The Application of Mechanics to Geometry (Popular Lectures in Mathematics)

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6Sp.: for related docuaentsl see SE 030 q~'-465. Not a vailab le in ha rd cop y t1 \Ie t:): cop yright ,restr lct ion s. rranslated and adapted from the Rus~ian edition. The University of Chicago Press, Chica~o, IL',60637, , (Order No. 450163: $~.50'. " i, ,. l

, '8P01 Plus postage.' peNot 'Available fios ED~S. DES:RrPrORS 4cColleqe !!athematics: Force: Geometric conc~pts-: " *Geolletry: Bigher Education: Lecture '!!et:hod; "*Mat~e.aticaI Applications: *Mathema~ic~: *Mech!ni:s

BDRS PIU CE

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Pr,esented in thi-s trl\nsHltion are three chapters. , Chapter I discusses the comp"os!~l~n of forces !ind several theorels !)f g~OIetry ~re pr!)vpd asing the'fundamental concepts and c~rtain l~ws of statics. Cha~pter II discusses the ~rpetu9.1 mot.ion postqlate; sever!l ;eometri:::. th20rems are proved, u$ing th~ postulate th~t p~rp~ual motion is impossib~e. !nCnapter IIl, the center o~ ... Grav~, Potential E!lergy. an4work. are, discussed. (HK) , . '

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A projeCt conducted oy lzaak Wirszup, Department of Mathetltatics. the University of Chicago, under a .grant from the National Science Foundation .

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.Mechatiics" to .Geometry

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Translated and •

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David J. Sookne and Robert A. Hummel

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lC 1974 by The University of Chicago All rights reserved. Published 1974 Printed in the Un.it~d States of America Intornational Standard Book Number: (}-"'226--450!6-3 • Libiafy of Congress Cataloi Card Number: 73-89789 .

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The Composition of FOrceS

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.' obtain their rcsultant.Ru.' The fo~ P s and Rill. are now in equilibrium. -- But this is possible only if they have a common line of action. Thus, the line of action of the force P s passes through the point O-that is, the' lin~ of action of all three forces ~t one another,at this point. " \ USing this propOsition;'we shall' now prove some theorems of geometrj. ~ ,

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'1.2. A,1bi:orem GO the ADale Bisectors of • ~

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, Let'us consider six equal forces Flo F~ •.. '~'FG acting along the sides -er a triangle, as shown iII figure 1.8. Since these forces cancel onc another in pairs, they are clearly in ~uilibrium, an~ therefore, the resultants ltl~' i.23 • and ~5 are 'also'1n equilibrium.'Bu( the forces RUh R~s! and ~~ arc, directed ~ong thebisect.ors of the interior angles 'T • A.,' B, and C. (The parallelograms are rhqm~i, and the' ~iagona1 is an , angle bisector.) This leads, consequently. to the following theorem: . , -\ THEoREllt 1.1. The bisectors of the interior. angles of a triangle inter-

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1.3. ADother Theorem on the Angle B'isectors of a Triangle

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Let us consider the six·e'qulli forces Flo Fill' , ., Fs shown in figure 1.9. These forces are in equilibriuIl! since each of the three' l!airs of forces, taken conse