the two equall Weights hanging at the Terms A and D, shall no longer weigh equally on that point C, because the distance of the Weight placed in D, is made lesse then it was when it hanged in B. For if we confider the Lines, along [or by] which the said Graves make their Impulse, and would descend, in case they were freely moved, there is no doubt but that they would make or describe the Lines AG, DF, BH: Therefore the Weight hanging on the point D, maketh it’s Moment and Impetus according to the Line D F: but when it hanged in B, it 40 41

pp. 163–164. Translation in [121]. p. 275.

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made Impetus in the Line BH: and because the Line DF is nearer to the Fulciment C, then is the Line BH Therefore we are to understand that the Weights hanging on the points A and D, are not equidistant from the point C, as they be when they are constituted according to their Right Line ACB [119].42 (A.5.31)

D

A

B

C F

H Fig. 5.18. The angled lever

In the passages above, Galileo leaves a certain ambiguity in the way the equality and moments should be understood. A modern reader based on the notion of static moment, and maybe after having read Jordanus, would be tempted to see a balance of two trends to go down. But some doubt remains because Galileo speaks of raising of a body and lowering of the other. In his Discorso intorno alle cose che stanno in su l’acqua Galileo specifies, or perhaps decides, that moment is also the resistance to gain speed up. So the equilibrium is not from the equality of two trends to go down, but from the balance of the impetus to go down and the resistance to go up, increased both by the speed (for comments on this shifting of meaning see [321]). Galileo treats his law of virtual work – the equality of moments – as a principle to study the equilibrium of fluids. For example, to justify the height of the fluid on both sides of a siphon of very different sections is the same. In reality, Galileo does not prove the equality, he limits himself only to substantiating the plausibility of the thing. The equality will be demonstrated by Pascal with the use of Torricelli’s principle (see Chapters 6 and 8). The justification, making reference to Fig. 5.19, assumes, by absurdity, that the fluid of the largest side of the siphon drops of the amount HO and at the same time the fluid of the small end of the siphon will rise of the height LA greater than HO – so that the volume ABL is equal to the volume QOH – and therefore much faster. It is plausible, therefore, that the motion of the water of the largest side (a great weight) be offset by the large velocity of the water on the smaller side (a small weight) and it is therefore justified that the water on both sides has the same level. And for very large confirmation, and clearer explanation of this, consider this figure (and, unless I am mistaken, it could be used to tear out fault some practical mechanicians, who on a false basis sometimes try impossible enterprises), in which the wide vessel EIDF, is continued by the thin barrel ICAB, and fill them with water up to LGH, which in this state is at rest, but not without some wonder, for who will not understand as soon as it is seen that the heavy burden of the large amount of water GD, pressing down does not raise and drive

42

pp. 164–165. Translation in [121].

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Fig. 5.19. The siphon (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome) away the small amount of the other contained within the barrel CL, by which the descent is disputed and prevented? But this will no longer be a wonder, if we begin to pretend to have lowered the water GD only up to QO, and then consider what water CL has made, which, to give place to the other that has diminished from the level GH to the level QO, must have in the same time been lifted up from the level L to AB, and the ascent LB is as much greater than the descent GQ, as much the amplitude GD of the vessel is greater than the width LC of the barrel, which in sum is what the water GD is more than the LC. But since the moment of the speed of motion in a mobile compensates for the gravity of another, what wonder if it will be the fastest ascent of a little water CL to resist the very slow greater quantity of water GD? [115]43 . (A.5.32)

It is worthwhile even to notice an exchange between Salviati and Sagredo, taken from Dialogo sopra i due massimi sistemi del mondo where the idea is emphasized that the velocity increases the effectiveness of the weights: SAGR. But do you think the speed compensates precisely the gravity? That is, that both the moment and the force of a mobile of four pounds of weight are as that of a hundred, when this had one hundred degrees of speed and that only four degrees? SALV. Of course yes, as I could show with many experiences, but for now it suffices to confirm this one of the balance, where you will see the little heavy roman be able to support and compensate the very heavy bale, when its distance from the centre over which the balance is sustained and rotates will be greater than the other lesser distance from which the bale hangs, as the absolute weight of the bale is greater than that of the roman. And the cause for which the bale cannot lift the roman, much less heavy, cannot be other but the difference of the movements of this and that. When the bale with the lowering of one finger did lift the roman of one hundred fingers [116].44 (A.5.33)

In the Discorsi e dimostrazioni matematiche Galileo adds other meanings to the term moment [321]. He does it for example on the second day where the strength of 43 44

pp. 77–78. p. 241.

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materials is considered. Here Galileo uses the concept of moment for the equilibrium of straight and angular levers with a language very close to that of modern textbooks of statics and the reader is tempted to assume the Galilean moment as the static moment (i.e. the product of force by its arm). In the evaluation of the resistance of a cantilever, Galileo introduces the moment of the strength to breaking, i.e. the moment of a ‘force’ with an ontological status similar to that of reactive forces and for which probably the concept of moment as propension to motion could hardly be applied. For details see [272]. Particularly interesting is the observation Galileo makes the fourth day, the efficacy of a very small weight h in lifting a very large one as illustrated in Fig. 5.20. A simple kinematic analysis shows that the relationship between the lowering e f of the weight h to the raising f i and f l respectively of the weights c and d is as great as you like. This implies that whatever the weight h, its velocity will compensate the weights c and d, however big they are, raising them. According to Duhem [305]45 the above considerations were suggested to Galileo by the reading of the Traité de mechanique of 1636 by Roberval, in particular by the demonstration of the rule of the parallelogram by means of a law of virtual work (see Chapter 7). This is possible, as the fourth day of Discorsi was added after the first edition in 1638 and Galileo before his death would certainly have got to know the ideas of Roberval.

Fig. 5.20. Large weights raised by a very small weight (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome) SAGR. You are quite right; you do not hesitate to admit that however small the force of the moving body be, it will overcome any resistance, however great, provided it gains more in velocity than it loses in force and weight. Now let us return to the case of the cord. In the accompanying figure ab represents a line passing through two fixed points a and b; at the extremities of this line, as you see, two large weights e and d hang, which stretch it with great force and keep it truly straight, being it merely a line without weight. Now I wish to remark that if from the middle point of this line, which we may call e, one suspends any small weight, say h, the line ab will yield toward the point f and on account of its elongation it will compel the two heavy weights c and d to rise. This I shall demonstrate as follows: with the points a and b as centres describe the two quadrants, eig and elm; now since the two semidiameters ai and bl are equal to ae and eb, the remainders f i and f l are the excesses of the lines a f and f b over ae and eb; they therefore determine the rise of the weights c and d, assuming of course that the weight h has taken the position f , which could happen whenever the line e f , which represents the descent of h had greater proportion than the line f i – associated to the rise of the weights c and d – of the heaviness of both the two weights to 45

vol. 1, p. 324.

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the heaviness of the weight h. But this will necessarily occur however large be the heaviness of weights c and d and little that of the weight h. Even when the weights of c and d are very great and that of h very small this will happen [118].46 (A.5.34)

5.6.2 A law of virtual displacements Galileo in the early stages of his studies considered the law of virtual work as a principle, that of virtual velocities. However, he did not disdain to also consider virtual displacements, coming to present the law of virtual work, as a theorem, proved for all simple machines. At the beginning of Le mecaniche Galileo seems to assign the law of virtual work based on displacements a fundamental role, given its immediate evidence, although based on every day experience. This passage anticipates in a form strikingly similar the reasoning of Descartes in his letters to Constantin Huygens and Mersenne in 1637–38: Of which mistakes I think I have found the principal cause to be the belief and constant opinion these Artificers had, and still have, that they are able with a small force to move and raise great weights; (in a certain manner with their Machines cozening nature, whose Instinct, yea most positive constitution it is, that no Resistance can be overcome, but by a Force more potent then it:) which conjecture how false it is, I hope by the ensuing true and necessary Demonstrations to evince [119].47 […] Now, any determinate Resistance and limited Force whatsoever being assigned, and any Distance given, there is no doubt to be made, but that the given Force may carry the given Weight to the determinate Distance; for, although the Force were extream small, yet, by dividing the Weight into many small parts, none of which remain superiour to the Force, and by transferring them one by one, it shall at last have carried the whole Weight to the assigned Term: and yet one cannot at the end of the Work with Reason say, that that great Weight hath been moved, and trans ported by a Force lesse then it self, howbeit indeed it was done by a Force, that many times reiterated that Motion, and that Space, which shall have been measured but only once by the whole Weight. From whence it appears, that the Velocity of the Force hath been as many times Superiour to the Resistance of the weight, as the said Weight was superiour to the Force; for that in the same Time that the moving Force hath many times measured the intervall between the Terms of the Motion, the said Moveable happens to have past it onely once: nor therefore ought we to affirm a great Resistance to have been overcome by a small Force, contrary to the constitution of Nature. Then onely may we say the Natural Constitution is overcome, when the lesser Force transfers the greater Resistance, with a Velocity of Motion like to that wherewith it self doth move; which we affirm absolutely to be impossible to be done with any Machine imaginable. But because it may sometimes come to passe, that having but little Force, it is required to move a great Weight all at once, without dividing it in pieces, on this occasion it will be necessary to have recourse to the Machine, by means whereof the proposed Weight may be transferred to the assigned Space by the Force given [119].48 […] And this ought to passe for one of the benefits taken from the Mechanicks: for indeed it frequently happens, that be ing scanted in Force but not Time, we are put upon moving 46 47 48

pp. 311–312. p. 155. Translation in [121]. pp. 156–157. Translation in [121].

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great Weights unitedly or in grosse: but he that should hope, and at tempt to do the same by the help of Machines without increase of Tardity in the Moveable, would certainly be deceived, and would declare his ignorance of the use of Mechanick Instruments, and the reason of their effects [119].49 (A.5.35)

The proof of the law of virtual work based on the displacements is exhibited for all simple machines after their operation has been explained by the law of the lever. For example, for the lever Galileo writes: D

G

C

B

L

M I

Fig. 5.21. Virtual displacement law for the lever And here it is to be noted, which I shall also in its place remember you of, that the benefit drawn from all Mechanical Instruments, is not that which the vulgar Mechanicians do persuade us, to wit, such, that there by Nature is overcome, and in a certain manner deluded, a small Force over-powring a very great Resistance with help of the Leaver; for we shall demonstrate, that without the help of the length of the Leaver, the same Force, in the same Time, shall work the same effect. For taking the same Leaver B C D, whose rest or Fulciment is in C, let the Distance C D be supposed, for example, to be in quintuple proportion to the Distance C B, & the said Leaver to be moved till it come to I C G: In the Time that the Force shall have passed the Space D I, the Weight shall have been moved from B to G: and because the Distance D C, was supposed quintuple to the other C B, it is manifest from the things demonstrated, that the Weight placed in B may be five times greater then the moving Force supposed to be in D: but now, if on the contrary, we take notice of the Way passed by the Force from D unto I, whilst the Weight is moved from B unto G, we shall find likewise the Way D I, to be quintuple to the Space B G. Moreover if we take the Distance C L, equal to the Distance C B, and place the same Force that was in D, in the point L, and in the point B the fifth part onely of the Weight that was put there at first, there is no question, but that the Force in L being now equal to this Weight in B, and the Distances L C and C B being equall, the said Force shall be able, being moved along the Space LM to transfer the Weight equall to it self, thorow the other equall Space B G: which five times reiterating this same action, shall transport all the parts of the said Weight to the same Term G: But the repeating of the Space L M, is certainly nothing more nor lesse then the onely once measuring the Space D I, quintuple to the said L M. Therefore the transferring of the Weight from B to G, requireth no lesse Force, nor lesse Time, nor a shorter Way if it wee placed in D, than it would need if the same were applied in L: And, in short, the benefit that is derived from the length of the Leaver C D, is no other, save the enabling us to move that Body all at once, which would not have been moved by the same Force, in the same Time, with an equall Motion, save onely in pieces, without the help of the Leaver [emphasis added] [119].50 (A.5.36)

In the previous quotation Galileo highlights two things: on the one hand that while a load p/n covers a path h, the load p will cover the path h/n. On the other hand 49 50

p. 157. Translation in [121]. pp. 166–167. Translation in [121].

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100 1 100

2

200

1 a)

1 b)

100

100

100

Fig. 5.22. Separate raising of a weight of 200 pounds to different heights

that the load p would be brought up to h/n by carrying n times a load p/n at h/n, as shown in Fig. 5.22, where it is clear that lifting a weight of 100 pounds to a height of two foot (Fig. 5.22a) is equivalent to lifting 200 pounds to a height of one feet (Fig. 5.22b). This operation can be done without the help of machines, from which the conclusion that the machines make the work of man more comfortable but do not allow any savings in work, or fatigue. In the case of the inclined plane it would seem that the law of virtual displacements is not true, because a weight E is shifted by a lower weight F of the same amount. This is true, but, says Galileo, the largest displacement of F should not be measured along the plane but on the vertical because “heavy bodies make no resistance to translational motions”. If the distance traveled on the vertical is measured, it can be seen that weights are in inverse ratio of changes in altitude. D C E F A

B

Fig. 5.23. Virtual displacement law for the inclined plane Lastly, we are not to pass over that Consideration with silence which at the beginning hath been said to be necessary for us to have in all Mechanick Instruments, to wit, That what is gained in Force by their assistance, is lost again in Time, and in the Velocity: which peradventure, might not have seemed to some so true and manifest in the present Contemplation; nay, rather it seems, that in this case the Force is multiplied without the Movers moving a longer way than the Moveable: In regard, that if we shall in the Triangle ABC suppose the Line AB to be the Plane of the Horizon, AC the elevated Plane, whose Altitude is measured by the Perpendicular CB, a Moveable placed upon the Plane AC, and the Cord EDF tyed to it, and a Force or Weight applyed in F that hath to the Gravity of the Weight E the same proportion that the Line BC hath to CA; by what hath been demonstrated, the Weight F shall descend downwards, drawing the Moveable E along the elevated Plane; nor shall the Move able E measure a greater Space when it shall have passed the whole Line AC, than that which the said [119].51 (A.5.37) 51

p. 185. Translation in [121].

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But here yet it must be advertised, that al though the Moveable E shall have passed the whole Line AC, in the same Time that the other Grave F shall have been abased the like Space, nevertheless the Grave E shall not have retired from the common Center of things Grave more than the Space of the Perpendicular CB. but yet the Grave F descending Perpendicularly shall be abased a Space equal to the whole Line AC. And because Grave Bodies make no Resistance to Transversal Motions, but only so far as they happen to recede from the Center of the Earth; There fore the Moveable E in all the Motion AC being raised no more than the length of the Line CB, but the other F being abased perpendicularly the quantity of all the Line A C: Therefore we may deservedly affirm that Way of the Force E maintaineth the same proportion to the Force F that the Line AC hath to CB; that is, the Weight E to the Weight F. It very much importeth, therefore, to consider by [or along] what Lines the Motions are made, especially in examine Grave Bodies, the Moments of which have their total Vigour, and entire Resistance in the Line Perpendicular to the Horizon; and in the others transversally Elevated and Inclined they feel the more or less Vigour, Impetus, or Resistance, the more or less those Inclinations approach unto the Perpendicular Inclination. [119].52 (A.5.38)

Galileo verifies the law of virtual displacements even for a block of pulleys. Referring to Fig. 5.24, after having demonstrated using the law of the lever that the relationship between the weight in H and force in I is 3 : 1, he concludes by noting that the shift of I is 3 times that of H, and the law of virtual displacement is verified.

E

F I D G

A

B

C H Fig. 5.24. Virtual displacement law for the pulley Which being demonstrated, we will pass forwards to the Pulleys, and will describe the inferiour Gyrils of ACB, voluble about the Center G, and the Weight H hanging thereat, we will draw the other up per one E F, winding about them both the Rope DFEACBI, of which let the end D be fastned to the inferiour Pulley, and to the other I let the Force be applyed: Which, I say, sustaining or moving the Weight H, shall feele no more than the third part of the Gravity of the same. For considering the contrivance of this Machine, we shall find that the Diameter A B supplieth the place of a Leaver, in whose term B the Force I is applied, and in the other A the Fulciment is placed, at the middle G the Grave H is hanged, and another Force D applied at the same place: so that the Weight is fastned to the three Ropes IB, FD, and EA, which with equal Labour sustain the Weight. Now, by what hath already been

52

pp. 185–186. Translation in [121].

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contemplated, the two Forces D and B being applied, one, to the midst of the Leaver A B, and the other to the extream term B, it is manifest, that each of them holdeth no more but the third part of the Weight H: Therefore the Power I, having a Moment equal to the third part of the Weight H, shall be able to sustain and move it: but yet the Way of the Force in I shall be triple to the Way that the Weight shall pass; the said Force being to distend it self according to the Length of the three Ropes I B, F D, and E A, of which one alone measureth the Way of the Weight H. [119].53 (A.5.39)

5.6.3 Proof of the law of the inclined plane The problem of the inclined plane did not appear, neither in Aristotle’s Mechanica problemata nor in Archimedes’ Aequiponderanti; the first known writing on the subject was reported on Hero’s Mechanica where was proposed a clever solution but only roughly approximated to the ‘real’ one. After Hero, Pappus of Alexandria (see Chapter 3), Leonardo da Vinci, Girolamo Cardano (with Nicola Antonio Stigliola in the wake of Hero) and some others did indeed formulate their own solutions different from each other but different also from the ‘correct’ one. At the end of the XVI century, the people who have stated ‘correctly’ this law could be counted on the fingers of one hand: Jordanus de Nemore and later Nicolò Tartaglia (see Chapter 4), Michel Varro (see Chapter 7), Simon Stevin, and Galileo Galilei. It is nearly impossible to evaluate the influence of the various scholars on each other. While there are no doubts that Tartaglia and Varro were inspired by Jordanus, Festa and Roux say instead that there are no external clues to affirm that Galileo knew his (or Tartaglia’s) writings [311], and also proofs of the contact between Stevin and Galileo are lacking.54 It must be noted however that it is very difficult to find a precise precursor to Galileo. For this there are also psychological reasons that a historian dissuade the possibility of crititizing a great scientist. Today the inclined plane is seen as a conceptual model different from that represented by the lever and essentially not reducible to it. The inclined plane is representative of virtual displacement laws and is somehow its geometric representation; the lever is representative of the virtual velocity laws. In the past, as it should appear clear from the short historical notes listed above, things however were not seen in this way. That the inclined plane had its peculiarities was understood by Aristotle who did not treat it and by Hero who treated it apart from the other machines. However after Pappus of Alexandria had reduced it to the lever, the difficulties in the study of the inclined plane seemed to vanish. In the Renaissance the problem reappeared because some scholars did not accept Pappus’ solution, because they considered both logically unconvincing and inadequate empirically. For example it featured an infinite value of the force required to lift a weight on a vertical plane, and this is patently absurd. Other scholars did not accept it because in contrast with Jordanus de Nemore’s solution, whose demonstration seemed more consistent, the principles adopted could appear not very obvious. 53 54

p. 172. Translation in [121]. pp. 202–203

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With Galileo the reductionist project, started with Pappus and strongly supported by Guidobaldo dal Monte, to reduce all simple machines including the inclined plane to the lever, was perfected. Note that Galileo’s attempt to reduce the inclined plane to the lever was accepted not because verified empirically – with the conceptions of experiment of the times also the results of Hero or Cardano were verified – but because he finally presented a rigorous reasoning and employed reasonable assumptions. Moreover Galileo’s result coincided with that of Jordanus and with that of Stevin more or less of the same period, very elegant and based on different assumptions as referred to in Chapter 7. Note again that if the reasoning of Galileo was corroborated by the result of Jordanus and Stevin, the reasoning of Jordanus and Stevin was corroborated by that of Galileo and from now on the problem of the inclined plane was considered by all the mathematicians to be definitively solved. In the section devoted to the mechanics of the screw, Galileo shows how the inclined plane can be reduced to the lever and furnishes a simple mathematical law. The proof reproduces the one that he had reported in De motu [113],55 differing mainly for the use of the word moment instead of gravitas. The present Speculation hath been attempted by Pappus Alexandrinus in Lib. 8. de Collection. Mathemat. but, if I be in the right, he hath not hit the mark, and was overseen in the Assumption that he maketh […]. Let us therefore suppose the Circle AIC, and in it the Diameter ABC, and the Center B, and two Weights of equal Moment in the extreams B and C; so that the Line AC being a Leaver, or Ballance moveable about the Center B, the Weight C shall come to be sustained by the Weight A. But if we shall imagine the Arm of the Ballance BC to be inclined downwards according to the Line B F, but yet in such a manner that the two Lines AB and BF do continue solidly conjoyned in the point B, in this case the Moment of the Weight C shall not be equal to the Moment of the Weight A, for that the Distance of the point F from the Line of Direction, which goeth accord ing to BI, from the Fulciment B unto the Center of the Earth, is diminished: But if from the point F we erect a Perpendicular unto BC, as is FK, the Moment of the Weight in F shall be as if it did hang by the Line KF [119].56 (A.5.40)

D

B M

A

K CH

O

F I N Fig. 5.25. The law of inclined plane 55 56

pp. 297–298. p. 181. Translation in [121].

L G

E

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See therefore that the Weight placed in the extream of the Leaver B C, in inclining downwards along the Circumference CFLI, cometh to diminish its Moment and Impetus of going downwards from time to time, more and less, as it is more or less sustained by the Lines BF and BL. […] If therefore upon the Plane HG the Moment of the Moveable be diminished by the total Impetus which it hath in its Perpendicular DCE, according to the proportion of the Line K B to the Line BC, and BF, being by the Solicitude of the Triangles KBF and KFH the same proportion betwixt the Lines KF and FH, as betwixt the said KB and BF, we will conclude that the proportion of the entire and absolute Moment, that the Moveable hath in the Perpendicular to the Horizon to that which it hath upon the Inclined Plane HF, hath the same proportion that the Line HF hath to the Line FK; that is, that the Length of the Inclined Plane hath to the Perpendicular which shall fall from it unto the Horizon. So that passing to a more distinct Figure, such as this here present, the Moment of Descending which the Moveable hath upon the inclined Plane CA hath to its total Moment wherewith it gravitates in the Perpendicular to the Horizon CP the same proportion that the said Line PC hath to CA. And if thus it be, it is manifest, that like as the Force that sustaineth the Weight in the Perpendiculation PC ought to be equal to the same, so for sustaining it in the inclined Plane CA, it will suffice that it be so much lesser, by how much the said Perpendicular CP wanteth of the Line CA: and because, as sometimes we sce, it sufficeth, that the Force for moving of the Weight do insensibly superate that which sustaineth it, therefore we will infer this universal Proposition, [That upon an Elevated Plane the Force hath to the Weight the same proportion as the Perpendicular let fall from the Plane unto the Horizon hath to the Length of the said Plane] [119].57 (A.5.41)

The key assumptions to demonstrate the law of the inclined plane are: a) for static purposes, moving on the inclined planes like NO or GH is the same as moving on the circumference described by the lever arms BL or BF of Fig. 5.25; b) the effectiveness of a heavy body on an angled lever is determined by the horizontal distance from the fulcrum. In essence, a weight p hanging in F from the angular lever ABF is balanced by a weight q = p BK/BA in A. But the q in A is also balanced by a force in F equal to q orthogonal to BF and then parallel to GH, because BA = BF. Therefore q is the force parallel to the plane necessary to support the weight. The second assumption is an accepted theorem of statics, but the first has a logic status not completely clear. It indeed appears quite intuitive, at least after its formulation, because to study the equilibrium it seems sufficient to verify that also very small displacements cannot occur. In this way the displacements at the extremity of the lever and on the inclined plane are the same, the two kinds of constraints are locally equivalent and can be replaced the one with the other. But this intuitive character stems more from empirical than logical considerations; it would be then a postulate which could even not be accepted, as will be shown further in Chapter 13.

57

pp. 182–183. Translation in [121].

6 Torricelli’s principle

Abstract. This chapter is devoted entirely to Evangelista Torricelli who formulated a principle of equilibrium without referring explicitly to dynamic aspects. In the first part some elements of centrobarica are introduced. In the central part Torricelli’s principle is introduced: The centre of gravity of an aggregate of heavy bodies cannot lift by itself. Following this law, which in a first reading does not seem to be a VWL, Torricelli and his successors derived the VWL based on virtual displacements for which any force that can lift a weight p to a height h can raise p/n of nh. In the final part generalization and simplification of Torricelli’s principle are presented. In the common interpretation, Torricelli’s principle is a criterion of statics which claims that it is impossible for the centre of gravity of a system of bodies in equilibrium to sink from any virtual movement of the bodies. This criterion had a vital role in the history of mechanics. It represents a generalisation of the ancient principle that a single body is in equilibrium if its centre of gravity cannot sink. The generalisation devised by Torricelli states that if the centre of gravity of an aggregate of rigid bodies, considering the aggregate as a whole, is evaluated according to Archimedean rules, then this point has effectively the physical meaning of a centre of gravity.1

6.1 The centrobaric The idea of the centre of gravity in antiquity can be found both in the writings of philosophers and mathematicians with a different value in the two cases. The first concentrate on a dynamic conception that can be traced back at least to Aristotle who, in De caelo, addressed issues relating to the shape of the earth. Here he specified the basic principle of his mechanics, that bodies move with natural motion toward the centre of the world, coincident, only accidentally, with the centre of the earth. According to Aristotle, it is clear that it is not enough that only the extreme part of 1

The first part of this chapter is taken from [275] which is summarized and largely revised.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_6, © Springer-Verlag Italia 2012

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a body reaches the centre, but the heavier parts must outweigh to the rest, to include with its centre the centre of the world: A short consideration will give us an easy answer, if we first give precision to our postulate that any body endowed with weight, of whatever size, moves towards the centre. Clearly it will not stop when its edge touches the centre. The greater quantity must prevail until the centre of the body occupies the centre [of the world]. For that is the goal of its impulse [12].2

In fact the part of a body which should be closer to the centre of the world is its centre. What then is this centre, Aristotle did not specify; only for the sphere and other regular solids can one sense that it coincides with the centre of the figure. The XIV century philosopher Albert of Saxony (c. 1316–1390), nicknamed Albertus Parvus by the Italian scholastics, pointed to the Aristotelian thesis. He assumed it is the centre of gravity and not simply the geometrical centre of a body that moves toward the centre of the world. Moreover he assumed that the gravity is concentrated in the centre of gravity [305].3 Thus he denied that the tendency to move downward belongs to the component parts of the body, because in his opinion this case would imply an interference between the parts and a slowdown in the free descent. Albert of Saxony argued that even in the case of an aggregate of heavy bodies it is the centre of gravity of the whole which tends toward the centre of the world. But he did not specify what type of constraint makes a series of bodies an aggregate. In particular, a heavy body tends to come down until the centre of gravity of the entire aggregate formed by it and the rest of earth is the centre of the world [305].4 From which the disturbing conclusion that the earth moves relentlessly because its centre of gravity moves toward the centre of the world for any shifting of bodies on the surface. A not very different approach was pursued by Arabic scholars; see for example [343]. Very interesting are the considerations by Muzaffar al-Isfizari about the way joined bodies tend to reach the centre of the earth, resulting in one of the most interesting and original ‘proofs’ of the law of the lever [247]. The idea that gravity is concentrated in the centre of gravity was taken up by mathematicians during the Renaissance. In particular Guidobaldo dal Monte wrote: All we said about the centre of gravity allows us to understand that a heavy body weighs, properly speaking, in its centre of gravity. The name itself seems to evoke this truth. All the force, all the gravity of the weight is concentrated in the centre of gravity; it seems to flow toward this point from all sides. Because of its gravities, indeed, the weight wants naturally to reach the centre of the world, but we said that what actually tends toward the centre of the world is the centre of gravity. Thus, when any weight is sustained by a whatever power from its centre of gravity, then the weight remains in equilibrium, and the whole gravity is perceived by senses. The same occurs if a weight is sustained from a point such that the straight line joining it to the centre of gravity passes from the centre of the world. In such a case, in effect, all is as if the weight were sustained from the centre of gravity. 2 3 4

II B 14, 297 b. vol. 2, p. 20. vol. 2, p. 25, in note 6.

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It is not the same if the weight is sustained from another point. In such a case the weight does not remain in equilibrium; before one could perceive its gravity, it turns, until, as in the previous case, the line joining the centre of gravity goes toward the centre of the universe [87].5 (A.6.1)

In the second part of the quoted passage it was derived almost as a logical consequence of the definition of the centre of gravity the criterion according to which a heavy body is in equilibrium if and only if its centre of gravity is prevented from lowering. It is however possible that historically the concept of gravity was directly derived from experiences of statics where attention is focused on equilibrium, easier to interpret, and only then was the concept extended to the case of motion. Greek mathematicians, engaged in research of the laws of equilibrium, reconnected to the static conception of centre of gravity and related the physical concept of centre of gravity to the geometrical concept of centre of a figure. The most remote terms and concepts which have been handed down to us on the centres of gravity are those, very far apart in time, of Archimedes and Pappus of Alexandria. With regard to Archimedes we received, unfortunately, neither the definition of centre of gravity nor the reasons for its introduction; we did however receive the axioms that determine the centre of gravity of composite bodies from elementary bodies of known centres of gravity (see Chapter 3). From Pappus we received instead the definition that fared well in the Middle Ages and the Renaissance. Here it is in the lesson of Federico Commandino: We say still centre of gravity of any body at its point, such that if the heavy body is imagined suspended from it and left at rest, it remains in the same position it had at the beginning, and does not tilt [75].6 (A.6.2)

After Pappus’ definition, Commandino adds one of his own that was then recovered by most of the mathematicians of the XVI century, including Guidobaldo dal Monte [86]:7 We can also define the centre of gravity of any solid body is one of its points, around which its parts have the same moment. If indeed a plane is drawn through this centre the plane splits the body in to two parts equally heavy [75].8 (A.6.3)

According to this last definition, the centre of gravity is identified by reference to the causes of motion which are in balance and not only through an empirically verifiable definition. On the idea of centre of gravity was based the centrobarica, a science in which the equilibrium problem of a body is reduced to the search for its centre of gravity and assurance that it is securely tied. One problem that was pressing scholars of the XVI century centrobarica, such as Commandino, Luca Valerio, Guidobaldo dal Monte, and Galileo was the difficulty of reconciling the rules to evaluate the centres 5 6 7 8

p. 10. p. 1. p. 1. p. 1.

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of gravity according to the Archimedean criteria, with the views of gravity of the time. Regardless of how the causes of gravity are conceived, all scholars agreed that bodies tend to reach the centre of the world. In other terms, using the modern categories to be understood, they believed that the weight forces were converging to the centre of the world and not just vertical and parallel. While the Archimedean rules of composition implicitly required parallelism. Simon Stevin (see Chapter 7) was the first in the history of mechanics to have some clear ideas on the subject. He did not accept the idea that gravity was concentrated in the centre of gravity of a body and thought instead it acted on all its components. On the one hand he showed that since the action of gravity converges toward the centre of the world, the centre of gravity in the sense of Pappus cannot exist for a body other than the sphere. On the other hand, however, he argued that this conclusion is only a theory and in practice because the action of gravity differs by a very small angle, the centre of gravity determined by the Archimedean rules meets the demand of Pappus to be the point of suspension of neutral equilibrium. The problem of the existence of centres of gravity was discussed again in the XVII century by René Descartes (1596–1650),9 Pierre de Fermat (1601–1665), Gilles Personne de Roberval (1602–1675) and Blaise Pascal (1623–1662), using a concept of distributed force [305].10 The end of the debate took place only when the distinction between mass and weight was clarified by Isaac Newton and the centre of gravity will no longer be the centre of weights but the centre of masses. Also interesting was the contribution by Girolamo Saccheri (1667–1683), who in his Neo-statica [212]11 showed that the centre of gravity also exists for central forces provided they vary in inverse proportion to their distance from the centre.

6.2 Galileo’s centrobaric To understand the thought of Torricelli it is useful to summarize the ideas of Galileo on the centres of gravity referred to mainly in Le mecaniche and in an Appendix to Discorsi e dimostrazioni matematiche [112].12 The various texts of Galileo show the insistence on the propensity to move toward “the common centre of heavy things”, while the drawings and the arguments only suggest lines of descent parallel of heavy bodies. He usually represents weights suspended from the lever by means of ropes, as also Luca Valerio did [236, 366], and the ropes are clearly parallel with each other, pointing out the implicit assumption of parallelism of the lines of descent of heavy bodies. Only a few times, the weights 9

In addition, Descartes will show that in the sphere, according to the theory of gravity force in his day, although there is a centre of gravity, it does not coincide with its centre but is lower [96], vol. 2, p. 245. 10 vol. II, pp. 156–186. 11 pp. 74–75. 12 pp. 187–208. It seems that the Appendix was prior to Le mecaniche. The composition of the work according to some historians dates from the period between 1585 and 1588 [123], vol. 1, pp. 181–182.

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are applied directly from the ends of the lever so the parallelism of the weight descent is not made explicit. This occurs, for example, in a manuscript by Vincenzo Viviani13 essentially reproducing the reasoning of dal Monte [86] for which the lever with equal arms and weights behaves as a single body with the gravity concentrated in the centre of gravity, coinciding with the fulcrum, and as such, in equilibrium regardless of its inclination. There are some aspects of the work of Galileo, which may have directly suggested the formulation of his principle to Torricelli, even if the conditional is a must. On the third day of the Discorsi e dimostrazioni matematiche, in the second edition of the work, the Medieval thesis is resubmitted that the centre of gravity of an aggregate, subject only to the weight, tends to move toward the centre of the world: Because, as it is impossible for a heavy body or a compound of heavy bodies to move naturally upward, departing from the common centre where all things tend, so it is impossible that it spontaneously moves, if with this motion his own centre of gravity does not approach to the said common centre [118].14 (A.6.4)

The interpretation of the text is not immediate, because it also depends on the meaning Galileo gives to the term compound. If it were a single body (more or less rigid) one could say that there was nothing new compared to the medieval view. If instead the compound or aggregate, were understood as a set of bodies also disconnected with each other, one would be in the presence of a new fact, i.e. it would be possible to identify the geometrical idea of centre of an aggregate of bodies unrelated to each other with a physical entity, the centre of gravity, as the point of application of the whole gravity. This is very important background knowledge for the further development of the ideas of Torricelli. But one must say that probably Torricelli did not know the over referred quotation, which appeared only in the edition of Discorsi e dimostrazioni matematiche of 1656 [120], before his arrival in Arcetri at the end of 1641. Though there is still the possibility that Torricelli knew the ideas of Galileo on the centre of gravity through Benedetto Castelli, to whom Galileo in 1639 sent a trace of the arguments given in the passage quoted above. In Le mecaniche, when the equilibrium of a body on an inclined plane is concerned, Galileo compared the weights and the motions of two bodies connected by a rope, one that moves on the inclined plane and another that moves on the vertical, to conclude that at equilibrium the lowering of both measured on the vertical are inversely proportional to their weights (see § 5.5.2). Note the proximity of this conclusion to the one given by Torricelli when studying the motion of two bodies on an inclined plane, to conclude that the centre of gravity of the whole does not lower or rise. Even here, however, it should perhaps be added that Torricelli did not know Le mecaniche.

13

This paper is considered as authentic of Galileo for sure by Favaro [123], vol. 8, p. 439 and with some doubt by Caverni [284] vol. 4, pp. 164–166. 14 p. 215.

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6.3 Torricelli’s joined heavy bodies Evangelista Torricelli was born in Rome in 1608 and died in Florence in 1647.15 As a boy he lived in Faenza with his maternal uncle Don Jacopo Torricelli of the order of Camaldolese. His uncle became his second father, giving him the possibility to study: humanities topics with him, sciences at the Jesuit school. Important were the years when he studied in Rome with Benedetto Castelli (1577–1644) and Michelangelo Ricci (1619–1682), in turn a pupil of Castelli. Galileo in 1641, thanks to Castelli, was able to read a Torricelli’s manuscript on the motion of bodies. He was so impressed that he invited him, the same year, to Arcetri as his disciple. Torricelli reached Arcetri October 1641 and remained there until the death of the master, which occurred in January 1642. Torricelli was then an effective Galilean disciple only for a few months. On the death of Galileo, the Grand Duke of Tuscany, Ferdinand II, appointed Torricelli official successor of Galileo in the Studio Fiorentino for the reading of mathematics. He was also appointed member of the Accademia della Crusca. In addition to the famous experiences around the barometer, fundamental are Torricelli’s works on geometry, Calculus, mechanics and optics [261, 320]. He organized a workshop [319]16 for the production of lenses and telescopes with a ‘secret’ method of work [294]. Torricelli alive had the satisfaction of seeing published only his greatest effort, the Opera geometrica [227]. Subsequently the Lezioni accademiche were printed [231, 232]. Other writings are published in [233, 234, 329, 392] and extensive excerpts from the manuscripts in [284]. The manuscripts of Torricelli are currently stored at the National Library in Florence in the collection of the Galilean manuscripts – volumes 21 to 44 of fourth division. His unpublished manuscripts had not an easy life. His premature death prevented the preparation and publication of his latest works that remained in the form of notes and memos. Not even his closest friends, able mathematicians, were able to continue his work as they also died shortly thereafter. For example, Bonaventura Cavalieri died in 1647 and Michelangelo Ricci in 1682. The historian Gino Loria mentions an amusing incident at the same time unforgivable. The cabinet with the manuscripts of Torricelli was sold to a grocer; one day the seller of sausage wrapped something into a sheet of a manuscript and sold it to Giovanni Battista Clemente Nelli. He recognized the importance of the quotations on the sheet and hastened to buy in bulk all them [392].17 Only in 1861 did Torricelli’s manuscripts find a safe home in the Libreria Palatina of the National Library of Florence. Evangelista Torricelli devoted much space to the study of mathematics and this favored a geometric Archimedean approach to mechanics. Most of his contributions, 15 It is now known with certainty that Rome was the birthplace of Torricelli. On Torricelli’s birth one can see the recent works by Bretoni [258, 263]. 16 pp. 84–95. 17 pp. 31–33.

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of pure mathematics, related to the determination of the centre of gravity of plane and solid figures, particularly complex. The most important conclusions are contained in the Opera geometrica, but results are scattered in manuscripts. Other contributions, perhaps less important quantitatively, but fundamental from my point of view, concern the physical aspects of the theory of gravity; in the following I will focus only on this part. The texts which I will refer to are the Opera geometrica, correspondence and published manuscripts. I consider these sufficient to provide an adequate full physical understanding of the theory of centres of gravity of Torricelli.

6.3.1 Torricelli’s fundamental concepts on the centre of gravity The Opera geometrica can be divided into three main parts, of which, the first two are divided in turn into books. A brief index follows: Part 1. De sphaera et solidis sphaearalibus, first book, pp. 3–46; second book, pp. 47–94. Part 2. De motu graviun naturaliter descendetium et proiectorum, first book, pp. 97–153; second book, pp. 154–243. Part 3. De dimensione parabolae solidique hyperboloci, pp. 1–84, not subdivided in books, which contains: Quadratura parabolae, De dimensione cycloidis, De solido acuto hypebolico, De dimensione cochlea. The fundamental ideas of Torricelli on the centres of gravity are brought back in the first book of the De dimensione parabolae. Though this book was written after De motu graviun naturaliter descendetium et proiectorum where Torricelli formulated his famous principle, the main concern of this chapter, it can be considered representative of young Torricelli’s conceptions. They are expressed with six items, definitions, theorem and postulates, under the undifferentiated name of Suppositions and Definitions. I refer to only those which seemed more important to me for this study. Suppositions and Definitions I. Suppose that the nature of the centre of gravitas is such that a magnitude freely suspended from whatever its part will be never at rest unless it centre of gravity reaches the lowest point of its sphere [228].18 VI. Equal heavy bodies, suspended at equal distances, are equilibrated both when the balance is parallel to horizon and when it is tilted. And weights having inverse proportion to distances are equilibrated both for the balance parallel to horizon and when it is tilted [228].19 (A.6.5)

Assertion I seems more a definition than a postulate because it is the first time the centre of gravity is named. If this is true, Torricelli’s definition is different from the traditional ones given by Pappus and Commandino and in some ways is more complete; with it the ‘empiric’ rule that a body is in equilibrium if its centre of gravity is on the vertical line from the centre of suspension becomes a trivial theorem. The assertion however deserves some comments. It can be written formally and synthetically with the following implication: Centre of gravity not at bottom → not equilibrium 18 19

p. 11. pp. 13–14.

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i.e. equilibrium → centre of gravity at bottom. However in the explanation of the proposition Torricelli uses also the implication: centre of gravity at bottom → equilibrium. It is then possible to argue that he could imply: Centre of gravity at bottom ↔ equilibrium Assertion VI is presented as a theorem. In fact, as clear from below, it is also a postulate which implicitly asks the reader to admit that the actions of gravity are vertical and parallel. At a first reading Torricelli seems to take a position on the thorny problem of the nature of the equilibrium of the lever with the same arms and weights: it would be neutral equilibrium. An analysis of the proof of the theorem shows that it is not exactly what Torricelli does. Torricelli was aware of the discussions in the international arena and especially in France, about the manner of variation of gravity with the assumption that it originates from the centre of the world [305].20 As clear from his correspondence with Benedetto Castelli and Antonio Nardi (1589–1649 ca.) [362],21 which date back at least to 1635 [284],22 [322]. But he shared with Galileo the idea that in physics it is possible to idealize the situation, and in the particular case of the lever, abstracting from the convergence of the actions of gravity. In this way it is possible to develop ex-supposizione exact reasoning that could also apply to reality, less than for small imperfections widely acceptable in practice. Torricelli exacerbates the reasoning of the substantial parallelism of the actions of gravity, by postulating the true parallelism, imagining a balance currently at infinite distance from the centre of the world: It is accepted even by sound scholars the objection that Archimedes made a false assumption, when he considered the wires through which the magnitude are suspended from a balance, as if they were parallel, when in fact they converge toward the centre of the earth. But I (pace of some illustrious men) believe that one must consider the principles of Mechanics in an entirely different way. I grant well that if physical magnitudes are suspended freely [without external forces] from a balance, the material wires of suspension will be converging, as they all tend toward the centre of the earth. However, if the same balance, even material is considered not on the surface of the earth, but high in regions beyond the orb of the sun, then the wires, although there also tend toward the centre of the earth, will be much less convergent with each other, and will be almost parallel. Imagine we take our mechanical balance, beyond the Libra, to an infinite distance. Who does not understand that now the wires of suspension will not be converging, but exactly parallel to each other? [228].23 (A.6.6)

Assuming the wires with which the weights hang from the balance as ‘really’ infinitely long appears a choice of infinity in act in the mathematical theory. Unlike Galileo, Torricelli, most interested in mathematical aspects, shows all his skills of abstraction and does not hesitate to claim daring theoretical positions:

20 21 22 23

vol. 2, pp. 24–26. p. 10. vol. 4, pp. 156–212. pp. 9–10.

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So it can be said that it is false the mechanical principle for which the ropes of the balance are parallel, where the magnitudes suspended from the balance are material and real and tend toward the centre of the earth. It will not instead be false in the case the magnitude, are they abstract or real, tend not to the centre of the earth or a point near to the balance, but to an infinitely distant point [228].24 (A.6.7)

Although Torricelli is dealing with the lines of descent as parallel, when he writes the Opera geometrica he has not yet made completely clear what were the issues involved in centrobarica if the lines of descent are assumed as convergent. One symptom of this lack of clarification can be found still in the proof of proposition VI of De dimensione parabolae, where after proving that a balance with weights inversely proportional to the arms is in equilibrium whether it is horizontal or tilted, he specifies: I don’t miss that in the dispute among the authors about the tilted balance, i.e. whether it comes back or it remains in its position, the centres of magnitudes are located into the balance. But because in the booklet we assume the magnitudes located below the balance [emphasis added], we prefer to follow our purpose instead to adapt our demonstrations to a controversy among other people [228].25 (A.6.8)

Torricelli says explicitly that when the centre of gravity of the weights is below the fulcrum, as it occurs when they are suspended by wires from the end of the arms, the balance is in equilibrium whatever its inclination. He avoids entiring the ‘dispute’ that still exists in the case of balances with weights attached to the ends where the centre of gravity of the weights coincides with the fulcrum. Thus revealing some uncertainty. After 1644 Torricelli comes back to consider the possibility of actions converging toward the centre of the world. They are only sporadic observations that show however a definitive understanding of the problem. For completeness, I consider it useful to report some considerations on the equilibrium of the lever. Torricelli first treats the lever with different arms and weights: When we admit that the weights of the balance have inclination towards the centre of the earth […] it will follow that there is no horizontal balance with unequal arms and weights in reciprocal proportion of the length of the arms, so that these weights equilibrate.26 (A.6.9)

The demonstration of what is said above, which I do not carry over, uses the idea of static moment developed by Giovambattista Benedetti that the effect of a force on a lever is proportional to its intensity and distance of its line of action from the fulcrum. Or better he reproduces the argumentation of Benedetti of § 5.5.2. Torricelli then deals with the lever with arms of equal length: Now place B as the centre, and AC a balance with arms of equal length with two equal weights in the extremities A and C, the moments or gravities of which are measured from the perpendiculars DF and DE, as Giov. Battista de Benedetti declares in his book of mathematical speculation, chapter III or IV [29]. It follows that the moment of the weight in A, to 24 25 26

pp. 9–10. p. 15. E. Torricelli, Galilean collection, manuscript n. 150, c. 112.

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D

A

F

C

E B

Fig. 6.1. The equilibrium of a lever with equal arms the moment of weight C, is reciprocally as the line BC is to the line AB, that is reciprocally as the distance of the weights from the centre of the Earth. And here we have not only that the weight closer to the centre, while it is in the balance, weighs more than the farther, but we yet know to what proportion it weighs more.27 (A.6.10)

The argument still uses the law of static moments – for which the ratio between weights A and C is as the ratio between the distances DE and DF also equal to that between AB and CB – is conclusive if it is accepted that the absolute gravity of bodies does not vary with distance.

6.4 Torricelli’s principle Torricelli applies his ideas on the centres of gravity in the De motu gravium [229], published as a book of Opera geometrica in 1644 but almost certainly based on a manuscript dating at least to 1641. The goal was ambitious: to rebuild the Galilean dynamics on a more solid foundation than the one Galileo had given in the first edition of the Discorsi e dimostrazioni matematiche of 1638. To do this it was necessary to clarify certain ‘assumptions’ of Galileo not sufficiently convincing, in particular what Torricelli called ‘theorem of Galileo’, which states: Proposition II Moments of equal heavy bodies over unequally inclined planes, having the same height, are in the reciprocal ratio with the length of planes [229].28 (A.6.11)

the demonstration of which was not reported in the first edition of the Discorsi e dimostrazioni matematiche. Torricelli had participated with Vincenzo Viviani in the last months of Galileo’s life, a period when the master, nearly blind, tried to shed light on the basic principles of his mechanics, and between them the proposition cited above. Correspondence between Torricelli, Nardi and Galileo bear witness to this work. In 1641 Galileo wrote to Torricelli that he has sent him the proof of his theorem by means of Nardi [276].29 When Torricelli wrote the Opera geometrica in 1644 he was therefore aware of the demonstration of Galileo. But because this was not yet 27 28 29

E. Torricelli, Galileian collection, manuscript n. 150, c. 112. p. 100. p. 16.

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printed,30

and because it was based on principles other than his own, he considers it useful to provide his own version: I know that Galileo during the last years of his life tried to prove this proposition. But because his argumentation was not published with his book on Motion, we assumed to place before our booklet these few things on the moments of heavy bodies. So it appears Galileo’s supposition can immediately be proved from the theorem he himself assumed for granted in his Mechanics [Discorsi e dimostrazioni matematiche] in the second part of the sixth proposition of accelerate motion, that is moments of equal bodies over unequally inclined planes are as the perpendiculars of equal parts of the planes [229].31 (A.6.12)

This quotation raises some doubts. Torricelli claims that Galileo gave as proved his theorem, and not that he proved it. This may mean either that Torricelli did not know exactly or did not approve Galileo’s demonstration. The two hypotheses are not mutually exclusive. It is possible that Torricelli did not know Le mecaniche of Galileo, published at that time only in French in 1634 [117]. Besides, it is also possible that Torricelli did not approve the proof of the law of the inclined plane that Galileo had sent to him and that will be published in the second edition of the Discorsi e dimostrazioni matematiche, because it was based on the questionable principle of virtual displacement [276].32 If Torricelli did not know Le mecaniche he could not have known that Galileo’s demonstration of the law of the inclined plane was based on the law of the lever, considered an unquestionable principle by all.33 And then one can justify his claim, at first sight unreasonable, that Galileo in the Le mecaniche only supposed that law. To prove Galileo’s theorem Torricelli puts a premise now known as the principle of Torricelli. The premise is followed by an explanation and a justification if not a demonstration, so somehow it retains the logic status of a principle. Premise Two equal bodies joined together cannot move by themselves if their common centre of gravity does not descend. When indeed two heavy bodies are joined together such that to the motion of one it follows the motion of the other, the two heavy bodies will be as one single body compound of the two, be it a balance, a pulley, or whatever mechanical devices. Such a heavy body will never move if its centre of gravity does not descend. When all is disposed so that the common centre of gravity cannot descend in any way, the heavy body will remain at rest, because otherwise it will move in vain, i.e. with a horizontal motion, which no way tends downward [229].34 (A.6.13)

30

It will appear in the second edition of the Discorsi e dimostrazioni matematiche [120]. It is worth noting that the proof of the law of the inclined plane shown in the Dialogo [116], pp. 215–218, is different from that reported in the Le mecaniche [119], p. 181. The first is based on the principle of virtual work, the second on the law of the lever. 31 p. 98. 32 p. 18. 33 It is interesting to note that Torricelli in De motu gravium refers also a demonstration of the law of the inclined plane, alternative to that developed with his principle, very similar to that of Galileo in the Le mecaniche. This, if not to commit the sin of plagiarism to Torricelli, could prove that he actually did not know Le mecaniche. 34 p. 99.

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The premise, made explicit with modern terms, states that two heavy bodies connected to each other in any way, cannot move by themselves from the configuration in which they are if [nisi] their overall centre of gravity did not sink for a generic virtual displacement of the two bodies compatible with constraints. Using a compact language: Common centre of gravity cannot sink → equilibrium It is then only a sufficient condition for equilibrium because it is not stated explicitly that there is not equilibrium when the common centre of gravity sinks. Torricelli begins the justification of his premise by stating that the two heavy bodies, as they are joined, have to be treated as a single body and returns as example cases of joined bodies of mechanics. Among them it is particularly significant the pulley where one has two weights connected with a wire inextensible but flexible. The assimilation of the two bodies to a single body is not at all obvious, in fact, the common centre of gravity of two bodies is a purely geometric point, with no substance, on which one can hardly think that gravity is applied, such as the Medieval scholars assumed for the individual body or for an aggregate of contiguous bodies. Torricelli has the ‘ingenius’ idea to extend analogically and unequivocally the reasoning valid for a body/centre of gravity to the aggregate. This analogical extension is made possible, perhaps even for the persistence of the idea of a global action of gravity, not divided in different parts of a body. I have however shown that in the analysis of the equilibrium of the lever for actions of gravity that converged, which the mature Torricelli developed (see above), there was clearly the idea that any portion of the body retains its individuality. Perhaps the mature Torricelli could not have conceived his principle. Once accepted that the mathematical centre of gravity of an aggregate behaves like a physical centre of gravity, the justification for the premise can be referred to the motion of centre of gravity of an ordinary body (grave autem). The locution “does not lower” (nisi descendant) referred to the centre of gravity is separated into two parts: (a) it rises, (b) it remains horizontal. The case (a) where the heavy body moves with the centre of gravity that rises is clearly impossible for the very definition of centre of gravity. Case (b) where the heavy body moves and the centre of gravity is kept horizontal is also impossible. According to the prevailing views on the gravity of Torricelli’s times, bodies moved down because they have the goal to reach the centre of the world. When a body is in a plane and cannot sink its movement is impossible, because it is without a goal. Since the goal is, according to the Aristotelian doctrine, a formal cause it can be said that the body moves without cause, which Torricelli considers absurd.

6.4.1 Analysis of the aggregate of two bodies Torricelli’s concept of aggregate differs in two fundamental aspects from Albert of Saxony’s vague concept mentioned at the beginning of this chapter, and also from Galileo’s slightly more defined concept. First, Torricelli presents a concrete case: the two spheres as heavy bodies. The centre of gravity of the aggregate is deter-

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C E A G G′ B D

Fig. 6.2. Equilibrium on the inclined plane of two joined bodies

mined by the Archimedean rules. In this way, Torricelli can check the condition of the aggregate equilibrium on the basis of the constraint imposed to lowering the centre of gravity. Second, the aggregate does not maintain the same shape as a single body does, but it is constrained in order to take different shapes. Note that Torricelli reinforces the concept of conjunction also with explicitly operational terms: Be connected even with an imaginary rope conducted through ABC, so that from the motion of one if follows the motion of the other [229].35 (A.6.14)

Torricelli suggests not only that a mechanism is needed because the motion of a moving body follows the motion of the other, but also explicitly states the instrument, the (imaginary) rope, with which it is connected and its motion. The implementation in a precise mathematical language puts Torricelli in a position to use his premise to prove the law of the inclined plane, the proposition I of the De motu gravium, which is preliminary to the proof of Galileo’s theorem. Proposition I If in two planes unequally inclined but with the same elevation, two heavy bodies are considered which are in the same ratio as the length of the planes, the heavy bodies will have the same moment [229].36 (A.6.15)

The mathematical core of the proof, which is interesting because it involved the concept of centre of gravity of heavy bodies, is in fact very simple. It proves that if the weights of two heavy bodies A and B are in the same proportion of the lengths of two inclined planes, their common centre of gravity G moves horizontally as shown in Fig. 6.2. The whole demonstration of proposition I, is divided into three steps, first Torricelli denies it (step 1) “They do not have, if possible, the same moment, but prevailing one, they move, and the heavy body A rises toward C, while heavy body B descends in D [229].37 Then he shows that the centre of gravity of the two bodies do not sink (step 2) so the two bodies have acted without cause, which is absurd, so (step 3) proposition I cannot be denied. In the words of Torricelli: 35 36 37

p. 99. p. 99. p. 100.

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G G1

G2

O Fig. 6.3. Lack of equilibrium for converging lines of gravity

Two equal bodies joined together were moved and their common centre of gravity did not descend. This is against the premised law of equilibrium [229].38 (A.6.16)

Note also that the proof of proposition I, about the truth of which Torricelli has no doubt, could be seen as an attempt to validate the premise of equilibrium. Proposition I so could be regarded as a methodological principle of the theory and not as a theorem to prove. The proof would not have succeeded if Torricelli had considered the ‘real’ situation in which the directions of the lines of gravity are converging toward the centre of the world. In fact, when the common centre G of two heavy bodies – assuming it exists – moves on a horizontal plane, it varies its distance from the centre of the world O (to not change it, the common centre of two heavy bodies should move on a sphere) and can even come close to it, as clear from Fig. 6.3 where one can see that a movement on the plane from G1 or G2 towards G leads to an approach toward the centre of the world. Having established Proposition I, Torricelli goes to show Proposition II, i.e. Galileo’s theorem, thus ending his project of re-foundation of the Galilean dynamics. It is outside my purpose to express my thoughts on the fact that the objective has been achieved.

6.4.2 Torricelli’s principle as a criterion of equilibrium I have shown how Torricelli introduced his principle for a very ambitious reason, to re-establish the mechanics of Galileo. He realizes that his principle furnishes also a criterion of equilibrium, but he does not take full advantage of the fact and limits himself to applying it on an occasional basis. Torricelli uses his principle in statics at least on two occasions. The first, already commented in another respect, in the Dimensione parabolae, where he intends to 38

p. 100.

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H

D M B

C

E L

A G

I

149

F

Fig. 6.4. A lever with suspended weights

show that a balance with weights inversely proportional to the arms is equilibrated regardless of its inclination. This case study is the balance of Fig. 6.4 from the ends of which weights with inextensible wires are hung (the body CBF is suspended in I from its centre of gravity L). Considering such a balance in any configuration, different from the horizontal, Torricelli determines the centre of gravity and establishes that it is below the fulcrum, on the vertical line from it, and concludes: From this reason the magnitudes suspended from the balance AC will equilibrate. Indeed if they moved, their common centre of gravity, which has been proved to be in the vertical DF, would rise. Which is impossible [228].39 (A.6.17)

i.e. for Torricelli the balance is in equilibrium according to his principle because in any possible motion the centre of gravity of the weight-rod system rises. The conclusion is true, but the argument is not perfectly developed. It seems wrong, because in reality the centre of gravity of the system of the two weights instead of rising remains at the same level as it is easy to see with simple calculations (in fact the centre of gravity does not change at all its position).40 Torricelli’s ‘mistake’, perhaps not just an oversight, suggests, that it was not so unnatural to him to think of two weights connected to each other as one body, and then apply to their centre of gravity, the same rules for the centres of gravity of monolithic bodies. Most important and challenging, is the second use of the principle, in a situation where one might think that Torricelli would not have thought possible the application. In fact, Torricelli’s principle, as it provides only a sufficient condition for the equilibrium, allows to recognize the equilibrium in cases now classified as stable (for example a ball placed on the lowest point of a gutter), but not in those classified as unstable (for example, a ball placed over a hump) where the centre of gravity can be lowered and yet the system is in equilibrium. In Torricelli’s times there did not exist a theory of stability – for this one must wait for at least Lagrange with his Mécanique analytique of 1788 – but for certain situations, such as the ball over a hump, 39

pp. 14–15. It seems strange that Torricelli could have made a such trivial error; there is the possibility that his text is simply imprecise. Indeed it is a mystery why Torricelli uses the body BFF with a side parallel to the beam AC, as if it were fixed to it. If this were actually fixed the centre of gravity of the whole could rise.

40

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A

n

B b

m

E

C

G

F

f

L

M

o O

s

Z Q r R

Fig. 6.5. Unstable equilibrium of an inclined shaft

common sense was enough to feel the problematic nature of equilibrium, that also seemed to subsist for the principle of sufficient reason. The case of equilibrium studied by Torricelli, reported in an undated manuscript [233],41 regards the very unstable equilibrium. An inclined shaft is supported without friction by two orthogonal walls, one vertical and the other horizontal, as shown in Fig. 6.5. The horizontal force should be determined which must push F toward E in order to maintain equilibrium in the shaft, heavy by itself or by a weight suspended from a point C. This issue was raised a century earlier by Leonardo da Vinci [284]42 who attempted a solution. Vincenzo Viviani and other mathematicians also wrestled with the job, not always with satisfactory results [284].43 Torricelli had already addressed this problem in a few letters from/to Michelangelo Ricci [233]44 and knew that there was a position of equilibrium and what it was. In these letters he provided a solution based on a principle of virtual work, limited to the case where to the ends A and C of Fig. 6.6 two forces concurring in B are applied: The power at A to the power at C, is as the line CB to the line CA. The proof [...] depends on the speed because moving the beam AC, along the two lines of the right angle ABC, the speed of the point A to the speed of the point B, is as BC to BA [233].45 (A.6.18)

In Torricelli’s letters anyway there is no mention of the fact that he considered problematic the equilibrium, or that he recognized what we now call its instability. Since the topic seems particularly interesting I carry out nearly in full Torricelli’s considerations [276]: 41

vol. 3, pp. 243–245. vol. 4, pp. 65–67. 43 vol. 4, pp. 65–67. 44 vol. 3, pp. 91–92; pp. 96–99; pp. 99–100. 45 vol. 3, p. 100. In [284], vol. 4, p. 64, there is the reference to a Torricelli’s manuscript which faces the same argument with the law of static moments, while maintaining a tone that suggests some doubt on the solution found. 42

6.4 Torricelli’s principle

151

A

B

C

Fig. 6.6. The equilibrium of an inclined bar Among the observed effects of mechanics, for what I know, there is one not yet felt by some people, even though from it notions of some interest and curiosity may derive. Assumed a vertical wall AE in the horizontal plane EF at which the line EF is normal, and also assumed the straight beam BCF the centre of gravity of which is C, that with the extremity B supports the wall above mentioned, and with the extremity F can slide freely on the floor EF, we look for the proportion of the weight of the beam to that force, which directly applied in F pushing in direction FE can balance the moment of the beam and to flow under its own weight in the direction EF. Suppose the required force be given by a weight attached at the point Z of a rope of given length FEZ, which we call λ, and that passes through the point E. From the centre of gravity C trace the normal CG over FE, and the ratio of BF to FC be the same as the ratio of 1 to x, we will have for the similar triangles BE : CG = BF : FC = 1 : x and consequently, assumed P the weight of the beam, the distance from the horizontal EF from the mentioned weight, which can be understood collected in the centre of gravity of the beam, will be equal to P · CG, or really writing x · BE instead of CG, the distance of the line EF to the weight P will be x · P · BE [233].46 (A.6.19)

In this first part the system to be analyzed is defined. In an intermediate position C of the shaft BF there is a weight P. Its position is defined by x so that xBF represents the distance from P to F. The shaft is held in equilibrium by a weight Q, connected by a horizontal string of length λ to the end F, which provides the inward thrust. Torricelli determines the distances of the weights P and Q from the horizontal wall, meaning in reality, the ‘weighted distance’, i.e. the distances multiplied by the weight associated with them. Let the searched weight be hung at point Z of the rope FEZ equal to Q, the distance of the named weight from the horizon will be equal to Q · ZE, and the distance from the common centre of gravity of the two weights P and Q from the horizon is equal to Q · ZE – P · CG = Q · ZE – x · P · BE. Consider the centre L and draw the quart of circle SMs with radius LM equal to BF and draw the ordinate OM, which necessarily will be equal to EF, and from point M trace the tangent to the circumference nM parallel to the straight line LrR and extend the ordinate MO until it meets the straight line LrR in R; the distance of the common centre of gravity of the two weights P and Q from the straight line EF, which is proved to be equal to Q · ZE - x · P · BE, still equal to Q · λ − Q · EF - x · P · BE, i.e. equal to Q · λ − Q · OM x · P · LO [233].47 (A.6.20)

The centre of gravity of the two weights P and Q is given by the algebraic summation of the weighted distances dP (= x · P · BE) of P and dQ (= (λ – EF) · Q) of Q, and then as the difference dP − dQ , then dividing for the summation P + Q. According to Torricelli’s terminology the difference dP − dQ is the ‘distance’ of the centre of gravity from the horizontal line EF. 46 47

vol. 3, pp. 243–244. vol. 3, pp. 243–244.

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6 Torricelli’s principle

Torricelli then assumes that P,Q and x are related by the expression xP/Q = EB / EF. This choice, which Torricelli knows is associated with equilibrium – this is the result reported by Torricelli to Ricci, commented on the previous pages – is not currently justified. With a simple but elegant reasoning, he shows that for the given values for P, Q and x, when P and Q move congruently with the constraints, the centre of gravity of the shaft moves along an arc around the point of maximum defined by xP/Q = EB / EF. Then he concludes: “So by virtue of the lemma above the centre of gravity of the two weights will remain motionless in the proposed case”. Here Torricelli makes one nontrivial step and not easily explainable. Basically he says that under the “previous lemma” the centre of gravity in our case in fact does not move downward, that is the system is in equilibrium. Unfortunately it is not clear what is the invoked lemma. Vassura believes it is a lemma that Torricelli would have written, but then it went missing [233].48 It seems to me that the lemma could be simply Torricelli’s principle, which is introduced by him as a premise of equilibrium, and therefore in some way as a lemma. In fact, if a differential approach of the modern type is adopted with the use of infinitesimal displacements it would be obtained that, for a very small change in the configuration of the system, the centre of gravity would not be lowered as if it moved in accordance with the horizontal tangent of an arc of circle and then one can invoke Torricelli’s principle for the equilibrium. The hypothesis that Torricelli is working with infinitesimals, of course intuitively, does not necessarily represent an undue modernization of his thought. Apart from his studies on the indivisibles where the concept of infinitesimal is touched, this possibility is also corroborated by another passage contained in the undated manuscript. At one point, Torricelli argues that as a result of the infinitesimal (the term is mine) shift of the point F for which the shaft (Fig. 6.5) moves from the configuration BF to the configuration b f , the centre of gravity sinks of the quantity mn. This reasoning would be valid only if he neglects the change of slope of Lr with respect to LR. And this process will be typical for those who will manipulate the infinitesimals in the XVIII century. But Torricelli’s reasoning can be explained even without explicit reference to infinitesimals. For example he might refer to the work of Descartes that a few years earlier had reached a conclusion similar to that of Torricelli, by applying the principle of virtual work to the equilibrium of a heavy material point which moves along an inclined plane of curved profile, also subject to an external force [96].49 According to Descartes (see Chapter 7) it is necessary to consider the displacement along the tangent and not the actual displacement. Torricelli could have known the ideas of Descartes by Mersenne with whom he was in contact. Considering the motions in their birth, however, was implicit in the modus operandi of many mathematicians of the XVI century. For example, Galileo and Roberval often replace the real constraints with other constraints, simpler, giving rise to the same infinitesimal displacements (see Chapters 5 and 7).

48 49

p. 234. vol. 1, p. 233.

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws

153

C C E A

F

B

A

D

F

B

G D

Fig. 6.7. Application of Torricelli’s principle to the pulley

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws Vincenzo Viviani was the first to highlight the potential of Torricelli’s principle, as the foundation of statics [284].50 Then the principle was extended and generalized and Torricelli’s principle was going to indicate a fundamental principle of statics and the fundamental role in dynamics assigned to it by Torricelli was completely forgotten. Viviani applied Torricelli’s principle to study the behaviour of weight suspended from a pulley and showed that only if the two weights are equal can their common centre of gravity not be lowered, and then they are in equilibrium; this result moreover is trivially known. If the two equal weights A and B (Fig. 6.7) are attached to a wire passed over a pulley or other support which can run, these will stay in balance, wherever they are located. (A.6.21) Because if they moved the one that descends as much acquires as the other which rises loses, since their ways are equal and for perpendicular lines. And if it were possible they can move from sites A and B to sites C and D, it is clear that because their centres of gravity move in a straight line, the common centre of A and B will be in the middle, that is in F, and the centre of gravity of C and D will be in the middle, that is in F. Then because AC and BD are equal and parallel to each other, CD and AB joined meet in the same point, i.e. in the middle, so the common centre will not be moved, and will not have acquired anything, so that heavy bodies A and B will not move from the site where they were situated. (A.6.22) But if the weight B that will sink will be greater than the weight A, because their common centre is not in the middle of BA, as in E, but more close to B it can sink along the perpendicular EG [284].51 (A.6.23)

To point out the use by Alfonfo Borelli (1608–1679) of Torricelli’s principle, or better, the idea of evaluating the mechanical behavior of an assembly of bodies with reference to the common centre of gravity [46].52 50 51 52

vol. 5, pp. 21–22. vol. 5, p. 22. pp. 311–312.

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The subsequent evolution of Torricelli’s principle, occurring for insensitive steps, consisted in replacing the language of proportions with the algebraic one. If p1 and p2 indicate two weights connected to each other and with Δh1 and Δh2 their height variations for a possible motion allowed by the constraints, Torricelli’s principle requires that for balance it should be satisfied that: p1 Δh1 + p2 Δh2 ≥ 0,

(6.1)

that expresses in formulas the fact that the centre cannot fall and which for infinitesimal displacements assumes the expression: p1 Δh1 + p2 Δh2 = 0.

(6.2)

The above formula can be given the mechanical meaning that if, in a system of constrained heavy bodies, one weight descends another must rise and the relationship between the descent and ascent is inversely proportional to the ratio of the weights. The principle of Torricelli is then a virtual work law, equivalent to Jordanus’ theorem from a mathematical point of view. Note that the mechanical nature of Torricelli’s principle is hidden by the Archimedean approach which seeks to reduce the physics as much as possible. For sure Torricelli would not have had problems to extend his principle to a system of n weights. If he had done this and had written the results with an equation he would have obtained the relation: n

∑ pi Δhi = 0.

(6.3)

i=1

To the reader the opinion on the level of anachronism is introduced with these considerations.

6.5.1 A restricted form of Torricelli’s principle In some textbooks of physics Torricelli’s principle is evoked also for a single body, according to which it is impossible for the centre of gravity of a body in equilibrium to sink from any possible movement. It must be said however that this statement is almost a logical consequence of the definition of centre of gravity and it does not require the degree of abstraction necessary to formulate a statement similar in form but valid for a set of bodies, as does Torricelli. Despite its poverty, the restricted Torricelli’s principle has some interesting applications; among these is the proof of the containment polygon theorem, which shows that a solid based on a horizontal plane is in equilibrium if and only if the vertical line from its centre of gravity falls within the (convex) perimeter of the base. This criterion has been exposed in a formal manner by Juan Bautista Villalpando (1552–1608) [241] in 1604 and Bernardino Baldi in his Aristotelis problemata exercitationes. Elio Nenci believes that Baldi knew the work of Villalpando and then he was inspired by him [20]. More dramatically Duhem believed that both Villalpando and Baldi were inspired by Leonardo [305].53 53

vol. 2, p. 119; p. 133.

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws F

G

A I

H

155

D

E

L

B

MC

K

Fig. 6.8. The containment polygon A H E B

F

K C

Fig. 6.9. A measure of stability against tilting

In the following I refer to what Baldi wrote in the Aristotelis problemata exercitationes in problem XXX, where he studied the equilibrium of the body shown in Fig. 6.9: Assume it moves, and from the semidiameter BE with centre in B the arc of circle EH is described which cuts BG in H and BF in I. And because EH perpendicular to the semidiameter BK does not pass through the centre B, EM is shorter than BK, i.e. of BI. Cut LB equal to EM from BI. Point L will be then below point I, i.e. closest to the centre of the world of I. Because the wall would collapse it is necessary that the centre of gravity E, after the rotation around B, reaches I before arriving eventually in H. But I is farther from the centre of the world than E and L, so the heavy body will rise against its nature, but this is impossible; this is what to prove [18].54 (A.6.24)

The body is in equilibrium. Indeed assume that it is not, and perform a rotation around B (the same applies to C); the centre of gravity describes the arc EIH and thus moves away from the centre of the world. But this is absurd, then the body is in equilibrium. In problem VIII Baldi takes as a measure of stability in the degree of raising of the centre of gravity in the rotation around a corner: Let the triangles ABF and ACF [Fig. 6.9] be equal and of the same weight, with AFB a right angle. Join F and C with EC greater than EF. Rotate the triangle around the point C and let EC become perpendicular to the horizontal plane as CH, and from E draw the parallel EK to the horizontal plane. Moving the triangle, the centre of gravity E will displace in H, but KC is equal to EF, less than CH, then the centre of gravity will be raised from E to H, i.e. above K for the whole space KH. This raising makes the motion difficult [18].55 (A.6.25)

54 55

p. 176. p. 2.

7 European statics during the XVI and XVII centuries

Abstract. This chapter presents Dutch and French contributions to mechanics, with a nod to the English in the XVI and XVII centuries. The first part describes the contribution of Gille Personne de Roberval who proved the rule of composition of forces with Jordanus’ displacement VWL. The central part describes the contribution of René Descartes who was among the first to base statics on a VWL according to the idea of Jordanus de Nemore. Descartes also introduces the idea of virtual displacements as the birth of virtual motion (draft of infinitesimal displacements). An important part is devoted to Simon Stevin. Given his role as the founder of modern statics I did not limit myself to presenting his contribution to the VWLs, which is controversial, but I also present some of his other contributions that are less documented in the manuals of the history of mechanics. The final part shows that Isaac Newton does not avoid important considerations on VWLs based on velocities, despite the fact that his mechanics is normally considered an alternative to them.

7.1 French statics In the early XVII century, when Italy was still the leading nation in Europe, the only text in French about mechanics was a translation of Cardano’s De subtilitate by Richard le Blanc [55]. In 1615 Salomon de Caus (1576–1626) who worked as a hydraulic engineer and architect under Louis XIII, published Les raisons des forces mouvantes avec diverses machines, a book having as subject the functioning of machines more than their equilibrium and which concentrated on a steam-driven pump similar to one developed by Giovanni Battista della Porta (c. 1538–1615) fourteen years earlier [90]. The text of de Caus is quoted by Pierre Duhem [305]1 and René Dugas [308]2 giving significance to the fact that he used (for the first time?) the word work (travail) to indicate precisely what today is called work. There is not however 1 2

vol. 1, pp. 290–292. p. 124.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_7, © Springer-Verlag Italia 2012

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any historical evidence that de Caus’ use influenced Coriolis who was for sure the person who spread the use of the term work (see Chapter 16). In the following I refer to a passage in which the word work is used, which also shows how de Caus takes a law of virtual work to explain the operation of machinery, at least the lever and pulley: Vitruvius mentions this kind of machine, called by Greeks trochlea [pulley, block and tackle], in which the motion has its way by means of pulleys […] one end will be attached to the pulley and the other will serve to raise the burden. As it may be seen from the figure if one pulls the said piece of rope marked G one foot down, the burden attached to the pulley E simultaneously will rise half a foot and then, since the rope is doubled in the pulley, if one pulls 20 feet of rope, the burden will move 10 feet. So a man will raise a heavy body with this machine, as he would be two, if the machine were simple but the two men together will draw twice the height i.e. 20 feet, before the other had pulled 10, and if in the trochlea there were two pulleys, as shown in figure M [Fig. 7.1], the force would be quadruple, but the burden would rise only 5 feet by pulling 20 feet of rope. The toothed wheels have still the same ratios as the previous ones, because the force increases proportionally as the time increases […] so that a single man, will use equal force pulling a load in this machine as eight […] but as the eight men took one hour to lift their weight, a man will take eight hours to lift his [90].3 (A.7.1)

Fig. 7.1. Work of a machine (reproduced with permission of Master and Fellows of St. John’s College, Cambridge)

In 1634, two translations and a new text, quite important, were published. Albert Girard (1595–1632) translated the Tomus quartum mathematicorum hypomnematum de statica by Stevin [217], Martin Mersenne (1582–1648) translated Le mecaniche by Galileo [117], Pierre Herigone wrote Cursus mathematici tomus tertius, a course of mechanics with text both in Latin and French [130]. Pierre Herigone was born in France in 1580 and died probably in Paris about 1643. Mathematician and astronomer, he taught in Paris for a long period [290]. His 3

p. 7.

7.1 French statics

159

book was a success and helped to spread Stevin’s ideas on mechanics. Important influences though not made explicit are those of Guidobaldo dal Monte and Jordanus de Nemore. Herigone did not use a single principle for his demonstrations. For example he demonstrated the law of the lever with an Archimedean approach; the law of the inclined plane was studied as in Stevin but also as in Jordanus. According to Duhem, Descartes received from Herigone Jordanus’ ideas on the law of virtual work. Of some interest is the use of a compact mathematical notation to indicate relations and operations; among these the sign ⊥ for orthogonality, the sign π for the ratio; 2/2 to indicate the equality, 2/3 to indicate less than, 3/2 greater than, < angle. To illustrate the use of a law of virtual work in Herigone I limit myself to demonstration of the laws of the lever and the inclined plane: C

D A

E F

G B

Fig. 7.2. The inclined plane of Herigone

Herigone proves the law of the lever with an Archimedean approach, close to that used by Stevin, then he proves as a theorem the following law of virtual work: For weights in equilibrium, the space of the lighter is to the space of the heavier as the heavier is to lighter; the same holds for the motion in vertical direction of the lighter to the vertical motion of the heavier [130].4 (A.7.2)

In the demonstration of the law of the inclined plane Herigone uses this theorem as a principle of general validity: For the same time the weight G descends from point C to point B, the weight D rises from point A to point E and BC will consequently be the perpendicular of weights G and EF that of weight D. But since D is to G, the perpendicular BC to the perpendicular EF, the weights D and G are balanced [130] .5 (A.7.3)

Note, however, that Herigone’s use of the law of virtual work is different from that of Jordanus in at least two respects. The first because instead of assuming the equivalence of rising p to h and p/n to hn, he assumes the equivalence of rising p to h and lowering p/n to hn. The second because he derives from this result the equilibrium and not just an equivalence. This allows Herigone to arrive directly at its result without recourse to an indirect (absurd) reasoning.

4 5

p. 290. pp. 301–302.

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7.1.1 Gille Personne de Roberval Gille Personne de Roberval was born in Roberval, near Senlis, probably in 1602 and died in Paris in 1675. His name was originally Gilles Personne, that of Roberval, by which he is known, comes from the place of his birth. Roberval was one of those mathematicians who occupied their attention with problems only soluble by methods of infinitesimals. Since only few writings were published by Roberval during his life he was for long eclipsed by Fermat, Pascal, and, above all, by Descartes his irreconcilable adversary. Serious research on Roberval dates from approximately the end of the XIX century, and many of his writings still remain unpublished. In 1666 Roberval was one of the charter members of the Académie des Sciences in Paris [290]. The best-known contribution of Roberval to statics is his Traité de méchanique of 1636 [209]. The treatise was entered by Mersenne in his Harmonie universelle [170] and Duhem believes there existed an even more extensive edition in Latin [305].6 Roberval at the beginning of his treatise cites only Archimedes, dal Monte and Luca Valerio. Nevertheless, he knew for sure Stevin’s work very well, and his treatise is almost a completion of Stevin’s, needed to remove some imperfections, such as that in the proof of the parallelogram of forces law. And of course there is the influence of Galileo, at least in the demonstration of the law of the inclined plane. One thing quite interesting and perhaps new is the assimilation of the weights with the muscle forces (powers), which is more evident than in previous authors, Stevin included. Roberval’s treatise is very formal in some places, for the author will not want to leave any doubt on the validity of the proof. If this pedantry can be criticized, it should be said that Roberval reaches his goal. 7.1.1.1 The inclined plane law The proof of the law of the inclined plane follows the same line of thought pursued by Galileo, and is substantially equivalent to it when a force parallel to the plane is considered. The key point of the demonstration of Roberval is the substitution of the constraint offered by the inclined plane with the arm ac of a lever with fulcrum c, as shown in Fig. 7.3. Unlike Galileo, the substitution is made by Roberval not so much because for small displacements the two types of constraint (plane and lever) are equivalent, but rather because the inclined plane is able to provide support along the direction ca equivalent to that of the lever. The motivation of this equivalence, which culminated in his axiom IV [209],7 is perhaps the least clear part of Roberval’s treatise. The determination of the force acting in directions not parallel to the inclined plane, which was given without proof by Stevin, begins with a similar argument. 6 7

vol. 1, pp. 322–323. p. 6.

7.1 French statics

e

b

h

g

c

o

1

f x

f 2

k z

161

a d l

u y

r s

p q

n m

Fig. 7.3. Inclined plane with force parallel to it

Fig. 7.4. A multiface figure

The inclined plane is still being replaced by a lever, only now the powers are not both orthogonal to the arms. Fig. 7.4 illustrates the situation; note that Roberval concentrated in this illustration more figures, including the case of a body lifted by two ropes presented in the subsequent section. 7.1.1.2 The rule of the parallelogram Roberval presents two proofs of the rule of the parallelogram of forces. The first exploits the law of the inclined plane, or at least brings everything back to the lever, the second uses a law of virtual work. The problem is reduced to determine the forces

162

7 European statics during the XVI and XVII centuries

C

F

Q

E B K

2

A N

L Fig. 7.5. The rules of the parallelogram and the lever

of two ropes inclined at any given angle supporting a weight A. Fig. 7.5, extracted from Fig. 7.4, illustrates the situation for the first proof. The reasoning is developed (not completely without error) by imagining one of the two ropes that support the weight, for example the AC, as fixed at one end. According to Roberval it may be replaced either by the rigid arm AC of a lever or the inclined plane LN2: Now for the second proposition [that of the inclined plane] we have seen that if CA is the arm of a balance on which the weight A is retained by the rope AC so it does not slide along the arm AC, and as CB is to CF, the weight A is to the power Q or E pulling with the rope QA, the power Q and E will hold the balance CA in equilibrium, and the rope QA being attached to the center of the weight A, the balance remains unloaded. The weight A will be supported partly by the power Q or E, partly by the plane LN2 perpendicular to the balance AC, or by the rope CA, by the fourth axiom of the Scholium [209].8 (A.7.4)

In this way the determination of the force of the rope AQ is reduced to a known case, the lifting of a weight on a plane with a force of given direction. The same can be made for the rope AC. The second Roberval’s proof requires the use of a law of virtual work. It resembles that of the angled lever of Jordanus de Nemore being based on the impossibility that the sum of the product of the weights that go up by the values of the ascent is different from the sum of the weights that go down by the values of the descent. Scholium VIII. [...] the weight is located in A on the ropes CA and QA sustained by the powers C and Q or K and E; with the weight that is to the powers as the perpendiculars QG and CB are to the lines CF and QD [i.e. P : K = QG : CF; P : E = CB : CQ]. […] If below the weight A, in its line of action, one considers the line AP, if the weight A descends to P, dragging the ropes and making the powers K and E to rise, the ratio of the path the powers will make in raising to that made by the weight in descending will be greater than 8

p. 22.

7.1 French statics

163

Q

D C F

E G K

V A

B

P Fig. 7.6. The rule of the parallelogram and virtual work the ratio of the weight to the two powers considered together. So the powers will raise more in proportion than the weight will descend, what is against the common order. If above the weight A, in its line of action, one considers the line AV, and the weight rises until V, the ropes rise because of the powers K and E that descend. It will be a greater ratio of the path the weight will make in raising to the path that the powers will make in descending than the ratio of the two powers considered together to the weight. So the weight will rise more than the powers will descend; and this is also against the common order. Now that the ratio of the weight A and the powers in rising or descending are such as we said, and against the common order, is proved in our Mechaniques,9 because it is too long to be reported here. Concluding as the weight A will remain in its place, for the reasons of the 3rd proposition, all goes according to the natural order. What is wanted to remark [209].10 (A.7.5)

In essence Roberval considers a weight A balanced through two ropes by the two weights K and E, inversely proportional respectively to GQ and CB of Fig. 7.6. He proves the equilibrium by a reduction to the absurd. Suppose there is not equilibrium, for example the weight P falls and K and E rise. Roberval said, without reporting the details, that the sum of the products – a modern interpretation – of the weights E and K for their ascent is greater than the product of the weight A for its descent. But this is impossible, then there cannot be motion. Hence the absurd. In previous demonstrations Roberval gave two different criteria for determining the tension in the ropes, but he did not explicitly set out the rule of the parallelogram of forces. He does this explicitly in a Scholium, by affirming: “If for any point made in the line of the direction of the weight, the line parallel to one of the strings and to the other are drawn, the side of the triangle thus formed will be homologous to the weight and the two powers” [209].11

9

It seems there exists another treatise where the geometrical conclusions on the variation of ropes by Roberval are proved in detail. 10 pp. 35–36. 11 p. 28.

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7.1.2 René Descartes René Descartes was born in La Haye en Touraine (now Descartes) in 1596 and died in Stockholm in 1650. Much modern Western philosophy is a response to his writings. Descartes was the greatest mathematicians of the first half of the XVII century and one of the founders of analytic geometry, the bridge between algebra and geometry. A most important writing for science is his Principia philosophiae [96]. Here he depicted his view of the world, where matter has only a geometrical nature and the whole amount of motion, the force, is conserved forever. I.e. he considered the philosophy of nature as coinciding with mechanics. The ideas of Descartes on statics are contained in three letters, two of which are quite long and with titles. The first is a letter dated October 5th 1637 to Constantin Huygens (father of Christiaan), entitled Explication des engins par l’ayde desquels on peut, avec une petite force, lever un fardeau fort pesant [96], the two other letters are written to Mersenne in July and September 1638 [96], the first, July 13th, entitled Examen si un corps pese plus ou moins, estant proche du centre de la terre qu’en estant esloigné, is the most interesting. 7.1.2.1 The concept of force The contribution of Descartes to the development of laws of virtual work consists mainly of a framing of the problem; still important are even some of his more strictly technical considerations. He was the first to give a mechanical sense to the product of the weight for the vertical displacement. This coincides essentially with the modern work, which he calls action, or more frequently still force, with a little unhappy term because Descartes also calls force the muscular effort, the power and in dynamics, the absolute value of the quantity of motion. He repeatedly says that it takes the same ‘force’ to lift a weight to a certain height, that to raise a double weight to half height. For instance he writes to Mersenne in July 1638: It needs neither more nor less force to lift a weight to a certain height than to raise a lower weight to a height as greater as the weight is less heavy, or to lift a heavier weight to a less height. […] This will be given easily, if it is considered that the effect must always be proportionate to the action that is necessary to produce it, and thus if it is necessary to use the force by which one can raise a weight of 100 pounds to the height of two feet, to lift one at a height of one foot only, it means that it weighs 200 pounds [96].12 (A.7.6)

And also: Above all it must be noted that I have spoken of the force that is used to lift a weight at some height, force that always has two dimensions, and not the force used to hold the weight at any point, which always has only one dimension. These two forces differ from one another 12

vol. 2, pp. 228–229.

7.1 French statics

165

ds

f

ds p

Fig. 7.7. Different ways to see the force of Descartes

as a surface differs from a line. In fact the same force a nail needs to support 100 pounds for a moment of time is sufficient, when it does not lessen, to support them for a whole year. But the same quantity of that force used to lift that weight up to a foot is not enough to lift it to the height of two feet, which is not less obvious than two plus two makes four, be it is clear it would need a double force [96].13 (A.7.7)

Descartes does not define his force in an algebraic way, explicitly as the product of weight for shifting, but rather in a geometrical way, as the area of a rectangle. The force which Descartes talks about concerns weight; with a modern language it is the work made to raise a weight and its value is measured by the product of the weight and the space covered. It is probably not far from the ideas of Descartes – not explicit in this regard – to represent the ‘force’ as the product of a muscle force by the motion of its point of application, which can be in any direction. For example, with reference to Fig. 7.7, the force is given by the rectangle p · ds but also from f · ds. Descartes clarifies his concept of force by adding that the equality of the work of the ‘forces’ can only be accomplished with the use of machines that transform rectangles of equal area in different forms: Because I did not simply say that if the force can lift a weight of 50 pounds a height of 4 feet, it shall be able to raise 200 pounds a height of one foot, but I said that it could be when it is applied. Now it is impossible to apply [this force], but by some other machine or invention that makes this weight [200 pounds] to rise one foot, while the force will act along the length of four feet, and transforms the rectangle by which the force required to lift that weight of 200 pounds to the height of one foot into another that is equal and similar to the one that represents the force required to lift a weight of 50 pounds to a height of 4 feet [96].14 (A.7.8)

In substance Descartes says a man cannot raise indifferently a weight of 200 pounds and one of 50 pounds, because probably he cannot exercise the muscular force necessary to raise the greater one, he can however choose opportune machines that can perform this operation. The above passage is followed by an application to the case of the inclined plane. 13 14

vol. 2, p. 353. vol. 2, p. 357.

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7 European statics during the XVI and XVII centuries P

H

C

L A

N

O

F

D

G

E A

D

D L

C

B

Fig. 7.8. The conservation of force on an inclined plane

Consider the inclined plane ABC of Fig. 7.8 with AB = 2 AC. The two-dimensional force to lift D along AB is represented by the rectangle FGH, with GH = AB. Consider then the weight L required to lift D along AB. The force in two dimensions to raise L to the height of AC is equal to that FGH required to lift D along AB. The force needed to lift the weight D without the intermediary of the plane AD is still equal to FGH but one of its dimensions, AC, is half of AB then the other dimension, which represents the force required to lift the weight will be double. Hence the weight L is half of the weight D. Descartes believes that the rule laid down by him, namely that the force needed to raise p to h is the same as that required to raise p/2 to 2h should be considered the only principle of statics. Principle because it can explain the operation of all machines. Principle because evident since one cannot challenge the simple consideration according to which: It is the same to lift 100 pounds to the height of one foot, and again another 100 to the height of one foot, as to raise 200 to the height of one foot, and the same also to raise one hundred to two feet [96].15 (A.7.9)

In fact, the justification above, coinciding with that of Galileo referred to in § 5.6.2, though ingenious, does not withstand critical analysis as noted by Mach [355].16 Indeed, the admission that to lift 100 pounds in two stages is equivalent to 200 in one, although intuitive, is not logically deductible and it is not contradictory to imagine that it is not true. Descartes believes that his principle gives a causal explanation, i.e. that it allows one to understand the why. It is natural to ask whether, given the importance of this principle, Descartes does not count it among the laws of nature found in the Principia philosophiae of 1644 written after the letters to Huygens and Mersenne. According to Sophie Roux [211] this is because the law is formulated by involving the weight and not the categories of matter and motion, the only ones that can give rise to clear and distinct concepts. 15 16

vol. 2, pp. 228–229. p. 84.

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It must be said that Descartes is not always consistent with his statements. In the applications presented, in particular the law of the pulley, but also the law of the lever, he does not use his principle as such, but rather recognizes equilibrium by other means. From this point of view, that of Galileo who reduced actually and clearly all the machines to a single principle, the lever, was a clearer approach, which also had causal value. The unifying approach of Descartes is considered attractive by modern historians of science because, being of algebraic nature, one has to apply always the same formula, and it is easier to use in intricate situations than that of geometric nature of Galileo, which can take a significant technical skill and imagination. 7.1.2.2 The application to simple machines The letter of 1637 to Constantin Huygens looks like a small treatise on mechanics in which all the simple machines are analyzed. Here Descartes refers to forces of gravity – or lines of descent – parallel to each other, though he admits that this is only an approximation. The treatise opens with a passage similar to that reported in the letter to Mersenne: The invention of all those engines is based on one single principle, that if the same force that can lift a weight, for example of 100 pounds to a height of two feet, it can also raise a weight of 200 pounds to one foot, or a weight of 400 pounds to the height of 1/2 foot, and others. […] Now the engines used to make this application of a force acting on a large space to a weight that it raises with a minor space, are the pulley, the inclined plane, the wedge, the wheel with the shaft, the screw and some more [96].17 (A.7.10)

The explanation of simple machines follows: The pulley. Let ABC be a rope passing around the pulley D, in which the weight E is applied and suppose first that two men support or raise equally each of the two ends of the rope, it is clear that if the weight weighs 200 pounds, each of the two men take, to support or lift, the force required to support 100 pounds, because each holds only one half. Let then that A, one end of the rope, being attached to a nail, the other C is still supported by a man, it is clear A C

C H

A D

K D

B E

Fig. 7.9. The pulley

17

vol. 2, pp. 435–436.

B E

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that this man in C would need, as before, only the force required to support 100 pounds, because the support is to do the same service of the man who was supposed before. Finally, suppose that this man in C pulls the rope to lift the weight E; it is clear that if he uses the force required to raise 100 pounds to the height of two feet, he will raise this weight which weighs 200 pounds to the height of one foot. This because the rope ABC being wrapped as it is, it must be pulled by two feet from the head C to raise the weight E as two men pulled it, one from the head A the other from the head B, each the length of a foot only. There is however something that prevents this calculation being correct, such as the weight of the pulley and the difficulty to make the rope to slide and hold it. But, this will be negligible compared to what it gets [96].18 (A.7.11)

It is worth noting that Descartes proves the law of the pulley directly with simple considerations of equilibrium, the same of Hero, and then verifies that it complies with the law of virtual work, contrary to the declared intentions of wanting to take a single law of statics. In his letter to Mersenne of September 1638, there is an interesting comment on the result for the pulley, which clarifies even further Descartes’s concept of ‘force’: So to not deny that the nail A [Fig. 7.10a] supports half the weight of B, one can only conclude that by this application [of the pulley], one of the dimensions of the force, that must be in C to lift that weight, is one half, the other therefore double. Thus, if the line FG [Fig. 7.10b] represents the force required to hold the weight B at some point without the help of any machine, and the rectangle GH which is enough to lift the height of a foot, the support of the nail A reduces to one half the dimension represented by the line FG, and doubling the rope BDC it doubles the other dimension represented by the line FH, so the force which must be in C to lift the weight E to the height of a foot, is represented by the rectangle IK [see Fig. 7.10b]. And since it is known in geometry that a line added or removed to an area neither enlarges nor diminishes, here you will observe that the force with which the nail in A supports the weight B, having only a single dimension cannot ensure that the force in C, seen in its two dimensions, should be less greater to lift the weight in this manner than to lift it without the pulley [96].19 (A.7.12)

D

C

K

E H

A

a)

b) B

Fig. 7.10. Two dimensional force

18 19

vol. 2, pp. 437–348. vol. 2, p. 356.

F

I

G

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In the following I will report only the demonstration of the law of the lever. That of the inclined plane is the same as that already shown in the letter to Mersenne but more concise and the demonstrations of the laws of the wedge, the screw and the axis and the wheel do not present elements of particular interest. Different is the situation of the lever for which Descartes says, not without surprise to a modern reader, that it is the case more complicated to prove. The lever considered by Descartes is that of Fig. 7.11. The weight is applied at the end H, the force at the end C. It is OC = 3 OH. When the lever is in the position GB, Descartes admits with Galileo, that the constraint of the lever is equivalent in G with the inclined plane GM tangent in G to the circle KHF. With the law of the inclined plane Descartes can determine the apparent gravity or relative gravity, as opposed to the absolute gravity of a body free from constraints, which acts perpendicular to the lever and then parallel to power in B. At this point, Descartes takes for granted the law of the lever with powers perpendicular to it and determines the power in B saying that it is equal to one third of the relative gravity of the weight. From the arguments of Descartes it is clear that the difficulty lies not in the law of the lever in itself, which is presupposed, but in the fact that at one end of the lever the power acts perpendicularly to it while at the other end the weight acts vertically, and that the efficacy of this weight depends on the inclination of the lever. Lever. And to accurately measure this force which must be at each point of the curved line ABCDE, it is known that it works the same way as if the weight moved on an inclined circular plane, and that the slope in each of the points of this circular plane is to be measured by that of the straight line that touches the circle at this point. Such as when the force is at point B, to find the proportion that the heaviness of the weight that is at that moment in G must have, it must draw the tangent GM and consider that the heaviness of this weight is proportional to the force required to drag it on this plane, and thus to rise it according to the circular arc FGH, as the line GM is to the line MS. Then, since BO is three times OG, it is sufficient that the force in B is to this weight in G as the third part of the line SM is to the whole GM. The same is true when the force is at point D [96].20 (A.7.13) A B K O

C

S

F M D E

Fig. 7.11. Descartes’ lever with a force and a weight

20

vol. 2, p. 445.

I N

P

H G

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Descartes, therefore, studies, or at least considers, the lever starting from the inclined plane, with a procedure completely opposite from that taken by Galileo. This is largely due to the fact that because of his concept of work, the inclined plane, which consider changes in height, naturally lends itself to play a paradigmatic role. 7.1.2.3 The refusal of virtual velocities In the above analysis on the simple machines it seems to have more to do with the motion than with the equilibrium. Descartes’ argument, however, is contrary to the use of virtual velocities, and it is not suitable for him to establish a principle of statics, because the velocity (real) depends on many factors, such as the resistance of the medium, the velocity of application of the force and so on: Many people regularly confused the consideration of the space with that of time, or speed, so that, for example, in the lever, or equally, in the balance ABCD [see Fig. 7.12a], having assumed that the arm AB is twice as long as BC, and that the weight in C is twice the weight in A, and that they are in equilibrium, instead of saying that the cause of this equilibrium is that because the weight C lifts or is lifted by the weight A, it will not go for half the space of it [the weight A] they say that it moves half more slowly, and this is a mistake, among the most insidious to be recognized, because it is not the difference in speed that determines the weight to be in equilibrium, but the difference of displacements [96].21 (A.7.14)

C

H

B D G a)

b)

F

A Fig. 7.12. The effect of velocity

For Descartes it is not the difference of velocity which determines that one of the two weights must be double the other, but the difference of displacement: As it is shown, for example, that to lift the weight F with your hand to G [Fig. 7.12b], it is not necessary, if one wants to lift it with twice the speed, to use a force that is exactly twice that otherwise required. It is required instead a force that is more or less double, according to the variable ratio that can have the speed to the factors that will resist, but to raise it at the same speed of twice the height, up to H, a force is needed that is exactly twice, I say that is exactly twice, just as one plus one makes two: in fact a certain amount of that force must be used to lift the weight from F to G and then again the same amount to raise it up from G to H [96].22 (A.7.15) And to the contrary, take a fan in your hands, you can lift it with the same speed with which it could descend by itself in the air when you leave it to fall, without any effort except that 21 22

vol. 2, pp. 353–354. vol. 2, p. 354.

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necessary to sustain it. But for lifting and lowering two times more fast it will be necessary that you employ a force greater than double the other, unless it be zero [96].23 (A.7.16)

He continues to justify his position, without being very convincing, saying that he knows very well that the ratio of velocities can be equal to that of displacements, but this is not enough to accredit a principle of statics: Now the reason because I am criticizing those who use speed to explain the force of the lever and the like, is not that I deny that the same proportion of speed does always occur, but because the speed does not explain why the force increases or decreases, as does the amount of space [motion], and that there are many other things to consider for the speeds that have not been explained [96].24 (A.7.17)

The criticism of Descartes, with considerations similar to those of Stevin, is directed also against the traditional formulations of the laws of virtual work, in which it is conceptually irrelevant to consider virtual velocities or virtual displacements, because both are hypothetical and not real. This fact was perfectly clear to Galileo, who for his ideas was the subject of an attack by Descartes, shown below, ungenerous toward a man who for sure had understood the problem of equilibrium better than him: What Galileo wrote about the balance and lever [in Le mecaniche], he explains how but not why, as I do with my principle. And for those who say that we must consider the speed, as Galileo, instead of the space to explain the machines, I believe, between ourselves, they are people who speak only by fantasy, without knowing anything about this subject [96].25 (A.7.18)

7.1.2.4 Displacements at the very beginning of motion An important contribution, of technical character, offered by Descartes to laws of virtual work is surely to have guessed, but not fully developed, the concept of its infinitesimal character in the case of bodies constrained to move on a curved path. And paradoxically, this character makes it easier to talk of virtual velocities, the use of which Descartes opposed, than of virtual displacements. Indeed velocities in a constrained motion can always be really possible, while displacements generally only approximate the real motion, the smaller they are. In the letter to Mersenne of July 1638, Descartes modifies in part the concept of gravity accepted in the letter to Huygens. There the lines of action of weights were considered as parallel, here converging. This choice complicates things unnecessarily – at least in the eyes of a modern scholar – however, it leads Descartes to say that the relative gravity of a body is measured by reference to the motion in its birth and it can vary along the same inclined plane. The relative weight of each body, or that is the same thing, the force that must be used to support it and prevent it from descending when it is in a certain position, shall be measured from the beginning of the motion that the force that sustains [the weight] should make, 23 24 25

vol. 3, p. 614. vol. 3, p. 614. vol. 2, p. 433.

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either to raise or to follow if it sinks. The proportion that exists between the straight line [the tangent] describing the motion and that which defines the approach of the body to the centre of the earth is the same as that which exists between the absolute and relative weight [96].26 (A.7.19)

In his analysis of the descent of a body on an inclined plane Descartes first argues that the relative gravity varies along an inclined plane as the angle between the directions of force of gravity and motion. In the generic point D of Fig. 7.13 the ratio between the relative and absolute gravity is given by the ratio between the sides FN and NP. A

H F E

D N

B

P

G

C

K

M Fig. 7.13. Motion over a curve path Let AC be an inclined plane on the horizon BC and AB tend directly to the centre of the earth. Those who write about mechanics shall ensure that the heaviness of the weight F, when it is on the plane AC, has the same proportion with its absolute weight as the line AB to the line AC. […] Which is not entirely true, however, except when it is assumed that the heavy bodies tend downward along parallel lines, an assumption that is commonly made when Mechanics is considered to be useful, since the little difference that can cause the inclination of these lines, that tend toward the centre of the earth, is not sensitive. […] And to know how much it weighs in every one of the other points of the plane with regard to this power, for example at point D, we must draw a straight line, as DN, toward the centre of the earth, and from the point N, arbitrarily assumed on this line, draw NP perpendicular to DN, which meets AC in P. As DN is to DP, so the relative gravity of the weight F in D is at its absolute gravity [96].27 (A.7.20)

Descartes then considers what would happen if one admitted that the weight was falling not on an inclined plane but on the curved surface EDG: Note that I say, begin to descend, not just descend, because it is at the beginning of the descent to which it is necessary to refer. So if, for example, the weight F is not supported at 26 27

vol. 2, p. 229. vol. 2, pp. 232–233.

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the point D on an inclined plane, as ADC is supposed, but on a spherical surface, or curve in any other manner, such as EGD, provided that the flat surface, which is imagined tangent at point D, it is the same as ADC, it will not weigh any more or less, for the power H, which must be applied to the plane AC. Because, although the motion that would make this body go up or down from point D to E or to G on the surface EDG, would be completely different than it would be on the flat surface ADC, however, being in the point D of EDG, it will be forced to move as if were on ADC, toward A or C. It is evident that the change of position resulting in the motion, when the body has ceased to touch the point D, cannot change the weight that it has when it touches it [96].28 (A.7.21)

What is reported above very clearly by Descartes can be repeated with a bit more modern language. With reference to Fig. 7.13 it can be seen that if the weight F is in equilibrium on the inclined plane AK, this equilibrium will not be upset if the plane turns, in the points where it is not in immediate contact with the weight, with the surface EDG, or any other surface. To check the equilibrium it is sufficient to consider infinitely small displacements which the smaller they are the more they are parallel to the tangent to the curved surface (in technical language, infinitesimal displacements). If one considers finite displacements the weight would be moving on a surface of different slope than the inclined plane in which he had found the equilibrium and this would no longer exist. 7.1.2.5 A possible precursor It has been said beforehand that Descartes could have borrowed his ideas on statics from Herigone. But it is also possible that there was no direct personal influence and that Descartes drew from formulations, more or less defined, of the laws of virtual work which were part of the background knowledge of the period. In this respect it seems interesting to refer to the little known work of Genevan Michel Varro. Not much is known of this author who is considered a minor scientist and there are few specific studies on him [363, 266]. Of his writings a treatise on mechanics (tractatulum) in Latin is known, entitled De motu tractatus [239], which studies equilibrium, making reference to a virtual work law based on speed. Varro’s treatise is very slim, less then fifty pages, and considers just general aspects. From this point of view it is quite different from the treatises of the period which almost all concentrated on explanation of the operation of simple machines, an approach that to some extent did not escape even Galileo and Stevin (to a lesser extent for the latter) with writings substantially contemporary to Varro. Varro says he was inspired by Archimedes in his mathematical approach to statics, not so much in the principles he uses but rather for the method. Like Archimedes, Varro argues that a treatise on mechanics should not have to deal with special cases but should report the general theory: It is for this that I think it is necessary to insist first in the theory, because what is applied could be considered without any difficulty. In some respects there is the danger that if we stop to deal with special cases, people are satisfied with these, and so it happens that the universal knowledge is neglected and the search for causes and science end [239].29 (A.7.22) 28 29

vol. 2, pp. 233–234. Preface.

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In a series of definitions concerning the nature of forces – not just weight and motion – Varro, despite his thinking being in part still inside Scholasticism with the use of concepts like natural motion etc., sets the stage for a quantitative study of the virtual work laws of equilibrium and reaches out to formulate his virtual work laws as ‘theorems’ of equilibrium, as the following for instance: Theorem I Two forces connected so that their motions will be inversely proportional do not move but are in equilibrium. [239].30 (A.7.23)

This theorem is enough for Varro to formulate a correct law of the inclined plane, although he gives the result no particular emphasis [239].31 Historians of mechanics [363],32 credit Varro with having given some contribution to the rule of composition of forces [239],33 however, the text of Varro is not so clear on the subject and I will not comment on the fact. As far as this chapter is concerned, it is interesting to note the following comment that Varro adds at the end of his treatise, which recalls the letters of Descartes to Constantin Huygens and Mersenne: To therefore conclude this small treatise, or to close in a summary, to produce the motion, three things must be considered: the force by which we want to make the motion, the force that we want to move, and the motion with which we want to move. Any two of them determine the third. Indeed if we want to move a large force with a small one, we can move it by a small motion, if on the contrary we want to move some force by a large motion, it requires a large driving force. For example, if we want to move 100 pounds with the aid of 1 pound, the motion must be by 100 times. If we want to use 1 pound to move another force so that it is driven 100 times faster than the weight of 1 pound, that force must be 100 times smaller. If we want to move 1 pound so that it is moved 100 times faster than the force that moves, it will need a force 100 times greater. Nature does not allow that in all these cases a new force arises. Indeed, if the proportion were by any means violated, there would be perpetual motion, or as it is named, perpetual motion in perpetual matter [239].34 (A.7.24)

Certainly there is a substantial difference in the use of speed, contrasted, instead of displacement, accepted, by Descartes, but the tone is the same and also the numerical values of weights referred to in the quotation are the same, so it would not be difficult for Descartes, supposing he had read Varro’s text, to translate it in his metaphysics. Among other things it is probable that Descartes had read the De motu tractatus. It was known in France as it is mentioned in Varignon Nouvelle mécanique ou statique [238]35 where some of Varro’s statements are flanked to Descartes’. It should also be noted that Descartes’ claim to use the law of virtual work as the only law of statics emerges also from the reading of Varro’s text, even if it is not specifically stated and there are no applications to the various cases. I will not insist on this reconstruction, 30 31 32 33 34 35

p. 19. pp. 35–37. p. 121. pp. 37–38. pp. 42–43. p. 321.

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which is certainly not fully founded, but certainly it offers matter to reflect on how the laws of virtual work were entrenched in mechanics of the late XVI century. Finally I would like to point out the introduction of the concept in an embryonic form of potential energy, which I think is one of the first in the history of mechanics. Towards the end of his work Varro says “that a lot of things can be raised beyond their natural place to use them when it is needed to produce motion”[239].36

7.1.3 Blaise Pascal Blaise Pascal was born in Clermont-Ferrand in 1623 and died in Paris in 1662. He was a mathematician, physicist, writer and Catholic philosopher. Pascal’s earliest work was in the natural and applied sciences where he made important contributions; he wrote a significant treatise on projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. Pascal also wrote in defense of the scientific method. About mechanics his most significant contribution is on hydrostatics where he made important studies on fluids and clarified the concepts of pressure and vacuum. In 1654, he had a ‘serious’ conversion, abandoned his scientific work, and devoted himself to philosophy and theology. Of this period are his most famous writings, the Lettres provinciales and the Pensées [354]. At the beginning of his Traité de l’equilibre de liqueurs [185]37 Pascal presented a new type of simple machine to add to the lever, the inclined plane and so on. To explain its functioning he assumed the virtual law according to which “the motion is increased with the same proportion as the force”: From which it appears that a reservoir of water is a new principle of mechanics and a new machine to multiply the forces to the degree one wants, so that a man by means of it could raise any weight he would. And it should be appreciated that in this new machine is found the constant order found in the ancient machines, i.e. the lever, the wheel with the shaft, the screw and so on, which is that the motion is increased with the same proportion of the force [emphasis added]. Because it can be seen when one of these holes [Fig. 7.14] is one hundredth the other, if the man who pushes the small piston by one inch, the other will move only one hundredth: because it depends on the incompressibility of the water which is common with the two pistons [185].38 (A.7.25)

To explain in detail the operation of the new machine, known today as the hydraulic press, Pascal curiously assumed another principle, that of Torricelli. He, however, did not mention Torricelli of whom almost certainly he knew the work, perhaps because he followed the fashion of the period to not give much evidence of the 36 37 38

p. 45. It seems however that Pascal’s treatise was composed in 1653. p. 183–184.

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sources. Perhaps because he considered now Torricelli’s principle as background knowledge. I take for granted that never does a body move for its weight if its centre of gravity does not descend. From it I prove the two pistons of the figure [Fig. 7.14] are in equilibrium and their common centre of gravity is a point which divides the line joining their individual centres of gravity, in the inverse ratio of their weights. Assume for absurdity that they move. Their motions will be inversely proportional to their weights as we assumed. Now if one takes the common centre of gravity in this second situation, it will be exactly at the same point as before, because it will be always at the point which divides the line joining the individual centres of gravity in the inverse ratio of their weights. So, because of the parallelism of the line of their displacements, it will be in intersection of the two lines which join their individual centres of gravity in the two situations. Then the two pistons considered as a unique body are moved without their common centre of gravity having moved down. This is against the principle; then they cannot move, they will be at rest, that is in equilibrium. As it was to prove [185].39 (A.7.26)

Fig. 7.14. The hydraulic press (reproduced with permission of Biblioteca e Archivio Accademia Nazionale delle Scienze, Torino)

Pascal above has considered a parallelepiped reservoir on which there are two circular holes, one great on the right and one smaller one on the left (see Fig. 7.14). After filling the container with fluid, a weight proportional to the area of the holes is applied to the two pistons. The proof of equilibrium is for reduction to the absurd. Suppose there is not equilibrium and a piston moves up and the other down. For the geometry of the pistons, for their weights and for the incompressibility of the fluid it is easy to infer that their common centre of gravity is not lowered. Then the motion cannot occur, hence the absurd.

7.1.4 Post Cartesians After Pascal and Descartes in France there were no longer written texts of value for statics. The only works of some importance were those of de Challes, Pardies, Rohault, Lamy. French statics emerged again with Pierre Varignon (1654–1722), to which I will refer in Chapter 8. 39

pp. 186–187.

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Claude François Milliet de Challes (1621?-1678) combined the talents of mathematician, teacher, and writer. His Cursus seu mundus mathematicus [91] is a remarkable and well-written course on mathematics and subjects such as optics, magnetism, mechanics, navigation, pyrotechnics, astronomy and music. De Challes was to incorporate the works of previous mathematicians into a coherent system and to explain the intricacies of the mathematical sciences with ease and accuracy. He assumed some form of virtual work law in his argumentations. Jacques Rohault (1620–1673) was a mechanistic Cartesian and experimental physicist. His Traité de physique [210] was a standard text for nearly fifty years. John Clarke and Samuel Clarke, rather than writing a Newtonian physics, translated Rohault’s work into Latin and English [70] and added Newtonian footnotes to correct Rohault’s mistakes. Ignaces Gaston Pardies (1636–1673) was a Jesuit. His collected mathematical works were published in French and in Latin. Of particular interest is his La statique ou la science de forces mouvantes [184], where there are contributions also to the strength of materials. Bernard Lamy (1608–1679) was professor of classics at the Jesuit Collège de César in Vendome. His major publication in statics was his Traité de mécanique, de l’équilibre des solides et des liqueurs in which the parallelogram of forces law is given [155]. Pierre Varignon discovered the parallelogram of forces law independently, at about the same time, and he saw more consequences of it than Lamy did.

7.2 Nederland statics In 1581, seven of the seventeen Low Countries refused to recognize Philip II as their king and originated the so-called Republic of the Seven Provinces, partially coinciding with the modern Nederland. Thus began a period of great political and religious changes and a large cultural and economic development; it usually is referred to as the Golden Century of Nederland. Great stimulus to the development of sciences in general and of mathematics and mechanics in particular came from commercial needs of the new state. The republic promoted the dissemination of scientific knowledge with the creation of new schools at the local level. Also higher studies were enhanced and the university of Leiden, founded in 1575, became a very important school. A special role for the development of mathematics was played by surveyors, who faced complex problems for the preparation of reliable nautical and land charts required for the trade policies of the new state. This fervor of scientific activity was rooted in an important cultural tradition. Just remember that, in the city of Deventer, Nicholas Cusanus (1401–1468) and Erasmus of Rotterdam (1466–1536) appeared on the scholary scene. Certainly a notable influence in the development of Dutch science was also due to the long stay of Descartes, started in 1617, with the fruitful collaboration of Isaac Beeckman (1588–1637). There is therefore no wonder that in this land, florid and tolerant, then as now, people with genes as unique as Simon

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Stevin and Christiaan Huygens were born, however, separated by a vast amount of time. In this section I illustrate with some detail only the contribution these two Dutch scientists made to the development of laws of virtual work; the contribution that albeit has involved only a small part of their production is nevertheless important [384].

7.2.1 Simon Stevin Simon Stevin was born in Bruges in 1548 and died in Leiden (or maybe in Den Haag) in 1620. He was for some years book-keeper in a business house at Antwerp; later he secured employment in the administration of the Franc of Bruges. In 1583 he entered the university of Leiden. From 1604 Stevin was an outstanding engineer who advised on building windmills, locks and ports. Author of many books, he made significant contributions to trigonometry, mechanics, architecture, musical theory, geography, fortification, and navigation. He introduced the use of decimals in mathematics in Europe [384]. Inspired by Archimedes, Stevin wrote important works on mechanics. His books De Beghinselen der Weegconst (Principles of the Art of Weighing) and De Beghinselen des Waterwichts (Principles of the Weight of Water), published in 1586, deal mainly with equilibrium. Although he undertook his mathematical work earlier in his life, Stevin collected together some of his mathematical writings and edited and published them during the years 1605 to 1608 in Wiskonstighe Ghedachtenissen (Mathematical Memoirs, in Latin Hypomnemata mathematica) [215, 216, 217, 218]. As a custom of the times he did not quote his predecessors, with the exception of Archimedes, Commandino and Cardano but in the last case only to criticize his (wrong) result for the inclined plane; for some comment on the matter see [346].40 Assessing Stevin’s contribution to the history of mechanics is not simple because his ideas were originally written in Dutch and then read by few. When they were translated into Latin (1605) and French (1636) the state of mechanics was already changed. He is indeed, in any case, the founder of statics in the modern sense. The name statics is in the title of his major work in mechanics Tomus quartus mathematicorum hypomnematum de statica, at least in the Latin version. And although he defines statics as the science of weights: Definition I. Statics41 its the science of the reasons, proportions, qualities and heaviness of heavy bodies [215].42 (A.7.27)

in fact he often introduces forces applied by ropes that can be tightened by weights or by human hands (muscle force). The Tomus quartus is divided into five books, plus an Appendix and some Additions to the Dutch edition of 1586. The approach is of Euclidean type, in the sense 40 41 42

p. 94. In the Dutch text instead of ‘statica’ there is written ‘art of weighing’ [218], p. 97. p. 5.

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that for every book there is a different topic; first there are definitions, then postulates and finally theorems, that are linked together. In the first part of the first book Stevin demonstrates the law of the lever, with an argument similar to that used by Galileo in Le mecaniche. Starting from a continuous prismatic body with geometric considerations in the wake of Archimedes he finds the law of inverse proportionality between weight and arm length. In the second part of the same book Stevin gives his famous demonstration of the law of the inclined plane, determining the value of the force parallel to the slope enough to maintain a heavy body in balance. Stevin extends his result to the case where the uplifting force is not parallel to the inclined plane. Gilles Personne de Roberval (see previous sections) found Stevin’s proof not satisfactory and gave a much more convincing proof; in a subsequent section I will discuss the legality of Stevin’s extension. Based on the law of the inclined plane generalized to a force of any direction, with a rather complex argument that is developed with many theorems and corollaries, Stevin puts the groundwork for the proof of the rule of the parallelogram of forces which is satisfactory if the generalized law for the inclined plane is accepted. The second book of the Tomus quartus regards the evaluation of the centres of gravity of plane and solid figures, and it is definitely less interesting. The third book is on practical statics in which lifting of bodies more complex than those treated in the first two books is considered. The fourth and fifth books are dedicated to hydrostatics. They are fundamental texts on the subject that however I do not comment on because they are not related to the subject of my work. The Appendix contains various comments, including perhaps the most interesting about the criticism of the principle of virtual velocities to be discussed below. In the Additions Stevin considers and devises demonstrations for pulleys, and treats with some generality the case of forces applied by means of ropes in a section called spartostatica. In this section statics has already became the science of equilibrium of force and no longer of weights. It contains the wording of the rule of the parallelogram which is a rule of composition of forces, even though it is presented as a way to determine the tension of two ropes which sustain a weight [215]. This change of attitude is a fundamental Stevin’s contribution to modern statics, and it does not matter if the rule of composition of forces is given an imperfect proof; it is however a rule which works. In the final part of the spartostatica Stevin considers for the first time fundamental arguments that can be conceived only in the new frame of reference, i.e. the funicular polygon of forces, the weight sustained by more than two ropes in the plane, and the non-coplanar ropes. The reading of Stevin’s mechanical work offers a much more modern view than that of Guidobaldo dal Monte (1577) [86] and Galileo (1594) [119]. The approach of Archimedean kind is equally rigorous, but less verbose. Unlike Galileo, Stevin does not bother to set up statics on a single principle, that of the lever. He uses the Archimedean geometric proof for the lever, but then he relies on the law of the inclined plane using an empirical principle, then in part still controversial, the impossibility of perpetual motion. Stevin among other things, is among the first to realize that the centre of gravity of a heavy body is not unique if one admits that the lines of descent of bodies are

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converging toward the centre of the world. He first shows that since the actions of gravity converge toward the centre of the world, the centre of gravity in the sense of Pappus and Commandino cannot exist for a body other than the sphere: From this it follows there is no body in Nature, speaking mathematically, other than the sphere which can be suspended from its centre of gravity and maintains any position. Or such that the plane passing from it splits the body in equally weighting parts. But for the various and infinite configurations there will be various and infinite centres of gravity [215].43 (A.7.28)

On the other hand, however, he argues that this is only a theoretic conclusion and in practice, because the actions of gravity differ by a very small angle, the centre of gravity determined with the Archimedean rules meets the demand of Pappus to be the point of suspension of neutral equilibrium. But this difference is not observable for the practice of men and the beam should be some miles long because it can be detected. So we postulate that the verticals be parallel each other [emphasis added] [215].44 (A.7.29)

L N

F

G

H K

M O

B A

E

C

D Fig. 7.15. The centre of gravity of a body

7.2.1.1 The rule of the parallelogram of forces Demonstration of the rule of the parallelogram for composition of forces was carried out by Stevin with a long series of theorems and corollaries (about twenty) that leave the modern reader a little upset . Also because the demonstration of each theorem and corollary is carried out with rather slender mathematical reasonings, very close to the modern sensibility, it is difficult to understand the reason for Stevin’s prolixity. A part of this difficulty might be overcome by assuming that Stevin’s objective originally was not to formulate the rule of composition of forces, of which 43 44

p. 11. p. 11.

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Plate 3. A Latin and a Dutch edition of Stevin’s books on mechanics (reproduced with permission, respectively, of Biblioteca Nazionale Centrale, Rome, and of Max Planck Institute for the History of Science, Berlin)

perhaps he did not understand the full extent, but only to make a series of comments on the way weights can be lifted. In fact, the explicit formulation of the rule of the parallelogram is in the section of the Additions named spartostatica. Below I refer with some detail to Stevin’s demonstration, although it in no way affects the laws of virtual work. This for two reasons: the first to illustrate the difficulties of the proof of the composition of the forces, a rule that was and still is alternative to laws of virtual work. The second reason is that normally Stevin’s demonstration is not reported so faithfully in the textbooks of history of science, perhaps because it is too complex. The starting point is the law of the inclined plane. For reasons that will appear clear later he refers to a prism that is moved along an inclined plane as shown in Fig. 7.16. In corollary V to the law of the inclined plane reported in the second half of the first book [216]45 , it is easy for Stevin to show that the ratio between the weight M of the prism, i.e. the force to lift it, called the direct uplifting force, and the force E needed to move it on the inclined plane, called the oblique uplifting force, is equal to the ratio of the segments LD and DI identified by the intersection of the ropes with the prism (because M : E = AB : BC = LD : DI).

45

p. 36.

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7 European statics during the XVI and XVII centuries

F H L I D O

M

E

B P

Q A

C

N

Fig. 7.16. Equilibrium of a prism over an inclined plane

In corollary VI Stevin considers a horizontal uplifting force measured by weight P in Fig. 7.16. Imagining a rotation of ninety degrees, the horizontal uplifting force becomes vertical and the plane ABC turns into a tilted plane whose slope is as NB of the triangle NCB. Following this rotation the ratio between direct and oblique uplifting forces is equal to that between the segments DO and DI. Stevin believes that this relationship is maintained even when the rope carrying the load P is effectively horizontal. At this point, he can say that in the vertical, in the inclined and in the horizontal directions, the values of the forces necessary to keep the prism in balance are proportional to the length of the segments DL, DI, DO, intercepted by the ropes on the prism, to conclude (improperly) that this fact applies to all directions. Stevin’s argument is interesting only for its strong rhetorical value, at least for the generalization to the case of any direction. The belief of the reader is made possible by the choice of a prism as the body to be lifted. It should be stressed however, that even if the reasoning cannot convince the result is correct. Below Stevin’s proof of corollary VI follows, to allow the reader to judge the lawfulness of the reasoning: Let BN be conducted cutting AC and extended to N, and the same DO cutting in O the extension of LI, so that the angle IDO is equal to the angle CBN, and then let the uplifting force P be applied along DO, taking the column in its position (with weights M and E balanced); then as LD is homologous to BA in the triangle BAC and DI with BC, it follows that BA is to BC as the weight on BA is to the weight on BC, by the second corollary. And also DL is to DI as the weight belonging to DL is to that to DI, i.e. M to E. Similarly the three lines LD, DI, DO being homologous to the three segments AB, BC, BN, then BA is to BN as the weights that belong to them, and also LD to DO will be like the weights that belong to them, i.e. M to P. Because this proportion is invalid not only at that elevation where DI is perpendicular to the axis, but for all sorts of angles [216].46 (A.7.30)

Stevin continues his argument with corollary VIII, which states that the relations found for the prism that moves on the inclined plane remain valid if the constraint 46

pp. 36–37.

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H L F M

D O G

P

Fig. 7.17. Prism supported by a fixed point

of the inclined plane is replaced with that provided by a fixed point, as shown in Fig. 7.17. Even in this case the relationship between the segments intercepted by the various ropes that support the cylinder is proportional to the forces necessary to balance the cylinder. In particular in the case of Fig. 7.17 the ratio between DL and DO is equal to the ratio of the direct and horizontal uplifting. Stevin does not pause to justify the lawfulness of the replacement of the inclined plane with the fixed point G. Reading between the lines it can be understood that, because for every inclination of the rope the intercept with the side of the prism provides the force necessary to maintain the equilibrium whichever is the inclination of the inclined plane, the inclined plane can be replaced with a constraint that performs its essential function, i.e. to offer a support to the prism. The result of Stevin, namely the determination of the force necessary to support the prism constrained to a fixed point, could have been extended quite easily to the case of a body of any shape to get a rule of equilibrium as efficient as the vanishing of the static moments. But Stevin does not do it. The next step, basically the definitive one, consists in the analysis of the situation of Fig. 7.18 for which Stevin states the following theorem XVIII: If a column is maintained in equilibrium by two oblique uplifting forces as the line of the oblique uplifting force is to the line of the direct uplifting force, so each oblique uplifting force is to its direct uplifting force [216].47 (A.7.31)

Notice that if points E and F have the same distance from the centre of gravity of the prism the vertical uplifting force I and K will be the same, so LE and FM have the ratio of G and H. From this theorem, of which I do not give a demonstration, it is very easy to arrive at the parallelogram rule. Stevin does this in the Additions. To get the rule of the parallelogram from theorem XVIII it suffices to consider the case where the two points E and F of Fig. 17.18 coincide with each other and with the centroid as shown in Fig. 7.19a. In this case it can be affirmed that the proportion between segments CI, DC, CE is the same as the direct, and inclined forces (corollary III of the Additions); but this is the rule of the parallelogram. The 47

p. 48.

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7 European statics during the XVI and XVII centuries

A L C

N

I

K

E

G

O F

M

H D B

Fig. 7.18. The prism sustained by two ropes

A

D D

A

I

I

E

H C

C

E

G a)

B

b) B

Fig. 7.19. The law of the parallelogram

proof is perfected by translating downward the prism as in Fig. 7.19b, which does not change the value of uplifting inclined forces (corollary 4) and finally replaced with a weight of any shape (corollary 5). 7.2.1.2 The law of virtual work On Stevin and the law of virtual work different opinions have been reported, some like those of Mach and Duhem make him one of the most modern supporters, others like Dijksterhuis deny this claim. Stevin’s considerations on the law of virtual work are listed in the Appendix and Addition of his Tomus quartus de mathematicorum hypomnematum de statica. In chapter I of the appendix, he writes: The cause of the equilibrium of the lever, as the chapter title says, does not lie in the arcs of a circle described by its ends. Common sense is enough to prove that equal weights suspended at equal distances are in equilibrium with a lever. But to say that different weights suspended at different distances are in equilibrium when these weights are in inverse proportion to the distances from which they are suspended, does not seem so obvious. The ancients thought that the reason for this was in the arcs described by the end of the lever. This view can be

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found in the Mechanica of Aristotle and the work of his followers. We will try the inaccuracy of this view as follows: A That what is in equilibrium does not describe a circle, E two weights in equilibrium are motionless, A so two weights in equilibrium do not describe a circle, So there is no circle. Once one deletes the circle, the cause that should reside in it also disappears. So the cause of the equilibrium cannot be found in the circle [215].48 (A.7.32)

Stevin uses, probably ironically, the syllogistic notation of medieval treatises on logic, directly derived from Aristotle. The letter A indicates the universal positive proposition (it is the first vowel of the latin verb adfirmo), the letter E indicates the universal negative proposition (it is the first vowel of the verb nego). Stevin’s reasoning is very natural, because the concept of virtual motion in a situation of equilibrium is far from intuitive. Essentially then Stevin denies that the laws of virtual velocities have explanatory value in mechanics. Indeed in his mechanical theory he does not make any use of them: a large part of Stevin’s mechanics is based on the theory of centres of gravity, another part on the theory of forces. Although Stevin declares his opposition to the principle of virtual velocities for which the equilibrium of a body depends on its possible motion, in at least one important situation he seems to contradict himself. In the proof of the law of the inclined plane Stevin considered a chain that wraps around it, as shown in Fig. 7.20. Stevin claims that the chain must be in balance in a given configuration otherwise, because the relative configuration of the chain cannot change, if it is not equilibrated in one configuration it is not equilibrated in any other configuration, then would occur perpetual motion, which is impossible: It is not possible that a given motion has not end [215].49 (A.7.33)

The law of the inclined plane was immediately followed by a comparison of weights of the chain that rely on the two opposing inclined planes (see Fig. 7.20).

Fig. 7.20. The chain of spheres on an inclined plane (reproduced with permission of Biblioteca Nazionale Centrale, Rome) 48 49

p. 151. p. 35.

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Notice that Stevin considers as unproblematic negating the perpetual motion, and does not assume it explicitly as a principle of statics though it is as fundamental for his mechanics at least as the law of the lever. The simple justification is that probably Stevin did not want his book to appear too new by introducing at the beginning a non-standard statement. Note also that the perpetual motion of which Stevin speaks, although he is not precise in this regard, is not the inertial motion, ideally possible, the physical perpetual motion of Leibniz [158],50 but a perpetual motion where one can always get work, the mechanical perpetual motion [158],51 actually impossible. So the reasoning of Stevin seems safe from such criticisms made against him by Dijksterhuis [292].52 Stevin returns to the subject in the Addition, where he speaks of Trochleostatica, and shows that for a simple system of pulleys as in Fig. 7.21 the power F is half the weight B because B is carried by two ropes: Proposition. To search the quality of lifting of weights with the pulley. Before starting to speak of the subject, we will say in general that when we speak of a given weight, we will assume a weight suspended from the lower pulley; regarding the weight of the rope we will neglect it. Examen of weight raised along a straight line: Let A in the first figure [Fig. 7.21] be a pulley, from which the weight B is suspended, and the rope CDEF, the part FE and CF of which are parallel and vertical. This posited, the weight B will be sustained equally by the two parts EF and CD, because the pulley acts equally on both. So if one would sustain the weight B with his hand in F, by keeping the weight in this position, he will sustain one half of B. From this it results that it is easier to lift a weight with a pulley than without it [215].53 (A.7.34)

Fig. 7.21. The pulley (reproduced with permission of Max Planck Institute for the History of Science, Berlin)

50 51 52 53

p. 472. p. 472. IV, p. 65. p. 171.

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And he comments: The following axiom of statics is valid: as the space of the agent to the space of the patient, so the power of patient to the power of agent [215].54 (A.7.35)

Here Stevin states unambiguously a law of virtual work. His remarks, according to Mach, are more mature than these purely geometric statements of Guidobaldo dal Monte, reported in the previous chapter [355].55 So to say that Stevin had a role in the development of the law of virtual work or to deny it, it’s just a matter of wording. The law of virtual work for Stevin is not a foundational principle of mechanics, but a theorem, surely relevant. Stevin with his consistency in considering only motion and not speed, will contribute to the development of the principle of virtual displacements, operationally equivalent but conceptually not to that of virtual velocities. In this way it is easier to separate the law of virtual work from Aristotelian dynamics, destined to be abandoned. Descartes and Wallis are the natural successors of Stevin in wanting to deny the dignity of an equilibrium law based on virtual velocity and to develop the approach of virtual displacements.

7.2.2 Christiaan Huygens Christiaan Huygens was born in Den Haag in 1629, and died in Den Haag in 1695. He generally wrote his name as Hugens, but I follow the usual custom in spelling it as Huygens. He was probably, with Newton, the greatest scientist of the XVII century. The most important of Huygens’s work was his Horologium oscillatorium published in 1673. The increasing intolerance of the Catholics led him to remain in Nederland where he devoted himself to the construction of lenses of enormous focal length: three of these of focal lengths were 123 feet, 180 feet, and 210 feet. In 1690 Huygens published his treatise on light in which the undulatory theory was expounded and explained [384]. It must be added that almost all his demonstrations, like those of Newton, are rigidly geometrical, and he would seem to have made no use of the differential or fluxional calculus, though he admitted the validity of the methods used therein. Huygens in his vast production also used some form of law of virtual work in statics. The originality and the importance of his contribution lies in the fact he introduced infinitesimal virtual displacements and forces. He gave his thoughts on some papers joined together today in his Oeuvres, in the chapter called Spartistatique [135].56 Is doubtful that the writings of Huygens were known by contemporaries, and then it may not have had any impact on developments of the law of virtual work, especially on Bernoulli. His considerations are in any case important, as they 54 55 56

pp. 171–172. p. 49. vol. 6.

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Fig. 7.22. Equilibrium of ropes (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

witness the level of maturation of the ideas on the law of virtual work at the end of the XVII century. The first Huygens’s application refers to three ropes converging together into a knot, illustrated in Fig. 7.22 on the left. He is looking for a relationship between the forces in the ropes in the state of equilibrium. The text of Huygens is laconic, just a note, in which Latin and French are mixed. To notice the adoption of an algebraic language see [135].57 Let the node be in E. And PE be shortened by DE QE by AE SE elongated by BE Then it is necessary DE in p + AE in q - BE in t = 0. (A.7.36)

In the final equation ‘in’ means multiplication, p, q, and t are the forces that pull the ropes P, Q and S, while DE, AE, BE are the respective variations of lengths. In modern terms, the equation expresses the vanishing of virtual work. The second application, more complex, refers to the equilibrium of the four ropes PRST shown in Fig. 7.22 on the right. Huygens had to determine the forces prst that pull the ropes, so that they are balanced. He considered separately four infinitesimal virtual displacements, each perpendicular to a rope. He then writes the virtual work depending on the other three ropes. In modern terms, his calculations are equivalent to the projection of three forces, one at a time on the straight lines orthogonal to the fourth [135].58 CA in P - CD in R -CO in S = 0 sin TEP in P – sin TER in R - sin TES in S sin PER in R – sin PES in S - sin PET in T sin RES in S – sin RET in T - sin REP in P sin SET in T + sin SEP in P - sin SER in R

57 58

vol. 19, p. 51. vol. 19, p. 52.

=0 =0 =0 =0

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on es + at f r − ct r= e d d p − f tt ap − br t= . s= e b p=

(A.7.37)

In the above equations ‘in’ again means multiplication, ‘sin’ is for sinus, the virtual displacement is simplified. Note that the forces p, r, s, t issued by the last four equations are not exactly univocally determined, but only the one over the others. In modern terms it can be said that the system is statically indeterminate and admits a simple infinity of solutions. Fig. 7.23 shows, for clarity, the case of virtual displacements orthogonal to the directions T and P, corresponding to the first two equations of virtual work, that with the symbols of the figure become: p sin α − r sin β − s sin γ = 0 r sin δ − s sin  − t sin α = 0. T P S s

γ

α E

p

T t

ε S s

P α

E

π−δ A2

β

a)

R

r b)

R

r

Fig. 7.23. Equilibrium of four ropes

7.3 British statics Also in England, as in France, the XVII century saw a revival of the sciences in general and the exact ones in particular. Not that there had been no great British scientists – William Harvey (1578–1657) and William Gilbert (1544–1603), to name just two – but these were sporadic cases. The real flowering of British science started from 1640, with the beginning of the Puritan Revolution, until the restoration of 1658 [395, 361, 338]. For Webster [395] reference should be made at least to 1626, the year in which the rise of the Puritan movement started, among other things, coinciding with the year of the death of Francis Bacon. Not all historians however agree in attributing a close connection between the Puritan movement and scientific development; some argue that the fact that British science developed during and after

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the Puritan Revolution is just a coincidence. But in fact, this coincidence exists and the conception of science and mathematics of the Puritan movement was certainly favourable to its development. The following list of scientists, to which should be added Isaac Newton, sufficiently show the impressive way British science grew in the second half of the XVII century: John Wallis (1616–1703), Robert Boyle (1627–1691), Isaac Barrow (1630–1677), Christopher Wren (1632–1723), Robert Hooke (1635–1703). With respect to statics the only direct contribution came from John Wallis who, from a certain point of view, can be seen as the last important representative of the ancient statics, where the lifting of weights was still an important aspect. Newton made a contribution mainly from a methodological, or philosophical, point of view by subordinating statics to dynamics.

7.3.1 John Wallis John Wallis was born in Ashford in 1616 and died in Oxford in 1703. He learned Latin, Greek, Hebrew, logic, and arithmetic during his early school years. In 1610 he received the degree of master of art and was ordained a priest; shortly afterward he exhibited his skill in mathematics by deciphering a number of cryptic messages from Royalist partisans that had fallen into the hands of the Parliamentarians. In 1645 Wallis moved to London. His appointment in 1649 as Savilian professor of geometry at the university of Oxford marked the beginning of intense mathematical activity that lasted almost uninterruptedly to his death. He also discovered methods of solving equations of degree four. Wallis contributed substantially to the origins of Calculus and was the most influential English mathematician before Newton. He studied the works of Kepler, Cavalieri, Cardano, Roberval, Torricelli and Descartes, and then introduced ideas of differential analysis going beyond these authors [290]. Wallis’s most famous work was Arithmetica infinitorum which he published in 1656. In 1670 he wrote an important treatise on mechanics, Mechanica sive de motu [245] where he made important contributions to mechanics in general and statics in particular. Wallis’ exposition is axiomatic deductive, based on definitions (very detailed), principles (not clearly marked) and theorems, as is typical of most mathematicians; the influence of Descartes is evident. The book is not easy to read, however it had a good success among scientists of the XVII century. Wallis considers gravity to be a force as the others, directed toward the centre of the earth. But he makes no comment on its causes, for which he proposes a few hypotheses: Gravity is the motive force, i.e. toward the centre of the Earth. Here we do not discuss what is the principle of the Gravity from a physical point of view, or which quality it has, or passion of the body, or with whatever name it could be called. Either it is innate in the body, or comes from the common tendency toward the centre of the Earth, or from an electric exhalation which attracts like chains, of from something else (of

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what it is not here the case to speak about). It is enough that with Gravity we intend what we know from senses. The force which moves downward either for the heavy body itself or for the less constraint toward the centre of the Earth. With weight I mean the measure of Gravity [245].59 (A.7.38)

One of the founding points of Wallis’ statics is the idea that the relationships of equilibrium occurring between weights also apply to ordinary forces, with appropriate adjustments. Every time he establishes a theorem or a definition for weights, he repeats it for ordinary forces: Prop. I Heavy bodies gravitate according to their weight. And in general, motive forces act according to the law of forces [245].60 (A.7.39) Prop. II Heavy bodies, unless constrained, descend, or get closer to the centre of the Earth. And in general any motive force, [moves] according to its direction, if there are not constraints [245].61 (A.7.40)

Wallis calls Descensus et Ascensus the virtual descent and ascent of heavy bodies; for forces he introduces the terms Progressus for displacement in the direction of the force and Regressus for displacement in the opposite direction: Prop. III For Heavy bodies Descensus is greater when [the body] becomes closer to the centre of Earth, Ascensus when it becomes farther. And in general, Progressus of the motive force is greater if [the body] moves according to its direction, and inversely for Regressus [245].62 (A.7.41)

But Descensus and Ascensus are not measured only by motion but also by weight, in the sense that they are proportional to them. And this holds also for Progressus and Regressus: Prop. V Descensus of Heavy bodies, compared among them, is proportional to the ratio of their weights and the value of descent. The same for the Ascensus. This is so if the weights are equal, [are proportional to] the ratio of the values of displacement, and if the displacements are equal to the ratio of weights. If weights and displacements are equal, or are in inverse proportion, [Ascensus or Descensus] are equivalent. And in general for the motive forces. Progressus and Regressus are proportional to the ratio of forces and to the regress and progress according to the line of action [245].63 (A.7.42)

With these definitions Wallis is able to introduce his law of virtual work, where the role of work is played by Ascensus, Descensus, Progressus and Regressus: a body is equilibrated if in a virtual motion the Ascensus (Regressus) and the Descensus (Progressus) are equal. 59 60 61 62 63

Chapter 1, pp. 3–4. Chapter 2, p. 33. Chapter 2, p. 33. Chapter 2, p. 34. Chapter 2, p. 37.

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In an aggregate of bodies Wallis’ virtual law says that there is equilibrium if the sum of Ascensus equals that of Descensus. Prop. VI Descensus and Ascensus of aggregates: if the Descensus is prevalent, simply it occurs a downward motion; otherwise if the Ascensus prevails an upward motion [occurs]. If they are equal, there is no motion. When many ascending or descending bodies are joined it is their summation which is relevant [245].64 (A.7.43)

The above is exemplified in some cases in which Wallis introduces the algebraic notation of Ascensus and Descensus, as the product of weight and displacement. With his symbols: For example […], compare the Descensus of a weight 2P for the displacement 3D, with the Descensus of a weight 3P for the displacement 2D: they are equivalent (because 2 × 3 = 3 × 2), in a virtual motion, so they are balanced. But the Descensus of a weight 2P for a displacement 4D is prevalent aver the Descensus of a weight 3P for a displacement 2D (because 2 × 4 > 3 × 2) it in a virtual motion; then it prevails [245].65 (A.7.44)

Basically Wallis generalizes the law of Descartes, or if one wants Torricelli’s principle, from the case of two weights to the case of n weights or n forces. Using an algebraic language, as Wallis does although in an embryonic form, for n forces fi and n motions ui , his results are summarized in the following relation: n

∑ fi ui = 0

(7.1)

i=1

where the signs are positive in the case of Progressus, negative in the case of Regressus. The above is valid for constant forces that move always parallel to themselves. For displacements along curved paths, however, Wallis suggests the solution already proposed by Descartes. One should consider motions in the direction of the tangent to the curve on which the heavy body moves. Prop. XV The slope of the descent of a curve line in a point is given by the tangent, and for a surface by the tangent plane [245].66 (A.7.45)

64 65 66

Chapter 2, p. 38. Chapter 2, p. 39. Chapter 2, p. 47.

7.3 British statics

193

7.3.2 Isaac Nevton Isaac Newton was born in Woolsthorpe, near Grantham in 1642 and died in London in 1727. Mathematician and physicist, one of the foremost scientific intellects of all time. Lucasian professor of mathematics in 1669 at the university of Cambridge [290]. His most famous book is the Philosophiae naturalis principia mathematica of 1687 [175]. Newton has been regarded for almost three hundred years as the founder of modern physical science, his achievements in experimental investigation being as innovative as those in mathematical research. With equal, if not greater, energy and originality he also plunged into chemistry, the early history of Western civilization, and theology. For Newton, statics is a special case of dynamics. And it is a trivial theorem of mathematical analysis to prove that a material point is in equilibrium when the resultant of its forces is zero. To check the balance it is therefore enough to dispose of a rule of composition of forces. This is provided by the rule of the parallelogram, which is a corollary to the second law of motion: Corollary I A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time thai it would describe the sides, by those forces apart [176].67

The study of equilibrium of a constrained material point is more complicated. Not covered in the Principia, it is customary to solve it with the assumption that the constraints exert reactive forces. One thus has the criteria of balance set out in Chapter 2 of this text It is clear even from these observations that Newtonian statics is quite different from that based on laws of virtual work. The differences are epistemological, ontological, mathematical. But there is a contact point, the idea that the equilibrium is a dynamic concept, a balance of tendencies contrary to the motion. To Newton the force expresses a tendency to motion; this trend, however, is not evaluated by observing motion, it is estimated before. To the contrary the virtual velocity, which is required in the application of the laws of virtual work, cannot be measured before the motion is imagined. Newton however was never disconnected from the idea that the tendency to motion and thus the force is measured by velocity, virtual or real. In the introductory part of the scholium to the Principia, Newton refers to the law of virtual velocities, presenting it as a special case of his third law of motion, I must say somewhat disconcerting to a modern reader: And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate force, so in the use of mechanic instruments those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces [176].68 67 68

p. 84. p. 93.

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A long passage follows in which Newton says more or less explicitly that forces and weights are more or less effective depending on their velocity. So there is a difference in the effectiveness and the force that Newton introduced in technical sense, as magnitude measured by the acceleration it imposes. To this efficacy one would be tempted to give the name of work, or better power (product of force for speed), but Newton does not it. So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upwards and downwards; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis of the balance; but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are reciprocally as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards. And in like manner in the pulley, or in a combination of pulleys, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. The force of the screw to press a body […]. The form by which the wedge presses or drives the two parts […]. The power and use of mechanics consist only in this, that by diminishing the velocity we may augment the force, and the contrary: from whence in all sorts of proper machines, we have the solution of this problem; To move a given weight with a given power, or with a given force to overcome any other given resistance. For if machines are so contrived that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant, but with a greater disparity of velocity will overcome it […] But to treat of mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law of motion [emphasis added]. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjunctly from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon the resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction [176].69

69

p. 94.

8 The principle of virtual velocities

Abstract. This chapter is almost entirely devoted to Johann Bernoulli, who considers the equilibrium for a set of infinitesimal displacements and forces. In the first part the different conceptions of the force of the XVIII century, including those of dead and living forces, are summarized. In the central part Bernoulli’s VWL is presented which enforces equality of positive and negative energies, the energy being defined as the scalar product of the force by the displacement of its application point, named virtual velocity. He offers a number of applications to the various cases including the fluid but does not provide any demonstration. In the final part a comparison of Varignon’s mechanics, based on composition of forces and Bernoulli’s mechanics based on his VWL is considered.

8.1 The concept of force in the XVIII century At the beginning of the XVIII century the concept of force had not yet a shared status. There was the static force measured by weight, there was the confusion concept of Newtonian force, there was the Cartesian concept associated to bodies in motion and the Leibnizian concept of living and dead forces [340, 304]. Before considering with some detail Bernoulli’s concept of force, the Newtonian and Leibnizian ones are presented.

8.1.1 Newtonian concept of force Newton assumed the following principles of mechanics which he referred to as laws, probably to emphasize that he considered them of experimental nature: Law I. Every body perseveres on its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. Law II. The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. Law III. To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts [176].1 1

p. 83.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_8, © Springer-Verlag Italia 2012

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These laws are quite familiar to a modern reader even though some particularity both formal and substantial does not escape, mainly for Law II. First there is nowhere the famous formula f = ma, commonly known as the second law of Newton, or better yet, no formula is referred to. Mass is not named explicitly but it is absorbed in [quantity of] motion; in the end no reference is made to acceleration. A scrutiny shows that also the impressed force, apparently the only familiar element in the second law, cannot be identified with the modern concept of force. Indeed, the integration of the law of motion, considered in modern sense as f = ma, over a finite interval T of time gives:  T 0

f dt = m

 T 0

adt = mΔv,

(8.1)

where the second part is the variation of the quantity of motion, or according to Newton’s terminology, the “alteration of motion”. Comparison of the analytical expression just obtained with the Law II of motion, shows that what Newton calls force  must be equal to 0T f dt. Newton chose the use of the word force to indicate a founding quantity of dynamics, but he did not reconnect it to any of the concepts today named in the same way. Newton’s force is likened to the whole force introduced by previous scientists such as Descartes or Torricelli. This concept today is scarcely used and anyway is not referred to with this name; the most common name for it is the impulse of the force f . In the scholium which follows the three laws of motion, Newton said verbatim about the force of gravity considered as an example of a force acting continuously: When a body is falling, the uniform force of its gravity acting equally, impresses, in equally particle of time, equal forces upon that body, and therefore generates equal velocity; and in the whole time impresses a whole velocity proportional to the time [176].2

That is the whole variation of velocity is proportional to the whole force, which is proportional to time. In the Principia the whole force can also represent the intensity of a pulse, and the action of continuum force, as the gravity for instance, is described usually as a sequence of pulses, divided by a constant time step Δt, which in the limit turns to zero. Regarding the ontology of force, Newton was quite ambiguous. He introduced force as a dominating concept together with that of absolute space and time, at the beginning of the Principia: Definition IV An impressed force is the action exerted upon a body to change its state, either of rest, or of moving uniformly forward in a right line This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires by its vis inertiae only. Impressed force are of different origins as from percussion, from pressure, from centripetal force [176].3

2 3

p. 89. p. 74.

8.1 The concept of force in the XVIII century

197

So the impressed force is what different kinds of ‘forces’ have in common from a mechanical point of view. Here it seems that Newton wanted to say that a force can be measured only by its effect.

8.1.2 Leibnizian concept of force Leibniz considered two types of force, the dead force and the living force, which are somehow related to our concept of force and kinetic energy respectively. These concepts cannot be understood without the comprehension of that of conatus which in Leibniz derived from Hobbes. The Hobbesian conatus is defined as the motion made in the shortest possible time and space; its use is associated not always to motion but also to the efficient cause of any change. Leibniz did not maintain a choerent position; at the beginning he assumed conatus, according to the theory of Cavalieri’s indivisibles, as the distance traveled in an indivisible element of time; thereafter the indivisible became an infinitesimal. In the Specimen dynamicum, he gave the following definitions: “Velocity taken together with direction is called conatus, while impetus is the product of the mass (moles) of a body and its velocity” [159].4 The sentence is consistent with previous formulations only if ‘velocity’ is the infinitesimal velocity dv. Already in a letter dated 1673, to Edme Mariotte, Leibniz used the terms force mort and force violent ou aimeé. In the Essay de dynamique he used the term vis viva, as opposed to vis mortua. The following passage of the Specimen dynamicum provides explanations in terms of non-quantitative relationship between dead and living forces: Hence force is twofold: the one elementary, which I call also dead, because motion (motus) does not yet exist in it, but only a solicitation to motion (solicitatio ad motum), such as that of the ball in the tube, or of the stone in the sling, even while it is held still by that chain; the other however, is the ordinary force, united with actual motion, which I call living. And an example of dead force indeed is the centrifugal force itself, and likewise the force of gravity or centripetal force, the force also by which the tense elastic body (elastrum) begins to restore itself. But in percussion, which arises from a heavy body falling already for some time, or from a similar cause, the force is living force, which has arisen from an infinite number of continued impressions of dead force [159].5 (A.8.1)

The relation between dead and living forces is commented on also in an important letter of Leibniz to Burchard de Volder (1643–1709) of 1699: Consequently, in the case of a heavy body receiving an increase of speed equal and infinitely small at every moment of its fall, the dead and the living force can be calculated at the same time. The speed increases uniformly with time but the absolute force as the square of the time, that is, as the effect. So according to the geometric  analogy or our analysis, the solicitations are as dx, the speed is as x, the forces as xx, or xdx [160].6 (A.8.2)

In this letter, in the definition of dead and living forces, the mass of the body was left in the shadow and according to a use of the times was regarded as a constant of 4 5 6

p. 237. p. 238. Translation in [161], p. 674. p. 156.

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proportionality. By making explicit mass, the previous quotation says that the dead force – actually Leibniz says solicitation – is proportional to the mass (m) multiplied by the infinitesimal speed (dv) and the living force is proportional to the mass multiplied by the square of speed (v), i.e. mv2 . Speed and force are linked by a simple  integration xdx, where time does not appear. There is some disagreement among historians about the relationship between dead and living force, simply because while living force is defined by Leibniz with a mathematical expression (mv2 ), dead force is not. Some argue that the living force is the integral of the dead force over an infinitesimal distance, for example René Dugas [308]7 believes that the relationship dead-living force expresses the theorem of living forces. A similar position was held by Ernst Cassirer. For them, where Leibniz seems to explicitly refer to integration in time, it would be an inaccurate language and, for example when talking about a heavy body which has fallen for some time, to simply report a qualitative description of the phenomenon. Other authors, including Westfall [396], argue that Leibniz has not grasped the true link between dead and living force. According to them, the natural Leibniz’s concept of variation would be with respect to time (the monads evolve over time) and then he should integrate the dead force overtime and this would simply give speed and not its square. What is certain is that Leibniz says in several places that the dead and the living force are in the same ratio as points to straight lines and then the dead force is infinitesimal and the living force finite. They are related by a simple integration or differentiation: The equilibrium consists of a simple effort (conatus) before the motion, and that is what I call a dead force that has the same relationship as respects the living force (which consists in the simple motion) as the point to the line. Now at the beginning of the descent, when the motion is small, the motion, the velocity or rather the elements of velocity are like the descents, instead after the integration, when the force has become alive, the descents are as the square of the velocity [158].8 (A.8.3)

Note that Leibniz does not say that the descent are as the speed, but that the elements of speeds are as the descents and then, at the rising of motion, the elementary displacements are proportional to the elementary speeds. I know of no other passages in which Leibniz presents the concept of dead and living force in a different way. In particular, there are no passages in which Leibniz ‘calculates’, or puts in relation with an explicit formula dead and living forces, or simply gave an analytical expression of the dead force. The concept of dead force should allow a direct link from statics to dynamics; of course the price is the acceptance of the metaphysical hypothesis of conservation of forces, according to which the dead force becomes the living force without any loss, a hypothesis that is not very different from that adopted by Newton that a cause produces its effect, by conserving in some way.

7 8

p. 211. p. 480.

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199

8.2 Johann Bernoulli mechanics Johann Bernoulli was born in Basel in 1667 and died in Basel in 1748. At Basel university Johann took courses in medicine but he also studied mathematics with his brother Jakob. Jakob was lecturing on experimental physics at the university when Johann entered the university; after two years of studying together, Johann became the equal of his brother in mathematical skill. Johann Bernoulli’s first publication was on the process of fermentation in 1690, apparently not a mathematical work, but it was really on an application of mathematics to medicine, being on muscular movement. In 1692 Bernoulli met Pierre Varignon, who later became his disciple and close friend. This tie also resulted in a voluminous correspondence. In 1693 Bernoulli began his exchange of letters with Leibniz, which was to grow into the most extensive correspondence ever conducted by the latter. In 1713 Bernoulli became involved in the Newton-Leibniz controversy on Calculus. He strongly supported Leibniz and added weight to the argument by showing the power of his calculus in solving certain problems which Newton had failed to solve with his methods. Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes’ vortex theory over Newton’s theory of gravitation. This in fact delayed acceptance of Newton’s physics on the Continent. Bernoulli also made important contributions to mechanics with his work on living forces, which, not surprisingly, was another topic on which mathematicians argued over for many years. Johann Bernoulli attained great fame in his lifetime. He was elected a fellow of the academies of Paris, Berlin, London, St. Petersburg and Bologna. He was known as the ‘Archimedes of his age’ and this is indeed inscribed on his tombstone [290].

8.2.1 Dead and living forces according to Bernoulli Johann Bernoulli in the Discours sur les loix de la communication du mouvement declared to have adhered to the Leibnizian concepts since 1714 [35],9 but it is for sure that his ideas had not matured at the time. As it will be clear from the following sections, in that period he reformulated the concept of dead force, replacing it with that of energy. Dead force identifies the force in the usual meaning, while the energy of a force f is an infinitesimal pulse defined explicitly by the relation f dx, where dx is the infinitesimal virtual displacement of the point of application of f in its direction. In static situations there is equilibrium when the energies of various forces are balanced; in dynamic situations the energies add up to give the living force. Bernoulli was the first to introduce an analytic relationship between living and dead forces (actually his energy). The following passage illustrates quite well the ideas of Bernoulli: 9 p. 40. More precisely he says twenty eight years after the publication of Leibniz’s famous Brevis demonstratio erroris memorabilis Cartesii [157].

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8 The principle of virtual velocities

N

B

D

O

K M

L

V

T

H P

D

J

A

C

G E

F

Fig. 8.1. Dead and living forces 1. I suppose two whatever given straight lines AC, BD, which I assume to represent two sets of small equal springs and equally compressed. I suppose also that two equal balls start to move from the points C and D toward F and J, when the springs begin to dilate. Let CML, DNK two curve lines, the ordinates GM and HN of them express the speed acquired at points G and H. I name BD = a , the abscissa DH = x, its differential HP, ou NT, = dx, the ordinate HN = v, its differential dv. I assume the abscissas CG and CE of the curve CML, such they are to the abscissas of the curve DNK as AC is to BD, or, which is the same, I make BD : AC = DH : CG = DP : CE. Supposing AC = na, it will be CG = nx, GE = ndx; let also GM = z. All this supposed I reason in this way. 2. When the balls arrive at points G and H, each spring, both that contained in the interval AC and the interval BD, will be equally extended, because AC : CG = BD : DH. Each of these springs will have lost, on both sides, an equal portion of elasticity and each spring will maintain the same elasticity. So the pressure or dead force [emphasis added] the balls have received are equal to each other. I name this pressure as p. But the elementary increasing of the speed in H, i.e. the differential TO or dv, is for the known law of acceleration, in a compound ratio of the motive force, or the pressure p, and of the little time the mobile takes to pass the differential HP, or dx, that can be expressed by HP : HN = dx  : v. It will be then dv = [pdt =]pdx : v, or vdv = pdx, which by integration gives 12 vv = pdx. For the same  reason it is dz = p× GE: GM =p × ndx :z, or zdz = npdx and by integration 12 zz = n pdx, from which it follows vv : zz = pdx : n pdx = 1 : n = a : na = BD : AC. But BD is to AC as the living force acquired in H is to the living force acquired in G. Then the two forces are to each other as vv to zz; so the living forces of bodies with equal mass are as the square of their speeds, and the speeds themselves are as the square root of the living forces [35].10 (A.8.4)

Bernoulli starts from the metaphysical assumption: as in the cause so in the effect, translating it in the statement that the action of the pulse of the dead force-energy becomes living force. Each pulse of dead force-energy has the expression pdx; their summation is the integral pdx which is shown to furnish the living force: 

pdx ∝ mv2 .

(8.2)

It should be noted that after Johann Bernoulli the previous relation was generally treated only as a mathematical theorem, and not as a principle of conservation. This is the case of Euler, d’Alembert and even perhaps of Lagrange. They, while somehow could give a physical meaning to the second member, because the idea of living force was certainly familiar to them, did not seem to know how to deal with the first member, to which neither a name nor a mechanical meaning was given. Especially 10

pp. 46–47.

8.2 Johann Bernoulli mechanics

201

for the scientists of the XVIII century it had not the meaning of mechanical work, understood as a physical quantity that can be converted into various forms of energy. Bernoulli’s idea of conservation of mechanical energy will be resumed with success only in the XIX century.

8.2.2 The rule of energies It is more or less universally acknowledged that the wording now used for the laws of virtual work has its core in the regle d’energie of Johann Bernoulli, better known after Lagrange as the principle of virtual velocities. Until a few years ago little was known about the origin of this principle. Even the date of its first statement was reported incorrectly. In fact, in the Nouvelle mécanique ou statique of 1725 by Pierre Varignon a letter dated 1717 is reported which sets out Bernoulli’s famous principle. There’s actually a misprint and the correct date of the letter is February 26th, 1715 [238].11 With the occasion of the new edition of the works of the Bernoullis [38] the correspondence of Johann Bernoulli, of which only a fraction of the letters were published, has been reconsidered. From it Patricia Radelet de Grave, one of the curators, made a more complete reconstruction. In what follows, I draw inspiration from the study of De Grave to reconstruct the development of the concepts of Bernoulli, not so much from the chronological point of view, but rather as an evolution of contents.12 All started with the publication of De la theorie de la manoeuvre des vaisseaux in 1689, by the naval engineer Bernard Renau d’Elizagaray [205] and from the criticisms about it by Christiaan Huygens. Johann Bernoulli joined in the discussion, initially taking the side of Renau, then that of Huygens. The debate between Bernoulli and Renau is embodied in some letters written in 1713, published in 1714 as addendum of the booklet Essay d’une nouvelle theorie de la manoeuvre des vaisseaux [33], in other letters to Renau and especially in some letters to Pierre Varignon in 1715, including the one above cited. In the absence of a shared concept of force, the debate between Bernoulli and Renau was heated and difficult to disentangle. In showing that Renau’s solution contradicts the fundamental principles of statics, especially the composition of forces, Bernoulli put the debate on a methodological level. Renau distinguished forces for which this principle is valid – for weights for instance – and forces for which it is not – the forces of wind for example. Bernoulli did not accept such a distinction; in a letter to Renau, November 9th 1713, he began to reflect on the nature of the force exerted by the wind, and concluded that it had nothing special compared to other forces acting in a continuous way, for example he mentions the magnetic force, but also the force of gravity. In this way the various problems involved in the theory of the vessels can be reconnected to statics:

11

vol. 2, pp. 175–176. Prof. Radelet De Grave sent me some typewritten Bernoulli’s letters; part of them are reproduced below [39]. 12

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The distinction you make between the forces of weight and of wind gives no reasons to admit the principle of statics for those [forces of weight] and to reject it for these [forces of wind], because the distinction concerns only the causes of the two forces. But it does not matter how the forces are produced, it is enough they exist, by any cause they derive they will have always the same action, and consequently the same effect, when the forces are applied the same way [33].13 (A.8.5)

The assimilation of the force of wind to a generic force can be a consequence or cause of Leibniz’s idea of dead and living force that he is discussing and elaborating just in this period. In the case of wind the existence of pulses is evident, in the case of gravity or magnetic forces there still are pulses, even if they are not so evident. Among the topics under discussion between Bernoulli and Renau, two had a special role in the formulation of the principle of virtual velocities. The first, associated to Fig. 8.2, refers to the determination of the speed of a vessel bound by the string BZ of infinite length, which would move if it were not constrained in the direction BQ of the wind speed. The second associated to Fig. 8.3, brings to statics the case of a vessel called by the wind in two directions. Q

C

E

B

D Z Fig. 8.2. A vessel constrained by a rope

G

F

M γ R

ε L

N

B C

R P

Fig. 8.3. A vessel constrained by a rope. Static model 13

p. 212.

O

8.2 Johann Bernoulli mechanics

203

Bernoulli introduces the word puissance perhaps because he feels force to be imprecise, a term he would likely refer to the living force. I use the word power here instead of force to make myself more intelligible by showing that the force of winds has no prerogative to another kind of power continuously and uniformly applied [33].14 (A.8.6)

In a letter to Varignon of June 1714 [39], Bernoulli discusses the problem of Fig. 8.2 in the special case where the water’s resistance to motion is zero. Bernoulli argued that the ship, without the rope, would be ready to move in the direction of the wind with its speed. In fact if the speed of the ship were less than that of the wind the wind would push the ship, conversely, if more it would restrain it. To Varignon who holds that the velocity should be as BE, as it would be dealing with the rule of the parallelogram, Bernoulli replies that in this case of velocity composition the rule of the parallelogram cannot be applied: I may be one of the most zealous defenders of the composition of forces, as you have seen in my book and in other occasions, but let me tell you that here you are abusing of this great principle of Mechanics. You do not make a good application to our subject. To show it to you, let us see what this principle says. There are mainly two cases: the first is when two dead forces acting together, but in different directions, originate a third medium force, the second of such cases is when two living forces are to apply immediately and in a short time following different directions on a moving body, which each separately would generate certain velocities. These forces would produce in the mobile if they act together, an average velocity, which will be as in the case of dead forces the diagonal of the parallelogram. […] To get to our subject, the first of our two cases does not apply, because we are not concerned with dead forces, the second there cannot be applied either, because the ship is not pushed by the wind like a ball by a single instant shock, but by a force applied continuously [39]. (A.8.7)

The speech is not entirely clear as Bernoulli seems to limit the validity of the rule of the parallelogram, but in fact it is not so. Bernoulli simply says that the rule of the parallelogram applies to forces and not necessarily to velocities. Very interesting is the following passage, which comes closely after the previous one: However, as the wind acts very differently on the sail by its continuation, we can consider its action as repeated bursts at any time, each of which adds a new level of speed infinitely small to the vessel until the overall speed of the vessel is so large that the wind can add nothing more to it. This happens when the ship, as I said, flees across the wind with the whole speed of the wind [39]. (A.8.8)

Above Bernoulli seems to apply the Leibnizian language to the transformation of dead force into living force, by means of subsequent pulses. The first time Bernoulli refers to a law of virtual work is however in a letter to Renau of August 12th, 1714, after the publication of Essay d’une nouvelle theorie de la manoeuvre des vaisseaux. The reference is to the diagram of Fig. 8.2, which now has lost any reference to navigation and is reduced to an ordinary problem of statics. 14

pp. 217–218.

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Bernoulli had already studied this case in another letter to Renau of July 12th, 1713 [33],15 by requiring that the centre of gravity of the three weights was as low as possible, taking up a similar analysis of Huygens [135].16 Now Bernoulli introduces the terms ernegie and vitesse virtuelle. Energie is not given a precise mechanical meaning. However it is not the dead force (Leibniz meaning) but the dead force (Bernoulli meaning) multiplied by the infinitesimal velocity and as such it has little to do with the modern concept of energy: In the demonstration you make about the equilibrium of weights you say the powers or the forces are like the mass multiplied by the velocity and this is very true in a sense, but consider in the application you made in the equilibrium of the three sails whether you confuse force or power with the energy of the power or the force [emphasis added] and you confuse the current velocity of the wind, which multiplied by the mass produces the absolute force, with the virtual velocity, which multiplied with the absolute force produces the momentum or the energy of this force [376]. 17 (A.8.9)

Immediately after, Bernoulli specifies that virtual velocity is identified with the infinitesimal displacement, energy with the product of the power or force multiplied by the virtual velocity. Note that at this stage forces and virtual velocities have the same direction and Bernoulli makes no distinction between force and power. I mean with virtual velocity the only tendency to move the forces have in a perfect equilibrium, where they do not move actually. So in your figure [Fig. 8.3], which is here the second, if the weight B inseparable from the line MB is in equilibrium with the weights N and O, the virtual velocity is the small line BP, and the virtual velocity of N and O are CP and RP, and then the product of the weight B by BP, which is the energy of the weight B, is equal to the products of weight N multiplied by PC, and the weight O multiplied by RP, which are their energies. Wherefore to avoid ambiguity, instead of saying that their powers or forces are as the products of the masses by their velocity you might have done better, to express yourself well, to say that the energies of powers or forces are as the products of these powers or forces by the virtual velocity [376]. 18 (A.8.10)

Bernoulli will come back on this in a letter to Varignon of November 12th, 1714. The essential point [of the divergence with Renau] can be put on half a page, but this is precisely where Mr. Renau grossly errs in that it merges the forces of winds with the energy of forces, forgetting that to have energy that the Latins called momentum [emphasis added] of the wind, it is not enough to take, as he does, the square of the wind speed, which would give the sheer force of the wind, but it is necessary to multiply the square of the velocity multiplied by its virtual velocity, i.e. by the distance from the centre of support, about which the applied force tends to move [39]. (A.8.11)

The last step is the famous letter to Varignon of February 26th, 1715. Here Bernoulli specifies his principle and affirms its generality, in the sense that he sees the principle of virtual velocities as the possible and only foundation of all statics, including hydrostatics. He is argumentative with Varignon who proposes to establish statics 15 16 17 18

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on the law of composition of forces, an intention declared in the Project d’une nouvelle mechanique [237] and completed with publication of the Nouvelle mécanique ou statique [238], after his death (see last section of this chapter): Conceive several different forces acting along different trends or directions to balance a point, line, surface, or body; conceive also to impress on the whole system of these forces a small motion either parallel to itself in any direction, or around a fixed point whatsoever: you will be glad to understand that with this motion each of these forces will advance or retire in its direction, unless someone or more forces had their trends perpendicular to the direction of the small movement, in which case this force or these forces, neither advance nor retire anything. These advancements or retirements, which are what I call virtual velocities, are nothing but what each direction increases or decreases by the small movement. These increases or decreases are found by drawing a perpendicular to the end of the line of action of any force. This perpendicular will cut in the same line of action, displaced in a close position by the small motion, a small part that will measure the virtual velocity of this force. Take, for example, any point P in the system of forces that is in equilibrium, F one of those forces which push or pull the point P in the direction FP or PF; Pp a small straight line that the point P describes because of the small motion, for which the trend FP takes the direction f p, which will be exactly parallel to FP if the small motion is made in all parts of the system along a given line, or will have, being prolonged, an infinitely small angle with FP if the small motion of the system is around a fixed point. So draw the perpendicular PC to f p, and you will have Cp for the virtual velocity of the force F, so that Cp × F is what I call energy. Note that Cp is negative or positive relative with respect to the others: it is positive if the point P is pushed by the force F, and the angle FPp is obtuse and is negative if the angle FPp is acute, but otherwise, if the point P is pulled, Cp will be negative when the angle FPp is obtuse, and positive when acute. All this being understood, I form this general proposition: In any equilibrium of any forces in any way they are applied and following any directions, either they interact with each other indirectly or directly, the sum of the positive energies will be equal to the sum of the negative energies taken positively [39] [238].19 (A.8.12)

f

F

P

C p

Fig. 8.4. Definition of the virtual velocity

In the passage above some things should be underlined; the first one is that the “general proposition” expresses a necessary but not sufficient condition for equilibrium. Secondly, the small movement is a rigid motion of all points and forces of the system – in particular, though not explicitly, a plane rigid motion, because it reduces to a translation or a rotation, to which it is natural to associate a system of plane forces – but these forces are not necessarily applied to a single rigid body. Thirdly, 19

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the motions are supposed to be small so not to affect, or to affect in an infinitesimal way, the position of the forces participating in the motion, as clear from Fig. 8.4 where FP and f p are the forces before and after the displacement Pp. The idea that the forces are involved in the virtual motion is typical of the whole literature of the XVIII century. While highlighting the infinitesimal character of virtual velocities, Bernoulli does not stress the need that such motion is according to internal and external constraints of the system of bodies. Bernoulli then has never commented on the importance of the principle to eliminate the constraint forces from the equations of equilibrium. And in fact sometimes he will consider also virtual velocities incompatible with constraints and the work of the reactions. The letter to Varignon continues with applications of the energy rule to all cases of simple machines and also to fluids. In each case the results obtained are compared with known ones. The applications refer only to single degree of freedom systems; in this case the vanishing of the sum of the energies is also a sufficient condition for equilibrium. Riccati, Angiulli and Lagrange (Chapters 9 and 10) will clarify that the equation of energy becomes a sufficient condition for equilibrium if the validity for all possible virtual velocities is imposed; Servois (Chapter 12) will add interesting comments on the difference between necessary and sufficient conditions. In the following I will present only a few applications to show the difference of Bernoulli’s formulation with Descartes’ or Wallis’. In the case of the inlined plane of Fig. 8.5; the two weights A and B are connected by an inextensible wire. The force which equilibrates the weight B laying on the inclined plane is furnished by the weight A. Supposing a virtual motion of the two bodies with A that moves into a and B into b , the virtual velocity of weight A is Aa, that of weight B is the line BC (i.e. the component of Bb in the direction of the weight force in B). Assuming equal energies leads to LN A × Aa = B × BC = B × , (8.3) LM a well-known result. The procedure is more or less the same of that of Descartes. However now an algebraic equation is written down, and no recourse is made to an absurd reasoning for the sufficiency of equilibrium.

P L A

B

a C N Fig. 8.5. Equilibrium on the inclined plane

b M

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a

A

B

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b

m p

P n

C

c

Fig. 8.6. The composition of forces

More or less the same is true for the pulley, where the kinematical analysis of the points of application of power and resistance is enough. Some more attention should be devoted to two particular cases. The first is the proof of the composition of the forces, the other the evaluation of the pressure a body exerts on the support. For the first case, Bernoulli considers the situation of Fig. 8.6, where there are three forces A, B, C converging into P. Imagine a horizontal translation of P and these forces – notice that also the forces are moved. The virtual velocity of forces A and B are pm and pn respectively, the virtual velocity of the force C is zero: So we will have A × pm = B × Ppn + C × 0 = B × pn, i.e. A : B = pn : pm = sinus of the angle pPn : sinus of the angle pPm [39]. (A.8.13)

Bernoulli says that similar relationships are obtained by imagining motions of the system of forces in the directions Pm perpendicular to A and Pn perpendicular to B. The relations obtained are the same as that obtained by applying algebraically Varignon’s rule of force composition. The second Bernoulli’s case refers to Fig. 8.7. The goal is to find the impression that each of the two inclined planes CA and CD receives from the ball of weight P. Bernoulli determines an impression at a time, thinking of replacing one of two planes

A

BD

a C

n Fig. 8.7. The impression on a support

f R b

d e c

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with a force. For example the plane CD with a force R orthogonal to it. This is the classic approach in statics to replace a constraint with forces and apply the rules of equilibrium to the resulting system as if the body were not constrained. Bernoulli imagines the displacement Cc along the plane AC. The virtual velocity of the weight B which moves to b is represented by Cn (the projection of Cc along the vertical), the virtual velocity of the force R that replaces the constraint of the plane Cd is represented by Ce (the projection of Cc along R). It is not hard to find the relationship of these two virtual velocities as a function of the angles ACD – the angle formed by the two planes – and Ccn, which is the slope of the plane CD: P : R = sinus ACD : sinus Ccn.

(8.4)

For fluids, assumed as incompressible, Bernoulli considers the case of the siphon and the hydraulic paradox. For simplicity, I will refer only to the latter. Bernoulli considers the tube SNns of Fig. 8.8, that extends into the cylinder SDABEs. The base AB can move inside the cylinder without allowing the fluid to drain. The weight P is at the end of a scale, to the other end of which there is the base AB. The system is filled with a fluid until F f , in order to equilibrate the weight P. By neglecting the weight of the base, it is found that the weight P necessary for the equilibrium equals the weight of the column of water with base AB and height JF, and not only the weight of the portion in gray of Fig. 8.8, which seems paradoxical. The mystery is explained by applying the rule of energies. Bernoulli imagines to divide the fluid into n layers of the same height, p in the cylinder and n − p in the tube, the weight P is also imagined divided into n equal parts. The virtual velocity of the weight P is equal to Aa as that of the base AB. Then each of the p layers of the cylinder has the same energy, Aa× AB, as each element of the weight P. The virtual velocity of a layer of the tube is given by f n which for the preservation of the

J

C

N

n

F

f

G

g

a A

Fig. 8.8. The hydraulic paradox

P

s

S

D

M

E b

J

B

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volume of incompressible fluids is related to Aa by the relation f n = Aa × AB/Gg, then the energy of each of the n − p layers of the cylinder being proportional to the product f n × Gg = Aa × AB is it the same as one of the remaining n − p elements of the weight P still not considered in the equilibrium. Thus the energy of the load P and the water column is the same and so the paradox is explained. After this exchange of letters with Varignon in the years 1714–1715, Bernoulli returned to his principle just once, a few years later, in 1728 in his Discours sur les loix de la communication du mouvement [35]. At the beginning of chapter III he defines virtual velocity: I call virtual velocities, those that two or more forces brought into equilibrium acquire when a small movement is impressed to them, or if these forces are already in motion. The virtual velocity is the element of velocity, that every body gains or loses, of a velocity already acquired during an infinitesimal interval of time, according to his direction [35].20 (A.8.14)

The above definition is not equivalent to that contained in the letter to Varignon. A true velocity is considered rather than a displacement, moreover, it is in general the variation dv of a motion. This new point of view is justified by the fact that Bernoulli is now considering the motion of bodies and not just their equilibrium. No reference, or comment is made to his earlier definition of the virtual velocity, as if he had never written anything about it. Slightly further down Bernoulli continues, with the title of Hypothesis I: Two agents are in equilibrium, or have the same moments, when their absolute forces are in the mutual relationship of their virtual velocities, either the forces acting on each other are in motion or at rest. This is a normal principle of Statics and Mechanics, I do not stop to prove it, I prefer rather to show how the motion is produced by the force of a pressure which acts continuously, and without further resistance in addition to those resulting from the inertia of the mobile [35].21 (A.8.15)

One must wonder about the very little weight Bernoulli now attaches to his principle, which in 1715 he had seen as a key of statics, and to the few references he made to it, declaring it, inter alia, to be a principle of ordinary mechanics and therefore well known. At the same time time he should have known that not all scientists accepted it. Moreover, Varignon’s Nouvelle mécanique, reporting his famous letter, was not released before 1725.

20 21

p. 23. p. 23.

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8.3 Varignon: the rule of energies and the law of composition of forces Pierre Varignon was born in Caen in 1654 and died in Paris in 1722. Educated at the Jesuit college and the university in Caen, he received his master’s in 1682 and holy orders the following year. He became professor of mathematics at the Collège Mazarin in Paris in 1688 and was elected to the Académie des sciences in Paris in the same year. He was elected to the Berlin academy in 1713 and to the Royal society in 1718 [354] Varignon was in touch with Newton, Leibniz, and the Bernoulli family. His principal contributions were to mechanics. With l’Hôpital, Varignon was the earliest and strongest French advocate of differential calculus. He simplified the proofs of many propositions in mechanics that were based on the composition of forces. An interesting publication of his concerned the application of differential calculus to fluid flow and to water clocks.

8.3.1 Elements of Varignon’s mechanics In 1687 Varignon published the Project d’une nouvelle mechanique [237], which gave rise to the Nouvelle mécanique ou statique of 1725 [238], after Varignon’s death (1722). In the premise of the Project Varignon explained the reasons that led him to undertake his work. He declared to have been very impressed by Descartes’ claims for whom there was no sense in reducing the pulley to the lever, as dal Monte and Galileo did. This led him to conclude that it made not much more sense to reduce the inclined plane to the lever, or to reduce one machine to another machine. To Varignon it was better to find a single simple principle which explained the operation of all machines. For him Descartes’ approach with the virtual displacement was interesting but it had the inconvenience of considering more the necessity than the sufficiency for equilibrium and of not furnishing a causal explanation. For Varignon the examination of all cases of equilibrium studied made it clear the need for a causal principle that serves to explain the reasons of equilibrium: I remain in the opinion that to understand equilibrium it is necessary to know how it is established and to see in it all the proprieties that all the other principles prove at most as a necessary condition [237].22 (A.8.16)

He found this causal mechanism in the law of composition of forces by means of the rule of the parallelogram which he assumed to be the only principle of statics. The law of composition of forces according to the rule of the parallelogram, known in the XVIII century as Stevin’s theorem, was reformulated by Varignon and demonstrated on a dynamic basis as Newton did in the same year in his Principia, with the difference that Varignon assumes proportionality between forces and 22

Preface.

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velocities instead of forces and accelerations. I do not want to comment here on the legality of this transaction, I will only refer to the wording of the law of composition of forces as proposed by Varignon: To prepare the imagination to compound motions, conceive [Fig. 8.9] the point A with no weight moving uniformly toward B along the straight line AB, while this line moves uniformly toward CD, along AC by remaining always parallel to itself, i.e. by making always the same angle with this fixed line AC. Of two motions started at the same time let the velocity of the first to the velocity of the second be as the sides AB of the parallelogram ABCD along which they [the motions] occur. Whatever the parallelogram ABCD be, I say that for the effect of the two forces producing these two motions in the mobile A, this point will pass the diagonal AD of this parallelogram, during the time that each of these [forces] would have make to pass along each of the correspondent sides AB and AC [238].23 (A.8.17)

Fig. 8.9. The composition of forces according to Varignon (reproduced with permission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

The Nouvelle mécanique had a great influence on statics for nearly a century. This influence did not so much derive from the theoretical content of the text, but rather from the large number of applications. Results of such applications obtained with geometric considerations on various parallelograms of forces were expressed by means of formulas, mainly proportions, which gave Varignon’s treatment a partially algebraic aspect that made it easier to solve static problems than the rigidly geometric approach of the lever. Varignon applied the composition of force rule also to constrained systems, replacing the constraints with equivalent forces. Among the algebraic relations that he established between the forces is an important one that we now call the Varignon theorem, which in modern terms says that the static moment of the resultant with respect to a pole is the sum of the static moments of the components with respect to the same pole [238].24 23 24

vol. 1, p. 13. pp. 84–85.

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Varignon in his book, probably for the first time, introduced the modern French term moment from the latin word momentum, with the meaning of static moment, i.e. the product of a force by its distance from a reference point (the arm):25 Definition XXII The product of a weight or absolute power by their distances from the fulcrum of the lever to which it is applied is called in latin, Momentum […] So we will continue to call it moment to remain close with the ordinary use. The reason of this name stems without any doubt from the fact that these products are equal or different as the actions of two powers in a lever [238].26 (A.8.18)

At the end of the first volume of the Nouvelle mécanique ou statique, for the lever subjected to various forces at various points of application and with different directions, Varignon presents as a theorem (really a corollary to a more general theorem), proved with the composition of forces, the rule of equilibrium based on the vanishing of static moments: The contrary expressions of moments will always be equal to each other, i.e. the sum of the moments conspiring to turn the lever in a sense about its support will always be equal to the sum of moments conspiring to rotate in the opposite sense on this support, as we have already seen in Corol. 9 of Th 21[238].27 (A.8.19)

The semi-algebraic approach of Varignon evolved toward a purely algebraic approach, for which the balance of forces results in forcing to zero the sum of the components of the forces and static moments, which today are called cardinal equations of statics. There is still no precise reconstruction in the literature of the way in which the modern form of cardinal equations of statics was obtained. D’Alembert is commonly credited as the first to give these equations, in the Recherches sur la précession des equinoxes in 1749 [82] followed by Euler [101, 106]. To the best of my knowledge they were Fossobroni with his Memoria sul principio delle velocità virtuali in 1794 [109],28 Prony with his Sur le principe des vitesses virtuelles in 1797 [202]29 and Lagrange, and only in the second edition of the Mécanique analytique of 1811 [148],30 which collectively gave the first modern expression to the cardinal equations of statics. But it was only with the Mémoire sur la composition des moments en mécanique by Poinsot in 1804 [193] that they were fully understood.

25 In specialised treatises of mechanics, static moment is not a term of mechanics but rather of geometry, like area or moment of inertia. The product of a force by its arm is simply called moment (of a force). Historians of mechanics however are used to speak about static moments to distinguish them from the Galilean moment. In the book I will follow this use. 26 p. 304. 27 pp. 385–386. 28 pp. 86–87. 29 p. 194. 30 pp. 46–58.

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8.3.2 The rule of the parallelogram versus the rule of energies In his letter of February 1715, Bernoulli declared the superiority of his principle: Your project of a new Mechanics is filled with a great number of examples, some of which, to judge from the figures, seem very complex. But I challenge you to propose one at your choice, and I will solve it on the field and as for a joke with my rule [39]. (A.8.20)

After having asserted that with his rule it is possible to solve all the problems of statics, Bernoulli added: The principle that you pretend to substitute to mine, and which is based on the composition of forces, is nothing but a little corollary of the energy rule. I have so the right to consider as the first and great principle of statics that on which I based my rule: in any equilibrium there is an equality between the energies of the absolute forces; i.e. between the product of the forces multiplied by their virtual velocities [39]. (A.8.21)

and suggested that Varignon replace the rule of the composition of forces with the rules of energies: I beg you to think, you will find in it an inexhaustible fund to enrich mechanics and to make the study incomparably more comfortable and simple than it was in the past. The complete treatise of this science, that you promise so long, could appear much more estimable, if it will be founded over a principle so universal, so simple, so clear and so certain, like that it is concerning and of which I showed so many advantages [39]. (A.8.22)

In March 1715 Varignon replied that, yes it is true that the rule of energy is interesting, but that the rule of composition of forces is easier and more fruitful: But mechanics, from this proposition and from the general one you added to your last letter, far from being the great and first principle of statics, is in my opinion only a corollary of compound motions, or of another principle, which proves this proposition, i.e. your equality of the sum of energies, by deducing with its aid or by supposition, the incipient motions of Mr. Descartes, which you call virtual velocities, that with the powers elsewhere evaluated, with the assumption of their equilibrium, it is all needed for the equality of the sum of the energies, of which one could ever have the right to think that it could be derived from one of these principles [39]. (A.8.23)

He saw the law of energy rather as a corollary of the composition of forces. In order to take the rule of energy as a principle of statics, for him it must be proved with the law of composition of forces or with some other principle – presumably Varignon thought the law of the lever. Varignon, rightly, traced the law of energy to Descartes: Cartesians, according to the letter I cited of their Master,31 had already deduced from his principle the same equality of Moments or energies, or the quantity of motion, that you use, for two powers in equilibrium on simple machines, and in fluids, from the incipient motion that Mr. Descartes prescribes in this letter. But you are the only one, for what I know, who extended the equality of energies to as many powers as you like, acting in any direction and in equilibrium with themselves. This point is very nice, but (as I have already said) it supposes the equilibrium among them and does not prove it [39]. (A.8.24) 31

[94], letter 73, vol. 1, pp. 327–346.

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and asserted that Bernoulli stated his principle only as a necessary condition, that is if there is equilibrium then the energies are the same for virtual motions, as the Cartesians that demonstrated the sufficiency of the balance with an ad absurdum argument: The equilibrium from non-equilibrium, they make only a demonstration ad absurdum [39]. (A.8.25)

Bernoulli replied, arguing the logic superiority of the rule of energies, stating that it applies equally well to solids and fluids, while this is not true for the composition of forces. Diplomatically he then ended by asserting that what counts for him is that his rule is correct and works very well: I am afraid of falling into a long verbosity if I try to discuss all you are saying regarding my rule of energy, that I pretend to be general for the whole of mechanics, both for fluids and solids […] let us avoid that verbosity. It is only sufficient to establish the truth and the universality of my rule of energies against your objections. That this rule be a principle or a theorem of another rule, it does not matter; it is enough that it is true, general and comfortable, without any exceptions, uniform and simple to use. Advantages that the composition of forces does not possess [39]. (A.8.26)

To the dispute that Descartes preceded him, Bernoulli added, with a touch of controversy, that Varignon too was not very original: You cite Mr. Descartes’s letter to prove that this author has already had the idea to explain the equilibrium of powers by means of the equality of energies by considering their incipient motion, that I call virtual velocities. I reply that I am not proud to be the first inventor of this idea; no more should you be proud to be the first to explain the equilibrium by means of the composition of forces [39]. (A.8.27)

To Varignon, who asked permission to report Bernoulli’s principle in the book he was writing – i.e. the Nouvelle mecanique –, Bernoulli had no difficulty in granting the permission in a letter of July 1715, provided he did not present it as subordinate to the rule of composition of forces: You can make what you like of my rule of energies, adding or not adding it to your mechanics. I allow both of them. But to pretend that it is a corollary of the principle of the composition of motions or forces, I may still hold the reasons given in my previous letters, to prove the contrary, if you want to engage me in a challenge that will cost us time and troubles. So I will prefer to leave to you the pleasure to believe that the principle of the composition of forces should precede that of energies, to try a long and lengthy contestation. It is enough that the second could be applied both to fluids and solids, it is more general than the first that is useful only for solids, moreover it will need one more principle from which it could be deduced, because the composition of forces is not so clear as to be assumed as an axiom. Then it looks to me more reasonable that the principle of energies, as the more general and at least as clear as the composition of motions, contains the last as less general [39]. (A.8.28)

Varignon however did not respect the desire of Bernoulli. Indeed in the second volume of the Nouvelle mécanique ou statique, he wrote a chapter titled Corollaire général de la théorie précédente, where he claims that the rule of energies is nothing

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but a corollary of the rule of composition of forces, qualified as Theorem XL [238].32 Actually Varignon was not successful in proving the rule of energies in general, but only for various cases: a weight supported by many strings, the pulley, the wheel, the inclined plane with the weight pulled by a force with any direction, the lever, the screw and the wedge, by checking that values of forces and displacements, evaluated respectively with the rule of composition of forces and with simple kinematical analysis, respect the equation of energies. The applications considered by Varignon, probably to avoid being accused of plagiarism, did not coincide with the applications suggested by Bernoulli in his letter of February 1715, in particular there are no applications to fluids.

32

pp. 174–176.

9 The Jesuit school of the XVIII century

Abstract. This chapter is devoted to the principle of action by Vincenzo Riccati and Vincenzo Angiulli, whose VWL is similar to that of Bernoulli. In the first part the contribution of Angiulli and his demonstration of VWL is presented in the foundational route. This is perhaps the first convincing demonstration of a VWL. In the second part the contribution of Vincenzo Riccati is presented. Unlike France which, as reported in the previous chapter, saw a revival of interest in mechanics, Italy in the second half of the XVII century started a slow decline, except for some recovery in the second half of the XVIII century, that will stop only after the unification of the nation in the late XIX century. This decline is particularly evident in the so-called exact sciences, including mathematics and mechanics. The last great Italian scientist in this field was Giovanni Alfonso Borelli (1608–1679). Not that clever and educated people were missing, except that the lines of research pursued in Italy were no longer included in those being conducted in Europe by Huygens, Newton, Leibniz, the Bernoullis, for example. The reasons for this delay were numerous and probably outside the nature of science itself, largely due to economic backwardness and independent political status compared to the national states that were consolidating in Europe, where scientists were asked to solve pressing practical problems, related for example to navigation and military activity. In this climate, Italian scientists found themselves almost compelled to think about old problems using old categories. To give an idea of this type of studies it is useful to cite Girolamo Saccheri (1667–1733) who dedicated his efforts to the study of classical geometry, providing interesting contributions to non-Euclidean geometry, that however will be resumed in Europe only toward the end of the XVIII century. One of the mathematicians who fitted well in the mainstream of European Calculus was Jacopo Francesco Riccati (1676–1754), who studied the equation that bears his name. The schools of Bologna and Naples [367] must also be referenced with regard to studies of mechanical theory, and yet another Riccati (Vincenzo) made a very important contribution, influenced in part by his father’s studies.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_9, © Springer-Verlag Italia 2012

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Vincenzo Riccati supported the ideas of Leibniz, following the Italian attitude where Newton’s ideas were not yet widespread. This was not actually much different from what was happening in Europe where, besides Leonhard Euler (1707–1783), who developed Newton’s mechanics to make it an effective instrument operating through the use of differential equations, there was still Johann Bernoulli who, with his principles of virtual velocities in statics and the theorem of living forces in dynamics, had proposed an alternative approach to mechanics, close to that desired by Leibniz. In Italy the tradition of Galilean mechanics was still alive and in it the laws of virtual work were very important. For sure the continuity with the approach of Galileo also influenced Vincenzo Angiulli, an intelligent Riccati’s pupil, to assume a law of virtual work as the basis of statics. But the greatest ‘Italian’ mathematician and mechanician of the middle of the XVIII century was Ruggiero Giuseppe Boscovich (1711–1787). Boscovich left no writings of statics, but in some of his works he made use of a law of virtual work, applied to a new area, the mechanics of structures, which will provide fertile ground for modern laws of virtual work. Probably it is not a coincidence that Boscovich and Riccati were both Jesuit and the two greatest mathematicians of the order. It is likely that Boscovich had knowledge of the research of the ‘brother’ Riccati, even though the publication of Riccati’s studies on laws of virtual work [207] was six years subsequent to Boscovich’s first applications [138].

9.1 Vincenzo Angiulli and Vincenzo Riccati The work of Vincenzo Riccati and Vincenzo Angiulli presented in this chapter should be seen as an interesting attempt to defend the approach of mechanics based on virtual work laws, which in Italy had its roots in the works of Galileo and Torricelli. The former showed his ideas in the Dialogo di Vincenzo Riccati della compagnia di Gesù dove ne’ congressi di più giornate delle forze vive e dell’azioni delle forze morte si tien discorso of 1749 [207] and the De’ principi della meccanica of 1772 [208], the latter in the Discorso intorno agli equilibri in 1770 [4]. Both took on as a foundation of mechanics, with various reasons, the principle of action, which is a possible version of virtual work laws. In the following I will set out a summary of the thought of the two Italian scholars, spending more time on Angiulli, who will be considered first. This is because, although Riccati’s contribution is more original, he first proposed the principle of action, the production of Angiulli and his attempts at justification are more effective.

9.1.1 The principle of actions of Vincenzo Angiulli Vincenzo Angiulli was born in Ascoli Satriano (Foggia) in 1747 and died in Naples in 1819. Very young he got a degree in law in Naples where he attended the enlightenment circles of the city. Only twenty three he became member of the Accademia Clementina, of the Istituto delle science of Bologna and professor of mathematics at the Real accademia of Nunziatella in Naples. On the occasion of the death of Charles

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III of Borbone he wrote and read in Ascoli Satriano an interesting funeral oration. In 1800 he was jailed for the role he had during the French occupation. He knew Riccati in Bologna where he followed his courses on mathematics. The Discorso intorno agli equilibri was the only scientific work of Angiulli written on the occasion of his professorship at the Nunziatella when he was a young man. After this experience Angiulli dedicated himself to administrating his properties and to a consistent political activity influenced by the Enlightenment [5]. 9.1.1.1 The action of a force The basic concept of Angiulli is that of action of a force, as developed by Riccati. To introduce this concept, Angiulli begins to point out that although the presence of a ‘force’ is a necessary condition for changing the state of equilibrium, it is not even sufficient. For example, in a heavy body suspended by a thread, although it is subject to gravity, there is no change in state. Therefore one must distinguish between the force and its action, only the latter can produce the change in state. From the foregoing it is evident that in Angiulli’s mechanics, like that of most scholars of the XVIII century, there is no room for constraint reactions and the ‘forces’ are only those now classified as active forces. After stating that Galileo was the first to distinguish between force and action, when he introduced the idea of the force of blow, Angiulli presents, by means of examples, his concept of force, which he calls power: For power, therefore, we do not mean anything but the pure and simple pressure, or that effort, which the gravity or other force makes against some invincible obstacle, as precisely it is what a ball of lead makes against a fixed table, or against the hand that sustains it [4].1 (A.9.1)

In this quotation the power does not appear as a purely static force, able to balance a weight, as it was mainly up to Galileo, it always makes an effort against an obstacle and therefore it plays an active role. In the next quotation Angiulli recalls the definition of Leibniz’s living force and its relationship with the dead force which is identified with the power: So if a ball, for example of lead, will be located above a fixed table, the gravity, which resides in it, will be the only pushing force, and therefore dead force. But if the obstacle is removed, that is the table, in the ball will soon be a change of state, […] the mechanicians imagined the force to give the body a boost, which, however, just born, was destroyed by the invincible obstacle, and so according to their method mathematicians represented the dead force with the idea of an infinite small impulse […]. But because the mechanicians could form a clearer idea of the action of the force, as they represented it under the idea of a pulse, which in the process of its birth is extinct and destroyed by the invincible obstacle, thus removing the invincible obstacle, conceived that all pulses […] were conserved in the same body, and then thought the action of the power not to be but the sum of all the pulses accumulated, and stored in the body. Then the amount of energy generated in the body for the action of the force [...] is called living force [4].2 (A.9.2)

1 2

p. 5. pp. 5–7.

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Power or dead force is thus presented as an infinitesimal pulse that is constantly being renewed, by gravity or other causes, and continuously destroyed by constraints. With constraints removed, pulses can accumulate and the action of power lies precisely in the effect of cumulative pulses that are not destroyed. The action of the power generated then the living force. In the following quotation it is possible to see how Angiulli conceives the relationship between power, action, living force and change of state: But if it is as real as it is said, that the power should be considered as a pulse less than any other, and that the action of the power is the sum of all the pulses communicated to the body and stored in the body, there will certainly be the same proportion of the power to the action of the power, as that passing between an infinitesimal quantity and a finite one [4].3 From above it appears, that the power acting in the body to which it is applied generates in it the living force, and this produces the state change. So the living force must be considered as an effect of the power and as a cause of the change of state that is induced in the body. And as in this case we speak of entire and total causes, the Ontological axiom will take place for which the causes must be proportional to the effects and the effect to the causes. So these are the two ways of measuring the living force, i.e. either with their effect, which is the change of state, or with the extent of their cause, which is the action of the force [4].4 (A.9.3)

Note that the action is identified as the cause not of motion but only of the living force that is the true cause of the motion. The analysis of the text clearly shows that Angiulli is linked to the school of Leibniz and Bernoulli, but it shows also the call to the Italian school; Angiulli strives to link the results of Galileo to those of Leibniz. For example, in his way of explaining the concepts of power and its action it is possible to see Torricelli’s language of the Lezioni accademiche, published only in 1715 [276]. After the presentation of the power and its action in the book of Angiulli, there follows some definitions connected to power, useful for the introduction of the principle of action, reported in the next paragraph. A fundamental concept is that of the centre of the power P. This is conceived as that point, considered fixed, where the source of the power itself is located. For example in the case of gravity the centre of power is the centre of the earth, in the case of a force due to an elastic spring the centre is the fixed extremity of the spring. The line joining the centre of the power to its point of application is the direction of the power. In addition, the space of access and space of recess are the amount by which the point of application of the power approaches or moves away from the centre P. In this nomenclature the influence of John Wallis is clear (see Chapter 7). In Fig. 9.1 there are given by way of example, three powers, AD, CD and DB, of centres A, B and C, respectively, applied at point D with the direction indicated. Assuming that D moves in a, the distance Da is the space of recess of power in A, the distance Db is the space of access of power in B and the distance Dc is the space of recess of power in C. For small motions, the access or recess spaces coincide with the projections of the displacement in the direction of power, both in the starting position or in that varied, because they differ by infinitesimal quantities from each other. 3 4

p. 7. p. 22.

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A

D

c

b a

C

B Fig. 9.1. Access and recess spaces

9.1.1.2 The principle of actions In his definition of the action of the power, Angiulli has left as indefinite one important aspect: he does not specify how the pulses should be counted. Notice that at this point Angiulli, as Bernoulli, because of the need to give a mathematical expression to the dead force, is obliged to redefine it. This is made by distinguishing power, dead force and infinitesimal action, i.e the action associated to a pulse. Up to now, on the one hand dead force and power were considered as synonymous, on the other hand infinitesimal actions and dead force are assumed to be both pulses, so there is no difference between them. From now on, dead force will be considered to be a power in the traditional sense, i.e. not a pulse, while the infinitesimal action will be considered as a pulse. If f denotes the measure of the power, x the measure of space s or time t, the measure of the infinitesimal action is, according to Angiulli, given by f dx. He argues also that it cannot be said in advance if pulses of action replicate in space or time. The indecision between space and time is not only due to a rhetorical need for objectivity, but it also has another origin. The definition by Angiulli of the action of a power, by his own admission, is not original but goes back to that of Vincenzo Riccati of 1749 which contains the same problem of choice. Vincenzo Riccati introduces the power and the action of the power in a way formally similar to Angiulli’s, but he attributes to them a different ontological status. Riccati’s power is the Newtonian force, acting to produce any change of motion of a body. The action of the power is still an aggregate of pulses, but the aggregate must be regarded simply as an associated mathematical quantity. In this purely mathematical sense it is reasonable to consider both f ds and f dt as equally representative of the infinitesimal. The choice of either option is left to their more or less usefulness in establishing the laws of equilibrium. For Angiulli instead the power is the Leibnizian dead force. The pulses associated with it are hypostatized: they are created and destroyed and accumulate only when they are not destroyed. With their accumulation, which represents the action of the power, they generate the living force (and it is not the action of the Newtonian force, as for Riccati) which is responsible for the change of motion.

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Prior to leaving the reservation whether to measure the replicas of the pulse of power in space or time, Angiulli tackles the problem of defining a criterion for equilibrium, reaching the conclusion that it is provided by the equality of the infinitesimal actions. He begins by stating that a policy of equilibrium, to be metaphysically well founded, must be based on some equality, without which motion emerges. In considering the various possibilities, Angiulli immediately discards the criterion that assumes the equality of powers. Although desirable, it is generally not empirically verified; for example the equilibrium of the lever with different arms is accomplished with two different powers between them. Then he criticizes the approach of the ‘ancient mechanicians’, which had placed equality of static moments – defined by him in the modern sense of force multiplied by arm – as the basis of equilibrium. It does not provide a causal explanation, since the moments have no physical reality, they are not beings – as powers are – able to act causally: So by saying with the Ancients, that the cause of the equilibrium is in equality of the moments, they seem having said, nothing but the equilibrium depends on the equality of those quantities from the equality of which the equilibrium depends [4].5 (A.9.4)

and then one encounters in a petition of principle. Angiulli concludes that since equality must relate in some way to powers, there is nothing but to consider the actions and define the equilibrium as a result of equality of actions of the powers that act one way and another. Then, if the actions which are the causes are prevented, the effects, i.e. motions, also are prevented, and there is equilibrium: The equilibrium comes from the fact that the actions of the powers which must be equilibrated, if born, would be equal and opposite, and therefore the equality, and opposition, of the actions of the powers is the actual cause of equilibrium. […] The equilibrium is nothing but that the impediment of the motions, that is of the effects of the powers, and it is not surprising if it matches the prevention of the causes, i.e. of the actions themselves [4].6 (A.9.5)

To explain better the contents of this quotation, Angiulli states that the balance of the actions of powers should not be considered as a cause in the strict sense of equilibrium. Each action by itself (through the intervention of the living force) is the cause of a motion; if there is equality between contrary actions, there is equality between the possible motions, and then there is no motion, and then balance. If at the beginning of a virtual motion the actions are equal to each other, then the motion is impossible, but if they are different, nothing will prevent the greater to make its effect and there will be motion. Therefore, Angiulli concludes, the general criterion of equilibrium is contained in the following theorem: Then we establish a principle, that is a general criterion to know when it will happen that between the forces there is balance, and it is what is contained in the following theorem: The forces will be in balance if they are in such circumstances that if an infinitesimal motion was born, their infinitesimal actions would be the same. And this principle must take place in all equilibria [4].7 (A.9.6) 5 6 7

p. 15. pp. 16–17. p. 18.

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Note the presence in the last passage of the term infinitesimal motion, which underlines the idea of motion in the process of being born, whose introduction is essential to arrive at a correct formulation of the criterion of equilibrium. Angiulli has qualified as a theorem his statement because he believes to have demonstrated it with metaphysical considerations. He followed trying to bring other arguments in favor of the principle of action, which for the moment is still quite general, not specified in its magnitude. The principle does not “Only administers a general method to examine the equilibrium, but it is also the real way in which the equilibria will go to establish themselves”. In fact, he says, suppose to put a ball between two opposing inclined planes, it just will not be placed in equilibrium, because there are no absolutely rigid bodies in nature, and the inclined planes will yield to pressure of the ball, and provide an elastic power, whose action is opposed to the gravity of the ball. The elastic powers give rise to actions, which being equal and opposite to that of weight ensure that the ball remains in equilibrium [4].8 However these considerations confuse rather than clarify. Indeed Angiulli introduces the elastic forces, trying to give physical sense to constraint reactions, that otherwise may remain obscure, a mere fiction. The concept of elastic constraint reaction, however, is incompatible with the concept of hard body, i.e. a body with only passive function, capable of absorbing all dynamic actions; a dominant concept in the mechanics of the XVIII century, which Angiulli accepts on a number of occasions. 9.1.1.3 The measure of actions Finally Angiulli switches to solve the problem of the concrete measure of the action, i.e. to decide if the pulses of powers are replicated in space or in time. He recognizes at this point that this choice is the object of the dispute between Cartesian and Leibnizian, which in Italy was still alive in his time: Because the famous dispute of the living forces, which is to establish whether these are measured by the mass multiplied by the speed, or by the mass multiplied by the square of the speed, reduces to this other question, namely whether the action of the force should be proportional to time rather than space [4].9 (A.9.7)

That the choice of the measure of the actions conditions the selection of the measure of living forces, is justified by claiming that there are two ways of measuring the living force: or by measuring the cause, which is the action, or by measuring the effect, that is the change in velocity. Since causes and effects should be proportional, so the measure will be reflected on each other. If it is proved that the action of a power should be measured with the space it would also be proved that the living force is measured by the square of the velocity. Angiulli refers to a law of motion established by Galileo [118],10 to state that the power multiplied by the space is proportional to the mass multiplied by the square of 8

p. 19. p. 22. 10 pp. 287–288. Galileo says simply that the space of descent is proportional to the square of velocity. 9

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velocity, but this subject was largely explored by other scholars. Of some interest are Boscovich’s considerations on his De virus vivis dissertation of 1745 [47],11 where he considers the possibility to integrate the equation of motion in time, to obtain the velocity, or the expression of force in space, to obtain the square of velocity [358]. Boscovich also names Vincenzo Riccati among Leibniz supporters. The arguments with which Angiulli arrives soon after to what he considers a fair measure of the power, are only partly convincing, as I shall explain later. He says that the action of powers cannot be obtained by combining powers with time because if the action was based on the time, and is the same time for all actions, the actions would be proportional to the power and the criterion of equality of action coincides with that of the equality of powers. This is not acceptable, because the equality of powers is not a universally valid criterion of balance, as argued above, so one must combine the power with the space: Not being able therefore to measure the action of a power by the power multiplied by time, is necessary to turn to space. In all known balances it is true, as discussed in the following chapters, that making an infinitesimal motion, the powers are in the inverse ratio of their respective space of access, or recess from the centre of the same powers […]. So whether the action of the force will be measured by the power multiplied by the distance, to which the force acting carries the body, making it closer to the centre, or making it away from the centre, it will be saved in the equilibria the equality between the actions of powers […]. So the action of the power has truly to be measured by the power multiplied by the space according to the method of Leibniz [4].12 (A.9.8)

Angiulli claims to have shown in this passage that the measure of the action as the power multiplied by the space of access or recess is a consequence metaphysically certain, calculated with exact reasoning from the premises. In fact, even accepting the evidence that the extent of the actions should be based on space instead of on time, it cannot be seen why the powers should be multiplied simply for displacement (and not their square, for example) and because these displacements should coincide with the previously introduced spaces of access and recess. The operational statement of the criterion of balance is eventually provided by the following theorem (the term is Angiulli’s): The powers are in balance, if they are in the situation that making an infinitesimal motion, some powers become as close to their centres, some others move away from their centres, the sum of products of positive powers multiplied by the respective spaces of access or recess is equal to the sum of similar negative products [4].13 (A.9.9)

With a choice that is questionable to a modern reader, he will then call the above ‘proved’ theorem by the name of the principle of action. The criterion of equality of the action is then considered by Angiulli as a theorem, when viewed from a metaphysical point of view, as a principle, when viewed from a purely mechanical point of view, as non-deductible by other laws of mechanics. The principle of the actions 11 12 13

p. 4; pp. 13–14; pp. 28–29. pp. 25–26. p. 28.

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stated above can essentially be considered as equivalent to Bernoulli’s rule of energies. Angiulli is also conscious of this and according to him “it does not seem to differ” from Bernoulli’s principle of which he only refers to the brief mentions in the definition of 1728 (see § 8.2). According to Angiulli the coincidence between Bernoulli’s rule and his principle appears immediately as soon as it is recognized that virtual velocity and access or recess space are the same thing. Angiulli however does not declare that he referred to Bernoulli, but rather to Galileo, Descartes, Borelli and other sublime mechanicians. His position should not be seen as a disavowal of the merits and priority of Bernoulli, it should be seen rather as the realization that his principle of action is one of the possible expressions of the principle of virtual work, to which formulation the Italian school has greatly contributed. Since Angiulli deals with fundamentals, it is right that he calls more on Galileo than on Bernoulli, because Galileo in Le mecaniche and the Discorsi sulle cose che stanno in su l’acqua had dealt with the basic concepts of virtual work laws, while Bernoulli contributed mainly from a technical point of view, recognizing the infinitesimal nature of the virtual displacements. 9.1.1.4 The principle of action and the principles of statics Angiulli had already said that the criterion of balance of moments, on which the law of the lever is often based, had no clear metaphysical evidence and therefore the law of the lever, referred to as the principle of ancients, is not ‘obvious’. Angiulli, following Riccati’s steps, believes that the law of the lever in the past was not achieved with metaphysical certitude, or only by recourse to a priori categories, and it was a ‘simply’ experimental principle, i.e. endowed with only physical evidence. He claims that the attempts carried out by Aristotle, Archimedes, Galileo, Stevin, Huygens, to give the law of the lever the metaphysical certitude have failed. Analogous criticisms are expressed about the demonstrations concerning the wedge and the screw. Even when passing over the difficulty of its proof, the law of the lever is not sufficiently general, it cannot be used for the equilibrium of fluids and even in simple machines such as the pulley, the inclined plane and then the wedge and the screw. For the pulley, for example, the law of the lever cannot be applied strictly because it is not possible to identify a priori the centre of rotation of the pulley and thus the fulcrum of the lever itself; this difficulty had already been removed by Descartes

K

X

P

B E G

M

F L I

Fig. 9.2. The inclined plane reduced to the lever

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b C

A

G E

F a

B N

M Z

X

Fig. 9.3. The law of the lever

[4].14 But according to Angiulli the law of the lever cannot be applied always to the inclined plane but only in the case of a rolling such body as a sphere of Fig. 9.2, which rolls having L as a fixed point. In such a case it is possible to assume a lever with fulcrum in L, the weight concentrated in E and a power P applied to any point B of the sphere. When instead the body slides on the inclined plane, for Angiulli, it is not possible to see any lever. This position could indicate that Angiulli did not know Galileo’s Le mecaniche, where the lever is applied also – probably only – to sliding bodies. According to Angiulli the principle of equality of the actions, instead, is generally valid and true in all cases. In the following, with reference to Fig. 9.3, I report the preliminaries of the proof of the law of the lever. The thesis: I say, from the principle of actions it can be deduced that when in the rod ABC there is equilibrium, the power Z is to the power X as CN : CM, i.e. that the equation Z · CM = X · CM is valid [4].15 (A.9.10)

And the proof: Let the points Z and X be the centres of the powers Z and X. Conceive now an infinitesimal motion be born in the rod ACB, so that the points A and B describing the arches Aa, Bb come in a and b. From point b to point X draw the line bX, and from point A to point Z the line aZ, then with the centre Z, and the interval aZ describe the arc that matches AZ in F, and similarly with the centre X […]. This makes it evident that AF is the minute space of access of the power of Z, and bG the minute space of recess from the centre of the power X. The principle of action requires, that to have equilibrium in the rod ABC, the power Z is to the power of X as bG : AF [4].16 (A.9.11)

The text allows comparison of the proof as given by Angiulli with that reported in Chapter 2. In Angiulli the virtual displacement is in real time with the points A 14 15 16

pp. 35–37. p. 42. pp. 42–43.

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and B that describe the arc of a circle; the directions of the powers depend on the motions of A and B. In a modern discussion the virtual displacements of A and B are considered to occur along the tangent, i.e. along the perpendicular to the line AB; also it is not assumed that such motions really act on the points A and B and therefore the direction of the forces shall be deemed unchanged. The two approaches are equivalent if one assumes, as Angiulli does, infinitesimal displacements. Then the change of direction of the forces is negligible, the arc can be confused with the tangent and the virtual displacements projected on the forces are indistinguishable from the spaces of access and recess, so the virtual work, calculated as the sum of the products of virtual displacements multiplied by the components of forces along them, coincides with the action, measured as the sum of the products of forces by the access or recess spaces. The fifth chapter of the Discorso intorno agli equilibri is dedicated to the principle of equivalence, i.e. the composition and decomposition of forces by the rule of the parallelogram, a principle which had been assumed ‘recently’ by Pierre Varignon in his Nouvelle mécanique ou statique, as the foundation of statics. Angiulli first criticizes as not metaphysically obvious the demonstrations of the principle as given by Newton and Varignon, because they assume that two forces produce the same motion both when they are applied together and when they are applied separately. He enhances instead that of Daniel Bernoulli, appreciated by Riccati [339]. Also the principle of equivalence of course can be proved with the principle of action. The proof refers to Fig. 9.4 and considers three powers AD, AC and AB forming the sides and diagonal of a parallelogram respectively. To demonstrate the equivalence between the powers AD and AC with AB, Angiulli demonstrates the equilibrium, assuming that the power AB is directed towards A, while the others start from A. He considers a general virtual displacement of the point of application of the three powers from A to R. For this motion there

F A R o

C

N

D M B Fig. 9.4. The rule of the parallelogram

p

q

F

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are the small spaces of access oR, qR, and recess pR (notice pR is opposite to -AB). The principle of action ensures the balance provided it is CA ×oR + AD ×qR = AB ×pR, but for a purely geometrical lemma, shown previously [4],17 this condition can be satisfied whatever is the displacement of A, if and only if AC, AD and AB are the sides and the diagonal of a parallelogram. The demonstration follows a fairly faithful proof by Bernoulli and Varignon [267] and is less general than that reported by Riccati in his De’ principi della meccanica, which considers the equality of the actions accounted for two types of virtual displacements, one along AB and the other along the perpendicular to AB, reaching two equations of equilibrium [208].18 After showing the fertility of the principle of action, however, Angiulli concludes somewhat surprisingly: Note secondly, that making comparisons between the principle of equivalence and that of actions, both of them must be estimated to be equally fruitful and extended, with that difference, for which in some cases the principle of equivalence can be used with more skill and elegance, in other cases, it is more convenient and appropriate to use the principle of actions. And finally it is to be noted that the method of composition, and resolution of the forces is not the true method of nature, but it is a method Geometers have developed for the easiest and quickest solution of their problems. Nature in its work never composes or resolves the forces, but always uses actions, that being equal and opposite, causing the equilibrium to be produced [4].19 (A.9.12)

Therefore in part he gives up the claim to make of the analytical principle of actions the cornerstone of statics, holding that in practice the approach based on the law of composition of forces is often convenient. The principle of actions remains the honor of being the general principle from which all methods of solving static problems can be derived. This position of Angiulli, that after Lagrange’s Mécanique analytique seemed unjustified given the high fertility of both theoretical and applicative shown by the principle of virtual velocities, is today supported by most scholars of applied statics, who while recognizing the theoretical importance of the modern virtual work principle prefer, in applications, to introduce directly the cardinal equations of statics which are the analytical counterpart of the principle of composition of forces, in which constraint forces are introduced as unknown quantities. 9.1.1.5 The applications to simple machines In the final part of his book devoted to applications, Angiulli considers the equilibrium of the simple machines: the lever, the shaft with the wheel, the pulley, the wedge, the screw and the inclined plane. The principle of action is used as a necessary criterion and provide the results of equilibrium in a very simple way, showing its great advantage in dealing with constraints. I refer as an example to the analysis of the equilibrium of the simple pulley: In the fixed pulley, to have equilibrium, it is asked the equality between power and weight. Let [Fig. 9.5] AB be a fixed pulley, which has around it the rope EABD, to the end D of 17 18 19

pp. 56–57. pp. 26–29. pp. 63–64.

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which the weight P is attached, to the other end E the power is applied, which supports the weight. I say that for there to be equilibrium it is necessary, that the power applied in E is equal to the weight P [4].20 (A.9.13)

The proof is very simple and is developed linearly by Angiulli: Be an infinitesimal motion in the direction of the power applied in E, so that the end of the rope E comes in G, while the end D comes in H. It is too obvious, that EG is the minute space of access to the centre of the power, and DH the minute space of recess from the centre of the weight. So because there is equilibrium between power and weight, it is necessary that the first is to the second as DH : EG. But DH = EG, for supposing that the rope does not practice any strain, but remains always of the same length, the length DAE will be equal to the length HAG; then, DH and EG will remain equal between themselves. So because there be equilibrium in the pulley it is asked the power be equal to the weight. What is needed to demonstrate [4].21 (A.9.14)

B

A C E G

H D

P

Fig. 9.5. The equilibrium of the pulley

In the statics of fluids, treated at the end of the applications, there is the influence of Galileo [115] and of Riccati [208]. The principle of equality of actions is sometimes used as the principle of equality of Galileian moments, as when Angiulli shows that the free surface of a fluid is horizontal. More articulated is the discourse on the calculation of the pressures on the bottom of a container of any shape and the demonstration of equality in the level of communicating vessels that I refer to as an example. Let [Fig. 9.6] GHPQ be a whatever trap, if an arm of it GH is filled with homogeneous fluid [...]. Given that the fluid poured in the trap will be in equilibrium, it will raised to the same height in one arm and another of the trap [4].22 (A.9.15)

The demonstration takes an infinitesimal motion for which GH will drop up to IK and at the same time, in the other arm, the surface PQ is brought into RS; in the above there is implied an admissible motion of the fluid congruent with the constraints 20 21 22

p. 89. pp. 89–90. p. 126.

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R H

G I

dx

V

dz

V

P

K

S Q

x

V

z

V T

U

Fig. 9.6. The equilibrium of fluids

imposed by the walls and its incompressibility. The infinitesimal motion of all the fluid occurs by translation, along the communicating vessels, of a fluid volume equal to the elementary volume indicated below with ΔV. Named x the height of the single fluid layer of the vessel GH and z that corresponding to the vessel PQ, given that the weight of the volume ΔV is proportionalto the same ΔV, the total actions in the two  vessels are proportional respectively to ΔV dx and ΔV dz, where dx and dz are changes in the level of the elementary volumes ΔV into which the fluid is supposed split and the integrals are extended along the two vessels. Imposing the equality of  the actions and in view of the constancy of volume ΔV, one has dx = dz. That is to say that the fluid being in equilibrium, their perpendicular GT and QU have to be the same and that is what was to demonstrate [4].23 The demonstration of Angiulli is a generalization of that reported by Galileo [115] and Bernoulli [39] in which cylindrical communicating vessels were considered with uniform sections. The generalization is made easy by the use of Calculus.

9.1.2 The principle of actions of Vincenzo Riccati Vincenzo Riccati was born in Castelfranco Veneto in 1707 and died in Treviso in 1775. Riccati was the fourth son of Jacopo Riccati. He began his studies at the College of St. Francesco Saverio in Bologna, run by the Society of Jesus under the guidance of the mathematician Luigi Marchenti. In 1734 he went to teach Latin and Italian literature at the College of St. Caterina of Parma and in 1735 began the study of theology, before in the Educandato of San Rocco in Parma, then (1736–1739) in the Institute of St. Ignazio in Rome. From 1739 he taught mathematics in the College of St. Francesco Saverio in Bologna, succeeding Marchenti. In February 1741 he took his vows. Vincenzo 23

pp. 126–128.

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Riccati remained in Bologna until 1773 when, due to the suppression of the Society of Jesus, he returned to Treviso, host of his brothers Montino and Giordano and in the same year refused the chairs of mathematics at the universities of Bologna and Pisa. After less than two years he died in Treviso. He was, with Ruggiero Giuseppe Boscovich, one of the greatest mathematicians of the Society of Jesus [252]. In what follows I will refer mainly to the work the De’ principi della meccanica del 1772 [208], which incorporates the ideas of the previous Dialogo di Vincenzo Riccati della Compagnia di Gesù dove ne’ congressi di più giornate delle forze vive e dell’azioni delle forze morte si tien discorso [207], but exposes them more clearly and in more mature form. Riccati argues that although many ‘writers’ who dealt with the method of statics make use of Bernoulli’s rule of energies, the principle of action is no different from it, and is clearer and tested more solidly. The other scholars, according to Riccati their conclusion based only on the success of the method of energies, in some cases to extend it into more general situations. Here is a comment by Riccati on Johann Bernoulli’s version of the principle of virtual velocities, limited to what was reported by Varignon; Riccati for sure did not know in full the letter of Bernoulli to Varignon of 1715. To notice that Riccati does not cite Lagrange who in 1763 introduced the principle of virtual velocity in the study of the libration of the lune (see Chapter 10): I only warn that the famous theorem of the incomparable Johann Bernoulli, who was shown in all the machines by the most learned Mr. Varignon, is simply a consequence of the equality of contrary actions, which is necessary in any equilibrium. Bernoulli’s theorem is as follows: In any equilibrium of how many and various powers they want, in any way applied, and agents for any direction, the sum of positive energies is equal to the sum of negative energies, as long as you take them as affirmative. By name of energy Mr. Bernoulli does not mean but the product of the power and the virtual velocity of the same power, which will be positive if it follows the direction of the power, negative if it follows the opposite direction. And who does not see that the virtual velocity of the power is proportional to the space, of which the body, or the power, approach the centre of the forces, or whether the powers are elastic ropes, to the contraction or relaxation of the ropes. So Bernoulli’s energy is the same, or at least proportional, to what is called by us action of the power [208].24 (A.9.16)

Riccati begins his enunciation of the principle of action, in a way that was taken almost exactly by Angiulli. He notes that the nature of the equilibrium requires equality between quantities dependent on forces which he, as Angiulli, indicates with the name of power, but the equality cannot be directly between the powers themselves. The equilibrium depends on how the powers act, i.e. on their action. To explain the difference between the powers and their actions Riccati refers to a weight suspended by a thread, even as presumed by Angiulli and writes: To declare as the powers and their actions are distinct, I conceive a heavy body suspended by a wire, which prevents him to descend, and to approach the earth. So far I do not mean but the power of gravity applied to the body, which is contrary to the elasticity of the wire, which contrasts it, and does not leave any chance to have effect. I cut the wire, and the elasticity contrary to gravity is removed. Now the power subsequently and continuously replicates its impulses or stress to the body, which is obliged to change its state. The sum and the aggregate of pulses is named the action of this power, and the effect, i.e. the mutation of 24

p. 237.

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state is proportional not to the power but to the aggregate of its pulses. Therefore it can be distinguished three quantities, namely the power considered in itself, that is usually called pressure, the action, which is the aggregate of its impulses, with which the power pushes the body, composed of the power and the number of pulses, and the effect, i.e. the mutation of state of the body, effect that has not proportion to power, but to its action [208].25 (A.9.17)

He takes from Leibniz the concept of dead force, the power that generates a sequence of infinitesimal impulses continuously replicated. If the pulses of this type are not destroyed, they can be added and should become the action of the power, which is responsible for the change in state. The constraint does not exert a force then, as it is accepted in modern statics, but has merely the role to destroy the power of pulses. Note that Riccati says that the action of the power is directly responsible for the change of state, and the language is here closer to Newton than to Leibniz, as was for Angiulli. According to Leibniz, the action of the dead force produces the living force which – and not the action of the force – is responsible for the change in state. Also the definition of the living force is separated from that of Leibniz and assumes a Newtonian notation; for Riccati the living force is merely the inertia of a body in motion which requires a force to be stopped: So that, therefore, although the centrifugal force is not really anything else than the inertia of the body in some considered circumstances, it is neither useless to introduce it in the reasonings, nor ought it be excluded from physics: in fact it will be profitable to fix its laws, thus the theorems elaborated around such a force by the learned and deep Christiaan Huygens, will be recognized to be true and beautiful. I will answer similarly around the living force. It is by no means so distinct from the force of inertia, rather it is the same force of inertia with some special conditions changed: and it will be useful to consider it with this name, and to fix the laws, which in many problems and research can be of great benefit [208].26 (A.9.18)

I pass over the treatment of Riccati which is substantially the same as that reported by Angiulli to comment on an important observation that the latter will not refer to what Riccati writes: To put in good view our method, and the use of the principle, I must not omit an observation, that appears important to me. When only one motion is possible, as happens to bodies that rotate about any axis, then if the slightest motion is conceived, the spontaneous actions measured by the space of access and recess are found equal, and without hesitation the equilibrium can be deduced. But when more motions in different directions are possible, whether devising some arbitrary motion, I still find the equality of the above actions, but I cannot claim a full equilibrium, but only say that that motion is impossible, and that in that direction the powers are balanced [208].27 (A.9.19)

Riccati then realizes that, for the equilibrium, when one has a system of bodies with more than one degree of freedom, it is not sufficient to impose the vanishing of the action in a single degree of freedom, because then equilibrium would be assumed 25 26 27

p. 13. p. 26. pp. 23–24.

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only with respect to the motions allowed by this degree of freedom, but it is necessary to impose the vanishing of the action in all motions permitted by the degrees of freedom of the system. And he makes the verification by considering the equilibrium of three parallel forces applied perpendicularly to a straight line, considering the cancellation of the actions for displacements resulting from rotation about two separate points. If this condition is met, Riccati shows that the actions are canceled for the rotation of the system of forces around a generic point of the plan. More interesting is the application to the equilibrium of three forces applied to a point. This demonstrates that the rule of the parallelogram has some significance because it is part of the discussion on the validity of the proof of Daniel Bernoulli, ‘based only’ on a priori considerations. Riccati in 1746 had written a work specifically on the rule of the parallelogram entitled Causa physica compositionis ac resolutionis viribus [339], in which he was already using the principle of action. Proving the rule of the parallelogram, he proved to have validated the principle of actions, since, as Bernoulli, Riccati believed that the rule of the parallelogram could be proved a priori. Even Riccati, as Angiulli, uses the ‘demonstration’ of the validity of his principle of action, to join the controversy over the living forces and support the thesis of Leibniz. After presenting the principle of action Riccati passes then to the application of his principle to fluids and to all the large chapter on dynamics, which he studies based on the law f ds = mvdv, instead of f dt = mdv.

9.2 Ruggiero Giuseppe Boscovich Ruggiero Giuseppe Boscovich was born in Dubrovnik (former Ragusa) in 1711 and died in Milan in 1787, his mother was Italian, and his culture was Italian too; for this reason he is often considered as an Italian scientist. At fourteen he was sent to continue his studies in Rome at the Collegio Romano of the Society of Jesus, where in less than thirty years he became one of the most distinguished teachers in the chair of mathematics and geometry, dealing with a broad spectrum of disciplines, from natural philosophy – with the development of a new theory that unifies the physical and chemical forces in a single law – algebraic and geometric calculus problems posed by the type of application in construction engineering, optics, geodesy, meteorology, hydraulics. He was also the author of verse, both in the traditional scientific-didactic poem of Lucretius and in the context of the Arcadian academy, of which he was a member. As a Jesuit and an eminent scientist he was asked to perform delicate diplomatic tasks in a time when scientific and technical expertise were considered important in resolving conflicts over political boundaries, geodetic measurements, possession of watercourses, and so on. His main contribution to theoretical mechanics is the Theoria philosophiae naturalis [48]; note also some more technical works including studies on the motion of solid bodies.

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In late 1742 and early 1743, with Thomas Le Seur, François Jacquier, he gave two reports to the press on their views on the static conditions of the dome of St. Peter. They are the Parere di tre mattematici sopra i danni, che si trovano nella cupola di S. Pietro [137], referred in the following as the Parere, and the Riflessioni sopra alcune difficoltà spettanti i danni, e risarcimenti della Cupola di S. Pietro [138]. The three mathematicians were commissioned by Benedetto XIV, a pope learned and sympathetic to the new science, which would shed light on the health of his ‘residence’, worried by alarming rumors on the spread of cracks. The opinion of the three mathematicians, even for the drama of the conclusions related to a possible collapse, instead of putting an end to rumors, provoked a lively discussion among architects, mathematicians, and ‘gentlemen’, which ended with the appointment of Giovanni Poleni. His suggestions, in fact coinciding with the remedies proposed by the three mathematicians and consisting of the introduction of new metal rims, were finally accepted by the pope and put into practice by the architect Luigi Vanvitelli (1700–1773) in 1748.

9.2.1 A virtual work law for Saint Peter’s dome Of the two reports of the three mathematicians the most interesting is the Parere. It is important in the history of architecture because it represents one of the first attempts to set up on a mechanical basis the testing and design of a structure as complex as a dome. From my point of view, the report is important because it represents the first application of a law of virtual work for the study of a complex system, by introducing some innovation. While in the Parere the division of scientific expertise is not made explicit and no one knows who did what, a few years later Boscovich will consider himself the author of the particular form of the virtual work law used for the static analysis of the dome of St. Peter: I even used some of the research that I had already made twenty years ago on the great dome of St. Peter’s in Rome, and especially the theory which led me to know the force with which an iron ring pulled out by force applied perpendicularly to all points, resist, finding it greater a little more than six times than it would be for the same iron bar pulled directly into the direction of its length, i.e. in proportion of the radius to the circumference of the circle, whence then the Marquis Polini28 conceived of the idea of that experience where a wire of silk, octagon in shape, pulled out from all the angles to be broken, needs a force about six times greater than when another companion wire was pulled directly [49].29 (A.9.20)

The reasons for which Boscovich has used a law of virtual work for the static analysis of the dome of St. Peter are not known. It is possible that he has read the works on the principle of actions of Vincent Riccati, his contemporary and brother of the Society of Jesus.

28 29

Boscovich perhaps deliberately misspells the surname of Giovanni Poleni. p. 54.

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9.2.1.1 The mechanism of failure and the forces After an accurate description of the ‘sufferings’ of the dome in the first part of the Parere, the three mathematicians go to the second part to verify the degree of resistance of the dome. And this in two stages, first identifying the failure mechanism, namely the virtual motions that can support the system of the bodies of the structure, after they have identified the actual and potential failure points, then considering the resistant forces and those which tend to cause the failure of the dome. The failure mechanism, i.e. the ‘system’ is shown in Fig. 9.7. The dome is set on a drum called attico which in turn rests on another drum of much greater thickness, it is reinforced by sixteen great ribs which are constrained by sixteen buttresses and a number of metal chains. To analyze the failure mechanism, the whole structure is divided into sixteen slices, each corresponding to a rib, and the possible motion of each slice is analysed. The base of the drum, cracked vertically moves like a single rigid body with the buttresses of the drum. This complex is represented by the external rectangle ABF in Fig. 9.7b, which shows a section corresponding to the ribs and buttresses. The wall of the drum, the inside of the base and the attic behave as another rigid body, is identified by the internal rectangle BDHI. Each rib identified in Fig. 9.7b by IHMN acts like a single rigid body, bringing with it a piece of the dome. Note that the three mathematicians consider the sixteen ribs of the dome as the only structural elements, neglecting the resistance of the remainder of the dome.

Fig. 9.7. Mechanism of failure of the dome (a). Layouts of the three mathematicians (b, c) (modified from [279])

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Also note that the failure mechanism considered, although not stated explicitly, is three-dimensional, consisting of sixteen radial elements similar to that of Fig. 9.7a connected by round chains, and it is mainly for this link that it is not reducible to a plane system. Once the failure mechanism was determined, Boscovich applied its principle of virtual work, ensuring that the sum of the positive work, due to resisting action, is equal to the sum of the work of negative actions that lead to failure. Notice that Boscovich uses the Galilean term moment for virtual work. Two forces push out on the spring gi [HI in Fig. 9.7], i.e. the weight of the lantern and the weight of the ribs with the portions of the dome, and likewise the two forces, which resist the impulse, i.e. the circular chains, or circles, and the support […], reduced to two distinct components, the first of which is the drum HI, with the interior part of the base CDF and the second the buttresses with the outside part ABF of the detachment of the base itself. […] It is not possible to make a precise estimate of the detachment of the parts and their resistance. It depends largely on the quality of the concrete and the diligence of the work. To evaluate the forces and whether these are in equilibrium, it is convenient first to determine the absolute quantity of them, and then that which by mechanicians is called moment. To get the absolute amount of force, with which on one hand the lantern and the vault of the dome act, and on the other hand the base, the drum, the buttresses act to contrast the pressure, it is convenient to know their weights [138].30 (A.9.21)

For the evaluation of the moments of the weights Boscovich has no difficulty; it is enough to multiply the weight of each masonry mass (ribs, buttresses, etc..) for the vertical displacement of its centre of gravity. More complex is the calculation of the virtual work of the chains that connect the ribs. Boscovich does it elegantly, with a reasoning by analogy, by assimilating the chain to a straight bar of length equal to the circumference of the chain, subject to the same stress. In this way he finds that the moment of each chain is obtained by multiplying the length change of the radius, given by the horizontal displacement of point H in Fig. 9.7, by 6π. Assuming this principle, first it seemed to us, that the energy of a chain of iron, bent in a circle must grow above the absolute force, which would have if it were lying on the straight position, in the same proportion which has the circumference of the circle to the radius, that is a little more than six. Conceive a force distributed throughout the circumference of a circle which is forced to relax and dilate in the act of breaking up, and an iron rod of the same length pulled by another force, such as would do a weight hung vertically. In the latter case, the descent of the weight in tending the fibers would be equal to the sum of all of the extensions of the fibers arranged along the same rod, but in the first by expanding the circle, and growing so its circumference, the force that compels it advances as much as the radius of the circle grows, while the sum of the extensions of the same fibers arranged around would be equal to an increment of the entire circumference [138].31 (A.9.22)

The conclusion of Boscovich’s work is that the chains are not strong enough to ensure the resistance of the dome, and therefore they should be replaced with more robust chains. For more details, see [279, 280]. Here I will only add that Boscovich a few years later applied the yet to be born law of virtual work to study the resistance of the cathedral of Milan [279, 281, 257]. 30 31

pp. 23–24. pp. 26–27.

10 Lagrange’s contribution

Abstract. This chapter is entirely devoted to Lagrange’s VWL. In the first part the first introduction of the law of Lagrange is reported, which has a wording similar to that of Bernoulli. Lagrange calls his VWL and Bernoulli’s the principle of virtual velocities. In the central part the wordings of VWL in the two editions of the Mécanique analytique in terms of virtual displacements (following a foundational route) and the Théorie des fonctions analytique in terms of virtual velocity (following a reductionist route) are presented. In the final part an overview of D’Alembert’s mechanics is presented aimed at an understanding of the extensibility of VWLs to dynamics. A hundred years after the first edition in 1687 of Isaac Newton’s Philosophiae naturalis principia mathematica [175], in 1788 the first edition of Joseph Louis Lagrange’s Mécanique analytique [145] saw the light. Between the two publishing events that symbolize, respectively, adolescence and maturity of classical mechanics, there was an intense and fruitful process of understanding, systematization and generalization of the possible approaches to mechanics. During this period, the age of the Enlightenment, a huge effort was made: of systematisation and development of concepts elaborated in the previous century, especially in mathematics and in physics; of development of the Baconian sciences; of development of technology. In recent past historians of science assumed Isaac Newton’s contribution as expounded in his masterpiece as the climax of classical mechanics and that scholars of the Enlightenment added little to it. Today historians realise that this was misleading and that period far from being a dark century was filled with fundamental contributions and most concepts of mechanics were laid down then [356, 389, 388]. Newton’s mechanics was for sure incomplete; it allowed only the study of the equilibrium and motion of material points free in space, with a mathematical apparatus not completely developed, based on an uncertain Calculus. Problems related to systems of constrained points remained unapproachable, as did the study of continuum bodies either rigid or deformable. Moreover Newton had to face, mainly on the Continent, people scarcely disposed to follow the religious metaphysics behind his work.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_10, © Springer-Verlag Italia 2012

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Thus he was not well considered and Cartesianism was still dominant. This situation promoted a profound innovation in the mechanics as formulated by Newton. Newton’s main concepts, that of force included, remained dominant among scientists but a different approach based on work and energy became a serious contender. The competition was not only on the basis of a more or less appealing ontology but also of either a more simple or a more complex mathematical formulation. At the beginning of the XVIII century Newton was surely seen by his contemporaries as one of the most prominent mathematicians and physicists but not as the one who carried mechanics to its final form. This appears clear from the literature of the time. Newton’s mechanics was considered unsatisfactory by many scholars from both epistemological and ontological points of view, also because of the introduction of forces acting at a distance, which were considered occult entities. More fundamentally for scientists, Newton’s mechanics was considered to be incomplete, because it was limited essentially to material points free in space, and were unsuitable to solve problems raised by the technology of the times. As an example of the opinions of the period, some comments by Daniel Bernoulli and Leonhard Euler are reported below: Theories for the oscillations of solid bodies that up to now authors furnished presuppose that into the bodies the single points relative positions remain unchanged, so that they are moved by the same angular motion. But bodies suspended at flexible threads call for another theory. Nor it seems that to this purpose the principles commonly used in mechanics are sufficient, because clearly the mutual dispositions of points is continuously changing [30].1 (A.10.1) But as with all writings composed without analysis, and that mainly falls to be the lot of Mechanics, for the reader to be convinced of the very truth of these propositions offered, an examination of these propositions cannot be followed with sufficient clarity and distinction: thus as the same questions, if changed a little, cannot be resolved from what is given, unless one enquires using analysis, and these same propositions are explained by the analytical method. Thus, I always have the same trouble, when I might chance to glance through Newton’s Principia or Hermann’s Phoronomiam, that comes about in using these, that whenever the solutions of problems seem to be sufficiently well understood by me, that yet by making only a small change, I might not be able to solve the new problem using this method [100].2 (A.10.2)

So Newtonian principles alone did not seem enough; it was necessary to look for some other more fundamental principles. Problems faced by XVIII century scientists were less demanding from a philosophical point of view than those faced by Newton, nothing less than the search for the universal laws, but this notwithstanding they were not simpler. They concerned for example the search for the oscillation centre of a rigid body and the study of vibrations of a chain. The search for the centre of oscillation was quite a relevant and difficult problem; it is equivalent to finding the length of a simple pendulum with the same period. The problem was substantially solved by Christiaan Huygens in his Horologium oscillatorium of 1646 [135], by means of a first formulation of the theorem of living forces. Jakob Bernoulli came back to the subject in 1713, with a completely different and promising approach, in 1 2

p. 108. Preface, translation by I. Bruce.

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the paper Démonstration générale du centre du balancement a toutes sortes de figure tirée de la nature du levier [32] whose redaction preceded Newton’s Principia. In it one can find roots of both D’Alembert’s principle and the angular moment equation. Johann Bernoulli also faced these problems at the end of the XVII century; a relevant example is published in his Opera Omnia of 1742 [37], where he first introduced the concept of angular acceleration. The problem of a vibrating chain was studied by scientists such as Euler, D’Alembert, Johann and Daniel Bernoulli. Among the other problems that occupied the minds of scientists of the XVIII century it must be remembered the study of the equilibrium of elastic rods, the motion of bodies on mobile surfaces such as, for example, the motion of a heavy body upon an inclined plane were without friction. Johann Bernoulli studied the motion of a material point with a Newtonian approach by introducing among the external forces also the constraint reactions, calling them immaterial forces, as they were outside the bodies in touch [37]. Notice that the assimilation of constraint reactions to ordinary forces were quite common in statics, but in dynamics the problem was much more complex conceptually, because reactions should be endowed with activity. Euler himself, who developed principles of mechanics which made it easy to introduce constraint reactions, tried as much as possible to avoid their explicit use. To conclude, in the solution of the various problems no reference was made to a unique principle and were sought as analogies with already solved problems. This way of thinking is sufficiently documented by the introduction to part II of Lagrange’s Mécanique analytique, which sets forth the history of dynamics from Galileo up to the end of the XVIII century. The description of Newton’s contribution is relegated to the first part and occupies a small space, but in this regard Lagrange’s position is certainly not objective. Most of the considerations are devoted to the study of Jakob, Johann and Daniel Bernoulli, Euler, D’Alembert and Hermann. In the latter part of the introduction Lagrange presents the ‘principles’ most frequently used: the conservation of living forces, the conservation of the motion of the centre of gravity, the principle of the areas and the principle of least action. And it was on the principle of least action that had focused the attention of Lagrange as a young man who, in 1762 published the Application containing the results of his studies on the principle of the minimum action applied to dynamics, making a fundamental contribution to the development of this principle, then seen as the most promising way to solve complex mechanical problems. Despite earlier formulation by Maupertuis and Euler, Lagrange claims for himself the credit for having moved the principle from metaphysics to science.

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10.1 First introduction of the virtual velocity principle Giuseppe Lodovico Lagrangia, better known as Joseph Louis Lagrange, was born in Turin in 1736 and died in Paris in 1813. In 1755 he was appointed assistant professor at the Regie scuole di artiglieria of Turin. In 1757 with Giuseppe Angelo Saluzzo (1734–1810) and Giovanni Francesco Cigno (1734–1790) he founded the Società privata torinese with the aim of promoting research in mathematics and in sciences, which later became the Reale accademia delle scienze of Turin. In 1766 D’Alembert, who will become a great friend of Lagrange, knew that Euler was returning to St. Petersburg and wrote to Lagrange to encourage him to accept a post in Berlin. Lagrange finally accepted. He succeeded Euler as director of mathematics at the Berlin academy in 1766. For twenty years Lagrange worked at Berlin, producing a steady stream of top quality papers and regularly winning the prize from the Académie des sciences de Paris. During his years in Berlin his health was rather poor on many occasions, and that of his wife was even worse. She died in 1783 after years of illness and Lagrange was very depressed. Three years later Frederick II died and Lagrange’s position in Berlin became a less happy one. In 1787 he left Berlin to become a member of the Académie des sciences de Paris, where he remained for the rest of his career. Lagrange was made a member of the committee of the Académie des sciences to standardise weights and measures in 1790. He married for a second time in 1792; his wife, Renée-Françoise-Adélaide Le Monnier was the daughter of one of his astronomer colleagues at the Académie des sciences. In 1794 the École polytechnique was founded with Lagrange as its first professor of analysis. In 1795 the École normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. Lagrange was, with Euler, one of the greatest mathematicians of the second half of the XVIII century [265, 353, 289].

10.1.1 The first ideas about a new principle of mechanics Even before the publication of tome II of the Miscellanea philosophica mathematica societatis privatae Taurinensis, where he presented his version of the minimum action principle [140], there were indications that Lagrange thought of a more fundamental and general principle than that of least action. In a letter to Euler dated 24 November 1759 [318],3 Lagrange wrote that he had composed elements of differential calculus and mechanics and had developed the ‘true metaphysics of its principles’. This can be considered as a symptom of Lagrange’s comprehension of the fundamentality of the principle of virtual work [262].4 Today a virtual work principle is considered more general than that of minimum action, because it allows to take account also the non-conservative forces. But to Lagrange a virtual work principle 3 4

p. 107. pp. 216–218.

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would appear to be more fundamental, mainly because it could be assumed alone as the foundation for all of mechanics. The minimum action principle instead required support of the living force principle; a principle assumed as true but not completely evident To confirm Lagrange’s change of mind in his 1759 letter, I refer to a memoir published in the same tome II of the 1762 Miscellanea by one of his students, David de Foncenex (1734–1799), who was actually older than Lagrange. In this paper, as well as an attempt to rationalize and make precise D’Alembert’s mechanics, a simple vision of the principle of virtual velocities is presented, inspired by Lagrange, who had since 1760 understood the generality of his principle: The composition of forces just as it was made can be used to demonstrate the equilibrium of the lever, and conversely this last proposition once proved, can easily give the composition of forces. It also gives us a highly simple demonstration of the principle of virtual velocities, which can rightly be considered the most fertile and most universal of mechanics: indeed any other principles easily reduce to it, the principle of conservation of living forces, and generally any principle imagined by a few Mathematicians to facilitate the solution of several problems, are nothing but a purely geometrical consequence or better, are this same principle reduced to formulas [107].5 (A.10.3)

These are the same words Lagrange will use in the Mécanique analytique. For this reason it is generally felt that the Foncenex memoir was influenced by Lagrange and can be seen as a witness of the turn of Lagrange regarding the principle of mechanics. An interesting and fairly comprehensive analysis of times and ways of the transition from one to another principle is that of Galletto, who reports the named letter to Euler dated November 24th 1759. Here and elsewhere [314, 260], it is assumed that Lagrange had already developed his method during the preparation of his courses for the Regie scuole di artiglieria in Turin, where the approach to mechanics with the principle of virtual work would have been better understood by students, compared to that based on the principle of least action which required knowledge of analysis too complex for that time. Unfortunately there is no evidence that this is true, and maybe there will never be because the manuscripts of Lagrange’s lessons on mechanics are unavailable. This chapter aims to highlight and critically analyze the principle of virtual work law which is at the foundation of Lagrange’s masterpiece, the Mécanique analytique. The study is conducted with numerous references also to the first publication available on this subject, the Recherches sur la libration de la Lune; it will be considered in some detail also a few years later in a work, dealing with the same subject, the Theorie de la libration de la Lune. For a historical reconstruction of the events that accompanied the publication of these works, I have examined mainly the secondary literature [318, 260].

5

p. 319.

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10.1.2 Recherches sur la libration de la Lune The Recherches sur la libration de la Lune [142] is the first of many memoirs of astronomy written by Lagrange and the work he presented, winning, for the competition held by the Académie des sciences de Paris in 1764 on the topic: “If it can be explained by any physical reason why the moon always presents the same face toward us, and how it may be determined by observations and by theory, if the axis of the satellite is subject to some motion of its own, similar to what is known for the earth’s axis, which produces the precession of the equinoxes”. The work belongs to the Italian period of Lagrange. In [260] there are reported elements of some interest for the reconstruction of the scientific training of Lagrange; analyzing the works produced from 1755 to 1764, one can understand how the main ideas, that made him famous, were already broadly defined. In 1759 the first volume of the Miscellanea philosophica mathematica societatis privatae Taurinensis (Melanges de Turin) was published [139] with three Lagrange memoirs, of which the most important, Recherches sur la nature et la propagation du son, is a long work on the solution of the equation of vibrating strings which already showed signs of the greatness of Lagrange, the same work that Euler expressed a flattering opinion of it [260]. In 1762 the second volume of the Miscellany was published, in which there is another memoir about the nature of sound propagation, but mainly there are two memoirs: Essai d’une nouvelle méthode pour determiner les maxima et les formules des integrales minimum indéfinies [140] and Application de la méthode exposée dans le mémoire précédente à la solution des problèmes de dynamique differents [141]. These two memoirs are not minor works, but they should rather be regarded among the most important contributions of Lagrange to analysis and dynamics [255]. The first memoir lays the foundations for the calculus of variations and the second is a coherent treatise on mechanics, based on the principle of least action. The following year, 1763, Lagrange ended the Recherches sur la libration de la Lune and, in 1765, before finally leaving Turin for Berlin, he wrote a vital work on astronomy, Recherches sur le inégalités des satellites de Jupiter causées par leur attractions mutuelles [142, 143]. In the Recherches sur la libration de la Lune (hereinafter referred to as the Recherches), for the first time, Lagrange formulated the dynamic equations of motion using a “new principle of mechanics” alternative to the least action: a variant of Bernoulli’s rule of energies, named by him and after him the principle of virtual velocity. The originality of the presentation of the virtual velocity principle in the Recherches is not always fully understood [314, 254] . It is widely accepted among historians of science that the ingredients that allowed Lagrange the use of a virtual work law for the study of dynamic problems were already available at the time. In reality it is not. The principle of virtual velocity was not yet an operational tool. Theoretical difficulties remained open to the epistemology of time, to present it as a principle. It is likely that Lagrange until the time of the publication of the Mécanique analytique in 1788 was convinced of the substantial evidence of the virtual

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velocity principle. Most likely, until after that date, as a result of heated discussions that followed in which the positions that deny the evidence of the principle were prevalent, Lagrange changed his mind (see also § 10.3.2) and in the second edition of the Mécanique analytique, immediately after the extremely positive opinion of the first edition, he added the remarks that it is not obvious enough in itself to be elected as a founding principle. There were also technical difficulties such as those related to the types of admissible displacements to be considered, such as whether or not they should be compatible with the constraints. Moreover, the issue of the Recherches is dynamic and the use of the virtual velocity principle requires its generalization to dynamics. It should be “combined with the principle of dynamics due to D’Alembert, it is a species of general formula containing the solution of all problems concerning the motion of bodies” [145].6 The interpretation of the principle of D’Alembert by Lagrange, now classic, although not corresponding to the original formulation, is that the accelerating forces with reversed sign ‘balance’ the applied forces: “This method reduces all the laws of motion of bodies in those of their equilibrium, so reduces Dynamics to Statics” [145].7 Before entering the merits of the use of the virtual velocity principle in the Recherches it is useful to give a short account of Lagrange’s conception of force. The discussions among post-Newtonian scientists on the ontological and epistemological status of force were not yet over, comparing on the one hand the positions of Newton and Euler, who considered it as a primitive quantity, and for whom f = ma is a law, on the other hand the positions of D’Alembert and Lazare Carnot who sees force as a derived quantity, and for which f = ma is essentially a definition. Lagrange assumed a pragmatic and instrumentalist position, which will be followed by most scientists of the XIX century. To them force is a useful concept for mechanics and if one is not too ‘exacting’ it does not create any problem in the development of theories. It is symptomatic in this regard the comment of Poinsot, a few years after the publication of the Mécanique analytique: The force is therefore any cause of motion. Without considering the force in itself, however we conceive very clearly that it is acting in accordance with a direction and with a certain intensity [195].8 (A.10.4)

Lagrange explicitly addresses the idea of force essentially with two brief comments both contained in the introduction to the first part of the Mécanique analytique. Right at the beginning of the 1788 edition, Lagrange will simply say: In general, force or power is defined as the cause, whatever it is, that impresses or tends to give motion to the body to which it is supposed applied, and it is by the amount of motion impressed or about to be impressed, that the force should be estimated. In the state of equilibrium, the force does not have an active role, it produces only a mere tendency to motion, but it must always be measured by the effect it would have if it were not blocked. Taking any

6 7 8

p. 12. p. 179. p. 2.

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force, or its effect, as a unit, the expression of all other forces is only a ratio, a mathematical quantity that can be represented by numbers or lines. It is from this point of view that in mechanics the forces should be considered [145].9 (A.10.5)

As can be seen on the one hand there is the definition of the ontological status of force, conceived as the cause, to which, however, Lagrange does not appear to give special weight. On the other hand, it is shown how to measure it, and how it enters into mechanics as a physical quantity. Even D’Alembert (and to some extent Lazare Carnot), did not consider unlawful the use of the concept of force in mechanics, for the explanation of qualitative character. But for the quantitative determination of the forces it is necessary to refer only to the effects that – according to him – are the only ones able to be measured. In this sense, for D’Alembert, ma is more than a definition of force, it represents the only possible measure of the force-cause based on the effects, which for simplicity is referred to as force. The other comment on force by Lagrange is given in the second edition of the Mécanique analytique, where while comparing the principle of the composition of forces and the law of the lever, he wrote: It can however cannot but recognize that only the law of the lever has the advantage of being based on the nature of equilibrium considered in itself, [that is regarded] as a state independent of the motion, so there is a fundamental difference in the way the powers that are in equilibrium are considered in these two principles, so that, if they were not linked because of the results, one could reasonably doubt that it would be allowed to replace the fundamental principle of the lever to that resulting from independent considerations on the compound motions [148].10 (A.10.6)

This problematic position reflects Lagrange’s embarrassment that is always present to address the problems of statics when power (force) is defined as ma and vice versa when addressing dynamic problems starting from the static concept of power. D’Alembert and Carnot strive to bring their concepts of force to the pre-Newtonian one; Euler conversely tries to base his dynamics on the concept of power, identified as a force capable of producing acceleration. Lagrange instead does not analyze in depth the problem and makes a substantial dichotomy between dynamic force and static force without any concern for the inconsistencies that can be determined at a theoretical level. It is worth noting that in the historical introduction to statics in the Mécanique analytique, Lagrange, to indicate force, uses either the words power (puisssance) and force (in 8500 words or so, one has 78 times power and 87 force). In the historical introduction to dynamics instead he uses almost only the term force (in 8300 words or so, one has 4 times power and 127 force). The same applies basically for the properly scientific parts of the Mécanique analytique, where when dealing with statics the use of power is frequent, and when dealing with dynamics the use of force is almost exclusive. This asymmetry reflects an asymmetry of concepts. In statics, force has always been considered a primitive concept, lacking of very challenging metaphysical connotations, it is the muscle force, that can always be replaced by a weight and the 9

pp. 1–2. p. 17.

10

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word power is consolidated by a long tradition, previous to Galileo. In dynamics, things are different. Here, force has an ontological connotation not yet completely clear. In the ‘scientific’ parts on dynamics of the Mécanique analytique, Lagrange uses a language similar to that of D’Alembert, and then it would seem that for him the dynamical force is for definition equal to ma. It is classified as both accelerating force or driving force. The accelerating force is simply equal to the acceleration: “one can always determine the value of the force acting on bodies in each instant, by comparing the velocity gained in this time with the length of this time” [148].11 The driving force is instead equal to the product ma. Lagrange does not appear to confer a different ontological status to the two terms, although the former is preferred when dealing with a single particle. In the following passage from the second edition of the Mécanique analytique, Lagrange returns to his conception of force: As the product of the mass by velocity expresses the force over a body in motion, so the product of the mass by accelerating force, that we have seen to be represented by the velocity divided by the element of time, expresses the elemental or rising force, and this amount, when taken as a measure of the effort that the body can make because of the elementary velocity that has acquired or tends to acquire, is what is called pressure. But if it is considered as a measure of force or power to give the same velocity, then it is called motive force [148].12 (A.10.7)

The accelerating force, considered as elemental or rising force, is compared with finite force of Cartesian conception, that is the product of mass and velocity. In the final part of the passage it also seems to be a reference to Leibniz’s idea of dead force, according to which pressure is generated by the destruction of elementary impulses. The failure to merge concepts of static and dynamic forces becomes really embarrassing when the way Lagrange treats constraint forces, which are assimilated to ‘real’ forces that constraints exercises ‘really’, is considered. These forces cannot be framed neither in D’Alembert dynamics nor in Euler’s. In the first case because constraints do not exert forces and in the latter because they are placed in an imprecise way. 10.1.2.1 Setting of the astronomical problem In the study of the motion of the moon Lagrange assumes a coordinate system, X,Y, Z, centred in the lunar centre of gravity as shown in Fig. 10.1. As the first coordinated plane he chooses the plane τ parallel to the ecliptic, i.e. the orbit of the earth around the sun. The X axis is directed toward the first point of Aries ϒ, the Y axis is perpendicular to X and contained in τ, the Z axis is perpendicular to τ. The moon is considered as a rigid body, not necessarily spherical in shape. A generic element of it dm is subject to the forces of gravity of the earth and the sun which have the expression: T S dm, dm, (10.1) R2 R2 11 12

p. 240. pp. 245–246.

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10 Lagrange’s contribution Moon detail

dm

Z

M

M X R′

R E

S

Aries point Z

ϒ E

Ecliptic plane

Y

R′ R

X

M Y

Fig. 10.1. Moon, earth and sun configurations for two different instants

where T and S are the masses and R and R are the distances of the moon from the earth and sun respectively – Lagrange, as a custom of the times, avoids exhibiting the gravitational constant. In addition to these forces there should also be considered those quantities that, with a terminology borrowed from D’Alembert, are called accelerating forces, given by: d2X dm, dt 2

d 2Y dm, dt 2

d2Z dm, dt 2

(10.2)

with X,Y and Z that define the position of dm. Immediately Lagrange introduces the ‘principle of D’Alembert’, by quoting: “These accelerating forces taken in the opposite direction and combined with the forces T/R2 dm and S/R2 dm, keep balanced the system of all points dm, i.e. the entire mass of the moon, in equilibrium around its centre of gravity, supposed fixed”. The analysis of the reference system and of the forces is completed in the first two paragraphs. Note that the reference system chosen by Lagrange is not fully inertial and then in addition to the forces he listed there should be considered also the dragging forces. Since the reference system has axes that do not rotate with respect to the fixed stars, drag forces are reduced to a homogeneous field defined by aL dm, where aL is the acceleration of the centre of the moon in its motion around the earth and the sun. If one considers the moon as a rigid body, these forces have no effect on its motion with respect to the X,Y, Z system as they are equivalent to a single force applied in the origin M (the centre of the moon). It is unclear whether Lagrange, in ignoring the drag forces, was conscious of the above considerations, or he has simply let himself be guided by instinct.

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10.1.2.2 The symbolic equation of dynamics Paragraph III of the Recherches is certainly one of the most important of the work. It begins with the enunciation of the principle that lays at the basis of Lagrange’s mechanics: There is a generally true principle in statics, that if any system of bodies or how many points you wish, each solicited from arbitrary powers, is in equilibrium and if someone gives the system a little motion, arbitrary, because of which each point moves along an infinitely small space, the sum of each power multiplied by the distance traveled by the point where it is applied, in the direction of this power will always be zero [142].13 (A.10.8)

Expressed in modern language this principle states that if a system of particles is in equilibrium, the active forces fi to which it is subject have to satisfy the relation ∑ fi · dui = 0, being dui the generic infinitesimal displacement of the application point of fi and dot the scalar product, i.e. if a system is in equilibrium the virtual work of active forces fi shall be zero for any virtual displacement. Lagrange here does not specify the nature of the infinitesimal displacements, it will be understood from the context that they are compatible with constraints. With no other comment Lagrange begins to ‘calculate’ as follows: Imagine that for an infinitesimal variation of the position of the Moon about its centre, the lines X,Y, Z, R, R , assume the values: X + δX, Y + δY, Z + δZ, R + δR, R + δR it is easy to see that the differences: δX, δY, δZ, δR, δR express the distances passed at the same time by point dm in the opposite direction to that of the powers:14 d2X dm, dt 2

d 2Y dm, dt 2

d2Z dm, dt 2

T dm, R2

S dm R2

acting on that point. It will then hold, for the condition [necessary] of equilibrium, the general equation:15    2 d X d 2Y d2Z T S  dm(−δX) + dm(−δY ) + dm(−δZ) + dm(−δR) + dm(−δR ) 2 dt 2 dt 2 R2 R2 L dt

13

pp. 8–9. In the transcription I made a slight typographical variation of Lagrange’s formulas, from the beginning and then even in relation (A), instead to indicate the element of mass with α, I used the symbol dm, putting it after the force per unit mass rather than before. The same notation was used in Lagrange’s Théorie de la libration de la Lune. 15 The negative signs are due for the first three terms to the principle of D’Alembert, for which the accelerating forces must be treated with sign changed, for the last two terms, to the convention on solar and terrestrial gravity forces which are considered positive if attractive, while the change of distance is positive if there is an increment of distance. 14

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i.e., changing sign:   L

   d 2Y d2Z d2X δR δR δX + δY + δZ + T + S dm 2 2 dt 2 dt 2 dt 2 L R L R

(A)

[142].16 (A.10.9)

Lagrange’s comments on how he arrives at equation (A) are extremely scanty. He will remedy, but only in part, his terseness in the next paragraph. The language of formulas is however sufficiently clear. According to the principle “true in general” one has to sum (integrate) the products of the elementary gravity and accelerating forces multiplied by the motions of their points of application and then impose this sum equal to zero. It is not entirely clear whether Lagrange considers the balance of forces existing in the portion of space occupied by the moon, or considers the equilibrium of material points dm, rigidly connected to each other, which form the moon. Distinguishing one case from the other, allows one to see if Lagrange is applying the principle of virtual velocities in exactly the form given to him by Bernoulli, which affects the balance of forces alone or, if he enters in the classical tradition of virtual work laws, according to which the equilibrium concerns bodies more than forces. The analysis of the entire Lagrange’s text seems to lead to the second possibility, and when he speaks of the elements dm it seems implied that he refers to a material point. When he is proving the theorem of living forces he will apply the “principle true in general in statics” explicitly to a system of particles and finally when he is concerning the changes δX, δY, δZ, δR, δR he speaks about the motion of the elements dm and not the points of application of forces. On the other hand, this interpretation is put a bit in crisis by the fact that Lagrange did not mention at all the internal forces between individual elements dm. These internal forces are certainly compatible with the concept of force of Lagrange, who admits the physical reality of the constraint forces (see Section 10.2.1), and therefore can consider them originated from the interaction of individual particles. Lagrange’s silence can be explained, however, in two ways. With the first way, which I think is the more plausible, it is arguable that the concept of internal forces was not too clear to Lagrange, which can perform as Euler when, in his study on the motion of rigid bodies in 1750, decided not to consider the influence of internal forces based on the heuristic principle that they could not give a global contribution. With the second way it can be assumed that the internal forces are associated with constraint reactions and, in view of the application of the virtual velocity principle they can be ignored because their virtual work is zero. It should be noted that, for a comprehensive and satisfactory analysis of the internal forces, it is necessary to wait for the work of Cauchy in 1822 [63], in which the concept of tension in continua is presented, as the internal forces of contact. The above equilibrium equation (A) is now under the name of symbolic equation of dynamics. In the Mécanique analytique Lagrange will refer to this as the general 16

p. 9.

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formula of dynamics. I would like to stress that to get it, without highlighting the thing, he used two independent principles: the first, referred to as “true in general in Statics”, is clearly the virtual velocity principle, while the second, which consists in the use of the accelerating force with sign reversed, is, as will be explained further below, the version of the principle of D’Alembert given by Lagrange. It is absolutely not true that the virtual velocity and D’Alembert principle were widely known and shared by ‘geometers’, and indeed it can be argued with good reasons that Lagrange was the first to enunciate clearly and disseminate them. And Lagrange in the Mécanique analytique, had a different approach with the two principles, for the virtual velocity principle he presented a demonstration – at least in the second edition and in the Théorie des fonctions analytiques – and for the principle of D’Alembert he provided a comment with a rather extensive historical analysis. The paragraph IV of the Recherches is intended to clarify what stated in the previous paragraph. Given the clarity of Lagrange, it is best to report his comments in full: The principle of Statics that I outlined, in the end is nothing but a generalization of what is usually called the principle of virtual velocities [emphasis added], which has long been known by Geometers as a fundamental principle of equilibrium. Mr. Jean Bernoulli is the first, for what I know, who has seen this principle in general form and applied it to all matters of statics, as it can be seen in section IX of the new Mechanics of Mr. Varignon, where such skilled Geometer, after referring Bernoulli’s principle, shows, in various applications, it leads to the same conclusion as that of the composition of forces [142].17 (A.10.10)

Note that though Lagrange was the first to name Johann Bernoulli’s rule of energies principle of virtual velocities, to identify the virtual work he used instead of the term energie the Galilean term moment. The next part of the Recherches, paragraph V, is dedicated to making explicit the symbolic equation of motion and the solution of the differential equations that follow. Of some interest is the presentation, at the end of section IV, of probably the first ‘satisfactory’ proof of the theorem of living forces, obtained as an immediate application of the symbolic equation of motion [267]. In the following I will only expose some hints on how to make explicit the symbolic equation of motion, with the introduction of the Lagrangian coordinates. The symbolic equation of dynamics contains virtual displacements not yet analyzed; for them the space of admissible values is not defined. Lagrange expresses the idea of admissibility using the concept of independent variables (known today as Lagrangian variables or coordinates) already introduced in the Applications [282] and developed, after the Recherches, in the Théorie de la libration de la Lune and in the Mécanique analytique, by recognizing that the virtual velocity principle leads to many balance equations as there are independent variables. In the words of Lagrange a few years later: In the following, keeping in mind the equations of condition among the coordinates of the different bodies, given the nature of this system, the variation in those coordinates will be

17

p. 10.

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10 Lagrange’s contribution

reduced to the smallest possible number, so that the resulting variations are completely independent and totally arbitrary. Then if the sum of all terms with each of these variation will be equated to zero, it will be obtained all the necessary equations for determining the motion of the system [145].18 (A.10.11)

In the three-body problem that Lagrange is studying, consisting of earth, sun – represented by two material points located in their centre of gravity – and moon – treated as a rigid body – there are three degrees of freedom each for the earth and the sun (the three motions of their centre of gravity) and three degrees of freedom for the moon (the three rotations). In total there are nine degrees of freedom and the equation of symbolic dynamics is expected to deliver nine equilibrium equations. In the simplified treatment of the three bodies that Lagrange is considering, he takes as known – without ever explaining clearly that assumption – the positions of the earth and the sun and, therefore, virtual displacement involves only possible motions of the moon, identified by three angular coordinates, to which, through the symbolic equation of dynamics, are associated three balance equations. 10.1.2.3 The virtual velocity principle It is not entirely clear what Lagrange meant by the term ‘generalization’ used in the opening of paragraph IV: “The principle of statics that I come to expose in substance is nothing but a generalization of what it is usually called the principle of virtual velocities”. On the one hand, Johann Bernoulli’s principle of virtual velocity that refers to concentrated forces, is certainly extended to distributed forces, and its range of validity is specified more precisely. On the other hand, the generalization concerns the extension of the principle to dynamics by assimilating accelerating force, with sign reversed, to ordinary forces. Bernoulli’s statement does not justify in full the applications that Lagrange makes. Here it should be noted that Lagrange for sure knew Bernoulli’s rule of energies only from what was reported by Varignon in his Nouvelle mécanique, because the letter of Bernoulli to Varignon in 1715 was not published. From what was reported by Varignon it can be evinced as follows: • virtual velocities are not necessarily compatible with constraints. Or better Bernoulli gives no particular attention to the problem of constraints, and he considers a general system of forces in equilibrium, which may also contain constraint reactions; • the class of virtual velocities is limited to a single degree of freedom; • the wording of energy rules is expressed in the language of geometry and the displacement and force vectors are represented as oriented segments; • the rule is limited to systems of particles and concentrated forces. Lagrange will change all these points. • the virtual velocities, according to the classical formulations of virtual work laws are always compatible with constraints, implicitly assumed holonomic and independent of time; 18

p. 197.

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• the virtual velocities have a variation defined by the number of degrees of freedom of the system under examination; • applications are purely analytical, as Lagrange works with components he needs to not consider the projection of forces in the direction of the displacement, since the components of the forces and those of the displacements have the same direction; • forces are not necessarily concentrated, and indeed in the Recherches they are only distributed. The idea to consider distributed forces other than those due to weight, and in particular the accelerating forces, was not common also after 1750, when Leonhard Euler, in a memoir of the Academy of sciences of Berlin [101] proposed that Newton’s second law ( f = ma) could also apply to entities other than the material points, such as an infinitesimal element of a continuum (d f = dma). From this point of view the generalization from concentrated to distributed forces, did not seem trivial. For the generalization to dynamics see § 10.4.

10.1.3 The Théorie de la libration de la Lune The Théorie de la libration de la Lune is another of Lagrange’s great work of astronomy, published in the memoirs of the Academy of sciences in Berlin, for the year 1780 [144] and written partly in response to questions about motions of the moon that remained open after publication of the Recherches. Here Lagrange introduces and uses the virtual velocity principle, initially without any reference to the specific astronomical problem, but referring directly to the general case of an indefinite number of bodies. A significant improvement in the analytical aspects should also be registered, in particular in the calculation of the virtual work of inertia forces, so that the developments of the Théorie de la libration de la Lune are substantially similar to those of the Mécanique analytique and also contain a statement of the now famous Lagrange equations. At the beginning of section I there appears an introduction of the virtual velocity principle and the principle of D’Alembert: 1.The principle provided by Mr. D’Alembert reduces the laws of dynamics to those of statics, but the search for these laws by the ordinary principles of equilibrium, the lever and/or the composition of forces, is often long and painful. Fortunately there is another principle of Statics, more general, and, above all, that has the advantage that it can be represented by analytical equations, which alone contains the conditions of equilibrium of any system of powers. This is the principle known as the law of virtual velocities. It usually will be set this way: when two powers are in equilibrium, the velocities of the points to which they are applied, estimated according to the direction of these powers, are in inverse ratio to the powers themselves. But this principle can be made more general as follows. 2. If any system of bodies, reduced to some points subject to any forces, is in equilibrium and if this system is given any little motion for which each body moves along an infinitely small space, the sum of the powers each multiplied for the displacement of the point where it is applied along the direction of this power is always zero [144].19 (A.10.12) 19

pp. 15–16.

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Lagrange presents here, as he will also do in the Mécanique analytique, two different statements of the virtual work law. The first is indirect, for only two bodies, and resembles the wording of the law of the lever; the second refers to that of Johann Bernoulli. Saying that the first ‘principle’ can be made more general by the second, Lagrange wishes to emphasize that the statement of Bernoulli is somewhat implied by a description of the law of the lever generalized, scaling down in some way the originality of Bernoulli. To make explicit his virtual work law, following shortly after, Lagrange adopts from the outset an approach that foreshadows its application to dynamics. Instead of powers he speaks of accelerating forces – which this time as with D’Alembert are just accelerations. Although the virtual work law is applied to the case of any number of points and not only the earth-moon system, the accelerating forces are considered always acting toward a centre. If P, Q, R, ..., P , Q , R , . . . are the accelerating forces acting on the mass points m, m , . . . toward the centres p, q, r, ..., p , q , r , . . . , and if there is equilibrium, the following equation is obtained – the symbols are Lagrange’s: m(Pδp + Qδq + Rδr + . . . ) + m (P δp + Q δq + R δr + . . . ) + · · · = 0. (10.3) To get the values of the changes δp, δq, δr, . . . , δp , δq , δr , . . . the expression of the distances p, q, r, . . . , p , q , r should be differentiated considering the centres of forces as fixed [144].20 (A.10.13)

The Théorie de la libration de la Lune continues to consider the virtual work of the inertia forces, providing a general expression, and introducing technical refinements to the Recherches. I will limit myself here to indicating only the additional evidence Lagrange gives of independent variables: Furthermore, given the mutual positions of the bodies, there will be many constraint equations between the variables x, y, z, x , y , z , . . . by which it is possible to express all the variations one over the other or rather by other variables in small number and such that they are entirely independent and correspond to the various motions that the system can receive [144].21 (A.10.14)

10.2 Méchanique analitique and Mécanique analytique In 1788 the first edition of Lagrange’s masterpiece was published in one volume with the title Méchanique analitique; it was published in a second edition in two volumes with a slight change of title, Mécanique analytique, reflecting changes in the written French language; the first volume released in 1811 with Lagrange still alive, the second in 1815 [145, 148, 150]. The third edition was published in 1853– 1855 by Joseph Bertrand (1822–1900); it differed from the second edition only for 20 21

p. 16. p. 20.

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253

the different typeface and the addition of notes by Bertrand. The fourth edition is the one shown in the complete works of Lagrange edited by Joseph Alfred Serret and Gaston Darboux. It was printed in 1888–1889 and it is simply a reproduction of Bertrand’s edition. An English edition is also available [153]. In the following, as it was indicated previously, I shall refer to the work of Lagrange as to the Mécanique analytique when it is not necessary to specify the edition. Otherwise I will talk about the first edition of the Mécanique or the Méchanique, or of the second edition of the Mécanique analytique. According to the not uncommon use of time, the Mécanique analytique contained historical ‘hints’, except that these hints were quite substantial and well made, so that they are generally regarded as the first example of a history of mechanics, an example that has influenced modern history for a long time. The Mécanique analytique is divided into two parts, one dealing with statics and the second with dynamics; each of the two parts is preceded by a history. The history of the first part, the only one I will examine below, presents the virtual velocity principle as the most recent of the principles used in mechanics. It was preceded by the principle of the lever and the law of composition of forces. Lagrange’s task is precisely to implement the application of the virtual velocity principle to all problems of mechanics. Only after the publication of his work, prompted by criticism from his colleagues, will Lagrange commit himself to a demonstration of the principle. Those who have hitherto written on the principle of virtual velocities have dedicated themselves to prove the truth of this principle, by the conformity of its results with those of the ordinary principles of statics, rather than to show the uses that can be made to directly solve problems of this science. We have proposed to dedicate ourselves to that subject with all the generality of which it is liable and to deduce from the principle at issue the analytical formulas, which contain the solution to all problems of equilibrium of bodies, in the same way the formulas of subtangents, the osculating radius, etc. contain the determinations of these lines among all the curves [145].22 (A.10.15)

10.2.1 Méchanique analitique In the historical part of the first edition of the Mécanique analytique, after a brief historical summary, citing Galileo, Torricelli, Descartes, and Wallis, Lagrange sets out the virtual velocity principle in substantially the same form in which it was exposed in the Recherches, attributing its formulation to Johann Bernoulli: If any system of as many bodies or points one wishes, each solicited from any powers, is in equilibrium, and if this system is given an arbitrary small motion, under which each point passes along an infinitely small space, which will be its virtual velocity, the sum of powers, multiplied each by the space that the point where it is applied passes in the direction of that power, will always be zero, considering as positive the small spaces in the direction of power and as negative the spaces in the opposite direction [145].23 (A.10.16)

22 23

pp. 44–45. pp. 10–11.

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10 Lagrange’s contribution

Plate 4. Front page of two editions of Lagranges’s Mécanique analytique (reproduced with permission, respectively, of Accademia Nazionale San Luca, Rome, and of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

The historical introduction of statics ends with the following passage: I think I can say in general that all the general principles that can still be discovered in the science of equilibrium, will not be but the same as the principle of virtual velocities, given in a different way, and from which they differ only in form. But this principle is not only itself very simple and general, it has, in addition, the valuable and unique benefit to result in a general relation that contains all problems that can be posed on the equilibrium of bodies. We will expose this relation in all its extensions, we will also try to present it in an even more general way than what has been made to date, and provide new applications [145].24 (A.10.17)

Here the confidence of Lagrange, in both the capacity of the virtual velocity principle to solve whatever mechanical problem and its simplicity of use, is clear. Theoretical-technical aspects begin to be addressed after the historical part. Lagrange cannot bring himself to apply directly Bernoulli’s virtual velocity principle, and tries to justify it with a simpler and more traditional statement, as he made in the Théorie de la libration de la Lune: The general law of equilibrium in machines is that the forces or powers, are among them in inverse proportion to the velocities of the points where they are applied, estimated in the direction of these powers [145].25 (A.10.18)

This virtual work law, less general than Bernoulli’s, as it refers only to two forces, is more simple and intuitive. Before presenting Lagrange’s proof is worth noting 24 25

pp. 11–12. p. 12.

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that the two forces, to which the above statement refers, must not be thought of as applied to the same point; they may be and generally are applied to different points of a machine, which transfers forces from one point to another. The two points are moving in the direction set by the kinematics of the machine itself; the statement contains the law of the lever as a special case. Lagrange begins his demonstration from three forces, P, Q and R in equilibrium with each other, generally applied to three distinct points p, q and r, somehow connected. For the principle of solidification (see also § 14.2.1), the equilibrium is not disturbed if any of the points of application of the forces is supposed fixed. Assuming, for example, r as the fixed point, the forces P and Q are still balanced between them. Denote by d p and dq the virtual velocities, i.e. the infinitesimal displacements, of the points p and q estimated in the direction of P and Q, in the case of any act of motion. The above virtual work law gives: P dq =− Q dp

(10.4)

or Pd p + Qdr = 0. For the three powers, considered two by two, then it is: Pd p + Qdq = 0;

Pd p + Rdr = 0;

Qdq + Rdr = 0.

(10.5)

To obtain an equation of virtual work valid for the three forces, Lagrange uses an argument, not too sharp, which will resume and improve in the second edition of the Mécanique analytique [274].26 In essence he argues that because the points p, q and r influence each other, among their displacement d p, dq and dr there must exist a relationship, linear because infinitesimal displacements are concerned. For example, it can be written: d p = mdq + ndr,

(10.6)

where m and n are numerical values that depend on the constraints and the type of virtual displacements considered. If dr = 0 is assumed, as in the first of (10.5), it is d p = mdq; dq = 0, as in the second, it is d p = ndr. Substituting these values in the first two of (10.5), one obtains: Pmdq + Qdq = 0 Pndr + Rdr = 0,

(10.7)

which adding member to member give: P(mdq + ndr) + Qdq + Rdr = 0

(10.8)

Pd p + Qdq + Rdr = 0.

(10.9)

or, for (10.6):

26

p. 137.

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10 Lagrange’s contribution

A similar result is obtained by combining (10.5) according to the other two possible combinations (the first with the third and second with the first). The proof outlined above lends itself to a process of mathematical induction, in the sense that the virtual work law for n + 1 powers is valid knowing it is valid for n power. For any number of them it is then: Pd p + Qdq + Rdr + etc. = 0,

(10.10)

which is precisely the principle of virtual velocities according to Bernoulli. At this point Lagrange introduces the definition of moment: “We will call each term of this formula, the moment of force, using the word moment in the sense Galileo has given to it, i.e., as the product of force for the virtual velocity. So that the general formula of equilibrium will consist in the equality of the moments of all forces to zero” [145].27 The term moment will continue to be adopted for a long time until it is gradually replaced by virtual work introduced by Coriolis (see Chapter 16). Lagrange continues: ‘With the use of this formula, the difficulty is reduced to determine the values of the differentials d p, dq, dr, in accordance with the nature of the given system” [145].28 The following is a brief but precise explanation how to get the expressions of differentials: We replace the expressions of d p, dq, dr, &c. in the proposed equation, and because the equilibrium of the system holds in general and in every sense, this equation must be satisfied, regardless of all the indeterminate quantities. It will be equated to zero separately the sum of the terms affected by the same indeterminate quantity [emphasis added]. There will be as many particular equations as many indeterminate quantities. Now it is not hard to believe that their number must always be equal to that of the unknown quantities of the configuration of the system, then this method will give as many equations are those needed to determine the equilibrium state of the system [145].29 (A.10.19)

The indeterminate quantities are those today called Lagrangian coordinates, which Lagrange, as reported in the previous pages, had already introduced in the Addition of 1762. This brief explanation of how to obtain the equilibrium equations will be completed in the second edition of the Mécanique analytique where Lagrange also introduces the concept of generalized forces. In the third section of the Méchanique analitique, Lagrange starts applying the virtual velocity principle to determine the equations that are necessary – but that may not be sufficient – for the equilibrium of a system of material points p, p , p , etc. subject to the forces P, P , P , etc., constrained together in some way (for the rigid body the equations that are obtained are also sufficient for the equilibrium). Lagrange gets first the equations of equilibrium to translation, then to rotation. In both cases, he divides the generic virtual displacement of the system of material points into two parts: a global displacement of the system and relative displacements between the points. He chooses as global motion that of an arbitrary point p, while the relative motions are assumed equal to the difference of their total motion and that of p. In 27 28 29

pp. 15–16. p. 16. p.16.

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the following I report only the proof of the equilibrium to translation, because it is a bit simpler and equally explanatory of the approach used by Lagrange. Naming x, y, z the coordinates of the point p, x , y , z , those of the point p ,  x , y , z , those of point p , he poses: dx

dx = dx + dξ, dy = dy + dη, dz = dz + dζ;      = dx + dξ , dy = dy + dη , dz = dz + dζ ;

(10.11)

where dξ, dη, dζ represent the relative motions and dx, dy, dz the global motions. Applying the equation of moments he obtains: 0 = (P cos α + P cos α + P cos α + etc.) dx + (P cos β + P cos β + P cos β + etc.) dy + (P cos γ + P cos γ + P cos γ + etc.) dz + P (cos α dξ + cos β dη + cos γ dζ)  + P (cos α dξ + cos β dη + cos γ dζ ) + etc.,

(10.12)

where α, β, γ, α , β , γ , etc. are the direction cosines of the powers P, P , etc. Assuming that the system is isolated, it is clear that the constraints only result from the mutual relations between the material points, and then from ξ, η, ζ, ξ , η , ζ , etc. and not by x, y, z, etc. which may vary arbitrarily. So because the above equation is satisfied for all possible virtual displacements, it is necessary that the coefficients of dx, dy, dz vanish. Lagrange has therefore come to the following, already well known as the cardinal equations of equilibrium to translation [145]:30 P cos α + P cos α + P cos α + etc. = 0 P cos β + P cos β + P cos β + etc. = 0 P cos γ + P cos γ + P cos γ + etc. = 0.

(10.13)

The method of Lagrange to obtain the cardinal equations, and in particular the meaning of the terms where relative displacements appear, attracted the curiosity of Fossombroni which provided an alternative derivation also based on the virtual velocity principle [109].31 In section IV Lagrange introduces his method of multipliers. Let L =const., M = const., N = const., etc. be the constraint equations that govern a system of material points with coordinates x , y , z , x , y , z , etc. which differentiated lead to the conditions dL = 0, dM = 0, dN = 0, etc.; then the following considerations apply: Now since these equations must be used to remove an equal number of differentials in the equation of virtual velocities, after which the remaining coefficients of the differentials must all be matched to zero, it is not difficult to prove through the elimination of linear equations, that it will be the same if to the equation of virtual velocities the different constraint equations dLN = 0, dM = 0, dN = 0, & c. are added, each multiplied by an indeterminate coefficient, and then the sum of all terms that are multiplied by the same differential is equated to zero. 30 31

p. 28. The equations of moments are deduced on pp. 28–29. pp. 101–115.

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10 Lagrange’s contribution

This will provide as many particular equations as the number of differential equations. Finally, the indeterminate coefficients by which the constraint equations are multiplied, are eliminated from these equations [145].32 (A.10.20)

Applying the above ‘instructions’ Lagrange gets the equation of the type: Pd p + Qdq + Rdr + etc. + λdL + μdM + νdN = 0,

(10.14)

that he names general equation of equilibrium. Note that in this equation d p, dq, dr etc. can vary freely, as if there were no constraints attached. In the next sections of the Mécanique, Lagrange addresses problems of some interest in mechanics, including Maupertuis’s law of rest; they do not cover basic aspects of virtual work laws and therefore will be ignored. 10.2.1.1 Constraint reactions In mechanics, the idea of constraint reactions has evolved along with those of force and constraint. For Aristotle, a constraint was essentially an impediment for a body to reach its natural place. Removing the constraint leaves the body free to move. Even among the ancient Greeks, especially among engineers, it was clear that the effect of a constraint could be obtained with a power, a muscular force for instance. If a heavy body was fixed on a hook by means of a rope, it was clear that the role of the hook could be played by a muscular force, appropriate to support the weight. Therefore, the possibility of interchangeability between constraints and powers is seen from the beginning of mechanics, although they remain distinct concepts, the constraint does not exert a force on a body but it seems as if it does. One can talk about constraint reaction as the force that, for equilibrium, has the same effect of the constraint. By accepting the rule of the parallelogram as the primary tool for addressing the study of static problems, which occurred due to Varignon’s Nouvelle mécanique ou statique of 1725, the reaction forces in this sense began to appear explicitly as geometric or algebraic variables in the calculations. As the equilibrium reduces to the annulment of the sum of the forces, if also a constraint contributes to the equilibrium, there is nothing more natural than in the equations of equilibrium symbols appearing to represent the forces equivalent to constraints, e.g the reactive forces. Only after Newton, with the introduction of forces at a distance as physical magnitudes, the principle of action and reaction and the emergence of the corpuscular concept of matter and the theory of elasticity, did the ontological status of the reactive forces begin to change. The constraints are no longer, in general, impediments to motion but they become bodies composed of particles that are centres of forces and the constraint reactions are ‘real’ forces that the constraint-body exert over other bodies which are to interact with them. In the XVIII century, constraints are still generally modeled as hard bodies, that is, as being capable of absorbing motions and impulses acting at right angles to them, but toward the end of the century with the emergence of the Eulerian and Newtonian concepts of force, constraints are entities treated as dispensers of forces and thus 32

pp. 45–46.

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259

no longer merely passive. By the XIX century, this second point of view becomes prevalent, especially among the French scientists. Lagrange is situated in an intermediate position, on the one hand he considers constraint forces as the forces required to perform the functions of the constraints, on the other hand he gives them the ontological status of active forces, or powers. He introduces the reactive forces from the general balance equation given above, noting that to terms λdL, μdM, νdN one can give, by analogy, the mechanical meaning of moment and that “each equation of constraint is equivalent to one or more forces applied to the system in accordance with given directions, so that the equilibrium state of the system will be the same – either one uses the account of these forces, or one refers to the equation of constraint [145].33 The direction of the forces is orthogonal to the surfaces L = 0, M = 0, N = 0. Regarding the interpretation of the multipliers λ, μ, ν, etc. as forces, Lagrange states: Reciprocally, these forces can replace the constraint equations resulting from the nature of the given system, so that with the use of these forces the system may be considered as completely free and without any constraint. And the metaphysical reason [emphasis added] can be seen, because the introduction of the terms λdL + μdM + &c. in the general equilibrium equation, makes that this equation can then be treated as if all the bodies of the system were completely free. This is the spirit of the method of this section. Properly speaking, the forces in question shall take account of the resistance that bodies have to bear because of the mutual constraints, or by the obstacles which, by the nature of the system, may oppose to the motion, or rather those forces are not but the same forces of resistance, which must be equal and opposite to the pressure exerted by the bodies. Our method provides, as we see, the means to evaluate this resistance. This is not one of the minor benefits under this method [145].34 (A.10.21)

As it can be seen, he explicitly introduces the constraint reactions as ordinary forces: “those forces are nothing but the very forces of this resistance, which must be equal and opposite to the pressure exerted by the bodies”, and also says that it is important to determine the constraint forces. This position puts him at odds with the mechanics of D’Alembert and Lazare Carnot, where constraints destroy the motions but do not exert forces.

10.2.2 Mécanique analytique The second edition of the Mécanique analytique registers a significant number of changes. With regard to the basic aspects of the principle of virtual velocities, with some refinements and additions which I have already mentioned, the most notable addition is a demonstration. Apart from the fact itself, it is important to point out the awareness by Lagrange of the problematic nature of the principle, which is well expressed by the considerations he makes following the assessments reported in the first edition (see previous section): 33 34

p. 49. p. 49.

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10 Lagrange’s contribution

As to the nature of the principle of virtual velocities, it is not so self-evident that it can be assumed as a primitive principle, but it can be considered as the expression of the general law of equilibrium, deduced by the two principles that we set out [that of the lever and composition of forces]. So in the proofs that are given of this principle it is always considered due to one of these, more or less directly. But, in Statics there is another general principle independent of the lever and the composition of forces, although the mechanicians will commonly refer it to them, which would seem to be the natural foundation of the principle of virtual velocities: you can call it the principle of the pulleys [emphasis added] [148].35 (A.10.22)

According to Lagrange the principle of virtual velocities has its foundation in the principle of the pulley. This principle says that if one considers a system of two pulleys, consisting of a fixed and a movable one, and wraps around them an inextensible rope, the relationship between power P and resistance R is 1/n, with n the number of cords. The ‘cords’ are the whipping situated on either side of the pulley, which may be either an even number or an odd number, not to be confused with the number of laps. For example, with reference to Fig. 10.2, there are 4 cords on the left and three on the right, for a total number of 7 cords; the laps are only 3. Lagrange argued that the principle of the pulley is absolutely self-evident because it is clear that all the cords of the rope have the same tension – supposing the absence of friction – and it is also clear, with reference to Fig. 10.2, that the lower pulley is sustained by a force equal to the tension of the rope multiplied by the number of cords. Notice that Lagrange does not attribute to the principle of the pulley the status of a virtual work law because at the moment he avoids any kinematical analysis. In the following I briefly summarize Lagrange’s proof, trying to interpret it. Referring to Fig. 10.3, consider three couples of pulleys set out at points A, B, C – the number of pulleys is limited only for ease of exposition, the following arguments apply equally to any number of them. The movable pulleys are placed at A , B , C . An inextensible rope is wrapped P times around AA , Q times around BB , R times around CC , or rather P, Q, R, are equal to the number of the cords of the pulleys. After being wrapped around CC , to the rope is hung a weight Γ. This system can be taken to represent a system of three forces P, Q, R commensurable with each other, applied to points A , B , C of a generic body, or a system of bodies, linked together and directed as AA , BB , CC . To show this assume the

4

Fig. 10.2. A simple system of two pulleys 35

p. 23.

3

10.2 Méchanique analitique and Mécanique analytique

261

A P

Q

uA′ α P

A′

β

C uB′

R

B′ C′

γ uC′

Q

R

B Γ Fig. 10.3. The system of pulleys equivalent to the system of

forces36

weight Γ as the maximum common divisor of P, Q, R and a unit of measure so that it is unitary. The unitary weight Γ applied to the last system of pulleys will be able to move a resistance and then to exert a force equal to PΓ = P ×1 = P in the first system of pulleys, to Q = Q in the second and to R = R in the third (notice that an assembly of such pulleys may represent only the central forces of centres A, B and C; for forces with constant directions it is sufficient to take AA , BB and CC very large). Points A , B , C move in directions that are allowed by the constraints. Let α, β, γ be the components of the infinitesimal virtual displacements uA , uB , uC of these points in the directions of the forces P, Q, R, that is their virtual velocities coinciding with the variation of distance between fixed and mobile pulleys. Because the rope is inextensible, the virtual vertical displacement Δl of the unit weight Γ is given by: Δl = Pα + Qβ + Rγ,

(10.15)

which expresses the variation of length of the rope wrapped around the three couples of pulleys. According to Lagrange, for the system of bodies to be in equilibrium it is necessary that in the virtual motion the weight does not sink, otherwise it will actually sink and then there will not be a state of rest. So the relation should be valid: Pα + Qβ + Rγ ≤ 0.

(10.16)

However, reversing the direction of the virtual displacements, which Lagrange considers always possible, and repeating the argument, one must also have: P(−α) + Q(−β) + R(−γ) ≥ 0.

36

The figure is a variant of Fig. 54 in [355], p. 66.

(10.17)

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10 Lagrange’s contribution

The only way to to satisfy both relations (10.16) and (10.17) is to accept the sign of equality, that is to have: Pα + Qβ + Rγ = 0.

(10.18)

But Pα, Qβ and Rγ are the moments of the forces P, Q and R because α, β and γ are the virtual velocities of their points of application while the number of wrappings of P, Q, R, can be interpreted as P, Q, R, and then the expression (10.18) says that a necessary condition for the equilibrium of a system of bodies is that the sum of the moments of all powers vanish. Notice that the reversibility of the virtual velocities considered as not problematic by Lagrange is possible only for ‘regular’ constraints. In such a case if, for example, f (u, v, w, etc.) = 0 is a constraint equation, fu du + fv dv + fw dw = 0 is the constraint equations for infinitesimal displacements which are linear, and if it is satisfied by du, dv, dw, it is satisfied also by −du, −dv, −dw [152], [274].37 The relation (10.18), if satisfied by any possible value of α, β and γ is also a sufficient condition for the equilibrium. Suppose that indeed (10.18) is satisfied for a set of virtual velocities α, β and γ. Given the linearity, it is also satisfied by −α, −β and −γ. According to Lagrange, since there is no way to prefer one or the other of two possible motions of the system they must both be zero and so the system is in equilibrium. The proof can be easily extended to the case of powers P, Q, R, etc. not commensurable with each other, using a limit process because it is known that all propositions proved for commensurable quantities can be proved equally when these quantities are incommensurable by means the reductio ad absurdum [148].38 It is clear that the demonstration of the virtual velocity principle referred to above does not require only the principle of the pulley. For the proof of the necessary part of the principle, Lagrange assumes that the weight Γ naturally tends to sink if the constraints, as determined by ropes and pulleys, allow it to do so, and this certainly upsets the equilibrium, then the weight Γ cannot sink. This principle of natural descent of heavy bodies is probably the most natural of all of mechanics, but it is not deductible a priori and it is true because of everyday experience. It is more intuitive also of the principle of the impossibility of perpetual motion. For the sufficient part of the demonstration Lagrange then uses a principle that is generally accepted as valid a priori, i.e. the principle of sufficient reason, which states that if there is no reason that a motion should be done in one way or another, the motion is not realized at all. The application of this principle, however, does not leave one completely satisfied because it is not so obvious that there is no reason to prefer one or the other of the two motions. Lagrange’s proof of the second edition of the Mécanique analytique referred to above, reconnects to a demonstration of 1798, published in the Journal of the École polytechnique along with the three demonstrations by Fourier [147] (see Chapter 12). It differs only for formal aspects, such as, for example, to consider a weight of 1/ 2 instead of a unit weight and for a more extensive discussion. 37 38

p. 137. p. 26.

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10.2.2.1 Criticisms of Lagrange’s proof Lagrange’s proof for the virtual velocity principle as reported in the Mécanique analytique was the object of a series of indirect and direct criticisms, during Lagrange’s life and after. Indirect criticisms are certified by the series of attempts to furnish a new proof of the principle; see for example the argumentations of Prony, Fossombroni and Servois in subsequent chapters. The following comment by Bertrand, the editor of the third edition of the Mécanique analytique is instead an example of direct criticism: It is opposed, with reason, to this affirmation of Lagrange, the example of a heavy material point equilibrated on the highest point of a hill. It is clear that an infinitely small displacement will make it to descend; this notwithstanding this displacement does not occur at all. The first rigorous proof of the principle of virtual velocities is due to Fourier (Journal of the École polytechnique, Volume III, Year VII). The same volume of the Journal contains Lagrange’s proof reported here [148].39 (A.10.23)

According to Bertrand, the demonstration is imperfect and the criterion of equilibrium based on the assumption that a weight must necessarily go down if it is allowed by constraints, has exceptions. The same criticism of Bertrand was reported in more detail by Jacobi in his lectures on mechanics of 1847–1848 [375]. Perhaps the criticism of Bertrand and Jacobi applies to the reasoning of Lagrange, because it is likely that Lagrange’s conception of infinitesimal displacements is that assigned by them, and the situation of Fig. 10.4a would reveal the weakness of Lagrange’s reasoning because here there is a material point which is in equilibrium notwithstanding the possibility of a downward motion. In a few places, however, Lagrange seems to adopt the infinitesimal displacement as in Fig. 10.4b – as a modern mathematician would do – when he says that the displacements of points in the system are reversible (as already mentioned just above) and in the Théorie des fonctions analytiques where he interprets the virtual displacement as velocity, and a velocity in the case of Fig. 10.4 is directed horizontally and no downward motion is allowed. An apparently more relevant criticism is that by Mach, who sustains the claim that Lagrange’s proof is circular because it assumes the principle of the pulley, i.e. a simplified version of virtual work law – that is the law which is to be proved –

p

p

du P

a) Fig. 10.4. Various kinds of infinitesimal displacements

39

p. 24. Note by Bertrand.

du

P b)

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10 Lagrange’s contribution

[355].40

Mach’s criticism is however not convincing from two points of view, firstly because Lagrange uses the principle of the pulley not as a law of virtual work, but simply as a principle of equilibrium of parallel forces, secondly because in any case passing from a simple proposition to a complex one should be considered as a form of proof. The criticism that a modern reader – myself, for example – can turn to Lagrange refers to various aspects. The first criticism is formal of logical character; Lagrange does not specify that he is in fact assuming smooth constraints. The issue is not even touched. A second criticism is methodological and concerns the possibility of replacing the real system of forces with a series of pulleys and wires completely ideal, i.e. infinitely flexible, with no mass or weight and without frictions. One more criticism concerns the admission of linear constraints, at least for infinitesimal displacements. Lagrange’s proof seems quite convincing, but it leaves some points of dissatisfaction, on the other hand it is not possible to dismiss it in an easy way. In my opinion Lagrange’s proof is the most convincing of all attempted up to now, after that of Poinsot (see Chapter 14) which however has a completely different nature, because Lagrange’s proof has a foundational character, while Poinsot’s a reductionist character.

10.3 The Théorie des fonctions analytiques The Théorie des fonctions analytiques had two editions issued by Lagrange, the first in 1797, the second in 1813 shortly before his death. In the section devoted to mechanics of the first edition, Lagrange addresses the characterization of reaction forces. Lagrange assumes without any criticism the postulate P3 of Chapter 2 of the present book and tries to determine the reactions that come to be established as a result of internal constraints. He considers a constraint of the type f (x, y, z, ξ, ν, ζ, . . . ) = 0, between the coordinates of the material points p, p , …. According to the axiom P3 , the components of constraint forces R at the points p are orthogonal to the surface f (x, y, z, −, . . . ) = 0, where ξ, ν, ζ are assumed as fixed, i.e.: Π

∂f ∂f ∂f , Π , Π ∂x ∂y ∂z

(10.19)

and that R in p are orthogonal to the surface f (−, ξ, ν, ζ, . . . ) = 0, with x, y, y fixed, i.e.: Ψ

∂f ∂f ∂f , Ψ , Ψ ∂ξ ∂ν ∂ζ

(10.20)

with Π and Ψ arbitrary constants. Lagrange shows with the use of the virtual velocity principle that Π = Ψ, or which is the same, R = R : 40

p. 67.

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265

R ],

the effect of If one impresses to each body forces equal and contraries to those [R and these forces will be destroyed by the resistance we referred above; and consequently the system should remain in equilibrium. […] But for the principle of virtual velocities, the sum of the forces multiplied by the velocities of their points of application, according to the direction of the force, should vanish in the case of equilibrium, […] for the equilibrium of the concerned forces it will be satisfied the equation: −Π f  (x)x − Π f  (y)y − Π f  (z)z − Ψ f  (ξ)ξ − Ψ f  (ν)ν − Ψ f  (ζ)ζ − &c. = 0 which should be valid with the equation of constraint f (x, y, z, ξ, ν, ζ) = 0 […]. But if it is taken the derivative of this equation, with respect to time t, on which the variables x, y, z, ξ, &c., depend, it holds: x f  (x) + y f  (y) + z f  (z) + ξ f  (ξ) + ν f  (ν) + ζ f  (ζ) + &c. = 0 and it is clear that this relation is satisfied with the previous one for all values of x , y , &c. only if Π = Ψ = &c. [146].41 (A.10.24)

In the second edition of the Théorie des fonctions analytiques Lagrange changes completely the structure of the paragraph concerning the constraint reactions, taking into account Poinsot’s comments [197] who believed neither useful nor necessary the use of the virtual velocity principle to show that Π = Ψ. Lagrange virtually reverses the setting of the first edition; instead of using the virtual velocity principle to characterize constraints, he uses the characterization of constraints in order to demonstrate the principle. The demonstration takes on two fundamental principles: the law of the pulley and the rule of composition of forces. For the sake of brevity I will change the order of presentation of Lagrange’s arguments, keeping the same logic. He considers first the case of only two material points, identified by the coordinates x, y, z, ξ, η, ζ which are subject to a quite general condition of constraint represented in the form: F(x, y, z, ξ, η, ζ) = 0.

(10.21)

Actually, the generality is limited by the fact that taking an expression of the type (10.21), where F is an ordinary function, is the same as considering holonomic and independent of time constraints. To the above mathematical expression of constraints Lagrange associates a geometric-mechanical model consisting of two fixed and two mobile pulleys connected by a taut rope of fixed length, as shown in Fig. 10.5. The geometric-mechanical constraint can be made locally equivalent – in a way that gives rise to the same infinitesimal virtual displacements – to the analytical constraint defined by equation (10.21) provided the position of the pulleys and the number of rope turns are chosen in an appropriate way. In particular, line RM joining the centres of the first two pulleys, fixed and mobile, should be orthogonal to the surface defined by F = 0 with ξ, η, ζ assumed as fixed, while line SN of the second two pulleys should be orthogonal to the surface F = 0 with x, y, z assumed as fixed. Now it is easy for Lagrange to define the direction of the reactions, by introducing in a non-problematic way the assumption of smooth constraints. For the system of 41

pp. 255–256.

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10 Lagrange’s contribution

N F(

R

, x, h, z) = 0

n F(x, y, z,

T

m

)=0

S

M Fig. 10.5. Reaction of a constraint

pulleys the constraint reaction R in M is clearly directed along the line MR joining the centres, while the reaction R in N is directed along the line NS. For the equivalence it can thus be said that the reaction forces R and R associated with (10.21) are orthogonal respectively to the surfaces F(x, y, z, −) = 0 and F(−, ξ, η, ζ) = 0 and therefore defined by relations like these: ΠF  (x), ΠF  (ξ),

ΠF  (y), ΠF  (η),

ΠF  (z) ΠF  (ζ),

(10.22)

where Π is an arbitrary constant of proportionality. Lagrange thus obtains the same result which will be obtained in 1805–1806 by Poinsot [194], starting not by the law of the pulley but by the ordinary principles of statics (see Chapter 14): The derivatives of the same function [the constraint conditions] considered with respect to the different coordinates are always proportional to the forces that act according to these coordinates [in the points having these coordinates] and depend on the constraints expressed by this function [149].42 (A.10.25)

The extension of the above considerations to any number of points does not present any difficulty. A condition of constraint, for example with three points, like this: Φ(x, y, z, ξ, η, ζ, x, y, z),

(10.23)

admits a geometric-mechanical model with three fixed and three mobile pulleys. Similarly in the case with four points and so on. For more than one constraint condition Lagrange assumes, without any justification, and then without taking into account Ampère’s comments to the first edition of the Theorie des fonctions analytiques (see Chapter 13), that constraints do not affect each other. Therefore, two boundary conditions for a single material point: F(x, y, z) = 0,

42

p. 384.

Φ(x, y, z) = 0,

(10.24)

10.3 The Théorie des fonctions analytiques

267

may be replaced by two forces of components, using Lagrange’s symbols: ΠF  (x) + ΨΦ (x),

ΠF  (y) + ΨΦ (y),

ΠF  (z) + ΨΦ (z),

(10.25)

where Π and Ψ are arbitrary constants of proportionality. Now Lagrange may formulate the equations of equilibrium for any number of material points subject to any number of constraints and forces. In his own words: Let X,Y, Z be the forces applied to one of the bodies in the directions of coordinates x, y, z, Ξ, ϒ, Σ the forces applied to another body in the directions of coordinates ξ, η, ζ, and X, Y, Z the forces applied to a third body according to the direction of the coordinates x, y, z; from what said it results: X = ΠF  (x) + ΨΦ (x), Ξ = ΠF  (ξ) + ΨΦ (xi), X = ΠF  (x) + ΨΦ (x),

Y = ΠF  (y) + ΨΦ (y), ϒ = ΠF  (η) + ΨΦ (η), Y = ΠF  (y) + ΨΦ (y),

Z = ΠF  (z) + ΨΦ (z) Σ = ΠF  (ζ) + ΨΦ (ζ) Z = ΠF  (z) + ΨΦ (z)

(a)

and from the equilibrium equation it will result: Xx +Y y + Zz + Ξξ + ϒη + Σζ + Xx + Yy + Zz = ΠF(x, y, z, ξ, η, ζ, x, y, z) + ΨΦ(x, y, z, ξ, η, ζ, x, y, z) .

(b)

The second member of this equation is clearly zero as a consequence of the constraint equations, because the indeterminate quantities Π, Ψ are multiplied by the derivatives of these equations; it will be then: Xx +Y y + Zz + Ξξ + ϒη + Σζ + Xx + Yy + Zz = 0

(c)

the general equation of the principle of virtual velocities for the balance of the forces X,Y, Z, Ξ, ϒ, Σ, X, Y, Z, where the derivatives x , y , z , ξ , …express the virtual velocities of the points to which the forces X,Y, Z, Ξ …, estimated according to the directions of these forces are applied (see the first part of the Mécanique analytique). After all, one should not be at all surprised to see that the principle of virtual velocities becomes a natural consequence of the formulas which express the forces resulting from the constraint conditions, because the consideration of a thread that acts on all bodies through its uniform tension and induces forces assigned, is sufficient to lead to a general and direct proof of this principle, as I showed in the second edition of the work cited [149].43 (A.10.26)

The text of Lagrange is sufficiently perspicuous though perhaps the transition from (a) to (b) is a little fast. To obtain (b) one has to multiply the first of (a) for x , the second for y , the third for z , and so on and then add and to recognize in the second member the total derivative of two composed functions. It is worth noting that in his proof Lagrange uses virtual velocities and not infinitesimal virtual displacements, without commenting on the fact and in this he follows more or less the same approach of Poinsot in his proof of virtual velocity principle [194], without declaring the fact. Joseph Louis François Bertrand (1822–1900), who edited the third edition of the Méchanique analytique, wrote a comment in the second volume of it, where he denounced that absence of any reference to Poinsot: One might wonder that the illustrious author, usually so careful to know the origin of the ideas he presents, do not quote here anything. The passage we just read is, in effect, seven years later of the publication of the famous paper on the equilibrium and motion of systems, 43

pp. 384–385.

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10 Lagrange’s contribution

in which Mr. Poinsot proposes and solves precisely the same question of freeing mechanics from the principle of virtual velocities seeking forces that correspond directly to a given equation. This memoir much struck Lagrange, as evidenced by numerous autograph notes placed by him on the margins of a copy that I was able to consult. I reproduce here one of these notes, which can leave no doubt on the question of priority [151].44 (A.10.27)

Towards the end of the Théorie des fonctions analytiques Lagrange stresses the commonality of the proof above with that of the Mécanique analytique (second edition), but he believes that this is more direct and general. One can agree with him that it is more direct, as in the proof of the Théorie des fonctions analytiques he uses the law of the pulley and the rule of composition of forces while in the Mécanique analytique he uses almost only the law of the pulley. It is difficult to agree with the opinion about the generality. Indeed, the proof of the Théorie des fonctions analytiques seems more comprehensive because with it the virtual velocity principle can be extended to the dynamic case without any difficulty. Indeed the characterization of smooth constraints as made by Lagrange allows the extension of the virtual velocity principle from the static case to the dynamic case without having to go through the principle of D’Alembert. It is enough that instead of the equation of statics (a), the equation of dynamics is considered and then to follow the same procedure to pass from (a) to (b) and to (c). This possibility of the extension was not, however, ever remarked by Lagrange.

10.4 Generalizations of the virtual velocity principle to dynamics Although virtual work laws have historically always been considered as principles of statics, speaking of Lagrange it is impossible not to consider their extension to dynamics. As seen in § 10.1.1, he introduced the virtual velocity principle by beginning to study a dynamic problem with a generalization made possible thanks to D’Alembert’s ideas. With his usual laconic style he wrote: The principle of statics that I come to expose, combined with the principle of dynamics due to D’Alembert, is a kind of general formula containing the solution of all problems concerning the motion of bodies [142].45 (A.10.28)

The principle of dynamics due to D’Alembert is that of paragraph II of the Recherches de la libration de la Lune according to which accelerating forces taken in the opposite direction can be treated as ordinary forces. The meager statements of Lagrange raise at least two problems: what is the principle of D’Alembert? Is the interpretation of Lagrange permitted? A first answer to these questions comes by reading the Traité de dynamique [84] by D’Alembert. From here it would seem that the ‘principle of D’Alembert’ reported by Lagrange in the Recherches does not have much to do with the original principle, and therefore his

44 45

p. 366. p. 12.

10.4 Generalizations of the virtual velocity principle to dynamics

269

interpretation is not permissible. But this conclusion is puzzling, because Lagrange is familiar with D’Alembert’s work and may not have completely misunderstood it. To try to overcome the problems associated with the name of the principle of D’Alembert and with the true ideas of Lagrange it is necessary to refer to works subsequent to the Recherches and in particular the two editions of the Mécanique analytique and the Théorie de la libration de la Lune. References and interpretations of the principle of D’Alembert in the Théorie de la libration de la Lune (1780) and in the first edition of the Mécanique analytique (1788) are equivalent, but there is a shift in the second edition of the Mécanique analytique (1811). In the introduction to the second part of the 1788 edition of the Mécanique analytique, Lagrange wrote: If now it is assumed the system in motion, and it is considered the motion of each body at any given infinitesimal interval of time as consisting of two motions, one of which is that the body will have immediately after, it is necessary the other is destroyed by the mutual actions of bodies and by the motive forces by which they are animated at the moment. So there should be equilibrium among these forces and pressures or resistances that results from the motions that may be regarded as lost from the body, from one instant to another. It follows that to extend the formulas of the equilibrium to the motion of systems it will be enough to add the terms due to these forces [145].46 (A.10.29)

The passage is quite obscure and it is not much clarified by what Lagrange added in the following pages where he passes to derivation of the equations of motion. Much clearer exposure of the same concepts can be found instead in the Théorie de la libration de la Lune, written some years before, of which I quote a brief excerpt: It is clear that the motion or the velocity of the body m during the time dt could be regarded as composed by other three velocities expressed by: dy dz dx , , dt dt dt and parallel to the axes x, y, z. It is then evident, when the bodies are free and no external forces act on them, each of these three velocities will remain constant; but actually in the subsequent instant they change and become: dx dx dy dy dz dz +d , +d , +d dt dt dt dt dt dt so, if the previous velocities are assumed to be composed of these last and of the velocities: −d

dx , dt

−d

dy , dt

−d

dz dt

or, assuming dt as costant: d2x d2y d2z , − 2, − 2 2 dt dt dt it follows that they should be destroyed by the action of the force acting on the bodies. But these velocities are due to accelerating forces equal to: −

d2x , dt 2 46

p. 181.

d2y , dt 2

d2z dt 2

270

10 Lagrange’s contribution

and directed parallel to the axes x, y, z (expressing as usual, the accelerating force as the element of velocity divided by the element of time) or, which is the same, the forces equal to d2x , dt 2

d2y , dt 2

d2z dt 2

and direct to the contrary. […] It follows that there should be equilibrium among these different forces and the others acting on the bodies, and so the laws of motion of the system are reduced to those of his equilibrium, it is that the substance of the nice principle of dynamics of Mr. D’Alembert [148].47 (A.10.30)

In these passages Lagrange’s attempt to interpret D’Alembert’s principle in terms of forces instead of motions is reflected, and this is partly justified by the fact that in the Traité de dynamique D’Alembert is ambiguous in his use of the words motion and force, meaning for the latter sometimes ma and sometimes mv. The language of Lagrange is similar to that of D’Alembert, but the concepts are very different. Lagrange’s destroyed motions have nothing to do with D’Alembert’s lost motions due to constraints; they are the actual changes of motion with sign reversed, due to all forces. It is not entirely clear if "equilibrium among these different forces and the others acting on the bodies”, means equilibrium in the sense of absence of motion, or simply a balance of forces, though not made in accordance with the parallelogram rule but according to the principle of virtual velocities. The sentence of the Méchanique analytique,“It follows that to extend the formulas of the system’s equilibrium to its motion, would be enough simply to add the term due to these forces”, then gives the idea of what Lagrange intends with to “reduce dynamics to statics”. It does not mean eliminating dynamics as a discipline, which would be absurd, but simply to apply the same algebraic method for solving equilibrium problems or motion problems. Dynamics is implicit in the definition of mass, in the use of an inertial reference system and in the assumption of f = ma as a fundamental quantity, wether it is treated as a definition or as a law. In the second edition of the Mécanique analytique there is a different presentation of the principle of D’Alembert that, while still attached to the one of the first edition however, is farther from D’Alembert’s original exposure. To clarify the ideas of Lagrange it is enough to report what he says in the introduction to part II of his book: If motions are imposed to so many bodies that they [the motions] are forced to be modified by their interaction, it is clear that these motions can be seen as consisting of those the bodies would follow really and other motions that are destroyed by which it follows that these motions must be such that bodies on which only them are imposed are in equilibrium [148].48 (A.10.31) But the difficulty of determining the forces that must be destroyed and even the laws of the equilibrium among these forces often makes the application of this principle embarrassing and difficult. […] 47 48

pp. 17–18. p. 255.

10.4 Generalizations of the virtual velocity principle to dynamics

271

If one wants to avoid the decomposition of motions, that this principle requires, it will be enough to establish directly the equilibrium between forces and resulting motions, but taken in the opposite direction. For it is imagined that the motion is impressed on each body in the opposite direction to that which must follow, it is clear that the system will be reduced to rest [emphasis added]. Therefore it will be necessary these movements destroy those that had received the bodies and that they would have followed without mutual interaction; so there should be equilibrium among all these movements, or among the forces that can produce them. This way of reducing the laws of dynamics to those of statics is less direct than that resulting from D’Alembert’s [original] principle, but it offers greater simplicity in applications. It is similar to those of Hermann and Euler applied to the solution of many problems of mechanics and it is sometimes found in the treaties under the name of Principle of D’Alembert [148].49 (A.10.32)

The idea here seems much easier than in the first edition, at least for a modern reader, and perhaps reveals a different conception of force, closer to that of Euler than to that of D’Alembert. If to a system of bodies in motion appropriate forces are applied (if motions are impressed) in the opposite direction to the actual motion of each body, the system remains in equilibrium. The main difference compared to the first issue is that here the motions are not destroyed by the active force and constraints, but by the fictitious forces, −ma. Note that this time the reference is to equilibrium in the strict sense (“The system will be reduced to rest”), and not just to the balance of forces. For clarity, I refer for simplicity to a single material point constrained to move on a surface; if f is the active force and a the acceleration, assumed φ = −ma, the forces f and φ are balanced with each other, not in the sense that f + φ = 0, but rather in the sense that the material point of mass m is at rest under the action of the forces f and φ on the surface to which it is constrained (in modern terms, the balance equation is satisfied with intervention of the reactive forces). Unfortunately, Lagrange’s attempt to bring balance of forces to equilibrium, assuming that my interpretation is correct, gives rise to a rule that does not always work, and when the forces acting on a system depend on the velocity, the application of the forces φ = −ma may be unable to keep the system in equilibrium. In fact, if f (v) indicates the force dependent on the velocity v, the addition of the forces φ = −ma to the system leads to the balance f (v) + φ = 0, but not even that f (0) + φ = 0, between the force φ and that which would act on the system at rest (v = 0). The position of the first Lagrange formulation, for which equilibrium and balance seem identified, which is even more problematic from several points of view, is not subject to this criticism. If the idea of equilibrium is generalized to the dynamic case, as a balance of accelerating and active forces, all the arguments of supporters of the virtual work laws as a criterion of rest (Aristotle, Galileo, Riccati), fall. But Bernoulli’s arguments do not fall because his principle of virtual velocities is set out with reference only to a balance of forces. In this statement there is no difficulty in inserting accelerating forces as balancing active forces applied and therefore Lagrange is justified in the use of the virtual velocity principle. 49

p. 256.

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10 Lagrange’s contribution

x1

S

x2

x3

r1 m1 a1

r2 p1

m2 r3

a2 p2

m3 a3

p3

Fig. 10.6. Static analysis of an oscillating compound pendulum

Lagrange, in the Mécanique analytique, says that his idea of considering φ = −ma as forces, is similar to the approaches pursued by Hermann and Euler. He certainly refers to Hermann’s Phoronomia of 1716 [134] and to various of Euler’s works after 1750, on vibrating strings [102, 105]. He forgets to mention Clairaut who in his work Sur quelques principes qui donnent la solution d’un grand nombre de problèmes de dynamique of 1742 [69] introduced reaction forces to study the motion of a simple point system, requiring that they be self-equilibrated. In the study of a compound pendulum dynamics, Hermann considered the elementary masses mi subject to the weight of pi , the driving force mi ai and forces which constrain the masses to belong to a rod. Adopting as a criterion of balance the law of the lever, Hermann ignored the constraint forces and required the equivalence of the static moments with respect to the centre of suspension between pi and mi ai :

∑ mi ai × ri = ∑ pi × xi

(10.26)

where × is the ordinary product. From this equation it is then easy for Hermann to deduce the law of motion of the compound pendulum. Fig. 10.6, illustrates the situation. Euler, in the study of the dynamics of a taut string, considered as a set of elementary masses mi connected by the string, said that the accelerating forces mi ai of the elementary masses considered in the opposite direction must be balanced by the restoring force of the string: As currently it is the case to determine the movement of the rope due to the force stressing it, i.e. the accelerating force P for which the point M of the rope is accelerated toward the axis AB, it is clear that all those forces by which each element of rope is urged in the direction of AB, taken together, shall be equivalent to the force from which the rope is in fact tight, which we have indicated with AF = F. Well if we conceive forces opposing and equal to P,

10.4 Generalizations of the virtual velocity principle to dynamics

L

l

273

D

m

M F A

P

p

C

B

G Fig. 10.7. Static analysis of a vibrating string

applied according to ML on each point M of the rope, then they must be in balance with the force that stretches the rope [102].50 (A.10.33)

Even today, thanks to Lagrange’s remarks, the term principle of D’Alembert means different things: the original principle of the Traité de dynamique, D’Alembert principle of Lagrange and even the symbolic equation of dynamics. (It should be emphasized, however, that Lagrange is not responsible for the enunciation of the principle of D’Alembert in the form: the forces of inertia balance the active forces, in which the forces of inertia are hypostatized and treated as ‘true’ forces.) I cannot avoid thinking that Lagrange (and Euler and Hermann) though if he has not introduced a new principle has at least had a very good idea.

10.4.1 The calculus of variations Another aspect of Lagrange’s dynamic generalization is the introduction of the time factor in defining the pattern of a system of material points. Although at any moment the dynamic problem can be studied with the same formulas of statics, i.e. the use of the virtual velocity principle, it must be taken into account that the configuration of this system changes with time and consequently also the virtual displacements change, which, considering the motion in all its duration, become functions of time. The expression of the virtual velocity principle in dynamics, along the trajectory of the system, has then the form ∑ f (t) · u(t) = 0, where u(t) is the vector of virtual displacements in the configuration at time t. In principle, the vectors u(t) may be completely unrelated to each other at different times. Here Lagrange introduces the calculus of variations and treats the virtual displacements u(t) as the variation of certain functions of time that represent motion. The virtual displacements, indicated by δu and thought of as a function of time, are so endowed with some degree of regularity, in particular Lagrange acknowledges at least the existence of the firstorder derivative. The introduction of the calculus of variations with the consequent possibility of ‘regularization’ of motions does not limit the generality of the virtual velocity principle, because its validity does not depend on the absolute value of the motion of vir50

p. 73.

274

10 Lagrange’s contribution

tual velocity in an instant, and makes simpler some fundamental deductions. Among them there is the one regarding the theorem of living forces, where real changes are treated as virtual variations due to the fact that both du and δu are smooth functions of time (assuming constant intervals dt). This same conception of virtual displacements allows us then to discover the general procedure for calculating the virtual work of accelerating forces from the expression of kinetic energy, used in the Mécanique analytique, using the possibility of the permutation of δ with d in the expression d 2 δu, which is conceivable only if δu is a continuous and differentiable function. For continuous systems, such as those included in the Recherches, virtual displacements of each point are identified by their coordinates x, y and z, instead of by a label. Again the use of the symbol δ implies some regularity with respect to variables x, y and z (beside t).

10.4.2 Elements of D’Alembert’s mechanics Jean le Rond D’Alembert was born in Paris in 1717, and died in Paris in 1783. He was the illegitimate child of the chevalier Destouches. Being abandoned by his mother on the steps of the little church of St. Jean le Rond, which then nestled under the great colonnade of Notre Dame, he was taken to the parish commissary, who, following the usual practice in such cases, gave him the Christian name of Jean le Rond. He was boarded out by the parish with the wife of a glazier who lived near the cathedral, and here he found a real home. His father appears to have looked after him, and paid for his going to a school where he obtained a fair mathematical education. Nearly all his mathematical works were produced during the years 1743 to 1754. During the latter part of his life D’Alembert was mainly occupied with the great Encyclopédie, with Diderot. For this he wrote the introduction, and numerous philosophical and mathematical articles; on geometry and on probabilities. His style is brilliant and faithfully reflects his character, which was bold, honest, and frank. The most famous of his books is perhaps the Traité de dynamique, published in 1743 [84], in which he proposed laws of mechanics other than the Newtonian ones [335]. According to D’Alembert the principles of geometry and algebra, with the addition of the assumption of the impenetrability of bodies, were enough to develop mechanics, which appeared to be a completely deductive science marked by the seal of evidence; as geometry and algebra. Motion and its property are the main object of mechanics. To be assumed as foundations of mechanics, to be its principles, all the relevant concepts must be subject to scrutiny by the philosopher: Only concepts with a sufficient clarity and distinction (in a Cartesian sense) could be accepted. And D’Alembert identified only two fundamental concepts of the kind: those of space and time, which are the only elements and principles of mechanics. The laws of mechanics are theorems which can be deduced by the two principles. The various

10.4 Generalizations of the virtual velocity principle to dynamics

275

concepts of force, together with that of motive cause, are to be rejected as “dark and metaphysical beings, only capable to spread shadows in a science clear in itself” [84].51 Scepticism toward force does not originate directly in D’Alembert. Maupertuis too had this same conception. This scepticism was already present in Descartes and mainly in Malebranche. Descartes thought there were no forces in bodies, even though he conceived the concept of cause. According to Descartes, God is the prima causa, but after the creation there is only a cause which is mechanical: the impact of impenetrable bodies. Malebranche did not accept this model and assigned to God a greater space. The concept of force is avoided by Malebranche not only because its assumption reduces God’s power but also because it is not well-defined. According to Malebranche force cannot be observed or measured directly, it looks like a simple word, made-up by philosophers to hide their illiteracy. Berkeley wrote some sentences in which D’Alembert will recognize himself, though it is difficult to determine if D’Alembert knew Berkeley’s work, published only in England in 1720. Hume too was contrary to the concept of force; but also in this case it seems difficult there was any influence, notwithstanding that Hume was familiar the Enlightenment philosophers [271]. Without force, motion can be described by geometry alone. So D’Alembert left no space for what today is called Newton’s second law. Here is what he wrote on the matter: Why should we recur to this principle of which everybody recurs today, that the force accelerative or retarding is proportional to the element of velocity divided by time? […] We neither will examine in any way whether this principle is a necessary truth […] nor, as some Geometer [Daniel Bernoulli] a purely contingent truth […] we will limit ourselves to observe that, true or false, clear or dark, it is useless in mechanics and consequently has to be banished [84].52 (A.10.34)

According to D’Alembert there were two species of causes, and then of forces in mechanics: a) causes which derive from the mutual actions of bodies because of their impenetrability, which are the “main causes of the effects” we observe in nature; b) causes not immediately reducible to impulsion or pressure. These causes have to be equally considered as distinct, if one considers as possible their reducibility to impulse but cannot prove the fact. Causes of the first kind have well-known laws; this is not true for causes of the second kind. They are known only through their effects; one speaks about a cause because one sees an effect. Among the causes of the second kind there is gravity, which because it could not be reduced to the impact, and then to geometry, must be excluded from the necessary laws of mechanics and considered as a contingent truth. D’Alembert asserted that also the causes of the first kind which look evident, are so only improperly:

51 52

p. XVII. p. XII.

276

10 Lagrange’s contribution

What we call causes, also of the first kind, are such only improperly; they are effects which determine other effects. A body pushes another body, or a body in motion meets another body, one must then have necessarily a change in the state of bodies in this occasion [emphasis added]. Because of their impenetrability, the laws of these changes are determined by means of sure principles; and consequently impelling bodies are considered as causes of the impelled bodies. But this way to speak is improper. The metaphysical cause, the true cause is not known to us [81]. (A.10.35)

As a matter of fact, D’Alembert notwithstanding the rational framework of his mechanics, affirmed his empiric faith. One has to do only with effects. In mechanics it is called cause of an effect, another effect; the true causes remain hidden. A chain of explanations cause-effect is nothing but a relation among effects, which however can be connected by necessary laws. Anyway D’Alembert felt the need to introduce a quantity called force that, at least from a mathematical point of view, played the role of the force as commonly intended in statics and in dynamics by most physicists. Force is defined simply as the product of mass by accelerations, where both concepts are previously defined: So we will intend in general with motive force the product of mass multiplied by the element of velocity [acceleration], or, which is the same, multiplied by the small space it covers in a given time because of the cause which accelerates or retardates its motion; with accelerating force we will intend the element of velocity only [84].53 (A.10.36)

The definition of force by D’Alembert presupposed that of mass. D’Alembert, however, in his examination of the principles of mechanics passes over this concept, without realizing its problematic character and that the lack of its clarification makes mechanics incomplete. To D’Alembert, and also to Newton, mass is given by the quantity of matter; a concept which could appear clear to anyone who had a conception of matter based upon a crude atomism with all equal atoms. D’Alembert assumed as fundamental theorems of dynamics the theorem of inertia divided in two parts (I and II law), the theorem of composition of forces, and the theorem of equilibrium. Their statements are referred to in Table 10.1 [84].54 Table 10.1. D’Alembert’s mechanics. Laws or theorems I law II law

Theorem

Theorem

53 54

A body in rest will remain in rest unless an external cause will force it. A body once put in motion by whichever cause, must persevere uniformly and in straight line, unless a new cause, different from that has caused the motion, will act on it. If any two forces act together on a point A to move it, the former uniformly from A to B, during a given time, the latter uniformly from A to C [...] I say that the body A will cover the diagonal AD uniformly, in the same time it will cover AB or BC. If two bodies whose velocities are in inverse ratio of their masses, such that one cannot move without shifting the other, there is equilibrium between these two bodies.

p. 26. pp. 3, 4, 35, 50–51.

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277

10.4.2.1 D’Alembert principle From the foregoing it appears that the so called principle of D’Alembert is not a law of D’Alembert’s mechanics, and it is still the subject of any dispute regarding its interpretation and possible logic status [315, 316, 303]. In the following I will present the statement of D’Alembert as formulated by himself and then I will cite the opinions of some scholars from which one can at least partially justify the interpretation given by Lagrange. D’Alembert presents his ‘principle’ as a procedure for solving a problem where it uses, though not very clearly, the real ‘principles’ of mechanics. The principle of D’Alembert is reported far enough in the Traité de dynamique, after the presentation of the fundamental theorems. General Problem Let a system of bodies be disposed in any way with respect to each other, and suppose we give each of them a particular motion, that cannot be accomplished due to the interaction with other bodies. Find in these conditions, the motion that every body should have. Solution Let A, B,C, &c. be the bodies of the system and suppose they are impressed with the motions a, b, c, &c. and that they are forced by their interactions to change in the motion a, b, c, &c. It is clear that the motion a impressed to body A can be regarded as composed of the motion a and of another motion α. In the same way it is possible to consider the motions b, c, &c. composed of the motions b, β, c, κ, &c., from which it follows that the relative motions of the bodies A, B,C, &c. would be the same if instead to give them the impulse a, b, c it would be given the couples of impulses a, α; b, β; c, κ, &c. Now, because of supposition, bodies A, B,C, &c. took the motions a, b, c,&c, then the motions α, β, κ, &c. must not to disturb in any way the motions a, b, c, &c. That is if the bodies had received only the motions α, β, κ, &c. they should have destroyed themselves mutually and the system to remain at rest. From this it results the following principle to find the motion of any bodies that interact each to the other. Decomposed each of the motions a, b, c, &c. impressed to each body in other two motions a, α; b, β; c, κ, &c., such that if only the motions a, b, c, &c. were impressed to the bodies, the system should have remained at res, it is clear that a, b, will be the motions these bodies will assume because of their actions. This is what has to be proved [84].55 (A.10.37)

Before the introduction of his principle, D’Alembert had specified the frame of reference: his principle concerns impact among bodies, direct or mediated by rigid rods; the impacts are a result of imposed motions, an expression that D’Alembert uses in place of imposed velocities. The bodies are to be understood as material points and are ‘hard’, i.e. not deformable, which do not bounce in the collision and in fact behave as perfectly plastic bodies (apart from the change in shape that does not exist). In the following I refer to the interpretations of Ernst Mach and Louis Poinsot. They change the principle of D’Alembert from a principle on motions to a principle on forces in a similar manner, although not identical, as made by Lagrange and serve to justify the ‘a bit free’ interpretation by the latter. Mach’s interpretation is important because it is the most famous, Poinsot’s interpretation, is important because it is substantially contemporaneous (1806) with the drafting of the second edition of the Mécanique analytique.

55

pp. 73–75.

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According to Mach [355],56 in place of the impressed motions a, b, c, etc. one can consider the external forces Ui , instead of the actual motions a, b, c, etc., the forces Vi able to produce these motions and instead of the suppressed motions α, β, κ, etc. the forces or constraint reaction Wi . D’Alembert’s principle, according to Mach, requires that the constraint forces Wi are balanced. If the system is free, constraint forces are balanced in the sense that their vector sum is zero; if the system is constrained externally, the balance is expressed by the annulment of their virtual work. Then, if the virtual work done by the constraint forces Wi is zero, even that of the forces Ui −Vi is zero, because Wi = Ui −Vi . In this interpretation of the principle of D’Alembert, the identification of the accelerating forces with the actual motion, has a key role and can justify the name of the principle of D’Alembert to the assimilation of the accelerating forces with changed sign to ordinary forces. Note that if there were no accelerating forces the above interpretation would coincide exactly with the virtual velocity principle and D’Alembert would not have said anything new. Poinsot also provides an interpretation of the principle of D’Alembert like Mach’s, after establishing the equations of motion in form ma − r = f , where f are the active forces and r the vectors orthogonal to constraints (which can be interpreted as constraint reactions). He writes: It also can be seen that it is useless to refer to the famous principle of D’Alembert, which reduces dynamics to statics. Under this principle if each impressed motion is decomposed into two others, one of which is what the body will really take, all the others must balance between them. That is, if each impressed movement is decomposed in two others, one of which is that the body loses, the other that the body will take. But it follows immediately from what was just said, namely, that the actual motion of each point is the result of the impressed motion and the resisting forces that it receives because of its connection with other [points], and this is self-evident. Thus the principle of D’Alembert is basically that simple idea that is barely noticeable in the course of reasoning, and which takes the form of a principle only for the expression that it is given to it [194].57 (A.10.38)

It is useful to report a comment by Lagrange to this interpretation of Poinsot and its reply [197] that may serve to clarify the difference in viewpoint between the two scientists. Lagrange writes: The advantage of the principle of D’Alembert is to find the law of motion regardless of resistance or constraint forces [forces constraint] exerted against it [197].58 (A.10.39)

Poinsot replies: The forces of resistance of which it is discussed are nothing but forces capable of being balanced on the system, they are the same ones that employs D’Alembert. It is, if one wants, to shorten that they are called mutual resistance forces [197].59 (A.10.40)

56 57 58 59

pp. 335–337. p. 233. pp. 72–73, part II. p. 73, part II.

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279

Lagrange criticizes the fact that the concept of constraint forces appears; Poinsot agrees and states that he speaks of constraint forces “only to abbreviate” and that in fact this term means the forces that must be absorbed by the constraints, which are absolutely comparable to the destroyed motions of D’Alembert, or also bound to maintain balance in the system, and that the difference is only a matter of “words”. Interpretations of D’Alembert’s principle as that of Mach introducing (a) the forces in place of motion and (b) internal or external constraints with no impact were subject to many criticisms. I will not enter into the details and I just imagine myself in the role of Lagrange in responding to these objections. The first claim is correct, D’Alembert’s principle was translated from the language of a metaphysical system in which there are no forces, to that of a metaphysical system in which there are forces and constraint reactions. But I believe that the translation given by Mach would be probably natural for Lagrange who had no particular position on the ontological status of forces. Note also that applications of D’Alembert’s principle by D’Alembert, in some cases – see for example problem I of the Traité de dynamique [84]60 – which is then taken up by Mach, where D’Alembert studies the motion of a compound pendulum – the suppressed motions are described as ‘puissances’, that is, as forces and although D’Alembert for ‘puissances’ means something different from what we consider as force, the fact remains that the translation of his principle in terms of forces is inviting. The second objection is less serious. To suppose that a collision takes place through rigid rods without mass is equivalent to assume a system constrained for internal constraints, those for which the distances between points does not vary. The impressed motions instead of being real as in the case of direct impact are those which would be if there were no constraints and all the reasoning of D’Alembert runs well. Then D’Alembert applied his principle even when there are external constraints (see again the problem I, when the pendulum is suspended on an external constraint).

60

pp. 96–97.

11 Lazare Carnot’s mechanics of collision

Abstract. This chapter is devoted to Lazare Carnot’s mechanics of impact and to a formulation of a WVL generalized to dynamics. In the first part the mechanics of impact of hard bodies is presented. Through an appropriate definition of virtual velocity and motion (the geometric motion) Carnot succeeds in formulating a generalization of VWL that allows one to evaluate the velocities of a system of hard bodies each of which the initial velocity is known. In the second part his extension to gradually variable forces is presented with the introduction of virtual or real work, named moment of activity, as the fundamental magnitudes of applied mechanics. Lazare Nicolas Marguerite Carnot was born in Nolay, Côted’Or, in 1753 and died in Magdeburg in 1823. He was one of the very few men of science and of politics whose career in each domain deserves serious attention on its own merits. Nicknamed Organizer of Victory or The Great Carnot, in 1771 he entered the Mezieres school of engineering, where he had met and studied with the likes of Benjamin Franklin. It was here that he early made a name for himself both in the line of physics and in the field of fortifications. Although in the army, he continued his study of mathematics. Carnot entered politics in 1791 when he was elected a deputy to the Legislative assembly from the Pasde-Calais. In the military disasters in Belgium in the spring of 1793 Carnot had to override the demoralized generals and organize first the defence and then the attack to his own prescription. On August 1793, the Convention appointed Carnot a member of the Committee of public safety. As minister of war he reorganized the French army. In 1797 the leftist coup d’état displaced Carnot from government. He took refuge in Switzerland and Germany, returning in 1800 soon after Napoleon’s seizure of power. Throughout the Napoleonic period he served on numerous commissions appointed by the Institute to examine the merits of many mechanical inventions. Amid the crumbling of the Napoleonic system, he offered his services when the retreat Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_11, © Springer-Verlag Italia 2012

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from Moscow reached the Rhine. In those desperate circumstances Napoleon appointed him governor of Antwerp. Carnot commanded the defence. He rallied to the emperor again during the Hundred Days and served as his last minister of the interior (this testifies to the consistency of Carnot yet more decisively than his having voted death to Louis XVI some twenty-two years before). In 1816, following Napoleon’s final defeat at Waterloo, Carnot, the only general of Napoleon’s army to never be defeated, went into exile in the German city of Magdeburg, where he occupied himself with science [290, 332]. Lazare Carnot wrote four memoirs about mechanics, two of them have the same title, Mémoires sur la téorie des machines, written to compete for a prize of the Académie des sciences de Paris in the years 1779 and 1781; both are conserved in the Archive de l’Académie des sciences and Institut de France and partially transcripted by Gillispie [332].1 The other two memoirs are the Essai sur les machines en général of 17832 and the Principes fondamentaux de l’équilibre et du mouvement of 1803 [60]. If it is incorrect to argue that Lazare Carnot was a precursor of Lagrange, because the latter had already reported a mature exposition of the principle of virtual velocities in his essay on the libration of the moon in 1764, when Carnot was just eleven years old, still it is to be noted that the Essai sur les machines en général where he exposed and treated in depth a formulation of the virtual work law, was printed before the first edition of Mécanique analytique and the first proof of the principle of virtual velocities reported by Lagrange in 1798 in the Journal de l’École polytechnique [147]. For this reason, considering also that the contribution of Carnot is not widely known, I will report an extensive comment also taking aspects not immediately related to virtual work laws from the Essai sur les machines en général (hereafter Essai), a slender writing, slightly more than a hundred pages. Only at a few points will I refer to the Principes fondamentaux de l’équilibre et du mouvement, where he continued and ‘perfected’ the ideas of the Essai; notice that the way to treat virtual work laws is here influenced by Lagrange’s Méchanique analitique and less interesting, at least from my point of view. Understanding the role Carnot gave to laws of virtual work requires an understanding of his mechanics and also of his entire epistemology because his point of view differs from the traditional one. The empiric mind of Carnot is reflected by the title and introduction: It has given to this pamphlet the title of Essai sur les machines en général firstly because it has particularly in view machines as the most important part of mechanics and secondly because it does not treat any particular machine, but only the properties that are common to all machines [59].3 (A.11.1)

and well documented better below, where he comments on the two ways to approach science, the rationalist or synthetic and the empiric or analytic approaches: 1

p. 347; pp. 271–296; pp. 299–340. Actually the 1783 edition is very rare; reference is usually made to the second edition of 1786 [58] and to the version reported in the Oeuvres mathématiques du citizen Carnot [59]. 3 p. VII. 2

11 Lazare Carnot’s mechanics of collision

283

Among philosophers interested in the search of the laws of motion, some make of mechanics an experimental science, some others make of it a purely rational science. That is, the former compare phenomena of nature, decompose them to know what they have in common, and so to reduce them to a small number of main facts which serve in the following to explain all the others, and to anticipate what has to occur in any circumstance. Some others start from hypotheses, then, by reasoning according to their suppositions, arrive to discover the laws which regulate bodies in their motion; if their hypotheses conform to nature, they conclude that their hypotheses were exact; that is bodies actually follow the laws that at the beginning they had only supposed. The former of these two classes of philosophers, starts then in their researches from primitive notions which nature has impressed in us, from the experiences that it continuously offers [empiric approach]. The latter starts from definitions and hypotheses [rationalist approach]. For the former the names of bodies, of powers, of equilibrium, of motion are considered as primitive ideas; they cannot and must not define them; the latter, to the contrary, must attain all from themselves and are obliged to define exactly these terms and to explain clearly all their hypotheses. But if this method appears more elegant, it is more difficult than the other, because there is noting more embarrassing in most natural science and especially in this [mechanics] than to assume at the beginning definitions deprived of any ambiguity. I would throw myself in metaphysical discussions if I tried to deepen this argument. I will be happy only to examine the first and simpler. […] The two fundamental laws from which I started are then purely experimental truths, and I propose them as such. A detailed explanation of these principles is out of the spirit of this work and could serve only but to tangle things: sciences are as a beautiful river whose course is easy to follow, when it has acquired a certain regularity; but if one wants to sail to the source one cannot find it anywhere, because it is far and near; it is diffuse somehow in the whole earth surface. The same if one wants to sail to the origin of science, one finds nothing but darkness and vague ideas, vicious circles; and one loses himself in the primitive ideas [59].4 (A.11.2)

In the first part of the above quotation Carnot declared his preference toward the empiric approach; in the second part he declared the two principles assumed in the Essai (the equality of action and reaction and the conservation of momenta in the collision) as empirical laws. In the introduction of the Principes fondamentaux de l’équilibre et du mouvement Carnot was a little bit more vague. Here he reasserted his empiric faith: Ancients established as an axiom that all our ideas come from senses; and this is no longer object of dispute [60].5 (A.11.3)

Nonetheless, he also expressed the opinion that, notwithstanding the laws of mechanics drawing much from experience, they seem so evident and clear that there is the impression they could be derived from reasoning only: Yet sciences do not all draw equally the basis from experience. The pure mathematics will take less than all the others, then the mathematical physical sciences, then the physical sciences. It would no doubt be satisfactory in every science, to decide the point where it ceases to be experimental to become rational, that is to be able to reduce to the smallest possible number of truths which we are forced to draw from observation and that, once accepted, together are sufficient for the sole reasoning to embrace all branches of science. But this seems 4 5

pp. 120–124. p. 2.

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too difficult. Wanting to go up too high, with the only reasoning, you are exposed to [the risk] to give obscure definitions and vague and lax demonstrations. There are fewer problems in obtaining more information from the experience of what might be strictly necessary. […] It is therefore from the experience that men have gained the first notions of mechanics. Nevertheless, the fundamental laws of equilibrium and movement that are its basis, are presented on the one hand so naturally to reason, and on the other hand they are expressed so clearly through the known facts that it seems difficult to say that it is of one rather than another that we have full conviction of these laws [60].6 (A.11.4)

Results of experimental observation about equilibrium and motion can be expressed by means of laws to which Carnot attributed the name of hypotheses, instead of principles, to underline that they do not posses absolute evidence. He considered also the possibility they could be changed where not able to explain the empiric evidence. Now it has to establish upon given facts, and upon other observations which we still could have, some hypotheses [emphasis added] which are constantly in accord with these observations and which we can assume as general laws of nature. […] We will then compare the consequences resulting from them [the hypotheses], with phenomena, and if we find they agree, we will conclude that we can consider these hypotheses as the true laws of nature [60].7 (A.11.5)

and it is not necessary that hypotheses concern phenomena which are unrelated to each other: My objective is not to reduce them [the hypotheses] to the smallest number; it is enough for me that they were consistent and clear enough [...] but they are the most suitable to confirm the principle [the experimental facts], by showing that they are, as to say, nothing but the same truth which says all the same under different forms [60].8 (A.11.6)

Using the classification of the theories proposed by Antonino Drago [300], that of Carnot was an approach for problems, and his main problem, at least officially, was the study of the behaviour of machines: Knowing the virtual [initial] motion of any system of bodies (i.e., that each body would take if it were free) find the real motion that will take place immediately following, due to the mutual action between bodies, considering them as they exist in nature, that is, endowed with inertia, common to all parts of the matter [59].9 (A.11.7)

He did not follow, like Newton, an axiomatic approach – partly because he was not sufficiently accurate – giving at once a principle from which to deduce all the mechanics, but rather he sought to trace the principles from elements more or less obvious. The phenomenon Carnot considered as more immediate is that of collision and from it he built his mechanics, which was seen essentially as the science that studies the communication of motion. 6 7 8 9

pp. 3–5. pp. 46–49. p. 47. p. 14.

11.1 Carnot’s laws of mechanics

285

11.1 Carnot’s laws of mechanics Carnot begins his exposition with an historical comment on two principles hitherto most widely used in mechanics. The first principle claims that because the centre of gravity of the machines tends to move as low as possible, there is equilibrium when moving down is prevented. It is a generalization of the principle due to Torricelli, generally known as the Torricelli principle, though not so-called by Carnot. Although Carnot thinks impossible a rigorous proof of this principle without going back to ‘first principles’ of mechanics, nonetheless he believes that it can be given an intuitive justification, which is presented as follows. Imagine a machine, subject only to weight, in a certain arrangement of its constituent parts. If there is equilibrium the sum of the resistances of the fixed points or of any obstacles, estimated in the opposite direction to the weight, equals the total weight of the system. But, Carnot says, if a motion can originate, some of the weight will be used to produce the motion and the fixed points will be loaded only by the remaining part of the weight. The difference between the force of gravity and that of the fixed points will result in a force that will bring the system from top to bottom as if it were free, then the centre of gravity of the system will fall, then there will not be equilibrium. Such a ‘demonstration’ appears quite confused and not very intuitive to a modern reader. Meanwhile, it is not clear what Carnot means by the word ‘force’; mainly one does not understand what he means with the power of fixed points, when taking into account his criticism of the concept of force. The second principle is “the famous law of Descartes”, according to which two powers are in equilibrium with each other if they are inversely proportional to the velocity that arises when a small movement is caused by an infinitesimal prevalence of a force on the other, estimated in the direction of the force. Descartes certainly would not have recognized it as his own principle, at least because it refers to the idea of virtual velocities to which he was clearly contrary. Why Carnot chooses these principles as the most representative of ‘past’ mechanics is partly explained by his ideas on mechanics outlined above, which is incompatible for example with the Newtonian approach. The practical reference to machines is helpful to Carnot for the exposition of his own principles. A machine is defined as the agent used to communicate motion from one body to another, an intermediary which is always necessary, not recognizing Carnot’s remote actions. To simplify the problem, he admits to having to deal with ideal machines, massless and without any friction. Carnot begins to enunciate what he sees as his new principles, which are also referred to as laws to underline their empirical content. In the Essai there are only two laws: First law. The reaction is always equal and contrary to the action. Second law. When two hard bodies act each other, because of the collision or pressure, i.e. because of their impenetrability, their relative velocity immediately after the mutual action is always zero [59].10 (A.11.8) 10

pp. 15–16.

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As mentioned earlier, Lazare Carnot’s laws are different from those of Newtonian mechanics. Carnot’s paradigm is collision rather than continued action, which prevails in the Newtonian paradigm. He also believes that his laws are intuitive enough: This Essay on machines is not a treatise of mechanics, my goal is not to explain in detail or demonstrate the basic laws that I have reported, these are truths that everyone feels very good [59].11 (A.11.9)

The first Carnot’s law of the Essai expresses the fact that all bodies which change their state of rest or motion always do it from the action of some other body, to which a force equal and directly opposite is impressed at the same time. All bodies resist changes to their state and to refer to this resistance Carnot uses the term inertia force of which he defends the use, for example against Euler who considered it a confused concept, and contributes to its spread. In more precise terms the force of inertia of a body is “The result of its current motion and of a motion equal and contrary to which it should be in the next instant” [59].12 In the explanation of the first law, Carnot also provides the prevailing meaning he attaches to force: it is the change of quantity of motion F = mΔv. The second law refers to hard bodies, which according to D’Alembert, are perfectly rigid bodies deprived of any elasticity; to justify the statement Carnot refers to experience. Here he seems not very honest in considering the case of collision of plastic bodies which, among other things, are anything but hard, as the most representative. In any case, the hard body model in the sense employed by Carnot, was widespread in the XVIII century and its use is justified not so much by experiments, as by the need for a simple model of behaviour. However Carnot is aware, he openly declares it, that the second principle leaves out the elastic bodies and justifies its acceptance by noting that the case of elastic bodies can be explained by that of hard bodies by assuming the former as consisting of an infinite number of hard bodies separated from each other by elastic springs. It is clear that this explanation of Carnot’s is a forced justification; for example the way to treat the elasticity which is transferred from the bodies to the springs, remains unclear. From the two laws, Carnot ‘claims’ two other “secondary principles” relative to the collision of hard bodies, which are commonly used in mechanics, they are: The intensity of collision or of action between two colliding bodies, does not depend on their absolute motions, but only on their relative motion. The force or the quantity of motion they exert on one another, is always perpendicular to their common surface at the point of tangency [59].13 (A.11.10)

In the Principles fondamentaux de l’équilibre et du mouvement Carnot presents his laws or principles in a more organized way. They are qualified as hypotheses and are all on the same ground. Besides the four laws of the Essai referred to above, there are three others, assumed implicitly in the Essai [60].14 11 12 13 14

pp. 17–18. p. 64. pp. 16–17. pp. 49–50.

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287

The fundamental Carnot’s problem, i.e. the study of behaviour of machines, is reduced to assume as a fundamental problem of mechanics the evaluation of the motion of a system of hard bodies as a result of shocks between them. This is the same problem considered by D’Alembert in the Traité de dynamique (see Chapter 10). There is however an important difference. D’Alembert considers it certainly an important problem, but to be solved on the basis of the general laws of mechanics already formulated by him. Carnot considers it instead the fundamental problem of his own theory that will be defined in the attempt to solve it. In any case Carnot’s formulation of the problem of collision is close to D’Alembert’s. Even the terminology is similar, as will become clear hereafter; in particular the reference to lost motions and the decomposition between actual motions, virtual motions and lost motions.

11.1.1 The first fundamental equation of mechanics By applying his principles to a system of hard bodies or in any way to a system of bodies separated by inextensible rods, Carnot obtains a first general principle of mechanics, according to the following reasoning: For pairs of hard particles define: m and m Masses of two contiguous particles; V  and V  their velocity after the collision; F the action of m over m , or the force or quantity of motion the first of the particles exerts on the other; F  the reaction of m over m ;   q and q the angles between the directions of V  and F  and between V  and F  . For the second law, after (and during) the collision, the two bodies must have a zero relative velocity in the direction of the force (which is unique, see Fig. 11.1). It is then: V  cos q +V  cos q = 0.

(11.1)

Since for the first law F  = F  , by multiplying (11.1) by F  or F, it is: F V  cos q + F V  cos q = 0

V' F'

q' m'

F" q"

m" V"

Fig. 11.1. Impact of two masses

(11.2)

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and considering all the particles:

∑ F V  cos q + ∑ F V  cos q = 0.

(11.3)

This expression is then rewritten in a more effective way, by introducing new symbols and concepts. Let it be: • the mass of each particle of the system, m; • its virtual velocity, i.e. the velocity it could take if it were free (the velocity before collision) W ; • its real velocity (after the collision) V ; • the lost velocity U so that W will be the resultant of V and U; • the force or quantity of motion F which each of the adjacent particle impresses to m, and through which it receives all the motion transmitted by the system; • the angle X between the directions of W and V ; • the angle Y between the directions of W and U; • the angle Z between the directions of V and U; • the angle q between the directions of V and F. First, with these new symbols, relation (11.3) assumes the expression:

∑ FV cos q = 0.

(11.4)

This relationship is not easily interpreted with the modern categories of mechanics since the term F, that represents the force, maintains a certain ambiguity, or rather, it is not yet reported in one of its ‘classical’ meanings. From a formal point of view one can say that relation (11.4) has the form of a virtual work law. Second, to achieve a more convincing recognition, the expression (11.4) is given a different form. The quantity V −W cos X is the velocity ‘gained’ by m for the effect of the constraints, and then m(V −W cos X) is the component of the force F, Carnot’s meaning, acting on the particle m in the direction of V , i.e., F cos q (see Fig. 11.2). In place of (11.4) one can then write the expression:

∑ mV (V −W cos X) = 0, Y mU

mW

Z

X q F Fig. 11.2. Impact of two masses

mV

(11.5)

11.1 Carnot’s laws of mechanics

289

but because W is the resultant of V and U and because W cos X = V +U cos Z, Carnot obtains the following relation:

∑ mVU cos Z = 0,

(11.6)

to which he refers as the first fundamental equation of mechanics.

11.1.2 Geometric motions At this point Carnot introduced the concept of geometric motion. If a system of bodies moves from a given position, with arbitrary motions, but such it is possible the system could have an equal but contrary motion, any of these motions will be called geometric motion [59].15 (A.11.11)

In the Principes fondamentaux de l’équilibre et du mouvement the definition is slightly different: Any motion will be called geometric if, when it is impressed upon a system of bodies, it has no effect on the intensity of the actions that they do or can exert on each other when any other motion is impressed upon them [60].16 (A.11.12)

The first definition is purely geometric, that is geometric motions are reversible motions congruent with constraints; the second definition seems to refer to mechanical concepts, because the word action calls for concepts like force or work. However this is not the case and also the second is a kinematical definition, because Carnot’s mechanics deals with impact of bodies and the impact is characterized kinematically. In any case Carnot thinks the two definitions are equivalent and ‘proves’ a theorem for which the definition of the Principes fondamentaux de l’équilibre et du mouvement implies that of the Essai [60].17 From the examples Carnot gives it appears that geometric motions can also be infinitesimal [59],18 [60].19 From an operational point of view the finite or infinitesimal nature of geometrical motion makes no difference because what Carnot uses is the velocity u associated to the geometric motion, called geometric velocity and sometimes still simply geometric motion. In summary, geometric motions are those motions compatible with all constraints finite or infinitesimal. For unilateral constraints, not all compatible motions are geometric, but only those that when reversed do not violate the constraints. With reference to the infinitesimal motion of the material point constrained on the concave surface of Fig. 11.3, that of Fig. 11.3a is a geometric motion while that of Fig. 11.3b, which detaches the material point from the surface, is not, because the contrary motion is not permissible. 15 16 17 18 19

p. 23. p. 108. English translation from [332], p. 43. p. 119. p. 26. p. 130.

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a)

b)

Fig. 11.3. Geometric and non-geometric motions

In some parts of his writings Carnot reserves the term absolute geometric motions to motion defined as above and speaks of geometric motions by supposition referring to those motions that could violate the existing constraints but do not actual violate them in real motion. The use of geometric motions by supposition can be useful to simplify the mechanical problem under consideration. In substance, Carnot says, assume that a unilateral constraint behaves as a bilateral one, then verify whether the assumption made is correct by checking if there are reactive forces that the ‘true’ constraint cannot exercise. If this is the case the assumption of geometric motion was not eligible. For example consider a wire capable only of tension producing the constraint of Fig. 11.3; then assume a geometric motion (case a). If the actual motion produces a compression in the wire, it means that the assumption of geometric motion should be disregarded, but it also means that the constraint too can be disregarded; in both cases, or assuming a geometric motion or disregarding the constraint, the analysis is made simpler. This idea of Carnot anticipates modern iterative calculation procedures to address problems with unilateral constraints. It is not clear if he had full awareness of this fact, although it probably was just to deal with unilateral constraints that he introduced geometric motion by supposition. Carnot gives a great emphasis to geometric motions, considering their introduction as one of his major contribution to mechanics: The theory of geometric motions is very important; it is, as I have already noted, like a mean science between ordinary geometry and mechanics. […] This science has never been treated in details, it is completely to create, and deserves both for its beauty and utility any care by Savants [60].20 (A.11.13)

The property of Carnot’s geometric motions which attracted the attention of French scholars of mechanics, Poinsot and Ampère in particular, is the fact that they are purely geometric, independent of the forces acting on the system to which they refer. The virtual displacements and velocities, regarded as geometric motions, are purely imaginary motions taking place in an imaginary time and do not alter the position of bodies and forces. In studies by Drago [299] and Drago and Manno [302], some criticisms are expounded about the way Carnot presents geometric motions and the way the first and second (see next section) fundamental equations of mechanics are deduced.

20

p. 116.

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291

11.1.3 The second fundamental equation of mechanics With the use of the concept of geometric motion relation (11.6) can be rewritten in an even more significant way. If u is a generic geometric motion, U the lost velocity, W = u +U is a virtual motion (Carnot’s meaning, i.e. before the impact), of which U is still the lost motion because u cannot find contrast in the constraints, then Carnot can write:

∑ mUu cos z = 0,

(11.7)

where now z is the angle between u and U. Carnot calls relation (11.7) the second fundamental equation. This equation completes the solution to the problem of the impact of hard bodies suggested by D’Alembert in the Traité de dynamique. D’Alembert had come to formulate the principle that the solution to the problem of impact is obtained by decomposing the motion before the impact a, b, c into two other motions a, b, c and α, β, γ. The first motion if applied alone would not have caused internal or external impact, the second motion applied alone would have been completely lost in collisions. The motion a, b, c provides the solution of the problem, i.e. the motion after the collision. This motion is determined as soon as the lost motion is found, as it holds a = a − α, b = b − β, c = c − γ. Relation (11.7) when u is varied in the space of all possible geometric motions, can furnish all the equations necessary to derive all unknown quantities U and then to solve the D’Alembert/Carnot collision problem. The role of the second fundamental equation of mechanics is therefore the same as that of the virtual velocity principle in which, by assuming different virtual velocities, all the equilibrium equations are obtained. In this analogy the lost motions could be compared with a set of balanced forces, because both of them leave a mechanical system unvaried. Carnot then defines the moment of momentum or the moment of quantity of motion to indicate the scalar product between the momentum and the geometric motion. In particular he introduces the moment of momentum of the current system: ∑ muV cos y – with u the geometric motion, V the actual velocity and y the angle between u and V – the moment of momentum before the collision: ∑ muW cos x – with W the velocity before of the collision and x the angle between u and W – and the moment of momentum lost in the collision: ∑ muU cos z. With simple steps, keeping in mind that for relation (11.7) the moment of quantity of motion lost in the collision is zero, Carnot demonstrates:

∑ muV cos y = ∑ muW cos x,

(11.8)

i.e. the moment of momentum before collision is equal to that after the collision. More precisely he proves the following theorem: In the collision of hard bodies, either that collision be direct or be made by a machine without any flexibility, for any geometric motion, it is invariably: 1 – The moment of quantity of motion lost by the whole system is equal to zero. 2 – The moment of quantity of motion lost by any part of the system of bodies is equal to the amount of moment of quantity of motion gained by the other side.

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11 Lazare Carnot’s mechanics of collision

3 – The real moment of quantity of motion of the general system, immediately after the collision, is equal to the moment of quantity of motion of that system, immediately before the collision [59].21 (A.11.14)

This theorem is just the second fundamental equation (11.7) expressed in a different way. Furthermore it “has a lot to do with what is obtained by considering the static moments with respect to different axes, but it is more general”. The third proposition of the theorem shows that to remain unchanged it is not so much, as Descartes thought, the quantity of motion (understood in the sense of scalar quantity) and not even the living force, because it always decreases in case of collision among particles, but there is a different quantity that neither the obstacles that oppose the motion, nor machines that transmit it, can change and this quantity is the moment of quantity of motion From the above theorem Carnot obtains the first corollary: Of all the motions of which a system of hard bodies agent on each other, either through an immediate contact, or through a machine without any flexibility, is capable, the motion which will actually take place after the collision among hard bodies, will be that geometric motion such that the sum of the products of each of the masses by the square of the speed that it will take is a minimum, i.e. less than the sum of the products of each of these bodies by the speed that it would have lost if the system had taken a whatever geometric motion [59].22 (A.11.15)

The proof of the corollary, which, for the sake of brevity I do not reproduce here, shows that the differential of the lost quantity of motion is zero when the geometric motion coincides with the real one. Carnot pointed out that this theorem has many similarities with the principle of least action of Maupertuis. As second corollary he obtains: In the collision of hard bodies, either that some are fixed or they are all mobile (which is the same), either the impact is direct or it is made by means of any inelastic machine, the sum of the living forces before the impact is always equal to the sum of the living forces after the collision, plus the sum of the living forces that it would have taken place if the residual velocity of each mobile were equal to that lost in the collision [59].23 (A.11.16)

The proof of the second corollary is the following. Posit W the velocity before the collision, V the velocity after the collision (the actual velocity), U the lost velocity and Z the angle between U and V ; from geometry we know that: W 2 = V 2 +U 2 + 2UV cos Z,

(11.9)

according to what today is known in geometry as Carnot’s theorem.24 Summing up all the hard bodies and bearing in mind (11.6) one has the second corollary:

∑ mW 2 = ∑ mV 2 + ∑ mU 2 , 21

(11.10)

p. 42. pp. 44–45. 23 p. 48. 24 Carnot’s theorem is today proved in a few easy steps, based on the property of the inner product between vectors: W 2 = (V +U) · (U +V ) = U 2 +V 2 + 2U ·V = U 2 +V 2 + 2UV cos Z. 22

11.2 Gradual changing of motion. A law of virtual work

293

still known in mechanics as Carnot’s theorem, which is clearly valid only for systems of hard bodies. But according to the Geometers of the XIX century it seemed it could be extended to the more general case of a very sudden shock – see for example the works of Coriolis, Cauchy, and Sturm in the first half of the XIX century.

11.2 Gradual changing of motion. A law of virtual work A third corollary extends the mechanics of collision to actions and movements changing gradually, so it is of considerable importance in applied mechanics. It says: When a system of hard bodies changes its motion for imperceptible degrees [gradually], m is the mass of each body, V its velocity, p its moving force, R the angle between the direction of V and p, u the velocity which m would have if the system would take any geometric motion, r the angle formed by u and p, y the angle formed by V and u, dt the element of time, it will hold any of two equations [59].25 (A.11.17)

∑ mV pdt cos R − ∑ mV dV = 0 ∑ mupdt cos r − ∑ mud(V cos y) = 0. In this corollary there appears the concept of moving force; it coincides with force in the ordinary sense per unit of mass. By posing V dt = ds, with ds an infinitesimal displacement, in the first of the two equations, Carnot obtains:

∑ mpds cos R − ∑ mV dV = 0,

(11.11)

that clearly represents the differential form of the “principle of conservation of living forces”. In the examination of forces that act continuously, in particular those of inertia, Carnot is not completely consistent. Generally, for him, the force is provided by mΔV , with ΔV being the change in velocity of the body; sometimes it is provided by mΔV /dt, dt being the infinitesimal interval in which there is a change in velocity, and this is used as an example in another proof of the theorem of live forces [59].26 Before moving on to deal explicitly with machines, Carnot tries to adapt his language to that normally used by the ‘practical’ mechanicians, who talk about power and not of lost quantity of motion. According to him, the power is the effort exerted by the agent, i.e. the tension or pressure, which acts on the body. The different powers are compared with each other, without regard to agents that produce them, because “the nature of the agents does not change anything about the properties of powers, which are required to satisfy the different uses the machines themselves are required” [59].27 25

pp. 49–50. p. 84. 27 p. 64. Carnot also tries to extend his theory to systems with elasticity. First, he points out that (11.7) also applies to bodies that are not hard, while relation (11.6) is no longer valid. He also pro26

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11 Lazare Carnot’s mechanics of collision

But a force or power considered in this sense, according to Carnot, is just a quantity of motion lost by the agent that exerts it, whatever that agent be, that it pulls with a rope or pushes with a rod. This power F coincides with the lost quantity of motion mU. So if one denotes by z the angle between the force F and the generic geometric motion u, the second fundamental equation becomes:

∑ Fu cos z = 0.

(11.12)

“And it is under this form that henceforth we will use this equation” [59].28 He adds that this general principle is the one of Descartes, to which is given a greater generality. Carnot reformulates all the previous theorems and corollaries using the concept of power F, generally referred to as force or weight. The ‘principle of Descartes’ generalized can also be applied to systems in motion if the forces of inertia are added to other powers. Below the general formulation of the virtual work law according to Carnot is reported, which substantially fits that of Lagrange, D’Alembert’s principle included: Fundamental theorem General principle of equilibrium and motion in machines XXXIV. Whatever is the state of repose or of motion in which any given system of forces applied to a Machine, if it is given any geometric motion, without changing these forces in any respect, the sum of the products of each of them, by the velocity which the point at which they are applied will have in the first instant, estimated in the direction of this force, will be equal to zero [59].29 (A.11.18)

It is interesting also reading what Carnot writes in a footnote which specifies why and in which sense relation (11.12) still holds in dynamic situations: It would not be useless to prevent an objection that could be presented to the spirit of those people who have not paid attention to what has been said about the true meaning that must be given to the word force. Imagine, for example, they would say, a winch to the axle and wheel of which weights are suspended by ropes, either there is equilibrium or uniform motion, the weight attached to the wheel will stay to that attached to the axis as the radius of the axis to that of the wheel, and this is consistent with the proposition [(11.12)]. But it is not the same thing when the machine takes an accelerated or delayed motion. Thus it would seem then that the forces are not at all in the inverse ratio of their mutual estimated speed in the direction of force as it would follow from the proposition [(11.12)]. The answer to this is that, in the case where this motion is not entirely uniform, the weights in question are not the only forces applied to the system, because the motion of each body, constantly changing, opposes in each instant, because of its inertia, a resistance to this change of state and one must therefore take account of this resistance. We have already said how to evaluate this force, and we will see later how it has to enter in calculations. It is enough to point out that the forces applied to the machine in question, are not the weights but the quantities of motion lost by these weights. Which must be estimated from the tension of the rope with which they are suspended. That the machine is at rest or in motion, this motion is uniform or not, the poses an approximate method to deal with the case by introducing a multiplier – it seems invariable over time – of the force of impact F. 28 p. 66. 29 p. 73.

11.3 The moment of activity

295

tension of the string attached to the wheel, is to that of the string attached to the axis as the beam axis is to the radius of the wheel. I.e. these tensions are always in inverse relationship to the speeds of the weights they support, and this is consistent with the proposition [(11.12)]. But these tensions are not equal to their weights at all, they are the result of these weights and their forces of inertia, which are themselves the result of the current motion of bodies and of the motion equal and directly opposed that it will actually take the next time [59].30 (A.11.19)

So finally equation (11.12) is a classical formulation of a virtual fork law, with geometric motions u that play the role of virtual displacements. It is certainly admirable the way in which Carnot reached his conclusion from phenomena that appear to be very simple and well defined. According to the categories introduced in Chapter 2, Carnot’s ‘demonstration’ has a foundational approach, which does not require any preexisting equilibrium criterion. The arguments, however, are not strict, mostly the transition from the case of collision to that of the continuous variation of motion is not clear, i.e. the identification of lost motion with power. This creates a proliferation of lost motions. On the one hand there are motions lost by means of constraints, on the other hand those lost by the agents and it is not clear why one should apply (11.12) to only the latter type of motion. However Carnot’s move produces the ‘miracle’ to transform a law of motion into a law of equilibrium. With his formulation in terms of power Carnot may eventually prove Torricelli’s principle, which he states in relation to machines: When some weights fitted to a machine are in equilibrium with each other, if you give this machine any geometric motion, the velocity of the centre of gravity of the system evaluated along the vertical should be zero [59].31 (A.11.20)

Carnot’s proof is very simple. Let u be a geometric motion; theorem (11.12) applied to weights mg gives:

∑ mgu cos z = 0.

(11.13)

But for the geometric properties of the centre of gravity, if u = u cos z denotes the component of u along the vertical, it is: mgu = MuG = 0,

(11.14)

where M is the total mass of the system and uG the velocity in the vertical direction of the centre of gravity, and then ultimately one has uG = 0; end of proof.

11.3 The moment of activity Lazar Carnot was opposed to introducing force as a founding concept: There are two ways to deduce mechanics from its principle. The first is to consider it as the theory of forces, that is the causes which impress motion. The second is to consider 30 31

pp. 73–75. p. 77.

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11 Lazare Carnot’s mechanics of collision

mechanics as the theory of motion in itself. The first way is that generally pursued, as the simplest; but it has the shortcoming to be founded on a obscure metaphysical concept, that of force [60].32 (A.11.21)

Carnot preferred the second way. He was not however opposed to the word force which he used often, sometimes with a technical meaning: “it is the product of the mass multiplied by the velocity it could take if it were not impeded by bodies having motions incompatible with it” [60],33 some other times according to the sense of the common language, sometimes even with the meaning of work. Carnot anyway sustained that, when the motion of a machine is concerned, that of force was not the most important concept because the effect it produces depends also in the way it is applied. To take into account this way Carnot introduced a concept coinciding with the modern meaning of work. He was not the first to do this, but he was the first to give it an emphasis and an operational meaning as a foundation of mechanics, especially for applied mechanics. The term he used to indicate work is moment of activity: If a force P moves with a velocity u and the angle formed by P and u is z, the quantity P cos zudt, where dt is the element of time, is called moment of activity [emphasis added] consumed by force P during dt [59].34 (A.11.22)

The total moment of activity during a finite interval of time T is given by:  T

Pu cos zdt.

(11.15)

0

Note that here there is an important shift compared to what had been done so far. Carnot’s laws, the first and second fundamental equations of mechanics (11.6) and (11.7), and the relation (11.12) that is valid for slowly varying forces, are expressed in terms of the geometric motion u; consequently expressions like Fu cos z and mUu cos z can be given any numerical value because u is not a physical magnitude but an undefined quantity; they are virtual works. Instead, the expression Pu cos zdt has a definite numerical value for it depends on two physical magnitudes, the ‘true’ force P and the ‘true’ velocity u, which in any real situation have well-defined numerical values, so the moment of activity is a physical magnitude, the real work. The possibility to replace geometric motion with actual motion allows Carnot to prove some interesting theorems. In particular he can quite easily formulate, as a corollary, a fundamental result of his mechanics, the conservation of work: Fifth Corollary. In a machine where the motion changes for imperceptible degrees the moment of activity consumed during a given time by the insisting forces is equal to the moment of activity exerted during the same time by the resisting forces [59].35 (A.11.23)

which comes directly from relation (11.12) when the geometry motion ‘u’ is replaced with the true motion ‘u’. 32 33 34 35

p XI. p. 2. p. 69. pp. 82–83.

11.3 The moment of activity

297

I sense some relief in a theorem ‘proved’ by Carnot inductively, by showing it is valid in all cases he knows, which concerns the moments of activity and, using modern language, coincides with the theorem of minimum of potential energy applied to central forces. Let some bodies, subject to an attraction exerted according to any function of distance, either by these bodies one on another or by different fixed points, be applied to a machine. If this machine passes from any assigned position to that of equilibrium, the moment of activity consumed by the forces of attraction from which bodies are animated, will be a maximum in this passage [59].36 (A.11.24)

The position of Carnot in attributing an important role to the concept of work will have a considerable influence on later scientists and his son Sadi, who will develop a thermodynamic theory based on the concept of work, which he will call moving puissance. Towards the end of the Essai he addressed the problem of perpetual motion and showed how it is impossible, on the basis of his principles, for the presence of passive forces.37 But although the impossibility of perpetual motion is easily justifiable, it is not employed by Lazare Carnot as a principle of mechanics, as was done by his son Sadi Carnot for thermodynamics. In considering the way in which Lazare Carnot’s ideas of work influenced subsequent scientists, such as Petit, Coriolis, Navier, Poncelet, Saint Venant and so on, it is interesting to note the following observation of Gillespie, for whom Carnot’s writings were less significant than his interaction with other scientists and his prominence as a statesman: The failure of Lazare’s Essai sur les machines to attract contemporary attention has already been discussed. Although more often mentioned since, the Principes fondamentaux fared little better when it appeared in 1803. It fell into the same obscurity, lasting another fifteen years. The book was occasionally mentioned prior to 1818, but rather by way of noticing its existence than because its point of view affected the treatment of problems. […] The explanation cannot well be that either father or son was an obscure or neglected personality (except in Lazare’s early years). What seems likely, therefore, is that attention first to machines and then to heat and power developed in a largely verbal, pedagogical, and practical way. The subjects constituted a kind of engineering mechanics avant la lettre in which problems were posed, principles tacitly selected, and quantities employed because that was the way to get results. What further seems likely is that both Carnots participated in that development personally rather than through their books. Lazare in his latter years was a kind of Nestor of engineering busying himself judging inventions for the Institute [332].38

36

p. 114. Carnot wondered what passive forces are, what difference there is between them and active forces. He believed that this is an important issue to which no one has responded, nor even attempted to answer. The distinctive character of the passive forces, for him, is that they can never become actions, while the active forces can act either as active forces or as resistant forces. Those of walls and fixed points are passive forces because they cannot act as active forces. 38 p. 101. The text by Gillispie is presently probably the most exhaustive on a historical analysis of Carnot’s mechanics. 37

12 The debate in Italy

Abstract. This chapter is devoted to the debate in Italy on the principle of virtual velocities as presented in Lagrange’s Méchanique analitique of 1788. Both reductionist and foundational approaches are presented. In the first part those contributions that criticize the evidence of the principle of virtual velocities are introduced, which arrived at slightly different formulations of VWLs. In the second part those more technical contributions are presented, aimed to reformulate the principle of virtual velocity principle without the use of infinitesimals. Italy was one of the European countries where virtual work laws received the greatest attention, as evidenced by the long list of Italian scholars related to this subject. In previous chapters I have shown that, before Lagrange and after Galileo and Torricelli, other relevant contributions came from the Italian school in the XVIII century. In 1743 Ruggiero Giuseppe Boscovich, a Dalmatian mathematician deeply rooted in Italian culture, used a virtual work law in his analysis of some damage suffered by St. Peter’s dome. But perhaps the most interesting contribution was the introduction of the principle of actions by Vincenzo Riccati in 1749. He generalized the principle of virtual velocities presented by Johann Bernoulli in 1715. The same improvements were reiterated in 1770 by Vincenzo Angiulli, professor in a military school at Naples. Lorenzo Mascheroni published in 1785 a paper on statics of domes [166], where a virtual work law played a meaningful role; the study by Daviet de Foncenex, a Lagrange student, is also remarkable. The efforts of the Italian scholars of the early XIX century, after the publication of Lagrange’s celebrated book Méchanique analitique in 1788, are surely less interesting. They can be divided into two groups: a) those which are addressed to improve the proof of the virtual work principle, before the second edition of the Mécanique analytique and the proof presented by Lagrange himself; b) those which are addressed to improve the mathematical formulation, by discussing in particular the possibility of avoiding the use of infinitesimal displacements.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_12, © Springer-Verlag Italia 2012

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12 The debate in Italy

12.1 The criticisms on the evidence of the principle In the following I will consider the contributions by Vittorio Fossombroni, Girolamo Saladini and François Joseph Servois.

12.1.1 Vittorio Fossombroni Vittorio Fossombroni was born in Arezzo in 1754 and died in Florence in 1844. He was a scientist and a statesman of the Grand Duchy of Tuscany and was particularly active as a hydraulic engineer. Since 1815 he was the chief of the government of the Grand Duchy of Tuscany, achieving important results in the modernization of the region [294]. Fossombroni was an important representative of the Italian school of virtual work laws. He presented his ideas in the Memoria sul principio delle velocità virtuali of 1794 [109], six years after the publication of Lagrange’s Méchanique analitique. The work of Fossombroni did not have a great theoretic value, but it contained some interesting contributions: it showed, probably for the first time, a convincing demonstration of virtual velocity principle relatively to rigid bodies, on a purely analytical basis in a mechanics of reference based on the cardinal equations of statics, with an approach that will be taken up by Prony. Fossombroni also raised the question of verifying the extent to which the principle of virtual velocities remains valid when considering finite rather than infinitesimal displacements. In Italy, a country then scientifically provincial, Fossombroni’s work was hailed as an event of considerable importance; it is significant that a the note appeared on June 8th 1797 in the Décade philosophique, littérarie et politique, which stated that “It is the glory of Tuscany, which had the honor of being home to the famous Galileo, who discovered this principle, the repetition by another compatriot, the first demonstration”. In the following I report a long quotation from the preface of Fossombroni’s text, which in addition to the understanding of the character of the man and somehow also his cultural level, contains interesting comments on the history of mechanics: At the rebirth of Sciences, Galileo investigated the Theoretical Foundations of the equilibrium and motion, subjecting them to geometry, and with the Principle of Virtual Velocities spread a new, universal radiation to all simple and compound machines. […] In fact Mechanics by means of the Principle of virtual velocities, combined with the Geometry shared the same evidence and the privileges to the full extent which this synthesis could reach. Following the new Geometry (which swiftly fly through the space that the old measured slowly, and reached places that had never penetrated) has met the most flattering hopes, and Mr. La Grange’s first in his immortal work entitled Analytical Mechanics, not only showed that the principle of virtual velocities is due to Galileo, but showed also, that this principle has the advantage of being translated into algebraic language, i.e. to be expressed into an analytical formula, so all the resources of analysis will apply directly. That principle after Galileo was almost neglected, as a large sword hanging is useless, as

12.1 The criticisms on the evidence of the principle

301

long as their did not arise an arm capable of wielding it. In fact, Mr. La Grange, master of all the mathematical entity, was able to assess its importance and fruitfulness, creating by means of it a new science of Mechanics, that in the universal doctrine of equilibrium, and motion of solids and fluids, all those difficult problems that had led up to now to the thorny Problems for a thousand different ways, are reduced to regular and uniform procedures. And to give an idea of how the human mind has progressed, we can say that the motion and the equilibrium of the Heavenly Bodies, the shape and the orbits they describe, do not call in essence, for what belongs to Mechanics, to consider other laws than those which arise in calculating the motion and equilibrium of a lever of the first kind through the difficulties of pure calculation, and the multitude of objects to contemplate, need a larger and impressive apparatus. […] Some were employed to show, that this principle is true, showing the results of its compliance with those raised by other methods generally allowed. But really if one could not obtain other genuine proof, we would be far from the purpose for which routinely Geometers strive; in the same manner, that when the followers of Leibniz lacked a convincing demonstration of calculus, it was weak support for them to observe the uniformity of their results with those of the Geometry of the Ancients. […] That common faculty of primitive intuition, so everyone is easily convinced by a simple axiom of geometry, as for example, that the whole is greater than the part, certainly do not need to agree on the aforementioned mechanical truth, which is much more complicated than that of the common axioms, as the genius of the great Men who have admitted the axiom, exceeds the ordinary measure of human intelligence, and it is therefore necessary for those who are not satisfied to obtain a proof resting on foreign theories, such as Riccati likes (which with some metaphysical arguments, has considered this particular case in letters printed in Venice in 1772), or to rest on the faith of chief men despising the usual reluctance to introduce the weight of authority in Mathematics. And if indeed this tyranny of reason were to appear only once in the Temple of Urania, it could not follow less scandal, that it was between Galileo and La Grange [109].1 (A.12.1)

12.1.1.1 Invariable distance systems The most interesting part of Fossombroni’s work is on the distance invariant systems, i.e. the rigid body, the only ones to which I refer in the following, neglecting, for reasons of space, systems of many rigid bodies and fluids. Probably the most original parts of the work are the analyses of the validity of the virtual velocity principle in the case of finite displacement and its proof for a rigid body. When examining the kinematics of invariant distance systems, necessary for his demonstration, Fossombroni introduces a distinction and a notation unnecessarily complicated. He indicates with the symbol d the whole motions, infinitesimal or finite, with the symbol Δ for pure translations, and with the symbols δ, δ for the motion associated with rotation (finite or infinitesimal), where the presence or absence of the apex is used to specify which is the axis of rotation. For example δx represents a rotation around the y-axis, while δ x a rotation around the z-axis. He does not give a name to his displacements; for example he does not refer to them as the virtual velocities. Instead he uses the term virtual velocity in a ‘modern’ sense, meaning the vector and not the component in the direction of the force.

1

pp. 3–27.

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12 The debate in Italy

First Fossombroni examines the translational motion and the equations of equilibrium to translation, deriving easily an equation of virtual work. Using his symbols, by indicating the points with p , p , p , the following kinematic relations can be written: Δp = Δx cos α + Δy cos β + Δz cos γ Δp = Δx cos α + Δy cos β + Δz cos γ Δp = Δx cos α + Δy cos β + Δz cos γ ···

(12.1)

with cos αi , cos βi , cos γi the direction cosines of the forces Pi acting on pi ; Δxi , Δyi , Δzi the components of the virtual displacements Δui , finite or infinitesimal, in the directions of the coordinate axes; Δpi the components of the virtual displacements in the direction of forces. Note that in the purely translational motion Δxi , Δyi , Δzi are equal for all points, and thus the translational motion is defined simply, for example, by Δx , Δy , Δz . The equation of equilibrium to translation for the components of the forces P , P , P on coordinate axes are: P cos α + P cos α + P cos α + · · · = 0 P cos β + P cos β + P cos β + · · · = 0 P cos γ + P cos γ + P cos γ + · · · = 0.

(12.2)

By multiplying these equilibrium equations by Δx , Δy , Δz in the order, adding and considering the kinematical relations (12.1), Fossombroni obtains: P Δp + P Δp + P Δp + · · · = 0,

(12.3)

which “is the same equation of moments deduced by the Principle of Virtual Velocities”. Fossombroni notes that there is no mandatory reason to suppose that the motion be infinitesimal. 12.1.1.2 The equation of forces In the analysis of the rotational motion and equilibrium of rigid bodies, Fossombroni wants to see preliminarily if the law of virtual work can be extended to finite virtual displacements and introduces the distinction between equation of forces and equation of moments to refer to the first when the displacements are finite and to the second when they are infinitesimal. With reference to the various forces Pi applied to points A, B, C of the line AC, as shown in Fig. 12.1, parallel among themselves and perpendicular to the line AC, the equilibrium conditions to translation and rotation, give respectively: P + P + P + · · · = 0 y P + y P + y P + · · · = 0.

(12.4)

The finite virtual displacements in the direction of forces, are defined by a constant value Δx associated with a translation and values δx , δx , δx , etc., associ-

12.1 The criticisms on the evidence of the principle

A M

a

B

C B′

A′ b

303

c

C′

Y

A′′ B′′ X

E

C ′′

G

F

Fig. 12.1. Equation of forces. Orthogonality to the line where they are applied

ated with rotation around the point M. Fossombroni expresses the different values δx , δx , δx , etc. to δx by means of the relation: δxi =

δx i y. y

(12.5)

Substituting the value of yi obtained from this equation, in the second of the equations of equilibrium (12.4), he first obtains:  y    P δx + P δx + P δx + · · · = 0 (12.6)  δx or: P δx + P δx + P δx + · · · = 0. Then, multiplying the first of the equilibrium equations (12.4) by that Δx = Δx = Δx , etc., it is:

(12.7) Δx ,

considering

P Δx + P Δx + P Δx + · · · = 0. Δxi + δxi

is the total space By adding (12.8) and (12.7), recalling that by the force in its own direction, Fossombroni writes the equation: P d p + P d p + P d p + · · · = 0,

(12.8) d pi

covered (12.9)

that is valid without requiring the displacements be infinitesimal. Fossombroni can then conclude that in the case of forces applied to the points of a line, perpendicular to it and lying on a plane, the equation of forces is valid for any motion of the plane. He finds the same result when the forces of the previous case, while remaining parallel to each other are not perpendicular to the line AC, but slanted, as shown in Fig. 12.2 to further generalize with the following theorem: Theorem. The equation of the forces will also hold as that of the moments, when the bodies will be established in a straight line, and also though the forces anyway applied that have directions not parallel to each other, have at least parallel their projections in a plane passing through the line of bodies [109].2 (A.12.2) 2

p. 86.

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12 The debate in Italy

A M

B

C a A′ b

A′′ E X

F

B′

c

C′

Y

B′′ C′′ G

Fig. 12.2. Equation of forces for parallel forces

This theorem may seem a mere curiosity and it is certainly true that, from a practical point of view, it is not of much use even if it is considered in the most extensive form, obtained by Poinsot [197]. But it should be seen primarily as a demonstration of hardship and as a first step towards the elimination of the concept of infinitesimals. This topic will be taken up in Chapter 14, dedicated to Poinsot.

12.1.1.3 The equation of moments Fossombroni established his theorem after acknowledging that, in general, it is unlawful to use finite displacements; he then returns to a rigid body subject to any forces, for which he proceeds to prove Lagrange’s virtual velocity principle assuming infinitesimal displacements. This probably is the first proof of the principle for a rigid body in the reductionist approach, assuming as a pre-definite criterion of equilibrium the validity of cardinal equations of statics, three equations for translation and three for rotation: P cos α + P cos α + P cosα + · · · = 0 P cos β + P cos β + P cosβ + · · · = 0 P cos γ + P cos γ + P cosγ + · · · = 0 P (cos α y − cos β x ) + P (cos α y − cos βx ) + P (cos α y − cos βx ) + · · · = 0

(12.10)

P (cos α z − cos γ x ) + P (cos α z − cos γ x ) + P (cos α z − cos γ x ) + · · · = 0

P (cos β z − cos γ y ) + P (cos β z − cos γ y ) + P (cos β z − cos γ y ) + · · · = 0. First he proves the necessary part of the virtual velocity principle, i.e. if the cardinal equations are satisfied, the equation of moments holds true. In the following I do not report Fossombroni’s lengthy passage, also because of his not happy notation for the kinematics. I only signal that he multiplies the equation of equilibrium to translation

12.1 The criticisms on the evidence of the principle

305

by translational motions and the equilibrium to rotation by rotational motions and add all, obtaining the equation of moments: P d p + P d p + P d p + · · · = 0,

(12.11)

which is formally equivalent to the equation of forces (12.9), but here d p , d p , d p , etc. are infinitesimal. Fossombroni concludes by underlining the condition of validity of his result: It could be concluded that in each system where the equilibrium depends from the equations (1), (2), (3), (4), (5), (6) [12.10], of § LXXI, the property sum of moments = 0 is a property necessary and inseparable of the equilibrium [109].3 (A.12.3)

In the following sections from LXXXIII to XCVI Fossombroni also demonstrates the sufficiency of the vanishing of the sum of moments for equilibrium. He is not completely satisfied with the way Lagrange got the equation of equilibrium for rigid bodies [145]4 from the principle of virtual velocities and tries to bring some contribution, but he does it so confusingly, though correct, that I omit the analytic passages of the proof that apart from being very boring do not provide interesting information to that already given. Fossombroni poses the question: It is not possible to deny, that whenever equilibrium takes place, the equation of moments is necessarily true, but is it certain that whenever there is an equation of the moments there is always the equilibrium? [109].5 (A.12.4)

After having raised the doubt: It could be dubious that beside these six equations [the cardinal equations of statics] there could be some more[109].6 (A.12.5)

He is able to resolve it. The heavy treatment of Fossombroni can be justified, because in his times there was not available the symbolism of the vector calculation. With it, the proof of necessary and sufficient parts of the virtual velocity principle, assuming the criterion of equilibrium provided by the cardinal equations of statics, would result in very few steps. Fossombroni’s work fell into the hands of Lagrange, who in May 1797 wrote a letter full of praise, but did not discuss its merits. The only point that Lagrange underlined was Fossombroni’s idea to consider finite displacements. Here is the text of the letter: I read your book with pleasure. If there is still something to be desired in mechanics it is the reductio of principles, which serve as its basis, and perhaps even direct and rigorous proof of these principles. Your work is a new service for this science. You observe, correctly, there are cases where the equation of virtual velocities also occurs in relation to finite differences, 3 4 5 6

p. 97. pp. 26–30. p. 101. p. 112.

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12 The debate in Italy

so the system while changing the configuration still remain at rest. These kinds of equilibria are midway between the stable equilibria, where the system returns to its first state when it is disturbed and unstable equilibria, where the system, once disturbed from its state of equilibrium, tends to move away more and more [206].7 (A.12.6)

Lagrange’s letter is interesting because he shows that he had understood better than Fossombroni what are the cases for which the equation of forces is valid. They are those today classified as cases of neutral equilibrium. Lagrange returned to the matter some time later in another letter: I gave a demonstration of the principle of virtual velocities derived from the equilibrium of pulleys. An important principle can be proven in many ways. Your work on this subject, besides its own merit, has that of having motivated other works as the memoirs of Prony and Fourier, whose authors made homage to you [206].8 (A.12.7)

12.1.2 Girolamo Saladini Girolamo Saladini was born in Lucca in 1731 and died in Bologna in 1813. He was a student of Vincenzo Riccati and an important Italian mathematician of the end of the XVIII century. In 1808, eighty years old, he wrote a paper on the principle of virtual velocities [213] which has little relevance from a theoretical point of view, but is interesting as the evidence of the relative sterility of the mathematics of the time in Italy. Though Saladini was at the top of Italian mathematics, the quality of his memoir is not even remotely comparable to that of contemporary French mathematicians and even to that of Fossombroni. Saladini aims to prove the principle of virtual velocities from the rule of composition of forces. He considers the case of a free material point subject to three forces and proceed to show with simple and elegant reasoning that if the system of forces is in equilibrium then the equations of moments are fulfilled. No account is given of constrained material points. Saladini first proves two geometric theorems about parallelograms. The first theorem [213]9 relates to Fig. 12.3. Given the parallelogram ABCD and the point X, the relation holds: AC × OX = DC × MX + BC × NX.

(12.12)

The second theorem [213]10 relates to Fig. 12.4. Here Saladini considers the dual parallelogram XVYZ the sides of which are orthogonal to those of the parallelogram ABCD (e.g., XY⊥ AC). Theorem I applied to the parallelogram XVYZ, gives: CT × XY = CP × XZ + CR × XV

7

p. 10. See also [122], vol. 13, pp. XXIII–XXIV. p. 10. 9 pp. 403–405. 10 pp. 405–406. 8

(12.13)

12.1 The criticisms on the evidence of the principle

307

P N

C

B O

M F

X

D

A

R

Fig. 12.3. The parallelogram

T

P

R C

V

X

B

D

Z Y

A Fig. 12.4. Dual parallelograms

or AC × CT = DC × CR + BC × CP,

(12.14)

which is the second theorem. Interpreting ABCD as the parallelogram of balanced forces such as AC, BC, DC, and the segment CX as a virtual displacement, CP is the virtual velocity of force CB, CR that of the force CD and CT that of the force AC. Taking the appropriate signs, the previous relation provides the law of moments for three balanced forces converging in C. The paper of Saladini appears less interesting if one reflects that theorem I is nothing but the so-called Varignon theorems, and that Poinsot a few years before, gave a very similar proof for the law of moments expressed by theorem II. Maybe Saladini could not know Poinsot’s work, he does not mention him, but this is not a valid excuse, at most it indicates the isolation of Italian mathematics. For what concerns the proof of the inverse proposition, that is that from the equality of moments there follows the rule of composition of forces, Saladini refers to the paper by Vincenzo Riccati [339].

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12 The debate in Italy

In the application of the virtual velocity principle to a system of material points, Saladini is rather hasty. He says that in this case it is clear that one must rely upon the law of the lever: If we suppose that there are more points in any way connected, that are moving around any axis, who now does not consider that the theory of such motion depends on the principle of lever? [213].11 (A.12.8)

Saladini then takes on two principles underlying the static demonstration of the principle of virtual velocities. One is the law of the composition of forces, which for him is of “metaphysical and geometric certainty” [213],12 the other is the law of the lever to which Saladini associated a lower level of confidence, seeing it as certainly true, but as a matter of fact and not logically necessary. Although, as we noted, some are of the opinion that a rigorous proof of the theory of the lever studied by Archimedes and after him by other savant men still leave something to be investigated [213].13 (A.12.9)

By evoking two principles with different degrees of evidence, the principle of virtual velocities may not have more evidence of the less evident, i.e. the law of the lever: So we still have to be of the sentiment of those who have opined that the principle of resolution and composition of forces have a metaphysical infallibility; that of the lever only the patronage of continuous and constant experience, and finally that of the virtual velocities deduced from the two previous principles, cannot acquire higher degree of certainty of what has been identified in the principle of the lever [213].14 (A.12.10)

12.1.3 François Joseph Servois François Joseph Servois was born in Mont-de-Laval in 1767 and died in Mont-deLaval in 1847. He was ordained a priest at Besançon at the beginning of the Revolution, but in 1793 he gave up his ecclesiastical duties in order to join the army. His production is not abundant, but it is quite original. Though Servois was a French mathematician, I will present him here because his study on the virtual work of 1810 De principio velocitatum virtualium commentatis [214] was entered in a prize competition sponsored by the Reale accademia delle scienze of Turin. Curiously, his memoir was the only entry the academy received and, because Servois missed the deadline, nobody won the prize. However, the paper was deemed worthy, so the Reale accademia published it and elected him a corresponding member. The object of the prix was: Clarify the principle of virtual velocities in its full generality such as it was enunciated by Lagrange. Show if this principle should be considered as a truth evident by itself or if it requires a proof. Give this proof in the case it is felt necessary. (A.12.11) 11 12 13 14

p. 415. p. 415. p. 416. p. 417.

12.1 The criticisms on the evidence of the principle

309

Servois work is pretty interesting, but little known, perhaps because it was written in Latin and in my opinion is worthy of a specific in depth study. In keeping with aims of the prize, in the first part of his memoir Servois traces the history of the principle of virtual velocities in the XVIII century. He gives some hints too about the state of the art in England and Germany, countries where the principle of virtual work was considered less than in Italy and France. After the historical part follows the theoretical part. Servois distinguishes between a priori demonstrations, which prove the principle of virtual velocities from simple and clear, or at least acceptable, considerations of statics without the use of a criterion of equilibrium fixed a priori, and a posteriori demonstrations, in which there is a pre-established criterion. Among them, he cites the demonstrations of Varignon, Fossombroni and Poinsot, all depending on the law of composition of forces. In these demonstrations, the principle of virtual velocities becomes a theorem, or a corollary. Servois aims to provide a demonstration a priori. He sets out a series of principles and definitions on statics very easy to accept, with the exception of an ultimate principle, the 8th, which he considers his own: In whichever way two points A and A are joined together, if their virtual velocities v and v have the same intensity, then the forces P and P applied [to A and A ] are equilibrated if they are equal [214].15 (A.12.12)

According to Servois the first seven propositions arose from the elements of mechanics and should be considered as axioms. The last should be assumed as true or at least postulated as such. Moreover it is of little use to claim its evidence. This proposition does not refer only to the case where two forces are aligned, but is more generally applied to two points connected in some way and that can move in any direction. Previous seven propositions got from the elements of mechanics and should be considered as axioms. The last should be assumed as true or at least postulated as such. Moreover it is of little use to claim its evidence [214].16 (A.12.13)

P

A q

q

m

m′

p

p A′ P′

Fig. 12.5. The law of the pulley 15 16

p. 191. p. 191.

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12 The debate in Italy

On the basis of his principles Servois begins to show a series of theorems of gradually increasing complexity. The first theorem of some interest is the law Lagrange called law of the pulley. In the case of the situation in Fig. 12.5 he shows that the law of the moments holds: Pv = P v ,

(12.15)

with v/v = P /p = 2. To demonstrate this law Servois uses his 8th proposition to state that the tension p of the rope that wraps a pulley remains constant for all its length. Then with simple consideration of equilibrium he obtains P = 2p and with simple consideration of kinematics v = 2v . Based on the law of the pulley Servois gets to prove the validity of the law of moments in various situations. For example for a system of forces concurrent in a point. Note that though Servois uses the same model considered by Lagrange in his demonstration of the principle of virtual work. i.e. the pulley, he uses it in a completely different way. He claims the originality of his argument by stating that beside him only Lagrange and the British mathematician John Landen (1719 -1790) gave importance to the law of the pulley: I without problem admit that near the ancients the block and tackle and the pulley were celebrated less than the lever, as it can be known from sources among others the 8-th book of Pappus’ mathematical collections (the pulley was the third faculty near Hero) and recently still less considered, among the principles of the science of equilibrium, and probably only by Landen and Lagrange. When the choice of a principle is concerned, one should pay attention to the evidence and especially to the fertility of it and nobody will deny that the theory of the pulley is very useful [214].17 (A.12.14)

Though in all cases considered it has been proved only that given the equilibrium it follows the sum of moments is zero (necessary part of the virtual velocity principle), Servois claims that it is true also for the reverse, i.e. that by imposing the vanishing of the sum of moments the equilibrium follows (sufficient condition). Indeed the sufficient condition results by imposing the necessary conditions for all possible virtual motions. In this way a certain number of equations between forces are obtained that define completely their relations at equilibrium. Moreover, somebody will perhaps say that our demonstration is mutilated and incomplete, because it should be discussed not only that from the equilibrium of forces it follows the equation of moments, but also reciprocally, that from the equation of moments it follows the equilibrium of forces. Let us consider the meaning of the equation of moments, valid for the system of pulleys: it shows the equilibrium among forces and meantime, because of the concatenation among the equations, it is obtained the meaning of the moment equations in all possible cases. And it will be clear that this equation should hold because the equilibrium follows and not only because it follows from the equilibrium [214].18 (A.12.15)

This position seems to me important because it sets clearly a new way of connecting the necessity and sufficiency of equilibrium in the principle of virtual velocities. Once the necessary conditions corresponding to all eligible virtual displacements 17 18

p. 220. p. 221.

12.2 The criticisms on the use of infinitesimals

311

are imposed, if from them there result relationships among the applied forces such that they can be uniquely evaluated, these relationships are also sufficient. Indeed if they were satisfied and there would not be equilibrium, the equilibrium should be found with values of forces that do not meet those relationships, but this is not possible because they are necessary.

12.2 The criticisms on the use of infinitesimals In this section I will refer to comments by Giovanni Battista Magistrini, Geminiano Riccardi, and Gabrio Piola. Only to the latter, I will dedicate a quite large space. Giovanni Battista Magistrini (Maggiora 1777–Bologna 1849) in a paper of 1815 [165] in a section titled Del principio delle velocità virtuali, e del modo di evitarne l’uso, criticizes the principle of virtual velocity because it introduces concepts not well defined, such as that of virtual velocities: Because here it is necessary the use of the expedient of a mechanical motion, which though originated wonderful renowned truths, it notwithstanding leaves the desire for a clear, simple and unique demonstration which proves in a necessary way these properties. Demonstration of which it can be said it is not yet obtained, if considering the complication, variety and darkness of the attempts made to find it [165].19 (A.12.16)

Magistrini aims to clean up the virtual velocity principle by redefining the concept of virtual displacement. With arguments that I have to say are not unexceptionable, he suggests redefining the virtual displacement of a generic point indicated as δqi by Lagrange with the differential as defined in the Théorie des fonçtions analytique, i.e. the ‘aggregate’of first dimension terms resulting from substituting, in the constraint equation, x +i, y+i , z+i for the coordinate x, y, z, with i, i , i arbitrary quantities. Starting from the rule of composition for the forces Q1 , Q2 , etc., Magistrini regains the moment equation in a form which is formally similar to Lagrange’s: Q1 dq1 + Q2 dq2 + etc. = 0,

(12.16)

but which does not contain the concept of infinitesimal. The work of Magistrini goes no further; he merely limits himself to suggesting a redefinition of the concept of virtual displacement without bringing in-depth discussion of how to do so in the case of a system of constrained material points. Geminiano Riccardi (Modena 1794–1857), thirty years later, came back to the problem of virtual velocities, in this case in the defense of Lagrange and with a criticism toward the Russian mathematician Viscovatov who in an 1802 paper [242] suggested to substitute virtual displacement with virtual velocities (in the modern sense). The principle of the virtual velocity according to Viscovatov should be modified as follows:

19

p. 450.

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12 The debate in Italy

If many forces with any direction applied to a system of bodies or points are equilibrated, the sum of these powers, multiplied each for the velocity which it tends to impress [emphasis added] to the point to which it is applied, is necessarily equal to zero. One can see that this statement is included in what was exposed before, but it is purified from the infinitely small quantities [242].20 (A.12.17)

In his paper of 1842 [206] Riccardi criticizes the way Viscovatov conducts his proof, and in my opinion correctly because the paper by Viscovatov is not very clear. A part of the discussion on virtual velocities, Riccardi’s paper is interesting for what it says about the diffusion of the principle in the didactic of mechanics [206].21

12.2.1 Gabrio Piola Gabrio Piola, to whom I will return in Chapter 17, refers to his consideration on Lagrange’s principle of virtual velocities in a paper which won a prize from the Reale Istituto Lombardo delle Scienze in Milan in 1824 and was published in 1825 [187]. The object of the prize was to “Explain the application of the main items of the Analytic Mechanics by the immortal Lagrange to the principal mechanical and hydraulic problems, from which it appears the great utility and efficiency of the Lagrangian methods”. Piola thinks that the virtual velocity principle as formulated by Lagrange has two main drawbacks: it is not completely evident and it makes use of the not well-defined concept of infinitesimals. These reflections persuaded that it would be a poor philosopher who would insist to know the truth of the fundamental principle of mechanics [the principle of virtual velocities] as an axiom. So it would lack of the evidence the principle I will assume […] which is the same assumed by Lagrange in the third part of his theory on functions. But if the fundamental principle of mechanics cannot be made evident, it should be at least a truth simple to understand and convincing [187].22 (A.12.18)

12.2.1.1 Piola’s principles of material point mechanics To study the motion of a material point Piola believes that the only principle really evident is that of superposition of motions – displacements, not forces. The principle is empirical, nonetheless it is absolutely clear because it refers to the evidence of all times; the same cannot be said of the principle of virtual velocities. According to the principle of superposition of motions, for two motions due to two different ‘causes’, the resulting motion is the vector sum of the two motions. Piola is well aware that there are cases where this principle does not apply, for example for force depending on position: If a body attracted toward a fixed point passes in a straight line during the time t the space φ(t), when another motion is impressed to it […] α(t) […] for the simultaneous action of the two motions, it does not cover the space expressed by φ(t) + α(t) but by another function of time [187].23 (A.12.19) 20 21 22 23

p. 176. p. 8. p. XVI. p. 5.

12.2 The criticisms on the use of infinitesimals

313

He criticizes Lagrange for his lack of clarity in the Théorie des fonctions analytique [149] 24 where it seems that the law of the composition of motion is a purely geometric theorem. For him this is true for the decomposition of ideal motions, but not for the real ones. Although the composition of motions expressly permits non-trivial exceptions, Piola assumes it as a principle. So it seems that his problem is to understand how much mechanics can be explained assuming the composition of motions; acting more as a mathematician than as a physicist. Due to the uncertainty of the general validity of the principle he used, Piola avoids giving an axiomatic structure to his mechanics. The various concepts are introduced when they are useful, without attempting to reduce each to other. In the spirit of the Théorie des fonçtions analytique Piola assumes that the general motion of a material point can be developed into series: 1 x(t) = V θ + Xθ2 + etc., 2

(12.17)

where θ = t − t0 is the difference between a reference time t0 and the current time t. The coefficient V of expansion is said to be the velocity or force exerted, the coefficient X is said to be the accelerating force. Note the dynamic characterization of the velocity which with an apparently ‘Cartesian’ language is treated as a force probably because at the beginning of motion the velocity can be considered as proportional to the action of forces. With the aid of the principle of composition of motions, Piola not only solves the problem of motion but also that of equilibrium. A material point is in equilibrium if and only if the motion components cancel each other, that is, for each θ the following relation holds: 1 (V1 +V2 + etc.)θ + (X1 + X2 + etc.)θ2 + etc. = 0. 2

(12.18)

From (12.18) it is clear that a necessary and sufficient condition for equilibrium is that for each instant t0 it is: V1 +V2 + etc. = 0 X1 + X2 + etc. = 0 etc. = 0.

(12.19)

The condition becomes less restrictive if the motion is continuous. In this case, for example, the vanishing at all times of the sum of velocity implies the vanishing of all terms in the series and then the equilibrium. It would appear that the fundamental law of equilibrium for Piola is not that of the cancellation of forces but rather that of the cancellation of velocity. This approach, interesting and unusual, however, is complicated and in fact Piola leaves it, simply checking the vanishing of the sum 24

Part III, Chapter 2.

314

12 The debate in Italy

of the forces. The study of motion of a free material point is thus reduced to the classical form: x¨ − X = 0, y¨ −Y = 0, z¨ − Z = 0,

(12.20)

where X,Y, Z are the accelerating forces while x, y, z are the components of motion that are defined only if the initial values of velocity are provided. 12.2.1.2 System of free material points The study of systems of material points interacting with each other requires the introduction of new principles and concepts, in particular the concept of mass should be introduced. Piola is aware of the difficulties inherent to the dynamic implication of his choice. He solves the problem by admitting the existence of molecules of matter all equal to each other, so that the mass of an aggregate is proportional to the number of atoms. In addition to the concept of mass he must also introduce the principle of action and reaction. According to Piola, this principle, which is never named as such, may be regarded partly as a principle of reason, partly as an empirical principle, at least for material points with equal mass. It is an empirical fact that two material points produce motion to each other; it could be a principle of reason that they move on the straight line: “it is easy to be convinced that two points, removed any other action […], will move toward each other and this motion will be on the line connecting them” [187].25 The artifice of considering mass points equal to each other also allows extension of the principle of action and reaction to points with unequal mass. Consider in fact two material points of mass m1 and m2 to be composed respectively of m1 and m2 single mass points of unit mass. If all points exchange the same force H between them, the two points of mass m1 and m2 exchange with each other an equal force proportional to m1 m2 H. In the end, the equations of motion of a system of material points free from constraints can be written in the form: x¨i − Xi = 0, y¨i −Yi = 0, z¨i − Zi = 0.

(12.21)

According to Piola it is easy to see that these equations can be deduced from the single variational equation:

∑(x¨i − Xi )δxi + ∑(y¨i −Yi )δyi + ∑(¨zi − Zi )δzi = 0, i

i

(12.22)

i

where δxi , δyi , δzi are generic functions, variables with time, independent of each other and are not infinitesimal virtual displacements, as it was for Lagrange.

25

p. 33.

12.2 The criticisms on the use of infinitesimals

315

In the case when the forces Xi ,Yi , Zi can be derived from a function Π of x, y, z , the above equation can be obtained as a variation of the functional: U = Π+

1 (dx2 + dy2 + dz2 ). 2∑

(12.23)

In modern terms it can be said that U represents the total mechanical energy of the material point system, which today is well known to be constant with time. 12.2.1.3 System of constrained material points The solution to the constrained motion is obtained by analogy from the techniques of solving the problems of constrained minimum. If a stationary problem represented by a function like (12.23) is subject to geometric constraints such as: L = 0,

(12.24)

the solution is obtained by the method of Lagrange multipliers, making stationary the function: Π+

1 (dx2 + dy2 + dz2 ) + λL, 2∑

(12.25)

where λ is an arbitrary coefficient, and this corresponds to Lagrange’s principle of virtual velocities. Note that no use is made of infinitesimals. The reasoning by analogy of Piola is however entirely devoid of any physical basis. No one tells us that for a constrained problem the motion is provided by minimizing the same functional valid for the free motion. Piola is implicitly taking the idea of smooth constraints, assumptions that had shown all its problematic nature, in the attempts to demonstrate the principle of virtual velocities. Probably in his youthful work, Piola felt the Lagrangian principle of virtual velocities as indubitable. Only the need to relate to epistemology of the times led him to attempt the demonstration, then if this proof was valid only at the rhetorical level, it would not matter. Piola surely realized the weakness of his arguments, because in his subsequent memoirs on mechanics he never attempted to prove the law of virtual velocity which definitively became for him the indubitable principle of mechanics.

13 The debate at the École polytechnique

Abstract. This chapter is devoted to the debate at the École polytechnique on the principle of virtual velocities as presented in Lagrange’s Méchanique analitique of 1788, assuming a reductionist approach. In the first part, after a brief mention of Gaspard de Prony’s contribution, three interesting demonstrations by Fourier of the principle of virtual velocities are presented. Fourier considers the case of unilateral constraints also. In the second part the demonstration of Ampère is presented. Note the use of purely geometric virtual velocities, following the lead of Lazare Carnot. In the third part the probably less interesting demonstration of Pierre Simon Laplace is reported. After the excitement of 1789, the young French republic was in difficulty, having to fight against internal and external enemies. In early 1794 the situation became desperate and the state began to feel a dramatic lack of scientific and technical services. In March 1794 on the initiative of Monge and Lazare Carnot, the Committee of public safety appointed a commission of public works which formulated the institution of an École centrale des travaux publics. The École was established by December 1794 with headquarters in the old Palais Bourbon. Its teachers were chosen from among the greatest names in science and students were recruited with a contest, the notice of which was spread throughout France. The rules stated that the students admitted to the school were to be salaried in a dignified manner and housed outside the Palais Bourbon, with the general populace. The first year saw the enrollment of 400 students at different levels. A first round consisting of a three-month course allowed division into three groups: those who could immediately enter the service of the state, those who needed a year of study, and those who needed two years. Since its inception, the school, which will be called École polytechnique in September 1795, had a well-defined objective. It was to provide its students with a solid scientific training based on mathematics, physics and chemistry. The École polytechnique was preliminary to specialist schools such as the École du genie, the École de mines, and the École des ponts et chaussées. In ten years, from 1794 to 1804, many eminent mathematicians were produced by the École Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_13, © Springer-Verlag Italia 2012

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13 The debate at the École polytechnique

polytechnique, such as Poisson and Poinsot, physicists such as Malus and Biot, the chemist Gay-Lussac; later Cauchy, Ampère, and with some problems, Saint-Venant. The excellence of the École polytechnique impelled Napoleon Bonaparte to choose for his expedition to Egypt some of the most prestigious teachers, Monge and Berthollet, along with forty two students. To curb the political vocation of students who urged them to stand up and fight government decisions, Napoleon decided to give the École a military structure. The headquarters was moved to the Mount St. Genevieve, in the premises of the College of Navarre and the College of Boucurt, and reamined there until 1975. Its motto was: For the Country, the Science and the Glory. (A.13.1)

In 1814, despite disputes with the empire, with foreign troops on the outskirts of Paris, students who had followed only a few courses of artillery, defended with great courage the Barrière du Trone. With the arrival of Louis XVIII they returned to their homes. Unpopular measures, such as the removal of the old Gaspard Monge created serious disturbances, so that, in 1816, the king will suspend the school. Auguste Comte was one of the students suspended. The courses were resumed only in 1817, with half of the students. The École polytechnique was equipped with a new statute, the uniform became civil, the students were in boarding school and discipline, besides being heavy, also imposed religious obligations. However, the objective of training technicians and scientists for the state remained. Throughout the reign of Louis XVIII and still more that of Charles X, the students were in a sharp contrast with the government. Nevertheless they continued to study under the guidance of renowned teachers, mostly former alumni of the École: Arago, Cauchy, Petit and Gay-Lussac. They, however, participated actively in the risings of 1830. The arrival to power of Louis Philippe brought a little order. The École polytechnique regained its military status, but students continued to express disagreement, so that they were suspended in 1832, in 1834 and 1844. In 1848 they were still in the street, but this time as a mediating force between the regime and the insurgents. Even the prince president, who later became Napoleon III, had little sympathy for the École, whose students did not submit to the central power. However, the courses followed each other regularly on the premises of Mount St. Geneviève providing scientific and technical services. The army absorbed a large part of them and two soldiers of the École, Faidherbe and Denfert-Rochereau, saved the honor of the army in the disastrous 1870 war. During the Paris Commune and its bloody repression, the École polytechnique was moved to Bordeaux and Tours, following the advance of the Germans. After 1870, despite the recruitment of graduates being mainly dependent on the army, sciences were not yet abandoned. As an example we mention one of the graduates: Henri Becquerelle, Nobel prize winner in physics [256]. The École polytechnique is still active today, although its headquarters since 1975 is at Paliseau, in larger premises. The motto still is: For the Country, the Science and the Glory.

13.1 One of the first professor of mechanics, Gaspard de Prony

319

13.1 One of the first professor of mechanics, Gaspard de Prony Gaspard Clair François Riche de Prony was born in Chamelet in 1755 and died in Paris in 1839. Educated at the École des ponts et chaussées, he was appointed in 1794 a professor of the École polytechnique. Prony was professor to Poinsot who followed his courses in mechanics, most likely in 1797 [197] and the study of his writings is useful to better understand those of Poinsot. In 1821 he invented the Prony brake to measure the performance of machines and engines. Prony was concerned with the virtual velocity principle which was discussed in his best known work, the Mécanique philosophique [203], but the work in which he expresses his ideas more fully on the subject was the Mémoire sur le principe des vitesses virtuelles, which appeared in the Journal of the École polytechnique of 1797 [202] in which also Fourier and Lagrange published their contributions to the demonstration of the virtual velocity principle. One aspect of some interest is the reference to the work of Vittorio Fossombroni’s memoir on the principle of virtual work, which left some footprints also on Poinsot. Prony writes: I must also refer the students to a work of which they will find very useful addition to the lessons received at the École on the same matter: it is a memoir published in Italian in Florence in 1796, and headed by Mr. Fossombroni, Memoria sul principio delle velocità virtuali. This treatise will provide a number of exercises especially well suited to those who want to study the Mécanique analytique [202].1 (A.13.2)

Prony began to put the argument into his own hands. It is true, he says, that Fourier and Lagrange offered excellent demonstrations of the virtual velocity principle, however, these treatments are not appropriate for students and therefore there is the need to develop others that are easier. Prony’s demonstrations, which cover almost all cases of equilibrium of rigid bodies, are actually a little easier. He immediately shows correctly the necessary part of the virtual velocity principle, i.e. if a system is in equilibrium according to the criteria provided by the cardinal equations of statics, then the sum of moments (Lagrange’s meaning) is zero. But then he is not satisfied, because: The previous proofs leave nothing to be desired in rigor, but the equation of virtual velocities presented in this way is a consequence [a theorem] rather than a fundamental truth and it is necessary, because it retains the characteristics of a principle, to deduce it from theorems [principles] of mechanics even more elementary and closer to the truth that derive directly from the definitions than those that I have used. That is what I am going to do by supposing only the composition of powers applied to the same point and that of parallel powers [202].2 (A.13.3)

So basically he starts over again, only taking for granted the rule of composition of forces. For him, while the cardinal equations of statics could not be regarded as more immediate of the principle of virtual velocity, the rule of the parallelogram does. 1 2

p. 204. p. 196.

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Prony justifies the principle of virtual velocities by examining many cases of different complexity. He limits his analysis to proving the necessary part of the principle, i.e. if there is equilibrium the equation of moments holds true. After that he reversed the order followed at the beginning and got the cardinal equations of equilibrium for rigid bodies from the equation of moments. His speech on this part is not completely rigorous, which is partly justified by the teaching nature of the work. In what follows I will refer only to the proof of the necessary part of the virtual velocity principle starting from the rule of composition of forces.

13.1.1 Proof from the composition of forces rule Prony begins by assuming the rule of composition of forces, expressed in algebraic form, for the forces PI , PII , PIII , etc. balanced and all converging toward the point p. PI cos αI + PII cos αII + etc. = 0 PI cos βI + PII cos βII + etc. = 0 PI cos γI + PII cos γII + etc. = 0.

(13.1)

The component d pi of the generic virtual displacement of p in the direction of the force Pi is given by: d pi = e1 cos αi + e2 cos βi + e3 cos γi ,

i = I, II, etc.

(13.2)

where e1 , e2 and e3 are the components of the displacement d p along the coordinate axis X,Y, Z, and αi , βi , γi are the angles that the forces Pi form with X,Y, Z. With a procedure similar to that of Fossombroni, Prony multiplies the equilibrium equations (13.1) respectively by e1 , e2 , e3 , adds the three equations and in the light of (13.2) he gets the equation of moments: PI d pI + PII d pII + PIII d pIII + etc. = 0.

(13.3)

Then Prony moves on to the case of two sets of forces PI , PII , PIII , etc. and PI , PII , PIII , etc. applied at the ends of a rod that acts as a link between the two points that make up the ends, as shown in Fig. 13.1, so that the overall system is in equilibrium. P III P II -T PIII

T

dt PI

PII PI Fig. 13.1. Equilibrium of forces applied to a rigid rod

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321

This case is quite interesting because it shows how Prony addresses the issue of constraints. He does it in a way already traditional for those times, assuming the existence of ‘real’ forces exerted by constraints. Prony considers the ends of the rod as material points, where the balance results from the active forces applied and reaction forces. As for the single material point he has already shown, the equation of moments, he can write the two equations: PI d pI + PII d pII + etc. + T1 dt1 = 0 PI d pI + PII d pII + etc. + T2 dt2 = 0,

(13.4)

where T1 , T2 are the forces exerted by the rod at both ends, and dt1 , dt2 are the infinitesimal virtual displacements of both ends of the rod along its direction. At this point Prony assumes without any comment a principle of action and reaction for which T1 = −T2 = T , which is not completely evident and which will be criticized by Poinsot. Moreover, because the rigidity of the rod and the infinitesimal displacement is dt1 = dt2 , the previous relations can be written as: PI d pI + PII d pII + etc. + T dt = 0 PI d pI + PII d pII + etc. − T dt = 0.

(13.5)

By eliminating the reaction T between the two equations, relation (13.5) gives: PI d pI + PII d pII + etc. + PI d pI + PII d pII + etc. = 0

(13.6)

and then still the equation of moments. Finally, Prony considers a simple not deformable body in the plane, formed by a triangle, to the vertices of which forces are applied so that the triangle is in equilibrium and finds again the equations of moments. From the triangle to a rigid body then the passage is almost immediate. Here however he assumes for granted some other ‘principles’ of statics.

13.2 Joseph Fourier Jean Baptiste Joseph Fourier was born in Auxerre in 1768 and died in Paris in 1830. In 1795 he was appointed administrateur de police, or assistant lecturer, to support the teaching of Lagrange and Monge. In 1798 Monge selected him to join Napoleon’s Egyptian campaign. He became secretary of the newly formed Institut d’Egypte, conducted negotiations and held diplomatic posts as well as pursuing research. After his return to France in 1801, Fourier wished to resume his work at the École polytechnique but Napoleon had spotted his administrative genius and appointed him prefect of the department of Isère, centred at Grenoble and extending to what was then the Italian border. In 1808 Napoleon conferred a barony on him. In 1817 he was elected to the Académie des sciences, of

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which, in 1822, he became perpetual secretary [290]. He is mostly famous for his work on the transmission of heat [111]. Fourier’s studies on the virtual velocity principle are reported in the Mémoire sur la statique contenant la démonstration du principe des vitesses virtuelles, et la théorie des moments of 1797 [110]; they are among the most interesting of his works. The memoir opens by claiming the purpose is to prove a virtual work law, in particular Lagrange’s principle of virtual velocities, without any reference to the particular nature of the system under examination: I also thought it was not enough to prove in an absolute way, the truth of the proposition, but we must do so regardless of knowledge that we have of conditions of equilibrium in different kinds of bodies, since these conditions should be considered as consequences of the general proposition. This objective is fulfilled by the demonstrations that I am going to refer. It seems that they leave nothing to be desired both in respect of the scope and accuracy. We will assume as known the principle of the lever, as shown in the books of Archimedes, or Stevin’s theorem on the composition of forces, and some propositions easy to deduce from the previous [110].3 (A.13.4)

To satisfy his purpose Fourier assumes two principles that were considered indubitable, like axioms, at his time: the law of the lever and the rule of composition of forces, known as Stevin’s theorem. It must be said however that Fourier is not always completely rigorous, in particular he does not distinguish always the difference between necessity (equilibrium → vanishing of moments) and sufficiency of equilibrium (vanishing of moments → equilibrium). He reports three separate demonstrations. The first is essentially based on the rule of the composition of forces. The second and third are based instead on the law of the lever. In the following I summarize these demonstrations, and although the third is probably the most convincing, also the first and the second should be viewed, for a number of interesting observations, including those involving unilateral constraints. Here is how Fourier introduces the virtual velocity principle: If a body is moved by any cause, according to a certain law, each of the quantities that vary with its position, as the distance of one of its points from a fixed point or a fixed plane, is a given function of time, and can be considered as the ordinate of a plane curve in which time is the abscissa. The tangent of the angle this curve makes in the origin with the x-axis, or the first reason for the increment of ordinates compared to the x-axis, expresses the rate at which that amount begins to grow, or for the use of a name known, the fluxion of this amount. Bodies being subjected to the action of several forces, if one takes on the direction of each [force] a fixed point toward which the force tends to carry the point of the system to which it is applied, the product of this force for the fluxion of the distance between the two points is the moment of the force. The body can be moved in countless ways, and each has a corresponding value of the moment. If the moment of each force for a given displacement is taken, the sum of all these contemporary moments will be called the total moment, or the moment of the forces for this displacement. We will distinguish the displacement compatible with the system state, from what one cannot be undertaken without affecting the [constraint] to which it is subject, and assume these conditions, expressed as far as possible by equations. 3

pp. 21–22.

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Now, for the principle of virtual velocities, when the forces which act on a body, of whatever nature it may be, are supposed in equilibrium, the moment of these forces is zero for each of the displacements which satisfy the constraint equations. Bernoulli, in place of fluxions, considers the rising increment [infinitesimal displacements]. So each of the points of the system should be considered as describing a small space with rectilinear uniform motion during a time infinitesimally small. This small space, projected perpendicularly to the direction of force, is the virtual velocity: and if it is multiplied by the force, the product is the moment. I will adopt this happy abbreviation and all the usual procedures of differential calculus [110].4 (A.13.5)

Note the distinction Fourier makes between the method of fluxions – i.e. of the derivatives – which, with modern terminology is called the method of virtual velocities, and the method of infinitesimal displacements, which is called the method of virtual displacements. Based on this distinction Fourier defines the moment of a force in two ways which, though mathematically coincident for infinitesimal displacements, are formally different. In a way the moment is the product of the force by the velocity with which its point of application approaches its centre. In another way, moment is more classically defined as the product of the force by the virtual velocity, with Bernoulli’s meaning, i.e. the projection of infinitesimal displacement in the direction of the force. Fourier declares that he has adopted this second meaning of moment. For the understanding of the analytical developments it should be noted that for Fourier the moment of a force P and a virtual velocity d p is given by −Pd p and not, as usual, by Pd p. In fact Fourier does not speak explicitly of a negative sign, but he gives an implicit definition – if it is not a mistake – at the end of paragraph 4 where he says “If two forces tend to bring close the two points, their moment will be negative or positive, depending on whether these two points are nearer or farther” [110].5

13.2.1 First proof In the first proof Fourier takes as reference the rule of composition of forces, or Stevin’s ‘theorem’. From this theorem it can easily be proved that the total moment of n forces P in a general virtual displacement d p of their common point of application p equals that of the resultant force. According to another theorem, which Fourier proves, it also holds that the moments do not change by moving a force along its line of application. Based on these theorems it is easy to show that a rigid body is in equilibrium if and only if the total moment of forces acting on the body is zero for any virtual displacement. In fact, if there is equilibrium, it is possible to reduce the system of couples of forces to equal and opposite forces. In this situation, the moment is certainly zero, but as the operations carried out over the forces do not change the total value of the moment, it would be zero even for the effective forces. Conversely, if the total moment of forces acting on a rigid body is zero for all virtual displace4 5

pp. 22–23. p. 25.

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ments, it means that the forces can be reduced to equal and opposite forces, so there is balance. The system of constrained bodies is discussed in § 13. Fourier considers bodies linked by inextensible wires and solicited by any force that is in equilibrium, and reasons as follows. The forces that act on each body are not only those applied from outside, the active forces, but also those that come from the wires. It therefore can be said that if every body is in balance then the moment of the total forces – including constraint forces – is zero when considering any motion. Fourier then assumes that constraint forces of each wire are equivalent to two equal and opposite forces directed along the wire applied at its own ends. With this assumption, if infinitesimal displacements congruent with constraints are assumed, it is easy to show that the moment for all the constraint reactions due to the wires is zero. So if the system of points is in equilibrium, because the moment of all forces is zero, even the moment of the active forces must be zero. To complete the proof of the virtual velocity principle the reverse should also be demonstrated, i.e. if the moment of the active forces is zero for any virtual displacement compatible with the constraints, then even the moment of all forces is zero and the system is in equilibrium. Fourier does not do it. Fourier’s assumptions are then: a) a rigid body is in equilibrium if and only if the forces acting on it can be reduced to collinear forces equally and contrary; b) constraint forces have the same ontological status of the active forces; c) these forces have directions consistent with the direction that defines the constraint. This is the case of smooth constraint. In different parts of his work, Fourier reflects on what happens for virtual motions that produce shortening of wires, which are by definition unilateral constraints. I refer below to the view expressed in § 6. If one considers two forces in equilibrium, being applied to the two ends of an inextensible wire (but not resistant to compression), it is easy to know their moments for a total displacement compatible with the nature of the body in equilibrium. Following the previous article, the moment is zero whenever the distance is preserved, i.e. when the equation of constraint is satisfied. For all other possible displacements, the moment is positive [Fourier assumes signs contrary to the usual convention], and the system in equilibrium cannot be disturbed so that the total moment be negative [110].6 (A.13.6)

Although Fourier does not make these considerations on unilateral constraints clearer, they are worthy of emphasis because they are the first with a certain degree of organic unity. It therefore seems appropriate that today, under the name of Fourier’s virtual work law, it is meant the statement that says there is equilibrium if and only if La ≤ 0, La being the work made by the active forces for all virtual displacements compatible with constraints – unilateral and bilateral – though Duhem [99]7 ascribes to Gauss a thorough understanding of the meaning of the inequality La ≤ 0 [126]. 6 7

p. 26. pp. 44, 195.

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The consideration of Fourier on unilateral constraints were considered more in depth by Mikhail Vasilievich Ostrogradsky (1801–1862) in the paper Considerations générales sur les moments des forces presented in 1834 at the Academy of Sciences of St. Petersburg [178]. In this paper Ostrogradsky considered also the case of moving constraints, that he studied more in depth in two subsequent papers [179, 180]. The objections that can be made in this first Fourier’s demonstration are the assumption of smooth constraints and the use of different reasonings for different kinds of systems, i.e. rigid, deformable, solid, fluid. The first objection can be removed at almost all demonstrations of the laws of virtual work, but the recruitment of smooth constraints seemed so natural at the time that it is difficult to believe that Fourier has become aware of its problematic nature. The second objection is rather felt by Fourier, as he seeks an alternative proof which is based on the lever and no longer on the rule of composition of forces which, although it has the advantage of encompassing a fully algebraic treatment, should seem to Fourier not completely immediate.

13.2.2 Second proof Fourier begins his second proof in § 17, stating that: Instead of transforming the forces that urge the system, we replace this system, in which they operate, with a more simple body, but capable of being moved in the same way [110].8 (A.13.7)

The problem is set as follows Let P, Q, R, S, etc. be the forces applied at the points p, q, r, s, etc., and give an infinitesimal motion that move the points p, q, r, s, etc. along the directions p , q , r , s , etc. and resulting in the assigned virtual infinitesimal displacements d p, dq, dr, ds, etc. along P, Q, R, S, etc. To obtain an equivalent system first consider only two points, p and q, as in Fig. 13.2. Let π p be the plane perpendicular to p passing through p, πq the plane perpendicular to q and passing through q. From the point p the perpendicular h draw to the line ρ common to both planes π p , πq , and on the plane πq trace a perpendicular h to ρ from the intersection of h and ρ, and finally draw from the point q, h perpendicular to h that meets it in k. The two segments h and h can be considered as forming an angled lever which rotates around the axis ρ. The segment h may be regarded as the mobile arm of a lever with fulcrum a point o. If p moves along p with virtual displacement d p, the angular lever p − h − k will move the straight lever o − q − k which in turn will move q in the direction q . The fulcrum o of the lever oqk can be chosen so that the virtual displacement of q is equal to the assigned value dq. Notice that the point k moves in the direction orthogonal to the plane πq because this rotates around the axis p. As a consequence, q moves in the direction orthogonal to πq and therefore parallel to qq . The same operation is made by joining q with r, r with s and so on. In this way the original system is replaced by an assembly of levers, which will vary by changing the infinitesimal virtual displacement imposed on the system, but it does not matter. 8

p. 36.

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πp

P p

p'

dp h

q'

h' ρ

k πq

q h'' o

Fig. 13.2. Reduction of a system to a set of levers

Fourier says that the equilibrium conditions of the original system are equivalent to those of the assembly of levers and that in the latter case it is clear that the necessary and sufficient conditions for equilibrium are provided by the annulment of moments. It must be said however that this statement leaves much to be desired because it is not so obvious. If it is accepted that the equilibrium of a single lever is based on the cancellation of the total moment, it cannot be admitted with equal ease that this is true for an assembly.

13.2.3 Third proof Fourier is probably not fully convinced of the second proof and he tries another. Now he assumes that a generic force P acting at a point p, belonging to a system of particles, can always be thought of as due to a weight A applied at the end of a lever, the other end of which is attached to a wire which through a pulley is made to be parallel to the line of action of P, as shown in Fig. 13.3. To eliminate the dependence

P c

e

O b

a

A Fig. 13.3. Reduction of the forces on a system to two weights

p'

p

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of the proof on the infinitesimal nature of virtual displacements, Fourier conceived the weight A applied not directly to the end c of the lever, but through a ‘sector’, a curved element, which allows a non-uniform motion of the point p also when the weight A moves uniformly. Assume now a material system, fluid or solid, constrained in any way subject to forces P, Q, R, S, etc. applied to points p, q, r, s, etc. and assume that the equation of moments Pd p + Qd p + Rdr + etc. = 0 are verified for any value of the virtual displacement d p, dq, dr, etc. By repeating the previous reasoning, any weight P, Q, R, S, etc. can be replaced by a lever loaded by appropriate weights A, B,C, D, etc. Because the size of the levers and weights are arbitrary, it is possible to choose them so that, while maintaining the assigned forces P, Q, R, S, etc. and the virtual displacements d p, dq, dr, ds, etc. to the desired values, the weights A, B,C, D all fall with the same virtual displacement, either they come up or down. It is possible thus to admit that all the lowered weights can be replaced by a single weight E and the raised weights and by a single weight F, which through the rings of reference provide their actions by means of various wires. These, through other rings of type b, transmit their tension at the ends of type a of the lever, intended to apply the forces. By the law of the lever – this is a crucial Fourier assumption – the moment associated, for example, with the force P must be equal to the moment of the weight A, in its vertical motion. The same applies to the other forces, so the total moment of the forces P, Q, R, S, etc. must be equal to that of the two weights A, B,C, D, etc. Because, by assumption, the total moment of forces is zero, the same holds true for the total moment of weights and since the virtual infinitesimal displacement of E and F are equal and opposite, it can be said that the two weights are the same. To prove that in this condition the system is equilibrated, Fourier imagines to connect with a rigid rod fixed in the middle the two equal weights E and F which is assimilated – not very convincing I must confess – to the diameter of a pulley as shown in Fig. 13.4. This is allowed because the pulley assures equal and opposite displacements for E and F. Fourier spends some words to prove that this system is in equilibrium, with a reasoning ad absurdum, probably unnecessary because of the evidence of the fact [110].9

E Fig. 13.4. Impossibility of motion for E = F 9

p. 42.

F

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The third demonstration, except for some slight embarrassment in the end, I think is the most compelling of those submitted by Fourier. Notice however that he does not refer to the demonstration of the necessary part of the virtual velocity principle, that if the system is in equilibrium then the total moment is zero. This demonstration is however nearly implicit in Fourier’s reasoning. Of some interest are the considerations at the end of § 23, of historical nature: If we are content to replace each of the forces with a weight attached to a wire connected to a fixed pulley, we recognize that for each movement of the system at equilibrium, the moment of the weights that rise is equal to that of the weights that lower, and although this consideration cannot be regarded as a proof nevertheless it refers the principle of virtual velocities to that of Descartes, or the principle used by Torricelli. It is natural to think that Johann Bernoulli knew some similar construction. There are the same ideas in a work of Carnot in 1783 printed under this title: Essai sur les machines en général [110].10 (A.13.8)

After the three demonstrations reported in its parts I and II, the Mémoire sur la statique continues with three other parts; in part III there are interesting considerations on the stability of the equilibrium. In part IV Fourier presents his own demonstrations of the law of the lever and the composition of forces, in order to make selfreferential his work. In part V he makes concluding remarks on the generality of the law of virtual work, noting that it is also valid for fluids.

13.3 André Marie Ampère André Marie Ampère was born in Lyon in 1775 and died in Marseille in 1836. During the French Revolution, Ampere’s father stayed at Lyon expecting to be safer there. Nevertheless, after the revolutionaries had taken the city he was captured and executed. This death was a great shock to Ampère. In 1809 Ampère was appointed professor of mathematics at the École polytechnique. Ampère’s fame mainly rests on his establishing the relations between electricity and magnetism, and in developing the science of electromagnetism, or, as he called it, electrodynamics. Throughout his life, Ampère reflected the double heritage of the Encyclopédie and Catholicism. From this conflict came his concern for metaphysics, which shaped his approach to science; to point out his Essai sur la philosophie des sciences where among other things he introduced the term kinematics and classified mechanics in statics, kinematic and dynamics [290]. He treats the principle of virtual velocities in the same issue of the Journal of the École polytechnique of 1806 where the Théorie générale by Poinsot was published, with a memoir entitled Démonstation générale du principe des vitesses virtuelles, dégagée de la consideration des infinitament petite [2]. It counteracts the criticism of Poinsot and proposes a new demonstration. Just for curiosity, I remember that Prony and Laplace were the ‘referees’ of Ampère and Lagrange, Laplace and Lacroix those 10

p. 43.

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of Poinsot, and that Ampère had a better score than Poinsot. The demonstration of Ampère is interesting in itself, but perhaps even more interesting are his introductory remarks, indirectly criticizing the work of Poinsot (see Chapter 14), who probably he knew, and that of Lagrange. Ampère declared: The principle of virtual velocities, which serves as the basis in this admirable work [the Mécanique analytique] was considered by its author as a fact of which he then develops all the consequences, then the general proof of this principle has been looked for. Lagrange brings it in a very simple way to the principle of a system of pulleys, Mr. Carnot to the equilibrium of the lever.11 The proof of this principle has been deduced by Laplace by means of a more general way, but too abstract to be made easily understood by beginners. I set out to provide, as far as I can with the same generality, a proof that rests only on the theory of composition and decomposition of forces applied to the same point, and is free from the consideration of infinitely small quantities. This is the goal that I set out in the research that I have the honor to present to the class [2].12 (A.13.9)

He believes that he has provided a simple and convincing discussion, but in fact his work is difficult to read and not always well organized. He, like Poinsot, deems it necessary to avoid using the concept of infinitesimal and to start from first principles of mechanics, among which the most important is the rule of composition and decomposition of forces. No wonder because, as I have said elsewhere, most scholars considered the composition of forces as the most ‘rigorous’ principle. Moreover it was easiest to be expressed in formulas, a fact that, with no real valid logical reasons, made it preferable to those scientists who had a sound analytical culture. Ampère considers the hypothesis of no interaction between the various conditions of constraint and concludes that these cannot be justified a priori, but only in retrospect, starting from the equilibrium equations: The laws of equilibrium are deduced, in the most rigorous way, from considerations very simple when the forces are applied to one point and become more difficult to prove, especially if one wants to consider [the laws] in all their generality, when the forces act on points subject to constraint conditions that contribute to the mutual destruction of forces. The difficulty comes mainly from the need that these conditions of constraint intervene in the calculation. At first glance, it seems that they can be considered separately, and initially assume a single condition, then another and so on. But a little reflection evidences that we have to show a priori that the effects produced by the union of multiple conditions is the sum of effects arising from each specific condition, without that they have changed from their union. Truth that would appear to be rather a consequence of the equations of equilibrium than a means of obtaining them [2].13 (A.13.10)

Ampère criticizes the principle of solidification too (see §14.2.1). One more simplification that could be used in the research we are concerned, is to assume as fixed, then, all the points of the system, with the exception of two of them. This is particularly convenient because the total derivatives which are necessary are obtained with the union of the equations thus obtained, with the partial derivatives for each variable. But a very simple example seems sufficient to show that this assumption is not always eligible [2].14 (A.13.11) 11 12 13 14

Here perhaps Ampère confuses Carnot with Fourier. pp. 247–248. p. 248. p. 248.

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Ampère’s example is not very adequate and I do not carry-over it, except to say that the principle of solidification, at least in the form used by Poinsot (see § 14.2.1), appears not problematic. However, the question raised is worthy of consideration. In the proof of the virtual principle Ampère uses two basic principles: the composition of forces and the smooth constraints assumption: “Because it is clear that a force will have no action for or against a motion of its point of application, when it is perpendicular to the tangent line at precisely the point on the curve which it describes” [2].15 In the following passage Ampère exposes what is the essence of the virtual velocity principle according to him and what he intends to demonstrate: The principle known as the principle of virtual velocities, reduces to the fact that if it is made the sum of the moments of all forces applied to the system, taking with different sign those in which the projections of the forces fall on the same side and those in which forces and projections fall into opposite sides; in addition, to this sum there are added the equations deduced by all the conditions [of constraint] assigned, each multiplied by an arbitrary factor, and reduced so they contain in all terms the derivatives with respect to x, y, z to the first order, if the quantities that multiply each derivative are equated to zero, separately, and all arbitrary factors are eliminated, the remaining equation or equations express all the conditions for equilibrium [2].16 (A.13.12)

To understand better the above statement of the virtual velocity principle, in particular the meaning of moment and projection, the proof of Ampère has to be followed. I will display only a very brief summary that, besides explaining the terms, explains also in what sense the proof avoids the concept of infinitesimal. I will not go into any detail because Ampère’s approach would require a massive use of mathematical manipulations, which, if not complex, is at least boring. Ampère considers in the first instance a system of constrained points so that there is only one degree of freedom, which he identifies as the parameter s. Under this assumption each material point moves on a pre-definite curve γ parametrized by s. The tangents to the curves x(s), y(s), z(s) in the various points are defined by the vector t of components x (s), y (s), z (s), where the apex denotes the derivative with respect to s. The component of the vector t on the force applied at point P is called by Ampère projection. The product of the force and projection, with the appropriate sign, is the moment of the force. Ampère then eliminated the concept of infinitesimal displacement vector and replaced it by the ‘velocity’ t. It is clear why Ampère has chosen s as a parameter of motion instead of time, even if he does not say anything about it. He wants to completely eliminate the concept of time by a law of equilibrium, thus proving to be more demanding of Poinsot himself (see Chapter 14). The demonstration consists in refering, by means a system of rigid rods, the motion of a point m on the curve γ, subject to a force P, to the motion of another point μ moving in a straight line and to which another force S is applied, using the principle of composition and decomposition of the forces and the assumption of smooth constraints. An examination of Fig. 13.5 partially gives the idea. 15 16

p. 250. p. 253.

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P m

γ

σ S

μ

M

A B Fig. 13.5. Reduction of a motion from a curve to a straight line

The curve γ along which m moves is in general skewed; Ampère can prove that it is possible to trace back the motion, using a rigid rod mM, first to a curve σ of a plane π, then, by means of another rigid rod Mμ to a straight line AB still belonging to π, so when μ moves on AB under a force S, the point m moves on the curve γ by force P, preserving the moment. In the case of many material points m1 , m2 , . . . mn things can be arranged through appropriate curves σi , so that there are so many points μ1 , μ2 , . . . , μn moving all on the line AB with the same velocity and therefore they can be assumed as joined together. These points, subject to the forces S1 , S2 , ..., Sn , shall move the points m1 , m2 , . . . mn of the curves γi , with the forces Pi so that the relation which expresses the preservation of moments is:

∑ Pi ui = ξ ∑ Si ,

(13.7)

where ui is the projection of the vector ti associated with the material point that moves on γi , along the direction of the force Pi and Pi ui is the moment according to Ampère, ξ is the derivative of the common displacements of the points μi . Ampère assumes that there is equilibrium if and only if the resultant of the forces acting on the points μi is zero, i.e. ∑ Si = 0; the form in which it follows the law of moments:

∑ Pi ui = 0.

(13.8)

Ampère concludes: Now, it is a theorem of algebra easy to prove that the equation resulting from the elimination process is the same as they would be obtained by adding the constraint equations, each multiplied by an arbitrary factors to the sum of the moments and equating to zero the amount which multiply each derivative and eliminating the factors [2].17 (A.13.13)

17

p. 261.

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13 The debate at the École polytechnique

The ‘easy’ theorem of algebra is not really too easy. The speech of Ampère gets a little vague when it begins to consider systems with more than one degree of freedom. One can therefore conclude that his attempt to address the constraints preventing the superposition of constraints, accepted by Lagrange and Poinsot, was not perfectly successful.

13.4 Pierre Simon Laplace Pierre Simon, marquis de Laplace was born in 1749 in Beaumont-en-Auge and died in Paris in 1827. His career was important for his technical contributions to exact science, for the philosophical point of view he developed in the presentation of his work, and for the part he took in forming the modern scientific disciplines. Laplace’s biography tells of the story that he, at the age of nineteen, was given two different difficult problems which he would solve in one night each. The story may be apocryphal, but there is no doubt that Jean le Rond D’Alembert was somehow impressed and took Laplace up, securing his new protégé the appointment of professor of mathematics at the École Militaire. In 1773 Laplace became a member of the Académie des sciences de Paris. He was a member and even chancellor of the Senate, and great officer of the Legion of Honour and of the new Order of Reunion. After the downfall of Napoleon he was nominated Peer of France, with the right of a seat in the Chamber, and was raised to the dignity of marquis [290]. Laplace’s contribution to the discussion on the virtual velocity principle is cited by Ampère and Poinsot, for the prestige of scientist and of man of power as well as for a specific important contribution that left its mark in many subsequent demonstrations referred to in the handbooks of mechanics. Laplace is partially outside the École polytechnique and this is reflected in his writings that he processes without taking into account the discussions and clarifications already reached. He deals with the demonstration of the virtual velocity principle at the beginning of his Mécanique celeste of 1799 [156] and with assumptions that are still those of Prony, Poinsot and Ampère, namely the composition and decomposition of forces and the need for equilibrium of the orthogonality of active forces to the contact surface. The force of pressure of a point on a surface perpendicular to it, could be divided into two, one perpendicular to the surface, which would be destroyed by it, the other parallel to the surface and under what the point would have no action on this surface, which is against the supposition [156].18 (A.13.14)

Besides the two assumptions cited above, Laplace adopts the principle of action and reaction for which two material points m and m act on each other with two forces equal and opposite in the direction of the line joining them: 18

Tome I, p. 9.

13.4 Pierre Simon Laplace

333

Two material points with masses m and m

can act on each other only according the line that joins them. Indeed, if the two points are connected by a wire passing through a fixed pulley, their reciprocal actions are never directed according to that line. But the fixed pulley can be considered as having, at its core, a mass of infinite density acting on the two bodies, where the action of one on the other is only indirect [156].19 (A.13.15)

The last part of the above quotation raises however doubts about the way Laplace conceived of the principle of action and reaction and the reading of the whole memoir does not clarify the matter. One more principle that Laplace considers, without explicitly stating it, is the principle of solidification. Laplace’s demonstration of the virtual velocity principle starts from a free material point M, partially reproducing symbols and arguments of Lagrange in the first edition of the Mécanique [145].20 Consider a point M subject to many forces S applied to it. Denote by s the distance of M from an arbitrary point C on the line of action of each force S:  s = (x − a)2 + (y − b)2 + (z − c)2 , (13.9) where x, y, z are the coordinates of M and a, b, c the coordinates of C. Let V be the resultant of the various forces S , still applied to M, and u the distance of an arbitrary point D on the line of action of V from M. The components of V and S, according to the direction of the x coordinate are given respectively by: δu δs V , S , (13.10) δx δx as it can be seen, for example, for the components of S, considering that: δs x − a = , (13.11) δx s which is the director cosine of the force S and which multiplied by S provides its component along x. The same is true for the directions y and z. From (13.10) and similar relations associated with y and z directions, using the rule of composition of forces one obtains: δu δu δu δs δs δs V = S ; V = S ; V = S , (13.12) δx ∑ δx δy ∑ δy δz ∑ δz where the sum is extended to all forces. Multiplying equations (13.12) for δx, δy, δz respectively, adding and taking into account the total differential expression as a function of the partial derivatives, one obtains: V δu = ∑ Sδs,

(13.13)

where the meaning of symbols is clear. In order to represent the equilibrium, Laplace says, the resultant V has to be zero, so there is equilibrium if and only if: 0 = ∑ Sδs, 19 20

Tome I, p. 37. p. 20.

(13.14)

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13 The debate at the École polytechnique

which is an equation of moments. It is worth noting that, at this stage of his memoir, Laplace makes no mention of Bernoulli’s principle of virtual velocities and gives no name to the product Sδs – the moment – pointing out, too much I think, his originality. In the case of a material point constrained on a surface, Laplace denotes by R the reactive forces and δr the infinitesimal displacement of their points of application. From (13.14) he obtains: 0 = ∑ Sδs + Rδr,

(13.15)

but, as the point m remains on the surface, it is δr = 0 and (13.15) reduces to (13.14). The case of a system of constrained particles m , m , etc. is studied similarly by writing relation (13.15) for each point. Now, in addition to the forces S and R, internal forces p must be introduced. To them Laplace applies the principle he has stated for which the internal forces act along the line joining the material points. So indicating with f the distance between the material points m and m and with f  that between the points m and m and so on, Laplace can write the equation of equilibrium for any material point m: 0 = ∑ Sδs + pδI f + p δI f  + etc. + Rδr + R δr ,

(13.16)

in which δI indicates that only the position of m changes while m , m , etc. are treated as fixed (and then he applies the principle of solidification). In relation (13.16) Laplace has considered also the possibility that m is constrained to two surfaces that exert the forces R and R in the directions r and r . For the point m it will be similarly: 0 = ∑ S δs + pδII f + p δII f  + etc. + R δr + R δr ,

(13.17)

where now S and s are respectively the ‘active’ forces of m and the displacements in their direction; δII f represents the variation of f taking m fixed by varying m , δ f  is the variation of f  which is the distance between m and m , by varying m , R and R represent the constraint forces of the surfaces to which m is constrained. By adding (13.16) and (13.17), together with similar relations for points m , m , etc., taking into account that for example δ f = δI f + δII f , represents the total change in length of f , the relation 0 = ∑ Sδs + ∑ pδ f + ∑ Rδr

(13.18)

is obtained. Because for every single point it is δr = 0, the last term is reduced to zero. Then if the system is of invariable distance, i.e. a rigid body, it is also δ f = 0. So the equation of moments (13.14) is again obtained. If the points of the system have not invariable distance, Laplace believes it can be demonstrated that it is still ∑ pδ f = 0. For brevity I will not discuss his argument, I will only notice that it was not completely correct, as Poinsot pointed out in 1838, after the death of Laplace, in a note of Crelle’s Journal [196].

14 Poinsot’s criticism

Abstract. This chapter is devoted to Louis Poinsot’s criticisms toward Lagrange’s Méchanique analitique of 1788, assuming a reductionist approach. The first part introduces Poinsot’s mechanics which also takes account of the existence of constraints. Poinsot believes that VWLs have no interest for mechanics. The final part of the chapter is a report on the demonstration of a law of virtual work similar to that of Lagrange in which use is made of velocities and not infinitesimal displacements. Although Poinsot considers this demonstration only a trivial corollary of his mechanics, it had a remarkable success. Louis Poinsot was born in Paris in 1777 and died in Paris in 1859. He enrolled at the École polytechnique probably in 1794, without much mathematical background, and studied there for three years. The influence on him of Prony, who at that time was a professor at the École, was remarkable. Poinsot is generally considered as a minor figure in the history of mechanics, not comparable to the great ones: Euler, Lagrange, Laplace, Cauchy, etc. This may be true, but it does not mean that the attemption to understand the role of virtual work laws in mechanics, Poinsot’s position is not in the foreground. Among the scholars of some importance in the first half of the XIX century he was the only one who rather than enhancing the principle of virtual velocities, tried to prove that it was neither necessary nor useful for a coherent and efficient foundation of mechanics. According to him, once this mechanics was established, any demonstration of a virtual work law reduced to mere geometry. For this reason, and because his ‘demonstration’ has influenced most of the treatises of mechanics, I will dedicate a large space referring both to his work and to Bailhache’s interesting scientific biography [197]. Poinsot was not a prolific author; his main works reduce to: • Éléments de statique 1803 (first edition) [195]; • Mémoire sur le compositions des moments et la composition des aires, 1804 [193]; Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_14, © Springer-Verlag Italia 2012

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14 Poinsot’s criticism

• Mémoire sur la théorie générale de l’équilibre et du mouvements des systèmes, 1806 [194]; • Mémoire sur la composition des moments en mécanique, 1804 [193]; • Remarque sur un point fondamentale de la Mécanique analytique de Lagrange, 1846 [198].1 The most interesting of Poinsot’s contributions on virtual work principles is perhaps the Mémoire sur la théorie générale de l’équilibre, which was included in the latest editions of the Éléments de statiq ue. This work is derived from a review of a memoir, Sur la théorie générale de la mécanique, of the previous year which was read by Lagrange. The criticism was not completely in favor and demanded a radical revision. Poinsot accepted the request and sent a new draft to Lagrange. Lagrange sent it back to him, with a series of notes, but after it had already been published in the Journal de l’École polytechnique. Poinsot replied, even orally, point by point. The result of this discussion was that Lagrange realized the value of his interlocutor and had him appointed inspector general of the university. Poinsot was twenty nine years old. In the following I will examine first an unpublished work entitled Considerations sur le principe des vitesses virtuelles of 1797, reported in full in [197], then the Mémoire sur la théorie générale de l’équilibre and its previous version, along with Lagrange’s annotations to it [197].

14.1 Considérations sur le principe des vitesses virtuelles Poinsot wrote the Considérations sur le principe des vitesses virtuelles when he was twenty years old, and so when he had not yet fully developed his critical views on Lagrange’s virtual velocity principle. And, entering the École polytechnique cultural climate, he even provided a demonstration, largely following Prony’s approach (see Chapter 13). However, the critical insights that presage the development of Poinsot’s thought can already be seen. In this regard, an interesting note was reported at the beginning of the work, when Poinsot introduced the virtual velocities: Lines aa, bb, cc, &c, are what scholars call the virtual velocities of the points a, b, c, & c., but if one wants to have only the value of the moment, one multiplies the force estimated for these lines in the direction of the force, i.e. projected onto them. To shorten, it is therefore convenient to call these projections themselves the virtual velocities [197].2 (A.14.1)

This clarification indicates the attention Poinsot put on the virtual velocity concept, understood in the modern sense as a vector quantity. Poinsot begins his demonstration of his version of the virtual work law taking for granted, as did Prony, the rule of composition of forces, but he does so in greater detail, as follows. Let P, Q and R be three forces in equilibrium on a plane, applied 1 2

vol. 11, pp. 445–456. p. 4, part II.

14.1 Considérations sur le principe des vitesses virtuelles

337

Q P

q

q

a r

z p

a)

R

x

a

y

r

a'

p

a' z

R

x

P

y

Q

b)

Fig. 14.1. Reaction of a constraint

to a point a. Assume that a moves into a with an infinitesimal displacement. As demonstrated by Varignon, the static moments of the forces P and Q with respect to any point, for example a , is equal to the static moment of the resulting R – or that of the balancing force with sign changed – evaluated for the same pole, and with reference to Fig. 14.1a , it is: Px + Qy + Rz = 0,

(14.1)

where x, y and z are the normal to the directions of P, Q and R conducted from a . Indicating with p, q and r the components of aa on P, Q and R respectively, i.e. Bernoulli’s virtual velocities associated with aa , it may be obtained easily: Pp + Qq + Rr = 0.

(14.2)

In order to prove this, simply extend the lines x, y, z and build on them forces equal to P, Q, R, but rotated by a right angle and with origin in a (see Fig. 14.1b), for which now the normals are p, q, r. Because the equilibrium of P, Q, R persists also if they are rotated, from the balance of statics moments the relation (14.2) is obtained. This expresses the vanishing of the sum of the ‘moments’ – Lagrange and Galileo terminology. The demonstration of Poinsot deserves attention because it shows the close analogy between Lagrange’s ‘moments’ and static ‘moments’. They derive from a different way to observe forces. Poinsot then argues that the proof of the equation of moments is also valid in the case of any number of forces applied to a point and even for any number of free points, because a system of material points is in equilibrium if and only if all its points are in equilibrium In the case of a system of constrained material points, the equation of moments is still valid, provided that in addition to the active forces P, Q, R, etc. also the reaction forces H, M, N, etc. and the corresponding virtual velocities h, m, n, etc., are considered, so it can be written: Pp + Qq + Rr + etc. + Hh + Mm + Nn + etc. = 0.

(14.3)

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14 Poinsot’s criticism

Note the explicit introduction of the constraint reactions, perhaps according to the teachings of Prony. Poinsot in his later writings, will instead avoid the concept. To prove the equation of moment in case of a body with ‘immutable distances’ (a rigid body) Poinsot bases his argument on the idea that there is equilibrium if (and only if) all the forces, as a result of translations, compositions and decompositions with the rule of the parallelogram can be reduced to only three forces – including constraint forces – balancing each other. These three forces should satisfy the law (14.2) which requires the vanishing of moments. Exploiting the fact that the total moment of forces does not change with the operations of translation and composition, Poinsot can then claim that the criterion of equilibrium of a rigid body is expressed by the annulment of the sum of all moments of forces acting on the body, including the constraint forces. But, when considering virtual displacements compatible with constraints, the moments of individual reactions, and hence their sum, are zero. The moments of the constraint forces applied to fixed points, because their virtual velocities are zero and “those of the forces normal to the resistance surface, are also zero because the projection of the infinitely small arc, described by the root of normal itself is zero”. Note that Poinsot, in his demonstration, in line with the scholars of the time, takes for granted the assumption of smooth constraints. Ultimately it can then be concluded that the criterion of equilibrium of a rigid body can be traced back to the annulment of the sum of the moments of the only active forces. Instead of proceeding further, Poinsot warns: “We will also change the wording of the general principle of virtual velocities to avoid the idea of infinitely small motions and disturbance of the equilibrium, which are ideas foreign to the subject and leave something obscure in the spirit”. To clarify his position on the virtual displacement, I quote in full a note written on a separate sheet of the Considerations: This will exclude the ideas of the infinitely small and disrupting the equilibrium; ideas that are alien to the subject, and the principle of virtual velocities appear as a simple theorem of geometry by ignoring those considerations that always leave something dark in the spirit. But it should be noted that this property of equilibrium that we study was discovered by means of these little velocity [motion], because those offer themselves naturally when you perturb a machine in equilibrium. It seems that through these movements the energies of the forces in motion of the machine are estimated. If a system is in equilibrium, you know the absolute value of each force, but not the effect it exerts on account of its position. Disturbing a bit the system to see what are the simultaneous movements that can take the points where forces are applied, some of these points are moving in the same direction of the forces, others are moving in the opposite direction, and the energy is evaluated as the product of forces by the velocity of the points of application, it is found that the energies that achieve their effect are the same as the energies overcome [197].3 (A.14.2)

Poinsot will succeed fully in order to eliminate the concept of virtual displacement only in the later works. Now, he limits himself to see under what conditions the equation of moments can be extended to the case of finite displacements. Besides the well-known examples of a straight lever and the inclined plane, Poinsot refers to results found by Fossombroni for parallel forces applied to the points of a line; cases that he generalizes by showing that the equation of moments is also true for finite 3

p. 7, part II.

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339

displacements when the forces, parallel to each other, are applied to the points of a plane: If a free system of invariable form is in equilibrium under all forces that are applied to it, assuming that all the forces acting at the junction of their directions with a plane situated at will, the equation of the moments will be valid whatever was the displacement of the system [197].4 (A.14.3)

In addition to striving to eliminate the idea of infinitesimal virtual displacements, replacing these with the virtual velocities, Poinsot points to – or rather decides to follow in – Carnot’s footsteps, contending that the virtual velocities refer to changes of position occurring in a virtual time while the real time is frozen (i.e. that the virtual velocities and forces are not correlated, even when the forces depend on real motion). The balance is made with forces frozen at the instant in which the equilibrium should be studied: It must be noted further that the system is supposed to move in any way, without reference to forces that tend to move it: the motion that you give is a simple change of position where the time has nothing to do at all [197].5 (A.14.4)

Toward the end of his text Poinsot writes: “It would therefore be futile to search for the metaphysics of the principle of virtual velocities and to endeavor to understand what they are in themselves the moments of the forces. Everything comes from the parallelogram of forces, where it is seen as the moments combine among them.” Poinsot is not the only one who uses virtual velocity instead of virtual infinitesimal displacement. Fourier seems to put in the same plane the method of ‘fluxions’ (i.e. the velocity) and that of infinitesimal displacements. Poinsot, however, is the first to emphasize the need to use only virtual velocities, finally abandoning the infinitesimal displacements. In this he will be followed later by Ampère and Lagrange (in the second edition of the Théorie des fonctions analytique).

14.2 Théorie générale de l’équilibre et du mouvement des systèmes The Théorie générale de l’équilibre et du mouvement des systèmes is much more mature than the previous text; it begins with historical considerations on virtual work laws, then develops a mechanical theory completely independent of it to finish by reducing the virtual velocity principle itself to a trivial theorem of ‘Geometry’. I will refer mainly to the edition of 1806, published in the XIII Chaier of the l’École polytechnique, but when it will be necessary I refer also to the text of 1805 and to the notes on the text reproduced in [197]. For the first part of the present section, which from certain points of view is the most interesting for what concerns the virtual velocity principle, I refer instead to the version of 1834 published in the Éléments de statique [195]. In it, historical references and comments to Lagrange’s work are much more extensive and interesting; the wide passage below clearly expresses Poinsot’s ideas with no need of comment: 4 5

p. 12, part II. p. 13, part II.

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14 Poinsot’s criticism

The principle of virtual velocities was known for a long time as well as the majority of the other principles of mechanics. Galileo first noticed in the machines, the famous property of virtual velocities, that is the known relationship that exists between the applied forces and speeds that their points of application would take if the equilibrium of the machine should upset by an infinitely little amount. Johann Bernoulli saw in the full extent this principle that he enunciated with the great generality it has today. Varignon and the majority of the Geometers were careful to check it in virtually all matters of statics. And although there was no general proof, it was universally regarded as a fundamental law of the equilibrium of systems. But up to Lagrange, the Geometers were oriented more to prove or to extend the general principles of science, than to obtain a general rule for problem solving, or rather they had not yet put this great problem, which alone represents all the mechanics. It was then a happy idea by relying on the principle of virtual velocities as an axiom and, without stopping to consider it in itself, to be concerned only to get a uniform method of calculation to derive the equations of motion and balance in all possible systems. Thus overcoming all the difficulties of mechanics, avoiding, so to speak, to address the Science itself, one transforms it into a matter of calculation, and this transformation, the objective of Mécanique analytique [Méchanique analitique], appeared as a striking example of the power of analysis. Nevertheless, since, in this work one was at first careful only to consider this beautiful development of Mechanics, which seemed to derive everything from a single formula, it was believed that natural science was made and one just has to try to prove the principle of virtual velocities. But this research has highlighted the difficulties of the principle itself. This law is so general, where vague and strange ideas on infinitely small movements and the disturbance of the balance mix, does nothing but to become dark in his examination and Lagrange’s book is not giving anything more clear than the course of calculations, one sees well that the mists were not avoided in the way of mechanics, because they were, so to speak together, at the very origin of this science. A general proof of the principle of virtual velocities has basically to put the entire mechanics on another basis, because the demonstration of a law that encompasses a science cannot be other than the reduction of this science to another law so general, but obvious, or at least easier than before, thus making it useless. So for the reason that the principle of virtual velocities contains all the mechanics, and that needs a thorough demonstration, it cannot serve as a primary basis. Trying to prove it on the basis of such a happy use has made is to try to go through this use; either finding some other law just as fruitful, but more clear, or founding on the principles of an ordinary general equilibrium theory, from which then the virtual velocities becomes just a corollary. So the state in which Lagrange had brought the science was not a demonstration of the principle of virtual velocities, which might be sought immediately. The Mécanique analytique [Méchanique analitique], as the author conceived it, is basically what it should be, and the demonstration of the principle of virtual velocities is not lacking at all, because if one tried to put it at the beginning of this book in a general and well developed way, the work would be made, i.e. this demonstration would include already all the mechanics. It should therefore be considered that Lagrange placed himself with a single shot on one of the high points of science in order to discover some general rule to solve, or at least to put in the form of equations all problems of mechanics, and this objective has been fully achieved. But to form the science itself, one has to produce a theory that dominates equally all points of view. One needs to go straight, not to the obscure principle of virtual velocities, but to the clear rule that can be extracted from the solution of problems. And this natural and direct search, which alone can satisfy our spirit, is the main purpose of the memoir that is going to be read [195].6 (A.14.5)

6

pp. 427–430.

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The above introduction to the Théorie générale, contains an attack on the Méchanique analitique much stronger than that of the introduction to the same work in 1806, read by Lagrange. Poinsot argues that the principle of virtual velocities is obscure and unintuitive. The darkness comes from two factors: the first is contested by all opponents of the principle of virtual work, Stevin above all, that in the study of equilibrium it does not make sense to consider a perturbation, a virtual movement. The second factor concerns the nature of infinitesimals, the notion of infinitesimal was not so clear to Poinsot, or at least it was less clear than the concept of velocity that Poinsot will choose. For this ‘darkness’, according to Poinsot, the virtual velocity principle cannot be assumed as a principle of mechanics, and any attempt to prove it does not make sense because it means replacing this principle with another principle, equally general but more clear, which makes the virtual velocity principle unnecessary. Moreover, there is no practical advantage to introduce it. Poinsot’s position is incomprehensible to a modern reader, accustomed to handbooks of mechanics based on a highly formalized approach, in which axioms fall from nothing and there is no need for justification other than success in explaining mechanical phenomena. At the beginning of the discussion the equations of Lagrange themselves are assumed [148]7 – or those of Hamilton – often as axioms, and these equations are anything but intuitive. Poinsot’s position becomes clear when examined in the perspective of the epistemology of the XIX century, essentially in Aristotelian style. A principle must be self-evident, and even if it is not required with Aristotle, to be evident to the pure intellect, it must at least reflect the more immediate experience. According to Poinsot who embraces Aristotle’s opinion, a principle cannot be proved, otherwise it is not a principle; according to other scholars of his time, a principle is not necessarily obvious, but it can be proved and this can be done starting from metaphysical arguments – that is, with topics outside the science of which the ‘principle’ is a principle – or from within the discipline, but with very elementary arguments. Poinsot’s criticism of the possibility to regard Lagrange’s virtual velocity law as a principle, therefore, appears to be partly reduced to a linguistic fact, all depending on what it is understood by ‘principle’. Of different values is instead the criticism for which the demonstrations reported until now are unsatisfactory, or that the principle of virtual work is neither simple nor fundamental. This seems unfair. The proofs of Lagrange, Fourier and Carnot, who do not depart from the ‘usual’ principles of mechanics, are certainly very interesting. Lagrange connects the principle of virtual velocities to the pulley, Fourier to the lever. Carnot starts instead by the impact regarded as a phenomenon that can be characterized in a simple and obvious way. Even the demonstrations that originate from the usual principles of mechanics, such as Prony’s, do not seem less interesting than the demonstration reported by Poinsot himself. Then, in ease of use, if not in enunciation, the superiority of the virtual velocity principle compared to other seemingly simple principles is proved by its diffusion in treatises of mechanics. As to whether it is fundamental there seems to be no doubt, with some difficulty in dealing with friction forces, but, however, the criticism of Poinsot certainly was not referring to them. 7

p. 334.

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14 Poinsot’s criticism

14.2.1 Poinsot’s principles of mechanics I now pass to the exposition of the Théorie générale de l’équilibre de systèmes, according to the text of 1806, beginning by enumerating the principles assumed. From this exposition it soon becomes clear that it would just turn against Poinsot the same criticism of vagueness that he ascribes to his colleague scientists, because he does not set clearly and definitively the principles he uses. Although one can give the excuse that some of them were already submitted in the Elements de statique of 1803, also the others do not seem so obvious. The first principle, which Poinsot gives as well known, an axiom, is generally called the solidification principle. For this principle, if constraints, both internal and external, are added to a system of bodies in equilibrium, the equilibrium is not altered [194].8 The principle was used by Stevin, Clairaut and Euler for the study of fluids (by Lagrange, Laplace and Ampère in previous chapters) and will be used later by Piola and Duhem (see Chapter 17) to obtain the indefinite equilibrium equations of a three-dimensional continuum. The second principle is presented as the fundamental property of equilibrium, it asserts that a necessary condition for the equilibrium of a system of bodies free from external constraints is that all the forces applied at various points can be reduced to any number of pairs of collinear forces equal and opposite to each other. The condition becomes sufficient for a body with invariant distances, i.e. a rigid body [194].9 The third principle is required by the second, even if not explicitly, and concerns the possibility to decompose a force into other forces with the rule of the parallelogram and to move a force along its line of action. It is embarrassing that Poinsot does not state explicitly this principle, which, perhaps, is the most important and complex. He evidently takes it for granted even though it is difficult to argue that it is inherently more intuitive than the virtual velocity principle. Just as it is not very intuitive to accept the second principle, for which the reduction of forces to a number of pairs of opposing forces to each other is a necessary condition for the equilibrium [194].10 The fourth principle concerns constrained material points moving on a surface and asserts the need of the orthogonality of the active forces to the surface for equilibrium: In the equilibrium of systems, any force must be perpendicular to the surface of the curve on which its point of application would move if all the other points were considered as fixed [194].11 (A.14.6)

A fifth principle concerns the mechanical superposition for constraints, that is if in a system of bodies or points there are more constraints, they are able to absorb the sum of the forces that each constraint is capable of absorbing separately [194].12 8

p. 208. p. 209. 10 p. 208. 11 p. 234. 12 p. 225. 9

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Before considering in detail the constrained systems of material points, Poinsot explores the consequences of the second and third principles, i.e. some necessary conditions for equilibrium. Without going into detail he imposes the equivalence between the applied active forces Pi and a system of a pair of forces equal and opposite, an Ri j agent along the line joining the material points i j. Notice that Poinsot is not using a principle of action and reaction, as for example Prony and Laplace did, but only takes an algebraic position because the lines joining the material points are in general only imaginary and do not represent rods, for example. For a system of n material points there are 3n − 6 possible connections and then 3n − 6 components Ri j to be considered as unknowns and 3n equations that express the equivalence between the active forces Pi and their decomposition Ri j . So there is a surplus of six equations which must be verified by the assigned active forces. Poinsot does not exhibit these equations of equilibrium, perhaps considering them irrelevant, perhaps because they are well exemplified in current treatises on mechanics as the cardinal equations of statics. Fig. 14.2 illustrates the above for the case of four material points, where there are 3 × 4 − 6 = 6 connections. The four forces P1 , P2 , P3 , P4 are decomposed in the six – couples of – forces R12 , R13 , . . . , R34 , a priori unknowns. Among Pi and Ri j there are the twelve equivalency equations of the kind: X1 = R12 cos α12 + R13 cos α13 + R14 cos α14 Y1 = R12 cos β12 + R13 cos β13 + R14 cos β14 Z1 = R12 cos γ12 + R13 cos γ13 + R14 cos γ14 ··· X4 = −R14 cos α14 − R24 cos α24 − R34 cos α34 ···

(14.4)

where αi j , βi j , γi j are the angles that the forces Ri j form with axes x, y, z respectively, and Xi ,Yi , Zi are the components of forces Pi along the same lines. By eliminating

P4 4 R 24 R14

P3

R 34 3

R13 1 P1

Fig. 14.2. Decomposition of forces

R 23 R

12

2 P2

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14 Poinsot’s criticism

Ri j the six cardinal equations of statics are achieved. Notice that the existence of solutions for Ri j is only a necessary condition for equilibrium, as stated by the second principle of Poinsot’s mechanics. 14.2.1.1 System of material points constrained by a unique equation Then Poinsot passes to a more in-depth analysis of systems of constrained material points with the use also of his first and fourth principles. For the sake of simplicity he considers the system of four points shown in Fig. 14.3, the six mutual distances of which, indicated by m, n, p, q, r, s, are subject to the equation of constraint: L(m, n, p, q, r, s) = 0. Applying the principle of solidification, the equilibrium conditions on the external forces of a point, such as x1 , do not change if the other three points are assumed as fixed. If m, n, p are the distances of x1 from the other three points, the condition of constraint takes the form, depending only on m, n, p: L(m, n, p, q, r, s) = 0,

(14.5)

in which the values of q, r, s are assigned. Relation (14.5) defines a surface, whose normal at x1 has as components in the directions m, n, p the quantities L (m), L (n), L (p), where the apex denotes the partial derivative with respect to the variable in parentheses. Applying the fourth principle, i.e. the hypothesis of smooth constraints, when the point x1 is in equilibrium, it is necessary that the external force applied to it has the direction defined by the components L (m), L (n), L (p). The same holds true for the other points. Consider now two points x1 and x3 of the system joined by the line m of Fig. 14.3. As mentioned above, for the equilibrium, the components of the forces applied to the two points x1 and x3 in the directions that connect them to other points must have components in the form: αL (m), αL (n), αL (p) and βL (m), βL (q), βL (r)

x4

r x3 s

n

m q

x1

Fig. 14.3. Constrained material points

p

x2

(14.6)

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respectively, where α and β are constants of proportionality. But for the second principle of Poinsot’s mechanics, it is necessary that the components of the forces in the direction m which joins x1 and x3 are equal and opposite, so it should be α = β. It can be concluded that in order to have equilibrium, the components of external forces in six directions should be proportional to the partial derivatives of L (m), L (n), L (p), L (q), L (r), L (s). This result extends to any number of points connected by a single condition of constraint. After these considerations Poinsot encounters some difficulty in the use of the components of forces in the global reference frame. His difficulties come from having delivered the conditions of constraints by means of the distances between points rather than by means of their coordinates with respect to the coordinate system, as it would seem more natural, at least to a modern reader. The reasons for this are quite complex although may be not so interesting [197]. A few years later Cauchy [66] will resume the reasoning of Poinsot using constraint equations expressed by means of the coordinates of the points. I avoid referring to those aspects that do not have a very important conceptual value and, without giving the proof I pass to exposing the first conclusion of Poinsot which ensures that to have equilibrium in a system of any number of particles, subject to a constraint of type L = 0, the components of the forces applied to each point xi should be proportional to the quantities:

∂L ∂L ∂L , , , ∂ xi ∂ yi ∂ zi

(14.7)

with (xi , yi , zi ) the Cartesian coordinates of the i-th point. This result was already obtained by Lagrange with a different approach. Moreover according to Poinsot, relations (14.7) provide the directions the external forces need to have so they are equilibrated, instead of according to Lagrange, being directions of constraint forces that rise for the equilibrium. Developing derivatives of (14.7) it is then:

∂L ∂L ∂ p ∂L ∂q ∂L ∂r = + + + etc. ∂ xi ∂ p ∂ xi ∂ q ∂ xi ∂ r ∂ xi ∂L ∂L ∂ p ∂L ∂q ∂L ∂r = + + + etc. ∂ yi ∂ p ∂ yi ∂ q ∂ yi ∂ r ∂ yi

(14.8)

∂L ∂L ∂ p ∂L ∂q ∂L ∂r = + + + etc., ∂ zi ∂ p ∂ zi ∂ q ∂ zi ∂ r ∂ zi where p, q, r which represent the distances of the various points, should be considered as functions of their Cartesian coordinates x1 , y1 , z1 , x2 , etc. in a fixed frame of reference.

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14 Poinsot’s criticism

14.2.1.2 System of material points constrained by more equations Poinsot can then turn to the case with more than one constraint condition: L(m, n, p, q, r, s) = 0 M(m, n, p, q, r, s) = 0 N(m, n, p, q, r, s) = 0 etc.

(14.9)

In his words: First, the mere fact that the points of the system are linked together by the first equation L = 0 the forces:    ∂L 2 ∂L 2 ∂L 2 λ + + ∂x ∂y ∂z

2

2

 ∂L ∂L ∂L 2 λ + +    ∂x ∂y ∂z

2

2

 ∂L ∂L ∂L 2 λ + + ∂ x ∂ y ∂ z &c. can be applied to them, where λ designates any undetermined coefficient and being each force perpendicular to the surface L = 0, when one considers the three coordinates of the point of application as the only variables. Second, because the points of the system are linked together by the second equation M = 0, it is still possible to apply the respective forces:





 ∂M 2 ∂M 2 ∂M 2 μ + + ∂x ∂y ∂z

2

2

 ∂M ∂M ∂M 2 μ + + ∂ x ∂ y ∂ z

2

2

 ∂M ∂M ∂M 2 μ + + ∂ x ∂ y ∂ z &c. with μ a new indeterminate coefficient, and each of these forces being perpendicular to the surface represented by the equation M = 0, when the three coordinates of the point of application are considered as the only variables. […] It is clear that there will be equilibrium on the basis of all these forces, because there would be equilibrium in particular in each group of each equation [194].13 (A.14.7)

Poinsot has implicitly accepted that more constraints working at the same time do not interact with each other and that the overall effect is the sum of individual effects (it is the fifth principle of his mechanics). He realizes that this fact is not very evident and tries to overcome a little below, in a passage that I do not refer for lack of space. He 13

pp. 223–224.

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himself did not deem it good enough and then will return to this point in the Elements de statique since the eighth edition of 1841 (for further clarification on the issue see the work of Ampère on the virtual velocity principle examined in Chapter 13). Even Lagrange in the Théorie des fonctions analytique, made the same assumption of Poinsot on the superposition of constraints but he did not feel the need to justify the fact. The Théorie générale ends with the following theorem: Whatever the equations governing the coordinates of various points of the system are, for equilibrium, each of them requires that to these points are applied forces, along their coordinates, proportional to the derivatives of these equations with respect to these coordinates, respectively. Thus, representing with L = 0, M = 0, etc. any equation between the coordinates x, y, z, x , y , z , etc. of the different points, and with λ, μ, etc. any of the undetermined coefficients, the components of the forces that must be applied to these points should satisfy:



 dL dM X =λ +μ + &c. dx dx



 dM dL +μ + &c. Y =λ dy dy



 dM dL +μ + &c. Z=λ dz dz



 dL dM X = λ + μ + &c. dx dx



 dL dM +μ + &c. Y = λ dy dy



 dL dM +μ + &c. Z = λ dz dz &c. If the indeterminate λ, μ, etc. are eliminated from these equations, there will remain the equilibrium equations themselves, i.e. the relationships that must take place between the applied forces and the coordinates of their points of application [194].14 (A.14.8)

The equations above allow the solution of the static problem. Given a set of forces X, Y , Z, X  , Y  , etc. verify whether the system of material points is in equilibrium in a given configuration x, y, z, x , y , etc. This can be made as Poinsot suggests, by solving the equations obtained by eliminating the indeterminate multipliers, or more simply, by considering that previous equations define a linear system of algebraic equations in λ, μ, etc. with known coefficients. The linear system may be determined, undetermined or overdetermined; if it admits at least a solution then the system of material points is in equilibrium. Note that Poinsot is establishing the mechanics of constrained bodies without reference to the concept of constraint reaction though it was there accepted at the École polytechnique. He is even more rigorous than Lagrange and leaves no physical meaning to the ‘indeterminate’ ‘coefficients’ λ, μ, etc. that once known are generally interpreted as constraint forces. 14

pp. 228–229.

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14 Poinsot’s criticism

The memoir ends with an interesting conclusion and two notes: the first regards the comments on the role of constraints, which is not particularly illuminating. The second note concerns the demonstration of the virtual work principle and is of great interest, which is why I quote it in full.

14.3 Demonstration of the virtual velocity principle In the demonstration which follows, Poinsot replaces, for the first time unequivocally, the virtual displacements (infinitesimal) with virtual velocities, that he also calls ‘actual’ to emphasize that there are no assumptions of smallness. Poinsot declares he wants to prove Lagrange’s virtual velocity principle; actually, however, because of his use of velocity instead of infinitesimal displacements he is going to prove a slightly different principle, which however is still a virtual work principle. According to Poinsot, because this is almost an immediate consequence of the mechanics he has developed, which takes into account the role of constraints, and follows nearly immediately when the equations of equilibrium and constraints are written side by side, its proof has not a great value and also the principle in itself is of little interest. Personally I do not share Poinsot’s opinion and consider Lagrange’s proof very interesting and among the most cogent ever given. Note II Demonstration of the principle of virtual velocities. Identity of this principle with the general theorem object of the previous Memoir. In the Memoir we have been content to observe that from the theorem on the expression of the general equilibrium of forces, one could easily switch to the principle of virtual velocities. But this principle is so famous in the history of mechanics that one cannot fail to point out a few words with these steps. I am very happy to do this, since the principle of virtual velocities is not only a corollary of the general proposition stated above, but I think even identical to it when one looks at it from his own point of view, and sets it out in a comprehensive manner. Let the system be defined by the following equations between the coordinates of the bodies: f (x, y, z, x , y , z , &c.) = 0. φ(x, y, z, x , y , z , &c.) = 0. &c. Suppose to impress to all bodies any of the velocity that can actually occur without violating the terms of the constraints. The coordinates x, y, z, x , y , z , &c., will vary with time t, of which they must be considered functions, and because the impressed velocities: dx dy dz dx , , , , &c. dt dt dt dt

(A)

could be admissible by the constraints, as supposed, it will be necessary that they satisfy the equations: dx dy dz dx dy + f  (y ) + &c. = 0 f  (x) + f  (y) + f  (z) + f  (x ) dt dt dt dt dt dx dy dz dx dy φ (x) + φ (y) + φ (z) + φ (x ) + φ (y ) + &c. = 0 dt dt dt dt dt &c.

(B)

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obtained from the previous (A) and it will be sufficient to ensure that they meet them so that the constraint conditions are met. Now if one multiplies these equations for the undetermined coefficients λ, μ, &c. and adds, it follows that the velocities satisfy the sole following condition, no matter λ, μ, &c.

dy dx  + λ f (y) + μφ (y) + &c. + [λ f  (x) + μφ (x) + &c.] dt dt

dz dx     + [λ f  (z) + μφ (z) + &c.] + λ f (x ) + μφ (x ) + &c. dt dt  dy [λ f  (y ) + μφ (y ) + &c.] + &c. = 0. dt

(C)

But the functions which multiply the velocities, dx dy dz dx , , , , &c. dt dt dt dt are nothing but (after what has been proven) the general expressions of the forces which can be balanced on the system. Assuming therefore that the forces X,Y, Z, X  ,Y  , Z  , &c., are effectively balanced, it is: X

dy dz dx dx dy dz +Y + Z + X +Y  + Z + &c. = 0. dt dt dt dt dt dt

(D)

Instead of the three components X,Y, Z, multiplied for the corresponding velocities: dx dy dz , , dt dt dt it can be considered the resultant P, multiplied by the resulting velocity dx/dt, dy/dt, dz/dt, projected into the direction of P, which I will call ds/dt; the same can be done for the other forces, and it will be: P

ds ds ds + P + P + &c. = 0. dt dt dt

That is, if the forces are in equilibrium on any system, the sum of their products for the velocities, one wants to give their bodies, whatever they may be, but allowed by their constraints, will always be zero, by estimating these velocities along the directions of forces. One can see from this, it is possible to take any velocities of finite value, which are measured by any straight lines that would be described simultaneously by the body if their links are suddenly broken and each of them run away freely toward their part. Because of constraints among the bodies the velocities vary in each moment, when one wants to measure these velocities using the spaces themselves that the bodies actually describe, these spaces should be taken infinitely small, otherwise they no longer would measure the impressed velocities, and in this way one falls into the virtual velocities themselves, where the principle is to lose some of its clarity. In fact it follows from what we have said, that this beautiful property of equilibrium can be stated as follows: When the different bodies of a system run any of the movements which do not violate in any way the link established between them, i.e. the system is continuously in one of those configurations allowed by the constraint equations, it can be sure that the forces that will be capable of being balanced in these configurations, when the system passes in them, are such that multiplied by the velocity of the bodies projected onto their directions, the sum of all these products is necessarily equal to zero. In this way, the principle no longer maintains any trace of the ideas of the infinitely small movements and disturbance of the equilibrium, which seem extraneous to the issue and leave some darkness in the spirit.

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14 Poinsot’s criticism

When there is equilibrium, it is clear that the principle holds for all the systems of velocities that the points could have, passing through the configuration that is considered. But, when one wants to start from the principle enunciated in such a way that it ensures the equilibrium one should require that it holds for this infinite number of velocity systems. There is a plethora of conditions, and it is possible to show that it is enough to say that the equation (D) must be verified for all systems of velocity allowed by the constraint equations (B) or (bringing together, as we did above, all these equations in one (C) by means of indeterminate λ, μ, &c.), it suffices to say that the equation (D) of the moments must be verified for all systems of velocities: dx dy dz dx , , , , &c. dt dt dt dt But since, by definition, each of these systems of velocities must satisfy the equation (C), which amounts to say that all forces X,Y, Z, X  ,Y  , Z  , &c. that multiply the velocities: dx dy dz dx dy dz , , , , , , &c. dt dt dt dt dt dt in the equation (D), have to be proportional to the functions: 

dx 

dy 

dz λ f (x) + μφ (x) + &c. , λ f (y) + μφ (y) + &c. , λ f (z) + μφ (z) + &c. , dt dt dt    

dx dy λ f (x ) + μφ (x ) + &c. , λ f  (y ) + μφ (y ) + &c. , &c. dt dt that multiply the same velocities as in the general equation (C), which requires for them the only conditions of constraints. So the principle of virtual velocities well set out, i.e. where all the ideas that one can make are clear: it is perfectly identical to the general theorem which is the subject of this memoir. I say exactly the same thing, namely that for the equilibrium, the components of the forces applied to bodies, by virtue of each equation must be proportional to the derivatives of these [constraint] equations with respect to these coordinates, which was to be proved.15 Moreover, it would have been taken to recognize this identity by a description of the ordinary principle of virtual velocities, by making well aware of the true meaning that it needs to be given. In fact, the general problem of statics is not only to seek the relationship between the forces which are in equilibrium, on the system, but rather [to seek] the general expression of the forces that may be continually equilibrated in any configurations where it can go under the constraint equations. The general equation given by the principle of virtual velocities is not, if I may speak so, the relation of an instant; it in no way should consider simply the equilibrium of the system in the configuration where it is, but also throughout the sequence of configurations where it can be, for it is this sequence of configurations that characterizes its definition [emphasis added] So the equation of moments does not say that one has to take the forces of a magnitude sufficient so that it is satisfied, but (since these forces must vary with the configuration) [it says] how one must choose these functions of the coordinates, so that the equation of moments remain continually satisfied. Now, under the constraint conditions themselves, one knows that between the velocities that the bodies can simultaneously have, it must apply the linear equation (C), the coefficients of which are the derivatives of functions given with respect to the coordinates by which this velocity is estimated. The equation of moments says that the forces of equilibrium must be represented 15

Here Poinsot’s writing is somewhat confused. He means that to prove the sufficiency of the virtual work principle one should require the satisfaction of the equation of moments (D) for all possible sets of virtual velocities. Comparing (D) with (C), which also apply to any set of virtual velocity, one deduces the equality of their coefficients, the terms that multiply the virtual velocities – and thus the equation of moments.

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351

by the derivative of these functions, therefore, to prove it, it is necessary to show how these forces are actually equilibrated or it must look directly for what functions of the coordinates can represent the forces of equilibrium, as we did from the beginning. This is why most of the demonstrations which trace back the principle of virtual velocities either to other principles or to the known law of some simple machine as the lever, &c., seem to us more justifications than real demonstrations. All in fact, even the happiest, that of Mr. Carnot, do not refer at all to the general definition of the system, as if the machine was, so to speak, voila, and one does not see anything but the ropes where the powers are applied. It may well be proved or made clear through some construction more or less simple that if one perturbs a bit the equilibrium, these powers must be in a relationship with the extensions allowed to its ropes, but this cannot provide that the current ratios forces considered as numeric values, and does not show at all its forms of expressions that are peculiar to them. This disturbance of the balance would not know, in no event with which machine one has to do, and the same relationship between the applied forces, could occur even if the machines were of quite different constitution, and each of them, however, imposes to the expression of the forces that are generated, a different form that one should always see and find, if the difficulty of the theorem were fully resolved. So the property of virtual velocities remains not less mysterious, and there is no real demonstration. I mean an open and clear explanation, where one sees not only that it works well but that it is a consequence of the general definition of the considered system. It is perhaps in a similar way, and to get the equation of moments as an equation identical that Mr. Laplace considered only the equations representing the link of the various parts of the system, and has, moreover, used other principles besides the composition of forces and the equality of action and reaction, which can be considered as elements of the equilibrium theory. As it is, after all, either one wants to start from the principle of the virtual velocities to follow its significance up to the end, or he directly attacks the problem of mechanics, which is simpler, one is conducted to look for the functions of the coordinates that give the forces of equilibrium in all the configurations that can be obtained in the system, in obedience to the relationships between the coordinates of the different bodies. This is exactly the problem we set ourselves, and our goal clear and distinct was to resolve it through the first principles of statics and geometry [194].16 (A.14.9)

I do not see that the text of Poinsot needs comment. On the basis of his mechanical theory, in which the role of constraints is clearly explained, and on the basis of his definition of virtual velocity, he can easily demonstrate a virtual work law which is a variant of Lagrange’s virtual velocity principle, both for its necessary and sufficient parts. Poinsot maintains that the virtual velocity principle allows the study of the equilibrium not only in a given configuration: “but also in the entire sequence of configurations where it can be, for it is this sequence of configurations which characterizes its definition”. In such a way it can also lead to solution of another interesting problem of statics. Assigned a given configuration x, y, z, x , y , etc. find a set of forces X,Y, Z, X  ,Y  , etc. for with the equilibrium is satisfied. This can be made by solving the equations obtained from the virtual work law by eliminating the indeterminate multipliers.

16

pp. 237–241.

15 Complementary virtual work laws

Abstract. This chapter is devoted to a variant of VWL which goes under the name of law of virtual forces. In the first part the formulation of the law by Cauchy is presented and used to prove simple theorems of plane kinematics. In the final part a few theorems of spatial kinematics are enunciated. From the demonstrations of reductionist type as that of Poinsot it should be clear that any law of virtual work can be stated in dual form. One is the traditional principle of virtual work, for example in the form given to it by Lagrange, which studies the equilibrium of a system of constrained bodies subject to various forces fi by imposing the vanishing of the work for a system of infinitesimal displacements ui congruent with constraints; i.e.:

∑ fi ui = 0, ∀ui .

(15.1)

Another form can be achieved by focusing on a given set of infinitesimal displacements ui consistent with constraints, forcing the cancellation of virtual work for all systems of balanced forces fi ; i.e.:

∑ fi ui = 0, ∀ fi .

(15.2)

This second type of law goes by the name of the principle of complementary virtual work or the principle of virtual forces. It is virtually ignored in the treaties of rational mechanics for physicists and mathematicians while it is widely used in those addressed to engineers, for whom the principle of virtual forces is an essential tool in the analysis of elastic structures [385]. In the following I cite only the applications of this principle to the study of rigid body kinematics carried out by Cauchy.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_15, © Springer-Verlag Italia 2012

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15 Complementary virtual work laws

15.1 Augustin Cauchy formulation Augustin Louis Cauchy was born in Paris in 1789 and died in Sceaux in 1857. In mathematics he pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also researched in convergence and divergence of infinite series, differential equations, determinants, probability. In mechanics he contributed to the introduction of the concept of stress and made fundamental studies on elastic bodies, considered both as made up of particles and as continuous bodies. Laplace and Lagrange were visitors at the Cauchy’s family home. Lagrange in particular seems to have taken an interest in young Cauchy’s mathematical education; he is said to have forecast Cauchy’s scientific genius while warning his father against showing him a mathematical text before the age of seventeen. In 1805 he entered the École polytechnique; in 1807 graduated from the École polytechnique and entered the engineering school École des ponts et chaussées which he left (1809?) to become an engineer, first at the works of the Oureq Canal, then the Saint-Cloud bridge, and finally, in 1810, at the harbor of Cherbourg. In 1816 he become professor at the École polytechnique. Cauchy was a very devout Catholic and this attitude was already causing problems for him and for others. After the revolution of 1830 Cauchy refused to take the oath of allegiance and lost his chairs. When the revolution of 1848 established the second republic Cauchy resumed his academic position and was retained even when Napoleon III reestablished the oath in 1852, for Napoleon generously exempted the republican Arago and the royalist Cauchy [290]. Cauchy gave no decisive contribution to the understanding of virtual work laws, but he used them enough and thus contributed to their spread. Among the works where Cauchy made use of virtual work laws they should be named: Sur un nouveau principe de mécanique of 1829 [67], which addresses issues of impact among bodies and the Recherches des équations générales d’équilibre pour un système de points matériels assujettis à des liaisons quelconques [66]. In this latter text, which deals with the motion of systems of constrained particles Cauchy resumed, exposing it under a slightly different point of view, the discourse written by Poinsot in the Théorie générale de l’équilibre et du mouvement des systèmes several years before. The main difference is the way he dealt with constraints, that rather than being defined by mutual relations among distances of material points, were defined by relations among their coordinates with respect to a fixed system. The memoir Sur le mouvements que peut prendre un système invariable, libre ou assujetti à certaines conditions of 1827 [64] is however in my opinion the most original work. Here Cauchy applies, perhaps for the first time, the principle of virtual forces to determine the congruence of the acts of motion of a rigid body. The strength of the memoir is not so much to obtain theorems of kinematics of rigid bodies, which can be obtained more easily and convincingly with purely geometric methods, but to present an alternative way to use the law of virtual work. In the following I will refer

15.1 Augustin Cauchy formulation

355

briefly to the case of a plane rigid body, ignoring the long and difficult discussion of Cauchy on the three-dimensional rigid body. Cauchy does not formulate clearly the principle of virtual forces, for which a system of velocities, i.e. an act of motion, is congruent with constraints if and only if the work made against any system of balanced virtual forces is zero. In fact he uses only the necessary part of this principle: for an act of motion congruent with constraints the virtual work of a system of balanced forces must be zero. The lack of perception, by Cauchy and even by any not careful reader of his work, that he is using a principle of virtual forces rather than of virtual velocities or displacements derives from the coincidence of the necessary parts of the two principles. These necessary conditions, set out for the acts of motion, are in the order: For a congruent act of motion L = 0, for a system of balanced forces. For a system of balanced forces L = 0, for a congruent act of motion. A modern reader however feels that Cauchy is using the principle of virtual forces, because he considers ‘real’ the velocities and virtual the forces. Regardless of his awareness, Sur le mouvements que peut prendre un système invariable, libre au assujetti à certaines conditions should be considered as the first step towards the explicit formulation of modern principles of virtual forces. Here’s what Cauchy says: When an invariable system [a rigid body], free or subject to certain constraints, moves in the space, there are among the velocities of the different points certain relationships that in many cases are expressed very simply and can be deduced from the formulas for the transformation of coordinates. I will show in this article, that the same relationships can be drawn by the principle of virtual velocities. This principle is usually used to determine the forces capable of maintaining equilibrium in a system of particles subject to given constraints, assuming as known the velocities that these points can have in one or more virtual motions of the system, i.e. in motions compatible with the constraints in question. But it is clear that one can reverse the question [emphasis added], and after establishing the equilibrium conditions through any method, or if you want through the consideration of some virtual motions, one can use, to determine the nature of all other, the principle that we have mentioned. We add that it is useful in this determination, to replace the principle of virtual velocities with another principle to be drawn immediately from the first, and which is contained in the following proposition: Theorem. Suppose that two system of forces are applied consecutively to points subject to any constraints. For these two systems of forces are equivalent, it will be necessary and sufficient that, in a whatsoever virtual motion, the sum of the moments of the forces of the first virtual system is equal to the sum of the moments of the forces of the second virtual system [64].1 (A.15.1)

The theorem referred to at the end of the quoted passage replaces a criterion of equilibrium of forces with a criterion of equivalence. With it the necessary part of the principle of virtual forces becomes: for a virtual congruent act of motion the moments of two equivalent systems of forces must be equal. I will refer in the following to this statement as Cauchy’s principle. Note that the application of Cauchy’s principle and in particular the equation of virtual forces in general, it is necessary to presuppose a criterion of equilibrium or 1

pp. 94–95.

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15 Complementary virtual work laws

equivalence. If one does not want to fall in to a petition of principle, this criterion should be other that that given by the equations of moments. In fact, if the equation of moments was used, the congruency of virtual displacements with constraints should be taken for granted, but that is precisely the object of the principle of virtual forces. As a predefined criterion of equilibrium Cauchy adopts, without explicitly stating it, the rule of composition and decomposition of the forces and admits the possibility of transport of forces along their lines of action.

15.1.1 Kinematics of plane rigid bodies Based on his principle, Cauchy passes to an examination of the compatibility of motion for points of a rigid plane. The first theorem he proves is: Theorem I. If at any time of the motion, two points of the not deformable system have zero velocity, the velocities of all the other points will be reduced to zero [64].2 (A.15.2)

The theorem can be proved with the help of the following reasoning. Let A and A be the points the velocity of which is zero and ω be the velocity a third point A chosen arbitrarily. Apply to this last point a force P parallel to ω, and using the rule of the parallelogram of forces decompose P into two forces P and P directed as the straight lines AA and AA , as shown in Fig. 15.1a. Since the translation along its line of action does not alter the conditions of equivalence between the forces, it is possible to assume the force P applied at point A and the force A at point P . The sum of the virtual moments of these two forces will vanish as the velocity of A and A are zero by assumption. Therefore, since the force P is equivalent in construction to the other two P and P , by virtue of Cauchy’s principle, it will have a virtual moment Pω equal to that of P and P . But this is zero because A and A have zero velocity, so Pω = 0 and then ω must be zero, because P is not zero. Since A was chosen arbitrarily the theorem is proved. The above reasoning does not apply if point A were located on the line A A . But, then, replacing the force P with two equivalent parallel forces P and P applied, at

a)

b)

A'

P

P'

A

A''

A' A''

A P'' P' P

ω

Fig. 15.1. Two points with zero velocity 2

p. 96.

P''

ω

15.1 Augustin Cauchy formulation

P

357

ω '' A

A'' A'

P''

ω' P'

Fig. 15.2. No points with zero velocity

A and A , it is still possible to recognize that the product Pω must be reduced to zero (see Fig. 10.1b). The second theorem is proved in a similar manner: Theorem II. If at any time of the motion, the velocity of all points of the not deformable system are different from zero, these velocities are all equal and directed along parallel lines [64].3 (A.15.3)

The proof is carried out by reductio ad absurdum. Let ω and ω be the velocities of any two points A and A of Fig. 15.2; assume, by the absurd, that these velocities are not parallel and then the perpendiculars to their directions for the two points A and A meet at a third point A. Consider then a force P applied at A and directed in a random manner, equivalent to the system of two other forces P , P parallel to AA and AA applied respectively to points A and A . If ω indicates the velocity of the point A, by virtue of Cauchy’s principle, it is: Pω cos(P, ω) = P ω cos(P , ω ) + P ω cos(P , ω ),

(15.3)

the first member of the equality being the virtual moment of the force P and the second member the virtual moment of the forces P and P statically equivalent to P. Moreover, the lines AA and AA are perpendicular to the direction of the velocity ω and ω , and then the second member of the previous equality is canceled, so it is: Pω cos(P, ω) = 0,

(15.4)

which, because P cos(P, ω) cannot be always equal to zero because P has arbitrary direction, implies ω = 0. This is absurd because ω = 0 by assumption. This means that ω and ω are parallel because the contrary gives the absurdity. It remains to show that the velocities ω and ω are equal to each other and to those of all the other points. Cauchy considers first the case where the line A A is not perpendicular to the directions of these velocity now supposed to be parallel. A generic force P directed along this line, can be thought as applied either at A or at A , forming two equivalent systems consisting of the only force P, as shown in Figs. 15.3a and 15.3b. 3

p. 96.

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15 Complementary virtual work laws

a)

b) A'' A'

P

P

A'' A'

ω"

ω'

ω"

ω'

Fig. 15.3. No points with zero velocity

For Cauchy’s principle the moments in the two situations are equal: Pω cos(P, ω ) = Pω cos(P, ω )

(15.5)

and since the angles (P, ω ), (P, ω ) are identical in construction it is ω = ω . If the line A A is perpendicular to the direction of the velocity of A and A , choose any point A outside of the line A A with velocity ω, then, reasoning as above, the equations ω = ω and ω = ω can be obtained, from which is again obtained ω = ω . The proof of the subsequent theorem is a bit more complicated: Theorem III. If at any time of the motion one point in the rigid system has zero velocity, the velocity of a second arbitrarily selected point will be perpendicular to the radius vector from the first point to the second and proportional to this vector radius [64].4 (A.15.4)

Let O be the only point on which the velocity is zero by assumption and ω the velocity of another arbitrarily chosen point A, as shown in Fig. 15.4. If a force P is applied to the point O directed along OA, its virtual moment will be zero (because the velocity of O is zero). Since moving P from A to O gives another force equivalent to it, the virtual moment of this transported force will be zero for Cauchy’s principle, and then it is: Pω cos(P, ω) = 0,

(15.6)

from which observing that the quantities P and ω are not zero it is: cos(P, ω) = 0.

(15.7)

So the angle (P, ω) is right and the velocity ω perpendicular to the radius vector OA. It remains to show that the velocities vary in proportion to the distance from O. A r P

O

Fig. 15.4. Only one point with zero velocity 4

p. 98.

ω

15.1 Augustin Cauchy formulation

ω'

ω

A' Q'

359

A

r'

Q'

r Q

O Q Fig. 15.5. Reaction of a constraint

Let now be ω the velocity of a third point A , as shown in Fig. 15.5. This velocity will be itself perpendicular to the radius vector OA . Apply to points A and O two forces equal to Q and perpendicular to OA (and therefore parallel to ω), forming a couple of forces, the first of which is directed in the same sense of the velocity ω. The sum of the virtual moments of the two forces of the couple is reduced to the virtual moment of the first force, and consequently the product Qω, because in O the velocity is zero. Apply also to the points A and O two new forces equal to Q and perpendicular to the radius vector OA (and therefore parallel to ω ) such that they give a second couple equivalent to the first. For the equivalence between the two couples, if r and r designate the radius vectors OA and OA respectively, it is: Q r = Qr.

(15.8)

Moreover, the virtual moment of the forces of the second couple must be equal to the virtual moment of the forces of the first couple for Cauchy’s principle. So it is: Q ω = Qω.

(15.9)

From this relation, combined with the previous one: ω ω = . r r

(15.10)

So the velocity of the points A and A are not only perpendicular to the radius vectors r and r , but also proportional to them. In the motion just seen, the velocity of a point at unit distance from the centre O is called the angular velocity of the rigid plane system around this same centre. If γ designates the angular velocity, the velocity of point A, located at the end of the radius vector r, is defined by the relation: ω = γr.

(15.11)

Cauchy sums up his analysis with the comment: Theorems I, II, III show all the relationships that may exist between the velocities of the material points rigidly linked to each other, contained in a fixed plane that may not ever get out. These theorems show that the velocities in question are always those which the system would have taken in the state of rest, or translated in the direction of a fixed axis, or turned around a fixed centre. We add 10 that a translation, parallel to a fixed axis, can be obtained by a rotation around a fixed centre, where this centre is away at an infinite distance from

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15 Complementary virtual work laws

the origin of the coordinates. 20 that the centre of rotation is a point the position of which varies in general from time to time in the plane that is considered. It is for this reason that we designate the point at issue as the instantaneous centre of rotation [64].5 (A.15.5)

The section on the motion of plane systems concludes by studying the behavior of the instantaneous centre of rotation. The discussion is quite interesting but it is purely geometric with no mechanical implications, thus I limit myself to the conclusion: We will note that at the end of a designated time t, the different points on the moving surface of the space will occupy certain positions, and that one of them, the point O, for example, is the instantaneous centre of rotation. In addition, it is clear that at this time [t] it is possible that through O two separate curves pass, drawn to include, the first, all the points of the moving surface and, the second, all the points of space, which later will become the instantaneous centres of rotation [64].6 (A.15.6)

Cauchy proves that the arcs OA, OB, measured from the point O on the two curves mentioned above differ by an amount of an order higher than the first. So these two curves are tangent to each other. In addition, the first curve, delivered by the motion of the surface on which it is drawn, will cover a portion of the plane that will form by envelope the second curve. In the special case where one of the curves just considered reduces to a point, the same is true of the other. Then the instantaneous centre of rotation keeps the same position not only in space but also on the moving surface. For the three-dimensional rigid body, I report only the statements of the theorems that Cauchy demonstrates, using a process similar to that used for the twodimensional rigid body. Theorem VI. Whatever the nature of the motion of a solid, the relationships between the various points will always be as those that they would occur if the body was kept so it could only turn around a fixed axis and slide along this axis [64].7 (A.15.7) Theorem VII. Conceive a rigid body which moves in space by any means, and that at a given moment trace 1) in the body, 2) in the space, the different lines with which subsequently the instantaneous axis of rotation of this rigid body will coincide. While the ruled surface, having as generatrices the straight lines traced in the body, will be dragged by the motion of this, it will constantly touch the ruled surface having as generatrices the straight lines traced in the space, and consequently the second surface will be nothing but the envelope of the portion of space traveled by the first [64].8 (A.15.8) Theorem VIII. Posited the same things as in the theorem VII, if the instantaneous axis of rotation of the solid body becomes fix in the body it will become fix in the space and vice versa [64].9 (A.15.9)

5 6 7 8 9

p. 100. p. 101. p. 116. pp. 119–120. p. 120.

16 The treatises of mechanics

Abstract. This chapter is devoted to VWLs as presented in the main treatises of mechanics where a reductionist approach is assumed. In the first part, the treatises of Siméon Denis Poisson and Jean Marie Constant Duhamel are presented. In the second part the approach of Jean Marie Gustav Gaspard Coriolis is presented. To point out the introduction of modern term virtual work, the introduction of the problems associated with friction and finally the change in the ontological status of virtual work. From simple mathematical to physical magnitude. After the publication of Lagrange’s Mécanique analytique, following the reorganization of schools for higher education where regular courses in physics and mechanics began to be taken by a large number of students, the first textbooks of mechanics started to spread throughout Europe. Among the most famous were those of the professors of the École polytechnique in Paris, one of the first modern scientific institutions. A feature of these textbooks was the reproduction, in general terms, of the main topics of mechanics. They were concerned with generally accepted principles and procedures, and in their presentation often not even the names of authors and works from which they are drawn were reported, as if they had become a common heritage which is unthinkable to criticize. Given the high cultural and intellectual level of many of the writers, not infrequently in these manuals there were exposures of levels at least equivalent to that of publications in scientific journals. This was especially true for the laws of virtual work, because as of then, they were disclosed only through manuals. In this chapter I consider three authors who seemed to me the most significant: Siméon Denis Poisson, Jean Marie Constant Duhamel and Jean Marie Gustav Gaspard Coriolis.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_16, © Springer-Verlag Italia 2012

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16.1 Siméon Denis Poisson Siméon Denis Poisson was born in Pithiviers in 1781 and died in Sceaux in 1840. Poisson went to study mathematics at the École polytechnique in 1798. His teachers Laplace and Lagrange became his friends. After he was graduated, in 1802, he remained at the École Polytechnique as an assistant to Fourier, from whom he subsequently inherited the chair. In 1809 he was named professor at the newly founded faculty of sciences. His Traité de mécanique published in 1811 and again in 1833 was a standard manual of mechanics. Poisson took care of electricity with an important essay in 1812 and the theory of elasticity with various memoirs, taking the molecular model of matter. Gugliemo Libri said of him: “His only passion was the science, he lived and died for it” [290]. In his Traité de mécanique [200], after having set out the virtual work principle and verified it in some simple cases, Poisson reports two separate demonstrations. The first shows a clear influence of Laplace’s Mécanique celeste, the second demonstration instead reproduces that of Lagrange’s Mécanique analytique of 1798 and 1811. In the following I give a somehow extensive sketch of the first demonstration that contains original and interesting ideas. Poisson demonstrates first the necessary part of the virtual work principle, i.e. if there is equilibrium, the virtual work, or rather the total moment as he calls it, is zero for any virtual displacement. In order to do this, he takes as a pre-existing criterion of equilibrium the rule of composition of forces. From this criterion he proves the equation of the moments for a free material point and for material points constrained to move on a surface. In the latter case the assumption of smooth constraints is implicit, assuming the orthogonality of the reaction force to the surface. Then Poisson extends his proof to systems of particles constrained by inextensible wires that pass through apposite rings. There are no comments by Poisson on this choice of internal constraints, i.e. if he sees them as representative or not of every situation. It is worth noting that Poisson uses the term virtual velocity meaning a vector quantity: The infinitely small straight lines [infinitesimal displacements] that describe the motion of a point […], name that comes from the fact that they are considered simultaneously as the distances traveled by the points of the system in the first instant, when the equilibrium is broken [200].1 (A.16.1)

So he does not adopt Bernoulli’s definition of infinitesimal displacements in the direction of forces. Poisson also seems influenced by the definition of Carnot’s geometric motions, since he raises the question of reversibility of virtual velocities. He deals only with bilateral constraints, believing that a virtual work principle can be formulated only with reference to them without commenting on the observations of Fourier on this topic. 1

p. 660.

16.1 Siméon Denis Poisson

363

The following comment on the role of constraint reactions is interesting: The advantage of the virtual velocity principle is to furnish the equilibrium equations in each particular case, without the need to evaluate the internal forces. But because the demonstration we are going to show is based on the consideration of these forces, of unknown value, here is the notation which we will use to represent them [200].2 (A.16.2)

Poisson writes first the equation of moments of a single material point, subject to active forces, to external and internal constraint forces. Then he moves on to examine the whole system, following a reasoning similar to that of Laplace’s Mécanique celeste for the analysis of the relative displacements of the material points, but developed in a more rigorous way, with the clear awareness that the considerations apply only for infinitesimal displacements. For the system of constrained material points Poisson achieves the following equations of moments: Pp + P p + P p + etc.

+[M, M ]δ(M, M ) + [M, M ]δ(M, M ) + etc.

+[M , M ]δ(M , M ) + [M , M ]δ(M , M ) + etc. +etc. = 0,

(16.1)

where P, P , P , etc. are the active forces applied to the points of the system, p, p p , etc. the components of virtual velocities in the direction of the forces, [M, M ], etc. represent the tension in the wires that connect M, M , etc., δ(M, M ), etc. are the infinitesimal variations of distance between m, m , etc. In the event that the relative changes in distance are zero, δ(M, M ), etc. = 0, as for rigid bodies, Poisson obtains the classical expression of the virtual work principle: Pp + P p + P p + etc. = 0.

(16.2)

Then he shows that the same expression remains valid even if the distances between material points vary, provided that the wires, always joining the material points, sliding without friction on the rings, maintain intact their whole length. After the treatment of the necessary part Poisson passes to the sufficient part of the virtual work principle, i.e. that if the virtual work of the active forces is zero for any virtual displacement then there is equilibrium. The demonstration that he develops has a historical interest and because of this I quote it in full: It remains to prove that, conversely, when the equation (b) [equation (16.2)] applies to all infinitesimal motions of the system of points M, M , M , etc., the given force P, P , P , etc., are equilibrated. […] Suppose for a moment that the equilibrium does not take place. The points M, M , M , etc. or any part of them, are set in motion and at the beginning, simultaneously describe the straight lines MN, M N , M N , etc. All these points can be reduced at rest by applying appropriate forces, directed along these lines, in the opposite direction to the motions produced. Therefore, denoting these forces with the unknowns R, R , R , etc., the equilibrium will be achieved among the forces P, P , P , etc., R, R , R , etc., so that if r, r , r , etc., designate the virtual velocities projected on the directions of these new forces R, R , R , etc., for 2

p. 664.

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16 The treatises of mechanics

the principle of virtual work we have demonstrated, it is: Pp + P p + P p + etc. + Rr + R r + R r + etc. = 0 or simply:

Rr + R r + R r + etc. = 0

(c)

by virtue of equation (b) which is valid by assumption. Since this equation (c) applies to all motions compatible with the constraint conditions of the system of points M, M , M , etc.., it is possible to choose for the virtual velocities the space actually described MN, M N , M N , etc. But since these spaces are valued in the extension of the directions of R, R , R , etc. it follows that all the projections r, r , r , etc.. will be negative […]. Then, since all the terms in the equation (c) have the same sign, their sum can not be zero, unless each term is zero, then: R · MN = 0, R · M N = 0, R · M N = 0, etc. Now because the product R · MN is zero, it must be, or R = 0, or MN = 0, which means in either case, the point M can not take any motion. The same applies to all other points and therefore the whole system is in equilibrium, what we proposed to demonstrate [200].3 (A.16.3)

The proof is made by reductio ad absurdum, assuming first that there is not equilibrium and then by showing that any forces that should be applied to restore it must all be zero, and then there was equilibrium. The demonstration seems convincing; it implicitly assumes that the forces to be applied to restore the equilibrium should be all directed in the opposite direction to the motion allowed by assumption. This, although intuitive, is not demonstrable in the mechanics of reference considered by Poisson and it should be taken as an axiom. This assumption may be false as it can be seen easily by the following example. Consider two material points m and m in Fig. 16.1 rigidly constrained to turn around the point O. It is clear that any rotational motion that leads m and m to move on the same side can be balanced also by two forces one in one direction and the other in the opposite direction and not only by two forces in the same direction, both opposed to the motions of m and m . What matters is the total static moment of the forces that,

v' m' v m O Fig. 16.1. Equilibrium of forces 3

pp. 670–672.

16.2 Jean Marie Duhamel

365

in the case of the figure must be clockwise. It is true that Poisson considers systems consisting of wires, but he also looks into the possibility of rigid motions, and then the above example should be valid. From a logical point of view the asymmetry between the demonstrations of the necessary and the sufficient parts of the principle should be underlined. The necessary part is proved as a theorem of a reference mechanics in which there is a prefixed criterion of equilibrium provided by the balance of forces in accordance with the rule of the parallelogram. The sufficient part is instead proved by ignoring, in an uneconomic way, that criterion and the mechanics of reference on which it is based, introducing the principle of dynamic character, according to which motions can be destroyed by forces acting in the opposite direction to them. A similar reasoning is also found in many recent treatises of physics and rational mechanics (see § 2.2). It is interesting to compare Poisson’s proof of the virtual work law with that of Poinsot. In the latter, where it is taken as a criterion of equilibrium of forces, there is a perfect symmetry between the demonstration of the necessary or sufficient parts, without the explicit use of dynamic principles.

16.2 Jean Marie Duhamel Jean Marie Constant Duhamel was born in St. Malo in 1797 and died in Paris in 1872. He entered the École polytechnique in 1814 to graduate in 1816. Except for one year, Duhamel taught continuously at the École polytechnique from 1830 to 1869. He was first given provisional charge of the analysis course, replacing Coriolis. He was made assistant lecturer in geodesy in 1831, entrance examiner in 1835, professor of analysis and mechanics in 1836, permanent examiner in 1840, and director of studies in 1844. The commission of 1850 demanded his removal because he resisted a program for change, but he returned as professor of analysis in 1851, replacing Liouville. Duhamel also taught at the École normale supérieure and at the Sorbonne. He was known as a good teacher. He was elected to the Académie des sciences de Paris in 1840. Duhamel’s scientific contributions were not fundamental; however, they were important. He worked on partial differential equations and applied his procedures to the theory of heat, rational mechanics and acoustics. Studies related to acoustic concerned vibration of strings and air in cylindrical and conical tubes. The ‘principle of Duhamel’ in the theory of differential equations derived from his work on the distribution of heat in a solid with a temperature variable boundary [290, 354]. In his Cours de mécanique [97] Duhamel once again re-proposes Poisson’s demonstration of the virtual velocity principle. Of some interest are his comments in the introduction and some clarification in the text. In the introduction, Duhamel criticizes the approach of those who replace the real system with an equivalent system, with no consideration for the true internal structure of the real system. Duhamel writes:

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Most Geometers regard as obvious that if the forces are in equilibrium in a system of points, subject to constraints that allow it to make certain infinitesimal motions, these same forces will still be in equilibrium for the same system of points, subject to different constraints that allow the same motions. This principle […] has always seemed to me doubtful […] it appears to be based on a real confusion between physics and geometry. […] Therefore we changed the proof of the principle of virtual velocities as derived by Ampère and we adopted one that does not have the same problem and which is nothing but, after all, than that of Poisson’s Traité de mécanique [97].4 (A.16.4)

The principle of equivalence criticized by Duhamel, can be made more explicit as follows: consider a system of particles subject to forces F and constraints L. If one replaces the constraints L with other constraints l that enable the same virtual velocity (the same infinitesimal displacements) for the configuration of the system where the forces are applied, then there is equilibrium of F on L if and only if there is equilibrium of F on l. Duhamel’s doubts about the validity of this principle appear to be well motivated; indeed, selecting the equality of virtual displacements as a criterion of equivalence between the two systems seems to have attained some circularity, because the equivalence is established for quantities which are essential for the provability of the virtual work principle, which is then somehow assumed. The criticism applies to the second demonstration of Fourier and to the demonstration of Lagrange in the Théorie des fonctions analytiques. And one can go even further, for example, to Galileo and his demonstration of the law of the inclined plane from that of the lever. The following comment by Giovanni Vailati highlights the intimate connection between the possibility of replacing the equivalent systems and the virtual work laws: So, for what concerns Johann Bernoulli it is noteworthy that, by taking into account, in the famous letter to Varignon (1717),5 the relations between the displacements [virtual], infinitely small of the points of application of forces, he did nothing but at the end to apply and enunciate, in general form, a standard method which had been already frequently used by his predecessors, among others, Leonardo da Vinci and Galileo, in their attempts to infer from the principle of the lever, that of the inclined plane, and to include it in that of a heavy body supported by two not parallel wires. This rule consist to substitute, as regards the equilibrium, two sets of constraints when they allow the same initial displacement. It, as Duhem notes, is set out more explicitly by Descartes, in a letter to Father Mersenne (1638) […]. On the presence of similar considerations in the writings of Galileo see Mach (Mechanik, 4th ed., pp. 25–26) [391].6 (A.16.5)

In Jouguet [341]7 it is pointed out that under certain constraint conditions the uncritical application of the equivalence principle can lead to errors. In the introduction of Duhamel’s book details are given that are interesting from a historical point of view, on the approximations involved in the use of infinitesimals: If any system of points is in equilibrium, and if we consider an infinitely small displacement of all points, which is compatible with all the conditions of the constraints, the sum of the 4 5 6 7

pp. VI-VII. The date is wrong, it should be 1715. p. 267. pp. 170–174.

16.3 Gaspard Gustave Coriolis

367

virtual moments of all forces is zero, whatever this displacement is. And vice versa, if this condition is met for all virtual displacements, the system is in equilibrium. In this statement the infinitely small are treated in an ordinary way. The equation is exact only considering the limits of the ratios, after dividing by any one of the infinitely small quantities, in other words, the sum of the moments is infinitely small compared to the moments themselves [97].8 (A.16.6)

These statements rebuff the embarrassment still existing in the second half of the XIX century on the use of the concept of infinitesimal displacements, ‘which are treated in an ordinary way’.

16.3 Gaspard Gustave Coriolis Gaspard Gustave Coriolis was born in Paris in 1792 and died in Paris in 1843. He entered the École polytechnique in 1808 and finished second among all students of that year. After graduation he entered the École des ponts et chaussées. He worked as an engineer in the district of Meurthe-et-Moselle and the Vosges mountains. Coriolis became professor of mechanics at the École centrale des artes et manufactures in 1829. Despite his reluctance to teach, in 1832 he accepted a position at the École des ponts and chaussées. Here he worked with Navier, teaching mechanics. After the death of Navier, in 1836, he took his chair and also replaced Navier at the Académie des sciences de Paris. He continued to teach until 1838 at the École polytechnique, when he decided to stop teaching and to become director of studies. The name of Coriolis is famous for his work on the forces of drag, which showed that the laws of motion apply equally in a rotating frame of reference provided to add Coriolis’s forces [290]. Coriolis’s role is very important for mechanics applied to machines also for his books De calcul de l’effect de machines published in 1829 [78] and the Traité de mécaniques des corps solides et du calcul de l’effect des machines, published in 1844 [80], with the latter considered a reworking of the first. In the following I will first consider briefly the De calcul de l’effect de machines. In the more than twenty years which separate it from Carnot’s Principe fondamentaux de l’équilibre et du mouvement, no important intermediate work was published [332]. Coriolis’s book is a didactic book so most parts are greatly extended; though considering machines in general it gave a lot of space to a particular kind of machines; terms and concepts are clearly stated. In the first chapter of his book Coriolis defines the main concepts of mechanics, among them that of force, mass and work. As for most scientists of the time, the ontology of force is no longer a problem for Coriolis, he is not interested in its status but only in its use; moreover force does not imply impact:

8

p. 193.

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In what we are going to say the word force will apply only to what is analogous to weight, that is to what is called, in most cases, pressure, tension, and traction. With this meaning force could not make the direction and the value of velocity to change sharply without it passes through all the intermediate states [78].9 (A.16.7)

The mass of a body is defined as the ratio between force and acceleration and its measure was given by its weight at an assigned sea level. The concept of work is considered the most important one for the study of machines in motion, while force is important for machines in equilibrium. Coriolis introduces the word work (travail) to indicate what Carnot called moment of activity and others moment, mechanical power, quantity of action, energy, or even simply force. These various and quite vague expressions were not suitable to spread easily. We propose  the appellation dynamical work, or simply work, for the quantity Pds […]. This name will not be confused with any other mechanical denomination. It seems suitable to give the right idea of the thing, by maintaining its common meaning of physical work […] this name is then suitable to designate the union of these two concepts, displacement and force [78].10 (A.16.8)

Coriolis uses the term work also in subsequent studies, particularly in the Mémoire sur la manière d’differéns établi les principes pour des systèmes de mécanique des corps, comme en des assemblages de considérant the molecules of 1835 [79]. It is a use that he definitely will consolidate with his work Mécanique des corps solides of 1844 where, in the preface, he writes: I employed in this work some new nomenclature: I name work the quantity usually named puissance mécanique, quantité d’action ou effet dynamique, and I propose the name dynamode for the unity of measure of this quantity. I introduced also one more little innovation by naming living force the product of the weight times the height associated to the height. This living force is one half of the product that today is associated to this name, i.e. the mass times the square of speed [80].11 (A.16.9)

Notice Coriolis is introducing the factor 1/2 in front of the expression of living force (i.e. kinetic energy), because he suggests measuring the living force of a body of mass m and velocity v with the product mh, with h the height the body can reach if thrown upward with an initial velocity v (h = v2 /2). In a note to the passage quoted above, Coriolis writes: This term work is so natural in the sense that I use it, which, though it has never been either proposed or approved as a technical expression, nevertheless it was used accidentally by Mr. Navier in his notes on Belidor and Prony in his Mémoire sur les expériences de la machine du Gros-Caillou [80].12 (A.16.10)

Although Coriolis’s texts were fundamental to the spread of the term work, again, at the end of the XIX century propositions like: principle of virtual velocities, principle of moments and principle of virtual work, co-existed. See for this purpose a note by 9

pp. 2–3. p. 17. 11 p. IX. 12 p. IX. 10

16.3 Gaspard Gustave Coriolis

369

Saint-Venant in his translation of Clebsh’s text on the theory of elasticity, where he speaks of a “theorem of virtual work or virtual velocities [72].13 Notice that Coriolis himself used the term principle of virtual velocities. It is also interesting the way Coriolis introduced machines in general, very close to Carnot’s. Here after we will use the name machine to indicate the mobile bodies to which we will apply the equation of living forces: in this sense a single body which moves is a machine, so has a more complicated system. In each particular case, once we will know by what bodies in motion the machine is composed it will be enough to apply the principle previously established, to know precisely what are the masses which must be considered in the living forces evaluation, and what are the motive and resistant forces which must be considered to evaluate the amount of work [78].14 (A.16.11)

In the following I report some reflections on the application of the virtual work principle taken from the Mémoire sur la manière d’établir les différens principes de mécanique pour des systèmes de corps, en les considérant comme des assemblages de molécules [79], a principle which is at the basis of his mechanics. Coriolis stands on the ground of physical mechanics, advocated by Poisson, as opposed to the analytical mechanics of Lagrange. In physical mechanics everything is reduced to a Laplacian model of material points, or molecules, unlike analytical mechanics, where the bodies are treated essentially in their geometric aspect, for example as rigid bodies. According to Coriolis, concerning the statics of a particle, it is sufficient to assign the law of composition of forces. To switch to statics of extended bodies or systems of bodies it is necessary to add other principles, among which are those of action and reaction: For statics and dynamics of the systems of bodies it is enough to lean on the principle of equality between action and reaction. This principle is that if a molecule of a body produces a certain force of attraction or repulsion on a neighboring molecule, it likewise receives from it a force equal and directly opposite, so that all the sets of molecules that make up a body is formed only by pairs of equal and opposite forces. It is only with the help of this starting point we are going to determine all the principles of mechanics [79].15 (A.16.12)

Coriolis begins his analysis of statics of extended bodies by introducing first the virtual work principle for the single molecule: If it is accepted that a point, to which a force P is applied, moves by an amount δs in any direction, we will call element of virtual work the product of δs times the component of the  the angle of δs with the force P, the element force16 in the direction of δs. Denoted by Pδs of virtual work will be  P cos(Pδs) [79].17 (A.16.13) 13

p. 577. p. 20. 15 p. 94. 16 Note that Coriolis is defining, as today, work as a product of displacement and the component of force in the direction of motion and not of the force and the component of displacement in the direction of the force, as was the tradition and the way he will define it later. 17 p. 95. 14

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16 The treatises of mechanics

In this passage and throughout the rest of his work, the definition of virtual displacement is a little ambiguous. Coriolis never says that it is an infinitesimal displacement. But in fact, he treats it that way. For example, in the case of two molecules m and m of Fig. 16.2, if R is the common force that they exchange, in accordance with the principle of action and reaction, Coriolis argues that the work done by the force R is provided by the relation:  + R cos (Rδs   )δs = Rδr, R cos (Rδs)δs

(16.3)

where δr is the change of distance between m and m . But this relation is valid only if δs and δs are infinitesimally small as clear from Fig. 16.2. Because Coriolis still considers virtual displacements that occur in real time, with which the forces vary according to what was generally accepted before Poinsot, the force R moves in R as a result of the virtual displacements δs and δs . If these are infinitesimal, the displacement δr can be considered to occur in the old direction of R and then  )δs , from which relation (16.3) immediately follows.  + cos (Rδs  δr  cos (Rδs)δs Coriolis continues his exposition by saying that each molecule inside a solid body will be in equilibrium under the action of external forces P and internal forces R exchanged among the molecules, i.e. R + S = 0, and then for each molecule, the element of virtual work will be zero, and so will be the sum of the elements of virtual work of all the molecules. Then it is:

∑ Rδr + ∑ Pδp = 0,

(16.4)

where δr represents the change of distance between the molecules and δp the component of the virtual displacement of the point of application of P in the direction of P itself. In the case of an undeformable body δr = 0, so the previous relation gives:

∑ Pδp = 0.

(16.5)

It is assumed now that the virtual motions are limited to motions leaving the molecules in the state of invariability of the mutual distances, then the distance δr will not change in this motion and it will be δr = 0 and the equation above is reduced to: ∑ Pδp = 0 [79].18 (A.16.14)

R

m'

s' R

R m

s

R

Fig. 16.2. Action and reaction of two molecules

18

p. 97.

16.3 Gaspard Gustave Coriolis

371

Thus Coriolis found the equilibrium equation of the moments for the ‘solidified’ body, which is only necessary for the equilibrium of a deformable body and sufficient for a body that is actually rigid. The above reasoning is similar to that developed by Laplace in the Mécanique celeste. Both Coriolis and Laplace apply the principle of action and reaction, but Laplace applies it to constrained points without any justification, Coriolis applies it to free molecules, which in full respect of Newtonian mechanics exchange equal and opposite forces. In this context, the use of infinitesimal motion or velocity is important, not to take into account the conditions of constraint, which do not impose any limit, but rather to simplify calculations. After considering the equilibrium (and motion) of a single body, Coriolis passes to the examination of a system of bodies. Here the language is similar to that found in the treatises of practical engineers [381] and those based on thermodynamics, to appear in a few years, see for example Chapter 18. The virtual work assumes a degree of reality. It is more a physical quantity, observable and measurable in some way, than a purely mathematical definition as it appeared in the works of Lagrange and his immediate successors: If the equilibrium is obtained under the action of forces P, each molecule will be in equilibrium and, taking into account all the molecular actions, it will be:

∑ Rδr + ∑ Pδp = 0. If now a virtual motion of each body is considered that leaves its invariability or solidity, and yet in this motion the bodies are left to slip or turn over each other with the freedom of motion allowed by the machine constitution, it is found that a large portion of virtual works Rδr vanishes: it is that due to actions between molecules that have not switched away during the virtual motion, namely those belonging to the same body. In the equation above it will remain only the element of virtual work ∑ Rδr coming from the actions among the molecules of adjacent bodies, when in the virtual motion these bodies do not move together as one system, but they slip or roll on each other. The actions R that remain will be only due to molecules that are at a distance from the contact surface less than the extension of the molecular actions, or in other words, the radius of the sphere of action [79].19 (A.16.15)

This piece documents the way Coriolis conceives of virtual displacements. They are small possible displacements, and the virtual work is determined on the basis of forces that are assumed varying with them and not assumed frozen at the instant and the point where to study the equilibrium, as Poinsot and Ampère did, but they are though as function of the virtual displacements. Coriolis then says that the virtual work between the molecules in the contact zone can be calculated, assuming one of the bodies as fixed and considering for the other a virtual displacement equal for all the molecules, because of the small size of the contact area. Furthermore he proposes to decompose the forces of each molecule into a tangential component and a normal component, so “the elementary work of the normal component will be zero because the angle that this component makes with the virtual displacement is right. It will then only remain the element of work of the component in the direction of the sliding plane". Denoting with δ f the virtual 19

pp. 114–115.

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16 The treatises of mechanics

displacement common to all molecules of the contact zone and with F the forces exchanged among the molecules,20 the sum of the virtual work due to the action of the two bodies will be [79]:21 f ), δ f ∑ F cos(Fδ

(16.6)

with the summation extended only to the molecules in the area of contact which act on each other and δ f has been put into evidence because it is common. The equation of virtual work for all bodies will then be: f ), ∑ Pd p + ∑ δ f ∑ F cos(Fδ

(16.7)

where the two summations of the internal forces are made with respect to the number of bodies concerned and with respect to the molecules of the contact zone of each body. If it is assumed that the actions of two bodies in contact are reduced to a single normal force, i.e. if the tangential component vanishes: f ) = 0. F cos(Fδ

(16.8)

Then relation (16.7) reduces simply to:

∑ Pd p = 0.

(16.9)

Thus the principle of virtual work applies in this case between the only external f ) is not null and forces [80].22 But, Coriolis continues: “In fact, the sum F cos(Fδ then it is necessary to take this into account. The difficulty is to evaluate it. The directions and values of the actions F can only be seen by experimental consideration” [80].23 Next Coriolis examines the problems related to friction, using a language that was completely different from the classic terminology used for the virtual work laws. It is possible, for example, by the nature of bodies, to suggest that there is no possibility of sliding and that one body rolls on the another “then the virtual velocities become zero for the points of contact […] so that the sum of the work due to this rolling is zero.” Finally he concludes: We are led to realize that the principle of virtual velocities in the equilibrium of a machine, composed of more bodies, cannot take place without considering first the sliding friction, where the virtual displacements cause the slipping of the the bodies one on others, and finally that the rolling when bodies cannot take that virtual motion without deformation near the contact points. Frictions are recognized always, for experience, able to maintain equilibrium in a certain degree of inequality between the sum of the positive work and the sum of the negative work, 20 There is something unclear in Coriolis’ text. He declares that F is the tangential component of the forces exchanged among the molecules. But in this case F and δ f would be parallel each other f ) = 1. I assume Coriolis got confused and attributed to F the meaning of whole and then cos(Fδ force. 21 p. 115. 22 p. 116. 23 p. 116.

16.3 Gaspard Gustave Coriolis

373

here taking as negative the elements belonging to the smaller sum. It follows that the sum of the elements to which they give rise has precisely the value that can cancel the total sum and is equal to the small difference between the sum of the positive and negative elements [79].24 (A.16.16)

So friction contributes to the balance of the work by providing a negative term, since “for experience, it gives rise to a negative sum.” In the classification of attempts to demonstrate the law of virtual work given in Chapter 2, Coriolis should be placed among the reductionists since he frames the law of virtual work in the context of Newtonian physics. His mechanical theory is, essentially, the one I used in Chapter 2 to highlight the problems of the logic status of the law of virtual work. Coriolis can prove quite easily the virtual work law for the single unconstrained rigid body, but he finds it difficult to deal with several bodies constrained together or with the outside world. In fact he uses the law of virtual work in the form T1 and sometimes in the form T2 , when he mentions the possibility that the tangential component of the contact forces vanishes. But he never introduces the law of virtual work in the form T3 , assuming the principle P2 of smooth constraints. This is because he wants to address the problems with friction and not to limit himself to a situation that he considers an ideal only.

24

p. 117.

17 Virtual work laws and continuum mechanics

Abstract. This chapter is devoted to the use of the VWL in the mechanics of continuous media. In the first part, the pioneering works of Joseph Louis Lagrange and Claude-Louis Navier are presented. In the second part the use of the VWL in the theory of elasticity by Alfred Clebsch is presented. In the final part the Italian school treatments of internal forces as reactions of constrain are presented.

17.1 First applications The application of virtual work laws to continuous bodies goes back if not to ancient Greek, at least to the XVI century, when Galileo Galilei applied it to study the equilibrium of fluids in several cases, including that of communicating vessels, in the Discorsi intorno alle cose che stanno in sull’acqua e scritture varie in 1602 [115]. More sophisticated applications arose in the XVIII century, see for example Discorso intorno agli equilibri of Vincenzo Angiulli in 1770 [4], in which a law of virtual work is used in conjunction with the differential calculus.

17.1.1 Joseph Louis Lagrange A mature application however must await Lagrange’s Mécanique analytique in 1788 [145]. Lagrange considers both mono-dimensional and bi-dimensional continua, under static and dynamic cases. Here, for the sake of space, I will present with some detail only the equilibrium of flexible but inextensible wire, elastic and flexible wire [145],1 non-deformable solids, and incompressible fluids.

1

pp. 156–162.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_17, © Springer-Verlag Italia 2012

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17 Virtual work laws and continuum mechanics

17.1.1.1 Mono-dimensional continuum For flexible and inextensible wires Lagrange writes first the expression of the virtual work of the active external forces, assumed as distributed along the wire:

S(Xδx +Y δy + Zδζ)dm,

(17.1)

in which X,Y, Z are the components of the active forces for unitary mass, δx, δy, δz the virtual displacements, S the integral symbol and dm the element of mass. The quantities that appear under the integral sign should be considered as the function of an abscissa a measured in the unchanged configuration, which in the case of inextensible wire coincides with the curvilinear abscissa s on the deformed configuration. Virtual displacements occurring in (17.1) cannot be arbitrary because the various points while moving maintain the constraint of inextensibility. Indicating with ds the length of the infinitesimal element of the wire, Lagrange writes the equation of inextensibility in the form: δ ds = 0.

(17.2)

By following the technique he developed, introducing a multiplier λ, Lagrange can write the following equation of equilibrium, or with his terminology, the equation of moments:

S(Xδx +Y δy + Zδz)dm + λSδ ds = 0,

(17.3)

where δx, δy, δz can vary freely. By making explicit δ ds as a function of δx, δy, δz: dxδx + dyδy + dzδz (17.4) ds and integrating, Lagrange gets the following differential equations of equilibrium [148]:2 δ ds =

λdy λdz λdx = 0, Y dm − d = 0, Zdm − d = 0. ds ds ds He comments on the result as follows: Xdm − d

(17.5)

As λδ ds can represent the moment of a force λ tending to vary the length of ds, the term S λδ ds of the general equilibrium equation of the wire will represent the sum of the moments of all forces λ that it can be assumed to act on all elements of the wire; in fact each element resists for his inextensibility to the action of external forces and these resistance are usually considered as active forces [emphasis added] called tensions. So the quantity λ expresses the tension of the wire [148].3 (A.17.1)

Lagrange will be also more clear about the meaning of the multiplier λ in the study of the equilibrium of the elastic wire. In this case the moment of the external forces is still given by the expression (17.1). Next to the active external forces Lagrange presupposes the existence of active internal forces F due to the elasticity of the wire. The moment of these forces on an infinitesimal element of length ds is given by 2 3

p. 147. pp. 148–149.

17.1 First applications

377

Fδ ds, as a work of two equal and opposing forces acting in the direction of ds at the ends of the element ds, which undergoes a relative displacement δ ds. Hence the equation of moments is [148]:4

S(Xδx +Y δy + Zδζ) + F Sds dm = 0,

(17.6)

which is mathematically equivalent to (17.3). From this equation it can be seen that the multiplier λ of constraint conditions is identical with the tension F. 17.1.1.2 Three-dimensional continuum Rigid bodies In the study of three-dimensional rigid solids. Lagrange imposes, as in the case of the inextensible wire, the constraint of rigidity, requiring this time that the mutual distances of all points of the solid remain unchanged for any virtual displacement. He gets a set of differential equations of the form in the Cartesian coordinates of the points (x, y, z): d n xd n δx + d n yd n δy + d n zd n δz = 0,

(17.7)

of which only three are independent, for example those corresponding to n = 1, 2, 3. These expressions were already obtained by Euler in his work Decouvert d’un nouveau principe de mécanique [101],5 in the case of motion of a body fixed to its centre of gravity. By integration of equations (17.7) Lagrange gets the following expression of virtual displacements of a rigid body: δx = δl − yδN + zδM, δy = δm + xδN − zδL, δx = δn − xδM + yδL,

(17.8)

that he comments: Expressions found above for changes δx, δy, δz show that these variations are the result of the motions of translation and rotation that we considered in section III. […] The previous analysis leads naturally to these expressions and testify with a more direct and more general way than that of article 10 of section III, that when the different points of a system retain their relative positions, the system can have at any given instant only translational motion in space and rotation around three orthogonal axes [148].6 (A.17.2)

Substituting expressions (17.8) into δx, δy, δz and ds in the moment equation:

S(Xδx +Y δy + Zδζ) + λSds dm = 0, Lagrange obtains the cardinal equations of statics. 4 5 6

p. 157. pp. 197–201. pp. 187–188.

(17.9)

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17 Virtual work laws and continuum mechanics

Indeformable fluids To illustrate the ideas of Lagrange on the constitution of fluids and the possibility of applying the principle of virtual velocities to study their balance, I quote two excerpts from the historical introduction to hydraulics of the Mécanique: Although we ignore the internal composition of fluids, we cannot doubt that the particles that compose them are material and that, therefore, for them, as for solids, the general laws of equilibrium apply. In fact, the main property of fluids, the only one that distinguishes them from solid bodies, is that all their parts cannot resist to the smallest force and can move between them with all possible ease, whatever the constraints and mutual action between the parts. But, being this property easily translated into calculation, it follows that the laws of equilibrium of fluids do not require a particular theory and that they must be a special case of the general theory of statics [145].7 (A.17.3) Previous theories of equilibrium and pressure of fluids are, as we have seen, wholly independent of the general principles of statics, being based on empirical principles proper to fluids. This way of demonstrating the laws of hydrostatics, deducing from the experimental knowledge of some of its properties that of the others, has been adopted by most modern writers who made a science of hydrostatics completely different and independent from statics. However, it is natural to connect these two sciences and having them depend on the same principle. Now, among the different principles that can serve as a basis for the equilibrium and of which we have given a brief exposure in section I, it is clear that there is the principle of virtual velocities that applies naturally to the equilibrium of fluids [145].8 (A.17.4)

The use of the term particle at the beginning of the first passage suggests that Lagrange, unlike Euler, admits an atomic structure for fluids. This fact is even clearer in the following considerations, which are located toward the end of the historical introduction: Clairaut’s principle is nothing but a natural consequence of the principle of equality of pressure in all directions and we must recognize that this principle contains, in fact, the most simple and general properties that the experience allows to discover in the equilibrium of fluids. But, in the search for the law of the equilibrium of fluids, is the knowledge of this property essential? Cannot we directly derive this law from the nature of the fluids considered as assembly of molecules loosely joined, independent of each other and perfectly mobile in all directions? [145].9 (A.17.5)

The atomic concept of matter does not prevent Lagrange from treating fluids as if they were continuous media and replacing, in the mathematical aspects, the particles of matter with infinitesimals dm, resulting in expressions where integrals appear instead of summations, easier to be treated. This approach of adopting a continuous mathematical model and a discrete physical model will be followed by the French scientists of the XIX century. Taking advantage of Euler’s studies on the concept of strain and pressure in fluids, Lagrange considers both compressible and incompressible fluids. I summarize below the case of equilibrium of incompressible fluids listed in Section VIII of part I of the Méchanique analitique, because it is the most simple and sufficiently representative. 7 8 9

p. 122. pp. 126–127. p. 129.

17.1 First applications

379

Let X,Y and Z be the components of the specific forces acting on the elementary masses dm, constituting the fluid, occupying the position x, y and z. The virtual work of these forces (the sum of the moments in the language of Lagrange) is given by the integral: 

(Xδx +Y δy + Zδz) dm,

(17.10)

where δx, δy, δz represent the virtual displacements of dm. Lagrange considers (17.10) as the expression of the virtual work of all forces acting on the fluid and believes implicitly zero the contribution of forces within the fluid, i.e. the pressure. For the justification of this viewpoint see the considerations in Chapter 10 on the approach of the internal forces of the moon treated by Lagrange as a rigid body. If the fluid particles were free to move, that is not constrained one to another, the balance would be provided by the annulment of the integral (17.10) for each virtual variation δx, δy, δz. In fact, since as assumed, the fluid is incompressible, its particles are subject to the constraint of incompressibility and the virtual displacements δx, δy, δz must meet this constraint, which can be represented by the relations: L = dxdydz = const.;

δL = δ(dxdydz) = 0,

(17.11)

dxdydz being associated with the volume of the element dm. Following the theory of multipliers Lagrange can derive the following equation of equilibrium: 

(Xδx +Y δy + Zδz)) dm + λ



Ldm = 0,

(17.12)

where λ is a Lagrange multiplier and where now δx, δy, δz can vary arbitrarily. From now on, for Lagrange it is only a matter of applications of his calculus of variation which leads to recognition of the identity:

 ∂ δx ∂ δy ∂ δz δ(dxdydz) = dxdydz + + . (17.13) ∂x ∂y ∂z He achieves this result both directly, by developing consistently the variation of dxdydz, or with the use of kinematic relations already obtained by Euler [104],10 that in an elementary parallelepiped with sides dx, dy, dz the variation of their size is expressed by the relation, correct up to infinitesimals of second order: 





∂ δx ∂ δy ∂ δz , dy 1 + , dz 1 + , (17.14) dx 1 + ∂x ∂y ∂z that with a modern language, using the axial components x , y , z of the tensor of deformation, and making reference to Fig. 17.1, can be rewritten as: dx (1 + x ) , dy (1 + y ) , dz (1 + z ) .

10

p. 286.

(17.15)

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17 Virtual work laws and continuum mechanics

z

dz y

P

(1 +εz)dz (1 +εx)dx

dx dy

(1 +εy)dy

x Fig. 17.1. Deformation of an infinitesimal parallelepiped

Lagrange finally reaches the equilibrium equations of incompressible fluids, already obtained by Euler [103]11 and acknowledges that the multiplier λ represents the scalar value of the pressure in the individual points of coordinates x, y and z: dλ dλ dλ = ΔX, = ΔY, = ΔZ, dx dy dz

(17.16)

where Δ is the density of the fluid. After Lagrange, the application of the laws of virtual work to continuous bodies is still considering them as aggregates of molecules, and then as if it were a set of material points. For the mathematical aspects use was made of both the discrete model, in which the total virtual work is represented by summations, and the continuous model, in which it is represented by integrals. Applications are not limited to static problems, but also to dynamic ones, with the use of the living force concept. Of some interest are the considerations on the phenomena of shock. A brief review of the memoirs where continuous bodies are studied as aggregates of molecules can be found in the works of Clebsch with commentary by Saint Venant [72],12 Cauchy in 1829 [67] and Coriolis in 1835 [79].

11 12

p. 230 pp. 577–582.

17.1 First applications

381

17.1.2 Navier’s equations of motion The most interesting work of the early XIX century, which concerns the application of virtual work laws to continuous mechanics, is however Navier’s Mémoire sur la lois de l’équilibre et du mouvements des corps solides élastiques of 1821, published in 1827 [173]. Claude Louis Navier was born in Dijon in 1785 and died in Paris in 1836. In 1802 he enrolled at the École polytechnique and in 1804 continued his studies at the École nationale des ponts et chaussées, from which he graduated in 1806. He directed the construction of bridges at Choisy, Asnières and Argenteuil in the Department of the Seine, and built a footbridge to the Île de la Cité in Paris. In 1824, Navier was admitted into the Académie des sciences. In 1831 he became Chevalier of the Legion of Honor and succeeded Cauchy as professor of calculus and mechanics at the École polytechnique [290]. Navier formulated the general theory of elasticity in a mathematically usable form. His major contribution however remains the Navier-Stokes equations (1822), central to fluid mechanics. In the second part of Mémoire sur la lois de l’équilibre et du mouvements des corps solides élastiques, Navier, who is inspired explicitly by Lagrange’s Mécanique analytique, writes the equation of virtual work of internal and external force – the equation of moments in Navier’s terminology – acting on an elastic body thought of as an aggregate of particles, or molecules, that attract or repel with an elastic force varying linearly with their relative displacements and obtains the equations of equilibrium. To determine the virtual work of the internal forces Navier focuses his attention first on a material point M. The virtual work made on M is that of the internal force exerted on M by all points M  of the continuum in a general virtual motion. As virtual motion Navier considers the variation δ f of the relative displacement f of M and M  from the virgin state. Since the constitutive law is assumed linear elastic, the force exerted on M by M  is given by  f , where  is a constant depending on the mechanical properties of a continuum, and its moment by  f δ f , or as Navier notes, 1/2δ f 2 , that with a modern language represents the variation of elastic potential energy. If the expression of 1/2δ f 2 is integrated by varying M  on all points of the elastic continuum, the total virtual work of M is obtained. Expressing the relative displacement f by means of the absolute displacement (x, y, z) of the point of the continuum from the virgin state (a, b, c), after some mathematics, the following expression results:  



1 dx2 dx dz 2 dx dy 2 dx dy dx dz δ 3 2 + + + + + +2 +2 2 da db da da db dc da da dc  (17.17) 



dy2 dx dz 2 dy dz 2 dy dz dz2 3 2+ + + + +2 +3 2 . db dc db db dc dc da dc

382

17 Virtual work laws and continuum mechanics

Developing the variation δ, adding the virtual work of the volume forces of components X,Y, Z and surface forces of components X  ,Y  , Z  , and integrating on the whole continuum, the following global equation of virtual work is obtained: 0=



dadbdc

⎫ ⎧ dx δdx dx δdx dx δdy dy δdx dy δdy dx δdy dy δdx ⎪ ⎪ ⎪ 3 + + + + + + +⎪ ⎪ ⎪ ⎪ da da db db db da da db da da da db db da ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ dx δdx dx δdz dz δdx dy δdy dz δdz dx δdz dz δdx + + +3 + + + + ⎪ da dc dc da da dc db db da da da dc dc da ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dy δdy dy δdz dz δdy dz δdz dy δdz dx δdy dz δdz ⎪ ⎪ ⎩ + + + + + +3 −⎭ dc dc dc db db dc db db db dc dc db dc dc −



dadbdc (Xδx +Y δy + Zδz) −



ds (X  δx +Y  δy + Z  δz ) . (17.18)

Here the triple integrals are extended on the volume and the double to the surface s of the elastic body. Navier, starting from the variational equation (17.18), uses an approach that will be followed by many scholars. Performing an integration by parts – which for us corresponds to the application of Green’s formula – and considering arbitrary virtual displacements, obtains the indefinite equilibrium equations (for internal points of the continuum) and boundary equations. By way of example I only offer the indefinite equilibrium equations:

2  d x d2x d2x d2z d2y −X =  3 2 + 2 + 2 + 2 +2 da db dc da db da dc 

2 2 2 2 d2z d y d y d y d x (17.19) + 2 −Y =  + 3 + + 2 da2 db2 dc2 da db db dc 

2 d2y d z d2z d2z d2x + 2 . −Z =  + + 3 + 2 da2 db2 dc2 da dc db dc

17.2 Applications in the theory of elasticity

383

17.2 Applications in the theory of elasticity 17.2.1 Alfred Clebsch Perhaps the first organic appearance of a virtual work law for three-dimensional continua, in particular linear elastic, which uses the concept of tension developed by Cauchy in his works of the period 1823–1828 [267], is the Theorie der Elastizität fester Körper by Alfred Clebsch in 1862 [71, 386, 387]. Rudolf Friedrich Alfred Clebsch was born in Königsberg in 1833 and died in Göttingen in 1872. He was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the university of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe. In 1866, he and Paul Gordan published a book on the theory of Abelian functions. On the whole, to a modern reader a century later, the book may seem old fashioned; but it must be remembered that it appeared long before Weierstrass’ more elegant lectures on the same subject. His cited book on elasticity may be regarded as marking the end of a period. In it he treated and extended problems of elastic vibrations of rods and plates. His interests concerned more the mathematical than the experimental side of the physical problems [290]. In 1883 Adhémar Jean Claude Barré de Saint-Venant (1797–1886), more than eighty years old, translated Clebsch’s work on elasticity into French and published it as Théorie de l’élasticité des corps solides [72] making accessible Clebsch’s contribution to European scholars. Because of the importance of this book for the application of virtual work laws to continua I translate the entire § 16: Evaluation of work for a small deformation of a body. Relation deduced among the thirty-six factors that serve to define the behavior of a crystalline substance, or all non-isotropic solids. Imagine that a body subject to the action of any forces, undergoes a sequence of changes of shape so that the coordinates (x, y, z) of its points, become x + u, y + v, z + w parallel to x, y, z. The work produced throughout the body for this displacement is obtained by multiplying a [generic] element of its volume dxdydz for the components in the directions x, y, z of the forces acting to drive that volume respectively for small paths δu, δv, δz and then adding the three products and integrating their sum for the whole extension or for all the elements of the body. Now, the three components of forces, agent in the unit volume element dxdydz are only the second member of equation (5) of § 14, page 54, i.e.:

∂ txx ∂ txy ∂ txz + + +X ∂x ∂y ∂z in x direction and two similar quadrinomials in the directions y and z. The elementary work product, when the points run through the spaces the projections of which on the axes x, y, z are δu, δv, δw, is therefore expressed as: δW = δU + δV

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17 Virtual work laws and continuum mechanics

where: δU =

  

(Xδu +Y δv + Zδw) dxdydz

represents the work of external forces acting within the body and: ⎫

⎧ ⎪ ⎪ δu ∂ txx + ∂ txy + ∂ txz ⎪ ⎪ ⎪ ⎪ ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪    ⎨ ∂ tyx ∂ tyy ∂ tyz ⎬ +δv + + δV = dxdydz ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ∂ tzx ∂ tzy ∂ tzyz ⎪ ⎪ ⎪ ⎪ + + ⎩ +δw ⎭ ∂x ∂y ∂z represents the work that derives from the reciprocal actions of its particles, either from traction or compression forces acting on its surface. Let us now consider one of the nine terms of the latter triple integral, for example:   

δu

∂ txx dxdydz. ∂x

By integrating partially with respect to x, i.e. for the small portion of the body contained in a channel [a cylinder] which is considered infinitely thin, the section dydz of which is considered as constant, as well as the coordinates y and z. If the second term

∂ δu ∂u is replaced with δ ∂x ∂x which is the same, this integration by part gives the expression:  

[txx δu]dydz −

  

txx δ

∂u . ∂x

The square brackets mean that, instead of the expression txx δu they contain, one must put the difference of this expression at the opposite ends [the two terminal surfaces] of the channel in question. Denote now with dσ, dσ the elements that the channel cuts on the [external] surface of the body and with p, q, r, the angles that the normal to dσ, directed toward the outside of the body, makes with the coordinate axes x, y, z, and finally with p , q , r , the same angles of the normal to dσ . If dσ is the front end of the channel, namely the one located on the positive side of x, and dσ is its rear end, cos p is necessarily positive and cos p negative, so it is: dydz = dσ cos p = −dσ cos p . The difference of the limit values of txx δudydz therefore becomes the sum of the values that the expression txx δudσ cos p takes at the end of the channel. Instead of extending the double integral above to the ends of all channels that can be carried out in a similar way within the body, it is possible to integrate over the set of elements dσ, which include the elements dσ . Then:    [txx δu]dydz − txx δudσ cos p. […] So in all cases, the considered term δV will be replaced by: 

txx δudσ cos p −

  

txx δ

∂u dxdydz ∂x

where the first term is extended to the whole surface of the body. If it is made the same for all the terms δV , it is obtained: δV = δU1 − δU2

17.2 Applications in the theory of elasticity

385

where δU1 represents all the simple integrals [double integral] and δU2 all the triplicate integrals, like that of the binomial expression of txx we have written. And it is: 

δU1 = (txx cos p + txy cos q + txz cos r) δudσ 

+ (tyx cos p + tyy cos q + tyz cos r) δudσ 

+ (tzx cos p + tzy cos q + tzz cos r) δudσ. Now the expressions in brackets are those that because of the equations (25) are equivalent to the component T cos ω, T cos κ, T cos ρ, of the tensile forces T applied on the body surface [as shown in another paragraph]; δU1 is then the work of the external tensile forces and it is: 

δU1 = T (cos ωδu + cos κδv + cos ρδw) dσ. Similarly it will be found that the eight terms of δU2 other than that containing: txx

∂u ∂x

and which we wrote, are affected, under the integral, by other differential quotients:

∂u ∂u ∂v , , ,... ∂x ∂y ∂x of the displacements u, v, w and by the other components tyy , . . . of the tensions. Hence, introducing the elementary deformations13 in place of:

∂u ∂u ∂u ∂v , , ..., + ∂x ∂y ∂y ∂x after the expressions (28) it is: δU2 =



(txx δ∂x + tyy δ∂y + tzz δ∂z + tyz δgyz + tzx δgzx + tyx δgyx ) dxdydz.

For the total work is is then: δW = δU + δU1 + δU2 where δU and δU1 represent the work of external forces acting respectively on the points on the body surface. As a result δU2 is necessarily the work of the internal forces resulting from the molecular actions [72].14 (A.17.6)

At this point Clebsch abandons the exposition of the virtual work law because his objective was not in fact to obtain the equations of equilibrium which could be obtained trivially by requiring the vanishing of the forces acting on the element of volume, but rather to derive the expression of the work of internal forces and then move to the expression of the elastic potential upon which to make reasonings regarding the constitutive relationships. The virtual work principle is used as a bridge to link the ‘physical’ approach of the constitutive relationship, based on the mechanical intuition of tension and strain, to the ‘purely’ mathematical approach developed by Green in his work of 1839 on the law of reflection and refraction of light, in 13

The symbolism of Clebsch-Saint-Venant is still used with regard to the components of a stress tensor; it was however abandoned with respect to the components of a deformation tensor. 14 pp. 57–64.

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17 Virtual work laws and continuum mechanics

which to the internal forces are imposed the only restriction to be conservative and therefore with a potential [278]. The treatment of Clebsch is quite modern, although now it is preferred to use the formalism of the vector and tensor calculus, the concepts of gradient and divergence, which makes the discussion much more compact. With a greater satisfaction for a modern reader, Clebsch’s exposition could have been concluded by showing explicitly that equating to zero the total work δW = δU +δU1 +δU2 , under the condition of equilibrium of stresses and congruence of deformations, the expression is obtained immediately:   

=



(Xδu +Y δv + Zδw) dxdydz +



T (cos ωδu + cos κδv + cos ρδw) dσ

(txx δ∂x + tyy δ∂y + tzz δ∂z + tyz δgyz + tzx δgzx + tyx δgyx ) dxdydz.

(17.20)

One characteristic perhaps surprises in Clebsch’s treatment. His mathematical passages would have been much easier if he had directly used the formulas of Green. They were obtained more or less simultaneously by George Green (1783– 1841) who published them at his own expense in 1828 [128] and Michel Ostrogradsky who presented them at the Academy of sciences in St. Petersburg, also in 1828 [177]. Then Clebsch surely knew them. The only justification for their nonuse is perhaps the fact that they were not yet well known and Clebsch could assume that his text would have been more understandable if he had not made reference to them. It is appropriate to quote a comment by Saint Venant on an alternative possibility for the expression of virtual work of the internal forces: In 1858, Mr. Kirchhoff has been kind enough to indicate to me a simple and direct way to account for the sextinôme composition given above about the internal or molecular work U2 for the unit of volume of an element. Let dx, dy, dz be the three sides, parallel to x, y, z of this rectangular element: 10 , if the dilatation ∂x already suffered in the x direction is to be increased, the two equal and opposed faces yz will stretch of ∂x dx, the normal components of the tension exerted by the matter surrounding these faces produce the work yztxx xδ∂ x, this is basically the work txx δ∂x per unit of volume. 20 if, one of the two sides is assumed as fixed, the distortion gxy is increased by δgxy , and there is as a displacement xδgxy of on one side over the other, so that the shear stress txy , acting on the unit of surface in the direction y of this distortion, produces a work txy xδgxy . There are then two other faces on which it acts a tension tyx equal to txy ; i.e. the faces xz. In this motion, they rotate around the two sides in the z direction of the two faces yz that remained motionless. But the tensions tyx act in the sense x and not in the sense y which is that of the motion and have nothing to add to the work yztxy xδgxy of the stress txy , work that is only txy gxy , for unit of volume. Now the work of the six tensions over the faces of the element, so that there is equilibrium after this small displacement, has to be equal to the molecular work within the element. So this work per unit volume has its own value the expression of the sextinôme (txx δ∂x + · · · + txy δgxy ) defining δU2 [72].15 (A.17.7)

In Appendix V of Navier-Saint Venant’s text [174]16 there is a strange reversal of exposure. In the body of the text, the expression of the work of internal forces is 15 16

p. 61. pp. 712–715.

17.3 The Italian school

387

determined exactly in the way Saint Venant attributes it to Kirchhoff. Then, in a footnote, Navier-Saint Venant refers to an alternative way, attributed to Lamé,17 exactly what Clebsch shows in the body of the text to find the expression of the work of internal forces. It is worth noting that Kirchhoff’s approach, to calculate the virtual work of internal forces, which now seems completely natural, did not appear that way to the founders of the theory of elasticity. There was still some uncertainty on the ontological status to be attributed to the forces in general and in particular to tensions. Today, with the prevalence, especially in the areas of application, of the instrumentalist spirit of theories, there are no scruples and symbols that represent mechanical quantities are handled with greater indifference. In addition, the purely formal approach for the expression of the work of the internal forces must have appeared more strict than the heuristic one provided by Kirchhoff. Without thinking, however, that the expression of the forces acting on the element of volume, or alternatively, the internal equilibrium equations, were obtained currently, at least in those days, with a heuristic approach, based on infinitesimals, no more stringent than that used by Kirchhoff to obtain an explicit equation of the virtual work of internal forces.

17.3 The Italian school In Italy, at the beginning of the XVIII century the influence of Lagrange was relevant. To many Italian mathematicians and mechanicians, modernity was represented by Lagrange. This was partly because Lagrange, even after leaving Turin in 1766, had remained in contact with the Italian world of science, and partly because Italians considered him Italian and this was a period of rising nationalistic feelings. Vincenzo Brunacci (1768–1818), professor of ‘Matematica sublime’ (Calculus) in Pavia, was one of the main supporters of Lagrange’s ideas. Along with the fashionable purism of the time, he accepted Lagrange’s reduction of the differential calculus to algebraic procedures [146] and rejected the XVIII century concept of infinitesimal in both calculus and mechanics [53]. Brunacci transmitted these ideas to his pupils, including Ottaviano Fabrizio Mossotti (1791–1863), Antonio Bordoni (1788–1860) and Gabrio Piola (1794–1850), the brightest Italian mathematicians of the first half of the XVIII century. From them the modern Italian schools of mathematics originate. For example Enrico Betti and Eugenio Beltrami, the highest rank Italian mathematician of the second half of the XIX century were students, respectively, of Mossotti and Bordoni.

17

These are the developments in the Clapeyron theorem contained in [154], pp. 80–83.

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17 Virtual work laws and continuum mechanics

17.3.1 Gabrio Piola Count Gabrio Piola Daverio was born in Milan in 1794 in a rich and aristocratic family and died in Giussano della Brianza in 1850. He studied mathematics at Pavia university as a pupil of Vincenzo Brunacci [313]. Though Piola was one of the most brilliant mathematical physicists of the XIX century he has been for a long time neglected. His revival is due mainly to Clifford Truesdell, a great estimator of Italian scholars, and Walter Noll, who was well aware of the interaction between the Italian and German schools in the last year of the 1800s. Piola’s work on continuum mechanics concerned fluids and solids. These last were published in various years [187, 188, 189, 190, 191], with La meccanica de’ corpi naturalmente estesi trattata col calcolo delle variazioni of 1833 [188], probably the most relevant one. Lagrange in his Mécanique analytique had applied the principle of virtual work, in conjunction with the calculus of variations, to the study of the internal forces to one-dimensional elastic continua and fluids. Piola generalizes the approach to threedimensional elastic continua. In his papers, Piola questions the need to introduce uncertain hypotheses on the constitution of matter by adopting a model of corpuscles and forces among them, as the French mechanicians did. Piola states that it is sufficient to refer to evident and certain phenomena: for instance, in rigid bodies, the shape of the body remains unaltered. Then, one may use the ‘undisputed’ equation of balance of virtual work; only after one has found a model and equations based exclusively on phenomena, Piola says, is it reasonable to look for deeper analyses: Here is the great benefit of Analytical Mechanics. It allows us to put the facts about which we have clear ideas into equations, without forcing us to consider unclear ideas […]. The action of active or passive forces (according to a well known distinction by Lagrange) is such that we can sometimes have some ideas about them; but more often there remain […] all doubts that the course of nature is different […]. But in the Analytical Mechanics the effects of internal forces are contemplated, not the forces themselves; namely, the constraint equations which must be satisfied […] and in this way, bypassed all difficulties about the action of forces, we have the same certain and exact equations as if those would result from the thorough knowledge of these actions [188].18 (A.17.8)

The value of the internal forces, whose legality of use Piola does not question, is determined by means of Lagrange multipliers of appropriate conditions of constraint. In La meccanica de’ corpi naturalmente estesi trattata col calcolo delle variazioni [188], Piola is inspired by Lagrange but follows an inverse path with respect to him; he takes for granted the global equation of the rigid body (17.8) which for Lagrange was the terminal point; he derives them opportunely and obtains six differential equations which characterise locally the rigidity constraint. The material points of a body are labelled by two sets of Cartesian coordinates. The first refers to axes called a, b, c, as made by Lagrange in the Mécanique analy18

pp. 203–204.

17.3 The Italian school

389

tique [149],19

rigidly attached to the body – reference configuration – and the second to axes called x, y, z, fixed in the ambient space and to which the motion of the body is referred – current configuration. It was not difficult for Piola to prove the validity of relations:

2 2 2 2 2 2 dx dx dx dy dy dy + + = + + = da db dc da db dc

2 2 2 dz dz dz + + = 1, da db dc

          dy dx dy dx dy dx dz dx dz dx + + = + da da db db dc dc da da db db

        dx dz dy dz dy dz dy dz + = + + = 0. dc dc da da db db dc dc (17.21) To write down the balance equation Piola uses the technique developed by Lagrange in the Mécanique analytique, by equating to zero the virtual work of volume (density) forces, inertia forces included, integrated over the body volume:  2 

2 

2      d x d y d z da db dc ΓH − X δx + −Y δy + − Z δz = 0, dt 2 dt 2 dt 2 (17.22) where Γ is the mass density, H the Jacobian of the transformation from (a, b, c) to (x, y, z) and (δx, δy, δz), the virtual displacement of a material point of the body. At this point Piola reminds the reader that the virtual displacements (δx, δy, δz) are not free but they are constrained according to relations (17.21). To free (δx, δy, δz) from any constraints the Lagrange multiplier method can be used, by adding to the integral on the left side of the variational equation (17.22) the integral of constraint relations (17.21) each multiplied by appropriate Lagrangian multipliers (A, B, C, D, E, F) that, by reproducing the original Piola’s text, with S the symbold of integrals, gives:  

 

 

 dx dδx dx dδx dx dδx Sda Sdb Sdc · A + + da da db db dc dc  

 

  

dδy dy dδy dy dδy dy + + Sda Sdb Sdc · B da da db db dc dc  

 

  

dδz dz dδz dz dδz dz + + Sda Sdb Sdc ·C da da db db dc dc

19

Section XI, art. 4.

390

17 Virtual work laws and continuum mechanics



 

 

 dδy dx dδy dx dδy dx + + da da db db dc dc  

 

  

dδx dy dδx dy dδx dy + + + da da db db dc dc  

 

  

dδz dx dδz dx dδz dx + + Sda Sdb Sdc · E da da db db dc dc  

 

  

dδx dz dδx dz dδx dz + + + da da db db dc dc  

 

  

dδz dy dδz dy dδz dy + + Sda Sdb Sdc · D da da db db dc dc  

 

  

dδy dz dδy dz dδy dz + + . + da da db db dc dc 

Sda Sdb Sdc · F

After lengthy calculations Piola arrives at the following balance equations in the reference configuration (x, y, z): 

2  d x dA dF dE Γ X− + + =0 + dt 2 dx dy dz 

2  dF dB dD d y + + =0 + Γ Y− dt 2 dx dy dz 

2  dE dD dC d z + + = 0, (17.23) + Γ Z− 2 dt dx dy dz which compared with results by Cauchy [65, 267] and Poisson [199], gives a mechanical meaning to the Lagrangian multipliers (A, B, C, D, E, F): they are the stress components in an assigned coordinate system in the reference configuration. Pierre Duhem in his course of Hydrodinamique, elasticity, acoustique of 1890–91 [98], used a virtual work law to obtain the equilibrium equations in a way close to that used by Piola, without any reference to him [268].

17.3.2 Eugenio Beltrami Eugenio Beltrami was born in Cremona in 1836 and died in Rome in 1900. He studied at the university of Pavia from 1853 to 1856 where his teacher was Francesco Brioschi. In 1863 he was offered the chair of geodesy at the university of Pisa by Enrico Betti. In Pisa he met Bernhard Riemann. From 1891 he was in Rome for the latest teaching [294]. Beltrami differentials techniques influenced the birth of the tensor calculus, providing a basis for the ideas later developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita. Some works concern the mechanical interpretation of Maxwell’s equations. Beltrami’s contribution to the history of mathematics is also important; in 1889 he brought to light the

17.3 The Italian school

391

work of Girolamo Saccheri [212] on parallel lines, compared his results with those of Borelli, Wallis, Clavius, Bolyai and Lobachevsky and gave himself an important contribution to non-Euclidean geometry [23]. Beltrami followed fairly closely the approach of Piola, making it more general. For Beltrami the equation of virtual work was a relation between dual variables, forces and deformations, taking mechanical meaning leaning to one another in forcing the cancellation of the virtual work. In his first organic paper on the theory of elasticity Sulle equazioni generali dell’elasticità in 1880–1882 [24], Beltrami studies the equations of elastic equilibrium in a space with constant curvature where a body with volume S and surface σ is present. Beltrami’s work stems from the results obtained by Lamé with curvilinear coordinates and from some subsequent works by Carl Neumann and Borchardt. The latter simplified Lamé’s calculations with the use of a potential function in curvilinear coordinates. According to Beltrami their approach, though it led to correct results, can be improved. Lamé, Neumann and Borchardt formulated the problem in Cartesian coordinates, implicitly assuming the Euclidian space. He instead proves directly the elastic equations of equilibrium without any assumption on the nature of space. Beltrami has a more mature feeling with internal forces than Piola, or rather he is more ‘relaxed’ than him. He is not afraid to make explicit reference to them and assumes a position similar to that of Lagrange. The internal forces (tension) and the components of the constraint conditions (deformation) are treated as dual variables, implicitly defined by the fact that their product is a virtual work, regardless of the metric adopted. His central idea lies in a suitable metrics and from it of suitable infinitesimal strain measures; with reference to the infinitesimal element ds he writes: ds2 = Q21 dq21 + ds2 + Q22 dq22 + ds2 + Q22 dq22 ,

(17.24)

where q1 , q2 , q3 are curvilinear coordinates and Q1 , Q2 , Q3 are functions of q1 , q2 , q3 (notice, the metrics will be Euclidian for Q1 = Q2 = Q3 = 1). Beltrami considers six auxiliary quantities θ1 , θ2 , θ3 , ω1 , ω2 , ω3 , which are somehow related to q1 , q2 , q3 , Q1 , Q2 , Q3 and allows him to write the equation [24]:20 δds = λ21 dθ1 + λ22 dθ2 + λ22 dθ3 + λ2 λ3 dω21 + λ1 λ3 dω22 + λ1 λ2 dω23 . (17.25) ds Here λ1 , λ2 , λ3 are the cosines of the angles that the linear elements ds make with the coordinate axes. Then he introduces the following expression for the virtual work: 

(Θ1 dθ1 + Θ2 dθ2 + Θ3 dθ3 + Ω1 dω1 + Ω2 dω2 + Ω3 dω3 ) dS,

(17.26)

where Θ1 , Θ2 , Θ3 , Ω1 , Ω2 , Ω3 are not a priori specified functions of q1 , q2 , q3 . Previous expressions of virtual work allow us to give a mechanical meaning to the terms θ1 , θ2 , θ3 , ω1 , ω2 , ω3 , Θ1 , Θ2 , Θ3 , Ω1 , Ω2 , Ω3 . They are respectively strain

20

pp. 384–385.

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17 Virtual work laws and continuum mechanics

and stress components; this is made clear also from the equilibrium equation derived from the variational problem associated to the previous virtual work expression. The equations obtained by Beltrami are coincident with those given by Lamé: Beltramis results are however independent of Euclid’s V postulate.

17.3.3 Enrico Betti Enrico Betti was born in Pistoia in 1823 and died in Soiana in 1892. He studied at Pisa university as a student of Ottaviano Mossotti. In 1859 Betti was appointed professor in Pisa. In the following year Betti, along with Brioschi and Casorati, visited the mathematical centres of Europe: Göttingen, Berlin and Paris making many important mathematical contacts. In particular in Göttingen Betti met and became friendly with Georg Friedrich Bernhard Riemann (1826–1866). Back in Pisa he moved in 1859 to the chair of analysis and higher geometry. In 1865 Betti was appointed director of the Scuola normale in Pisa, a role that he maintained until his death. Since 1862 Betti was deputy and then senator of the Italian parliament [277]. Betti explored many aspects of mathematical physics; one of the most important was that regarding classical mechanics. At the beginning he assumed a mechanistic approach, where force and not energy was the founding concept and the virtual work the regulating law. In a first work on capillarity [42], Betti assumed bodies as formed by molecules which are attracted to each other at short distance, repelled at very short distance, and which do not practically interact at larger, but still very short distances. In his memoirs on Newtonian forces Betti [41] declared his Newtonian ideology. Indeed he introduced a potential function, but only on mathematical grounds, as a function from which forces can be obtained by derivation. Betti changed his attitude in a second memoir on capillarity [43], by giving the potential an energetic meaning. This change was once and for all in the 1874 Teoria della elasticità [44], where no reference is made to internal forces, even managing to avoid the explicit mention of stress. When Betti wrote the Teoria della elasticità, the theory of elasticity was already mature with known principles, though not completely shared. The exposition develops then as in modern handbooks, following the axiomatic approach. Betti’s principles are on one hand the concepts of potential energy and strains, on the other hand the principle of virtual work. Betti cites the work of William Thomson (1824–1907) to give a thermodynamical basis to potential of elastic forces [226]. Thermodynamics however, at the time, concerned only homogeneous thermal processes, while in the continuum mechanics heterogeneous processes are prevalent. To overcome this difficulty, after Thomson, Betti divides the continuum S into infinitesimal elements dS, each of them considered as homogeneous. The whole potential energy is expressed as a summation of all the infinitesimals. So if P is the density of elastic potential energy, the whole potential

17.3 The Italian school

energy for S is given by: Φ=

393



PdS.

(17.27)

Then Betti, like Thomson and Green, assumes that P depends on infinitesimal strains, components of which he neglects any power higher than the second, by obtaining the following quadratic expression: P = ∑∑ Ars ar as ,

(17.28)

where Ars are constitutive constants and ar , as are the generic components of strain. In the derivation of the equations of equilibrium, as indeed throughout the book, stresses are not introduced in any way. Betti writes down the equilibrium equation by means of the virtual work principle: Let X,Y, Z be the components of the accelerating forces [forces per unit mass] that act on each point of the body, L, M, N the components of the forces acting on each point on the surface of it, and ρ the constant density. Let any point of the body take a virtual motion and denote by δu, δv, δw the changes that will take for his motion u, v, w. The work made in this motion by the given forces will obviously be:  S

(Xδu +Y δv + Zδw)dS +

 σ

(Lδu + Mδv + Mδw)dσ

being S the space occupied by the body and σ the surface. The work made by the elastic forces will be equal to the increase of the potential of the whole body: Φ=



PdS

then, for the principle of Lagrange: δΦ +

 S

(Xδu +Y δv + Zδw)dS +

 σ

(Lδu + Mδv + Mδw)dσ = 0

[44].21 (A.17.9)

He then develops the variation δΦ to which he applies Green’s formula, without any comment or reference, to obtain an expression of the work of internal forces where strains are replaced by δu, δv, δw. He obtains in this way three internal equilibrium equations and three more boundary equations. For the sake of compactness I will write down only one of the first kind and another of the second: d dP d dP d dP ρX = + + dx da dy 2dh dz 2dg (17.29) dP dP dP α+ β+ γ. L= da 2dh 2dg Here x, y, z are the Cartesian coordinates of a generic point and a, h and g are the components of strain [44].22 Notice that these equations, as in Navier, depend on strains – as P is a function of strains – instead of on stresses as usual. 21 22

pp. 20–21. p. 19.

18 Thermodynamical approach

Abstract. This chapter is devoted to the use of the VWL where to virtual work is given a mechanical meaning. In the first part, notes on mechanics derived from Pierre Duhem’s thermodynamics are presented. In the second part the derivation of a VWL from the law of conservation of energy is shown. In the second half of the XIX century, mechanics no longer seemed to be the paradigm for all of physics. Not so much because of internal reasons due for example to the presence of contradictions and vagueness, but for external reasons as to its difficulty in explaining the new phenomena that were the subject of the nascent physical disciplines such as thermodynamics and electromagnetism. In particular, the idea of force as a fundamental quantity of physics found itself in difficulty. For many scientists it had to be replaced, or at least allied with energy. And the mechanistic explanatory model had to be replaced by a less challenging one at the metaphysical level, on the example of thermodynamics. Among the promoters of this tendency to be remembered are William John Macquorn Rankine (1820–1872) [204] Ernst Mach, Hermann Ludwig Ferdinand von Helmholtz (1821–1894), Pierre Maurice Marie Duhem (1858–1916), Wilhelm Ostwald (1853–1932) [371]. In particular, the latter author was a supporter of the energetics that took the form of a philosophical movement in which energy was seen as a substance. In this chapter there is no claim to comment on philosophical and scientific matters but only to see how the concepts of work and energy of thermodynamics can provide a new foundation of the laws of mechanics, in particular the laws of virtual work. To do this I will refer here only to the contribution of Duhem, beginning with a summary of his ideas about mechanics: The attempt that aims to reduce all Physics to rational Mechanics, which was always a futile attempt in the past, is it intended to pass a day? A prophet alone could answer affirmatively or negatively to this question. Without prejudging the direction of this response, it seems wiser to abandon, at least provisorily, these fruitless efforts toward the mechanical explanation of the Universe.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective. DOI 10.1007/978-88-470-2056-6_18, © Springer-Verlag Italia 2012

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We will try to formulate general laws for bodies to which all physical properties must obey, without assuming a priori that these properties are all reducible to geometry and local movement. The body of this general laws no longer will reduce to rational Mechanics. […] Rational Mechanics must therefore result from the body of general laws that we propose to hold; it must be what one gets when applying these general laws to particular systems where one considers only the figure of bodies and their local motion. The code of general laws of Physics is now known by two names: the name of Thermodynamics and the name of Energetics [99].1 (A.18.1)

18.1 Pierre Duhem’s concept of oeuvre Pierre Maurice Marie Duhem was born in Paris in 1861 and died in Cabrespine in 1916. He was one of the greatest scientists, philosophers and historians of his period. Duhem had the misfortune of being an enemy of the chemist Marcellin Berthelot (1827–1904) who became minister of public instruction, precluding him from a brilliant academic career. As a physicist, he championed ‘energetics’, holding generalized thermodynamics as foundational for physical theory, that is, thinking that all of chemistry and physics, including mechanics, electricity, and magnetism, should be derivable from first principles of thermodynamics. As a historian he was a supporter of the continuous development of science; in particular he gave great importance to medieval science. As a philosopher of science he argued that a scientific theory does not require a justification that goes beyond the control of its internal consistency and accuracy in the prediction of experimental results. Duhem’s interests fell roughly into periods. Thermodynamics and electromagnetism predominated between 1884 and 1900, although he returned to them in 1913–1916. He concentrated on hydrodynamics from 1900 to 1906. His interest in the philosophy of science was mostly in the period 1892–1906, and in the history of science from 1904 to 1916, although his earliest historical papers date from 1895 [290]. Duhem did not directly place a virtual work law as a principle of mechanics but rather as a theorem derived from the principle of conservation of energy rooted in a general physics called by him either thermodynamics or energetics. The term thermodynamics refers directly to the history of mechanical practice. Its two most basic principles, that of Sadi Carnot – to transform heat into work, there must be bodies at different temperatures – and the principle of conservation of energy, were discovered by studying the power of steam. The name of energetics is due to Rankine, energy being the first quantity to be defined, on which most other notions are based. Below I will present Duhem’s ideas as reported in the Traité d’énergétique ou de thermodynamique générale [99]. More precisely, for simplicity, I only will refer to 1

pp. 2–3.

18.1 Pierre Duhem’s concept of oeuvre

397

the first volume devoted to non-dissipative systems. The text of Duhem represents one of the first attempts to establish a physical theory on a modern axiomatic basis. The author believes that the principles of a physical theory do not require any justification beyond the assessment of the internal consistency. He considers, however, that the principles cannot be chosen at random, but that to arrive at a satisfactory theory one must benefit from formulations of past similar principles; in this way the history of science becomes an integral part of science. The most important and original concepts of Duhem in his energetic approach to mechanics, after those of space and time, are that of virtual transformation and activity (oeuvre). Below in commenting on the introduction of these concepts I will take for granted some basic notions of thermodynamics including that of state.

18.1.1 Virtual transformations Virtual transformations are defined as operations performed completely in the mind, which submit to a mathematical scheme that serves to represent the system under examination, imagining a continuous succession of states. The only requirement the transformations should satisfy is the respect of constraints affecting the essence of the system, while experimental laws may be violated. They occur in a hyperuranic time, or even better without time. Consider a continuous sequence of states of the same system, we fix the attention on these different states in the order that allows us to switch between them continuously. To identify this intellectual operation to which we submit all the mathematical schemes used to represent the set of concrete bodies, we say we impose on the system a virtual change. […] Changes in the numerical values of the variables used to define the state of the system must be compatible with the conditions that logically result from the definition of the system, but only with these conditions. And in particular, the changes in numerical values may well contradict the experimental laws that govern the system of all the concrete bodies that our abstract mathematical system has the duty to represent [99].2 (A.18.2)

Duhem uses another term to describe changes that meet the conditions of constraints but not necessarily the experimental laws and take place in time: ideal transformations: Do not confuse an ideal variation with virtual variation; virtual variation is composed of virtual configurations of the system that do not succeed in time, so that the change of configuration which is a virtual variation is not related to motion. In the virtual variation the variation of velocity has no reason to exist [99].3 (A.18.3)

Regarding velocity Duhem introduces the difference between local velocities, which are velocities in the standard meaning, i.e. the derivative of space with respect to time, and general velocities, defined as the quantity which, once the state of the system is known, allows one to evaluate all the local velocities and the derivative with 2 3

pp. 46–47. p.84.

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respect to time of all the state variables. For example for the mechanical system constituted by a rigid body, the general velocities are furnished by the three components of the velocity of the center of gravity and of the three Eulerian angles. Ideal transformations, when restricted to mechanics, coincide with virtual displacements as conceived before Lazare Carnot, virtual transformations with Carnot’s geometric motion (at least for bilateral constraints). In virtual transformations it makes no sense to talk about velocity (in the physical sense), in ideal transformation it does. From the above it is evident that ideal transformations are also virtual transformations (just ignore time); the contrary is not true. The example Duhem presents for virtual transformations concerns a mixture of hydrogen, oxygen and water. Suppose that the state of a system is defined by the temperature θ and a variable x, which expresses the percentage of water vapour. There is an experimental law that at chemical equilibrium x relates to θ; be it x = f (θ); in the system there are no conditions of constituent constraints. In a virtual variation it is possible to leave from a pair x0 , θ0 that satisfies the empirical law and to vary x above or below the value x0 while maintaining θ = θ0 . This transformation is purely intellectual and may not happen in practice.

18.1.2 Activity, energy and work The definition of activity is much more complex. It in fact is not an explicit definition, but rather an implicit one as usual in the modern axiomatic theories. Duhem formulates this ‘definition’: Thus, when a system is transformed in the presence of external bodies, we admit that these external bodies contribute to the transformation, either by causing it or facilitating it or blocking it, and this contribution we call the activity in the transformation of the system, by the bodies outside that system [99].4 (A.18.4)

The activity has not necessarily a mechanical nature. It can have any nature, for example, maybe the administration of electrical current. However Duhem himself recognizes that this definition is “too obscure, vague and mostly impregnated with anthropomorphism" [99].5 To eliminate these defects he declares that the activity should be considered as a scalar physical quantity to be represented with an appropriate algebraic symbol to perform calculations. He then makes a number of stipulations/conventions on how to assign it a numerical value. I quote here the first stipulation only to give an idea of the level of abstraction on which Duhem moves. First convention. The mathematical symbol which should represent the activity, in a real or ideal transformation, will be defined any time the nature of the system and of the transformation it undertook is known. It will not change if the place and time of the system and the external bodies, where the transformation has occurred, change [99].6 (A.18.5) 4 5 6

p. 81. p. 81. p. 87.

18.1 Pierre Duhem’s concept of oeuvre

399

Notice that Duhem’s ‘conventions’ refer to ideal and real changes but not the virtual ones, because they only take into account the general velocity. Although Duhem declares that his conventions should be considered as arbitrary and therefore do not require any justification, their reading makes it manifest that in introducing the concept of activity Duhem is generalizing that of mechanical work. In particular, he assumes the additivity and the independence of paths: if G1 , G2 , . . . Gn are the activities carried out on a system after the transformations M1 , M2 , . . . Mn and if a unique transformation M is imagined that starts from the initial state to the final due to these transformations, the overall activity, G, is the sum of all activities, G = G1 + G2 + · · · + Gn , and does not depend on the order in which the transformations occur but only on initial and final states. Note that this rule applied to a mechanical system subject to internal or external forces, when activity is identified with work, requires the field of forces to be conservative. Duhem does not give particular emphasis to this fact, or better he does not even make it explicit. Once the requirements which an activity must satisfy are defined, Duhem introduces the concept of total energy. If G(e0 , μ0 , e, μ) is the activity required of a system to pass from one arbitrarily chosen reference e0 and global velocity μ0 to the generic state e and global velocity μ, the total energy of the system in the state e and global velocity is μ, as in the expression: E(e, μ) = G(e0 , μ0 , e, μ).

(18.1)

He proceeds by saying that the following relation holds true: G(e1 , μ1 ; e2 , μ2 ) = E(e2 , μ2 ) − E(e1 , μ1 );

(18.2)

i.e. the activity to go from (e1 , μ1 ) to (e2 , μ2 ) is equal to the difference between the total energies. At this point Duhem can formulate a principle of conservation of energy, which he qualifies as a hypothesis: Principle of conservation of energy. When any system, isolated in the space, undergoes a whatever real variation, the total energy of the system maintain an invariable value [99].7 (A.18.6)

The principle of conservation of energy is not a convention such as all those introduced to define the activity; it receives its validity from the experience. The principle of conservation of energy does not apply to all the ideal changes but only to those that are also real. The total energy principle formulated above seems to Duhem too general; he then submits it to two restrictions that exclude its validity from important fields of physics, including electrical systems. The first restriction requires that the total energy of the system does not change as a result of a simple translation in space [99].8 The second 7 8

p. 93. p. 97.

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restriction requires that the total energy is composed of two terms [99]:9 U(e, μ) = U(e) + K(λ).

(18.3)

The first term which depends only on the state e, is named potential energy or internal energy; the second which depends on the local velocity λ (understood in the classical sense) of all system components, is named kinetic energy. For systems made of infinitely small parts, Duhem attains the traditional representation for kinetic energy: K=

1 2



(u2 + v2 + w2 )dm,

(18.4)

where u, v, w are the components of velocity of the mass element dm of the system. With these restrictions the previous theorem of the total energy takes the more traditional form: Restricted form of the principle of conservation of energy. In any real variation of an isolated system the following equality: U+

1 2

 M

u2 + v2 + w2 dm = const.

is verified [99].10 (A.18.7)

Consider now two independent mechanical systems Sa and Sb . The system Sa , taken alone in the configuration A, has the potential energy U(A), the system Sb taken alone in the configuration B, has the potential energy U(B). The two systems Sa and Sb , taken together to form the system S, have the total potential energy U, different from the sum of potential energies of Sa and Sb . In all generality it can be assumed that: U = Ua (A) +Ub (B) + Ψab (A, B)

(18.5)

where Ψab (A, B) is called mutual potential energy. Consider then a virtual infinitesimal variation δa , generic but limited only to the system Sa , while the state of the system Sb remains unchanged. There is the following variation of the total energy induced by bodies external to S: δaU = δaUa (A) + δa Ψab (A, B).

(18.6)

δa L = −δa Ψab (A, B)

(18.7)

The quantity:

represents the activity accomplished by the system Sa on the system Sb in the virtual infinitesimal variation δa ; to it Duhem refers as the infinitesimal virtual work that Sb makes on Sa . The work as defined above is very similar to the activity, because it is furnished by a variation of the energy. It however represents the variation of a limited part of 9

pp. 97–98. p. 113.

10

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401

energy. To reach the concept of force it is enough to consider that the infinitesimal virtual work can always be expressed as: δL = −δΨab (A, B) = ∑ fk δqk

(18.8)

where qk are the variables that define the system configuration and fk = ∂ Uab /∂ qb appropriate coefficients. Duhem refers to these coefficients as the actions of Sa on system Sb in order to reserve the term force for when qk have the meaning of displacements. In this case work is mechanical work. Note how Duhem has introduced the concept of mechanical work. This is not a primitive quantity, but one derived from the activity, which has a primitive character. It is difficult to agree that the concept of activity could be more primitive than that of work.

18.1.3 Rational mechanics 18.1.3.1 Free systems After these preliminaries Duhem proceeds to particularize the laws of energetics to mechanics. He does it in Chapter 5, entitled: La mécanique des solides invariables et la mécanique rationelle [99].11 The first consideration Duhem makes is that for a mechanical system, for example consisting of two bodies, the potential energies U(A) and U(B) are constant, then the potential energy of the system can be provided only by the mutual potential energy, i.e.: U = Ψ.

(18.9)

In case external forces act on the system, the principle of conservation of energy is no longer valid; it is replaced by Duhem with the following principle: T + τ − δΨ = 0,

(18.10)

where T is the virtual work made by the external forces, τ the virtual work made by the forces of inertia. On the basis of relation (18.10) Duhem can affirm: Comparison of the conditions […] provides the following statement, which is the principle of d’Alembert. To obtain for each moment the laws of motion of a system of rigid bodies without passive resistance, simply require that the system remains in equilibrium if it is placed motionless in the state is passing in that moment and submit it not only to external actions that are actually carrying on it, when it is in this state, but also to the fictitious external action equivalent to the inertia actions in the body at that time [99].12 (A.18.8)

In case equilibrium is concerned, the virtual work of the inertia forces is zero, therefore the previous relation provides: T − δΨ = 0. 11 12

pp. 183–246. p. 242.

(18.11)

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This expression represents a law of virtual work: For a system subjected to bilateral constraints [with no passive resistance], it is enough for the equilibrium that the virtual work of the external forces is at most equal to the increase in potential energy [i.e. the virtual work of internal forces][99].13 (A.18.9)

18.1.3.2 Constrained systems The introduction of constraints on the components of the system Sa does not significantly modify the framework outlined above. Duhem says to consider constraints in a purely geometric way (first restriction) “All constraints have this common character and to define them, it is useless to appeal to any notion alien to Geometry ” [99].14 But he is not consistent and equips them with the property of the absence of passive resistance (second restriction), in the sense that: The compulsion that the system receives from the constraints, in the case of its actual displacement is less than the compulsion that it receives from all the virtual displacements originated from the same state: 

2

MN dm
0; quindi, sommando tutti i lavori parziali relativi agli altri punti del sistema che effettivamente si muova, avremo anche:

∑ Fi dri > 0,

(1)

dato che la somma è formata tutta di termini non negativi e di cui almeno uno, per ipotesi, non è nullo. (a) Ma Fi = Fi + Ri , per cui riscriviamo la (1) come: (a)

∑(Fi

+ Ri ) · dri > 0.

(2) (a)

A questo punto si fa l”ipotesi dei vincoli lisci e si ottiene l’assurdo ∑ Fi contro l’ipotesi.

+ Ri ) · dri > 0, perché

2.3 En effet, on démontre que, si un point n’a d’autre liberté dans l’espace, que celle de se mouvoir sur une surface ou sur une ligne fixement arrêtée, il n’y peut être en équilibre, à moins que la résultante des forces qui le sollicitent se soit perpendiculaire a cette surface ou à cette ligne courbe. 2.4 La force de pression d’un point sur une surface lui est perpendiculaire, autrement elle pourrait se décomposer en deux, l’une perpendiculaire à la surface, et qui seroit détruite par elle, l’autre parallèle à la surface, et en vertu de la quelle le point n’aurait point d’action sur cette surface, ce qui est contre la supposition. 2.5 Or si l’on fait abstraction de la force P, et qu’on suppose que le corps soit forcé de se mouvoir sur cette surface, il est claire que l’action, o plutôt la résistance que la surface oppose au corps ne peut agir que dans une direction perpendiculaire à la surface.

A.3 Chapter 3

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A.3 Chapter 3 3.1 ‘Ambiguità’ che del resto si potrebbe riguardare come un documento glottologico della primordialità della credenza a una connessione, tra le diverse velocità compatibili dei vari punti le cui posizioni dipendono le une dalle altre, e la diversa facilità colla quale i punti stessi possono essere mossi a parità delle altre condizioni. 3.2 Percioché, se Aristotile risolve per qual cagione la leva lunga muove più facilmente il peso, dice avvenir ciò per la lunghezza maggiore dalla parte della potenza che muove; e ciò benissimo secondo il suo principio nel quale suppone, che quelle cose che sono in maggior distanza dal centro, si muovano più facilmente e con maggior forza: del che reca egli la causa principale nella velocità secondo la quale il cerchio maggiore supera il minore. È vera dunque la causa, ma indeterminata; perciochè non so io per tanto, dato un peso, una leva, et una potenza, come io li abbia da dividere la leva nel punto ove ella gira, acciochè la data potenza bilanci il dato peso. Ammesso dunque Archimede il principio d’Aristotile passò più oltre; nè si contentò che maggiore fosse la forza dalla parte della leva più lunga, ma determinò quanto ella deve essere, cioè con qual proportione ella deve. 3.3 Il secondo principio è, che il momento e la forza della gravità venga accresciuto dalla velocità del moto; sì che pesi assolutamente eguali, ma congiunti con velocità diseguali, sieno di forza, momento e virtù diseguale, e più potente il più veloce, secondo la proporzione della velocità sua alla velocità dell’altro. [...] Tal ragguagliamento tra la gravità e la velocità si ritrova in tutti gli strumenti meccanici, e fu considerato da Aristotile come principio nelle sue Questioni meccaniche: onde noi ancora possiamo prender per verissimo assunto che pesi assolutamente diseguali, alternatamente si contrappesano e si rendono di momenti eguali, ogni volta che le loro gravità con proporzione contraria rispondono alle velocità de’ moti, cioè che quanto l’uno è men grave dell’altro, tanto sia in constituzione di muoversi più velocemente di quello. 3.4 Ma per non confonderci dichiarerò prima il principio del quale parla Aristotele, et insieme delle machine delle quali habbiamo dato gli esempi, cioè delle taglie et della stanga. [...] Tutte le operazioni delle machine adunque consistono nel movimento loro, et per conseguente l’istessa machina farà maggiore et minore effetto quanto più propinquo sarà il movimento che se le farà fare al suo proprio. [...] Resta adunque per manifesto dalle dimostrazioni precedenti che quanto meno un peso obligato à muoversi in giro, ò una forza s’allontana dal centro, con tanto maggior velocità si moverà et la forza tanto maggiore effetto farà. 3.5 Il a dit que centre de gravité ou d’inclinaison est un point tel que, lorsque le poids est suspendu par ce point, il est divisé en deux portions équivalentes. A la suite de cela Archimède et les mécaniciens qui l’ont imité, ont scindé cette définition, et ils ont distingué le point de suspension du centre d’inclinaison. 3.6 Quelques-uns ont pensé à tort que la proportion existant dans l’état d’équilibre n’était plus vraie dans le cas d’un fléau irrégulier. Supposons un fléau de balance n’ayant pas partout même poids ni même épaisseur, et fait de matière quelconque; il est en équilibre lorsqu’on le suspend au point γ; nous entendons ici par équilibre l’arrêt du fléau dans une position stable, quand bien même il serait incliné dans un sens ou dans un autre. Suspendons ensuite des poids à des points quelconques du fléau; soient δ et  ces points; le fléau reprend une position d’équilibre après que les poids ont été suspendus; et Archimède a démontré que, dans ce cas encore, le rapport des poids est égal au rapport inverse des distances respectives. 3.7 Il est nécessaire d’expliquer comment on soutient, comment on porte et transporte les corps graves, avec les développements convenables pour une introduction. Archimède a traité cette matière avec un art très sûr dans son livre appelé Livre des Supports. 3.8 Supposons deux cercles avant un même centre α; soient leurs diamètres les deux lignes βγ, δ; ces deux cercles sont mobiles autour du point α, qui est leur centre commun, et perpendiculaires

408

Appendix. Quotations au plan de l’horizon. Suspendons aux deux points β, γ deux poids égaux, désignés par η et ζ. Il est évident que les cercles ne penchent ni d’un côté ni de l’autre, puisque les deux poids ζ et η sont égaux et les distances βα, αγ égales. Faisons de βγ un fléau de balance mobile autour d’un point de suspension qui est le point α. Si nous transportons en  le poids qui est appliqué en γ, le poids inclinera ζ vers le bas, et il fera tourner les cercles. Mais si nous augmentons le poids θ, il fera de nouveau équilibre au poids ζ et le rapport du poids θ au poids ζ sera égal au rapport de la distance βα à la distance α. Ainsi la ligne β joue le rôle d’un fléau de balance mobile autour d’un point de suspension, qui est le point α. Archimède a déjà donné cette proposition dans son livre sur l’équilibre entre les poids.

3.9 Les cinq machines simples qui meuvent le poids se ramènent à des cercles montés sur un seul centre; c’est ce que nous avons démontré sur les diverses figures que nous avons précédemment décrites. Je remarque pourtant qu’elles se réduisent encore plus directement à la balance qu’aux cercles; on a vu en effet que les principes de la démonstration des cercles ne nous sont, venus que de la balance; on démontre que le rapport du poids suspendu au petit bras de la balance, au poids suspendu au grand bras, est égal au rapport du grand bras au petit. 3.10 Imaginons au contraire un autre poids au point ζ, et fixons-y une poulie η; faisons entrer dans cette poulie une corde, et attachons-en les deux extrémités à un support fixe, en sorte que le poids ζ demeure suspendu. Chacun des deux brins de la corde sera tendu par la moitié du poids; et si l’on délie l’un des deux bouts de la corde, celui qui est attaché au point κ, et qu’on continue à maintenir la corde dans la même position, on aura à porter la moitié du poids. Le poids se trouve donc être double de la puissance qui le retient. 3.11 Cet instrument et toutes les machines de grande force qui lui ressemblent sont lents, parce que, plus est faible la puissance comparée au poids très lourd qu’elle meut, plus est long le temps que demande le travail. Il y a un même rapport entre les puissances et les temps. Par exemple, lorsqu’une puissance de 200 talents a été appliquée au tambour β, et quelle a mis le poids en mouvement, il faut un tour entier de β pour que le poids se meuve de la longueur de la circonférence de l’arbre γ. Si le mouvement est donné à l’aide du tambour δ, il faut que l’arbre γ tourne cinq fois pour que l’arbre fasse un seul tour, puisque le diamètre du tambour est cinq fois celui de l’arbre γ et que cinq tours de γ valent un tour de β. Cette remarque se renouvelle pour la suite des organes du train, soit que nous fassions les arbres égaux entre eux ainsi que les tambours, soit que nous leur donnions des rapports variés, connue ceux que nous avons choisis. Le tambour δ fait mouvoir le tambour β et les cinq tours que doit effectuer le tambour δ prennent cinq fois le temps d’un seul tour; 200 talents, d’autre part, valent cinq fois 40 talents. Ainsi le rapport du poids à la force motrice est égal à l’inverse du rapport d’ensemble des arbres et des tambours, quelque nombreux qu’ils soient. Cela achève la démonstration. 3.12 Cet instrument et toutes les machines de grande force qui lui ressemblent sont lents, parce que, plus est faible la puissance comparée au poids très lourd qu’elle meut, plus est long le temps que demande le travail. Il y a un même rapport entre les puissances et les temps. […] Le ralentissement de la vitesse a lieu aussi dans cette machine. 3.13 Supposons, par exemple, que le support. stable auquel le poids est suspendu soit α. La corde est la ligne αβ. Menons la ligne αγ perpendiculaire sur la ligne αβ, et marquons sur la ligne αβ deux points quelconques que nous désignons par les lettres δ, . […] Donc, quand nous tirons le poids à partir du point , il vient en κ, et quand nous le tirons à partir du point δ, il vient en η. Ainsi on élève davantage le poids en partant du point δ qu’en partant du point ; et pour porter le poids plus haut, il faut une plus grande force que pour le porter moins haut, parce que, pour le porter dans un lieu plus élevé, il faut un temps plus long. 3.14 Nous aurons recors à quelque puissance ou à quelque poids appliqué de l’autre côte, pour faire d’abord équilibre au poids donné, afin qu’un excès de puissance l’emporte sur de poids et le tire en haut.

A.4 Chapter 4

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A.4 Chapter 4 4.1 I. Dico ergo quod omnium duorum spaciorum que duo mota secant in tempore uno, proportio unius ad alterum est sic ut proportio virtutis motus eius quod secat spacium unum ad virtutem motus illius secantis spacium alterum. Et ponam ad illud exemplum. Dico duorum viatorum perambulat unus 30 miliaria et perambulat secundus 60 miliaria in tempore uno. Et notum est ergo quod virtus motus eius qui perambulat 60 miliaria dupla est virtutis motus eius qui perambulat 30 miliaria sicut spacium quod est 60 miliaria est duplum spacii quad est 30 miliaria. Hec est propositio recepta per se, inter quam et inter intellectum non est medium separans ea. 4.2 III. Cum ergo iam manifestum est istud, tunc dico quod omnis linea que dividitur in duas sectiones diversas et extimatur quod linea suspendatur per punctum dividens ipsam, et quod duorum ponderum proportionalium sicut [invers]1 proportionalitas duarum partium linee unius ad comparem suam secundum attractionem suspenditur unum in extremitate unius duarum sectionum et secundum in extremitate altera, tunc linea equatur super equidistantiam orizontis. 4.3 II. Et post hoc dico quod omnis linea que dividitur in duas sectiones, et figitur punctum eius secans et movetur linea tota penitus motu quo non redit ad locum suum, tunc ipsa facit accidere duos sectores similes duorum circulorum, medietas diametri unius quorum est linea longior et medietas diametri secundi est linea brevior et quod proportio arcus quem signat punctum extremitatis unius duarum linearum ad arcum quem signat punctum extremitatis linee secunde est sicut proportio linee revolventis illum arcum ad lineam secundam. 4.4 Iam diximus in duobus spaciis que secant duo mota in tempore uno quod proportio virtutis motus unius eorum ad virtutem motus alterius est sicut proportio spacii quod ipsum secat ad spacium alterum, et punctum A apud motum linee iam secavit arcum AT, et punctum B iam secavit etiam apud motum linee arcum BD et illud in tempore uno. Ergo proportio virtutis motus puncti B ad virtutem motus puncti A est sicut proportio duorum spaciorum que secuerunt duo puncta in tempore uno, unius ad alterum, scilicet proportio arcus BD ad arcum AT. Et hec proportio iam ostensum est quod est sicut proportio linee GB ad lineam AG. 4.5 Cuius hec est demonstratio, secabo ex BG longiore quod sit equale AG breviori quod sit GE. Si ergo suspendantur super duo puncta A. E duo pondera equalia, equidistabit linea AE orizonti, quoniam virtus motus duorum punctorum est equalis, secundum quod ostendimus, donec si inclinaverimus punctum A ad punctum T sufficiet cum eo pondus quod est ad punctum A donec redeat ad locum suum, et sit arcus AT. Et quando permutabimus pondus ex puncto E ad punctum B, et si voluerimus ut linea remaneat super equidistantiam orizontis est nobis necesse ut addamus in pondere quod est apud A additionem aliquam donec sit proportio eius totius ad pondus quod est apud B sicut proportio BG ad AG. Quoniam virtus puncti B superfluit super virtutem puncti A per quantitatem superfluitatis BG super AG, secundum quod iam ostendimus, pondus ergo quod est apud punctum fortioris est minus pondere quod est apud punctum debilioris secundum quantitatem qua proportionatur arcus arcui. Cum ergo est apud punctum B pondus et est apud A pondus secundum et est proportio ponderis a ad pondus b sicut proportio BG ad AG, equidistat linea ab orizonti. 4.6 Je dis que si l’on suspend ab au point g et que l’on applique à ses deux extrémités, en a et b, deux poids proportionnels et équivalents à ses deux segments, [ab] sera parallèle à l’horizon. En effet, prenons sur le [côté] le plus long ag un [segment] gd égal à gb. Si on applique en d un poids égal au poids appliqué en b, [ab] sera parallèle à l’horizon. Si on incline alors vers le bas [le poids qui est en d], le poids qui est en b le soulèvera et lui fera parcourir l’arc dd égal à l’arc bb, car gd est égal à gb. Si nous déplaçons alors le poids du point d au point a, [celui-ci étant dans la position] inférieure, et que nous voulions le soulever jusqu’à [la position] supérieure 1

In the Latin text here there is an “editing” mistake, corrected by Moody and Clagett.

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Appendix. Quotations de a, il nous faudra augmenter le poids qui est en b de telle sorte que le rapport du [poids] total [en b] au poids qui est en a soit égal au rapport de l’ara aa à l’arc dd. lesquels sont parcourus en même temps alors qu’ils sont inégaux. Or ce rapport est égal au rapport de l’un des deux segments de la droite a l’autre.

4.7 Si l’axe est pesant et s’il est divisé en deux segments inégaux, on augmente l’épaississeur du segment le plus court jusqu’a ce que l’axe soit parallèle a l’horizon. […] On est alors ramené au cas déjà traité de la ligne dépourvue de poids. 4.8 Considérons un levier où la puissance est α où la résistance est β; cette résistance se trouvait à une certaine distance du point d’appui, supposons que la puissance α la puisse mouvoir et lui faire décrire en un temps δ l’arc γ; elle pourra également mouvoir le poids α/2, placé à une distance double du point d’appui, car dans les même temps δ, et le lui fera parcourir l’arc 2γ. Il faut donc la même puissance pour mouvoir un certain poids, place a une certaine distance du point d’appui, et pour mouvoir un poids moitié placé à une distance double. De là, on tire aisément la justification de la théorie du levier donnée dan les Questions mécaniques. 4.9 Table 4.1 S1 Omnis ponderosi motum esse ad medium virtutemque ipsius esse potentia ad inferiora tendendi et motui contrario resistendi. S2 Quod gravius est velocius descendere. S3 Gravius esse in descendendo, quanto eiusdem motus ad medium rectior. S4 Secundum situm gravius esse, cuius in eodem situ minus obliquus descensus. S5 Obliquiorem autem descensum in eadem quantitate minus capere de directo. S6 Minus grave aliud alio secundum situm, quod descensum alterius sequitur contrario S7 Situm aequalitatis esse aequalitatem angulorum circa perpendiculum, sive rectitudinem angulorum, sive aeque distantiam regulae superficiei Orizontis. 4.10 Omnis ponderosi motum esse ad medium virtutemque ipsius esse potentia ad inferiora tendendi virtutem ipsius, sive potentia possumus intelligere longitudinem brachij librae, aut velociter eius quem probatur ex longitudine brachij librae, et motui contrario resistendi. 4.11 Patet ergo quod major est violentia in motu secundum arcum maiorem, quam secundum minorem; alias enim non fieret motus magis contrarius. Cum ergo apparet plus in descensu adquirendum impedienti, patet quia minor erit gravitas secundum hoc. Et quia secundum situationem gravium sic fit, dicatur gravitas secundum situm in futuro processo. Ita enim, sillogizando de motu tamquam motus sit causa gravitatis vel levitatis, potius per motum magis contrarium concludimus causam huiusmodi contrarietatis esse plus contrariam, id est, plus habere violentie. Quod quidem grave descendat, hoc est a natura; sed quod per lineam curvam, hoc est contra naturam, et ideo iste descensos est mixtus ex naturali et violento. In ascensu vero ponderis, cum ibi nihil sit secundum naturam, debet arguì sicut de igne, quoniam nihil naturaliter ascendit. De igne enim arguitur in ascensu, sicut de gravi in descensu; ex quo sequitur quod grave, quanto plus sic ascendit, tanto minus habet de levitate secundum situm, et sic plus habet de gravitate secundum situm. 4.12 Table 4.2 P1 P2 P3 P4 P5 P6

Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio. Cum equilibris fuerit positio equalis, equis ponderibus appensis ab equalitate non discedet: et si a rectitudine separetur, ad aequalitatis situm revertetur. Si vero inequalia appendantur, ex parte gravioris usque ad directionem declinare cogetur. Omne pondus in quamcunque partem ab aequalitate discedat, secundum situm fit levius. Cum fuerint appensorum pondera equalia, non faciet nutum in equilibri appendiculorum inequalitas. Si brachia librae fuerint inequalia, equalibus appensis ex parte longiore nutum faciet. Si fuerint brachia libre proportionalia ponderibus appensorum, ita ut in breviori gravius appendatur, eque gravia erunt secundum situm appensa.

A.4 Chapter 4 P7 P8 P9 P10

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Si duo oblonga per totum similia, et quantitate, et pondere equalia, appendantur ita ut alterum dirigatur, alterum orthogonaliter dependeat, ita etiam ut termini dependentis et medii alterius eadem sit a centro distantia, secundum hunc situm eque gravia fient. Si inequalia fuerint brachia libre, et in centro motus angulum fecerint, si termini eorum ad directionem hinc inde equaliter accesserint, equalia appensa in hac dispositione aequaliter ponderabunt. Equalitas declinationis identitatem conservat ponderis. Si per diversarum obliquitatum vias duo pondera descendant, fueritque declinationum et ponderum una proportio eodem ordine sumpta, una erit utriusque virtus in descendendo.

4.13 Table 4.3 Inter quaelibet gravia est velocitatis in descendendo et ponderis eodem ordine sumpta proportio, descensus autem et contrarii motus proportio eadem sed permutata. INTER QUAELIBET DUO GRAVIA EST VELOCITATIS IN DESCENDENDO PROPRIE, ET PONDERIS EODEM ORDINE SUMPTA PROPORTIO, DESCENSUS AUTEM ET CONTRARII MOTUS PROPORTIO EADEM SED PERMUTATA. Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio. 4.14 Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio. Sint pondera ab et c, levius c. Descendatque ab, in D, et c, in E; itemque pellatur ab, sursum in F, et c, in H. Dico ergo quod AD ad CE, sicut ab ponderis ad c pondus, quanta enim virtus ponderosi tanta descendendi velocitas. Atqui compositi virtus ex virtutibus componentium componitur. Sit ergo a aequale c, que igitur virtus a eadem et c. Si ergo proportio ab ad c, minor quam virtutis ad virtutem, erit similiter proportio ab ad a, minor proportio quam virtutis ab ad virtutem a. Ergo virtutis ab, ad virtutem b, minor proportio quam ab ad b; similium ergo ponderum minor et maior proportio, quam virtutum. Et quia hoc inconveniens erit, utrobique eadem; ideoque ab ad c, sicut AD ad CE, et econtrario sicut CH ad AF. 4.15 Ponatur item quod submittatur ex parte b, et ascendat ex parte c, dico quoniam redibit ad aequalitatem. est enim minus obliquus descensus a, ad aequalitatem, quam a, b, versus e. Sumantur enim sursum arcus aequales, quantumlibet parvi qui sint c, d, et b, g, et ductis lineis ad aequidistantiam aequalitatis, quae sint, c, z, l, et d, m, n. Item b, k, m, g, y, t, dimittatur orthogonaliter descendens diametrum quae sit f, r, z, m, a, k, y, e, erit quod z, m, maior k, y, quia sumpto versus f, arcu ex eo quod sit aequalis c, d, et ducta ex transverso linea x, r, s, erit r,z, minor z, m, quod facile demonstrabis. Et quia r, z, est aequalis k, y, erit z, m, maior k, y. Quia igitur quilibet arcus sub c, plus capiat de directo quam ei aequalis sub b, directo est descensus a, c, quam a, b, et ideo in altiori situ gravius erit c, quam b, redibit ergo ad aequalitatem. 4.16 Propositio VI Si fuerint brachia librae proportionalia ponderibus appensorum, ita ut in breviori gravius appendatur, eque grauia erunt secundum situm appensa. Sit ut prius regula ACB, appensa a et b; sitque proportio b ad a, tamquam AC ad BC. Dico quod non mutabit in aliquam partem libra. Sit enim ut ex parte B, descendat, transeatque in obliquum linea DCE, loco ACB. Et appensa d, ut a, et e, ut b, et DG, linea orthogonaliter descendat, et EH ascendat, palam autem quoniam trianguli DCG et ECH similes sunt, quare proportio DC ad CE que DG, ad EH. Atqui DC ad CE sicut b, ad a ; ergo DG ad EH, sicut b ad a. Sit igitur CL equalis CB, et CE, et l, equum b, in pondere, et descendat perpendicularis LM. Quia igitur LM et EH constant esse equales, erit DG ad LM sicut b ad a, est sicut l, ad a. Sed, ut ostensum est, a et l, proportionaliter se habent ad contrarios motus alternatim. Quod ergo sufficit attollere a in D, sufficiet attollere l, secundum LM. Cum ergo equalia sint l, et b, et LC equale CB, l, non sequetur b, contrario motu, neque a sequetur b, secundum quod proponitur. 4.17 Sit centrum c, brachia ac, longius bc, brevius, et descendat perpendiculariter ceg supra quam perpendiculariter cadant hinc, inde ag. et be, aequales. […] Pertranseant enim aequaliter a , et be, ad k, et z, et super eas fiant portiones circulorum mbhz,

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Appendix. Quotations kxal, et circa centrum c, fiat commune proportio kyaf, similis, et aequalis portionis mbhz, et sint arcus ax, al, aequales sibi atque similes arcubus mb, bh. Itemque ay, af. Si ergo ponderosius est a, quam b, in hoc situ descendat a, in x, et ascendat b, in m, ducantur igitur lineae zm, kx, yk, fl, et mp, super zbp stet perpendiculariter etiam xe, et fd, super kad, et quia mp, aequatur fd, et ipsa est maior xt, per similes triangulos erunt mp, maior xt, quia plus ascendit b, ad rectitudinem, quam a, descendit. quod est impossibile, cum sint aequalia

4.18 Sit linea abc, aequedistans orizonti, et super eam orthogonaliter erecta sit bd, a qua descendant hinc, inde lineae da, dc, sitque dc, maioris obliquitatis proportione igitur declinationum dico non angulorum, sed linearum usque ad aequedistantem resecationem, in qua aequaliter sumunt de directo. Sit ergo e, pondus super dc, et h, super da, et sit e, ad b, sicut dc, ad ad. Dico ea pondera esse unius virtutis in hoc situ, sit enim dk, linea unius obliquitatis, cum dc, et pondus super eam. ergo aequale est e, quae sit g. Si igitur possibile est, descendat e, in l, et trahat h, in m, sitque gn, aequale hm, quod etiam aequale est e. Et transeat per g et h, perpendicularis, super db; sitque ghy, et ab l, tl, t; et tunc super ghy, nz, mx, et super lt, erit er. Quia igitur proportio nz, ad ng, sicut dy ad dg, et ideo sicut db, ad dk, et quia similiter mx, ad mh, sicut db, ad da. Erit propter aequalem proportionalitatem perturbata mx, ad nz, sicut dk, ad da, et hoc est sicut g, ad h. Sed quia e, non sufficit attollere g, in n, nec sufficiet attollere h in m. Sic ergo manebunt.

A.5 Chapter 5 5.1 Non è dubbio che se a una semplice fune si attacca un peso, poniamo il caso di mille libbre, che tutta la fatica e forza non sia unitamente da quella fune sostenuta, che poi se la detta fune sarà raddoppiata e a quella una taglia d’un raggio appesa dove penda quel peso, che la fune non sia per avere il doppio meno di fatica e il doppio meno di forza non basterà ad alzare quel peso; or che sarà poi se ci saranno più taglie? […] Se’l primo raddoppiamento leva la metà del peso, il secondo al quale resta la metà, leverà via la metà di quella metà che sarà la quarta parte di tutto il peso, et dalla quarta parte della detta forza di prima sarà il peso levato. 5.2 Il segreto di tutti gli inventori delle machine de’ Molini, et altro è di cercare solo, come si disse, di poter accompagnare la forza con la velocità, cosa in vero difficilissima; perche dovendosi un istessa potenza multiplicare in molte, che possino l’una doppo l’altra alzare, overo portare un peso, è necessario, che similmente si multiplichi il tempo, come per essempio saria se si dovesse trasportare un peso di mille libre da un luogo all’altro, con la semplice forza d’un solo huomo, il quale ne porterà solo una parte che sarà cinquanta libre. 5.3 Table 5.3 Diffinitione III

Diffinitione IIII Diffinitione XIII

Diffinitione XIIII Diffinitione XVII

La vertu d’un corpo grave se intende, e piglia per quella potentia, che lui ha da tendere, over di andare al basso, e anchora da resistere al moto contrario, cioe à che il volesse tirar in suso. Li corpi se dicono de vertu, over potentia, equali, quando che quelli in tempi eguali di moto pertransiscono spacii eguali. Un corpo si dice essere piu grave, over men grave d’un’altro, secondo il luoco, over sito, quando che la qualita del luoco dove che lui se riposa, e giace, lo fa essere piu grave dell’altro anchor che fusseno simplicemente egualmente gravi. La gravita d’un corpo se dice essere nota, quando che il numero delle libre, che lui pesa nesia noto, over altra denomination de peso. Piu obliquo se dice essere quel descenso, d’un corpo grave, il quale in una medesima quantita, capisse manco della linea della direttione, overamente del descenso retto verso il centro del mondo.

A.5 Chapter 5 Petitione II

Petitione III

Petitione VI

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Simelmente adimandamo, che ne sia concesso quel corpo, ch’è di maggior potentia debbia anchora discendere piu velocemente, et nelli moti contrarii, cioe nelli ascensi, ascendere piu pigramente, dico nella libra. Anchora adimandamo, che ne sia concesso un corpo grave esser in el discendere tanto piu grave, quanto che il moto di quello è piu retto al centro del mondo. Anchora adimandamo, che ne sia concesso, niun corpo esser grave in se medesimo.

5.4 Table 5.4 I II

III

IIII

V

VI

VII

VIII

XIIII XV

La proportione della grandezza di corpi de un medesimo genere, e quella della lor potentia è una medesima. La proportione della potentia di corpi gravi de uno medesimo genere, e quella della lor velocita (nelli descensi) se conchiude esser una medesima, anchor quella delli lor moti contrarii (cioe delli lor ascensi) se conchiude esser la medesima, ma trasmutativamente. Se saranno dui corpi simplicemente eguali di gravita, ma ineguali per vigor del sito, over positione, la proportione della lor potentia, e quella della lor velocita necessariamente sara una medesima. Ma nelli lor moti contrarii, cioe nelli ascensi, la proportione della lor potentia, e quella della lor velocita se afferma esser la medesima, ma trasmutativamente. La proportione della potentia di corpi simplicemente equali in gravita, ma inequali per vigor del sito, over positione, e quella delle lor distantie dal sparto, over centro della libra, se approvano esser equali. Quando, che la positione de una libra de brazzi equali sta nel sito della equalita, e nella istremita de l’uno, e l’altro brazzo vi siano appesi corpi simplicemente equali in gravita, tal libra non se separara dal detto sito della equalita, e se per caso la sia da qualche altro peso in luno de detti brazzi imposto separata dal detto sito della equalita, overamente con la mano, remosso quel tal peso, over mano, tal libra de necessita ritornara al detto sito della equalita. Quando che la positione d’una libra de bracci eguali sia nel sito della egualita, e che nella istremita dell’uno e l’altro brazzo vi siano appesi corpi simplicemente ineguali di gravita, dalla parte dove sara il piu grave sara sforzata à declinare per fin alla linea della direttione. Se li brazzi della libra saranno ineguali, et che nella istremita di cadauno de quelli vi siano appesi corpi simplicemente eguali in gravita dalla banda del piu longo brazzo tal libra fara declinatione. Se li brazzi della libra saranno proportionali alli pesi in quella imposti, talmente, che nel brazzo piu corto sia appeso il corpo piu grave, quelli tai corpi, over pesi seranno equalmente gravi, secondo tal positione, over sito. La egualita della declinatione è una medesima egualita de peso. Se dui corpi gravi descendano per vie de diverse obliquita, e che la proportione delle declinationi delle due vie, e della gravita de detti corpi sia fatta una medesima, tolta per el medesimo ordine. Anchora la virtu de luno, e laltro de detti dui corpi gravi, in el descendere sara una medesima.

5.5 Siano li dui corpi .a.b. e .c. de uno medesimo genere, e sia .a.b. maggiore, e sia la potentia del corpo .a.b. la .d.e. e quella de corpo .c. la .f. Hor dico che quella proportione, che è dal corpo .a.b. al corpo .c. quella medesima è della potentia .d.e. alla potentia .f. Et se possibile è esser altramente (per l’aversario) sia che la proportione del corpo .a.b. al corpo .c. sia menore di quella della potentia .d.e. alla potentia .f. Hor sta del corpo .a.b. (maggiore) compreso una parte eguale al corpo .c. menore, quale sia la parte .a. e perche la vertu, over potentia del composito è composta dalla vertu di componenti. Sia adunque la vertu, over potentia della parte .a. la .d. e la vertu, over potentia del residuo .b. de necessita sara la restante potentia .e. et perche

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Appendix. Quotations la parte .a. è tolta egual al .c. la potentia .d. (per il converso della .7. diffinitione) sara eguale alla potentia .f. e la proportione de tutto il corpo .a.b. alla sua parte .a. (per la seconda parte della .7. del quinto di Euclide) sara, si come quella del medesimo corpo .a.b. al corpo .c. (per esser .a. egual al .c.) e similmente la proportione della potentia .d.e. alla potentia .f. sara, si come quella della detta potentia .d.e. alla sua parte .d. (per esser la .d. egual alla .f.). Adunque la proportione de tutto il corpo .a.b. alla sua parte .a. sara menore di quella di tutta la potentia .d.e. alla sua parte .d. Adunque eversamente (per la .30. del quinto di Euclide) la proportione del medesimo corpo .a.b. al residuo corpo .b. sara maggiore di quella di tutta la potentia .d.e. alla restante potentia .e. la qual cosa saria inconveniente, e contra la opinion dell’aversario, il qual vol che la proportione del maggior corpo al menore sia menore, di quella della sua potentia alla potentia del detto menore. Adunque destrutto l’opposito rimane il proposito.

5.6 Propositione VIII Se li brazzi della libra saranno proportionali alli pesi in quella imposti, talmente, che nel brazzo piu corto sia appeso il corpo piu grave, quelli tai corpi, over pesi seranno equalmente gravi, secondo tal positione, over sito. […] Sia come prima la regola, over libra .a.c.b. e vi siano appesi .a. e .b. et sia la proportione del .b. al .a. si come del brazzo .a.c. al brazzo .b.c. Dico, che tal libra non declinara in alcuna parte di quella, e se possibil fusse (per l’aversario) che declinar potesse, poniamo che quella declini dalla parte del .b. e che quella discenda, e transisca in obliquo, si come sta la linea .d.c.e. in luoco della .a.c.b. e attaccatovi .d. come .a. e .e. come .b. e la linea .d.f. descenda orthogonalmente, e simelmente ascenda la .e.h. […] e sia posto .l. equale al .b. in gravita, e descenda el perpendicolo .l.m. Adunque perche eglie manifesto la .l.m. e la .e.h. esser equale, la proportione della .d.f. alla .l.m. sara si come delle simplice gravita del corpo .b. alla simplice gravita del corpo .a. over della simplice gravita del corpo .l. alla simplice gravita del corpo .d. […] Onde se li detti dui corpi gravi, cioe .d. e .l. fusseno simplicemente equali in gravita, stanti poi in li medesimi siti, over luochi, dove, che al presente vengono supposti, el corpo .d. saria piu grave del corpo .l. secondo elsito (per la .4. propositione) in tal proportione, qual é di tutto il brazzo .d.c. al brazzo .l.c. e per che il corpo .l. è simplicemente (dal presupposito) piu grave del corpo .d. secondo la medesima proportione (cioe, si come la proportione del brazzo. d.c. al brazzo .l.c. adunque li detti dui corpi .d. e .l. nel sito della equalita veneranno ad essere egualmente gravi [...]. Adunque sel corpo .b. (per l’aversario) è atto ad ellevare il corpo .a. dal sito della equalita per fin al ponto .d. el medesimo corpo .b. saria anchora atto, e sofficiente ad ellevare il corpo .l. dal medesimo sito della equalita per fin al ponto, dove che al presente è, el qual consequente é falso, e contra alla quinta propositione […] distrutto adunque l’opposito, rimane il proposito. 5.7 Propositione XV Se dui corpi gravi descendano per vie de diverse obliquita, e che la proportione delle declinationi delle due vie, e della gravita de detti corpi sia fatta una medesima, tolta per el medesimo ordine. Anchora la virtu de luno, e laltro de detti dui corpi gravi, in el descendere sara una medesima. […] Sia adunque la lettera .e. supposta per un corpo grave posto sopra la linea .d.c. e un’altro la lettera .h. sopra la linea .d.a. e sia la proportione della simplice gravita del corpo .e. alla simplice gravita del corpo .h. si come quella della .d.c. alla .d.a. Dico li detti dui corpi gravi esser in tai siti, over luochi di una medesime virtu, over potentia. Et per dimostrar questo, tiro la .d.k. di quella medesima obliquita, ch’è la .d.c. e imagino un corpo grave sopra di quella equale al corpo .e. elqual pongo sia la lettera .g. ma che sia in diretto con .e.h. cioe equalmente distante dalla .c.k. Hor se possibel è (per l’aversario) […]. Anchora la proportione della .m.x. alla .n.z. sara si come quella della .d.k. alla .d.a. e quella medesima (dal presupposito) e dalla gravita del corpo .g. alla gravita del corpo .h. perche il detto corpo .g. fu supposto esser simplicemente, egualmente grave con el corpo .e. adunque tanto quanto, che il corpo .g. è simplicemente piu grave del corpo .h. per altro tanto il corpo .h. vien à esser piu grave per vigor del sito del

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detto corpo .g. e pero si vengono ad egualiar in virtu, over potentia, e per tanto quella virtu, over potentia, che sara atta à far ascendere luno de detti dui corpi, cioe à tirarlo in suso, quella medesima sara atta, over sofficiente à fare ascendere anchora l’altro, adunque sel corpo .e. (per l’aversario) è atto, e sofficiente à far ascendere il corpo .h .per fin in .m. el medesimo corpo .e.s aria adunque sofficiente à far ascendere anchora il corpo .g. à lui equale, e inequale declinatione, la qual cosa é impossibile per la precedente propositione, adunque il corpo .e. non sara de maggior virtu del corpo .h. in tali siti, over luochi, ch’é il proposito. 5.8 Post haec videndum est de ponderibus quae in libra constituuntur. Sit igitur libra, cuius trutina sit appensa in A, et finis ubi iunguntur latera lancis B, et lanx CD, et manifestum est quod CD movetur circa B, velut centrum quoddam, quia CD non potest separari ab ipso B: et sit angulus ABC, et ABD rectus. 5.9 Dico quod pondus in C constitutum erit gravius, quam si lanx collocetur in quocunque alio loco, ut puto quod constitueretur lanx in F. Ut autem cognoscamus quod C sit gravius in eo situ, quam in F […] Ideo ergo duplici ratione magis gravabit pondus lance posita ad perpendiculum cum trutina, quam in quoque alio loco […]. Primum igitur sic declaratur. Manifestum est in stateris, et in his, qui pondera elevant, quod quanto magis pondus ae trutina, eo magis grave videtur [...] manifestum est, libram quanto magis descendit versus C ex A, tanto gravius pondus reddere, et eo velocius moveri: at ex C versus Q, contraria ratione pondus reddi levius, et motum segniorem, quod et experimentum docet. 5.10 Secundum vero sic demonstratur. Quia enim CE est aequalis FG, sumatur CH aequalis CE [...] igitur BN maior OF, et ideo BM maior OP. Dum igitur libra movetur ex C in E pondus descendit per BM lineam, seu propinquius centro redditur quam esset in C, et dum movetur per spatium arcus FG, descenditque per OP, et BM, maior est OP. Igitur supposito etiam quod in aequali tempore transiret ex C in K, et ex F in G, adhuc velocius descendit ex C, quam ex F. Igitur gravius est in C, quam in F. 5.11 Quartum subtilitatis exemplum est in trochleis. Sed quia temporum proportio est, ut potentiarum per binos orbiculos, quadruplo per ternos, sexcuplo lentius trahet [...] quo sit ut puer ille vix in unius hore spacio idem pondus his trochleis trahet, quod sexcuplo robustior vir, unica, superius existes, levare potest illico fune. 5.12 Propositio quadragesima quinta Rationem staterae ostendere Si ergo ponantur loco lineae bd in e et f, et sit proportio e b ad bf, ut g ad h, dico, quod erit aequilibrium, per eandem enim h movebitur in k, scilicet ut perveniat in rectam a d, si enim non esset suspensum h, moveretur in recta e h per eandem, quia ergo retinetur, movetur per obliquam hk, et sumatur in propinquum punctum (m ?) in be, et n in aequali distantia in ef, quia ergo eb totum movetur eadem vi in singulis partibus, quia a pondere h, et in h movetur per hk in m per mp, ergo qualis est proportio magnitudinis hk ad mp, talis est vis in mp ad vim in hk, et ita in b erit pene infinita. 5.13 Il centro della grandezza di ciascun corpo è un certo punto posto dentro, dal quale se con la imaginatione s’intende esservi appeso il grave, mentre è portato sta fermo, et mantiene quel sito, che egli haveva da principio, ne in quel portamento si và rivolgendo. Questa diffinitione del centro della gravezza insegnò Pappo Alessandrino nell’ottavo libro delle raccolte Matematiche. Ma Federico Commandino nel libro del centro della gravezza de’ corpi solidi dichiarò l’istesso centro in questa maniera descrivendolo: Il centro della gravezza di ciascuna figura solida è quel punto posto dentro, d’intorno al quale le parti di momenti eguali da ogni parte si fermano. Perochè se per tale centro sarà condotto un piano, che seghi in qual si voglia modo la figura, sempre la dividerà in parti che peseranno egualmente.

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5.14 La bilancia egualmente distante dall’orizzonte, et che habbia nelle stremità pesi eguali, et ugualmente distanti dal centro collocato in essa bilancia. Se ella indi sarà mossa, o non, dunque ella sarà lasciata, rimarrà. […] Dico primariamente, che la bilancia DE non si muoverà, et rimarrà in quel sito. Hor percioche pesi AB sono eguali, sarà il centro della grandezza della magnitudine composta delli due pesi A et B in C. Per la qual cosa l’istesso punto C sarà il centro della bilancia, et il centro della gravezza di tutto il peso. Et percioche il centro della bilancia che è C, mentre la bilancia A B insieme co’ pesi si muove in DE, rimane immobile, non si muoverà ne anche il centro della gravezza, che è l’istesso C. 5.15 Hor perche dicono che il peso posto in D in quel sito è più grave del peso posto in E nell’altro sito dal basso: mentre i pesi sono in DE, non sarà il punto C più centro della grandezza, imperochè non stanno fermi se sono attaccati al C, ma sarà nella linea CD per la terza del primo di Archimede delle cose, che pesano ugualmente. Non sarà già nella CE per essere il peso D più grave del peso E: sia dunque in H, nel quale se saranno attaccati, rimaranno. Et percioche il centro della gravezza de’ pesi congiunti in AB stà nel punto C: ma de’ pesi posti in DE il punto è H: mentre dunque i pesi AB si muovono in DE, il centro della grandezza C moverassi verso D, et s’appresserà più da vicino al D, il che è impossibile, per mantenere i pesi una medesima distanza fra loro: peroche il centro della gravezza di ciascun corpo stà sempre nel medesimo sito per rispetto al suo corpo. 5.16 Pongansi le cose istesse, et da i punti DE siano tirate le linee DHEK a piombo dell’orizonte, et sia un’altro cerchio LDM, il cui centro sia N, il quale tocchi FDG nel punto D, et sia eguale ad FDG. […] Ma la proportione dell’angolo MHD all’angolo HDG è minore di qual si voglia altra proportione, che si trovi trà la maggiore, et minore quantità: Adunque la proportione de i pesi DE sarà la minima di tutte le proportioni, anzi non sarà quasi neanche proportione, essendo la minima di tutte le proportioni. […] dalle quali troveremo sempre la proportione minore in infinito: et così segue, che la proportione del peso posto in D al peso posto in E non sia tanto picciola, che non si possa ritrovarla sempre minore in infinito. Et perche l’angolo MDG si puote dividere in infinito, si potrà anche dividere quel più di grandezza che ha il D sopra lo E in infinito. 5.17 Percioche il peso posto in L libero, et sciolto si muoverebbe verso il centro del mondo per LS, et il peso posto in D per DS . Ma perche il peso messo in L grava tutto sopra LS, et quello che è in D sopra DS, il peso in L graverà più sopra la linea CL, che quello, che sta in D sopra la linea DC. Adunque la linea CL sosterrà più il peso, che la linea CD, et nel mondo istesso quanto più il peso sarà da presso ad F, si dimostrerà più esser sostenuto dalla linea CL per cotesta cagione, peroche sempre l’angolo CLS sarebbe minore, la qual cosa etiandio è manifesta; perché se le linee CKL, et LS s’incontrassero in una linea, il che aviene in FCS, all’hora la linea CF sosterrebbe tutto il peso, che è in F, et lo renderebbe immobile, né havrebbe niuna gravezza in tutto nella circonferenza del cerchio. 5.18 Se dunque il peso posto in E è più grave del peso posto in D, la bilancia DE non starà giamai in questo sito, la qual cosa noi habbiamo proposto di mantenere, ma si moverà in FG. Alle quali cose rispondiamo che importa assai, se noi consideriamo i pesi overo in quanto sono separati l’uno dall’altro, overo in quanto sono tra loro congiunti: perche altra è la ragione del peso posto in E senza il congiungimento del peso posto in D, et altra di lui con l’altro peso congiunto, si fattamente che l’uno senza l’altro non si possa movere. Imperoche la diritta, et naturale discesa dal peso posto in E, in quanto egli è senza altro congiungimento di peso, si fa per la linea ES, ma in quanto egli è congiunto col peso D, la sua naturale discesa non sarà più per la linea ES, ma per una linea egualmente distante da CS percioche la magnitudine comporta de i pesi ED, et della bilancia DE il cui centro della gravezza è C, se in nessun luogo non sarà sostenuta, si muoverà naturalmente in giù nel modo che si trova, secondo la grandezza del centro per la linea diritta tirata dal centro della gravezza C al centro del mondo S, finche il centro C pervenga nel

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centro S […] Ma se i pesi posti in ED sono l’un l’altro fra se congiunti, et gli considereremo in quanto sono congiunti, sarà la naturale inclinazione del peso posto in E per la linea MEK, percioche la gravezza dell’altro peso posto in D fa si, che il peso posto in E non gravi sopra la linea ES, ma nella EK. Il che fa parimente la gravezza del peso posto in E, cioè, che’l peso posto in D non gravi per la linea reta DS, ma secondo DH impedirsi ambedue l’uno l’altro, che non vadino a propri luoghi […]. Adunque il peso posto in D non moverà in su il peso posto in E. Dalle quali cose segue che i pesi posti in DE, in quanto tra loro sono congiunti, sono egualmente gravi. 5.19 Sia la leva AB, il cui sostegno C, et sia il peso D attaccato al punto B, et sia la possanza in A movente il peso D con la leva AB. Dico lo spatio della possanza in A allo spatio del peso essere cosi come CA à CB. Ma sia la leva AB il cui sostegno B, & la possanza movente in A, & il peso in C. Dico lo spatio della possanza mossa allo spatio del peso trasportato essere, come BA a BC. 5.20 Corollario Da queste cose è manifesto, che maggiore proportione ha lo spatio della possanza, che move allo spatio del peso mosso, che il peso alla medesima possanza. Percioche lo spatio della possanza allo spatio del peso ha la medesima proportione, che il peso alla possanza, che sostiene il detto peso. Ma la possanza, che sostiene è minore della possanza che move, però il peso havrà proportione minore alla possanza che lo move, che alla possanza, che lo sostiene. Lo spatio dunque della possanza che move allo spatio del peso haurà proportione maggiore, che il peso all’istessa possanza. 5.21 Proportio ponderis in C ad idem pandus in F erit quemadmodum totius brachii .BC. ad partem .B.U. 5.22 Ad cuius rei evidentiam imaginemur filum .F.u. perpendiculare, & in cuius extremo .U. pendere pondus, quod erat in .F. unde clarum erit quod eundem effectum gignet, ac si fuisset in .F. […] Idem assero si brachium esset in situ .e.B. […] Quia tantum est quod ipsum sit appensum filo Q pendet ab .u. quantum quod ab ipso liberum appensum fuisset .e. brachii .B.e. & hoc procederet ab eo quod partim penderet a centro .B. & si brachii esset in situ .B.Q. totum pondus centro .B. remaneret appensum, quem admodum in situ B.A. totum dicto centrum anniteretur. 5.23 Et quamuis appellem latus .BC. orizontale, supponens illud angulum rectum cum .C.O: facere, unde angulus C.B.Q. sit ut minor sit rector, ob quantitatem unius anguli equals ei, quem duae .C.O. et B.Q. in centro regionis elementaris constitutu, hoc tamen nihil refert, cum dictus angulus insensibilis sit magnitudinem. 5.24 Ex iis quae nobis hucusque sunt dicta, facile intelligi potest, quantitatis B.u. quae fere perpendicularis es a centro .B. ad lineam .F.u. inclinationis, ea est, quae non ductis in cognitionem quantitatis virtutis ipsius F in huiusmodi situ constituens videlicet linea .F.u. cum brachio .F.B. angulum acutum. 5.25 Ut hoc tamen melius intelligamus, imaginemur libram .b.o.a. fixam in centro .o. ad cuius extrema sint appensa duo pondera, aut duae virtutes moventes .e. et .c. ita tamen, linea inclinationis .e. idest .be. faciat angulum rectum cum .o.b. in puncto .b. linea vero inclinationis .c. idest .a.c. faciat angulum acutum, aut obtusum cum .o.a. in puncto .a. Imaginemur ergo lineam .o.t. perpendicularem lineae .c.a. inclinationis […] secetur deinde imaginatione .o.a. in puncto .i. ita ut .o.i. aequalis. sit .o.t. & puncto .i. appensum sit a pondus aequale ipsi .c. cuius inclinationis linea parallela sit linea inclinationis ponderis .e. supponendo tamen pondus aut virtutem .c. ea ratione maiorem esse ea, quae est .e. qua .b.o. maior est .o.t. absque dubio ex 6 lib. primi Archi. de ponderibus .b.o.i. non movebitur situ, sed si loco .o.i. imaginabimur .o.t. consolidatam cum .o.b. & per lineam .t.c. attractam virtute .c. similiter quoque contingent ut .b.o.t.; communi quadam scientiam, non moveatur situ.

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5.26 Sed in secunda parte quinte propositionis non videt vigore situs eo modo, quo ipse disputat, nulla elicitur ponderis differentia quia si corpus .B. descendere debet per arcum .IL. corpus .A. ascendere debet per arcum .V.S. Haec autem quinta propositio Tartalea est secunda quaestio a Iordano proposita. 5.27 Momento è la propensione di andare al basso, cagionata non tanto dalla gravità del mobile, quanto dalla disposizione che abbino tra di loro diversi corpi gravi; mediante il qual momento si vedrà molte volte un corpo men grave contrapesare un altro di maggior gravità: come nella stadera si vede un picciolo contrapeso alzare un altro peso grandissimo, non per eccesso di gravità, ma sì bene per la lontananza dal punto donde viene sostenuta la stadera; la quale, congiunta con la gravità del minor peso, gli accresce momento ed impeto di andare al basso, col quale può eccedere il momento dell’altro maggior grave. È dunque il momento quell’impeto di andare al basso, composto di gravità, posizione e di altro, dal che possa essere tal propensione cagionata. 5.28 Momento, appresso i meccanici, significa quella virtù, quella forza, quella efficacia, con la quale il motor muove e ’l mobile resiste; la qual virtù depende non solo dalla semplice gravità, ma dalla velocità del moto, dalle diverse inclinazioni degli spazii sopra i quali si fa il moto, perché più fa impeto un grave descendente in uno spazio molto declive che in un meno. Il secondo principio è, che il momento e la forza della gravità venga accresciuto dalla velocità del moto: sì che pesi assolutamente eguali, ma congiunti con velocità diseguali, sieno di forza, momento e virtù diseguale, e più potente il più veloce, secondo la proporzione della velocità sua alla velocità dell’altro. Di questo abbiamo accomodatissimo esemplo nella libra o stadera di braccia disuguali, nelle quali posti pesi assolutamente eguali, non premono e fanno forza egualmente, ma quello che è nella maggior distanza dal centro, circa il quale la libra si muove, s’abbassa sollevando l’altro, ed è il moto di questo che ascende, lento e l’altro veloce: e tale è la forza e virtù che dalla velocità del moto vien conferita al mobile che la riceve, che ella può compensare altrettanto peso che all’altro mobile più tardo fosse accresciuto; sì che, se delle braccia della libra uno fosse dieci volte più lungo dell’altro, onde nel muoversi la libra circa il suo centro, l’estremità di quello passasse dieci volte maggiore spazio che l’estremità di questo, un peso posto nella maggiore distanza potrà sostenerne ed equilibrarne un altro dieci volte assolutamente più grave che non egli è; e ciò perché, muovendosi la stadera, il minor peso si moveria dieci volte più velocemente che l’altro. 5.29 Avendo noi mostrato come i momenti di pesi diseguali vengono pareggiati dall’essere sospesi contrariamente in distanze che abbino la medesima proporzione, non mi pare di doversi passar con silenzio un’altra congruenza e probabilità, dalla quale ci può ragionevolmente essere confermata la medesima verità. Però che, considerisi la libra AB divisa in parti diseguali nel punto C, ed i pesi, della medesima proporzione che hanno le distanze BC, CA, alternatamente sospesi dalli punti A, B: è già manifesto come l’uno contrapeserà l’altro, e, per conseguenza, come, se a uno di essi fusse aggiunto un minimo momento di gravità, si moverebbe al basso innalzando l’altro; sì che, aggiunto insensibile peso al grave B, si moveria la libra discendendo il punto B verso E, ed ascendendo l’altra estremità A in D. E perché, per fare descendere il peso B, ogni minima gravità accresciutagli è bastante, però, non tenendo noi conto di questo insensibile, non faremo differenza dal poter un peso sostenere un altro al poterlo movere. Ora, considerisi il moto che fa il grave B, discendendo in E, e quello che fa l’altro A, ascendendo in D; e troveremo senza alcun dubbio, tanto essere maggiore lo spazio BE dello spazio AD, quanto la distanza BC è maggiore della CA; formandosi nel centro C due angoli, DCA ed ECB, eguali per essere alla cima, e, per conseguenza, due circonferenze, BE, AD, simili, e aventi tra di sé l’istessa proporzione delli semidiametri BC, CA, dai quali vengono descritte. Viene adunque ad essere la velocità del moto del grave B, discendente, tanto superiore alla velocità dell’altro mobile A, ascendente, quanto la gravità di questo eccede la gravità di quello; né potendo essere alzato il peso A in D, benché lentamente, se l’altro grave B non si muove in E velocemente, non sarà maraviglia, né alieno dalla costituzione naturale, che la velocità del moto del grave B compensi la maggior resistenze del peso A, mentre egli in D pigramente si

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muove e l’altro in E velocemente descende. E così, all’incontro, posto il grave A nel punto D e l’altro nel punto E, non sarà fuor di ragione che quello possa, calando tardamente in A, alzare velocemente l’altro in B, ristorando, con la sua gravità, quello che per la tardità del moto viene a perdere. E da questo discorso possiamo venire in cognizione, come la velocità del moto sia potente ad accrescere momento nel mobile, secondo quella medesima proporzione con la quale essa velocità di moto viene augumentata. 5.30 Tal ragguagliamento tra la gravità e la velocità si ritrova in tutti gli strumenti meccanici, e fu considerato da Aristotele nelle sue Questioni meccaniche: onde noi ancora possiamo prender per verissimo assunto che pesi assolutamente diseguali, alternativamente si contrappesano e si rendono di momenti uguali, ogni volta che le loro gravità con proporzione contraria rispondono alle velocità dei lor moti. 5.31 Un’altra cosa, prima che più oltre si proceda, bisogna che sia considerata; e questa è intorno alle distanze, nelle quali i gravi vengono appesi: per ciò che molto importa il sapere come s’intendano distanze eguali e diseguali, ed in somma in qual maniera devono misurarsi. [...] Ma se, elevando la linea CB e girandola intorno al punto C, sarà trasferita in CD, sì che la libra resti secondo le due linee AC, CD, gli due eguali pesi pendenti dai termini A, D non più peseranno egualmente sopra il punto C; perché la distanza del peso posto in D è fatta minor di quello che era mentre si ritrovava in B. Imperò che, se considereno le linee per le quali i detti gravi fanno impeto, e discenderebbono quando liberamente si movessero, non è dubbio alcuno che sariano le linee AG, DF, BH: fa dunque momento ed impeto il peso pendente dal punto D secondo la linea DF; ma quando pendeva dal punto B, faceva impeto nella linea BH; e perché essa linea DF più vicina al sostegno C di quello che faccia linea BH, perciò doviamo intendere, gli pesi pendenti dalli punti A, D non essere in distanze eguali dal punto C, ma sì bene quando saranno constituiti secondo la linea retta ACB. 5.32 E per amplissima confermazione e più chiara esplicazione di questo medesimo, considerisi la presente figura (e, s’io non m’inganno, potrà servire per cavar d’errore alcuni meccanici prattici, che sopra un falso fondamento tentano talora imprese impossibili), nella quale al vaso larghissimo EIDF, vien continuata l’angustissima canna ICAB, ed intendasi in essi infusa l’acqua sino al livello LGH; la quale in questo stato si quieterà, non senza meraviglia di alcuno, che non capirà così subito come esser possa, che il grave carico della gran mole dell’acqua GD, premendo abbasso, non sollevi e scacci la piccola quantità dell’altra contenuta dentro alla canna CL, dalla quale gli vien contesa ed impedita la scesa. Ma tal meraviglia cesserà, se noi cominceremo a fingere l’acqua GD essersi abbassata solamente sino a QO, e considereremo poi ciò che averà fatto l’acqua CL. la quale, per dar luogo all’altra che si è scemata dal livello GH sino al livello QO, doverà per necessità essersi nell’istesso tempo alzata dal livello L sino in AB, ed esser la salita LB tanto maggiore della scesa GQ, quant’è l’ampiezza del vaso GD maggiore della larghezza della canna LC, che in somma è quanto l’acqua GD è più della LC. Ma essendo che il momento della velocità del moto in un mobile compensa quello della gravità di un altro, qual meraviglia sarà se la velocissima salita della poca acqua CL resisterà alla tardissima scesa della molta GD? 5.33 SAGR. Ma credete voi che la velocità ristori per l’appunto la gravità? Cioè che tanto sia il momento e la forza di un mobile verbigrazia, di quattro libbre di peso, quanto quella di un di cento, qualunque volta quello avesse cento gradi di velocità e questo quattro gradi solamente? SALV. Certo sì, come io vi potrei con molte esperienze mostrare: ma per ora bastivi la confermazione di questa sola della stadera, nella quale voi vedrete il poco pesante romano allora poter sostenere ed equilibrare la gravissima balla, quando la sua lontanaza dal centro sopra il quale si sostiene e volgesi la stadera, sarà tanto maggiore dell’altra minor distanza dalla quale pende la balla, quanto il peso assoluto della balla è maggior di quel del romano. E di questo non poter la gran balla co ’l suo peso sollevar il romano, tanto men grave, altra non si vede poter esser cagione che la disparità de i movimenti che e quella e questo far dovrebbero, mentre la balla con l’abbassarsi di un sol dito facesse alzare il romano di cento dita.

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5.34 SAGR. Voi ottimamente discorrete, e non mettete dubbio alcuno nel concedere, che per piccola che sia la forza del movente, supererà qualsivoglia gran resistenza, tutta volta che quello più avanzi di velocità, ch’ei non cede di vigore e gravità. Or venghiamo al caso della corda: e segnando un poco di figura, intendete per ora, questa linea ab, passando sopra i due punti fissi e stabili a, b, aver nelle estremità sue pendenti, come vedete, due immensi pesi c, d, li quali, tirandola con grandissima forza, la facciano star veramente tesa dirittamente, essendo essa una semplice linea, senza veruna gravità. Or qui vi soggiungo e dico, che se dal mezzo di quella, che sia il punto e, voi sospenderete qualsivoglia piccolo peso, quale sia questo h, la linea ab cederà, ed inclinandosi verso il punto f, ed in consequenza allungandosi, costringerà i due gravissimi pesi c, d a salir in alto: il che in tal guisa vi dimostro. Intorno a i due punti a, b, come centri, descrivo 2 quadranti, eig, elm; ed essendo che li due semidiametri ai, bl sono eguali alli due ae, eb, gli avanzi fi, fl saranno le quantità de gli allungamenti delle parti af, fb sopra le ae, eb, ed in conseguenza determinano le salite de i pesi c, d, tutta volta però che il peso h avesse auto facoltà di calare in f: il che allora potrebbe seguire, quando la linea ef, che è la quantità della scesa di esso peso h, avesse maggior proporzione alla linea fi, che determina la salita de i due pesi c, d che non ha la gravità di amendue essi pesi alla gravità del peso h. Ma questo necessariamente avverrà, sia pur quanto si voglia massima la gravità de i pesi c, d, e minima quella dell’h. 5.35 Dei quali inganni parmi di avere compreso essere principalmente cagione la credenza, che i detti artefici hanno avuta ed hanno continuamente, di poter con poca forza muovere ed alzare grandissimi pesi, ingannando, in un certo modo, con le loro machine la natura; instinto della quale, anzi fermissima constituzione, è che niuna resistenza possa essere superata da forza, che di quella non sia più potente. La quale credenza quanto sia falsa, spero con le dimostrazioni vere necessarie, che averemo nel progresso, di fare manifestissimo. […] Ora, assegnata qual si voglia resistenza determinata, e limitata qualunque forza, e notata qualsivoglia distanza, non è dubbio alcuno, che sia per condurre la data forza il dato peso alla determinata distanza; perciò che, quando bene la forza fusse picciolissima, dividendosi il peso in molte particelle, ciascheduna delle quali non resti superiore alla forza, e tranferendosene una per volta. Avrà finalmente condotto tutto il peso allo statuito termine: né però nella fine dell’operazione si potrà con ragione dire, quel gran peso essere stato mosso e traslato da forza minore di sé, ma sì bene da forza la quale più volte averà reiterato quel moto e spazio, che una sol volta sarà stato da tutto il peso misurato. Dal che appare, la velocità della forza essere stata tanta volte superiore alla resistenza del peso, quante esso peso è superiore alla forza; poiché in quel tempo nel quale la forza movente ha molte volte misurato l’intervallo tra i termini del moto, esso mobile viene ad aver passato una sol volta: né perciò si deve dire, essersi superata gran resistenza con piccola forza, fuori della costituzione della natura. Allora solamente si potria dire, essersi superato il naturale instituto, quando la minor forza trasferisse la maggiore resistenza con pari velocità di moto, secondo il quale essa cammina; il che assolutamente affermiamo essere impossibile a farsi con qual si voglia machina, immaginata o che immaginar si possa. Ma perché potria tal ora avvenire che, avendo poca forza, ci bisognasse muovere un gran peso tutto congiunto insieme, senza dividerlo il pesi, in questa occasione sarà necessario ricorrere alla machina: col mezzo della quale si trasferirà il peso proposto nell’assegnato spazio dalla data forza. […] E questa deve essere per una delle utilità che dal mecanico si cavano, annoverata: perché invero spesse volte occore che, avendo scarsità di forza, ma non di tempo, ci occorre muovere gran pesi tutti unitamente. Ma ci sperasse e tentasse, per via di machine far l’istesso effetto senza crescere tardità al mobile, questo certamente rimarrà ingannato, e dimostrerà di non intendere la natura delli strumenti mecanici e le ragioni delli effetti loro. 5.36 E qui si deve notare (il che anco a suo luogo si anderà avvertendo intorno a tutti gli altri strumenti mecanici) che la utilità, che si trae da tale strumento, non è quella che i volgari mecanici si persuadono, ciò è che si venga a superare, ed in un certo modo ingannare, la natura, vincendo

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con piccola forza una resistenza grandissima con l’intervento del vette; perché dimostreremo, che senza l’aiuto della lunghezza della lieva si saria, con la medesima forza, dentro al medesimo tempo, fatto il medesimo effetto. Imperò che, ripigliando la medesima lieva BCD, della quale sia C il sostegno, e la distanza CD pongasi, per esempio, quintupla alla distanza CB, e mossa la lieva sin che pervenga al sito ICG, quando la forza avrà passato lo spazio DI, il peso sarà stato mosso dal B in G; e perché la distanza DC si è posta esser quintupla dell’altra CB, è manifesto, dalle cose dimostrate, poter essere il peso, posto in B, cinque volte maggiore della forza movente, posta in D. Ma se, all’incontro, porremo mente al camino che fa la forza da D in I, mentre che il peso vien mosso da B in G, cognosceremo parimente il viaggio DI esser quintuplo allo spazio BG: in oltre, se piglieremo la distanza CL eguale alla distanza CB, posta la medesima forza, che fu in D, nel punto L, e nel punto B la quinta parte solamente del peso che prima vi fu messo, non è alcun dubbio, che, divenuta la forza in L eguale a quasto peso in B, ed essendo eguali le distanze LC, CB, potrà la detta forza, mossa per lo spazio LM, trasferire il peso a sé uguale per l’altro eguale intervallo BG; e che reiterando cinque volte questa medesima azione, trasferirà tutte le parti del detto peso al medesimo termine G. Ma il replicare lo spazio ML niente per certo è di più o di meno che il misurare una sol volta l’intervallo DI, quintuplo di esso LM: adunque il trasferire il peso da B in G non ricerca forza minore, o minor tempo, o più breve viaggio, se quella si ponga in D, di quello che faccia di bisogno quando la medesima fosse applicata in L. Ed insomma il commodo, che si acquista dal benefizio della lunghezza della lieva CD, non è altro che il potere muovere tutto insieme quel corpo grave, il quale dalla medesima forza, dentro al medesimo tempo, con moto eguale, non saria, se non in pezzi, senza il benefizio delle vette, potuto condursi. 5.37 Finalmente non è da passare sotto silenzio quella considerazione, la quale da principio si disse esser necessaria d’avere in tutti gl’instrumenti mecanici: cioè, che quanto si guadagna di forza per mezo loro, altrettanto si scapita nel tempo e nella velocità. Il che per avventura non potria parere ad alcuno così vero e manifesto nella presente speculazione; anzi pare che qui si mutliplichi la forza senza che il motore si muova per più lungo viaggio che il mobile. Essendo che se intenderemo, nel triangolo ABC la linea AB essere il piano dell’orizonte, AC piano elevato, la cui altezza sia misurata dalla perpendicolare CB, un mobile posto sopra il piano AC, e ad esso legata la corda ADF, e posta in F una forza o un peso, il quale alla gravità del peso E abbia la medesima proporzione che la linea BC alla CA; per quello che s’è dimostrato, il peso F calerà al basso tirando sopra il piano elevato il mobile E, né maggior spazio misurerà detto grave F nel calare al basso di quello che si misuri il mobile E sopra la linea AC. 5.38 Ma qui però si deve avvertire che, se bene il mobile E averà passata tutta la linea AC nel tempo medesimo che l’altro grave F si sarà per eguale intervallo abbassato, niente di meno il grave E non si sarà discostato dal centro comune delle cose gravi più di quello che sia la perpendicolare CB; ma però il grave F, discendendo a perpendicolo, si sarà abbassato per spazio eguale a tutta la linea AC. E perché i corpi gravi non fanno resistenza a i moti transversali, se non in quanto in essi vengono a discostarsi dal centro della terra, però, non s’essendo il mobile E in tutto il moto AC alzato più che sia la linea CB, ma l’altro F abbassato a perpendicolo quanto è tutta la lunghezza AC, però potremo meritamente dire, il viaggio della forza F al viaggio della forza E mantenere quella istessa proporzione, che ha la linea AC alla CB, cioè il peso E al peso F. Molto adunque importa il considerare per quali linee si facciano i moti, e massime ne i gravi inanimati: dei quali i momenti hanno il loro total vigore e la intiera resistenza nella linea perpendicolare all’orizonte; e nell’altre, trasversalmente elevate o inchinate, servono solamente quel più o meno vigore, impeto, o resistenza, secondo che più o meno le dette inclinazioni s’avvicinano alla perpendicolar elevazione. 5.39 Il che avendo dimostrato, faremo passaggio alle taglie, e descrivendo la girella inferiore ACB, volubile intorno al centro G, e da essa pendente il peso H, segneremo l’altra superiore EF; avvolgendo intorno ad ambedue la corda DFEACBI, di cui il capo D sia fermato alla taglia inferiore, ed all’altro I sia applicata la forza; la quale dico che, sostenendo o movendo il peso H, non sentirà altro che la terza parte della gravità di quello. Imperò che, considerando la

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5.40 È la presente speculazione stata tentata ancora da Pappo Alessandrino nel’8° libro delle sue Collezioni Matematiche; ma, per mio avviso, non ha toccato lo scopo, e si è abbagliato […]. Intendasi dunque il cerchio AIC, ed in esso il diametro ABC, ed il centro B, e due pesi eguali momenti nelle estremità A, C; sì che, essendo la linea AC un vette o libra mobile intorno al centro B, il peso C verrà sostenuto dal peso A. Ma se c’immagineremo il braccio della libra BC essere inchinato a basso secondo la linea BF, in guisa tale però che le due linee AB, BF restino salde insieme nel punto B, allora il momento del peso C non sarà più eguale al momento del peso A, per esser diminuita la distanza del punto F dalla linea della direzione che dal sostegno B, secondo la BI, va al centro della terra. Ma se tireremo dal punto F una perpendicolare alla BC, quale è la FK, il momento del peso in F sarà come se pendesse dalla linea KB. 5.41 Vedesi dunque come, nell’inclinare a basso per la circonferenza CFLI il peso posto nell’estremità della linea BC, viene a scemarsi il suo momento ed impeto d’andare a basso di mano in mano più, per esser sostenuto più e più dalle linee BF, BL. […] Se dunque sopra il piano HG il momento del mobile si diminuisce dal suo totale impeto, quale ha nella perpendicolare DCE, secondo la proporzione della linea KB alla linea BC o BF; essendo, per la similitudine de i triangoli KBF, KFH, la proporzione medesima tra le linee KF, FH che tra le dette KB, BF, concluderemo, il momento integro ed assoluto che ha il mobile nella perpendicolare all’orizzonte, a quello che ha sopra il piano inclinato HF, avere la medesima proporzione che la linea HF alla linea FK, cioè che la lunghezza del piano inclinato alla perpendicolare che da esso cascherà sopra l’orizonte. Sì che, passando a più distinta figura, quale è la presente, il momento di venire al basso che ha il mobile sopra il piano inclinato FH, al suo totale momento, con lo qual gravita nella perpendicolare all’orizonte FK, ha la medesima proporzione che essa linea KF alla FH. E se così è, resta manifesto che, sì come la forza che sostiene il peso nella perpendicolare FK deve essere ad esso eguale, così per sostenerlo nel piano inclinato FH basterà che siano tanto minore, quanto essa perpendicolare FK manca dalla linea FH. E perché, come altre volte s’è avvertito, la forza per muover il peso basta che insensibilmente superi quella che lo sostiene, però concluderemo questa universale proposizione: sopra il piano elevato la forza al peso avere la medesima proporzione, che la perpendicolare dal termine del piano tirata all’orizonte, alla lunghezza d’esso piano.

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A.6 Chapter 6 6.1 Ex ijs omnibus, quae hactenus de centro gravitatis dicta sunt, perspicuum est, unum quod [que] grave in eius centro gravitatis proprie gravitare, veluti nomen ipsum centri gravitatis idipsum manifeste praeseferre videtur. ita ut tota vis, gravitasque ponderis in ipso gravitatis centro coacervata, collectaque esse, ac tanquam in ipsum undique fluere videatur. Nam ob gravitatem pondus in centrum universi naturaliter pervenire cupit; centrum vero gravi tatis (exdictis) est id, quod proprie in centrum mundi tendit. in centro igitur gravitatis pondus proprie gravitat. Praeterea quando aliquod pondus ab aliqua potentia in centro gravitatis sustinetur; tunc pondus statim manet, totaque ipsius ponderis gravitas sensu percipitur. quod etiam contingit, si susteneatur pondus in aliquo puncto, a quo per centrum gravitatis ducta recta linea in centrum mundi tendat. hoc nam [que] modo idem est, ac si pondus in eius centro gravitatis proprie sustineretur. Quod quidem non contingit, si sustineatur pondus in alio puncto. ne [que] enim pondus manet, quin potius antequam ipsius gravitas percipi possit, vertitur uti [que] pondus, donec similiter a suspensionis puncto ad centrum gravitatis ducta recta linea in universi centrum recto tramite feratur. 6.2 Dicimus autem centrum gravitatis uniuscuiusque corporis punctum quoddam intra positum, a quo si grave appensum mente concipiatur, dum fertur quiescit; et servat eam, quam in principio habebat positionem: neque in ipsa latione circumvertitur. 6.3 Centrum gravitatis uniuscuiusque solidae figurae est punctum illud intra positum, circa quod undique partes aequalium momentorum consistunt. Si enim per tale centrum ducatur planum figuram quomodocunque secans semper in partes aequeponderantes ipsam dividet. 6.4 Poiché, sì come è impossibile che un grave o un composto di essi si muova naturalmente all’in su, discostandosi dal comun centro verso dove conspirano tutte le cose gravi, così è impossibile che egli spontaneamente si muova, se con tal moto il suo proprio centro di gravità non acquista avvicinamento al sudetto centro comune. 6.5 Suppositiones, et definitiones I. Ponatur eam esse centri gravitatis naturam, ut magnitudo libere suspensa ex quolibet sui puncto nunquam quiescat nisi cum centrum gravitatis ad infimum suae sphaerae punctum pervenerit. VI. Aequalia gravia ex aequalibus distantijs aequiponderant, sive libra ad horizontem parallela fuerit, sive inclinata. Et gravia eandem reciproce rationem habentia, quam distantiae, aequiponderant, sive libra sit ad horizontem parallela, sive inclinata. 6.6 Vulgatissima est etiam apud gravissimos viros obiectio illa, videlicet. Archimedem supposuisse aliquod falsum, dum fila magnitudinum ex libra pendentium consideravit tanquam inter se parallela, cum tamen re vera in ipso terrae centro concurrere debeant. Ego vero, (quod pace clarissimorum virorum dictum sit) crediderim fundamentum Mecanicum longe alia ratione esse considerandum. Concedo si Fisicae magnitudines ad libram libere suspendantur, quod fila materialia suspensionum convergentia erunt; quandoquidem singula ad centrum terrae respiciunt. Verum tamen si eadem libra, licet corporea, consideretur non in superficie terrae, sed in altissimis regionibus utra orbem solis; tum fila (dummodo adhuc ad terrae centrum respiciant) multo minus convergentia inter se erunt; sed quasi aequidistantia. Concipiamus iam ipsam libram Mecanicam ultra stellatam libram firmamenti in infinitam distantiam esse provectam; quis non intelligit fila suspensionum iam non amplius convergentia, sed exacte parallela fore? 6.7 Tunc itaque falsum dici poterit fundamentum Mecanicum, nempe fila librae parallela esse, quando magnitudines ad libram appensae Fisicae sint, realesque, et ad terrae centrum conspirantes. Non autem falsum erit, quando magnitudines (sive abstractae, sive concretae sint) non ad centrum terrae, neque ad aliud punctum propinquum librae respiciant; sed ad aliquod punctum infinite distans connitantur.

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6.8 Non me latet auctorum controversiam, circa libram inclinatam, an redeat, maneatve supponere centra magnitudinum in ipsa libra esse collocata [il corsivo è nostro]. Nos tamen, quia in libello, semper considerabimus magnitudines infra libram appensas, maluimus rei nostrae servire, quam aliorum controversiae demonstrationem accomodare. 6.9 Quando noi ammettiamo che i pesi della libbra abbiano inclinazione verso il centro della Terra […] ne seguirà che non ci sia libbra orizzontale con braccia disuguali e con pesi in reciproca proporzione della lunghezza delle braccia, sicché detti pesi facciano equilibrio. 6.10 Ora posto che B figuri il centro, ed AC una Libbra di braccia uguali con due pesi uguali nelle estremità A, C, i cui momenti o gravità son misurate dalle perpendicolari DF, DE, siccome dichiara Giov. Battista de’ Benedetti nel suo libro Delle speculazioni matematiche, capitolo III ovvero IV; ne segue che il momento del peso in A, al momento del peso in C, sia reciprocamente come la retta BC alla retta AB, cioè reciprocamente come la distanza dei pesi dal centro della Terra. E qui abbiamo, non solamente che il peso più vicino al centro, mentre è nella Libbra, pesa più del meno vicino, ma sappiamo ancora in qual proporzione più pesa. 6.11 Propositio II Momenta gravium aequalium super planis inaequaliter inclinatis, eandem tamen elevationem habentibus, sunt in reciproca ratione cum longitudinibus planorum. 6.12 Scio Galileum ultimis vitae suae annis suppositionem illam demonstrare conatum, sed quia ipsius argomentatio cum lib. de Motu edita non est pauca haec de momentis gravium libello nostro praefigenda duximus; ut appareat quod Galilei suppositio demonstrari potest, et quidem immediate ex illo Theoremate quod pro demonstrato ex Mechanicis ipse desumit in secunda parte sextae Propositionis de motu accelerato, videlicet., esse inter se ut sunt perpendicula partium aequalium eorumdem planorum. 6.13 Praemittimus Duo gravia simul coniuncta ex se moveri non posse, nisi centrum commune gravitatis ipsorum descendat. Quando enim duo gravia ita inter se coniuncta fuerint, ut ad motum unius motus etiam alterius consequatur, erunt duo illa gravia tamquam grave unum ex duobus compositum, sive id libra fiat, sive trochlea, sive qualibet alia Mechanica ratione, grave autem huiusmodi non movebitur unquam, nisi centrum gravitatis ipsius descendat. Quando vero ita constitutum fuerit ut nullo modo commune ipsius centrum gravitatis descendere possit, grave penitus in sua positione quiescet: alias enim frustra moveretur; horizontali, scilicet latione, quae nequa? quam deorsum tendit. 6.14 Connectantur etiam aliquo imaginario funiculo per ACB ducto, adeo ut ad motum unius motus alterius consequatur. 6.15 Proposition I Si in planis inaequaliter inclinatis, eandem tamen elevationem habentibus, duo gravia constituantur, quae inter se eandem homologe rationem habeant quam habent longitudines planorum, gravia aequale momentum habebunt. 6.16 Duo ergo gravia simul colligata mota sunt, et eorum commune centrum gravitatis non descendit. Quod est contra praemissam aequilibrij legem. 6.17 Propterea magnitudines aequiponderabunt etiam dum ad libram AC suspenduntur: alias, si moverentur, commune centrum gravitatis ipsarum, quod demonstratum est esse in perpendiculo DF, ascenderet. Quod est impossibile. 6.18 La potenza in A alla potenza in C, sta reciprocamente come la retta CB alla CA. La dimostrazione […] dipende dalle velocità perché muovendosi così la stanga AC, radente le due linee dell’angolo retto ABC, le velocità nelle quali sta costituito il punto A alla velocità nella quale sta costituito il punto B, sta come la BC alla BA.

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6.19 Fra gli effetti della meccanica osservati uno se ne trova non avvertito ancora da alcuno che io sappia e pur da esso possono derivare cognizioni di qualche momento e di molta curiosità. Supposta una muraglia verticale AE alla quale nel piano orizzontale EF sia normale la retta EF, e supposta inoltre la trave diritta BCF il di cui centro di gravità sia C, la quale coll’estremità superiore B si appoggi alla muraglia accennata, e possa coll’estremità F scorrere liberamente sopra il Piano EF, si cerca la proporzione del peso della trave a quella forza, la quale applicata in F e spingendo direttamente per la direzione FE può equilibrare il momento della trave a scorrere in virtù del suo peso per le direzione EF. Si supponga la forza richiesta eguale ad un peso attaccato nel punto Z a una corda di data lunghezza FEZ, la cui lunghezza chiameremo λ, e che passi per il punto E. Dal Centro di gravita C si abbassi sopra la FE la normale CG, e sia la ragione di BF ad FC la stessa che la ragione di 1 ad x, avremo per la similitudine dei triangoli BE : CG = BF : FC = 1 : x e conseguentemente, posto P il peso della trave, sarà la distanza dell’orizzontale EF dal peso accennato, che può intendersi raccolto bel centro di gravità della trave, eguale a, P· CG, o veramente scrivendo invece di CG l’eguale x ·BE sarà la distanza della retta EF dal peso P eguale ad x· P ·BE. 6.20 Sia il peso che si cerca attaccato al punto Z della corda FEZ eguale a Q, sarà la distanza del peso accennato dall’orizzontale eguale a Q· ZE, e la distanza del centro di gravità comune dei due pesi P e Q sotto l’orizzontale eguale a Q · ZE − P · CG = Q · ZE − x · P · BE. Si intenda descritto il centro L e col raggio LM eguale a BF il quarto di una circonferenza circolare SMs. Sia la retta Ls parallela all’orizzontale e riponga l’ascissa LO eguale a BE e si tiri l’ordinata OM, la quale necessariamente sarà eguale a EF, e dal punto M si conduca la tangente alla circonferenza circolare nM alla quale sia parallela la retta LrR e si prolunghi l’ordinata MO fino che incontri la retta LrR in R, sarà la distanza dal centro di gravità comune dei due pesi P e Q dalla retta EF, che si è dimostrata eguale a Q ·ZE − x · P· BE, eguale ancora a Q · λ − Q · EF −4 x · P · BE, cioè a dire eguale a Q · λ − Q · OM − x · P · LO. 6.21 Se i due pesi eguali A, B sono legati ad un filo, passato sopra una carrucola o altro sostegno, che possano scorrere questi staranno in equilibrio, dovunque si saranno situati. 6.22 Perché se si movessero tanto acquisterebbe l’uno che scendesse, quanto acquisterebbe l’ altro che salisse, essendo i loro modi eguali, e per linee perpendicolari. E se possibile si muovano dal sito A , B nel sito C, D; è manifesto che, giunti li centri di gravità in linea retta, il centro comune di A, B verrà in mezzo, cioè in E, ed il centro di gravità di C, D verrà in mezzo, cioè in F; perché essendo CA, BD uguali tra toro e parallele, congiunte CD, AB si segano nella medesima proporzione e nel mezzo, onde il centro comune non si sarà mosso, e non avrà acquistato niente, sicchè i gravi A, B non si moveranno dal loro sito, in che furono posti. 6.23 Ma se il peso B sarà maggiore del peso A, quello scenderà, perché il centro comune loro è fuori del mezzo della BA, come in E, più vicina al centro B, ed è in luogo può può scendere sempre per la linea perpendicolare EG. 6.24 Moveatur autem et ex semidiametro BE centro B portio circuli describatur EH, quae secet BG in H, et BF in I; Et quia EM semidiametro BK perpendicularis per B, centrum non transit, erit EM ipsa BK, hoc est, BI brevior. Abscindatur ex BI, ipsi EM aequalis LB. Erit igitur punctum L infra punctum I, hoc est, ipso I, mundi centro propius. Necesse igitur erit ad hoc ut murus corruat, centrum gravitatis E facta circa B, conversione aliquando fieri in I, ut demum transferri possit in H, sed I remotius est a mundi centro ipsis E, L, ascendet igitur grave contra sui naturam ex E in I, at hoc est impossibile; quod fuerat demonstrandum. 6.25 Sunt autem trianguli ABF, ACF, aequales et aequeponderantes. angulus vero AFC rectus. lungatur EC, erit igitur maior EC, ipsa EF. Rotetur iraque triangulum circa punctum C, fiatque; EC horizonti perpendicularis, sit que GH, et per E horizonti parallela ducatur EK, moto igitur triangulo, centrum gravitatis E translatum erit in H, sed KC aequalis est EF, minor autem ipsa CH, elevatur ergo centrum gravitatis ab E in H, nempe supra K, totum spatium KH. ex qua elevatione fit in motu difficultas.

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A.7 Chapter 7 7.1 Vitruve fait mention de cette sorte de machine, dite des Grecs troclearum, la quelle a son mouvement par le moiey de poulies [...] un bout sera attache à la mousse e l’autre bout servira pour tirer le fardeau, comme il se peut voir en la figure si l’on tire le dit bout de corde marqué G un pied en bas, le fardeau qui sera attaché à la mousse E en mesme temps levera un demi pied, et ce d’autant que la corde est passee double aux polies, ainsi si l’on tire 20. pieds de corde, le fardeau ne levera que 10 aussi un homme tirera aussi pesant avec cette machine, comme en seroient deux, si la machine estoit simple mais les deux hommes tireront en mesme temps le double de la hauteur savoir 20 pieds, avant que l’autre en aye tiré plus de dix, et si aux mousses il y avoit deux poulies, comme la figure M, la force seroit quadruple, mais aussi ne monteroit le fardeau que 5 pieds en tirant 20 pieds de corde. Les roues dentelees se font encores avec la mesme raison comme les precedentes, car en augment tant la force, l’on augmente proportionnellement le temps [...] tellement qu’un homme seul, fera autant de force tirant un fardeau par cette machine comme huit homme [...] ma aussi si les huit hommes son une heure à lever leur pois, l’homme sera huit heures à lever le sien. 7.2 Aux poids equilibres comme le plus pesant est au plus leger, ainsi l’espace du plus leger est à l’espace du plus pesant, ainsi aussi est la perpendiculaire du mouvement du plus leger à la perpendiculaire du mouvement du plus pesant. 7.3 Car en mesme tens que le poids G descend du point C au point B, le poids D monte du point A au point E et par consequent BC sera la perpendiculaire des poids G et EF du poids D: pourtant puisque D est à G, comme la perpendiculaire BC à la perpendiculaire EF, les poids D et G seront equilibres à raisons des leurs situations. 7.4 Maintenant par la 2. prop. nous avons veu que si CA est le bras d’une balance sur le quel soit le poids A retenue par la chorde CA qu’il ne glisse le long du bras CA, et comme CB est a CF, ainsi soit le poids A a la puissance Q ou E tirant par la chorde QA, cette puissance Q ou E tiendra la balance CA en equilibre, et la chorde QA estant attache au centre du poids A, la balance demeura deschargee, et le poids A sera soustenu partie par la puissance Q ou E, parte par le plain LN2 perpendiculaire à la balance CA; ou en la place du pla LN2 par la chorde CA, par le Scholie du 4 axiome. 7.5 Scholie VIII. [...] le poids est posé en A sur les chordes CA et QA soustenues par les puissances C, Q ou K, E, le poids estant aux puissances comme les perpendiculaires CB et QG sont aux lignes CF et QD. […] Si au dessus du poids A, dans sa ligne de direction, on prend quelque ligne comme AP, il arrivera que si le poids A descend jusques en P, tirant avec soy les chordes et faisant monter les puissances KE, il y aura reciproquement plus grande raison du chemin que les puissances feront en montant, au chemin que le poids fait en descendant, que du mesme poids aux deux puissances prises ensemble; ainsi les puissances monteroient plus à proportion, que le poids ne descendroit en les emportant, qui est contre l’ordre commun. Que si au dessus du poids A, dans sa ligne de direction, on prend une ligne, comme AV, que le poids monte jusques en V, les chordes montants aussi emportees par les puissances KE qui descendent, il y aura reciproquement plus grande raison du chemin que le poids sera en montant, au chemin que les puissances seront en descendant, que les deux puissances prises ensemble, au poids; ainsi le poids monteroit plus à proportion que les puissances ne descendroient en l’emportant, e qui est encore contre l’ordre commun, dans lequel le poids ou la puissance qui emporte l’autre, sait toujours plus de chemin à proportion, que le poids ou la puissance qui est emportee. Or que les raisons des chemins que seroient les poids A et ses puissances en montant, et descendant, soient telles que nous venons de dire, et contre l’ordre commun, on

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en, trouvera la demonstration dans nos Mechaniqes, car elle est trop longue pour estre mise icy. Partant le poids A en subsistant et demeurant en son lieu, par les raisons de la 3. Prop. demeure ainsi dans l’ordre commun, ce que nous voulions remarquer. 7.6 Qu’il ne faut ny plus ny moins de force, pour lever un cors pesant a certaine hauteur, que pour en lever un autre moins pesant a une hauteur d’autant plus grande qu’il est moins pesant, ou pour en lever un plus pesant a une auteur d’autant moindre. […] Ce qu’on accordera facilement, si on considere que l’effect doit tousiours estre proportionné a l’action qui est necessaire pour le produire, et ainsy que, s’il est necessaire d’employer la force par la quelle on peut lever un poids de 100 livres a la hauteur de deux pieds, pour en lever un a la hauteur d’un pied seulement, cela tesmogne que cetuy pese 200 livres. 7.7 Il faut sur tout considerer que j’ai parlé de la force qui sert pour lever un poids a quelque hauteur, la quelle force a toujours deux dimensions & non de celle qui sert en chasque point pour le soutenir, la quelle n’a jamais qu’une dimension, en sorte que ces deux forces differerent autant l’une de l’autre q’une superficie differe d’une ligne. Car la mesme force que doit avoir un clou pour soustenir un poids de 100 livres un moment de tems, luy suffit pour soutenir un an durant, pourvu qu’elle ne diminue point. Mais la mesme quantité de cete force qui sert a lever ce poids a la hauteur d’un pied ne suffit pas eadem numero pour le lever a la hauteur de deux pieds, & il n’est pas plus clair que deux & deux font quatre, qu’il est clair qu’il y en faut employer le double. 7.8 Car je ne dis pas simplement que la force qui peut lever un poids de 50 livres a la hauteur de 4 pieds, en peut lever un de 200 livres a la hauteur d’un pied, mai je dis qu’elle le peut, si tant est quelle lui soit appliquée. Or est-il qu’il est impossible de l’y appliquer par le moyen de quelque machine ou autre invention qui face que ce poids ne se hausse que d’un pied, pendant que cete force agira en tout la longueur de quatre pieds, & ainsy qui transforme le rectangle par lequel est representée la force qu’il faut pour lever ce poids de 200 livres a la hauteur d’un pied, en un autre qui soit egal & semblable a celuy qui represente la force qu’il faut pour lever un poids de 50 livres a la hauteur de 4 pieds. 7.9 Car c’est le mesme de lever 100 livres a la hauteur d’un pied, et derechef encore 100 a la hauteur d’un pied, que d’en lever 200 a la hauteur d’un pied, et le mesme aussy que d’en lever cent a la hauteur de deux pieds. 7.10 L’invention de tous ces engins n’est fondée que sur un seul principe, qui est que la mesme force qui peut lever un poids, par exemple, de cent, livres a la hauteur de deux pieds, en peut aussy lever un de 200 livres, a la hauteur d’un pied, ou un de 400 a la hauteur d’un demi pied, & ainsy des autres, si tant est qu’elle luy soit appliquée. […] Or les engins qui servent a faire cete application d’une force qui agist par un grand espace a un poids qu’elle fait lever par un moindre, sont la poulie, le plan incliné, le coin, le tour ou la roue, la vis le levier; et quelques autres. 7.11 La poulie. Soit ABC une chorde passée autour de la poulie D, a laquelle poulie soit attaché le poids E. Et premierement supposant que deux hommes soutienent ou haussent egalement chascun un des bouts de cete chorde, il est evident que si ce poids pese 200 livres, chascun de ces hommes n’employera, pour le soutenir ou soulever, que la force qu’il faut pour soutenir ou soulever 100 livres; car chascun n’en porte que la moitié. Faisons apres cela qu’A, l’un des bouts de cete chorde, estant attaché ferme a quelque clou, l’autre C soit derechef soutenu par un homme; & il est evident que cet homme, en C, n’aura besoin, non plus que devant, pour soutenir le poids E, que de la force qu’il faut pour soutenir cent livres: a cause que le clou qui est vers A y fait le mesme office que l’homme que nous y supposions auparavant. Enfin, posons que cet homme qui est vers C tire la chorde pour faire hausser le poids E; & il est evident que, s’il y employe la force qu’il faut pour Iever 100 liures a la hauteur de deux pieds,

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Appendix. Quotations il fera hausser ce poids E, qui en pese 200, de la hauteur d’un pied car la chorde ABC estant doublée comme elle est, on la doit tirer de deux pieds par le bout C pour faire autant hausser le poids E que si deux hommes la tiroient, l’un par le bout A & L’autre par le bout C, chascun de la longueur d’un pied seulement. Il y a toutefois une chose qui empesche que ce calcul ne soit exact, a scavoir la pesanteur de la poulie, & la difficulté qu’on peut voir a faire couler la chorde & a la porter. Mais cela est fort peu a comparaison de ce qu’on leve, & ne peut estre estimé qu’a peu pres.

7.12 Ainsy donc, pour ne point faillir, de ce que le clou A soutient la moitié du poids B, on ne doit conclure autre chose sinon que, par cete application, l’une des dimensions de la force qui doit estre en C, pour lever ce poids, diminué de moitié, & que l’autre en suite devient double. De façon que, si la ligne FG represente la force qu’il faudroit pour soutenir en un point le poids B, sans l’ayde d’aucune machine, & le rectangle GH, celle qu’il faudroit pour le lever a la hauteur d’un pied, le soutien du clou A diminué de moitié la dimension qui est représentée par la ligne FG, & le redoublement de la chorde ABC fait doubler l’autre dimension, qui est representée par la ligne FH; & ainsy la force qui doit estre en C, pour lever le poids B a la hauteur d’un pied, est representée par le rectangle IK. Et comme on sçait en Geometrie qu’une ligne estant adioustée ou ostée d’une superficie, ne l’augmente ny ne la diminué de rien du tout, ainsy doit on icy remarquer que la force dont le clou A soutient le poids B, n’ayant qu’une seule dimension, ne peut faire que la force en C, considérée selon ses deux dimensions, doive estre moindre pour lever ainsy le poids B que pour le lever sans poulie. 7.13 La levier. Et pour mesurer exactement qu’elle doit estre cete force en chasque point de la tigne courbe ABCDE, il faut scavoir qu’e!le y agit tout de mesme que Ii elle trainoit le poids fur un plan circulairement incliné, & que l’inclination de chascun des poins de ce plan circulaire se doit mesurer par celle de la ligne droite qui touche le cercle en ce point. Comme par exemple quand la force est au point B, pour trouver la proportion qu’elle doit avoir avec la pesanteur du poids qui est alors au point G, il faut tirer la contingente GM, & penser que la pesanteur de ce poids est a la force qui est requise pour le trainer sur ce plan, & par consequent aussy pour le hausser suivant le cercle FGH, comme la ligne GM ella SM. Puis a cause que BO est triple de OG, la force en B n’a besoin d’estre a ce poids en G, que comme le tiers de la ligne SM est a la toute GM. Tout de mesme quand la force est au point D. 7.14 Plusieurs ont coustume de confondre la consideration de l’espace avec celle du tems ou de la vitesse, en sorte que, par exemple, au levier, ou, ce qui est le mesme, en la balance ABCD, ayant suppose que le bras AB est double de BC, & que le poids en C est double du poids en A, & ainsy qu’ils sont en equilibre, au lieu de dire que ce qui est cause de cet equilibre est que, si le poids C soulevoit ou bien estoit soulevé par le poids A, il ne passeroit que par la moité d’autant d’effect que luy, il disent qu’il iroit de la moitié plus lentement, ce qui est une faute d’autant plus nuisible qu’elle est plus malaysée a reconnoistre; car ce n’est point la difference de la vitesse qui fait que ces poids doivent estre l’un double de l’autre, mais la difference de l’espace. 7.15 Comme il paroist de ce que, pour lever, par exemple, le poids F avec la main iusques à G, il n’y faut point employer une force qui soit iustement double de celle qu’on y aura employée le premier coup, si on le veut lever deux fois plus viste; mais il y en faut employer une qui soit plus ou moins grande que la double, selon la diverse proportion que peut avoir cete vitesse avec les causes qui luy resistent; au lieu qu’il faut une force qui soit justement double pour le lever avec mesme vitesse deux fois plus haut, a sçavoir jusques a H. Je dis qui soit justement double, en contant qu’un & un sont justement deux: car il faut empolyer certaine quantité de cete force pour lever ce pois d’F jusques a G, & derechef encore autant de la mesme force pour le lever de G jusques a H. 7.16 Et au contraire, prenant un eventail en vostre main, vous le pourrez hausser, de la mesme vistesse qu’il pourrait descendre de soy mesme dan l’air, si vous le laissez tomber, sans qu’il vous y faille employer aucune force, excepté celle qu’il faut pour le soustenir; mais pour le

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hausser ou baisser deux fois plus viste, il vous faudra employer quelque force qui sera plus que double de l’autre, puis qu’elle estoit nulle. 7.17 Or la raison qui fait que je reprens ceux qui se servent de la vitesse pour expliquer la force du levier, & autres semblantes, n’est pas que je nie que la mesme proportion de vitesse ne s’y rencontre tousiours; mais pouce que ceste vitesse ne comprend pas la raison pour laquelle la force augmente ou diminue, comme fait la quantité de l’espace, & qu’il y a plusieurs autres choses à considerer touchant la vitesse, qui ne sont pas aysées à expliquer. 7.18 Pour ce qu’a écrit Galilee touchant la balance & le levier, il explique fort bien quod ita sit, mais non pas cur ita sit, comme je fais par mon Principe. Et pour ceux qui disent que le devois considerer la vitesse, comme Galilée, plutost que l’espace, pour rendre raison des Machines, je croy, entre nous, que ce sont des gens qui n’en parlant que par fantaisie, sans entendre rien en cette matiere. 7.19 Que la pesanteur relative de chaque cors, ou ce qui est le mesme, la force qu’il faut employer pour le soutenir & empescher qu’il ne descende, lors qu’il est en certaine position, se doit mesurer par le commencement du mouvement que devroit faire la puissance qui le soustient, tant pour le hausser que pour le suivre s’il s’abaissoit. En sorte que la proportion qui est entre la ligne droite que descriroit ce mouvement, & celle qui marqueroit de combien ce cors s’approcheroit cependant du centre de la terre, est la mesme qui est entre la pesanteur absolute & la relative. 7.20 Soit AC un plan incliné sur l’horizon BC, et qu’AB tende a plomb vers le centre de la terre. Tous ceux qui escrivent des Mechaniques assurent que la pesanteur du poids F, en tant qu’il est appuié sur ce plan AC, a mesme proportion a sa pesanteur absolue que la ligne AB a la ligne AC. […] Ce qui n’est pas toutefois entierement vray, sinon lorsqu’on suppose que les cors pesans tendent en bas suivant des lignes paralleles, ainsy qu’on fait communement, lors qu’on ne considere les Mechaniques que pour les rapporter a l’usage; car le peu de difference que peut causer l’inclination de ces lignes, entant qu’elles tendent vers le centre de la terre, n’est point sensible. […] Et pour scavoir combien il pese en chascun des autres points de ce plan au regard de cete puissance, par exemple au point D, il faut tirer une ligne droite, comme DN, vers le centre de la terre, et du point N, pris a discretion en cete ligne, tirer NP, perpendiculaire sur DN, qui rencontre AC au point P. Car, comme DN est a DP, ainsy la pesanteur relative du poids F en D est a sa pesanteur absolue. 7.21 Notez que le dis commencer a descendre, non pas simplement descendre, a cause que ce n’est qu’au commencement de cete descente a laquelle il faut prendre garde. En sorte que si, par exemple, ce poids F n’estoit pas appuié au point D sur une superficie plate, comme est supposée AD C, mais sur une spherique, ou courbée en quelque autre facon, comme EDG, pourvu que la superficie plate, qu’on imagineroit la toucher au point D, sur la mesme que ADC, il ne peseroit ny plus ny moins, au regard de la puissance H, qu’il fait estant appuié sur le plan AC. Car, bien que le mouvement que seroit ce poids, en montant ou descendant du point D vers E ou vers G sur la superficie courbe EDG, sust tout autre que celuy qu’il seroit sur la superficie plate ADC, toutefois, estant au point D sur EDG, il seroit determiné a se mouvoir vers le mesme costé que s’il estoit sur ADC, a scavoir vers A ou vers C. Et il est evident que le changement qui arrive a ce mouvement, sitost qu’il a cessé de toucher le point D, ne peut rien changer en la pesanteur qu’il a, lorsqu’il le touche. 7.22 Itaque theoriae magis insistendum puto, in qua si quis exercitatus fuerit, nullo negotio illam in opus educere poterit, idque sine periculo fiet, cum vulgo non pateat. Alioqui periculum est, ne si particularia tradantur iis contenti homines, ut fieri solet universalem cognitionem & causarum inquisi tionem negligant, pereatque scientia.

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7.23 Theorema I Duarum virium connexarum, quarum (si moveantur) motus erunt ipsis >antipeponjwz ˜ proportionales neutra alteram movebit, sed equilibrium facient. 7.24 Ut igitur hunc tractatulum concludamus, ac velut in summam contrahamus: In motibus ciendis tria sunt consideranda. Vis qua motum ciere volumus, vis quam movere volumus, & motum quo movere volumus: duo enim quælibet ex illis tertium determinant. Si enim vi parva vim magnam movere volumus, id nonnisi parvo motu facere possumus: si vero vim aliquam magno motu movere velimus, vi magna movente ad id opus est [...]: ut puta, si libra una centum libras movere velimus, oportet motum illius, motu huius centuplo maiorem esse. Si vero velimus libra una aliam vim ita movere, ut ea centuplo citius moveatur, quam librae illius pondus, illam centesimam tantum librae unius partem esse necesse est: si vero libram unam ita movere velimus, ut centuplo citius moveatur, quam vis quae illam movebit, vi centum libris maiore ad id opus erit. Neque patitur natura sibi in his vim fieri: si enim eiusmodi proportio aliquo modo infringi posset, statim daretur a>ut’wma >end’elecez, vel ut vocant, motus perpetuus in perpetua materia. 7.25 D’où il paroît qu’un vaisseau plein d’eau est un nouveau principe de mécanique, et une machine nouvelle pour multiplier les forces à tel degré qu’on voudra, puisqu’un homme, par ce moyen, pourra enlever tel fardeau qu’on lui proposera. Et l’on doit admirer qu’il se rencontre en cette machine nouvelle cet ordre constant qui se trouve en toutes les anciennes; savoir, le levier, le tour, la vis sans fin, etc., qui est, que le chemin est augmenté en même proportion que la force. Car il est visible, comme une de ces ouvertures est centuple de l’autre, si l’homme qui pousse le petit piston, l’enfonçoit d’un pouce, il ne repousseroit l’autre que de la centième partie seulement: car comme cette impulsion se fait a cause de la continuité de l’eau, qui communique de l’un des pistons à l’autre. 7.26 Je prends pour principe, que jamais un corps ne se meut par son poids, sans que son centre de gravité descende. D’où je prouve que les deux pistons figurés en la figure 7, sont en équilibre en cett sorte; car leur centre de gravité commun est an point qui divise la ligne, qui joint leurs centres de gravité particuliers, en la proportion réciproque de leurs poids; qu’ils se meuvent maintenant, s’il est possible: donc leurs chemins seront entre eux comme leurs poids réciproquement, comme nous avons fait voir: or, si on prend leur centre de gravité commun en cette seconde situation, on le trouvera précisément au même endroit que la première fois; car il se trouvera toujours au point qui divise la ligne, qui joint leurs centres de gravité particuliers, en la proportion réciproque de leurs poids; donc à cause du parallélisme des lignes de leurs chemins, il se trouvera en l’intersection des deux lignes qui joignent les centres de gravité dans les deux situations: donc le centre de gravité commun sera au même point qu’auparavant: donc les deux pistons considérés comme un seul corps, se sont mus, sans que le centre de gravité commun soit descendu; ce qui est contre le principe: donc its ne peuvent se mouvoir: donc its seront en repos, c’est-à-dire, en équilibre; ce qu’il falloit démontrer. 7.27 I Definitio. Statica est quae ponderis et gravitatis corporum rationes, proportiones, et qualitates interpretatur. 7.28 Ex his consequens est nullum corpus sive solidum in rerum Natura esse, ut mathematice loquar, praeter Globum, quod est suae gravitatis centro cogitatione suspensum, quemlibet datum situm retinet, sive per quod planum quodlibet ipsum corpus in partes situ aequipondias dividit, verum propter varios et infinitos situs, varia etiam et infinita gravitatis centra erunt. 7.29 Verum, quandoquidem discriminem illud, in iis quae ab hominibus ponderantur, nullum, saltem inobservabile est, iugum enim aliquot milia longum esse debet, antequam deprehendi posset, perpendiculares parallelas habendas esse concedi nobis postulamus. 7.30 BN ducatur, secans AC continuatam in N, consimiliter D O secans continuatam LI, hoc est, latus columnæ in O, ut angulus IDO aequalis sit angulo CBN. Appendatur quoque ad DO pondus

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P oblique attollens, quod (amotis M, E ponderibus) columnam in suo situ conservet. Quia vero DL & BA, item DI & BC latera triangulorum DLI & BAC homologa sunt, hujusmodi conclusio inde deducitur. Quemadmodum BA ad BC: ita sacoma lateris B A ad anti sacoma lateris BC (per 2 consectarium) item quemadmodum DL ad DI: ita sacoma lateris DL ad antisacoma lateris DI, hoc estita M ad E. sed homologa latera triangulorum similium ABN, LDO sunt AB & DL, item BN, & DO. Itaque ut supra, quemadmodum BA ad B N: ira sacoma B A ad anti sacoma B N (per 1 consectarium) Et quemadmodum DL ad DO: ita illius sacoma ad hujus anti sacoma, id est, M ad P. si linea BN à puncto B aliovorsum; A scilicet versus, ultra BC fuisset ducta, etiam recta DO à D ultra DI cecidisset, hoc est, ut nunc citra: ita tunc ultra cecidisset, & praecedens demonstratio etiam isti situi accommoda fuisset, hoc est, quemadmodum BA ad BN ita sacoma lateris BA, ad anti sacoma lateris BN esset: & quem-admodum DL ad DO: ita sacoma lateris DL, ad anti sacoma lateris DO. hoc est M ad P. Ut ista proportio non tantum in exemplis valeat, in quibus linea attollens, ut DI, perpendicularis est axi, sed etiam in aliis cuiusmodi cunque sint anguli. 7.31 Si columna, & duo pondera oblique extollentia situ aequilibria sunt, erit quemadmodum linea oblique extollans, ad lineam rectè extolletem: ita ponderum quodque obliquum ad suum pondus rectum. 7.32 Causam aequilibritatis situs non esse in circulis ab extremitatibus radiorum descriptis. Cur pondera aequalia in aequalibus radiis situ aequiponderent, communi notione scitur: at non perinde patet causa aequamenti ponderum inaequalium in radiis disparibus, quique ponderibus suis reciproce proportionales sint hanc veteres circuli de circinatis a radiorum extremitatibus in esse crediderunt, quemadmodum apud Aristotelem in Mechanicis eiusque spectatores videre licet, quod falsum esse hoc pacto redarguimus. Quiescen nullum describit circulum, Duo situ aequilibria quiescunt, Itaque duo situ aequilibria nullum describunt circulum. Et consequenter nullus erit circulus; atqui sublato circulo etiam causa tollitur quae ipsi subest, quae causa aequilibrium situs in circuli hic non latet. 7.33 Ipsique globi ex sese continuum et aeternum motum efficient, quod est falsum. 7.34 Propositio. Ponderum trochleis sublime tractorum formas inquirere. Priusquam rem ipsam exordimur generaliter intelligito, et cogitatione concipito, datum pondus hic constitui a trochlea infima cum pondere ipsi alligato: praeterea differentiam gravitatis quae a funibus existit, nullius momentia nobis nunc aestimari. I Exemplum ponderum quae recta attolluntur. Esto in primo hoc diagrammate trochlea A, ex qua dependet pondus B, funis CD F, cuius duae partes CD, FE parallelae sint, et utraque horizonti perpendicularis, Quibus positis, totoque pondere B ita è duabus istis partibus CD, FE suspenso, ututraque pars pari potentia afficiatur, etiam singulis propter orbiculi volubilitatem cedet semissis ponderis B. quamobrem si quis manu sua funem in F sustineat, is ferret gravitatem dimidii ponderis B, ex quo liquet, cur etiam unica trochlea facilius, quam sine ea pondus attollatur. 7.35 Notato autem hic illud Staticum axioma etiam locum habere: ut spatium agentis, ad spatium patients: Sic potentia patientis, ad potentiam agentis. 7.36 Sit [nodus] in E. Ergo PE raccourci de DE QE de AE SE enlongé de BE Optortet DE in p + AE in q − BE in t = 0.

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7.37 CA in P − CD in R − CO in S = 0 sin TEP in P − sin TER in R − sin TES in S sin PER in R − sin PES in S − sin PET in T sin RES in S − sin RET in T − sin REP in P sin SET in T + sin SEP in P − sin SER in R

=0 =0 =0 =0

ou es + at f r − ct r= e d d p − f tt ap − br t= . s= e b p=

7.38 Gravitas, est vis motrix, deorsum; sive ad Centrum Terrae. Quodnam sit, in consideratione physica, Gravitatis principium, non hic inquisimus. Neque etiam, an Qualitas dicit debeat, aut, corporis Affectio; aut, quo alio nomine censeri par sit. Sive enim ab innata qualitate in ipso gravi corpore; sive a communi circumstatium vergentia ad centrum; sive ab electrica vel magnetica Terrae facultate, quae gravia ad se alliciat; et effluviis suis, tamquam catenulis, attrahat; sive alias undecunque; (de quo non est ut hic moveamus litem) sufficit ut Gravitas nomine, eam intelligamus, quam sensu deprehendimus, Vim deorsum movendi, tum ipsum Corpus grave, tum quae obstant minus efficacia impedimenta. Per pondus intelligo gravitatis mensuram. 7.39 Prop. I Gravia, caeteris paribus, gravitant in ratione Ponderum. Et, universaliter, Vires Motrices, quaelibet, agunt pro Virium ratione. 7.40 Prop. II Grave, quatenus non impeditur, Descendit; seu propius ad Terrae Centrum appropinquat. Et universaliter, vis quaevis Motrix, secundum Directionem suam, quatenus non impeditur, procedit. 7.41 Prop. III Grave tantumdem Descendit, quanto sit Terra Centro propius: Tanto ascendit, quanto remotius. Et Universaliter, Cuiusvis Vis Motricis progressus tantus est, quantum secundum directionem suam movetur; Regressus, Contra. 7.42 Prop. V Gravium Descensus, invicem comparati, in ea ratione pollent, quae ex Ponderum ratione et ratione Altitudinum Descensuum componitur. Atque Ascensus similiter. Adeoque; Sit Pondera sint aequalia, in ratione Altitudinum: si Altitudines sint aequales; in ratione Ponderum: si Pondera et Altitudines, vel utraque sint aequalia, vel sint reciproce proportionalia; Aequipollent. Et, universaliter, Virium Motricium, quarumcunque Progressus Regressive, pollent in Ratione, qua ex ratione Virium, et Progressuum Regressuumve secundum lineam Directionis Virium Aestimatorum, componitur. 7.43 Prop. VI Conjunctis invicem Descensu et Ascensu; si praepollet Descensus, pro Descensu simpliciter habendi sunt: Pro Ascensu vero, si Ascensu Praepollet: (ET quidem utrubique tanto, quanta est praepollentia:) Sin aequipollent, pro Neutro. Si vero, vel plures Ascensus conjuncti sint, vel plures Descensus: tantundem simul pollent atque eorundem summa. 7.44 Exempli gratia […] puta, Descensus Ponderis 2P per Altitudinem 3d, cum Descensu Ponderis 3P per altitudinem 2D comparatus; Aequipollebit, (propter 2 × 3 = 3 × 2;) adeoque quae sic movenda sunt, Aequiponderabunt. At descensus Ponderis 2 P per Altitudinem 4 D, Descensui

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Ponderis 3 P per Altitudinem 2D, praepollebit, (propter 2 × 4 > 3 × 2;), Adeoque, quod sic movendum erit, praeponderabit. Et in aliis similiter. 7.45 Prop XV Lineae curvae Declivitas, in singulis respective punctis; eadem est atque rectae ibidem contingentis. Et superficiei curvae, eadem atque ibidem contingenti plani. Quod aliis perinde atque gravium motibus accommodabitur.

A.8 Chapter 8 8.1 Hinc Vis quoque duplex: alia elementaris, quam et mortuam appello, quia in ea nondum existit motus, sed tantum solicitatio ad motum, qualis est globi in tubo, aut lapidis in funda, etiam dum adhuc vinculo tenetur; alia vero vis ordinaria est, cum motu actuali conjuncta, quam voco vivam. Et vis mortuae quidem exemplum est ipso vis centrifuga itemque vis gravitatis seu centripeta, vies etiam qua Elastrum tensum se restituere incipit. Sed in percussione, quae nasciter a gravi iam aliquamdiu cadente, ant ab arcu se aliquamdiu restituente, aut a simili causa vis est viva, ex infinitis vis mortua impressionibus continuatis nata. 8.2 Eodem modo etiam fit, ut gravi descendente, si fingatur ei quovis momento nova aequalisque dari celeritatis accessio infinite parva, vis mortuae simul et vivae aestimatio observetur, nempe ut celeritas quidem aequabiliter crescat secundum tempora, sed vis ipsa absoluta secundum statis seu tempore quadrata, id est secundum effectus. Ut ita secundum analogiam  Geometriae seu analysis nostrae solicitationes sint ut dx, celeritates ut x, vires ut xx seu ut xdx. 8.3 Et est à propos de considerer que l’équilibre consiste dans un simple effort (conatus) avant le mouvement, et c’est que j’appelle la force morte qui a la mesme raison à l’égard de la force vive (qui est dans le mouvement mesme) que le point à la ligne. Or, au commencement de la descente, lorsque le mouvement est infiniment petit, les vitesses ou plutôt les élémens des vitesses sont comme les descentes, au lieu qu’après l’accélération, lorsque la force est devenue vive, les descentes sont comme les carré des vitesses. 8.4 Je suppose deux lignes droites quelconques données AC, BD, que je prends pour deux rangs de petits ressorts égaux & également bandez. je suppose de plus, que deux boules égales commencent à se mouvoir des points C & D, vers F & J, lorsque les ressorts commencent à se dilater: Soient CML, DNK deux lignes courbes, dont les appliquées GM, HN expriment les vitesses acquises aux point G & H: je nomme BD = a , l’abscisse DH = x, sa differentielle HP, ou NT = dx, l’apliquée HN= v, sa differentielle dv: Je prends ensuite les abscisses CG, CE de la courbe CML, telles qu’elles soient aux abscisses de la courbe DNK, comme AC est à BD, ou, ce qui est, la même chose, je fais BD : AC = DH: CG = DP: CE. Suposant donc AC = na, on aura CG = nx, GE = ndx; soit enfin l’apliquée GM = z. Tout ceci supposé, je raisonne ainsi. 2. Les boules étant parvenues aux points G & H, chaque ressort, tant de ceux qui étoient resserrez dans l’intervalle AC, que de ceux qui l’étoient dans l’interval BD, sera dilaté également, parce que AC : CG = BD : DH; chacun de ces ressorts aura donc perdu, de part & d’autre, une partie égale de son élasticité, & il leur en restera par conséquent à chacun également. Donc les pressions & les forces morte, que les boules en reçoivent, sont aussi égals entr’elles: je nomme cette pression p. Or l’accroissement élementaire de la vitesse en H, je veux dire la differentielle TO, ou dv, est, par la loi connue de l’acceleration, en raison composée de la force motrice, ou de la pression p, & du petit tems que le mobile met parcourir la differentielle HP, ou dx, lequel tems s’exprime par HP : HN = dx : v; On aura donc dv = pdx : v, & partant vdv = pdx, ce qui donne par l’intégration 12 vv = pdx. Par la même raison on a dz = p× GE  : GM = p× ndx : z, par conséquent zdz = npdx; & en intégrant 12 zz = n pdx, d’où il suit que vv : zz = pdx : n pdx = 1 : n = a : na = BD : AC. Or B D est à AC, comme la force vive acquise en H est à la force vive acquise en G. Donc ces deux forces sont entr’elles comme vv à

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Appendix. Quotations zz; ainsi les forces vives des corps égaux en masses ont comme les quarrez de leurs vitesses, & les vitesses elles mêmes sont en raison sousdoublée, ou comme les racines quarrées des forces vive. C.Q.F.D.

8.5 La distinction que vous faites entre la force des poids et celle des vents n’est point une raison d’admettre le principe de statique pour ceux-là et de le rejeter pour ceux-ci, car cette distinction ne regarde que les causes productrices des forces. Or il n’est pas question de savoir comment les forces sont produites, il suffit qu’elles soient existantes; de quelque cause qu’elles proviennent, elles feront toujours la même impression, la même action, par conséquent le même effet pourvu que ces forces soient appliquées de la même manière. 8.6 Je me sers ici du mot de puissance an lieu de celui de force, afin de me rendre plus intelligible en faisant voir que la force des vents n’a rient de singulier pour la distinguer sur un autre genre de puissance continuellement et uniformément appliquée. 8.7 Je suis peut être un des plus zelés defenseurs de la composition des forces, comme Vous l’aurez vi dans mon livre et en ben d’autres occasions; mais permettez moi que je Vous dise que Vous abusez ici de ce grand principe de Mechanique: Vous n’en faites pas une bonne application à notre sujet; pour Vous le faire voir, voyons en quoi consiste ce principe; c’est principalement en deux cas: le premier est, lorsque deux forces mortes agissantes ensemble, mais suivant différentes directions, elles en font naitre une troisième moyenne; le second de ces cas est, lorsque deux forces vivantes s ’appliquent immediatement et dans un moment suivant differentes directions sur un corps mobile, qui lui imprimeroient chacune séparément de certaines vitesses, ces forces produiront dans le mobile, si elles agissent ensemble, une vitesse moyenne, qui sera comme dans le cas des forces mortes la diagonale du parallelogramme. […] Pour en venir à notre sujet, le premier de nos deux cas n’y fait rien, car il ne s ’agit pas ici de forces mortes; le second n’y scauroit être appliqué non plus, car le vaisseau n’est pas poussé par le vent comme une bille par un seul choc instantané, mais par une force continuellement appliquée. 8.8 Cependant comme le vent agit tout autrement sur la voile par sa continuation, on peut considérer son action comme des bouffées reiterées à tout moment, dont chacun ajoute un nouveau degrez infiniment petit de vitesse au vaisseau, jusqu’à ce que la vitesse totale du vaisseau soit si grande que le vent ne lui en puisse plus rien ajouter, ce qui arrive quand le vaisseau, comme je l’ai, dit, fuit le vent de toute la vitesse absolue du vent. 8.9 Dans la demonstration que vous faites de l’equilibre des poids vous dites que les puissances ou les forces sont comme les produits des masses par les vitesses; cela est tres vrai dans un bon sens, mais prenez garde si dans l’application que faites à l’equilibre des trois voiles vous ne confondez pas la puissance ou la force avec l’energie de la puissance ou de la force; et si vous ne confondez pas la vitesse actuelle du vent la quelle multipliée par la masse produit la force absolue, avec la vitesse virtuelle, laquelle étant multipliée avec la force absolue produit le momentum ou l’energie de cette force. 8.10 J’entends par vitesse virtuelle la seule disposition à se mouvoir que les forces ont dans un parfait equilibre, où elles ne se meuvent pas actuellement. Ainsi dans votre figure 1 qui est ici la seconde, si ce poids B inseparable de la ligne MB, est en equilibre avec les poids N et O, sa vitesse virtuelle est la petite ligne BP, et les vitesses virtuelles de N et O sont CP et RP; et alors le produit du poids B par BP, ce qui fait l’energie du poids B, est egal aux deux prodnits du poids N par CP, et du poids O par RP lesquels sont leur energies; C’est pourquoy eviter l’equivoque, an lieu de dire que leurs puissances ou les forces sont comme les produits des masses par leurs vitesse vous auriez peut-être mieux fait de vous exprimer ainsi, les energies des puissances ou des forces sont comme les produits de ces puissances ou de ces forces par les vitesses virtuelles.

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8.11 Le point essentiel en pouvant être mis sur une demi page; mais c’est là justement où M. Renau se trompe grossierement, en ce qu’il confond les forces des vents avec les energies des forces, oubliant que pour avoir l’energie ou ce que les Latins appellent momentum de la force du vent, il ne suffit pas de prendre, comme il fait, le quarré de la vitesse du vent, ce qui ne donneroit que la simple force du vent, mais qu’il faut multiplier ce quarré de la vitesse avec sa vitesse virtuelle, c’est à dire avec l’eloignement du centre d’appui, autour duquel la force appliquée tend à se mouvoir. 8.12 Concevez plusieurs forces différentes qui agissent suivant différentes tendances ou directions pour tenir en équilibre un point, une ligne, une surface, ou un corps; concevez aussi que l’on imprime a tout le système de ces forces un petit mouvement, soit parallèle a soi-même suivant une direction quelconque, soit autour d’un point fixe quelconque: il vous sera aise de comprendre que par ce mouvement chacune de ces forces avancera on reculera dans sa direction, a moins que quelqu’une ou plusieurs des forces n’ayent leurs tendances perpendiculaires a la direction du petit mouvement; auquel cas cette force, ou ces forces, n’avanceroient ni ne reculeroient de rien; car ces avancemens ou reculemens, qui sont ce que j’appelle vitesses virtuelles, ne sont autre chose que ce dont chaque ligne de tendance augmente ou diminue par le petit mouvement; et ces augmentations ou diminutions se trouvent, si l’on tire une perpendiculaire a l’extremite de la ligne de tendance de quelque force, la quelle perpendiculaire retranchera de la meme ligne de tendance, mise dans la situation voisine par le petit mouvement, une petite partie qui sera la mesure de la vitesse virtuelle de cette force. Soit, par exemple, P un point quelconque dans le système des forces qui se soutiennent en équilibre; F, une de ces forces, qui pousse ou qui tire le point P suivant la direction FP on PF; Pp, une petite ligne droite que décrit le point P par un petit mouvement, par le quel la tendance FP prend la direction fp, qui sera ou exactement parallele a FP, si le petit mouvement du système se fait en tous les points du système parallèlement a une droite donnée de position; ou elle fera, étant prolongee, avec FP, un angle infiniment petit, si le petit mouvement du système se fait autour d’un point fixe. Tirez donc PC perpendiculaire sur fp, et vous aurez Cp pour la vitesse virtuelle de la force F, en sorte que F × Cp fait ce que j’appelle Energie. Remarquez que Cp est on affirmatif on négatif par rapport aux autres: il est affirmatif si le point P est poussé par la force F, et que l’angle FPp soit obtus; il est négatif, si l ’angle FPp est aigu; mais au contraire, si le point P est tire, Cp sera négatif lorsque l’angle FPp est obtus; et affirmatif lorsqu’il est aigu. Tout cela étant bien entendu, se forme cette Proposition generale: En tout équilibre de forces quelconques, en quelque maniere qu’elles soient appliquées, et suivant quelques directions qu’elles agissent les unes sur les autres, on mediatement, on immediatement, la somme des Energies affirmatives sera egale a la somme des Energies négatives prises affirmativement. 8.13 Donc nous aurons A × pm = B × pn + C × 0 = B × pn c’est a dire A : B = pn : pm = sinus de l’angle pPn : sinus de l’angle pPm. 8.14 J’appelle vitesses virtuelles, celles que deux ou plusieurs forces mises en équilibre acquirent, quand on leur imprime un petit mouvement; ou si ces forces sont déja en mouvement. La vitesse virtuelle est l’élement de vitesse, que chaque corps gagne ou perde, d’une vitesse déja acquise, dan un tems infiniment petit, suivant sa direction. 8.15 Deux agens sont in équilibre, ou ont des momens égaux, lorsque leurs forces absolues sont en raison reciproque de leurs vitesses virtuelle; soit que les forces qui agissent l’une sur l’autre soins en mouvement, ou en repose. C’est un principe ordinaire de Statique & Méchanique; que je ne m’arrêterai pas à démontrer: j’aime mieux l’employer à faire voir la maniere dont le mouvement se produit par la force d’une pression qui agit sans interruption, & sans autre opposition que celle qui vient de l’inertie du mobile. 8.16 Me confirma encore dans l’opinion ou j’étais qu’il faut entrer dans la génération de l’équilibre pour y voir en soi, & pour y reconnaitre les propriétez que tous les autres principes ne prouvent, tout au plus, que pour nécessité de conséquence.

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8.17 Pour préparer l’imagination aux mouvemens composez, coincevons le point A sans pesanteur uniformement mû vers B le long de la droite AB, pendant que cette ligne se meut uniformement vers CD le long de AC en demeurant toujours parallele a elle-même, c’est à dire, faisant l’angle toujours le même quelconque avec cette ligne immobile AC: de ce deux mouvemens commencez en même tems, soit la vitesse du premier a la vitesse du second, comme les cotez AB, AC, du parallelogramme ABCD, le long des quels ils se font, Quel que soit ce parallelogramme ABCD, je dis que par le concourse des deux forces productrices de ces deux mouvemens dans le mobile A, ce point parcourra la diagonale AD de ces parallelogramme, pendant le tems que chacune d’elles lui en auroit fait parcourir seule chacun des cotes AB, AB, correspondans. 8.18 Definition XXII Les produit de chaque poids ou puissance absolue par sa distance à l’appui du Levier auquel elle est appliquée s’appelle en Latin Momentum […] nous ne laisserons pourtant pas de l’appeller aussi moment, pour nous moins éloigner du langage ordinaire. La raison de ce nom vient sans doute de ce que ces produits son égaux ou inégaux comme les impressions de deux puissances sur un Levier. 8.19 Les formules contraire de momens en seront toujours égales entr’elles, c’est à dire que la somme de leurs momens conspirans à faire tourner le Levier en un sens sur son appui sera toujours alors égale à la somme des conspirans à le faire tourner en sens contraire sur cet appui, ainsi qu’on l’à déjà vu dans le Corol. 9 du Th. 21. 8.20 Votre project d’une nouvelle Mécanique fourmille d’un grand nombre d’exemples, dont quelque uns à juger par les figures paroissent assez compliquées; mai je vous deffie de m’en proposer un à votre choix, que je ne resolve sur le champe et comme en jouant par ma dite regle. 8.21 Celle que vous pretendez substituer à la mienne et qui est fondée sur la composition des forces, n’est elle meme qu’un petit corollaire de la regle d’energie. J’ai donc raison d’appeller le grand et le premier principe de statique sur le quel j’ai fondé ma regle qui est que dans chaque equilibre il y a une egalité d’energie des forces absolues, c’est à dire entre le produict des forces absolues par les vitesses virtuelles. 8.22 Je vous prie d’y panser, vous y trouverez sans doute un fond inepuisable pour enrichir la mecanique et pour en rendre l’etude incomparablement plus commode et plus aisée qu’elle n’a eté pur le passé, le traité complet de cette science que vous promettez depuis si longtems ne pourra que paroitre d’autant plus estimable, qu’on le verra fondé sur un principe aussi universelle, aussi simple, aussi intelligible et aussi certain que celui dont il s’agit ici et dont je vous ai montré tant d’avantages. 8.23 Mais la mecanique de cette même proposition, & de la generale que vous ajoutez dans votre derniere, bien loin d’être le grand & le premiere principe de statique, ne me paroit pouvoir être qu’un corollaire des mouvemens composée, ou de quelqu’autre principe, qui demontre cette proposition, c’est a dire votre egalité des sommes d’energies, en deduisant des directions donnée par lui, ou par supposition, les chemine instantanées de M. Descartes, que vous appellez vitesses virtuelles, qui avec les puissances trouvées d’ailleurs, à la supposition de leur equilibre, son tout ce qui entre dans cette egalité de sommes des energies, de laquelle on aouroit droit de douter si elle n’etoit pas prouvés par quelqu’un de ces principes. 8.24 Les Cartesiens conformément à la lettre que je viens de citer (art. 1.) de leur Maistre, avoient desja deduit de son principe la même egalité de Momens, ou d’energie, ou de quantités de mouvement, que vous employez pour deux puissances en equilibre sur les machines simples, & dans le fluides, par les commencemens de mouvement que M. Descartes prescrit dans cette lettre; mais vous etes le seul, que je sçache, qui ait étendu cette égalité d’energies à tant de puissances qu’on voudra supposer en équilibre entr’elles suivant des directions quelconques. Cette Remarque est fort belle; mais (comme j’ay desja dit) elle suppose équilibre entre des puissances donnée & de directions données sans le prouver.

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8.25 L’équilibre du non équilibre, ils ne font proprement qu’une demonstration ab abdsurdo. 8.26 J’ai peur de tomber dans une logomachie si j’entreprens de m’entendre sur tous ce que vous me dites touchant ma regle des energie, que j’ai pretendue être generale pour toute la mecanique, tant des fluides que des solides [...] evitons donc la logomachie et ne prenez pas le change, il ne s’agissoit uniquement que d’etablir la verité et l’universalité de ma regle des energies contre votre objection; que cette regle soit un principe ou un corollaire d’un autre, qu’importe, il suffit qu’elle soit vraie, generale et commode, sans exception, uniforme et facile à en faire l’application; avantage que la composition des forces n’a pas. 8.27 Vous me citez la lettre de M. Descartes pour me prouver que cet Autheur avoit deja l’idée d’expliquer l’équilibre des puissances par l’egalité des energies en considerant leurs chemins instantanée, que j’appelle vitesses virtuelles, je respons, que je ne me vans pas d’être le premier inventeur de cette idée, non plus que vous vous vanterez d’être celui d’expliquer les équilibres par la composition des forces. 8.28 Vous ferez de ma regle d’energie ce que vous voudrez en l’ajoutant ou en ne l’ajoutant pas à votre mechanique; je vous permete l’un comme l’autre; mais de prétendre qu’elle soit un corollaire du principe de la composition des mouvemens ou forces, j’aurois peutetre encore de quoi faire valoir les raisons données dans mes precedentes pour en prouver le contraire, si je voulais m’engager dans une dispute qui nous couteroit du temps et de la peine: ainsi j’aime mieux vous laisser le plaisir de croire que le principe de la composition deive préceder celui d’energie que de hazarder une longue et ennuyeuse contestation, il suffit que le dernier pouvant être appliqué aux fluides comme aux solides, soit plus general qua le premier qui ne sert que pour les solides, outre que celui ci demanderoit encore un autre principe dont il doit être deduit, puisque la composition des forces n’est pas claire par elle même comme un axiome. Il est donc ce me semble raisonnable que le principe d’energie comme le plus general et pour le moine aussi clair pur lui même que le principe de la composition contienne celui ci comme le moins general.

A.9 Chapter 9 9.1 Per nome di potenza dunque non altro intendiamo, che la pura, e semplice pressione, o sia quello sforzo, che fa la gravità, o altra forza contro qualche ostacolo invincibile, come è per l’appunto quello, che fa una palla di piombo contro una tavola immobile, oppure contro la mano che la sostiene. 9.2 Sicché se una palla, a cagion d’esempio di piombo, sarà collocata sopra di una tavola immobile, la gravità, che in essa risiede, sarà forza soltanto premente, e perciò forza morta. Ma se si rimuoverà l’ostacolo, cioè la tavola sottoposta, nella palla si indurrà tosto cangiamento di stato. […] i meccanici per loro metodo si immaginarono la potenza dar al corpo un impulso, il quale però appena nato, fosse dall’invincibile ostacolo distrutto, e così secondo il metodo dei matematici si rappresentarono la forza morta sotto l’idea di un impulso infinitamente picciolo [...] Ma poiché i meccanici più chiara idea formar potessero dell’azion della potenza, siccome s’avevano rappresentata la potenza sotto l’idea di un impulso, che nel procinto del suo nascere resta per l’invincibile ostacolo estinto, e distrutto, così, rimosso l’ostacolo invincibile, concepirono tutti gli impulsi […] conservarsi nel corpo medesimo, e quindi si avvisarono l’azione della potenza non esser, che la somma di tutti gli impulsi accumulati, e conservati nel corpo. Quel tanto poi d’energia che per l’azion della potenza si genera nel corpo [...] viene chiamata forza viva. 9.3 Imperocché s’egli è vero come si è detto, che la potenza considerar si deve come un impulso minore di ogni altro dato, e che l’azion della potenza è la somma di tutti gli impulsi comunicati al corpo, e nel corpo stesso conservati, quella dovrà certamente esser la proporzione della potenza e l’azion della potenza, che passa tra una quantità infinitesima, ed una quantità finita.

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Appendix. Quotations Imperocché dalle cose sopra divisate apparisce, che agendo la potenza nel corpo, a cui è applicata, genera in esso la forza viva, e che questa produce il cambiamento di stato. Sicché la forza viva considerarsi deve come un effetto dell’azion della potenza e come causa del cangiamento di stato che nel corpo si induce; e poiché in questo caso si parla di cause intere, e totali, avrà luogo l’assioma Ontologico, che le cause debbono essere proporzionali agli effetti e gli effetti alle cause. Quindi nascono due modi di misurar la forza viva; cioè, o con misurar il di lei effetto, che è il cangiamento di stato, o con misurar la di lei causa, che è l’azione della potenza.

9.4 Sicché col dire degli Antichi, che la causa degli equilibri consiste nell’uguaglianza de’ momenti, non altro sembran aver detto, che l’equilibrio dipende dall’uguaglianza di quelle quantità, dall’uguaglianza delle quali l’equilibrio dipende. 9.5 L’equilibrio nasce da ciò, che le azioni delle potenze, che equilibrar si devono, se nascessero, sarebbero uguali, e contrarie; e perciò l’uguaglianza, e la contrarietà delle azioni delle potenze è la vera causa dell’equilibrio. […] L’equilibrio non è altro, che l’impedimento de’ moti, cioè degli effetti dell’azione delle potenze, a cui non è meraviglia se corrisponde l’impedimento delle cause, cioè delle azioni stesse. 9.6 Quindi stabiliamo un principio, cioè un criterio generale per conoscere quando tra le potenze succeder debba l’equilibrio, ed egli è quello che si contiene nel seguente teorema: Le potenze saranno in equilibrio qualora trovansi in tali circostanze, che se nascesse un moto infinitesimo, le di loro infinitesime azioni sarebbero uguali. E tal principio deve aver luogo in tutti gli equilibri. 9.7 Perché la celebre controversia delle forze vive, che consiste nel definire se quelle si debbano misurar per la massa moltiplicata per la velocità, oppure per la massa moltiplicata per il quadrato della velocità stessa, riducesi a quest’altra quistione, cioè se l’azione della potenza debba esser proporzionale al tempo piuttosto che allo spazio. 9.8 Non potendosi dunque l’azion della potenza misurare per la potenza moltiplicata pel tempo, uopo è rivolgersi allo spazio. In tutti gli equilibri conosciuti si trova vero, come si vedrà nei seguenti capitoli, che facendosi un moto infinitesimo, le potenze sono in ragion reciproca de’ loro rispettivi spazietti d’accesso, o di ricesso dal centro delle potenze stesse […] Sicché se l’azione della potenza si misurerà per la potenza moltiplicata per lo spazio, per cui la potenza agendo trasporta il corpo, facendolo avvicinare fino al centro, o dal centro facendolo allontanare, si salverà negli equilibri l’uguaglianza tra le minime azioni delle potenze […]. Dunque l’azione della potenza dee veracemente misurarsi per la potenza moltiplicata per lo spazio secondo il metodo dei Leibniziani. 9.9 Le potenze sono in equilibrio, qualora trovansi in tali circostanze costituite, che facendosi un moto infinitesimo, onde alcune potenze si avvicinino al suo centro, alcune altre dal suo centro si allontanano, la somma dei prodotti positivi delle potenze moltiplicate per gli rispettivi spazietti d’accesso o di recesso, sia uguale allo somma de’ simili prodotti negativi. 9.10 Dico dal principio delle azioni dedursi, che qualora nella verga ABC si ha l’equilibrio, la potenza Z sia alla potenza X come CN : CM, cioè, che vale l’equazione Z · CM = X · CM. 9.11 I centri delle potenze Z, X sian i punti Z, X. Si concepisca ora nella verga ACB nascer un moto infinitesimo, cosicché i punti A, B descrivendo gli archetti Aa, Bb vengano in a, b. Dal punto b al punto X si tiri la retta bX, e dal punto a al punto Z la retta aZ; indi col centro Z, e coll’intervallo aZ intendasi descritto l’archetto aF che incontri la AZ in F, e similmente col centro X, […] Fatto ciò è manifesto esser AF lo spazietto di accesso al centro della potenza Z, e bG lo spazietto di recesso dal centro della potenza X. Il principio delle azioni richiede, che avendosi nella verga ABC l’equilibrio, sia la potenza Z alla potenza X come bG : AF.

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9.12 Si noti in secondo luogo, che facendosi comparazione tra il principio dell’equivalenza, e quello delle azioni, debbono amendue stimarsi egualmente fecondi, ed estesi, con quella lor differenza, che in alcun casi con maggiore facilità, ed eleganza si adopra il principio dell’equivalenza, in altri casi poi riesce più comodo, ed opportuno l’adoperare il principio delle azioni. È finalmente con attenzion da notarsi, che il metodo della composizione, e risoluzion delle forze non è il vero metodo della natura, ma è un metodo che si han formato i Geometri per la più facile e spedita solutione de’ lor problemi. La natura nelle sue opere non va’ giammai a comporre, e risolvere le forze, ma adopera sempre azioni, le quali essendo uguali, e contrarie, fan sì, che si producano gli equilibri. 9.13 Nella puleggia stabile, poiché si abbia equilibrio richiedesi tra la potenza, e il peso la ragion d’uguaglianza. Sia AB una puleggia stabile, che abbia intorno a sé la fune EABD, alla di cui estremità D sia attaccato il peso P, all’altra estremità E sia applicata la potenza, che il peso stesso sostiene. Dico, che per aversi l’equilibrio in quella macchina bisogna, che la potenza applicata in E sia al peso P affatto uguale. 9.14 Si facci un moto infinitesimo secondo la direzione della potenza applicata in E, cosichè l’estremità E della fune giunga in G, mentre l’estremità D giungerà in H. Egli è troppo manifesto, che è EG lo spazietto d’accesso al centro della potenza, e DH lo spazietto di recesso dal centro del peso. Sicché acciò s’abbia l’equilibrio tra la potenza, e il peso, convien, che quella stia a questa come DH : EG. Ma è DH = EG; poiché supponendosi, che la fune non pratica alcuna distrazione, ma che resti sempre della stessa lunghezza, sarà la lunghezza DAE uguale alla lunghezza HAG; onde, resteranno DH, EG uguali tra di loro. Dunque nella puleggia perché si abbia l’equilibrio, richiedesi che la potenza sia uguale al peso. Che è quel, che bisognava dimostrare. 9.15 Sia GHPQ un sifone qualunque, se in un braccio di esso GH si verserà una quantità di fluido omogeneo […]. Posto che sarà in equilibrio il fluido versato nel sifone, nell’un braccio e nell’altro del sifone stesso si troverà elevato alla medesima altezza. 9.16 Solamente io avvertirò che il famoso teorema dell’incomparabile Giovanni Bernoulli, il quale è stato dimostrato in tutte le macchine dal dottissimo sig. Varignon, non è altro che una conseguenza dell’equalità delle azioni contrarie, che è necessaria in ogni equilibrio. Il teorema Bernoulliano è il seguente. In ogni equilibrio di quante e quali potenze si vogliano, in qualunque maniera applicate, e agenti per qualsiasi direzione, la somma delle energie positive è uguale alla somma delle energie negative, purché come affermative si prendano. Per nome d’energia il sig. Bernoulli altro non intende se non se il prodotto della potenza e della velocità virtuale della stessa potenza; la quale sarà positiva, se seguita la direzione della potenza, sarà negativa, se seguita la direzione opposta. E chi è che non veda, che la velocità virtuale della potenza è proporzionale allo spazio, per cui il corpo, o la potenza si avvicina al centro delle forze; ovvero se le potenze siano corde elastiche alla contrazione o diffrazione delle corde. Dunque l’energia bernoulliana è la stessa, o almeno proporzionale a quella che per noi chiamasi azione della potenza. 9.17 Per dichiarare siccome si distinguano le potenze e l’azioni loro, io concepisco un corpo grave sospeso da un filo, che gl’impedisce di discendere, e d’avvicinarsi alla terra. Fin ora altro non intendo, ch’una potenza di gravità applicata al corpo, a cui è contraria l’elasticità del filo, che la contrasta, e non le lascia produrre effetto di sorte alcuna. Io tronco il filo, e levo l’elasticità contraria alla gravità. Ora oltre la potenza intendo, ch’essa successivamente e continuamente replica i suoi impulsi o sollecitazioni contro il corpo, il quale è obbligato di cangiare stato. La somma e l’aggregato di cotai impulsi si vuol chiamare l’azione di tal potenza; e l’effetto ossia la mutazione di stato non alla potenza, ma all’aggregato dei suoi impulsi è proporzionale. Tre quantità pertanto si vogliono distinguere, cioè la potenza considerata in se stessa, la qual pressione ancor si suol chiamare; l’azione, che è l’aggregato dei suoi impulsi, onde la potenza spinge il corpo; la quale è in ragion composta della potenza e del numero degli impulsi; e

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9.18 Sicché dunque sebben la forza centrifuga non ha propriamente altra cosa, che l’inerzia del corpo in alcune circostanze considerata, non è inutile l’introdurla ne’ raziocini, ne si dee bandir dalla fisica: anzi sarà profittevole il fissar le sue leggi, e si riconosceranno per veri, e belli i teoremi prodotti intorno a cotal forza dal dotto, e profondo Cristian Ugenio. Similmente risponderò io intorno alla forza viva. Essa non è per verun modo distinta dalla forza d’inerzia, anzi è la medesima forza d’inerzia da alcune particolari condizioni modificata: contuttociò sarà utile il considerarla con questo nome, e il fissarne le leggi, che in molte quistioni, e ricerche potranno essere di non picciolo giovamento. 9.19 Per mettere in buona vista il nostro metodo, e l’uso del principio, non devo omettere una osservazione, che sembrami importante. Quando non sia possibile, se non se un movimento, come avviene a’ corpi, che si raggirano intorno a un asse, allora se concepito un minimo movimento, la azioni spontanee e sforzate misurate dallo spazio di accesso e di recesso si ritrovano eguali, ma senza cautela si deduca l’equilibrio. Ma quando liberi sieno più movimenti e in più direzioni, se concependo un qualche movimento ad arbitrio, io ritrovo come sopra l’egualità delle azioni, non posso affermare un equilibrio pieno e compito, ma soltanto pronunciare che quel movimento è impossibile, e che in quella direzione equilibrate sono le potenze. 9.20 Mi sono ancora servito di qualche parte delle ricerche, che io avea già fatte vent’anni addietro sulla gran Cupola di S. Pietro in Roma, e principalmente dalla teoria che mi condusse a conoscere la forza con cui un cerchio di ferro spinto in fuori da forza applicata perpendicolarmente in tutti i suoi punti, resiste, trovandola maggiore di quella che sarebbe la stessa spranga di ferro tirata direttamente nella direzione della sua lunghezza un poco più che a sei doppi, cioè in proporzione della circonferenza del circolo al raggio, d’onde poi il Marchese Polini ricavò l’idea di quella esperienza, in cui un filo di seta ottagono, tirato in fuora in tutti gli angoli per esser rotto, ebbe bisogno di una forza incirca a sei doppi maggiore, che quando un altro filo suo compagno era tirato direttamente. 9.21 Due sono le forze, che spingono in fuori verso l’imposta gi, cioè il peso del cupolino, e il peso de’ costoloni con gli spicchi delle cupole; e due parimente le forze, che resistono a tale spinta, cioè le catene circolari, o cerchi LL, ed il sostegno […] ridotto a due distinti, il primo de quali è il tamburo HI, col pezzo interior della base CDF; il secondo i contraforti mGF colla parte esteriore ABE della base medesima […]. Il distacco, delle parti quanto fosse difficile, e che resistenza, abbia fatto non è possibile l’esaminarlo a minuto. Dipende esso in gran parte dalla qualità del cemento, e dalla diligenza del lavoro. Per metter’in conto le forze, e vedere se queste stanno in equilibrio convien’in prima determinare la quantità assoluta delle medesime, e poi quello che da’ Mecanici chiamasi il Momento. Per avere la quantità assoluta della forza, con cui agisce da una parte il Cupolino, e la volta della Cupola co’ costoloni per spingere, e dall’altra la base, il tamburo, i contrafforti per ritenere la spinta, conviene averne il peso. 9.22 Supposto questo principio, in primo luogo parci, che l’energia di una catena di ferro, curvata in cerchio debba crescere sopra quella forza assoluta, che avrebbe se distesa fosse in dirittura, in quella medesima proporzione, che ha la circonferenza del circolo al raggio, cioè poco più che a sei doppj. Imperocchè, si concepisca distribuita una forza per tutta la circonferenza di un cerchio, che da essa venga costretto a distendersi, e dilatarsi fino all’atto di rompersi, ed una verga di ferro uguale distesa in dirittura venga tirata da un’altra forza, come farebbe un peso attaccatole verticalmente, che la riduca al medesimo estremo. In questo secondo caso la discesa del peso nel tender le fibre di quella sarebbe uguale alla somma delle tensioni di tutte quante le fibre disposte lungo la stessa verga, ma nel primo dilatandosi il cerchio, e crescendo così la sua circonferenza, la forza che lo costringe, a dilatarsi non si avanzerebbe, se non quanto cresce il raggio del circolo, mentre la somma delle tensioni delle medesime fibre disposte in giro sarebbe uguale all’accrescimento di tutta quanta la circonferenza.

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A.10 Chapter 10 10.1 Theoriae oscillationum, quas adhuc Auctores pro corporibus dederunt solidis, invariatum partium situm in illis ponunt, ita ut singula communi motu angulari ferantur. Corpora autem, quae ex filo flexili suspenduntur, aliam postulant theoriam, nec sufficere ad id negotium videntur principia communiter in mechanica adhiberi solita, incerto nempe situ, quem corpora inter se habeant, eodemque continue variabili. 10.2 Sed quod omnibus scriptis, quae sine analysi sunt composita, id potissimum Mechanicis obtingit, ut Lector, etiamsi de veritate eorum, quae proferuntur, convincatur, tamen non satis claram et distinctam eorum cognitionem assequatur, ita ut easdem quaestiones, si tantillum immutentur, proprio marte vix resolvere valeat, nisi ipse in analysin inquirat easdemque propositiones analytica methodo evolvat. Idem omnino mihi, cum Neutoni Principia et Hermanni Phoronomiam perlustrare coepissem, usu venit, ut, quamvis plurium problematum solutiones satis percepisse mihi viderere, tamen parum tantum discrepantia problemata resolvere non potuerim. 10.3 La composition des forces suffit comme l’on fait pour démontrer l’équilibre du levier, & réciproquement cette dernière proposition une fois prouvée, on peut facilement en déduire la composition des forces. Elle nous fournit d’ailleurs une démonstration fort-simple du principe des vitesses virtuelles, qu’on peut avec raisons considérer comme le plus fécond & le plus universel de la Mécanique: tout les autres en effet s’y réduisent sans peine, le principe de la conservation des forces vives, & généralement, tous ceux que quelques Géomètres on imaginés pour faciliter la solution de plusieurs Problèmes, n’en sont qu’une conséquence purement géométrique, ou plus tost ne sont que ce même principe réduit en formule. 10.4 La force est donc une cause quelconque de mouvement. Sans connaitre la force en elle-même, nous concevons encore très clairement qu’elle agit suivant une certaine direction, et avec une certaine intensité. 10.5 On entend en général par force ou puissance la cause, quelle qu’elle soit, qui imprime ou tend à imprimer du mouvement au corps auquel on la suppose appliquée; & c’est aussi par la quantité du mouvement imprimé, ou prêt à imprimer, que la force ou puissance doit s’estimer. Dans 1’état d’équilibre la force n’a pas d’exercice actuel; elle ne produit qu’une simple tendance au mouvement; mais on doit toujours la mesurer par l’effet qu’elle produiroit si elle n’étoit pas arrêtée. En prenant une force quelconque, ou son effet pour l’unité, l’expression de toute autre force n’est plus qu’un rapport, une quantité mathématique qui peut être représentée par des nombres ou des lignes; c’est sous ce point de vue que l’on doit considérer les forces dans la Méchanique. 10.6 On ne peut cependant s’empêcher de reconnaitre que le principe du levier a seul l’avantage d’être fondé sur la nature de l’équilibre considéré en lui-même, et comme un état indépendant du mouvement: d’ailleurs il y a une différence essentielle dans la manière d’estimer les puissances, qui se font équilibre dans, ces deux principes. De sorte que, si l’on n’était pas parvenu, à les lier par les résultats, on aurait pu douter avec raison s’il était permis de substituer au principe fondamental du levier celui qui résulte de la considération étrangère des mouvements composes. 10.7 De même que le produit de la masse et de la vitesse exprime la force finie d’un corps en mouvement, ainsi le produit de la masse et de la force accélératrice que nous avons vu être représentée par I’élément de la vitesse divisé par l’élément du temps, exprimera, la force élémentaire on naissante; et cette quantité, si on Ia considère comme la mesure de l’effort que le corps peut faire en vertu de la vitesse élémentaire qu’il a prise, ou qu’il tend à prendre, constitue ce qu’on nomme pression; mais si on la regarde comme la mesure (le la force ou puissance nécessaire pour imprimer cette même vitesse, elle est alors ce qu’on nomme force.

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10.8 C’est un principe généralement vrai en Statique que, si un système quelconque de tant de corps ou de points que l’on veut, tirés chacun par des puissances quelconques, est en équilibre, et qu’on donne à ce système un petit mouvement quelconque, en vertu duquel chaque point parcoure un espace infiniment petit, la somme des puissances, multipliées chacune par l’espace que le point où elle est appliquée parcourt suivant la direction de cette même puissance, sera toujours égaie à zéro. 10.9 Dans la question presente, si l’on imagine que les lignes X,Y, Z, R, R deviennent, en variant infiniment peu la position de la Lune autour de son centre X + δX, Y + δY, Z + δZ, R + δR, R + δR il est facile de voir que les différences δX, δY, δZ, δR, δR exprimeront les espaces parcourus en même temps pat le point α dans des directions opposées a celles des puissances α

d2X d 2Y d2Z T S , α 2 , α 2 , α 2 , α 2 dm 2 dt dt dt R R

qui sont censées agir sur ce point; on aura donc, pour les conditions de l’équilibre, l’équation générale    d 2Y d2Z T S d2X α 2 (−δX) + α 2 (−δY ) + α 2 dm(−δZ) + α 2 (−δR) + α 2 (−δR ) dt dt dt R R L savoir, en changeant les signes,  

  d 2Y d2Z δR δR d2X α 2 δX + α 2 δY + α 2 δZ + T α 2 + S α 2 . dt dt dt L L R L R 10.10 Le principe de Statique que je viens d’exposer n’est, dans le fond, qu’une généralisation de celui qu’on nomme communément le principe des vitesses virtuelles, et qui est reconnu depuis longtemps par les Géomètres pour le principe fondamental de l’équilibre. M. Jean Bernoulli est le premier, que je sache, qui ait envisagé ce principe sous un point de vue général et applicable à toutes les questions de Statique, comme on le peut voir dans la Section IX de la nouvelle Mécanique de M. Varignon, ou cet habile Géomètre, après avoir rapporté d’après M. Bernoulli le principe dont il s’agit, fait voir, par différentes applications, qu’il conduit aux mêmes conclusions que celui de la composition des forces. 10.11 Ensuite en ayant égard aux équations de condition, donnes par la nature du système proposé, entre les coordonnées des différens corps, on réduira les variations de ces coordonnées au plus petit nombre possible, ensorte que les variations restantes soient tout-à-fait indépendantes entr’elles & absolument arbitraires. Alors on égalera à zéro la somme de tous les termes affectés de chacune de ces dernières variations; & l’on aura toutes les équations nécessaires pour la détermination du mouvement du système. 10.12 1. Le principe donné par M. d’Alembert réduit les lois de la Dynamique à celles de la Statique; mais la recherche de ces dernières lois par les principes ordinaires de l’équilibre du levier, ou de la composition des forces, est souvent longue et pénible. Heureusement il y a un autre principe de Statique plus général, et qui a surtout l’avantage de pouvoir être représenté par une équation analytique, laquelle renferme seule les conditions nécessaires pour l’équilibre d’un système quelconque de puissances. Tel est le principe connu sous la dénomination de loi des vitesses virtuelles; on l’énonce ordinairement ainsi: Quand des puissances se font équilibre, les vitesses des points où elles sont appliquées, estimées suivant la direction de ces

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puissances, sont en raison inverse de ces mêmes puissances. Mais ce principe peut être rendu très-général de la manière suivante. 2. Si un système quelconque de corps, réduits a des points et tirés par des puissances quelconques, est en équilibre, et qu’on donne à ce système un petit mouvement quelconque en vertu duquel chaque corps parcoure un espace infiniment petit, la somme des puissances multipliées chacune par l’espace que le point où elle est appliquée parcourt suivant la direction de cette puissance est toujours égale a zéro. 10.13 Pour avoir les valeurs des variations ou différences δp, δq, δr, . . . , δp , δq , δr , . . . on différentiera à l’ordinaire les expressions des distances p, q.r, . . . , p , q , r mais en regardant les centres des forces comme fixes. 10.14 De plus, en ayant égard à la disposition mutuelle des corps, on aura une ou plusieurs équations de condition entre les variables x, y, z, x , y , z par le moyen desquelles on pourra exprimer toutes ces variables par quelques-unes d’entre elles, ou bien par d’autres variables en moindre nombre et telles, qu’elles soient entièrement indépendantes et répondent aux différents mouvements que le système peut recevoir. 10.15 Ceux qui jusqu’à présent ont écrit fur le Principe des vitesses virtuelles, se sont plutôt attachés à démontrer la vérité de ce principe par la conformité de ses résultats avec ceux des principes ordinaires de la Statique, qu’à montrer l’usage qu’on en peut faire pour résoudre directement les problèmes de cette Science. Nous nous sommes proposé de remplir ce dernier objet avec toute la généralité dont il est susceptible, & de déduire du Principe dont il s’agit, des formules analitiques qui renferment la solution de tous les problèmes sur l’équilibre des corps, à-peu-près de la même manière que les formules des soutangentes, des rayons osculateurs, &, renferment la détermination de ces lignes dans toutes les courbes. 10.16 Si un système quelconque de tant de corps ou points que l’on veut, tirés chacun par des puissances quelconques, est en équilibre, et qu’on donne à ce système un petit mouvement quelconque, en vertu duquel chaque point parcoure un espace infiniment petit qui exprimera sa vitesse virtuelle, la somme des puissances multiplies chacune par l’espace que le point ou elle est appliquée parcourt suivant la direction de cette même puissance, sera toujours égale à zéro, en regardant comme positifs les petits espaces parcourus dans le sens des puissances, et comme négatifs les espaces parcourus dans un sens opposé. 10.17 Et en général je crois pouvoir avancer que tous les principes généraux qu’on pourroit encore découvrir dans la science de l’équilibre, ne seront que le même principe des vitesses virtuelles, envisagé différemment, & dont ils ne différeront que dans l’expression. Au reste, ce Principe est non seulement en lui même très simple & très général; il a de plus l’avantage précieux & unique de pouvoir se traduire en une formule générale qui renferme tous les problèmes qu’on peut proposer sur l’équilibre des corps. Nous allons exposer cette formule dans toute son étendue; nous tâcherons même de la présenter d’une manière encore plus générale qu’on n’est pas fait jusqu’à présent, & d’en donner des applications nouvelles. 10.18 La loi générale de l’équilibre dans les machines, est que les forces ou puissances soient entr’elles réciproquement comme les vitesses des points où elles sont appliquées, estimées suivant la direction de ces puissances. 10.19 On substituera ensuite ces expressions de d p, dq, dr, &, dans l’équation proposée, & il faudra que cette équation ait lieu, indépendamment de toutes les indéterminées, afin que l’équilibre du système subsiste en général & dans tous les sens. On égalera donc séparément à zero, la somme des termes affectés de chacune des mêmes indéterminées; & l’on aura, par ce moyen, autant d’équations particulières, qu’il y aura de ces indéterminées; or il n’est pas difficile de se convaincre que leur nombre doit toujours être égal à celui des quantités inconnues dans la

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Appendix. Quotations position du système; donc on aura par cette méthode, autant d’équations qu’il en faudra pour déterminer l’état d’équilibre du système.

10.20 Maintenant comme ces équations ne doivent servir qu’à éliminer un pareil nombre de différentielles dans l’équation des vitesses virtuelles, après quoi les coefficiens des différentielles restantes, doivent être égalés chacun à zéro, il n’est pas difficile de prouver par la théorie de l’élimination des équations linéaires, qu’on aura les mêmes résultats si on ajoute simplement à l’équation des vitesses virtuelles, les différentes équations de condition dL = 0, dM = 0, dN = 0, &, multipliées chacune par un coefficient indéterminé, qu’ensuite on égale à zéro la somme de tous les termes qui se trouvent multipliés par une même différentielle; ce qui donnera autant d’équations particulières qu’il y a de différentielles; qu’enfin on élimine de ces dernieres équations les coefficients indéterminés par lesquels on a multiplié les équations de condition. 10.21 Réciproquement ces forces peuvent tenir lieu des équations de condition résultantes de la nature du système donné; de manière qu’en employant ces forces, on pourra regarder les corps comme entièrement libres & sans aucune liaison. Et de-là on voit la raison métaphysique, pourquoi introduction des termes λdL + μdM + &c., dans l’équation générale de l’équilibre, fait qu’on peut ensuite traiter cette équation comme si tous les corps du système étoient entièrement libres; c’est en quoi consiste l’esprit de la méthode de cette section. A proprement parler, les forces en question tiennent lieu des résistances que les corps devroient éprouver en vertu de leur liaison mutuelle, ou de la part des obstacles qui, par la nature du système, pourroient s’opposer à leur mouvement, ou plutôt ces forces ne sont que les forces mêmes de ces résistances, lesquelles doivent être égales & directement opposées aux pressions exercées par les corps. Notre méthode donne, comme l’on voit, le moyen de déterminer ces forces & ces résistances; ce qui n’est pas un des moindres avantages de cette méthode. 10.22 Quant à la nature du principe des vitesses virtuelles, il faut convenir qu’il n’est pas assez évident par lui-même pour pouvoir être érigé en principe primitif; mais on petit le regarder comme l’expression générale des lois de l’équilibre, déduites des deux principes que nous venons d’exposer. Aussi, dans les démonstrations qu’on a données de ce principe, on l’a toujours fait dépendre de ceux-ci, par des moyens plus on moins directs. Mais ii y a, en Statique, un autre principe général et indépendant du levier et de la composition des forces, quoique les mécaniciens l’y rapportent communément, lequel parait être le fondement naturel du principe des vitesses virtuelles; on peut l’appeler le principe des poulies. 10.23 On a objecté, avec raison, à cette assertion de Lagrange l’exemple d’un point pesant en équilibre an sommet le plus élevé d’une courbe; il est évident qu’un déplacement infiniment petit le ferait descendre, et, pourtant, ce déplacement ne se produit pas. La première démonstration rigoureuse du principe des vitesses virtuelles est due à Fourier (Journal de École Polytechnique, tome II, an VII). Le même Cahier du Journal contient la démonstration que Lagrange reproduit ici. 10.24 Si donc on imprimant à chaque corps des forces égales et directement contraire a à celles-là, l’effet de ces forces serait détruit par la résistances dont nous venons de parler; par conséquent, le système devrait demeure en équilibre. […] Or, par le principe des vitesses virtuelles, la somme des forces multipliée chaque par la vitesse que le point où elle est appliquée aurait, suivant la direction de la force, si on donnant au système un mouvement quelconque, doit être nulle dans le cas de l’équilibre […] on aura pour l’équilibre des forces dont il s’agit, l’équation: −Π f  (x) − Π f  (y) − Π f  (z) − Ψ f  (ξ) − Ψ f  (ν) − Ψ f  (ζ) − &c. = 0

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f (x, y, z, ξ, ν, ζ) = 0 […]. Or, si on prend l’équation prime de cette équation, relativement au temps t, dont les variables x, y, z, ξ, &c., sont censées être fonctions, on a: x f  (x) + y f  (y) + z f  (z) + ξ f  (ξ) + ν f  (ν) + ζ f  (ζ) + &c. = 0 et il est visible que cette équation ne peut subsister avec la précédent, indépendamment des valeurs des vitesses x , y , &c., à moins qu’on n’ait Π = Ψ = &c. 10.25 Les fonctions primes de la même fonction, prises par rapport aux différentes coordonnées, sont toujours proportionnelles aux forces qui agissent suivant ces coordonnées, et qui dépendent de la condition exprimée par cette fonction. 10.26 Soient X,Y, Z les forces appliquées à l’un des corps suivant les directions des coordonnées x, y, z prolongées, Ξ, ϒ, Σ les forces appliques à un autre corps suivant le prolongement de ses coordonnées, ξ, η, ζ, et X, Y, Z les forces appliquées a un troisième corps suivant le prolongement de ses coordonnées x, y, z; on aura, par ce qu’on vient de démontrer, X = ΠF  (x) + ΨΦ (x), 



Ξ = ΠF (ξ) + ΨΦ (ξ), 



X = ΠF (x) + ΨΦ (x),

Y = ΠF  (y) + ΨΦ (y), 

Σ = ΠF  (ζ) + ΨΦ (ζ)



Z = ΠF  (z) + ΨΦ (z)

ϒ = ΠF (η) + ΨΦ (η), 

Z = ΠF  (z) + ΨΦ (z)



Y = ΠF (y) + ΨΦ (y),

et de là on tirera immédiatement Xx +Y y + Zz + Ξξ + ϒη + Σζ + Xx + Yy + Zz = ΠF(x, y, z, ξ, η, ζ, x, y, z) + ΨΦ(x, y, z, ξ, η, ζ, x, y, z) . Le second membre de cette équation est évidemment nul, en vertu des équations de condition, puisque les quantités indéterminées Π, Ψ, se trouvent multipliées par les fonctions primes de ces équations; donc on aura Xx +Y y + Zz + Ξξ + ϒη + Σζ + Xx + Yy + Zz = 0 équation générale du principe des vitesses virtuelles pour I’équilibre des forces X,Y, Z, Ξ, ϒ, Σ, X, Y, Z, dans laquelle les fonctions primes x , y , z , ξ , … expriment les vitesses virtuelles des points auxquels soit appliquées les forces X,Y, Z, Ξ … estimées suivant les directions de ces forces. Au reste, on ne doit pas être surpris de voir le principe des vitesses virtuelles devenir une conséquence naturelle des formules qui expriment les forces d’après les équations de condition, puisque la considération d’un fil qui par sa tension uniforme agit sur tous les corps et y produit des forces données suffit pour conduire à une démonstration directe et générale de ce principe, comme je l’ai fait voir dans la seconde. 10.27 On peut s’étonner que l’illustre auteur, ordinairement si soigneux de faire connaitre l’origine des idées qu’il expose, ne fasse ici aucune citation. Le passage qu’on vient da lire est, en effet, postérieur da sept années à la publication du célèbre Mémoire sur l’Equilibre et le mouvement des systèmes, dans lequel M. Poinsot se propose et résout précisément la même question, d’affranchir la mécanique du principe des vitesses virtuelles en cherchant directement les forces qui correspondent à une équation donnée. Ce Mémoire avait vivement frappé Lagrange, comme le prouvent des notes autographes nombreuse placées par lui sur les marges d’un exemplaire qu’il m’a été permis de consulter. Je me bornerai à reproduire ici une de ces notes, qui ne peut laisser subsister aucun doute sur la question de priorité.

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10.28 Au reste le principe de Statique que je viens d’exposer, étant combine avec le principe de Dynamique donné par M. d’Alembert, constitue une espèce de formule générale qui renferme la solution de tons les Problèmes qui regardent le mouvement des corps. 10.29 Si maintenant on suppose le système en mouvement, & qu’on regarde le mouvement que chaque corps a dans un instant comme composé de deux; dont l’un soit celui que le corps aura dans l’instant suivant, il faudra que l’autre soit détruit par l’action réciproque des corps, & par celle des forces motrices dont ils sont actuellement animés. Ainsi il devra y avoir équilibre entre ces forces & les pressions ou résistances qui résultent des mouvemens qu’on peut regarder comme perdus par les corps d’un instant à l’autre. D’où il suit que pour étendre au mouvement du système la formule de son équilibre il suffira d’y ajouter les termes ds ces dernieres forces. 10.30 Il est clair que le mouvement ou la vitesse du corps m dans l’instant dt ou peut être regardée comme composée de trois autres vitesses exprimées par: dx , dt

dy , dt

dz dt

et dirigées parallèlement aux axes des x, y, z. Il est de plus évident que si le corps était libre et qu’aucune force étrangère n’agit sur lui, chacune de ces trois vitesses demeurerait constante; mais dans l’instant suivant elles se changent réellement en celles-ci dx dx +d , dt dt

dy dy +d , dt dt

dz dz +d dt dt

donc, si l’on regarde les vitesses précédentes comme composées de ces dernières et des vitesses dy dz dx −d , −d −d , dt dt dt ou bien (en prenant di constant) −

d2x , dt 2

−

d2y , dt 2

−

d2z dt 2

il s’ensuit que celles-ci doivent être détruites par l’action des forces qui agissent sur les corps. Mais ces vitesses sont dues à des forces accélératrices égales à d2x , dt 2

d2y , dt 2

d2z dt 2

et dirigées parallèlement aux axes des x, y, z (en exprimant, suivant l’usage reçu, la force accélératrice par l’élément de la vitesse divisé par l’élément du temps), on, ce qui revient au même, à des forces égales à d2x d2y d2z , , dt 2 dt 2 dt 2 et dirigées en sens contraire. […] D’ou il suit qu’il doit y avoir équilibre entre ces différentes forces et les autres forces qui sollicitent les corps, et qu’ainsi les lois du mouvement du système se réduisent à celles de son équilibre; c’est en quoi consiste le beau principe de Dynamique de M. d’Alembert. 10.31 Si l’on imprime à plusieurs corps des mouvements qu’ils soient forcé de changer à cause de leur action mutuelle. il est claire qu’on peut regarder ces mouvements comme composés de ceux que les corps prendront réellement, et d’autres mouvements qui sont détruites: d’ou il suit que ces deniers doivent être tels, que les corps animés de ces seuls mouvements se fassent équilibre.

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10.32 Mais la difficulté de déterminer les forces qui doivent être détruites, ainsi que les lois de l’équilibre entre ces forces, rend souvent l’application de ce principe embarrassant et pénible. […] Si l’on voulait éviter les décompositions de mouvements que ce principe exige, il n’y aurait qu’à établir tout de suite l’équilibre entre les forces et les mouvements engendrés, mais pris dans des directions contraires. Car, si l’on imagine qu’on imprime à chaque corps, en sens contraire, le mouvement qu’il doit prendre, it est clair que le système sera réduit au repos; par conséquent, il faudra que ces mouvements détruisent ceux que les corps avaient reçus et qu’ils auraient suivis sans leur action mutuelle; ainsi il doit, y avoir équilibre entre tous ces mouvements, ou entre les forces qui peuvent les produire. Cette manière de rappeler les lois de la Dynamique à celles de la Statique est à la vérité moins directe que celle qui résulte du principe de d’Alembert, mais elle offre plus de simplicité dans les applications; elle revient à celle d’Herman et d’Euler qui l’a employée dans la solution de beaucoup de problèmes de Mécanique, et on la trouve dans quelques Traités de Mécanique. sous le nom de Principe de d’Alembert. 10.33 Comme il s’agit présentement de déterminer le mouvement de la corde par les forces sollicitantes, soit la force accélératrice, par laquelle le point M de la corde est accéléré vers l’axe AB = P, & il est clair que toutes ces forces, par lesquelles chacun des élemens de la corde est pressé vers l’axe AB prises ensemble doivent être équivalentes à la force, par laquelle la corde est actuellement tendue & qui nous avons posée AF = F; ou bien, si nous concevons des forces contraries & égales à P, appliquées suivant ML dans chacun des points M de la corde, alors elles devront se trouver en équilibre avec la force qui tend la corde. 10.34 Pourquoi donc aurions-nous recours à ce principe dont tout le monde fait usage aujourd’hui, que la force accélératrice est proportionnelle à l’élément de vitesse? […] Nous n’examinerons point si ce principe est de vérité nécessaire […] non plus, avec quelque Géometres, comme de vérité contingente […] nous nous contenterons d’observer, que vrai ou douteux, clair ou obscure, il est inutile à la Méchanique, & que par conséquent il doit être banni. 10.35 Ce que nous appelons causes, même de la première espèce, n’est tel qu’improprement; ce sont des effets desquels il résulte d’autres effets. Un corps en pousse un autre, c’est-à-dire ce corps est en mouvement, il en rencontre un autre, il doit nécessairement arriver du changement à cette occasion dans l’état des deux corps, à cause de leur impénétrabilité; l’on détermine les lois de ce changement par des principes certains, & l’on regarde en conséquence le corps choquant comme la cause du mouvement du corps choqué. Mais cette façon de parler est impropre. La cause métaphysique, la vraie cause nous est inconnue. 10.36 Ainsi nous entendrons en général par la force motrice le produit de la masse qui se meut par l’élement de sa vitesse, ou qui est la même chose, par le petit espace qu’elle parcurroit dans un instant donné en vertu de la cause qui accélere ou retarde son Mouvement; par force accélératrice nous entendrons simplement l’élément de la vitesse. 10.37 Probléme Général Soit donné un système de corps disposés les uns par rapport aux autres d’une manière quelconque; & supposons q’on imprime à chacun de ces Corps un Mouvement particulier, qu’il ne puisse suivre à cause de l’action des autres Corps; trouver le Mouvement que chaque Corps doit prendre. Solution Soient A, B,C, &c. les corps qui composent le système, & suppose qu’on leur ait imprimé les mouvemens a, b, c, &c. qu’ils soient forcés, à cause de leur action mutuelle, de changer dans le mouvemens a, b, c, &c. Il est clair qu’on peut regarder le mouvement a comme imprimé au Corp A comme composé du mouvement a, qu’il a prise, & d’un autre mouvement α; qu’on peut de même regarder le mouvemens b, c, &c. comme composé de mouvemens b, β, c, κ, &c. d’ou il s’ensuit que le mouvement des corps A, B,C, &c. entr’eux auroit été le même, si au lieu de leur donner les impulsions a, b, c on leur donné à la fois les doubles impulsions

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Appendix. Quotations a, α; b, β, c; κ, &c. Or par la supposition, les corps A, B,C, &c. ont prix d’eux mêmes les mouvemens a, b, c,&c. Donc les mouvemens α, β, κ, &c. doivent être tels qu’ils ne déranger rien dans les mouvemens a, b, c, &c. c’est-à-dire que si les corps n’avoient reçu que les mouvemens α, β, κ, &c. ces mouvemens auroient du se détruire, le système demeurer en repos. De là résulte le principe suivant, pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres. Décomposé le mouvemens a, b, c, &c. imprimés à chaque corps, chacun en deux autres a, α ; b, β; c, κ, &c., qui soient tels, qui si l’on n’eut imprimé aux corps que les mouvemens a, b, c, &c. ils eussent pu conserver ces mouvemens sans se nuire réciproquement; & que si on ne leur eut imprimé que les mouvements α, β, κ, &c. le système fut demeure en repos; il est claire que a, b, c seront les mouvemens que ces corps prendront en vertu de leur action. Ce qu’il falloit trouver.

10.38 On voit encore qu’il a été inutile de rappeler le fameux principe de D’Alembert, qui réduit la Dynamique a la statique. En vertu de ce principe, si l’on décompose chaque mouvement imprimé en deux autres, dont l’un soit celui que le corps prendra réellement, tous les autres doivent se faire équilibre entre eux; c’est-à-dire, que si l’on décompose chaque mouvement imprimé en deux autres dont l’un soit celui que le corps perd, l’autre sera celui qu’il prendra. Mais cela revient immédiatement à ce qu’ion vient de dire, savoir que le mouvement réel de chaque point est la résultant de son mouvement imprimé, et de la résistance qu’il éprouve par sa liaison avec les autres; ce qui est évident de soi-même. Ainsi le principe de D’Alembert n’est au fond que cette idée simple qu’on remarque à peine dans la suite du raisonnement, et qui ne revêt la forme d’un principe que par l’expression qu’on lui donne. 10.39 L’avantage du principe de D’Alembert consiste à trouver les lois du mouvement indépendamment de la considération des résistances ou forces de tension qu’on employait avant lui. 10.40 Les forces de résistance dont en parle ne sont autre chose que les forces capables d’être en équilibre sur le système, ce sont les mêmes que celles qui emplois D’Alembert. C’est, si l’on veut pour abréger, qu’on les appelle forces de résistance mutuelles.

A.11 Chapter 11 11.1 On a donné a cet opuscule le titre d’Essai sur les machines en général, premièrement, parce que ce sont principalement les machines qu’on y en vue, comme étant l’objet le plus importante de la mécanique; et en second lieu, parce qu’il n’y est question d’aucune machine particulière, mais seulement des propriétés qui sont communes à toutes. 11.2 Parmi les philosophes qui s’occupent de la recherche des loix du mouvement, les uns font de la mécanique une science expérimentale, les autres, une science purement rationnelle; c’està-dire, que les premiers comparant les phénomènes de la nature, les décomposent, pour ainsi dire, pour connoître ce qui, ont de commun, et le réduire ainsi a un petit nombre de faits principaux, qui servent en suite à expliquer tous les autres, et à prévoir ce qui doit arriver dans chaque circonstance; les autres commencent par des hypothèses, puis raisonnant conséquemment à leurs suppositions, parviennent à découvrir les loix que suivirent les corps dans leurs mouvements, si leur hypothèses étoient conformes à la nature, puis comparant leurs résultats avec les phénomènes, et trouvant qu’ils s’accordent, en concluent que leur hypothèse est exact, c’est à dire, que les corps suivent en effet les loix qu’ils n’avoient fait d’abord que supposer. Les premiers de ces deux classes de philosophes, partent donc dans leurs recherches, des notions primitives que la nature a imprimées en nous, et des expériences qu’elle nous offre continuellement; les autres partent de définitions et d’hypothèses; pour les premiers, les noms de corps, de puissance, d’équilibre, de mouvement, répondent à des idées premières; ils ne peuvent ni ne doivent définir; les autres au contraire ayant tout a tirer de leur propre lands, sont

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obligé de définir ces termes avec exactitude, et d’expliquer clairement toutes leurs suppositions; mais si cette méthode paroit plus élégante, elle est aussi bien plus difficile que l’autre; car il n’y a rien da embarrassant dans la plupart des sciences rationnelles, et sur-tout dans celle-ci, que de poser d’abord d’exactes définitions sur les quelles il ne reste aucune ambiguité: ce seroit me jeter dans des discussions métaphysiques, bien au dessus de mes forces, que de vouloir approfondir toutes celles qu’on a proposées jusqu’ici: je me contenterai d’examiner la première et la plus simple. […] Les deux loix fondamentales dont je suis parti (Xl), sont donc des vérités purement expérimentales; et je les ai proposées comme telles. Une explication détaillée de ces principes n’entroit pas dans le plan de cet ouvrage, et n’auroit peut-être servi qui embrouiller les choses: les sciences sont comme un beau fleuve, dont le cours est facile à suivre, lorsqu’il a acquis une certaine régularité; mais si l’en veut remonter à la source, on ne la trouve nulle part, parce qu’elle est par-tout; elle est répandue en quelque sorte sur toute la surface de la terre; de même si l’on veut remonter à l’origine des sciences, on ne trouve qu’obscurité, idées vagues, cercles vicieux; et l’on se perde dans les idées primitif. 11.3 Les anciens établirent en axiome que toutes nos idées viennent des sens: et cette grande vérité n’est plus aujourd’hui un sujet de contestation. 11.4 Cependant les sciences ne tirent pas toutes un même fonds de l’expérience: les mathématiques pures en tirent moins que toutes les autres; ensuite les sciences physico-mathématiques; ensuite les sciences. Il séroit sans doute satisfaisant de pouvoir assigner su juste dans chaque science, le point où elle cesse d’etre expérimentale pour devenir entièrement rationnelle: c’est-à-dire, de pouvoir réduire au plus petit nombre possible les vérités qu’on est obligé de tirer de l’observation, et qui une fois établies, suffisent pour qu’étant combinées par le seul raisonnement, elles embrassent toutes les ramifications de la science: mais cela paroit très-difficile. En voulant remonter trop haut par le seul raisonnement, on s’expose à donner des définitions obscures, des démonstrations vagues et peu rigoureuses. Il y a moins d’inconvénient à tirer de l’expérience plus de données qu’il ne seroit peut-etre strictement nécessaire. […] C’est donc dans l’expérience que les hommes ont puisé les premières notions de la mécanique. Cependant les lois fondamentales de l’équilibre et du mouvement qui lui servent de base s’offrent d’une part si naturellement à la raison, et de l’autre, elles se manifestent si clairement par le faits les plus communs, qu’il semble d’abord difficile de dire, si c’est à l’une plutôt qu’aux autres que nous devons la parfaite conviction de ces lois. 11.5 Maintenant il s’agit d’établir sur ces faits, et sur les autres observations qui peuvent encore s’offrir, des hypothèses qui se trouvent constamment d’accord avec ces observations, et que dès-lors on puisse regarder comme des lois générales de la nature. […] Nous comparons ensuite les conséquences qui en résultent, avec les phénomènes, et si nous trouvons qu’ils s’accordent, nous conclurons que nous pouvons considérer ces hypothèses comme les véritable lois de la nature. 11.6 Mon objet n’a pas été de les réduire au plus petit nombre possible; il me suffit qu’elles ne soient point contradictoires et qu’elles soient clairement entendues […] mais elles sont peutêtre plus propre à confirmer les principes, en faisant voir comment ils ne sont, pour ainsi dire, que les mêmes vérités qui reparoissent toujours sous des formes différentes. 11.7 Connaissant le mouvement virtuel d’un système quelconque de corps, (c’est-â-dire, celui que prenderoit chacun de ces corps, s’il étoit libre) trouver le mouvement réel qui aura lieu l’instant suivant, à cause de l’action réciproque des corps, en les considérant tels qu’ils existent dans la nature, c’est-à-dire, comme doués de l’inertie commune è toutes les parties de la matière.

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11.8 Première loi. La réaction est toujours égale et contraire à l’action. Seconde loi. Lorsque deux corps durs agissent l’un sur l’autre, par choc ou pression, c’està-dire, en vertu de leur impénétrabilité leur vitesses relative, immédiatement après l’action réciproque, est toujours nulle. 11.9 Cette Essai sur les machines n’étant point un Traité de mécanique, mon but n’est pas d’expliquer en détail, ni de prouver les loix fondamentales que je viens de rapporter; ce sont des vérité que tout le monde sent très-bien. 11.10 Que l’intensité du choc ou de l’action qui s’exerce entre deux corps qui se rencontrent, ne dépend point de leurs mouvements absolus, mais seulement de leur mouvement relatif. Que la force ou quantité de mouvement qu’ils exercent l’un sur l’autre, par le choc, est toujours dirigée perpendiculairement à leur surface commune au point de contingence. 11.11 Si un système de corps part d’une position donnée, avec un mouvement arbitraire, mais tel qu’il eut été possible aussi de lui en faire prendre un autre tout-à-fait égal et directement opposé: chacun de ces mouvements sera nommé mouvement géométrique. 11.12 Tout mouvement, qui imprimé à un système de corps ne change rien à l’intensité de l’action qu’ils exercent ou pourroient exercer les uns sur les autres si on leur imprimoit d’autres mouvemens quelconques, sera nommé mouvement géométrique. 11.13 La théorie des mouvemens géométriques est très-importante; c’est, comme je l’ai déjà observé ailleurs (Géométrie de position, page 337), une espece de science intermédiaire entre la géométrie ordinaire et la mécanique. […] Cette science n’a jamais été traitée spécialement: elle est entièrement à créer, et mérite, tant par sa beauté en elle-même que par son utilité, toute l’attention des Savans. 11.14 Dans le choc des corps durs, soit que ce choc soit immediat, ou qu’il se fasse par le moyen d’une machine quelconque sans ressort, il est constant qu’à l’égard d’un mouvement quelconque géométrique: 1 – Le moment de la quantité de mouvement perdue par tout le système, est égal à zéro. 2 – Le moment de la quantité de mouvement perdue par une partie quelconque des corps du système, est égal au moment de la quantité de mouvement gagné par l’autre partie. 3 – Le moment de la quantité de mouvement réelle du système générale, immédiatement après le choc, est égal au moment de la quantité de mouvement du même système, immédiatement avant le choc. 11.15 Parmi tous les mouvements dont est susceptible un système quelconque de corps durs agissant les uns sur les autres, soit par un choc immédiat, soit par des machines quelconques sans ressort, celui de ces mouvements qui aura lieu réellement, l’instant d’après, sera le mouvement géométrique, qui est tel que la somme des produits de chacune des masses, par le carré de la vitesse qu’elle perdra, est un minimum, c’est à dire, moindre que la somme des produits de chacun de ces corps, par la vitesse qu’il auroit perdue, si le système eut pris un autre mouvement quelconque géométrique. 11.16 Dans le choc de corps durs, soit qu’il y en ait de fixes, ou qu’ils soient tous mobiles (ou ce qui revient au même) soit que ce choc soit immédiat, ou qu’il se fasse par le moyen d’une machine quelconque sans ressort; la somme des forces vives avant le choc, est toujours égale à la somme des forces vives après le choc, plus la somme des forces vives qui auroit lieu, si la vitesse qui reste à chaque mobile, étoit égale à celle qu’il a perdus dans le choc. 11.17 Lorsqu’un système quelconque de corps durs change de mouvement par degré insensibles; si pour un instant quelconque on appelle m la masse de chacun corps, V sa vitesse, p sa force motrice, R, l’angle compris entre les directions de V et p, u la vitesse qu’auroit m, si on faisoit prendre au système un mouvement quelconque géométrique, r l’angle formé par u

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et p, y l’angle formé par V et u, dt l’élément du temps; on aura ces deux équations

∑ mV pdt cos R − ∑ mV dV = 0

∑ mupdt cos r − ∑ mud(V cos y) = 0. 11.18 Théorème fondmental Principe général de l’équilibre et du mouvement dans les machines. XXXIV. Quel que soit l’état de repos ou de mouvement où se trouve un système quelconque de forces appliquées à une machine, si l’on fait prendre tôt-à-coup un mouvement quelconque géométrique, sans rien changer à ces forces, la somme des produits de chacune d’elles, par la vitesse qu’aura dans le premier instant le point ou elle est appliquée, estimée dans le sens de cette force, sera égale à zéro. 11.19 Il ne sera peut-étre pas inutile de prévenir une objection qui pourroit se présenter à l’esprit de ceux qui n’auroient pas fait attention à ce qui a été dit sur le vrai sens qu’on doit attacher au mot force: imaginons, par exemple, dira-t-on, un treuil à la roue et au cylindre duquel soient suspendus des poids par des cordes; s’il y a équilibre, ou que le mouvement soit uniforme le poids attaché à la roue, sera à celui du cylindre, comme le rayon du cylindre est au rayon de la roue; ce qui est conforme è la proposition. Mai il n’est pas de même lorsque la machine prend un mouvement accéléré ou retardé; il paroit donc qu’alors les forces ne sont pas en raison réciproque de leurs vitesses estimées dans le sens de ces forces, comme il suivroit de la proposition. La réponse à cela est, que dans le case où ce mouvement n’est pas uniforme, les poids en question ne sont pas les seules forces exercées dans le système, car le mouvements de chaque corps, changeant continuellement, il oppose aussi à chaque instant, par son inertie, une résistance à ce changement d’état; il faut donc aussi tenir compte de cette résistance. Nous avons déjà dit, comment cette force doit s’évaluer, et nous verrons plus bas, comment on doit la faire entrer dans le calcul. En attendant, il suffit de remarquer que les forces appliquées à la machine dont il est ici question, ne sont pas les poids même, mais les quantité de mouvement perdues par ces poids, lesquelles doivent s’estimer par les tensions des cordons auxquels ils sont suspendus: or, que la machine soit en repos ou en mouvement, que ce mouvement soit uniforme ou non, la tension du cordon attaché à la roue, est à celle du cordon attaché au cylindre, comme le rayon du cylindre est au rayon de la roue, c’est-à-dire, que ces tensions sont toujours en raison réciproque des vitesses des poids qu’ils soutiennent; ce qui est d’accord avec la proposition. Mais ces tension ne sont pas égales aux poids; elles sont les résultantes de ces poids et de leurs forces d’inertie, lesquelles sont elles-mêmes les résultantes des mouvements actuels de ces corps, et des mouvements égaux et directement opposés à ceux qu’ils prendront réellement l’instant d’après. 11.20 Lorsque plusieurs poids appliqués à une machine quelconque, se font mutuellement équilibre, si l’on fait prendre à cette machine un mouvement quelconque géométrique, la vitesse du centre de gravité du système, estimée dans le sens vertical sera nulle au premier instant. 11.21 Il y a deux manières d’envisager la mécanique dans ses principes. La première est de la considérer comme la théorie des forces; c’est-à-dire des causes qui impriment les mouvemens. La seconde est de la considérer comme la théorie des mouvemens eux-mêmes. La première est presque généralement suivie, comme la plus simple; mais elle a le désavantage d’être fondée sur une notion métaphysique et obscure qui est celle des forces. 11.22 Si une force P se meut avec la vitesse u, et que l’angle formé par le concours de u et P soit z, la quantité P cos zudt dans laquelle dt exprime l’élément du temp, sera nommé moment d’activité, consommé par la force P pendant dt. 11.23 Dans une machine dont le mouvement change par degrés insensibles, le moment d’activité consommé dans un temps donné par les forces sollicitante, est égal au moment d’activité exercé en même temps par les forces résistantes.

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11.24 Plusieurs corps soumis aux loix d’une attraction exercée en raison d’une fonction quelconque des distances, soit par ces corps mêmes les uns sur les autres, soit par différents points fixes, étant appliqué à une machine quelconque; si l’on fait passer cette machine d’une position quelconque donnée, à celle de l’équilibre, le moment d’activité consommé dans ce passage par les forces attractives dont ces corps seront animés pendant ce mouvement, sera un maximum.

A.12 Chapter 12 12.1 Al rinascere delle Scienze Galileo investigò i Teorici Fondamenti dell’equilibrio, e del moto assoggettandoli alla guida della Geometria, e col Principio delle Velocità Virtuali sparse una nuova, universale radiazione, in tutte le macchine semplici e composte. […] Infatti la Meccanica per mezzo del Principio delle Velocità Virtuali, unita alla Geometria partecipò della medesima evidenza, e ne godè i privilegi, per tutta l’ampiezza, in cui poteva spaziare la sintesi. In seguito la nuova Geometria (la quale con rapido volo percorre lo spazio, che l’antica era obbligata a misurare con lento passo, e giunge ove quella non si sa che sia mai penetrata) ha corrisposto alle più lusinghiere speranze, ed il Sig. La Grange il primo nell’immortale sua Opera intitolata Meccanica Analitica, non solo mostrò che il Principio delle Velocità Virtuali è dovuto a Galileo, ma rilevò ancora, che questo Principio ha il vantaggio di potersi tradurre in linguaggio algebraico, cioè di essere espresso per una formula analitica, onde tutte le risorse della analisi vi si applicano direttamente. Quel Principio dopo inventato da Galileo era rimasto, quasi negletto, come penderebbe inutile una grande spada, fino a tanto che non nascesse un braccio atto a brandirla. Infatti il Sig. La Grange padrone di tutto l’Ente matematico, ha saputo valutarne l’importanza, e la fecondità facendo per mezzo di esso della Meccanica una scienza nuova a segno, che nella universale dottrina dell’equilibrio, e del moto dei solidi, e dei fluidi, tutti quei difficili Problemi, che avevano condotto fino ad ora i Geometri per mille diverse spinosissime strade, sono ridotti ad un procedere regolare ed uniforme. E per dare un’idea di quanto abbia progredito lo spirito umano, si può dire, che il moto, e l’equilibrio dei Corpi Celesti, la figura di essi e le orbite, che descrivono, non richiamano in sostanza, per quanto appartiene alla Meccanica, a considerare altre leggi oltre quelle, che hanno luogo nel calcolare il moto, e l’equilibrio di un Vette del primo genere quantunque le difficoltà di puro calcolo, e la moltitudine degli oggetti da contemplare presentino un apparato più vasto, ed imponente. […] Alcuni si sono occupati nel far vedere, che questo Principio è vero, mostrando la conformità dei risultati di esso con quelli dedotti da altri metodi universalmente ammessi. Ma veramente non se ne potesse ottenere altra autentica, saremmo ben lontani dallo scopo, a cui mirano ordinariamente i Geometri; nella stessa guisa, che allor quando i seguaci di Leibnitz mancavano di una convincente dimostrazione del Calcolo Infinitesimale, era debole appoggio per essi l’osservare l’uniformità de’ suoi resultati, con quelli della Geometria degli Antichi. […] Quella comune facoltà di primitiva intuizione, per cui ognuno si convince facilmente di un semplice assioma Geometrico, come per esempio, che il tutto sia maggior della parte non serve certamente per convenire della sopraccennata verità meccanica, la quale è tanto più complicata di quello che sia uno degli ordinari assiomi, quanto il genio di quei grandi Uomini, che l’hanno ammessa per assioma, supera l’ordinaria misura dell’ingegno umano; ed è in conseguenza necessario per coloro, che non ne restano appagati, il procurarsene una dimostrazione dipendentemente da estranee teorie, come è piaciuto al Riccati (che con qualche soccorso tutto metafisico, si è ristretto presso a poco a questo caso particolare in alcune Lettere stampate in Venezia nel 1772) ovvero riposarsi sulla fede d’uomini sommi disprezzando l’abituale ripugnanza ad introdurre in Matematica il peso dell’autorità. E se veramente questa tiranna della ragione dovesse per una sol volta apparire nel Tempio d’Urania, non potrebbe seguir ciò con minore scandalo, che trovandosi essa in mezzo a Galileo, e a La Grange.

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12.2 Teorema. L’equazione delle forze avrà luogo egualmente che quella dei momenti, quando i corpi saranno stabiliti in linea retta, ed inoltre le forze comunque applicatevi, se non avranno le direzioni parallele tra loro, le avranno almeno tali, che sieno parallele le proiezioni di esse, fatte in un piano passante per la linea dei corpi. 12.3 Si potrà dunque concludere, che in ogni sistema, in cui l’equilibrio dipenda dalle equazioni punti (1), (2), (3), (4), (5), (6) [4.3–8], del § LXXI, la proprietà della somma dei momenti = 0 è una proprietà necessaria e indivisibile dall’equilibrio. 12.4 Non è dunque possibile di negare, che qualunque volta abbia luogo l’equilibrio, esista necessariamente l’equazione dei momenti; ma gli è sicuro che qualunque volta esiste l’equazione dei momenti abbia sempre luogo l’equilibrio. 12.5 Potrebbe dubitarsi che oltre a queste sei equazioni se ne potessero dare altre. 12.6 J’ai lu votre Ouvrage avec plaisir. S’il a encore quelque chose à désirer dans la Mécanique, c’est le rapprochement, et la réunion des principes, qui lui servent de base, et peut-être même la démonstration rigoureuse et directe de ces principes. Votre travail est un nouveau service rendu a cette science. Vous observez avec raison, qu’il y a des cas, où l’équation des vitesses virtuelles a lieu aussi par rapport aux différences finies, le système alors en changeant de situation ne cesse par d’être en équilibre. Ces sortes d’équilibres tiennent le milieu entre les équilibres stables, où le système revient de lui mime à son premier état, lorsqu’il en est dérange, et les équilibres non stables, où le système, une fois dérangé de son état d’équilibre, tend à s’en éloigner de plus en plus. 12.7 J’ai donné une démonstration du principe des vitesses virtuelles tirée de 1’équilibre des mousses. Un principe si important ne peut-étre prouvé de trop de manières. Votre travail sur ce sujet a, outre son propre mérite, celui d’avoir fait éclore d’autres ouvrages, et on lui doit les Mémoires de Prony et de Fourier, et dont 1es auteurs ont dû vous faire hommage. 12.8 Se poi supponghiamo che siano più punti in qualunque modo insieme connessi, e muovansi ancora all’intorno d’un asse qualsiasi, chi è che subito non ravvisi che la teoria di siffatto moto dal principio del vette dipende necessariamente. 12.9 Benché, come notammo, alcuni son d’avviso che una dimostrazione rigorosa della teoria del vette da Archimede e dopo di lui da altri uomini prestatissimi investigata lasci tuttor desiderio di sè. 12.10 Dunque dovremo essere ancor noi del sentimento di quelli che opinano avere il principio di risoluzion delle forze e di composizione per invisibil compagnia l’infallibilità metafisica; quello del vette, il patrocinio soltanto della continua e costante esperienza, e finalmente quello delle velocità virtuali che da’ due precedenti deducesi, non poter maggior grado di certezza acquistare di quello che si ravvisa nel principio del vette. 12.11 Eclaircir le principe des vitesses virtuelle dans toute sa généralité tel qu’il a été énoncé par M. Lagrange: Faire voir, si ce principe doit être regardé, comme une vérité évident par la seul exposition du principe même, ou s’il exige une démonstration: fournir cette démonstration dans le case qu’on la juge nécessaire. 12.12 Quocumque modo secum invicem connectantur duo puncta A et A , si velocitates eorum virtuales v, v sint semper intensionis aequales, vires P, P respective iis applicate et in rectis velocitatum oppositae in aequilibrio constant. 12.13 Propositiones septem priores ex prolegomenis mechanices depromptae ut axiomata teneri debent; ultimam vero ut concedatur saltem postulamus. Caeterum eius evidentiam paucis declarare iuvat.

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12.14 Libenter fateor Polyspastum apud antiquos, licet ab iis cognitum, uti , inter alia , ex Pappi collectionum libro 8 col1igitur, (erat enim Polyspaston tertia facultas mechanica apud Heronem) minus celebratum quam vectis et in novissimi temporibus, inter aequilibrii scientiae principia, praedicatum non fuisse nisi a solis fere Landen et Lagrange: ast ubi de delectu principii agitur, attendendum videtur ad ipsius evidentiam et foecunditatem praesertim: porro commoda haec in summo gradu prae se fert Polyspasti theoria at nemo non diffitebitur. 12.15 Dicet insuper aliquis forsan mancam aut incompletam esse nostram demonstrationem, ratus cum quibusdam proeliandum esse non solum ab aequilibrii inter vires hypothesi momentorum aequationem dimanare, sed etiam reciproce, ex hypothesi momentorum aequationis, aequilibrium inter vires sequi: verum attendatur momentorum aequatione (6) (11°), quae in aequilibrio systematis Polyspastis instructi valet, exprimi evidenter aequilibrium adesse inter vires; eandemque, propter propositionum concatenationem , aequationis momentorum significationem obtinere in omni systematum genere et liquebit aequationem hanc haberi debere ut aequilibrium adesse declarantem non vero tantum ut aequilibrium Concomitantem. 12.16 Perciocché vedesi costretta a mettere in campo il ripiego di certo meccanico movimento fittizio infinitesimale, che diede occasione bensì alla scoperta d’insigni verità maravigliose, ma che lascia nel tempo stesso sussistere tutt’ora il desiderio di una chiara semplice ed unica dimostrazione del vincolo primitivo, e necessario, che ad esso lega siffatte proprietà, dimostrazione che può dirsi non ancora conseguita, se si considera l’incostanza, la complicazione, e l’oscurità dei tentativi, che per essa sono stati fatti. 12.17 Si plusieurs forces, ayant des directions quelconques, sont appliques à un système de corps ou de points et se font équilibre; la somme de ces puissances multipliées chacune par la vitesse qu’elle tend à imprimer au point auquel elle est appliquée est nécessairement égale à zero. On voit évidemment que cet énoncé rentre dans celui qui à été exposée ci-dessus, mais qu’il en seulement degagé des quantités infiniment petits. 12.18 Queste riflessioni persuadono che sarebbe un cattivo filosofo chi si ostinasse a volere conoscere la verità del principio fondamentale della meccanica in quella maniera che gli riesce manifesta l’evidenza degli assiomi. Però dovrà necessariamente mancare di questa evidenza il principio che assumerò […] il quale è lo stesso assunto da Lagrange nella parte terza della teorica delle funzioni. Ma se il principio fondamentale della meccanica non può essere evidente, dovrà essere non di meno una verità facile a intendersi e a persuadersi. 12.19 Se un corpo attratto verso un punto fino a passare in linea retta nel tempo t lo spazio φ(t), qualora gli venga impresso un altro moto […] αt […] per l’azione simultanea dei due moti non percorre uno spazio espresso da φ(t) + αt ma da un’altra funzione del tempo.

A.13 Chapter 13 13.1 Pour la Patrie, pour les Sciences et la Gloire. 13.2 Je dois aussi indiquer aux élèves un ouvrage dont il leur sera très-utile de réunir la lecture et l’étude, aux instructions qu’ils reçoivent à l’École sur fa même matière c’est un, mémoire italien publié à Florence en 1796, par M. Fossombroni, et intitulé Memoria sul principio delle velocità virtuali. Ce traité leur offrira une foule d’exercice très-profitables sur-tout à ceux qui veulent étudier la Mécanique analytique. 13.3 La démonstration précédent ne laisse rien à désirer pour la rigueur; mais l’équation des vitesses virtuelles, présentée de cette manière, offre une conséquence plutôt qu’une vérité fondamentale; et il est nécessaire, pour lui conserver le caractère de principe, de la déduire de théorèmes de Mécanique encore plus élémentaire et plus près des vérités de définition que

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ceux dont je me suis servi: c’est que je vais faire, en ne supposant que la composition des puissances appliquées à un point unique et de celle des puissances parallèles. 13.4 J’ai pensé aussi qu’il ne suffisait pas de prouver, d’une manière absolue, la vérité de la proposition, mais qu’on devait le faire indépendamment de la connaissance que nous avons des conditions de de l’équilibre dans les différentes espèces de corps, puisqu’il s’agit de considérer ces conditions comme des conséquences de la proposition générale. Cet objet se trouve rempli par les démonstrations que nous allons rapporter; il nous semble qu’elles ne laissent rien à désirer sous le double rapport de l’étendue et de l’exactitude. Nous supposerons connue le principe de levier, tel qu’il est démontré dans les livres d’Archimède, ou ce qui revient au même le théorème de Stevin sur la composition des forces, et quelques propositions qu’il est aisé de déduire des précédentes. 13.5 Si un corps est déplacé par une cause quelconque suivant une certaine loi, chacune des quantités qui varient avec sa position, comme la distance d’un de ses points à un point ou à un plan fixe, est une fonction déterminée du temps, et peut être considérée comme l’ordonnée d’une courbe plane dont le temps est l’abscisse, la tangente de l’angle que fait cette courbe à l’origine avec la ligne des abscisses, où la première raison de l’accroissement de l’ordonnée à l’abscisse exprime la vitesse avec laquelle cette quantité commence a croitre, ou, pour nous servir d’une dénomination reçue, la fluxion de cette quantité. Le corps étant soumis à l’action de plusieurs forces, si l’on prend sur la direction de chacune un point fixe dont la force tende a rapprocher le point du système où elle est appliquée, le produit de cette force par la fluxion de la distance entre les deux points est le moment de la force: le corps peut être déplacé d’une, infinité de manières, et à chacune répond une valeur du moment. Si l’on prend le moment de chaque force pour un même déplacement, la somme de tous ces momens contemporains sera appelée le moment total, ou le moment des forces, pour ce déplacement. Nous distinguerons d’abord les déplacemens compatibles avec l’espèce et l’état du système, de ceux qu’on ne petit lui faire éprouver sans altérer les conditions auxquelles il est assujetti; et nous supposons ces conditions exprimée, autant qu’il est possible, par des équations. Maintenant le principe des vitesses virtuelles consiste en ce que les forces qui sollicitent un corps de quelque nature qu’il puisse être, étant supposées se faire équilibre, le moment total des forces est nul pour chacun des déplacemens qui satisfont aux equations de condition. Jean Bernoulli considère au lieu des fluxions les accroissements naissans. Il faut alors regarder chacun des points du système comme décrivant un petit espace rectiligne d’un mouvement uniforme pendant un instant infiniment petit. Cet petit espace projeté perpendiculairement sur la direction de la force, est la vitesse virtuelle; et si on la multiplie par la force, le produit représent le moment. J’adopterai cette heureuse abréviation, et tous les procédés usités du calcul différentiel. 13.6 Si l’on considère deux forces qui se font équilibre étant appliquées aux exterminés d’un fil inextensible, il sera facile de connaitre leur moment total pour un déplacement compatible avec la nature du corps en équilibre. Il suit de l’article précédent, que le moment est nul toutes lés fois que la distance est conservée; c’est-à-dire, lorsque l’équation de condition est satisfaite. Pour tous les autres déplacemens possibles, le moment est positif, et le système en équilibre ne peut être trouble de manière que le moment total soit négatif. 13.7 Au lieu de transformer, comme nous l’avons fait jusqu’ici, les forces qui sollicitent le système, nous substituerons à ce système, sur le quel elles agissent, un corps plus simple, mais susceptible d’être déplacé de la même manière. 13.8 Si l’on se contentait de substituer à chacune des forces un poids attaché a un fil renvoyé par une poulie fixe, on reconnaitrait que pour chaque déplacement du système en équilibre la quantité de mouvement des poids que s’élèvent est égale à celle des poids qui s’abaissent; et quoique cette remarque ne puisse pas être considérée comme une démonstration, néanmoins elle ramène le principe des vitesses virtuelles à celui de Descartes, ou au principe, employé

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Appendix. Quotations par Toricelli. Il est naturel de penser que Jean Bernoulli connaissait quelque construction analogue. On trouve les memes idées dans un ouvrage de Carnot, imprimé dès 1783, sous ce litre: Essai sur les machines en général.

13.9 Le principe des vitesses virtuelles qui sen de base à cet admirable ouvrage, fut considéré par son auteur comme un fait dont il ne se proposait alors que de développer toutes les conséquences; on s’est occupé depuis de la démonstration générale de ce principe M. Lagrange l’a ramené, d’une manière très-simple, au principe de l’équilibre des mousses, M. Carnot à celui de l’équilibre du levier. La démonstration de ce même principe a été déduite par M. Laplace, de considérations plus générales, mais trop abstraites peut-être pour être mises facilement à la portée des commençans. Je me suis proposé de donner, autant qu’il me sera possible, la même généralité à une démonstration qui reposât uniquement sur la théorie de la composition et de la décomposition des forces appliques à un même point, et qui fut dégagé de la considération des quantités infiniment petite: tel a été le but que je me suis proposé dans les recherches que j’ai l’honneur de présenter à la classe. 13.10 Les lois de l’équilibre se déduisent, de la manière la plus rigoureuse, de quelques considérations fort simples, lorsque les forces sont appliquées à un même point; mais elles deviennent plus difficiles à démontrer, surtout lorsqu’on se propose de les considérer dans toute leur généralité, dès que les forces agissent sur différens points: assujettis à des conditions qui contribuent à la destruction mutuelle des forces. La difficulté vient surtout de la nécessité de faire entrer, d’une manière générale, ces conditions dans le calcul. Il semble, au premier aspect, qu’on peut les considérer séparément, et ne supposer d’abord qu’une des conditions, puis une autre, et ainsi de suite; mais un peu de réflexion fait voir qu’il faudrait alors pouvoir démontrer à priori, que les effets produits par la réunion de plusieurs conditions, se composent des effets qui résultent de chaque condition en particulier, sans qu’elles soient modifiées par leur réunion; vérité qui parait plutôt devoir une conséquence des équations de l’équilibre, qu’un moyen de les obtenir. 13.11 Une autre simplification qu’on pourrait employer dans la recherche dont nous nous occupons, consiste à supposer successivement tous les points du système fixes, à l’exception de deux d’entre eux, ce qui est d’autant plus commode que, par l’addition des équations ainsi obtenues, on compose précisément avec les dérivées partielles relatives à chaque variable, les dérivées totales dont on a besoin; mais un exemple très-simple me parait suffisant pour faire voir que cette supposition n’est pas toujours admissible. 13.12 Le principe connu sous le nom de principe des vitesses virtuelles, se réduit à ce que si l’on fait une somme des momens de toutes les forces appliquées an système, en prenant avec des signes contraires, ceux dont les forces et les projections tombent du même coté, et ceux dont les forces et les projections tombent de cotés opposés; que l’on ajoute à celle somme celle des équations déduites de toutes les conditions données, multipliées chacune par un facteur arbitraire; et réduites à contenir dans tous leurs termes les dérivées x , y , z , à la première puissance; qu’on égale séparément a zéro les quantités qui multiplient chaque dérivée, et qu’on élimine tous les facteurs arbitraires, l’équation ou les équations restantes expireront mutes les conditions de l’équilibre. 13.13 Or, c’est un théorème d’algebre aisé à démontrer que l’équation résultant de cette élimination est identiquement la même que celle qu’on obtiendrait en ajoutant à la somme des momens les équations des conditions multipliées par des facteurs arbitraires, en égalant séparément à zéro les quantities qui multiplient chaque dérivée, et en eliminant les facteurs. 13.14 Or la force du pression d’un point sur une surface lui est perpendiculaire, autrement elle pourroit se décomposer en deux, l’une perpendiculaire à la surface, et qui seroit détruit par elle, l’autre parallèle à la surface, et en vertu de laquelle le point n’auroit point d’action sur cette surface, ce qui est contre la supposition.

A.14 Chapter 14

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m,

ne peuvent agir l’une sur l’autre, que suivant la droite 13.15 Deux points dont les masses m et qui le joint. A la vérité, si les deux points sont liés par un fil qui passe sur une poulie fixe, leur action réciproque peut n’être point dirigée suivant cette droite. Mais on peut considérer la poulie fixe, comme ayant à son centre, une masse d’une densité infinie, qui réagit sur les deux corps m et m , dont l’action l’un sur l’autre n’est plus qu’indirecte.

A.14 Chapter 14 14.1 Les lignes aa , bb , cc , &c, sont ce qu’on appelle dans les auteurs le vitesses virtuelles des points a, b, c, &c., mais si l’on veut avoir le valeur des moments, on multiple les forces par ces lignes estimées suivant les directions des forces, c’est à dire projetées sur elles. Il est donc convenable, pour abréger, d’appeler ces projections elles-mêmes les vitesses virtuelles. 14.2 De cette manière on exclurait les idées de mouvemens infiniment petits et de perturbations d’équilibre qui sont des idées étrangères à la question; et le principe des vitesses virtuelles paraitrait comme un simple théorème de géométrie dégagé de ces considérations qui laissent toujours dans l’esprit quelque chose d’obscur. Mais il est bon d’observer que cette propriété de l’équilibre dont nous nous occupons ne fut découverte que par la considération de ces petites vitesses, par ce qu’elle s’offre naturellement lorsqu’on dérange une machine en équilibre. Il semble que par ce dérangement on estime les énergies des puissances pour mouvoir la machine. Lorsqu’un système est en équilibre, on connait bien la valeur absolue de chaque force, mais non pas l’effort quelle exerce su egard à sa position. En dérangeant un peu le système, on voit quels sont les mouvemens simultanés que peuvent prendre les points où les forces sont appliquées, quelques uns de ces points se mouvant du même cote que tirent les forces, les autres étant entrainés dans le sens contraire, et si l’on estime les énergies proportionnelles aux produits des forces par les vitesses des points d’application, on trouve que les énergies qui obtiennent leur effet sont égales aux énergies vaincues. 14.3 Si un système libre de figure invariable est en équilibre en vertu de forces quelconques qui lui sont appliquées, en supposant que les forces agissent toutes aux points de rencontre de leurs directions avec un plan situé comme on voudra, l’équation des moments aura lieu quel que soit le déplacement fini qu’on donne au système. 14.4 Il faut encore remarquer qu’on suppose le système déplacé d’une manière quelconque, sans aucun égard à l’action des puissances qui tend à le déplacer; le mouvement qu’on lui donne est un simple change de position où le temps n’entre pour rien. 14.5 Le principe des vitesses virtuelles est connu depuis long-temps, aussi bien que la plupart des principes généraux de la mécanique. Galilée observa le premier, dans les machines, cette fameuse propriété des vitesse. virtuelles, c’est-à-dire, cette relation si connue qui existe entre les forces appliquées et les vitesses que prendraient leurs points d’application si l’on venait à troubler infiniment peu l’équilibre de la machine. Jean Bernouilli vit toute l’étendue de ce principe, et l’énonça avec cette grande généralité qu’on lui donne aujourd’hui. Varignon et la plupart des géomètres prirent soin de le vérifier dans presque toutes les questions de la statique; et quoiqu’on n’en est point de démonstration générale, il fut universellement regardé comme une loi fondamentale de l’équilibre des systèmes. Mais jusqu’à M. Lagrange, les géomètres s’étaient plus appliqués à démontrer ou I’étendre les principes généraux de la science, qu’à en tirer une règle générale pour la solution des problèmes; ou plutôt ils ne s’étaient pas encore propose ce grand problème qui est à lui seul tonte la mécanique. Ce fut alors une heureuse idée de partir sur-le-champ du principe des vitesses virtuelles comme d’un axiome, et sans s’arrêter davantage à le considérer en lui-même de ne songer qu’à en tirer une méthode uniforme de calcul pour former les équations de l’équilibre et du

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Appendix. Quotations mouvement dans tous les systèmes possibles. On franchit par là toutes les difficultés de la mécanique: évitant pour ainsi dire, de faire la science elle-même, on la transforma en une question de calcul: et cette transformation, l’objet et le résultat de la Mécanique analytique, parut comme un exemple frappant de la puissance de l’Analyse. Cependant, comme dans cet Ouvrage, on ne fut d’abord attentif qu’à considérer ce beau développement de la Mécanique qui semblait sortir tout entière d’une seule et même formule, on crut naturellement que la science était faite, et qu’il ne restait plus qu’à chercher la démonstration du principe des vitesses virtuelles. Mais cette recherche ramena toutes les difficultés qu’on avait franchies par de principe même. Cette loi si générale, où se mêlent des idées vagues et étrangères de mouvemens infiniment petits et de perturbation d’équilibre, ne fit en quelque sorte que s’obscurcir l’examen: et le livre de M. Lagrange n’offrant plus alors rien de clair que la marche des calculs, on vit bien que les nuages n’avaient paru levés sur le cours de la Mécanique, que parce qu’ils talent pour ainsi dire rassembles à l’origine même de cette science. Une démonstration générale du principe des vitesses virtuelles devait au fond revenir à établir la mécanique entière sur une autre base. Car la démonstration d’une loi qui embrasse tout une science ne peut être autre chose que la réduction de cette science à une autre loi aussi générale, mais évidente, ou du moins plus simple que la première, et qui partant la rende inutile. Ainsi, par cela même que le principe des vitesses virtuelles renferme toute la mécanique, comme il a besoin d’une démonstration approfondie, il ne peut lui servir de base première. Chercher à le démontrer pour l’heureuse usage qu’on en a fait, c’est chercher à s’en passer pour cet usage même; soit en trouvant quelque autre loi aussi féconde mais plus claire, soit en fondant sur les principes ordinaires une théorie générale de l’équilibre, dont la propriété des vitesses virtuelles ne devient plus alors qu’un simple corollaire. Ainsi, dans cet état où M. Lagrange avait porté la science, ce n’était point la démonstration du principe des vitesses virtuelles qu’il fallait chercher immédiatement. La Mécanique analytique, telle que l’auteur l’a conçue, est au fond ce qu’elle doit être: et la démonstration du principe des vitesses virtuelles n’y manque point, puisque, si l’on essayait de la mettre à la tête de ce livre d’une manière générale et bien développée, l’ouvrage se trouverait fait deux fois; je veux dire que cette démonstration comprendrait déjà toute la mécanique. Il faut considérer que M. Lagrange s’est placé tout d’un coup sur un des points élevés de la science, afin de découvrir quelque règle générale pour résoudre, ou du moins pour mettre en équations, tous les problèmes de la mécanique; et cet objet est parfaitement rempli. Mais pour former la science elle-même, il faut élever une théorie qui domine également tous les points de vue d’où l’on peut l’envisager. Il faut aller directement, non pas au principe obscur des vitesses virtuelles, mais à cette règle claire qu’on en a pour la solution des problèmes: et cette recherche directe, la sente propre à satisfaire notre esprit, fait l’objet principal du Mémoire qu’on va lire.

14.6 Dans l’équilibre des systèmes, chaque force doit être perpendiculaire à la surface ou à la courbe sur la quelle le point d’application n’aurait plus que la liberté se se mouvoir si tous les aitres points devenaient fixes. 14.7 Premièrement, par cela seul que les points du système sont liés entre eux par la première équation L = const., on peut leur appliquer les forces respectives:

2 2 2 ∂L ∂L ∂L λ + + ∂x ∂y ∂z

λ

∂L ∂ x



2 +

∂L ∂ y



2 +

∂L ∂ z

2

A.14 Chapter 14

λ

∂L ∂ x



2 +

∂L ∂ y &c.



2 +

∂L ∂ z

459

2

λ désignant un coefficient quelconque indéterminé, et chaque force étant normale à la surface représenté par l’équation L =const., lorsq’on y regarde les trois coordonnées du point d’application comme seules variables; et l’on est sur que ces forces se feront équilibre sur le système. En second lieu, parce que les points sont liés entre eux par la seconde équation M = const., on peut leur appliquer encore les forces respectives





 ∂M 2 ∂M 2 ∂M 2 μ + + ∂x ∂y ∂z

2

2

 ∂M ∂M ∂M 2 μ + + ∂ x ∂ y ∂ z





 ∂M 2 ∂M 2 ∂M 2 μ + +    ∂x ∂y ∂z &c. μ étant un nouveau coefficient indéterminé; et chacune de ces forces étant normale à la surface représenté par l’équation M =const. lorsqu’on y regarde les cordonnées du point d’application comme seules variables. […] Il est bien manifeste qu’il y aura équilibre en vertu de toutes ces forces, puisqu’il y aurait équilibre en particulier dans chaque groupe relatif à chaque équation. 14.8 Quelles que soient les équations qui règnent ente les coordonnées des différens points du système, chacune d’elles pour l’équilibre, demande qu’on applique à ces points, le long de leurs coordonnées, des forces quelconques proportionnelles aux fonctions primes de cette équation, relativement à ces coordonnées respectives. Ainsi, en représentant par L = 0, M = 0, &c. des équations quelconques entre les coordonnées x.y, z; x , y , z , &c. des différens points, et par λ, μ, &c. des coefficients quelconques indéterminés, on aura, pour les forces totales X,Y, Z; X  ,Y  , Z  , &c., qui doivent être appliquées à ces points suivant leurs coordonnées:



 dL dM X =λ +μ + &c. dx dx



 dM dL +μ + &c. Y =λ dy dy



 dM dL +μ + &c. Z=λ dz dz



 dL dM + μ + &c. X = λ dx dx



 dL dM +μ + &c. Y = λ dy dz



 dL dM +μ + &c. Z = λ dz dz &c.

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Appendix. Quotations Si l’on élimine de ces équations les indéterminées λ, μ, &c., il restera les conditions de l’équilibre proprement dites, c’est à dire les relations qui doivent avoir lieu entre les seules forces appliquées et les coordonnées de leurs points d’application pour l’équilibre du système.

14.9 Note II Démonstration du principe des vitesses virtuelles: Identité de ce principe avec le théorème général qui fait l’objet du Mémoire précédent. On s’est contenté d’observer dans le Mémoire, que du théorème où l’on est parvenu sur l’expression générale des forces de l’équilibre, on pouvait passer aisément au principe des vitesses virtuelles. Mais ce principe est si célèbre dans l’histoire de la Mécanique, que je ne puis m’empêcher de marquer en peu de mots ce passage; et j’y reviens d’autant plus volontiers, que, non -seulement le principe des vitesses virtuelles est un corollaire de la proposition générale établie ci-dessus, mais qu’il me parait encore identique avec elle lorsqu’on le regarde sous son vrai point de vue, et qu’on l’énonce d’une maniere complète. Soit le système défini par les équations suivantes entre les coordonnées des corps, f (x, y, z, x , y , z , &c.) = 0. φ(x, y, z, x , y , z , &c.) = 0.

(A)

&c. Supposons qu’on imprime a tous ces corps des vitesses quelconques qu’ils puissent avoir actuellement sans violer les conditions de la liaison; ses coordonnées x, y, z; x , y , z , &c. varieront avec le temps t, dont il faudra les regarder comme fonctions; et, pour que les vitesses dy dz dx imprimées dx dt , dt , dt , dt , &c. soient permises par la liaison, comme on le suppose, il faudra qu’elles satisfassent aux équations dx dy dz dx dy f  (x) + f  (y) + f  (z) + f  (x ) + f  (y ) + &c. = 0 dt dt dt dt dt dx dy dz     dx   dy  + φ (y ) + &c. = 0 φ (x) + φ (y) + φ (z) + φ (x ) dt dt dt dt dt &c.

(B)

tirées des précédentes (A); et il suffira qu’elles y satisfassent pour que les conditions de la liaison soient observées. Ou bien, si l’on multiplie ces équations par des coefficiens quelconques indéterminés, λ, μ, &c., et qu’on les ajoute, il suffira qu’elles satisfassent à la seule équation suivante, indépendamment de λ, μ, &c.

dy dx  [λ f  (x) + μφ (x) + &c.] + λ f (y) + μφ (y) + &c. + dt dt

dz dx + [λ f  (z) + μφ (z) + &c.] + λ f  (x ) + μφ (x ) + &c. dt dt dy     [λ f (y ) + μφ (y ) + &c.] + &c. = 0. dt

(C)



dy dz dx Or les fonctions qui multiplient les vitesses dx dt , dt , dt , dt , &c., ne sont autre chose (d’après ce qui a été démontré ) que les expressions générales des forces capables d’être en équilibre sur le système. Supposant donc des forces X,Y, Z, X  ,Y  , Z  , &c. qui se feraient actuellement équilibre, on aurait:

X

dy dz dx dx dy dz +Y + Z + X +Y  + Z + &c. = 0. dt dt dt dt dt dt

(D)

dy dz Au lieu des trois composantes X,Y, Z, multipliées par les vitesses respectives dx dt , dt , dt on peut mettre la résultante P, multipliée par la vitesse résultante, projetée sur la direction de P,

A.14 Chapter 14 et que je nommerai

ds dt ;

461

et de même pour les autres; et l’on aura: P

ds ds ds + P + P + &c. = 0 dt dt dt

c’est-à-dire que si des forces se font équilibre sur un système quelconque, la somme de leurs produits par les vitesses, quelles qu’elles soient, qu’on voudra imprimer aux corps, mais que leur liaison permet, sera toujours égale à zéro, en estimant ces vitesses suivant les directions des forces. On voit par-là qu’on peut prendre des vitesses quelconque finies, que l’on mesurerait par des droites quelconques qui seraient simultanément décrites par les corps, s’ils venaient tout-àcoup à rompre leur liaison et à s’échapper librement chacun de son coté. Quand on veut mesurer ces vitesses par les espaces mêmes que les corps décrivent réellement, comme elles varient à chaque instant par la liaison des corps, il faut prendre ces espaces infiniment petits, sans quoi ils ne mesureraient plus les vitesses imprimées; et c’est ainsi qu’on tombe dans les vitesses virtuelles proprement dites, où le principe vient perdre une partie de sa clarté. Il résulte, en effet, de ce que nous venons de dire, que cette belle propriété de l’équilibre peut s’énoncer de la manière suivante: Lorsqu’on voit suivre aux différens corps d’un système des mouvemens quelconque qui ne violent point la liaison établie entre eux, c’est- à-dire, qui nous présentent continuellement le système dans des figures où les équations de condition subsistent, on put être sûr que les forces qui seraient capables de se faire équilibre sur une de ces figures, dans le moment où le système y passe, sont telles que, multipliées par les vitesses actuelles des corps projetées sur leurs directions, la somme de tous ces produits est nécessairement égale à zéro. Le principe de cette manière n’offre plus aucune trace de ces idées de mouvemens infiniment petits, et de perturbation d’équilibre: qui paraissent étrangères à la question, et qui laissent dans l’esprit quelque chose d’obscur. Lorsqu’il y a équilibre, il est clair que le principe a lieu pour tous les systèmes de vitesses que les points pourraient avoir en passant par la figure que l’on considère. Mais, quand on veut partir du principe, et l’énoncer de manière qu’il assure l’équilibre, faut-il dire qu’il a lieu pour ce nombre infini de systèmes de vitesses. Il y aurait surabondance de conditions, et l’on voit qu’il suffit de dire que l’équation (D) doit se vérifier pour autant de systèmes de vitesses que les équations de condition (B) en laissent d’indépendantes; ou bien (en réunissant comme on l’a fait ci-dessus, toutes ces équations en une seule (C), au moyen des indéterminées, λ, μ, &c.), il suffit de dire que l’équation (D) des momens doit se verifier pour dy dz dx autant de systèmes de vitesses qu’il y a de vitesses dx dt , dt , dt , dt , &c. Mais comme chacun de ces systèmes de vitesses doit satisfaire a l’équation (C), par hypothèse, cela revient à dire que dy dz dx toutes les forces appliquées X,Y, Z, X  ,Y  , Z  , &c. qui multiplient les vitesses dx dt , dt , dt , dt , &c., dans l’équation (D), doivent être toutes proportionnelles aux fonctions

dx 

dy λ f  (x) + μφ (x) + &c. , λ f (y) + μφ (y) + etc.. , dt dt  

dz  

dx λ f (z) + μφ (z) + &c. , λ f (x ) + μφ (x ) + &c. , dt dt  

dy , &c. λ f (y ) + μφ (y ) + &c. dt qui multiplient les mêmes vitesses dans l’équation générale (C). qui fait régner entre elles les seules conditions que la liaison exige. Donc le principe des vitesses virtuelles, bien énoncé, c’est-à-dire avec toutes les idées qui peuvent le faire comprendre, est parfaitement identique avec le théorème général qui fait l’objet du Mémoire. Il dit exactement la même chose; savoir, que, pour l’équilibre, les forces appliquées suivant les coordonnées des corps, en vertu de chaque équation, doivent être proportionnelles aux fonctions primes de cette équation rela-

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Appendix. Quotations tivement a ces coordonnés respectives: mais c’est là précisément ce qu’il fallait démontrer. Au reste, on serait encore conduit à reconnaitre cette identité en partant de l’énoncé ordinaire du principe des vitesses virtuelles, et se rendant bien compte, avant tout, du vrai sens qu’on y doit attacher. En effet, le problème général de la statique n’est pas seulement de chercher les rapports des forces qui se font actuellement équilibre, sur le système, mais bien l’expression générale des forces qui peuvent s’y faire continuellement équilibre, dans toutes les figures où il peut passer en vertu des équations de condition. L’équation générale donnée par le principe des vitesses virtuelles n’est donc pas, s’il est permis de parler ainsi, la relation d’un moment; elle ne doit pas simplement considérer l’équilibre du système dans la figure où il est, mais encore dans toute la suite des figures où il peut être, puisque c’est cette suite de figures qui le caractérise et en constitue la définition. Ainsi l’équation des momens ne dit pas qu’il faut prendre pour les forces de tels nombres qu’elle en soit satisfaite, mais (puisque ces forces doivent varier avec la figure) qu’il faut choisir pour les représenter de telles fonctions des coordonnées, qu’elle demeure continuellement satisfaite, ou soit identique. Or, en vertu des conditions mêmes, on sait qu’il doit régner entre les vitesses simultanées que pourraient avoir les corps, une équation linéaire identique (C), dont les coefficiens sont les fonctions primes des fonctions données par rapport aux coordonnées suivant lesquelles on estime ces vitesses. L’équation des momens dit donc que les forces de l’équilibre doivent être représentées par ces fonctions; et par conséquent, pour la démontrer, il faut faire voir comment de telles forces se font effectivement équilibre; ou bien il fallait chercher directement quelles fonctions des coordonnées peuvent représenter les forces de l’équilibre, comme nous l’avons fait d’abord. C’est pourquoi la plupart des démonstrations par lesquelles on a ramené le principe des vitesses virtuelles, ou à d’autres principes, ou a la loi connue de quelque machine simple, telle que le levier, &c., nous, paraissent bien plutôt des preuves que de véritables démonstrations. Toutes, en effet, même la plus heureuse, qui est de M. Carnot, e font sans rien emprunter de la définition générale du système, comme si la machine était, pour ainsi dire voilée, et qu’on n’en vît sortir que les cordons où sont appliquées les puissances. On peut bien prouver ou rendre sensible par quelque construction plus ou moins simple, que si l’on trouble un peu l’équilibre, ces puissances doivent être dans un certain rapport avec les allongements permis de ces cordons; mais cela ne peut offrir que les rapports actuels des forces considérées comme nombres, et ne montre point du tout la forme d’expression qui leur est propre propre. Cette perturbation de l’équilibre n’apprendrait, dans aucun cas, à quelle machine on aurait affaire, et les mêmes rapports pourraient s’offrir entre les forces appliquées, quoique les machines fussent de constitution tout-à–fait différente, et que chacune d’elles imprimât pourtant à l’expression des forces qui lui conviennent, une forme différente qu’on y devrait voir et retrouver sans cesse, si la difficulté du théorème était entièrement consommée. Ainsi, la propriété des vitesses virtuelles n’en reste pas moins mystérieuse, et l’on n’a pas de véritable démonstration, veux dire une explication ouverte et claire, où l ’on voie non-seulement que la chose se passe ainsi, mais qu’eIle est encore une suite de la définition général que soi-même on a donnée au système que l’on considère. C’est peut-être par mie vue semblable, et pour arriver à l’équation des moment comme à une équation identique, que M. Laplace n’a considéré que les équations qui représentent la liaison des parties du système, et n’a d’ailleurs employé d’autres principes que celui de la composition des forces et de l’égalité entre I’action et la réaction; ce qu’on peut regarder comme les élémens de la théorie de l’équilibre. Quoi qu’il en soit, au reste, soit qu’on veuille partir du principe des vitesses virtuelles pour en suivre jusqu’au bout la signification intime, soit qu’on attaque directement le problème de la mécanique, ce qui est plus simple, on se trouve amené sur-le-champ à chercher quelles sont les fonctions des coordonnées qui donnent les forces de l’équilibre dans toutes les figures que peut affecter le système, en obéissant aux équations qui règnent entre les coordonnées des différens corps. Tel est exactement le problème que nous sommes proposé; et notre objet bien net et bien distinct a été de le résoudre par les premiers principes de la statique et de la géométrie.

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A.15 Chapter 15 15.1 Lorsqu’un système invariable, libre ou assujetti à certaines conditions, se meut dans l’espace, il existe entre les vitesses des différents points certaines relations qui, dans beaucoup de cas, s’expriment très simplement, et que l’on déduit des formules relatives à la transformation des coordonnées. Je vais montrer, dans cet article, que les mêmes relations peuvent être tirées du principe de vitesses virtuelles. Ordinairement, on se sert de ce principe pour déterminer les forces capables de maintenir en équilibre un système de points matériels assujetti à des liaisons données, en supposant connues les vitesses que ces points peuvent acquérir dans un ou plusieurs mouvements virtuels du système, c’est-à-dire dans des mouvements compatibles avec les liaisons dont il s’agit. Mais il est clair qu’on peut renverser la question, et qu’après avoir établi les conditions d’équilibre par une méthode quelconque, ou même, si l’on veut, par la considération de quelques-uns des mouvements virtuels, on pourra se servir, pour déterminer la nature de tous les autres, du principe que nous venons de rappeler. Ajoutons qu’il est utile, dans cette détermination, de substituer au principe des vitesses virtuelles un autre principe que l’en tire immédiatement du premier, ci qui se trouve renfermé dans la proposition suivante. Théorème. Supposons que deux systèmes de forces soient successivement appliqués à des points assujettis à des liaisons quelconques. Pour que ces deux systèmes de forces soient équivalents, il sera nécessaire, et il suffira que, dans un mouvement virtuel quelconque, la somme des moments virtuels des force du premier système soit égale à la somme des moments virtuels des forces du second système. 15.2 Théorème I. Si, à une époque quelconque du mouvement, deux points du système invariable ont des vitesses nulles, les vitesses de tous les autres points se réduiront à zéro. 15.3 Théorème II. Si, à une époque quelconque du mouvement, les vitesses de tous les points du système invariable sont différentes de zéro, ces vitesses seront toutes égales et dirigées suivant des droites paralleés. 15.4 Théorème III. Si, à une époque quelconque du mouvement, un seul point du système invariable a une vitesse nulle, la vitesse d’un second point choisi arbitrairement sera perpendiculaire au rayon vecteur mené du premier point au second, et proportionnelle à ce rayon vecteur. 15.5 Les théorèmes I, II et III indiquent toutes les relations qui peuvent exister entre les vitesses de points matériels liées invariablement les uns aux autres, et compris dans un plan fixe dont ils ne doivent jamais sortir. Ces théorèmes prouvent que les vitesses dont il s’agit sont toujours celles que présenterait le système pris dans l’état de repos, ou transporté parallèlement à un axe fixe, ou tournant autour d’un centre fixe. Ajoutons: 1) que le mouvement de translation, parallèlement à un axe fixe, se déduit du mouvement de rotation autour d’un centre fixe, quand ce centre s’éloigne à une distance infinie de l’origine des coordonnées; 2) que le centre de rotation est un point dont la position, déterminée à chaque instant, varie en général d’un moment à l’autre dans le plan que l’on considère. C’est pour celle raison que nous désignerons le point dont il s’agit sous le nom de centre instantané de rotation. 15.6 Nous observerons d’abord que, à la fin d’un temps désigné par t, les différents points de la surface mobile occuperont dans l’espace des positions déterminées, et que l’un d’eux, le point O, par exemple, sera le centre instantané de rotation. De plus, il est clair que, à cette époque, on pourra faire passer par le point O deux courbes distinctes tracées de manière à comprendre, la première, tous les points de la surface mobile, et, la seconde, tous les points de l’espace qui deviendront plus tard des centres instantanés de rotation. 15.7 Théorème VI. Quelle que soit la nature du mouvement d’un corps solide, les relations existantes entre les différents points seront toujours celles qui auraient lieu, si le corps était retenue de manière à pouvoir seulement tourner autour d’un axe fixe et glisser le long de cet axe.

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15.8 Théorème VII. Concevons qu’un corps solide se meuve d’une manière quelcomque dans l’espace, et qu’à un instant donné on trace: 1) dans le corps; 2) dans l’espace, les différentes droites avec lesquelles coïncidera successivement l’axe instantané de rotation de ce corps solide. Tandis que la surface réglée, qui aura pour génératrices les droites tracées dans le corps, sera entrainée par le mouvement de celui-ci, elle touchera constamment la surface réglée qui aura pour génératrices les droites tracées dans l’espace, et, par conséquent, la seconde surface ne sera autre chose que l’enveloppe de la portion de l’espace parcourue par la première. 15.9 Théorème VIII. Les mêmes choses étant posées que dans le théorème VII, si l’axe instantané de rotation da corps solide devient fixe de position dans le corps, il sera fixe dans l’espace; et réciproquement.

A.16 Chapter 16 16.1 Les droites infiniment petites MN, M  N  , M  N  , etc., sont ce qu’on appelle les vitesses virtuelles des points M, M  , M  , etc.; dénomination qui provient de ce qu’elles sont considérées comme les espaces qui seraient parcourus simultanément par les points du système, dans le premier instant où l’équilibre viendrait à se rompre. 16.2 L’avantage du principe des vitesses virtuelles est de donner l’équation d’équilibre dans chaque cas particulier, sans qu’on ait besoin de calculer ces forces intérieures; mais comme la démonstration que nous allons donner est fondée sur la considération de ces forces, de grandeur inconnue, voici la notation dont nous ferons usage pour les représenter. 16.3 Il faut encore démontrer que, réciproquement, quand l’équation (b) a lieu pour tous les mouvemens infiniment petits qu’on peut faire prendre au système des points M, M , M , etc., les forces données P, P , P , etc., sont en équilibre. […] Supposons pur un moment que l’équilibre n’ait pas lieu. Les points M, M , M , etc., ou une partie d’entre eux, se mettront en mouvement, et, dans le premier moment, ils décriront simultanément des droites telles que MN, M N , M N , etc.; on pourra donc réduire tous ces points au repos, en leur appliquant des forces convenables, diriges suivant les prolongements de ces droites, en sens contraire des mouvemens produits; per conséquent, si nous désignons ces forces inconnues par R, R , R , etc. l’équilibre aura lieu entre les forces P, P , P , etc., R, R , R , etc.; en sorte que r, r , r”, etc., désignant les vitesses virtuelles projetées sur les directions de ces nouvelles forces R, R , R”, etc., on aura, d’apres le principe des vitesses virtuelles qui vient d’être démontré, Pp + P p + P p + etc. + Rr + R r + R r + etc. = 0 ou simplement:

Rr + R r + R r + etc. = 0

(c)

en vertu de l’équation (b), qui a lieu par hypothèse. Cette équation (c) existant pour tous les mouvemens infiniment petits compatibles avec les conditions du système des points M, M , M , etc., nous pouvons choisir pour leurs vitesses virtuelles les espaces réellement décrits MN, M N , M N , etc., dans un même instant; mais comme ces lignes sont comptées sur les prolongemens des directions de R, R , R , etc., il s’ensuit que toutes les projections r, r , r , etc., seront négatives, et égales, abstraction faite du signe, à ces mêmes lignes MN, M N , M N , etc. Alors, tous les termes de l’équation (c) étant de même signe, leur somme ne peut être nulle, moins que chaque terme ne soit séparément

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égal à zéro; on aura donc R · MN = 0, R · M N = 0, R · M N = 0, etc. or, pou que le produit R · MN soit nul il faut qu’on ait, ou R = 0, ou MN = 0; ce qui signifie, dans l’un et l’autre cas, que le point M ne peut prendre aucun mouvement: il en est de même égard de tous les autres points; par conséquent, le système entier est en équilibre; et c’est ce que nous nous proposions de démontrer. 16.4 La plupart des géomètres regardent comme évident que si des forces sont en équilibre sur un système de points, soumis à des liaisons qui leur permettent de prendre certains mouvements, ces mêmes forces seraient encore en équilibre sur le même système de points, soumis à des liaisons différentes qui permettraient identiquement les mêmes déplacements. Ce principe […] nous avait toujours paru un peu hypothétique […] qui nous semble fondé sur une véritable confusion de la Géométrie et de la Mécanique. […] En conséquence, nous avons changé la démonstration du principe des vitesses virtuelles que nous avions empruntée à Ampère, et nous en avons adopté une qui n’offre pas le même inconvénient, et n’est autre chose au fond que celle qui se trouve dans le Traité de Mécanique de Poisson. 16.5 Perciò che riguarda Giovanni Bernoulli è da notare che, col prendere in considerazione, nella celebre lettera a Varignon (1717), le relazioni tra gli spostamenti (virtuali), infinitamente piccoli, dei punti d’applicazione delle forze, egli non fece in fondo che applicare ed enunciare, in forma generale, una norma di metodo di cui era stato fatto già frequentemente uso dai suoi predecessori, tra gli altri da Leonardo da Vinci e da Galileo, nei loro tentativi di dedurre, dal principio della leva, quello del piano inclinato, e di far rientrare sotto quest’ultimo il caso di un grave sostenuto da due fili non paralleli. Tale norma è quella che consiste nel riguardare come sostituibili, per quanto riguarda l’equilibrio, due sistemi di vincoli quando essi permettono gli stessi spostamenti iniziali. Essa, come nota a proposito il Duhem, si trova enunciata, sotto la forma più esplicita, da Descartes, in una lettera al Padre Mersenne (1638) […]. Sulla presenza di considerazioni analoghe negli scritti di Galileo è da vedere quanto dice il Mach (Mechanik, 4° ediz., pag. 25-26). 16.6 Si un système quelconque de points est en équilibre, et que l’on conçoive un déplacement infiniment petit de tous ses points, qui soit compatible avec toutes les conditions auxquelles il est assujetti, la somme des moments virtuels de toutes les forces est nulle, quel que soit ce déplacement. Et réciproquement, si cette condition a lieu pour tous les déplacement virtuels, le système est en équilibre. Dans cet énoncé, les infiniment petits sont considérés de la manière ordinaire. L’équation n’est exacte qu’en considérant les limites des rapports, après avoir divisé par l’une quelconque des quantités infiniment petites; en d’autres termes, la somme des moments est infiniment petite par rapport à ces moments eux-mêmes. 16.7 Dans ce qui nous allons dire, le mot de force s’appliquera doc seulement à ce qui est analogues aux poids, c’est-à-dire à ce qu’on appelle, dans plusieurs cas, pression, tension, ou traction. En ce sens, une force ne peut jamais faire changer sensiblement la direction et la grandeur d’une vitesse sans le faire passer par tous les états intermédiaires. 16.8 Ces diverses expressions assez vagues ne paraissent pas propre à se répandre facilement. Nous proposerons la dénomination de travail dynamique, ou simplement travail, pour la quantité Pds […]. Ce nom ne fera confusion avec aucune autre dénomination mécanique; il parait très propre à donner une juste idée de la chose, tout en conservant son acception commune dans le sens de travail physique […] ce nome est donc très propre à designer la réunion de ces deux éléments, chemin et force.

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Appendix. Quotations

16.9 J’ai employé dans cet ouvrage quelques dénominations nouvelles: je désigne par le nom de travail la quantité qu’on appelle assez communément puissance mécanique, quantité d’action ou effet dynamique, et je propose le nom de dynamode pour l’unité de cette quantité. Je me suis permis encore une légère innovation en appelant force vive le produit du poids par la hauteur due à la vitesse. Cette force vive n’est que la moitié da produit qu’on a désigné jusqu’à présent par ce nom, c’est-à-dire de la masse par le carré de la vitesse. 16.10 Ce mot de travail vient si naturellement dans le sens où je l’emploie, que, sans qu’il ait été ni proposé, ni reconnu comme expression technique, cependant il a été employé accidentellement par M. Navier, dans ses notes sur Bélidor, et par M. de Prony dans son Mémoire sur les expériences de la machine du Gros-Caillou. 16.11 Dorénavant nous nous servirons de la dénomination de machine pour designer les corps mobiles auxquels nous appliquerons l’équation des forces vives: en ce sens. un seul corps qui se meut serait une machine tout comme un ensemble plus compliqué. Dans chaque cas particulier, une fois qu’on saura bien de quels corps en mouvement se compose la machine dont on veut s’occuper, il suffira pour y appliquer les principes précédemment établis, de bien connaître quelles sont les masses qui doivent entrer dans le calcul des forces vives, et quelles sont les forces mouvantes et résistants qui doivent entrer dans le calcul de la quantité de travail. 16.12 Pour passer à la statique et à la dynamique des systèmes des corps on n’aura besoin que de s’appuyer sur le seul principe de l’égalité entre l’action et la réaction. Ce principe consiste in ce que, si une molécule d’un corps produit une certaine force d’attraction ou de répulsion sur une molécule voisine, elle recevra en même temps de celle-ci une force égale et directement opposée: en sorte que les forces qui se produisent dans l’ensemble des molécules qui forment un corps n’existent que par couples d’action égales et opposées. C’est à l’aide de ces seuls points que nous allons donner tous les principes de mécanique. 16.13 Si l’on conçoit qu’un point auquel est appliquée une force P vienne à se déplacer d’une quantité δs dans une direction quelconque, nous appellerons élement de travail virtuel le produit  l’angle de δs par la componente de la force dans la direction de δs; nous désignerons par Pδs de δs avec la force P, en sorte que l’élément de travail virtuel sera  P cos(Pδs). 16.14 Si l’on suppose maintenant que les mouvemens virtuelles soient restreints à des mouvemens opérée, en laissant l’ensemble des molécules dans l’état d’invariabilité des distances mutuelles, qu’on peut appeler de solidification; alors les distances r ne variant pas dans ce mouvement, on aura δr = 0, et l’équation ci-dessus se réduit à: ∑ Pδp = 0. 16.15 L’équilibre ayant lieu sous l’action des forces extérieures P, chaque molécuIe sera en équilibre, et l’on aura, en tenant compte de toutes le actions moléculaires R,

∑ Rδr + ∑ Pδp = 0. Si maintenant on prend un mouvement virtuel qui laisse à chaque corps son invariabilité de forme ou sa solidité, et que néanmoins dans ce mouvement on fasse glisser et rouler les corps les uns sur les autres avec toute la latitude dans ces mouvemens que permet la construction même de la machine; il y aura une grande partie des élemens de travail virtuels Rδr qui s’en iront ce seront tous ceux qui sont dus des actions entre des molécules qui n’ont pas changé de distance pendant le mouvement virtuel, c’est-à-dire entre celles qui appartiennent à un même corps. Il ne restera donc dans l’équation ci-dessus que ceux des élémens de travail virtuel ∑ Pδr qui proviennent des actions entre les molécules de deux corps contigus, lorsque dans le mouvement virtuel ces corps ne se mouvront pas ensemble comme un seul système, mais qu’ils glisseront ou rouleront l’un sur l’autre. Les actions R qui resteront ainsi ne seront dues

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qu’à des molécules qui seront è une distance de la surface de contact qui sera moindre que l’étendue des actions moléculaires, ou en d’autres termes, que le rayon de la sphère d’activité. 16.16 Nous sommes conduits ainsi à reconnaître que le principe des vitesses virtuelles dans l’équilibre d’une machine composée de plusieurs corps solides ne peut avoir lieu qu’en considérant d’abord les frottemens de glissement, lorsque les déplacemens virtuels peuvent faire lisser les corps les uns sur les autres, et en outre ceux de roulement lorsque les corps ne peuvent prendre de mouvement virtuel sans se déformer près des points de contact. Les frottemens étant reconnus par expérience toujours capables de maintenir l’équilibre dans de certaines limites d’inegalité entre la somme des élémens de travail positif et la somme des élémens de travail negatif, en prenant ici pour négatifs les élémens appartenant à la somme la plus petite; il s’ensuit que la somme des élémens auxquels ils donnent lieu a préciément la valeur propre à rendre nulle la somme totale et se trouve égale à la petite différence qui existe entre les sommes des élémens positifs et des élémens négatifs.

A.17 Chapter 17 17.1 Comme λδ ds la quantité peut représenter le moment d’une force λ tendante à diminuer la longueur de l’élément ds le terme S λδ ds de l’équation générale de l’équilibre du fil représentera la somme des moments de toutes les forces λ qu’on peut supposer agir sur tous les éléments du fil: en effet chaque élément résiste par son inextensibilité à l’action des forces extérieures, et l’on regard communément cette résistance comme une force active qu’on nomme tension. Ainsi la quantité λ exprimera la tension du fil. 17.2 Les expressions trouvé plus haut pour les variations font voir que ces variations ne sont que les résultats des mouvements de translation et de rotation que nous avons considérés en particulier dans la section III. […] L’analyse précédente conduit naturellement à ces expressions et prouve par là, d’une manière encore plus directe et plus générale que celle de l’article 10 de la Section III, que lorsque les différents points d’un système conservent leur position relative, le système ne peut avoir à chaque instant que des mouvements de translation dans l’espace et de rotation autour de trois axes perpendiculaire entre eux. 17.3 Quoique nous ignorions la constitution interne des fluides, nous ne pouvons douter que les particules qui les composent ne soient matérielles, & que par cette raison les loix générales de l’équilibre ne leur conviennent comme aux corps solides. En effet, la propriété principale des fluides & la seule qui les distingues des corps solides, consiste en ce que toutes leurs parties cèdent à la moindre force, & peuvent se mouvoir entr’elles avec tonte la facilité possible, quelle que soit d’ailleurs la liaison & l’action mutuelle de ces parties. Or cette propriété pouvant aisément être traduite en calcul, il s’ensuit que les loix de l’équilibre des fluides ne demandent pas une théorie particulière, mais qu’elles. ne doivent être qu’un cas particulier de la théorie générale de la Statique. 17.4 Les théorie précédentes de l’équilibre & de la pression des fluides sont, comme l’on voit, entièrement indépendantes des principes généraux de la Statique, n’étant fondées que sur des principes d’expérience, particuliers aux fluides; & cette manière de démontrer les loix de l’Hydrostatique, en déduisant de la connaissance expérimentale de quelques-unes de ces loix, celle de toutes les autres a été adopté depuis par la plupart des Auteurs modernes, & a fait de l’Hydrostatique une science tout-à-fait différente, & indépendante de la Statique. Cependant il étoit importante de lier ces deux sciences ensemble, & le faire dépendre d’un seul & même principe. Or parmi les differens Principes qui peuvent servir de base à la Statique, & dont nous avons donné une exposition succinte dans la premier Section, il est visible qu’il n’y a que celui des vitesses virtuelles qui s’applique naturellement à l’équilibre des fluides.

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17.5 Le principe de M. Clairaut n’est que une conséquence naturelle du Principe de l’égalité des pression en tout sens. Aussi M. d’Alembert a-t-il déduit immédiatement de ce dernière principe, les mêmes équations différentielles que M. Clairaut avoit trouvées par le sien; & il faut avouer que ce principe renferme en effet la propriété la plus simple & la plus générale que l’expérience ait fait découvrir dans l’équilibre des fluides. Mais la connaissance de cette propriété est-elle indispensable dans la recherche des loix de l’équilibre des fluides? Et ne peut-on pas dériver ces loix directement de la nature même des fluides considérées comme des amas de molécules très-déliées, indépendant les unes des autres, & parfaitement mobiles en tout sens? 17.6 Détermination du travail pour une petite déformation d’un corps. Relations qu’on en déduit entre les trente-six coefficients qui servent a definir la manière de se comporter dune substance cristalline, ou de toute substance solide non isotrope. lmmaginons qu’un corps, soumis à l’action de forces quelconques, subisse une suite de changements dans sa forme, en sorte que ses divers points (x, y, z) dont les coordonnées sont devenues x + u, y + v, z + w, continuent de se déplacer, et franchissent, pendant un temps infiniment petit des espaces élémentaires δu, δv, δw parallèlement aux x, aux y et aux z. Le travail produit dans tout le corps par ce petit mouvement s’obtiendra en multipliant un élément dxdydz de son volume par les composantes, dans les directions x, y, z des forces agissant sur l’unité de ce volume ci respectivement par les petits espaces parcourus δu, δv, δw, puis aoutant dans les trois produits et intégrant leur somme pour toute l’étendue ou pour tous les élémens du corps. Or, les trois composantes de forces agissant sur l’unité de volume de l’élement dxdydz ne sont autre chose que les seconds membres des équations (50) du § 14, page 54, c’est-à-dire ∂ txx ∂ txy ∂ txz + + +X ∂x ∂y ∂z dans le sens X, et deux quadrinomes analogues dans les sens y et z. L’élément de travail produit pendant que les points parcourent les espaces dont les projections suir les x, y, z, sont δu, δv, δw, se présente donc sous la forme δW = δU + δV où δU =

  

(Xδu +Y δv + Zδw) dxdydz

représente le travail des forces extérieures agissant sur l’intérieur du corps, et ⎧   ⎫ ∂t ⎪ ⎪ + ∂∂txzz δu ∂∂txxx + ∂ xy ⎪ ⎪ ⎪ y    ⎨ ⎬  ⎪  ∂ tyx ∂ tyy ∂ tyz dxdydz δV = +δv ∂ x + ∂ y + ∂ z ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ∂ t ∂ t ∂ t zy zyz zx ⎩ +δw ⎭ ∂x + ∂y + ∂z représente le travail des tensions qui proviennent des actions réciproques de ces molécules, que des forces quelconques de pression et de traction pouvant solliciter sa surface. Considérons d ’abord un seul des neuf termes de cette dernière intégrale triple, par exemple   

δu

∂ txx dxdydz ∂x

integrant par rapport à x partiellement, à savoir pour la petite portion du corps qui est contenue dans un canal infiniment délié dont nous considérons la section dydz comme constante ainsi que les cordonnées y et z. Cette integration par parties, si l’on replace, dans le second term

∂ δu ∂u avec δ ∂x ∂x

A.17 Chapter 17

469

qui lui est identique, nous donne l’expression :  

[txx δu]dydz −

  

txx δ

∂u . ∂x

La parenthèse carré signifie qu’au lieu de l’expression txx δu qu’elle renferme, on doit mettre la différence des valeurs que prend cette expression aux deux extrémité du canal considéré. Désignons maintenant par dσ, dσ les élémens que découpe, sur la surface du corps, se canal a ses extrémités, et par p, q, r les angles que forme avec les axes coordonnés la normale à dσ menée vers l’extérieur du corps, enfin par p , q , r , les mêmes angles pour la normale à dσ. Si dσ est l’extrémité antérieure du canal, c’est-à-dire celle qui se trouve le plus du côté positif des x, et dσ son extrémité postérieure, cos p est nécessairement positif, et cos p négatif, en sorte qu’on a dydz = dσ cos p = −dσ cos p . La différence des valeurs limites de txx δudydz devient donc la somme des valeurs que prend l’expression txx δudσ cos p pour les extrémités du canal. Au lieu d’étendre l’intégrale double ci-dessus aux extrémités de tous les canaux parallèles à l’axe des x que l’on peut mener semblablement dans l’intérieur du corps, il est évidemment possible d’intégrer directement pour l’ensemble des éléments dσ qui comprennent les éléments dσ . Donc  

[txx δu]dydz −



txx δudσ cos p.

[…] Ainsi donc dans tous les cas, on replacera le terme de δV que nous avons considéré par 

txx δudσ cos p −

  

txx δ

∂u dxdydz ∂x

où la première intégrale doit être étendue à toute la surface σ du corps. Si l’on fait de même pour tous les autres termes de δV , on obtient δV = δU1 − δU2 δU1 représentant l’ensemble des integrales simples et δU2 l’ensemble des intégrales triples comme est celle de l’expression binôme en txx qu’on vient d’écrire. Et l’on a 

δU1 = (txx cos p + txy cos q + txz cos r) δudσ 

+ (tyx cos p + tyy cos q + tyz cos r) δudσ 

+ (tzx cos p + tzy cos q + tzz cos r) δudσ. Or les expressions entre parenthèses sont précisément celles qui d’aprés les équations (25), équivalent aux composantes T cos ω, T cos κ, T cos ρ des forces de traction T appliquées à la surface du corps; δU1 , n’est donc rien autre chose que le travail de ces forces de traction extérieures, en sorte que 

δU1 = T (cos ωδu + cos kδv + cos oδw) dσ. On trouvera de moine que les huit termes de U2 autres que celui qui contient txx δ ∂∂ ux , et que nous avons écrit, sont affectés, sous le triple signe d’intégration, des incréments δ des autres quotients différentiels ∂u ∂u ∂v , , ,... ∂x ∂y ∂x

470

Appendix. Quotations des déplacements u, v, w ci des autres composantes tyy , . . . des tensions; d’où résulte, en mettant, d’après les expressions (28) les déformations élémentaires ∂x , ∂y , . . . , gxy au lieu de

∂u ∂u ∂u ∂v , , ..., + . ∂x ∂y ∂y ∂x On a ainsi pour le travail total δW = δU + δU1 + δU2 où δU et δU1 représentent les travaux des forces extérieures agissant respectivement sur les points intérieurs ci sur la surface du corps. Par conséquent −δU2 est nécessairement le travail des forces internes qui procèdent des actions moléculaires. 17.7 M. Kirchhoff a eu l’obligeance de m’indiquer, vers 1858, une manière simple et directe de se rendre compte de a composition sextinôme e l’expression ainsi donnée du travail interne ou moléculaire U2 pour l’unité de volume d’un élément. Soient dx, dy, dz les trois cotes très petits, parallèles aux x, y, z, de cet élément rectangle; 10 si la dilatation ∂x déjà subie par son cote x vient à être accrue de δ∂x , les deux faces opposées et égales yz s’éloignent de xδ∂x ; les composantes normales de tension exercées par la matière environnante sur ces faces produisent un travail yztxx xδ∂x ; cela fait, par unité du volume xyz, le travail txx δ∂x ; 20 si, l’une des deux faces opposées yz restant immobile, le glissement gxy vient à augmenter de δgxy , il y a un cheminement xδgxy de l’autre face parallèlement à celle-ci; en sorte que la tension tangentielle txy qui agit par unité de sa surface yz dans le sens y de ce cheminement, produit un travail yztxy xδgxy . Il y a bien deux autres faces sur lesquelles agit une tension ou tyx égale a txy ; ce sont les faces xz. Elles ont, dans ce mouvement pivoté autour des deux cotes z de celle des deux faces yz qui est restée immobile; mais les tensions tyz s’y exercent dans le sens x et non dans le sens y qui a été celui du mouvement, elles n’ont donc rien ajouté au travail yztxy xδgxy des tensions txy , travail qui est ainsi, seulement, txy gxy par unité de volume de l’élément. Or, le travail des six tensions sur les faces de l’élément doit, pour que l’équilibre ait lieu. après comme avant ces petits mouvements, être égal (au signe près) au travail moléculaire de l’intérieur de l’élément. Donc ce travail a bien pour grandeur, par unité de volume, le sextinôme (txx δ∂x + · · · +txy δgxy ) de la parenthèse de l’expression de δU2 . 17.8 Ecco il maggiore vantaggio del sistema della Meccanica Analitica. Esso ci fa mettere in equazione i fatti di cui abbiamo le idee chiare senza obbligarci a considerare le cagioni di cui abbiamo idee oscure […]. L’azione delle forze attive o passive (secondo una nota distinzione di Lagrange) è qualche volta tale che possiamo farcene un concetto, ma il più sovente rimane […] tutto il dubbio che il magistero della natura sia ben diverso […]. Ma nella M. A. si contemplano gli effetti delle forze interne e non le forze stesse, vale a dire le equazioni di condizione che devono essere soddisfatte […] e in tal modo, saltate tutte le difficoltà intorno alle azioni delle forze, si hanno le stesse equazioni sicure ed esatte che si avrebbero da una perspicua cognizione di dette azioni. 17.9 Siano X, Y, Z le componenti delle forze acceleratrici che agiscono su ciascun punto del corpo; L, M, N, le componenti delle forze che agiscono su ciascun punto della superficie di esso, e ρ la densità costante. Diamo ad ogni punto del corpo un moto virtuale e denotiamo con δu, δv, δw le variazioni che prenderanno per questo u, v, w. Il lavoro fatto in questo moto dalle forze date sarà evidentemente:  S

(Xδu +Y δv + Zδw)dS +

 σ

(Lδu + Mδv + Mδw)dσ

A.18 Chapter 18

471

essendo S lo spazio occupato dal corpo e σ la sua superficie. Il lavoro fatto dalle forze elastiche sarà uguale all’aumento del potenziale di tutto il corpo dato da: Φ=



PdS

onde per il principio di Lagrange: δΦ +

 S

(Xδu +Y δv + Zδw)dS +

 σ

(Lδu + Mδv + Mδw)dσ = 0.

A.18 Chapter 18 18.1 La tentative qui se propose de réduire toute la Physique à la Mécanique rationnelle, tentative qui fut toujours vaine dans le passé, est-elle destinée à réussir un jour? Un prophète seul pourrait répondre affirmativement ou négativement à cette question. Sans préjuger le sens de cette réponse, il parait plus sage de renoncer, au moins provisoirement, à ces efforts, stériles jusqu’ici, vers l’explication mécanique de Univers. Nous allons donc tenter de formuler le corps des lois générales auxquelles doivent obéir toutes les propriétés physiques, sans supposer à priori que ces propriétés soient toutes réductibles à la figure géométrique et au mouvement local. Le corps de ces lois générales ne se réduira plus, dès lors, à la Mécanique rationnelle. […] La Mécanique rationnelle doit donc résulter du corps de lois générales que nous nous proposons de constituer; elle doit etre ce qu’on obtient lorsqu’on applique ces lois général à des systèmes particuliers où l’on ne tient compte que de la figure des corps et de leur mouvement local. Le code des lois générales de la Physique est connu aujourd’hui sous deux noms: le nom de Thermodynamique et le nom d’Energétique. 18.2 Imaginons qu’une suite continue d’états d’un même système isolé ait été formée: fixons notre attention sur ces divers états dans l’ordre qui permet de passer de l’un à l’autre d’une manière continue; pour designer cette opération tout intellectuel à laquelle nous soumettons le schème mathématique qui nous doit servir à représenter un ensemble de corps concrets, nous disons que nous imposons au système une modification virtuelle. […] Les variations des valeurs numériques des variables qui servent à définir un état du système doivent être compatibles avec les conditions qui résultent logiquement de la définition de ce système, mais avec ces conditions-là seulement. En particulier elles peuvent fort bien contredire aux lois expérimentales régissant l’ensemble de corps concrets que notre système abstrait et mathématique a pour objet de représenter. 18.3 Il ne faut pas confondre une modification idéale avec une modification virtuelle; une modification virtuelle se compose d’états du système qui ne se succèdent pas dans le temps; en sorte que le changement d’état qui constitue une modification virtuelle n’est pas lié à un mouvement; en la modification virtuelle, la notion de vitesse n’a point de place. 18.4 Ainsi donc quand un système se transforme en présence de corps étrangers nous considérons ces corps étrangers comme contribuant à cette transformation soit en la causant, soit en aidant, soit l’entravant; c’est cette contribution que nous nommons l’oeuvre accomplie, en une transformation d’un système par les corps étrangers à ce système. 18.5 Première convection. La symbole mathématique destiné a représenter le valeur de l’oeuvre accomplie, en une modification réal ou idéale d’un système, sera déterminé toutes les fois qu’on connaitre la nature du système et la modification qu’il a subie; il ne changera pas si

472

Appendix. Quotations l’on se borne à changer l’époque et le lieu où la modification a été produite ainsi quel les corps étrangers en présence desquelles elle a été accomplie.

18.6 Principe de la Conservation de l’Énergie. Lorsqu’un système quelconque, isolé dans l’espace, éprouvé une modification réelle quelconque, l’énergie totale du système garde une valeur invariable. 18.7 Forme restreinte du principe de la Conservation de l’Énergie. En toute modification réelle d’un système isolé, l’égalité U+

1 2

 M

u2 + v2 + w2 dm = const.

est vérifiée. 18.8 La comparaison de ces conditions […] fournit l’énoncé suivant, qui est celui du principe de d’Alembert: Pour obtenir, à chaque instant, les lois du mouvement d’un système de solides assujettis à des liaisons sans résistance passive, il suffit d’écrire que le système demeurait en équilibre si on le plaçait sans mouvement dans l’état qu’il traverse à cet instant, et si on le soumettait non seulement aux actions extérieures qui s’exercent réellement sur lui au moment où il se trouve en cet état, mais encore à des actions extérieures fictives équivalentes aux actions d’inertie qui le sollicitent à ce moment. 18.9 Dans le cas particulier où le système est assujetti exclusivement à des liaisons et bilatérales, il faut il suffit, pour l’équilibre, que le travail externe soit, tout déplacement en virtuel, égal à l’accroissement de l’énergie interne. 18.10 La contrainte que le système éprouve, de la part des liaisons, au cours de son déplacement réel est moindre que la contrainte qu’il éprouverait en toute autre déplacement virtuel issu du même état:   2 2 MN dm < PN dm that is what we mean when we say that the studied constraints have no passive resistance. 18.11 Die Bewegung eines Systems materieller, auf was immer für eine Art unter sich verknüpfter Punkte, deren Bewegungen zugleich an was immer für äussere Beschränkungen gebunden sind, geschieht in jedem Augenblick in möglich grösster Übereinstimmung mit der freien Bewegung, oder unter möglich kleinstem Zwangen, indem man als Maass des Zwanges, den das ganze System in jedem Zeittheilchen erleidet, die Summe der Producte aus dem Quadrate der Ablenkung jedes Punkts von seiner freien Bewegung in seine Masse betrachtet. Es seien m, m , m“ u. s. w. die Massen der Punkte; a, a , a u. s. w. ihre platze zur Zeit t; b, b , b u. a. w. die Plätze, weiche sie, nach dem unendlich kleinen Zeittheilchen dt, in Folge der wahrend dieser Zeit auf sie wirkenden Kräfte und der zur Zeit t erlangten Geschwindigkeiten und Richtungen, einnehmen wurden, falls sie alle vollkommen frei waren. Die wirklichen Plätze c, c , c u.s.w. werden dann diejenigen sein, für welke, unter allen mit den Bedingungen des Systems vereinbaren, m(bc)2 + m (b c )2 + m (b c )2 u.s.v. ein Minimum wird.

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Index

Abattouy, Mohammed, 68, 70, 482 Al-Isfizari, 66, 71, 74, 136 Alberti, Giovanni Battista, 95, 473 Ampère, Andreé Marie, 1, 8, 9, 12, 13, 266, 290, 317, 318, 328–332, 339, 342, 347, 366, 371, 473, Angiulli, Vincenzo, 7, 12, 13, 16, 206, 217–229, 231–233, 299, 375, 473 Apianus, Petrus, 75, 93, 473 Araldi, Michele, 473 Archimedes, 3, 34, 43, 45–54, 59, 64–67, 92, 93, 95–97, 103, 104, 108, 110, 114, 118, 131, 137, 142, 160, 173, 178, 179, 199, 225, 308, 322, 473, 475, 484, 486 Aristotle, 1–3, 10, 12, 34–38, 40–45, 51, 52, 56, 63–66, 77, 90, 92, 95, 104, 105, 123, 131, 135, 136, 185, 225, 258, 271, 341, 473, 483, 486, 486, 487 Bagni, Giorgio, 482 Bailhache, Patrice, 335, 482 Baldi, Bernardino, 43, 92–94, 108, 154, 155, 473 Barbaro, Daniele, 95–97, 474 Barroso, Filho, 482 Belhoste, Bruno, 482 Bellucci, Giovanni Battista, 95, 474 Beltrami, Eugenio, 387, 390–392, 474 Benedetti, Giovanni Battista, 84, 91, 92, 94, 96, 116–119, 121, 143, 474 Benvenuto, Edoardo, 482 Bernoulli, Daniel, 239 Bernoulli, Jakob, 239 Bernoulli, Johann, v, 5–7, 10, 12, 13, 15, 16, 50, 187, 195, 199–204, 206–210, 213–215, 217, 218, 220, 221, 225, 227, 228, 231,

233, 237–239, 242, 248–250, 252–254, 256, 271, 299, 323, 328, 334, 337, 340, 362, 366, 474 Bertoni, Giuseppe, 482 Bertrand, Joseph Louis François, 252, 253, 263, 267 Betti, Enrico, 387, 390, 392, 393, 474, 483 Bevilacqua, 483 Biringuccio, Vannoccio, 95, 474 Bordoni, Antonio, 387 Borelli, Alfonso, 5, 153, 217, 225, 391, 474 Borgato, Maria Teresa, 482 Boschiero, Luciano, 482 Boscovich, Ruggiero Giovanni, 218, 224, 231, 233, 234, 236, 299, 475, 483, 485 Bottecchia Dehò, Maria Elisabetta, 40, 473 Bradwardwine, Thomas, 64 Brioschi, Francesco, 390, 392 Brown, Joseph Edward, 75, 77, 83, 475 Brugmans, Anton, 51, 475 Brunacci, Vincenzo, 387, 388, 475 Buchner, Ferdinand, 482 Boudri, Christian, 482 Burzio, Filippo, 482 Camerota, Michele, 482 Capecchi, Danilo, 482, 483 Cardano, Girolamo, 91–94, 96, 104–107, 110, 131, 132, 157, 178, 190, 475 Carnot, Lazare, 8–12, 16, 243, 244, 259, 297, 281–297, 317, 328, 329, 339, 341, 351, 362, 367–369, 398, 475, 483–485 Carnot, Sadi, 297, 396 Cauchy, Augustin Louis, 8, 248, 293, 318, 335, 353–360, 381, 383, 390, 475, 482 Cavalieri, Bonaventura, 190

490

Index

Caverni, Raffaello, 103, 139, 483 Ceccarelli, 485, 486 Ceradini, Cesare, 483 Cesariano, Cesare, 93, 475 Clagett, Marshall, 56, 57, 68, 75, 77, 79, 483 Clairaut, Alexis Claude, 475 Clarke, John, 475 Clarke, Samuel, 177, 475 Clavius, Cristophorus, 120, 391 Clebsch, Alfred, 375, 380, 383, 385–387, 475 Clerke, Maxwell James, 177, 390 Cockle, Maurice James Draffen, 483 Commandino, Federico, 59, 92, 93, 96, 108, 109, 137, 141, 178, 180, 473, 475, 476 Comte, August, 318, 483 Coriolis, Gustave Gaspard, 10, 11, 158, 256, 293, 297, 361, 367–373, 380, 476 Cusanus, Nicholas, 177

Erasmus of Rotterdam, 177 Euclid, 3, 9, 12, 34, 45–47, 63, 65–67, 69, 74, 83, 97, 100, 108, 178, 217, 391, 392, 474, 487 Euler, Leonhard, 15, 200, 212, 218, 238–245, 248, 251, 271–273, 286, 335, 342, 377–379, 476

D’Alembert, Jean Baptiste le Ronde, 200, 212, 237, 239–241, 243–247, 249, 251, 252, 259, 268–271, 273–279, 286, 287, 291, 294, 332, 401, 482, 476, 484 D’Ayala, Mariano, 482 Da Cremona, Gerardo, 65, 68, 70 Dal Monte, Guidobaldo, 12, 13, 30, 43, 56, 64, 84, 91–93, 96, 108–110, 112–116, 120, 121, 132, 136, 137, 139, 159, 160, 179, 187, 210, 476, 486 Damerow, Peter, 486 De Brussel, Gerardus, 35, 477 De Caus, Salomon, 476 De Challes, Claude François Milliet, 176, 177, 476 De Marchi, Francesco, 95, 476 De Nemore, Jordanus, 12 Descartes, René, 4, 5, 12, 13, 114, 127, 138, 152, 157, 159, 160, 164–174, 176, 177, 187, 190, 192, 196, 199, 206, 210, 213, 214, 225, 253, 275, 285, 292, 294, 328, 366, 476 Di Giorgio, Francesco, 483 Dijksterhuis, Eduard Jan, 50, 95, 184, 186, 483 Drabkin, Israel Edward, 115, 483 Drago, Antonino, vi, 23, 284, 290, 483, 484 Drake, Stillman, 92, 104, 115, 483, 486 Dugas, René, 157, 198, 484 Duhamel, Jean Marie, 361, 365, 366, 476 Duhem, Pierre Maurice Marie, 11, 32, 40, 44, 56, 66, 75–77, 84, 90, 95, 104, 126, 154, 157, 159, 160, 184, 324, 342, 366, 390, 395–403, 476, 484

Galilei, Galileo, 2–5, 12, 13, 30, 31, 43, 44, 50, 58, 71, 72, 91, 93, 108, 120–132, 137–140, 142, 144–148, 152, 160, 166, 167, 169–171, 173, 179, 210, 218–220, 223, 225, 226, 229, 239, 245, 253, 256, 271, 300, 301, 340, 366, 375, 477, 483–485 Galletto, Dionigi, 50, 241, 484 Galluzzi, Paolo, 122, 484 Gatto, Romano, vi, 92, 93, 484 Gaukroger, S., 482 Gauss, Carl Friedrich, 8, 324, 403, 477 Gentile, Giuseppe, 484 Gerardo da Cremona, 65 Germain, Paul, 484 Ghinassi, G., 484 Giampaglia, Amedeo, 484 Giardina, Giovanna Rita, 484 Gille, Bernard, 484 Gillispie, Charles Coulston, 282, 297, 485 Giusti, Enrico, 56, 485 Green, George, 382, 385, 386, 393, 477 Gutas, Dimitri, 485

Ferriello, Giuseppina, 484 Festa, Egidio, 131, 484 Filoni, Andrea, 484 Folkerts, Menso, 484 Foncenex, Daviet de, 241, 299 Fontana, Domenico, 95, 477 Fossombroni, Vittorio, 9, 12, 16, 257, 263, 300–306, 309, 319, 320, 338, 477 Fourier, Jean Baptiste Joseph, 8, 9, 12, 13, 262, 317, 319, 321–329, 341, 362, 366, 477 Fraser, Craig, 484

Hankins, Thomas, 485 Helbing, Mario Otto, 485 Helmholtz, Hermann Ludwig, 11, 395, 477 Herigone, Pierre, 158, 159, 173, 477 Hermann, Jacob, 238, 239, 271–273, 478 Hero of Alexandria, 3, 4, 12, 33, 34, 45, 51–61, 65–67, 91–93, 95, 96, 104, 106, 109, 131, 132, 168, 310, 473, 476–478, 484, 486 Høyrup, Jensen, 485 Hunter, Michael, 485

Index Huygens, Chrstiaan, 5, 6, 50, 127, 166, 171, 178, 187, 188, 201, 204, 217, 225, 232, 238, 478, 486 Huygens, Constantin, 164, 167, 174 Indorato, Luigi, 485 Jacobi, Carl, 263 Jacquier, 478, 483 Jammer, Max, 485 Jaouiche, Khalil, 64, 68, 70, 73, 478 Jordanus, 4, 12, 13, 63, 65, 66, 75–87, 89–93, 97–100, 102, 103, 107–112, 114–116, 118, 119, 121, 124, 131, 132, 154, 157, 159, 162, 484, 485 Jouguet, François, 366, 485 Khanikoff, Nicolas, 485 Kirchhoff, Gustav, 386, 387 Klein, Felix, 485 Knorr, Wilbur Richard, 64, 68, 485 Koetsier, Teun, 485 Koyré, Alexandre, 485 Kraft, Fritz, 38, 485 Lagrange, Giuseppe Lodovico, v, 7–9, 11–13, 15–17, 22, 27, 31, 50, 149, 200, 201, 206, 212, 228, 231, 237, 239–273, 277–279, 282, 299, 300, 304–306, 308, 310–315, 317, 319, 321, 322, 328, 329, 332, 333, 335–337, 339–342, 345, 347, 348, 351, 353, 354, 361, 362, 369, 371, 375–381, 387–389, 391, 478, 480, 482, 483, 484, 486 Laird, Walter Roy, 44, 485 Lamé, Gabriel, 387, 391, 392, 478 Lamy, Bernard, 5, 176, 177, 478 Laplace, Pierre Simon, 8, 13, 17, 26, 328, 329, 332–335, 342, 343, 351, 354, 362, 363, 371, 478 Le Seur, Thomas, 234, 483 Leibniz, Gottfried Wilhelm, 6, 7, 186, 197–199, 202, 204, 210, 217–220, 224, 232, 233, 245, 301, 478, 479 Leonico, Tomeo, 94, 479 Libri, Guglielmo, 362, 485 Lorch, Richard, 484 Loria, Gino, 140, 485 Lorini, Bonaiuto, 95, 96, 479 Love, Augustus Edward Hough, 479 Mach, Ernst, 44, 48, 50, 166, 184, 187, 263, 264, 277, 278, 366, 395, 485 Magistrini, Giovanni Battista, 311, 479

491

Maltese, Giulio, 485 Manno, Salvatore Domenico, 290 Marcolongo, Roberto, 95, 485 Martinovic, 485 Mascheroni, Lorenzo, 299, 479 Maupertuis, Pierre Louis, 239, 258, 275, 292, 479 Maurolico, Francesco, 93, 94, 479 McCloskey, Michael, 485 Mersenne, Martin, 120, 127, 152, 158, 160, 164, 166–169, 171, 174, 366, 479 Merton, Robert King, 486 Monge, Gaspard, 317, 318, 321 Montfaucon, Bernard, 92, 479 Moody, Edward, 68, 75, 77, 79, 479 Moscovici, Serge, 486 Mossotti, Ottaviano Fabrizio, 387, 392 Mussini, Massimo, 486 Nagel, Ernest, 486 Napolitani, Pier Daniele, 486 Nastasi, Pietro, 485, 486 Navier, Claude Louis Marie Henri, 297, 367, 368, 375, 381, 382, 386, 387, 393, 479 Nenci, Elio, 154, 486 Nernessian, Nancy, 486 Newton, Isaac, 37, 138, 157, 187, 190, 193–199, 210, 217, 218, 227, 232, 237–239, 243, 251, 258, 275, 276, 284, 474, 476, 479, 482, 486 Ostrogradsky, Mikhail Vasilyevich, 8, 325, 386, 479 Ostwald, Wilhelm, 395, 486 Pappus of Alexandria, 33, 34, 45, 50, 51, 59–61, 64–66, 91, 96, 109, 114, 131, 132, 137, 138, 141, 180, 310, 479, 480 Pardies, Ignace Gaston, 5, 176, 177, 480 Pascal, Blaise, 124, 138, 175, 176, 480 Pearson, Karl, 486 Philoponus, John, 480 Piaget, Jean, 486 Piola, Gabrio, 12, 311–315, 342, 387–391, 480, 484 Pisano, Raffaele, vi, 486 Plutarcus, 480 Poincaré, Henri, 29 Poinsot, Louis, 1, 8–10, 12, 13, 16, 26, 27, 212, 243, 264–268, 277–279, 290, 304, 307, 309, 318, 319, 321, 328–330, 332, 334–339, 341–348, 350, 351, 353, 354, 365, 370, 371, 480 Poisson, Simon Denise, 8, 24, 318, 361–366, 369, 480

492

Index

Prony, Gaspard Clair François Riche de, 8, 212, 263, 300, 317, 319–321, 328, 332, 335–337, 341, 343, 368, 480 Pulte, Helmut, 486 Radelet, De Grave, Patricia, 201, 486 Rankine, William John Macquom, 11, 395, 396, 480 Renau, d’Elicagaray, Bernard, 6, 201–204, 480 Renn, Jürgen, 486 Riccardi, Geminiano, 311, 312, 480 Riccati, Jacopo, 217, 230 Riccati, Vincenzo, 7, 12, 13, 16, 206, 217–219, 221, 224, 225, 227–234, 271, 299, 301, 306, 307, 480, 482, 485 Riemann, Bernhard, 390, 392 Roberval, Gille Personne, 5, 126, 138, 152, 157, 160–163, 179, 190, 480 Rohault, Jaques, 176, 177, 480 Rose, Paul Lawrence, 486 Roux, 481, 484 Russo, Lucio, 486 Ruta, Giuseppe, 483 Saccheri, Girolamo, 138, 217, 391, 481 Saint Venant, Adhémar J.C. Barré, 297, 318, 369, 380, 383, 385, 386, 387 Saladini, Girolamo, 300, 306–308, 481 Scott, Wilson L., 486 Servois, François Joseph, 12, 206, 263, 300, 308–310, 481 Stevin, Simon, 12, 50, 56, 122, 131, 132, 138, 157–160, 171, 173, 178–187, 210, 225, 322, 323, 341, 342, 481, 485, 486 Stigliola, Nicola Antonio, 91–93, 131, 481, 484 Sturm, Charles François, 293, 481 Tartaglia, Niccolò, 12, 77–79, 83, 84, 91–93, 97–103, 108, 110, 116, 118–120, 131, 481

Tazzioli, Rossana, 483 Thabi, Ibn Qurra, 68, 72 Thabit, 12, 63, 65–72, 74, 75, 77, 81, 82, 84, 85, 106, 107, 482 Thomson, William (Lord Kelvin), 392, 393, 481 Timoshenko, Stephen Prokofievich, 486 Tocci, Cesare, vi, 483 Todhunter, Isaac, 486 Torricelli, Evangelista, 4, 5, 12, 109, 124, 135, 138–154, 175, 176, 190, 192, 196, 218, 220, 253, 285, 295, 299, 328, 481–484, 486 Truesdell, Clifford Ambrose, 95, 388, 486 Vailati, Giovanni, 40, 44, 50, 366, 486 Valerio, Luca, 93, 137, 138, 160, 481, 486 Valla, Lorenzo, 93 Varignon, Pierre, v, 5–7, 15, 174, 176, 177, 195, 199, 201, 203, 204, 206, 207, 209–215, 227, 228, 231, 249, 250, 258, 307, 309, 337, 340, 366, 481 Varro, Michel, 131, 173–175, 481, 482, 486 Vassura, Giuseppe, 152, 486 Venturi, Giovanni Battista, 95, 487 Venturoli, Giuseppe, 481 Vilain, Christiane, 40, 487 Villalpando, Juan Bautista, 154, 481 Viscovatov, B., 481 Vitruvio, 93, 96, 481, 486 Wallis, John, 5, 12, 59, 187, 190–192, 206, 220, 253, 391, 481 Webster, Charles, 189, 487 Westfall, Richard Samuel, 198, 487 Wiedemann, Eilhard, 68, 487 Winter, Thomas Nelson, 38, 487 Woepcke, Franz, 65, 487

End of printing: February 2012