Lenses and Waves

Archimedes Volume 9

Archimedes NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE AND TECHNOLOGY VOLUME 9

EDITOR JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology, Pasadena, CA, USA. ADVISORY BOARD HENK BOS, University of Utrecht MORDECHAI FEINGOLD, Virginia Polytechnic Institute ALLAN D. FRANKLIN, University of Colorado at Boulder KOSTAS GAVROGLU, National Technical University of Athens ANTHONY GRAFTON, Princeton University FREDERIC L. HOLMES, Yale University PAUL HOYNINGEN-HUENE, University of Hannover EVELYN FOX KELLER, MIT TREVOR LEVERE, University of Toronto JESPER LÜTZEN, Copenhagen University WILLIAM NEWMAN, Harvard University JÜRGEN RENN, Max-Planck-Institut für Wissenschaftsgeschichte ALEX ROLAND, Duke University ALAN SHAPIRO, University of Minnesota NANCY SIRAISI, Hunter College of the City University of New York NOEL SWERDLOW, University of Chicago

Archimedes has three fundamental goals; to further the integration of the histories of science and technology with one another: to investigate the technical, social and practical histories of specific developments in science and technology; and finally, where possible and desirable, to bring the histories of science and technology into closer contact with the philosophy of science. To these ends, each volume will have its own theme and title and will be planned by one or more members of the Advisory Board in consultation with the editor. Although the volumes have specific themes, the series itself will not be limited to one or even to a few particular areas. Its subjects include any of the sciences, ranging from biology through physics, all aspects of technology, broadly construed, as well as historically-engaged philosophy of science or technology. Taken as a whole, Archimedes will be of interest to historians, philosophers, and scientists, as well as to those in business and industry who seek to understand how science and industry have come to be so strongly linked.

Lenses and Waves Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century

by

FOKKO JAN DIJKSTERHUIS University of Twente, Enschede, The Netherlands

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

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Contents CHAPTER 1 INTRODUCTION – ‘THE PERFECT CARTESIAN’

1

A history of Traité de la Lumière Huygens’ optics New light on Huygens

2 4 8

CHAPTER 2 1653 - 'TRACTATUS'

11

2.1 2.1.1 2.1.2

The Tractatus of 1653 Ovals to lenses A theory of the telescope

12 13 16

The focal distance of a bi-convex lens Images Conclusion

17 20 24

Dioptrics and the telescope Kepler and the mathematics of lenses

24 26

Image formation Lenses Perspectiva and the telescope

28 29 33

2.2 2.2.1

2.2.2

2.2.3

The use of the sine law

35

Descartes and the ideal telescope After Descartes Dioptrics as mathematics

36 37 40

The need for theory

41

The micrometer and telescopic sights Understanding the telescope Huygens’ position

42 46 50

CHAPTER 3 1655-1672 - 'DE ABERRATIONE'

53

3.1 3.1.1

The use of theory Huygens and the art of telescope making

55 57

Huygens’ skills Alternative configurations Experiential knowledge

58 59 60

3.1.2

Inventions on telescopes by Huygens

63

3.2 3.2.1

Dealing with aberrations Properties of spherical aberration

67 67

Specilla circularia Theory and its applications

70 72

vi

3.2.2

CONTENTS

Putting theory to practice

77

A new design

80

3.2.3

Newton’s other look and Huygens’ response

83

3.3 3.3.1

Dioptrica in the context of Huygens’ mathematical science The mathematics of things

92 92

Huygens ‘géomètre’

3.3.2

100

The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument

104

CHAPTER 4 THE 'PROJET' OF 1672 4.1 4.1.1 4.1.2

95

Huygens the scholar & Huygens the craftsman

107

‘Projet du Contenu de la Dioptrique’

109

The nature of light and the laws of optics Alhacen on the cause of refraction Kepler on the measure and the cause of refraction

112 114 117

The measure of refraction 118 True measures 123 Paralipomena and the seventeenth-century reconfiguration of optics 124

4.1.3

4.2 4.2.1 4.2.2

The laws of optics in corpuscular thinking

126

Refraction in La Dioptrique Epistemic aspects of Descartes’ account in historical context Historian’s assessment of Descartes’ optics Reception of Descartes’ account of refraction Barrow’s causal account of refraction

128 130 132 134 136

The mathematics of strange refraction Bartholinus and Huygens on Iceland Crystal

140 142

Bartholinus’ experimenta Huygens’ alternatives

143 147

Rays versus waves: the mathematics of things revisited

152

The particular problem of strange refraction: waves versus masses

155

CHAPTER 5 1677-1679 - WAVES OF LIGHT

159

5.1 5.1.1

161 162

A new theory of waves A first EUPHKA The solution of the ‘difficulté’ of Iceland Crystal

168

Undulatory theory

172

Explaining strange refraction

176

5.1.3

Traité de la Lumière and the ‘Projet’

181

5.2 5.2.1

Comprehensible explanations Mechanisms of light

185 186

5.1.2

CONTENTS

5.2.2

5.3 5.3.1 5.3.2

vii

Hobbes, Hooke and the pitfalls of mechanistic philosophy: rigid waves

189

‘Raisons de mechanique’

195

Newton’s speculations on the nature of light The status of ‘raisons de mechanique’

196 200

A second EUPHKA Danish objections

204 205

Forced innovation

207

Hypotheses and deductions

209

CHAPTER 6 1690 - TRAITÉ DE LA LUMIÈRE

213

6.1 6.1.1

Creating Traité de la Lumière Completing ‘Dioptrique’

214 216

Huygens’ dioptrics in the 1680s

216

6.1.2 6.2

6.3 6.3.1

6.3.2

From ‘Dioptrique’ to Traité de la Lumière

219

The publication of Traité de la Lumière

222

Traité de la Lumière and the advent of physical optics

225

Mathematization by extending mathematics The matter of rays The mathematics of light

227 229 232

Traité de la Lumière and Huygens’ oeuvre Huygens’ Cartesianism

236 237

The subtle matter of 1669 Huygens versus Newton Huygens’ self-image

238 242 247

The reception of Huygens

249

CHAPTER 7 CONCLUSION: LENSES & WAVES A seventeenth-century Archimedes From mathematics to mechanisms Huygens and Descartes The small Archimedes

255 255 259 261 262

LIST OF FIGURES

265

BIBLIOGRAPHY

267

INDEX

285

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Preface “Le doute fait peine a l’esprit. C’est pourquoy tout le monde se range volontiers a l’opinion de ceux qui pretendent avoir trouvè la certitude.”1 This book evolved out of the dissertation that I defended on April 1, 1999 at the University of Twente. At the successive stages of its development critical readers have cast doubts on my argument. It has not troubled my mind; on the contrary, they enabled me to improve my argument in ways I could not have managed on my own. So, I want to thank Floris Cohen, Alan Shapiro, Jed Buchwald, Joella Yoder, and many others. Most of all, however, I have to thank Casper Hakfoort, who saw the final text of my dissertation but did not live to witness my defence and the further development of this study of optics in the seventeenth century. This book would not have been possible without NWO (Netherlands Organisation for Scientific Research) and NACEE (Netherlands American Commission for Educational Exchange) who supplied me with a travel grant and a Fulbright grant, respectively, to work with Alan Shapiro in Minneapolis. The book would also not have been possible without the willingness of Kluwer Publishers and Jed Buchwald to include it in the Archimedes Series, and the unrelenting efforts of Charles Erkelens to see it through. During the years this text accompanied my professional and personal doings, numerous people have helped me grow professionally and personally. I want to thank Peter-Paul Verbeek, John Heymans, Petra Bruulsema, Kai Barth, Albert van Helden, Rienk Vermij, Paul Lauxtermann, Lissa Roberts, and many, many others. Still, the idea to study Huygens and his optics would not have even germinated – let alone that this book would have matured – without my life companion, Anne, with whom I now share a much more valuable creation. Thank you. Fokko Jan Dijksterhuis Calgary, June 2004 This book is dedicated to Casper Hakfoort In memory of Lies Dijksterhuis

1

Undated note by Christiaan Huygens (probably 1686 or 1687), OC21, 342

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Chapter 1 Introduction – ‘the perfect Cartesian’ Christiaan Huygens, optics & the scientific revolution

“EYPHKA. The confirmation of my theory of light and refractions”, proclaimed Christiaan Huygens on 6 August 1679. The line is accompanied by a small sketch, consisting of a parallelogram, an ellipse (though barely recognizable as such) and two pairs of perpendicular lines (Figure 1). The composition of geometrical figures does not immediately divulge its meaning. Yet, it conveys a pivotal event in the development of seventeenthcentury science. What is it? The parallelogram is a section – the principal section – of a piece of Iceland crystal, which is a transparent form of calcite with extraordinary optical properties. It refracts rays of light in a strange way that does not conform to the established laws of refraction. The ellipse represents the propagation of a wave of light in this crystal. It is not an ordinary, spherical wave, as waves of light are by Figure 1 The sketch of 6 August 1679 nature, but that is precisely because the elliptical shape explains the strange refraction of light rays in Iceland crystal. The two pairs of lines denote the occasion for Huygens’ joy. They are unnatural sections of the crystal, which he had managed to produce by cutting and polishing the crystal. They produced refractions exactly as his theory, by means of those elliptical waves, had predicted. The elliptical waves were derived from the wave theory he had developed two years earlier, with the formulation of a principle of wave propagation. Like ordinary spherical waves, these elliptical waves were hypothetical entities defining the mechanistic nature of light. Now, seventeenth-century science was full of hypotheses regarding the corpuscular nature of things. But Huygens’ wave theory was not just another corpuscular theory. His principle defined the behavior of waves in a mathematical way, based on a theory describing the mechanics of light propagation in the form of collisions between ether particles. The mathematical character of Huygens’ wave theory is historically significant. Huygens was the first in the seventeenth century to fully mathematize a mechanistic explanation of the properties of light. As contrasted to the qualitative pictures of his contemporaries, he could derive the exact properties of rays refracted by

2

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Iceland crystal, including refractions that could only be observed by cutting the crystal along unnatural sections. The sketch records the experimental verification of Huygens’ elliptical waves and, with it, the confirmation of his theory of light and refractions. This brief synopsis explains what ‘actually’ happened on that 6th of August in 1679. The various terms and concepts will be explicated later on in this book. For this moment, it suffices to make clear the core of Huygens’ wave theory and its historical significance. For Huygens the successful experiment meant the confirmation of his explanation of strange refraction and his wave theory in general. In the context of the history of seventeenthcentury optics, and of the mathematical sciences in general, the importance of the event lies in the twofold particular nature of Huygens’ theory: a mathematized model of the mechanistic nature of light considered as a hypothesis validated by experimental confirmation. With the mathematical form of his theory, Huygens can be said to have restored the problematic legacy of Descartes’ natural philosophy, by defining mathematical principles for the mechanistic explanation of the physical nature of light. The hypothetical-deductive structure of his theory implied the abandonment of the quest for certainty of that same Cartesian legacy and of seventeenthcentury science in general. Huygens presented waves of light, the inextricable core of his account of optical phenomena, explicitly as hypothetical entities whose certainty is inherently relative. In so doing, he set off from Descartes in a direction diametrically opposite to Newton, the principal other restorer of mechanistic science. About a decade after the EUREKA of 6 August 1679, Huygens published his wave theory of light and his explanations of ordinary and strange refraction in Traité de la Lumière (1690). This book established his fame as a pioneer of mathematical physics as evidenced by the fact his principle of wave propagation is still known and used in various fields of modern physics under the name ‘Huygens’ principle’. Its historical importance is also generally acknowledged. According to Shapiro, Huygens stood out for his “…continual ability to rise above mechanism and to treat the continuum theory of light mathematically.”1 E.J. Dijksterhuis calls it the high point of mechanistic science and its creator the first ‘perfect Cartesian’: “In Huygens does Cartesian physics for the first time take the shape its creator had in mind.”2 How did this come about? How did Christiaan Huygens came to realize this historical landmark? Or more specifically, how did he arrive at his wave theory of light? That is the central question of this study. A history of Traité de la Lumière Unlike its eventual formulation in Traité de la Lumière, the development of Huygens’ wave theory has hardly been subject to historical investigation. The 1

Shapiro, “Kinematic optics”, 244. (For referencing see page 267) Dijksterhuis, Mechanization, IV: 212 & 283 (references to this book will be made by section numbers). It should be noted that Dijksterhuis mainly focuses on the mathematical model of wave propagation. 2

‘THE PERFECT CARTESIAN’

3

first step for such a study is to go into the papers documenting Huygens’ optics. The historian who does so on the basis of existing literature, awaits a surprise. There is much more to Huygens’ optics than waves. He elaborated a comprehensive theory of the dioptrical properties of lenses and their configurations in telescopes, that goes by the title of Dioptrica. A second surprise is in store when one takes a closer looks to these papers on geometrical optics. The papers on dioptrics cover the Huygens’ complete scientific career and form the exclusive content of the first two decades of his optical studies. The wave theory and related subjects are fully absent; not before 1672 do they turn up. In other words, the optics that brought Huygens future fame dates from a considerably late stage in the development of Huygens’ optics. In this way new and more specific questions arise regarding the question ‘how did Huygens arrive at his wave theory’? What exactly was his optics? How did he move from Dioptrica to Traité de la Lumière? And what does this teach us about the historical significance of his wave theory, Huygens’ creation of a physical optics, and the character of his science? The point is that Huygens’ dioptrics turns his seemingly self-explanatory wave theory into a historical problem. It did not develop from some innate cartesianism, for he was no born Cartesian, certainly not in optics at least. Fully absent from Dioptrica is the central question of Traité de la Lumière: what is the nature of light and how can it explain the laws of optics. Huygens first raised this question in 1672 – five years before he found his definite answer (which he confirmed another two years later in 1679). His previous twenty years of extensive dioptrical studies give scarcely any occasion to expect that this man was to give the mechanistic explanation of light and its properties a wholly new direction. In view of Dioptrica, the question is not only how Huygens came to treat the mechanistic nature of light in his particular way, but even how he came to consider the mechanistic nature of light in the first place. In the literature on Huygens’ optics, mechanistic philosophy has been customarily considered a natural part of his thinking. Only Bos, in his entry in the Dictionary of Scientific Biography, points at the relatively minor role mechanistic philosophizing played in his science before his move to Paris in the late 1660s.3 The question of how the wave theory took shape in Huygens’ mind thus becomes all the more intriguing. What caused Huygens to tackle this subject he had consistently ignored throughout his earlier work on optics? How do Dioptrica and Traité de la Lumière relate and what light does the former shed on the latter? Part of the answer is given by the fact that only at the very last moment, short before its publication, Huygens decided to change the title of his treatise on the wave theory from ‘dioptrics’ to ‘treatise on light’. In his mind the two were closely connected, questions now

3

Bos, “Huygens”, 609. Van Berkel further alludes to the influence of Parisian circles on the prominence of mechanistic philosophy in Huygens’ oeuvre: Van Berkel, “Legacy”, 55-59.

4

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are: how exactly and what does this mean for our understanding of Huygens’ optics? In addition to the question of where in Huygens’ oeuvre Traité de la Lumière properly belongs, a more general question may be asked: where in seventeenth-century science may this kind of science be taken to belong? Were the questions Huygens addressed in Traité de la Lumière part of any particular scientific discipline or coherent field of study? In the course of my investigation, it has became increasingly clear to me that the term ‘optics’ is rather problematic regarding the study of light in the seventeenth century, just like the term ‘science’ in general. Our modern understanding of ‘optics’ implies an investigation of phenomena of light much like Traité de la Lumière: a mathematically formulated theory of the physical nature of light explaining the mathematical regularities of those phenomena. Yet, optics in this sense was only just beginning to develop during the seventeenth century. The term ‘optics’ in the seventeenth century denoted the mathematical study of the behavior of light rays that we are used to identify with geometrical optics. This is what Huygens pursued in Dioptrica, prior to developing his wave theory. A general question regarding the history of optics raised by Huygens’ Traité de la Lumière is how a new kind of optics, a physical optics, came into being in the seventeenth century and how this related to the older science of geometrical optics. This transformation of the mathematical science of optics is manifest in the title Huygens eventually chose for his treatise. Huygens’ optics This book offers in the first place an account of the development of Huygens’ optics, from the first steps of Dioptrica in 1652 to the eventual Traité de la Lumière of 1690. The following chapters take a chronological course through his engagements with the study of light, whereby the historical connection of its various parts sets the perspective. Terms like ‘optics’ and ‘science’ are problematic historically. Nevertheless, for sake of convenience, I will freely use them to denote the study of light in general and natural inquiry, except when this would give rise to (historical) misunderstandings. When discussing their historical character and development specifically, I will use appropriately historicizing phrases. Chapter two discusses Tractatus – the unfinished treatise of 1653 on dioptrics that marks the beginning of Huygens’ engagement with optics. Tractatus contained an comprehensive and rigorous theory of the telescope and I will argue that this makes it unique in seventeenth-century mathematical optics. Huygens was one of the few to raise theoretical questions regarding the properties and working of the telescope, and almost the only one to direct his mathematical proficiency towards the actual instruments used in astronomy. Kepler had preceded him, but he had not known the law of refraction and therefore could not derive but an approximate theory of lenses and their configurations. Some four decades afterwards, and two decades after the publication of the sine law, Huygens

‘THE PERFECT CARTESIAN’

5

was the first to apply it to spherical lenses and remained so for almost two decades more. Chapter three discusses his practical pursuits in dioptrics leading into his subsequent treatise on dioptrics, De Aberratione of 1665. Huygens made a unique effort to employ dioptrical theory to improve the telescope. The contrast with Descartes is particularly conspicuous, for Huygens did not fit the telescope into the ideal mold prescribed by theory but directed his theory towards the instruments that were practically feasible. The effort was unsuccessful, for with his new theory of light and colors Newton made him realize the futility of his design. Taking into account Huygens’ background in dioptrics sheds, I will argue, new light on the famous dispute with Newton in 1672. These chapters are confined to what we would call geometrical optics and to its relationship to practical matters of telescopy.4 I try to explain what this science was about and what was particular about the way Huygens pursued it. These chapters offer a fairly detailed account of Dioptrica within the context of seventeenth-century geometrical optics, and as such open fresh ground in the history of science. At the turn of the century, Kepler had laid a new foundation for geometrical optics. Image formation now became a matter of determining where and how a bundle of diverging rays from each point of the object is brought into focus again (or not) instead of tracing single rays from object point to image point. Of old, the ray was the bearer of the physical properties of light, but in the course of the seventeenth century this began to be qualified and questioned. In the wake of Kepler and Descartes the mathematical science of optics gradually transformed into new ways of doing optics. The traditional, geometrical way of doing optics did not vanish, though. It was ray optics in which the question of the nature of light need not penetrate further than determining the physical properties of rays in their interaction with opaque and transparent materials. This is the mathematical optics Huygens grew up with and that set the tone in his earliest dealings with the physics of light propagation. Only on second thought did he focus on the new question what is light and how can this explain its properties. This transformation is the subject of the next chapters. In chapters four and five my focus shifts, along with Huygens’, to the mechanistic nature of light. In 1672 a particular problem drew his attention to the question what light is and how its properties can be explained: the strange refraction in Iceland Crystal which created a puzzle regarding the physics of refraction that Huygens wanted to solve. These two chapters discuss the three stages of his investigation, his first analysis of the mathematics of strange refraction in 1672 and his eventual solution by means of elliptical waves in 1677 and its confirmation in 1679. Huygens’ first attack on the problem of strange refraction is historically significant because he approached it along traditional lines of geometrical optics. Only in second 4

Preliminary results are published in: Dijksterhuis, “Huygens’ Dioptrica” and Dijksterhuis, “Huygens’s efforts”.

6

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instance did he turn to the actual question underlying the problem: how exactly do waves of light propagate. And only in third instance, and forced by critical reactions, did Huygens seek for experimental validation of the theory he initially had developed primarily rationally. These chapters offer a new, detailed reconstruction of the origin of the wave theory on the basis of manuscript material that has not been taken into account earlier. It is also a reconstruction of how Huygens got from Dioptrica to Traité de la Lumière, in which I compare his approach to questions pertaining to the physical nature of light in the mathematical science of optics to his predecessors and contemporaries. Central themes are the way the nature of light was accounted for in the mathematical study of light in the seventeenth century and to relationships between explanatory theories of light and the laws of optics. I will argue that Huygens was the first to successfully mathematize a mechanistic conception of light. He was rivaled only by Newton, but for epistemological reasons he kept his hypotheses private. Chapter six reviews the development of Traité de la Lumière, its significance for the history of seventeenth-century optics, and for our understanding of Huygens’ science. After discussing the publication history of Traité de la Lumière, which reveals that Huygens disconnected it from Dioptrica only at the very last moment, I sketch some lines for a new perspective of the history of seventeenth-century optics in which traditional geometrical optics is taken into account as an important root. The mathematico-physical consideration of light of Traité de la Lumière was a particular answer to a new kind of question. A kind of question also addressed by such diverse scholars as Kepler, Descartes and Newton. In this sense, my study of the development of Traité de la Lumière, in particular in relationship with Dioptrica, is also a study of the origins of a new science of optics, nowadays denoted by the term ‘physical optics’. Some instances of physical optics developed in the seventeenth-century, most notably by Huygens and Newton. But the primacy of the question ‘what is the physical nature of light and how may this explain its properties? first had to be discovered and this only gradually came about in the pursuit of the mathematical science of optics. In the case of Huygens this emergence was particularly quiet. While solving the intriguing puzzle of strange refraction, he developed a new way of doing mathematical optics but he seems to have been hardly aware of the new ground he was breaking. At the close of this chapter, I discuss his alleged Cartesianism and I will argue that Huygens stumbled into becoming a ‘perfect Cartesian’ rather than determinedly and systematically create it. This study is based on the optical papers in the Oeuvres Complètes and additional manuscript material. A large part of these have as yet not been studied. The Oeuvres Complètes split up Huygens’ optics in two parts – volume 13 for Dioptrica and volume 19 for Traité de la Lumière. This subdivision along modern disciplinary lines resounds in the historical literature. E.J. Dijksterhuis, for example, separates explicitly ‘geometrical optics’ – where he

‘THE PERFECT CARTESIAN’

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merely mentions Huygens – and physical theories of optics.5 No doubt all this has contributed its share to the fact that the relationship between Traité de la Lumière and Dioptrica – historical, conceptual as well as epistemical – has gone unexamined so far.6 As for Traité de la Lumière, most historical interpretations are based on and confined to the published text. The additional manuscript material published in OC19 has hardly been taken into account and no-one to my knowledge has used the original manuscript material in the Codices Hugeniorum in the Leiden university library.7 Huygens’ wave theory has been the subject of several historical studies. Each in their own way has been valuable for this study. E.J. Dijksterhuis gives an illuminating analysis of the merits of Traité de la Lumière as a pioneering instance of mathematical physics considered in the light of Descartes’ mechanistic program. Sabra includes an account of Huygens’ wave theory in his study of the historical development of the interplay of theory and observation in seventeenth-century optics. Shapiro offers a searching analysis of the historical development of the physical concepts underlying Huygens’ wave theory. I intend to add to our growing historical understanding of Traité de la Lumière by reconstructing its origin and development in the context of his optical studies as a whole and of that of seventeenth-century optics in general. Huygens’ lifelong engagement with dioptrics as such has hardly been studied.8 Even Harting, the microscopist who by mid-nineteenth century gives Huygens’ telescopic work a central place in his biographical sketch, mentions dioptrical theory only in passing.9 The editorial remarks in the ‘Avertissement’ of OC13 form the main exception and are one of the few sources of information on the history of seventeenth-century geometrical optics in general. Some topics pertaining to seventeenth-century geometrical optics have been studied in considerable detail, but for the most part in the context of the seventeenth-century development of physical science. These are Kepler’s theory of image formation, the discovery of the sine law and Newton’s mathematical theory of colors and they are integrated in my 5

Dijksterhuis, Mechanisering, IV: 168-171, 284-287. Hashimoto hardly goes beyond noting that “… two works were closely related in Huygens’s mind.”: Hashimoto, “Huygens”, 87-88. 7 Dijksterhuis, Mechanization, IV: 284-287 and Sabra, Theories, 159-230 are confined to Traité de la Lumière. Shapiro uses some of the manuscripts published in Oeuvres Complètes. Ziggelaar, “How”, draws mainly on OC19. Yoder has pointed out that the wave theory is no exception to the rule that in general, studies of Huygens’ work tend to focus on his published works. 8 Hashimoto has published a not too satisfactory article in which he discusses Huygens’ dioptrics in general terms. Apart from some substantial flaws in his analyses and argument, Hashimoto fails to substantiate some of his main claims regarding Huygens’ ‘Baconianism’. Hashimoto, “Huygens”, 75-76; 86-87; 89-90. For example, he reads back into Tractatus the utilitarian goal of De aberratione (60, compare my section 3.3.2), he thinks Huygens determined the configuration of his eyepiece theoretically (75, compare my section 3.1.2), maintains that Systema saturnium grew out of his study of dioptrics (89, compare my section 3.1.2) and that Huygens ‘went into the speculation about the cause of colors’ after his study of spherical aberration (89, compare my section 3.2.3) 9 Harting, Christiaan Huygens, 13-14. Harting based himself on manuscript material disclosed in Uylenbroek’s oration on the dioptrical work by the brothers Huygens: Uylenbroek, Oratio. 6

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accounts of, respectively, seventeenth-century dioptrics in chapter 2, the epistemic role and status of explanations in optics in chapter 4, and Huygens’ own dealings with colors in chapter 3 as well as his specific approach to mechanistic reasoning in chapter 5. Little literature on the history of the field of geometrical optics and its context exist.10 A substantial part of my argument is based upon comparisons with the pursuits of other seventeenth-century students of optics. In order to come to a historically sound understanding of what Huygens was doing, I find it necessary to find out how his optics relates to the pursuits of his predecessors and contemporaries. What questions did they ask (and what not) and how did they answer them? Why did they ask these questions and what answers did they find satisfactory? For example, in chapter 2 the earliest part of Dioptrica is compared with, among other works, Kepler’s Dioptrice and Descartes’ La Dioptrique. All bear the same title, yet the differences are considerable. Descartes discussed ideal lenses and did so in general terms only, rather than explaining their focusing and magnifying properties as Kepler had done. Huygens, in his habitual search for practical application, expressly focused on analyzing the dioptrical properties of real, spherical lenses and their configurations, thus developing a rigorous and general mathematical theory of the telescope. By means of such comparisons it is possible to determine in what way Huygens marked himself off as a seventeenth-century student of optics, or did not. These comparisons are focused on Huygens’ optics, so I confine my discussions of seventeenthcentury of optics to the mathematical aspects of dioptrics and physical optics. Other themes like practical dioptrics and natural philosophy in general will be treated only in relation to Huygens. This is an intellectual history of Huygens’ optics and of seventeenthcentury optics in general. The nature of the available sources – as well that of the man – are not suitable for some kind of social or cultural history. He operated rather autonomously, mainly because he was in the position to do so, and he was no gatherer of allies and did not try very hard to propagate his ideas about science and gain a following. New light on Huygens This study offers, in the first place, a concise history of Huygens’ optics. Yet it is not a mere discussion of Huygens’ contributions to various parts of optics. I also intend to shed more light on the character of Huygens’ scientific personality. The issue of getting a clear picture of his scientific activity and its defining features is an acknowledged problem. In summing up a 1979 symposium on the life and work of Huygens, Rupert Hall 10

For example, the precise application of the sine law to dioptrical problems, for example, has hardly been studied. Shapiro, “The Optical Lectures” is a valuable exception, discussing Barrow’s lectures and their historical context. The relationship between the development of the telescope and of dioptrical theory – essential to my account of Dioptrica – has never been investigated in any detail. Van Helden has pointed out the weak connection between both in general terms: Van Helden, “The telescope in the 17th century”, 45-49; Van Helden, “Birth”, 63-68.

‘THE PERFECT CARTESIAN’

9

concluded that “… it isn’t at all easy to understand how all the multifarious activities of this man’s life fit together.”11 Huygens has been called the true heir of Galileo, the perfect Cartesian, and also a man deftly steering a middle course between Baconian empiricism and Cartesian rationalism.12 Huygens himself has not been much of a help in this. He always was particularly reticent about his own motives. He was an intermediate figure between Galileo and Descartes on the one hand and Newton and Leibniz on the other but, lacking as he did a pronounced conception of the aims and methods of his science, he is difficult to situate among the protagonists of the scientific revolution. In 1979, the most apt characterization of Huygens seemed to be that of an eclectic, who took up loose issues and solved them with the means he considered appropriate without some sort of central direction becoming apparent.13 The original idea behind this study was that in Traité de la Lumière this eclecticism grew into a fruitful synthesis of mathematical, mechanistic and experimental approaches. This idea originates from the work my advisor, the Casper Hakfoort. In his study of eighteenth-century optics he formulated the idea when he pointed out the significance of natural philosophy for the development of optics, which in his view was hithertho neglected.14 I consider it an honor to have been able to pursue this idea and to have had it bear unanticipated fruit. By trying to understand how the multifarious aspects of his optics fit together, I hope to be able to shed light on the character of his science in general and on his place in seventeenth-century science. Dioptrica, while adding to the standing impression of the great versatility of Huygens’ oeuvre, has not changed my expectation that an understanding of the way the possible coherence of these aspects evolved may contribute to a better characterization of Huygens’ science and of his place in seventeenth-century science as a whole. In the final chapter of this book, I review my account of the development of Huygens’ optics to see what light this may shed on his scientific personality. By way of conclusion it offers a sketch of his science, based on the previous chapters that go into the details – often technical – of his optics and its development. This chapter can be read independently as an essay on Huygens. Huygens was a puzzle solver indeed, an avid seeker of rigorous, exact solutions to intricate mathematical puzzles. But these puzzles do have coherence, they all concerned questions regarding concrete, almost tangible subjects in the various fields of seventeenth-century mathematics. He was an eclectic, but only in comparison with the chief protagonists of the scientific 11

Hall, “Summary”, 311. Westfall, Construction, 132-154; Dijksterhuis, Mechanization, 212; Elzinga, Research program and Westman, “Problem”, 100-101. 13 Hall, “Summary”, 305-306. As regards his studies of motion, Yoder has further specified this characterization; Yoder, Unrollling time, 169-179. 14 Hakfoort, Optics in the age of Euler, 183-184. 12

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CHAPTER 1

revolution, that set up schemes to lay new foundations for natural inquiry. Huygens did not have such a program and, as a result, his science seems to lack coherence, unless a coherence is looked for on a different level of seventeenth-century science. The essay therefore first forgets about Huygens’ alleged Cartesianism to sketch the mathematician and his idiosyncratic focus on instruments. He was not a half-baked philosopher but a typical mathematician. A new Archimedes, as Mersenne foretold in 1647. The incomparable Huygens, as Leibniz said in 1695 upon the news of his death.15 Then I ask anew how his Cartesianism fits into the picture. We may have trouble getting a balanced idea of what he was doing, but it appears that he was hardly aware of the size of the new ground he had been breaking. He had, in fact, developed a new mathematical science of optics.

15

OC1, 47 and OC10, 721.

Chapter 2 1653 - 'Tractatus' The mathematical understanding of telescopes

“Now, however, I am completely into dioptrics”, Huygens wrote on 29 October 1652 in a letter to his former teacher in mathematics, Frans van Schooten, Jr.1 His enthusiasm had been induced by a discovery in dioptrical theory he had recently made. It was an addition to Descartes’ account of the refracting properties of curves in La Géométrie, that promised a useful extension of the plan for telescopes with perfect focusing properties Descartes had set out in La Dioptrique. During his study at Leiden University in 1645-6, Huygens had studied Descartes’ mathematical works, La Géométrie in particular, intensively with Van Schooten. Although Van Schooten was professor of ‘Duytsche Mathematique’ at the Engineering school, appointed to teach practical mathematics in the vernacular to surveyors and the like. Huygens was not the only patrician son he introduced to the new mathematics: the future Pensionary Johan de Witt and the future Amsterdam mayor Johannes Hudde. From 1647 Huygens and Van Schooten had to resort to corresponding over mathematics, when Huygens had to go to Breda to the newly established ‘Collegium Auriacum’, the college of the Oranges to which the Huygens family was closely connected politically. In 1649 Huygens had returned home to The Hague and now, in 1652, he was ‘private citizen’. He did not feel like pursuing the career in diplomacy his father had planned for him and, with the Oranges out of power since 1650, not many duties were left to call on him. Huygens could, in other words, freely pursue his one interest, the study of the mathematical sciences. An appointment at a university was out of order for someone of his standing and, as Holland lacked a centrally organized church and a grand court, interesting options for patronage were not directly available.2 So, with a room in his parental home at the ‘Plein’ in The Hague and a modest allowance from his father, he could live the honorable life of an ‘amateur des sciences’. He enjoyed to company of his older brother Constantijn, who joined him in his work in practical dioptrics (see next chapter), and dedicated himself to mathematics. Geometry and mechanics were his main focus in these years, with his theories of impact and of pendulum motion and his invention of the 1

OC 1, 215. “Nunc autem in dioptricis totus sum ...” Berkel, “Illusies”, 83-84. In the 1660s Huygens would start to seek patronage abroad, first in Florence and then, successfully in Paris.

2

12

CHAPTER 2

pendulum clock as the most renowned achievements. Yet, the discovery made late 1652 had sparked his interest in dioptrics, which largely dominated his scholarly activities the next two years. The letter to Van Schooten was the onset to a lifelong engagement with dioptrics, which nevertheless has little been studied historically.3 In the months following the letter to Van Schooten, Huygens elaborated a treatise that contained a mathematical theory of the dioptrical properties of lenses and telescopes. I will refer to this treatise as Tractatus and it is the subject of this chapter.4 In Tractatus Huygens treated a specific set of dioptrical questions, directed at understanding the working of the telescope. In the first section of this chapter the content and character of the treatise are discussed. In the second section Huygens’ approach to dioptrics is compared with that of contemporaries, by examining how other mathematicians dealt with the questions that stood central in Tractatus. In this discussion of seventeenth-century dioptrics the relationship between dioptrical theory and the development of the telescope is the central topic. I will argue that Huygens in his mathematical theory stood out for his focus on questions that were relevant to actual telescopes. In this he was the first to follow Kepler’s lead; other theorists were absorbed by abstract questions emerging from mathematical theory for which men of practice, in their turn, did not care. In the next chapter Huygens’ own telescopic practices are discussed. Now first the theoretical considerations of Tractatus. 2.1 The Tractatus of 1653 The background to Huygens’ letter to Van Schooten was a problem with the lenses used in the telescopes of those days. Lenses were spherical, i.e. their cross section is circular. As a result they do not focus parallel rays perfectly. Rays from a distant point source that are refracted by a spherical surface do not intersect in a single point, rays close to the axis are refracted to a more distant point on the axis than rays farther from the axis (Figure 2). This is called spherical aberration and results in slightly blurred Figure 2 Spherical aberration 3

The most thorough-going account still are the ‘avertissements’ by the editors of the Oeuvres Complètes. Southall, “Some of Huygens’ contributions” reported on Huygens’ dioptrics after the publication of volume 13. Harting, Christiaan Huygens had earlier discussed it briefly. In relationship with his astronomical work and his practical dioptrics, Albert van Helden, “Development” and Anne van Helden/Van Gent, The Huygens collection and “Lens production” discuss some topics. In the context of the history of seventeenth-century geometrical optics – which in its own right has little been studied – Shapiro, “’Optical Lectures’” mention Huygens’ contributions. They are remarkably absent from the Malet, “Isaac Barrow” and “Kepler and the telescope”. Hashimoto, “Huygens, dioptrics” is the only effort to discuss Huygens’ dioptrics in the context of his broader oeuvre. 4 OC13, 1-271. The editors of the Oeuvres Complètes have labeled it Dioptrica, Pars I. Tractatus de refractione et telescopiis. Its content stems from the 1650s. The original version of Tractatus does not exist anymore. A copy was made in Paris by Niquet – probably in 1666 or 1667, at the beginning of Huygens’ stay in Paris – on which the text of the Oeuvres Complètes is based. The editors assume Niquet’s copy of Tractatus is largely identical with the original 1653 manuscript; “Avertissement”, XXX.

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13

images. In La Dioptrique (1637), Descartes had explained that surfaces whose section is an ellipse or a hyperbola do not suffer this impediment. They are called aplanatic surfaces. Descartes could demonstrate this by means of the sine law, the exact law of refraction he had discovered some 10 years earlier. According to the sine law, the sines of incident and of refracted rays are in constant proportion. This ratio of sines is nowadays called index of refraction, it depends upon the refracting medium. The discovery Huygens made in late 1652 sprang from Descartes’ La Géométrie. Together with La Dioptrique and Les Météores, this essay was appended to Discours de la methode (1637). In La Géométrie, Descartes had introduced his new analytic geometry. In La Géométrie mathematical proof was given of the claim of La Dioptrique that the ellipse and hyperbola are aplanatic curves. In his letter to Van Schooten, Huygens wrote that he had discovered that under certain conditions circles also are aplanatic. This discovery implied that spherical lenses could focus perfectly in particular cases. Consequently, Huygens considered it of considerable importance for the improvement of telescopes. Huygens’ expectation that his discovery would be useful in practice, was fostered by the fact that Descartes’ claims had turned out not to be practically feasable. Around 1650, no one had succeeded in actually grinding the lenses prescribed in La Dioptrique.5 Apart from that, the treatise did not discuss the spherical lenses actually employed in telescopes. Descartes had applied his exact law only to theoretical lenses. When he made his discovery, Huygens must have realized that no-one had applied the sine law to spherical lenses yet. In the aftermath of his discovery, Huygens set out to correct this and develop a dioptrical theory of real lenses. 2.1.1 OVALS TO LENSES In his letter to Van Schooten, Huygens did not explain the details of his discovery. He did so much later, in an appendix to a letter of 29 October 1654 that contained comments upon Van Schooten’s first Latin edition of La Géométrie: Geometria à Renato Des Cartes (1649).6 In book two, Descartes had introduced a range of special curves, ovals as he called them. This was not a mere abstract exercise, he said, for these curves were useful in optics: “For the rest, so that you know that the consideration of the curved lines here proposed is not without use, and that they have diverse properties that do not yield at all to those of conic sections, I here want to add further the explanation of certain ovals, that you will see to be very useful for the theory of catoptrics and of dioptrics.”7

By means of the sine law, Descartes derived four classes of ovals that are aplanatic curves. If such a curve is the section of a refracting surface, rays 5

With the possible exception of Descartes himself. See below, section 3.1 OC1, 305-305. 7 Descartes, Geometrie, 352 (AT6, 424). “Au reste affin que vous sçachiées que la consideration des lignes courbes icy proposée n’est pas sans usage, & qu’elles ont diverses proprietés, qui ne cedent en rien a celles des sections coniques, ie veux encore adiouster icy l’explication de certaines Ovales, que vous verrés estres tres utiles pour la Theorie de la Catoptrique, & de la Dioptrique.” 6

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CHAPTER 2

coming from a single point are refracted towards another single point. In certain cases, the ovals reduce to the ellipses and hyperbolas of La Dioptrique. Huygens in his turn discovered that a particular class of these ovals may also reduce to a circle. In La Géométrie Descartes introduced the said class of ovals as follows (Figure 3). The dotted line is an oval of this class. If the right part 2X2 of the oval is the right boundary of a refracting medium, rays intersecting in point F are refracted to point G.8 The oval is constructed as follows. Lines FA and AS intersect in A at an arbitrary angle, F is an arbitrary point on FA. Draw a circle with center F and radius F5. Line 56 is Figure 3 Cartesian oval. drawn, so that A5 is to A6 as the ratio of sines of the refracting medium. G is an arbitrary point between A and 5, S is on A6 with AS = AG. A circle with center G and radius S6 cuts the first circle in the points 2, 2. These are the first two points of the oval. This procedure is repeated with points 7 and 8, et cetera until the oval 22X22 is completed.9 Huygens’ discovered that the oval reduces to a circle when the ratio of AF to AG is equal to the ratio of A5 to A6, the ratio of sines.10 This means that with respect to rays tending to F, a spherical surface 2X2 will focus them exactly in G. Van Schooten was a bit skeptical about Huygens’ claim. Could such a simple fact have escaped Descartes? Nevertheless, he included it in the second edition of Geometria à Renatio Des Cartes (1659).11 Discovering that a spherical surface is aplanatic in certain cases is one thing, applying it to lenses in practice is another. It remained to be seen what shape the second surface should have and how it might be employed in telescopes. For one thing, it does not seem useful for objective lenses, the front and most important lens of a telescope that receives parallel rays. It appears the usefulness of the discovery was limited, for Huygens never returned to it in his dioptrical studies.12 The historical importance of the discovery lies in the fact that it aroused Huygens’ interest in dioptrics. He did not exaggerate when he said he was engrossed in dioptrics. Not only its theory, practice too. Five days after his letter to Van Schooten, on 4 November, he wrote to Gerard Gutschoven, an acquainted mathematician in Antwerp.13 After some introductory remarks, 8

Descartes, Geometrie, 358-359 (AT6, 430-431). The left part 2A2 is a mirror that reflects rays intersecting in G so that they (virtually) intersect in F, provided that it diminishes the ‘tendency’ of the rays to a given degree. 9 Descartes, Geometrie, 353-354 (AT6, 424-426). The curve satisfies the equation F2 – FA = n(G2 – GA). 10 OC1, 305. See note 9: the equation becomes AF = nAG. 11 Reproduced in OC14, 419. 12 In Tractatus, he merely mentioned that a spherical surface is aplanatic for certain points: OC13, 64-67. 13 OC1, 190-192.

1653 - TRACTATUS

15

Huygens launched a series of questions on the art of making lenses. What material are grinding moulds made of, how is the spherical figure of a lens checked, what glue is used to attach the lenses to a grip, et cetera. Only after these questions did he explain to Gutschoven that he wanted to know all these things because he had discovered something that would greatly improve telescopes.

Figure 4 Focal distance of a bi-convex lens

The letter reveals that Huygens had already begun to investigate the dioptrical properties of spherical lenses. It contained a theorem on the focal distance of parallel rays refracted by a bi-convex lens CD (Figure 4).14 AC and DB are the radii of the anterior and posterior side of the lens. L and E are determined by DL : LB =CE : EA = n, the index of refraction. O is found by EL : LB = ED : EO. Rays parallel to the axis EL come from the direction of L. Without proof Huygens said that O is the focus of the refracted rays. He said that he could prove this and that he had found many more theorems. A month later, in a letter of 10 December to André Tacquet, a Jesuit mathematician in Louvain, he added an important insight.15 As a result of spherical aberration point O is not the exact focus. Nevertheless, it may be taken as the focus: “… since beyond point O no converging rays intersect with the axis.”16 In later letters to Tacquet and Gutschoven he called this point the ‘punctum concursus’.17 It is the where rays closest to the axis are refracted to. This definition would be fundamental to the theory of Tractatus, which apparently was well under way. Huygens told Tacquet that he had already written two books of a treatise on dioptrics: one on focal distances, another one on magnification. A third one on telescopes was in preparation. Within a month or two after his letter to van Schooten, Huygens’ understanding of dioptrics was rapidly developing. It was also developing in a particular direction. Huygens was studying the dioptrical properties of spherical lenses. He must have found out that little had been published on the subject. The only mathematical theory of spherical lenses was Kepler’s Dioptrice (1611), but it lacked an exact law of refraction. Only Descartes had applied the sine law to lenses, but he had ignored spherical lenses. Huygens had begun to develop an exact theory of spherical lenses by himself. He combined this theoretical interest with an interest in practical matters of telescope making. He reported to have seen a telescope made by the famous craftsman Johann Wiesel of Augsburg. He was impressed and regretted that 14

OC1, 192. OC1, 201-205. 16 OC1, 204. “… adeo ut nullius radij concursus cum axe contingat ultra punctum O.” 17 OC1, 224-226. 15

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CHAPTER 2

Holland did not have such excellent craftsmen.18 On 10 February, 1653, Gutschoven finally informed him on the art of lens making.19 Huygens did not put the information to practice right-away, he first elaborated his dioptrical theory. 2.1.2 A THEORY OF THE TELESCOPE Huygens had written to Tacquet that his treatise would consist of three parts: a theory of focal distances of lenses, a theory of the magnification produced by configurations of lenses, and an account of the dioptrical properties of telescopes based on the theory of the two preceding parts. The third part was not yet finished when Huygens wrote Tacquet, in fact he never elaborated in the form originally conceived. The third part of Tractatus as it is found in the Oeuvres Complètes is a collection of dispersed propositions collected by the editors. Only the first two seem to be from the 1650s.20 In the arrangements of manuscripts Huygens made in the late 1680s, part one of Tractatus appears for the large part as it is found in the Oeuvres Complètes.21 Judging from the various page numberings, Huygens has not edited it very much, except that he inserted – probably in the late 1660s – parts of his study of spherical aberration after the twentieth proposition. Part two of Tractatus has been reshuffled somewhat more, but the main line appears to be sufficiently original. In the following discussion of Tractatus, I follow the text of the Oeuvres Complètes in so far as it appears to reflect the original treatise. Huygens coupled his orientation on the telescope with the mathematical rigor typical of him. Although he singled out dioptrical problems that were relevant to the telescope, he treated these with a generality and completeness that often exceeded the direct needs of explaining the working of the telescope. Huygens’ rigorous approach is clear from the very start of Tractatus. Basic for his treatment of focal distances was the realization that spherical surfaces do not focus exactly. This had been noticed earlier and had been the rationale behind La Dioptrique. Nobody, however, had gone beyond the mere observation of spherical aberration. Huygens got a firmer mathematical grip on the imperfect focusing of lenses by defining which point on the axis may count as the focus. Although he only discussed focal points, Huygens took spherical aberration into account by consistently determining the focus as the ‘punctum concursus’.

18

OC1, 215. OC1, 219-223. 20 The decisions the editors made for the remaining propositions are sometimes somewhat mysterious. For example, the fourth proposition has been assembled of fragments from various folios. And from folio Hug29, 177 they put a diagram in part three of Tractatus, but they transferred the main contents to ‘De telescopiis’ (see section 6.1.2). 21 On this arrangement see page 221. By the way, the two first propositions of part three are inserted after part one. 19

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17

In part one of Tractatus he defined ‘punctum concursus’ as follows. In the third proposition, he defined the focus as the limit point of the intersections of refracted rays with the axis (Figure 5).22 ABC is a planoconvex lens and parallel rays are incident from the direction of D. Consequently, they are only refracted by the spherical surface. Huygens showed that the closer rays are to the axis DE, the closer to E they reach it. Beyond E no refracted rays crosses the axis. This limit point E he defined as the ‘punctum concursus’ of the spherical surface ABC. If the surface is concave, rays do not intersect at all after refraction, they diverge. In this case, the ‘punctum concursus’ is the virtual focus, the limit point of the intersections with the axis of the backwards extended refracted rays. Figure 5 Punctum concursus In the first part of Tractatus, Huygens derived focal distances of all types of spherical lenses by determining exactly the ‘punctum concursus’ in each case. Refraction of parallel rays by a lens consists in most cases of the two successive refractions by each side of the lens. Determining the focal distance thus consists of three problems. First, the refraction of parallel rays from air to glass by a spherical surface. Second, that of the refraction of converging or diverging rays from glass to air. Finally, combining both. Huygens built up his theory accordingly. He first derived theorems expressing focal distance of spherical surfaces for parallel rays in terms of their radii. Secondly, he derived theorems expressing the focal distance for non-parallel rays in terms of the radius and the focal distance for parallel rays. Finally, he expressed the focal distance of the various kinds of lenses in terms of the radii of their sides. In each case he took the thickness of the lens into account. Only afterwards did he derive simplified theorems for thin lenses, in which their thickness is ignored. I now sketch the typical case of a bi-convex lens, the theorem that Huygens included without proof in his letters to Gutschoven and Tacquet. The determination of the focal distances of other lenses – plano-convex, biconcave, etc. – went along similar lines. The focal distance of a bi-convex lens The eighth proposition of Tractatus dealt with parallel rays refracted at the convex surface of a denser medium (Figure 6). AC is the radius of ABP; Q is a point on the axis AC so that AQ : QC = n, where n is the index of refraction. Huygens demonstrated that Q is the ‘punctum concursus’ of parallel rays OB, NP. A refracted ray BL intersects axis AC in a point L between A and Q. With the sine law BL : LC = AQ : QC = n. For any ray OB, BL is smaller than AL and AL is smaller than AQ. Therefore no refracted rays intersect the axis beyond 22

OC13, 16-19

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Q. In order to prove that Q is the ‘punctum concursus’ of ABP, consider ray NP and its refraction PK. PK is found with the sine law and KQ is therefore a given interval. On KQ choose L and draw T, close to A, so that LT : CL = AQ : CQ = n. Now PL : LC < PK : KC = n. PL is smaller than TL, which in its turn is smaller AL. A circle with center L and radius TL intersects the refracting surface ABP between A and P in a point B. Draw BL and BC and it follows that BL : LC = TL : LC = n. Therefore BO is refracted to L. So, the closer a paraxial ray is to the axis, the closer to Q the refracted ray will intersect with the axis. Q is the limit point of these intersections and

therefore the ‘punctum concursus’. When the index of refraction = 3 : 2 – the approximate value for glass – AQ is exactly three times the radius AC. The refraction at the posterior side of the lens is dealt with in the twelfth proposition. This case is more complex as the incident rays are converging due to the refraction at the anterior side. Huygens dealt with eight cases of non-parallel rays. 23 For all cases, he expressed the focal distance of the non-parallel rays in terms of the focal distance of the surface for paraxial rays. The case at hand is the fourth part of the proposition (Figure 7).24 Rays Figure 6 Refraction at Figure 7 Refraction converge towards a point S, the anterior side of a at the posterior side bi-convex lens of a bi-convex lens. outside the dense medium bounded by a spherical surface AB with radius AC. Q is the ‘punctum concursus’ of paraxial rays coming from R. With SQ : SA = SC : SD, the ‘punctum concursus’ D of the converging rays LB is found. Huygens’ proof consisted of a reversal of the first case treated in this proposition: rays diverging from D are refracted so that they (virtually) intersect in S.25 This proof is similar to the one above. Finally, in the sixteenth proposition of Tractatus, Huygens determined the focal distance of a convex lens by combining the preceding results. It was equal to the theorem he put forward in his letters to Tacquet and Gutschoven. CD is a bi-convex lens with radii of curvature AC and BD (Figure 8). The foci for paraxial rays are respectively E and L. According to the eighth proposition CE : EA = DL : LB = n. With the twelfth proposition, AQ : QC

23

OC13, 40-79. OC13, 70-73. 25 OC13, 42-47. 24

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19

the ‘punctum concursus’ N for parallel rays from the direction of L is found with EL : ED = EB : EN. After the refraction at the surface C, the rays converge towards E; they are then refracted at the surface D towards N. In modern notation: DN

nAC ˜ BD  BC ˜ CD n 1 n( AC  BD )  ( n  1)CD

, where DN is the focal distance

measured from the anterior face of the lens. For rays coming from the other direction O is the ‘punctum concursus’.

Figure 8 Focal distance DN of a bi-convex lens

The case of a bi-convex lens was only one out of many cases Huygens treated in the fourteenth to seventeenth proposition of Tractatus. Taking both spherical aberration and the thickness of the lens into account, he derived exact theorems for the focal distance of each type of lens. In each case, he also showed how to simplify the theorem when the thickness of the lens is not taken into account. In the case of a bi-convex lens, he started by comparing the focal distances CO and DN when the radii of both sides of the lens are not equal. Their difference vanishes when the thickness of the lens CD is ignored and both refractions are assumed to take place simultaneously. The focal distance N is then easily found by first determining point L with DL : LB

= n and then AB : AD = DE : EA, or DN

2 AC ˜ BD 26 . In the case of AC  BD

a glass lens (n = 3 : 2) LB is twice BD and (AC + BD) : AC = 2BD : DN. It follows directly that the focal distance is equal to the radius in the case of an equi-convex lens. In the twentieth proposition of Tractatus, Huygens extended the results for thin lenses to non-parallel rays. In this case rays diverge from a point on the axis relatively close to the lens and are refracted towards a point P found by DO · DP = DC2 (DO is the focal distance for parallel rays coming from the opposite direction). Huygens had to treat all cases of positive and negative lens sides separately, but the result comes down to the modern formula 1  1 1 .27 p

p'

f

In the remainder of the first book of Tractutus, Huygens completed his theory of focal distances by determining the image of an extended object, rounded off in the twenty-fourth proposition (Figure 9). The diameter of the image IG is to the diameter of the object KF as the distance HL of the image 26

OC13, 88-89. Equivalent to the modern formula 1f

27

OC13, 98-109.

= (n -1)(

1 + R12 R1

).

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to the lens is to the distance EL of the object to the lens.28 The point L has a special property that Huygens had established in the preceding proposition. An arbitrary ray that passes through this point leaves the lens parallel to the incident ray.29 In the twenty-second proposition, Huygens had demonstrated that the focal distance LG of rays from a point K of the axis is more or less equal to that of a point E on the axis.30 The triangles KLF and GLI are therefore similar, which proves the theorem.31

Figure 9 Extended image.

Images

The theory of focal distances formed the basis of Huygens’ discussion of the properties of images formed by lenses and lens-systems in the second book of Tractatus. Huygens’ theory of images is once again both rigorous and general. The central questions in book two were how to determine the orientation of the image and the degree of magnification. For the time being, Huygens ignored the question whether an image is in focus. In this way he could derive general theorems on the relationship between the shape of lenses and their magnifying properties. He then showed how these reduced to simpler theorems in particular cases, for example for a distant object. In the third book he showed what configurations produced focused images.

Figure 10 Magnification by a convex lens.

In the second and third propositions of book two, Huygens discussed a convex lens. His aim was to determine the magnification of the image for the various positions of eye D, lens ACB and object MEN (Figure 10). In order to distinguish between upright and reversed images, Huygens defined the ‘punctum correspondens’ (later called ‘punctum dirigens’).32 This is the focus of rays emanating from the point where the eye is situated and is thus found by means of the theory of the first book. First, the eye D is between a convex 28

OC13, 122-125. OC13, 118-123. In modern terms, L is the optical center. 30 OC13, 114-119. 31 Huygens added that when the thickness of the less is taken into account, point V in the lens instead of L should be taken as the vertex of the triangle. 32 OC13, 176n1. 29

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lens ACB and its focus O. In this case, the object MEN is seen upright and magnified. The lens refracts a ray NBP to BD so that point N of the object is seen in B, whereas it would be seen in C without the lens. AB is larger than AC and on the same side of the axis. Huygens then showed that AB : AC = (AO : OD)·(ED : EP), which in the case of a distant object reduces to AB : AC = AO : OD.33 If, on the other hand, the eye is placed in the focus (so that AD = AO) and NB is taken parallel to the axis, AB : AC = EO : AO which becomes infinitely large when the object is placed at large distance. In the next proposition, Huygens considered the cases where the eye is placed beyond the focus O. In this case the ‘punctum correspondens’ P is on the other side of the lens and the image will be reversed when the object is placed beyond it. With the same degree of generality, in the fifth proposition Huygens discussed the images produced by a configuration of two lenses.34 He figured (Figure 11, most left one) two lenses A and B with focal distances GA and HB, the eye C and the object DEF, all arbitrarily positioned on a common axis. He then constructed point K on the axis, the ‘punctum correspondens’ of the eye with respect to lens B, the ocular lens. Next, he constructed point L on the axis, the ‘punctum correspondens’ of point K with respect to lens A, the objective lens. In this way, a ray LD will be refracted by the two lenses to the eye via points M on lens A and N on lens B. Without lenses, the eye sees point F of the object – where DE = DF – along line COF. The degree of magnification is therefore determined by the proportion BN : BO. In this general case, the magnification follows from BN : BO = (HB : HC)·(AG : GK)·(EC : EL). Huygens derived this proportion for the case of a concave ocular and a convex objective, but the same applied to a system of two convex lenses. In the adjoining drawings, Huygens sketched various positions of eye, lenses and object (Figure 11 gives four cases). These showed whether the image was upright or reversed. In addition, he showed how the general theorem reduced to a simpler one in particular cases. For example, when the ‘punctum correspondens’ of the ocular K and the focus of the objective G coincide, it reduces to (HB : HC)·(EC : AK). Likewise, the configurations used in practice were only a special case that Huygens discussed as he went along. If a concave ocular and a convex objective are positioned in such a way that BG = BH, where the ocular is between the objective and its focus, the magnification of a distant object is AG : BH. The same applies to two convex lenses that are positioned with their foci coinciding in between. In other words, the magnification is equal to the quotient of the focal distances of both lenses. In this roundabout way, Huygens proved what had been, and

33 34

OC13, 174-179. OC13, 186-197.

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Figure 11 Four of the cases discussed (additional lettering added).

continued to be, assumed for quite some time, as Molyneux was to remark in 1690.35 With the magnifying properties of lens-systems thus established in a most general way in parts one and two of Tractatus, Huygens’ subsequent account of actual telescopes came down to a rather straightforward application to a few specific cases. The state in which he left the third part of Tractatus in 1653 is hard to determine. It probably consisted of only two or three theorems. Huygens did not discuss optimal configurations of lenses in telescopes systematically, but only explained under what conditions ocular and objective produced sharp images. The solution was simple, as he stated in the first proposition. In order to see a sharp image, the rays from the 35

“This is the great Proposition asserted by most Dioptrick Writers, but hitherto proved by none (for as much as I know) …” Molyneux, Dioptrica nova, 161.

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object should leave the ocular parallel to the axis.36 The foci of the lenses should therefore coincide. For myopic people and those using a telescope to project images things are different. In these cases the rays should be brought to focus after they have passed the ocular and the foci of the lenses should not coincide. In the second proposition, Huygens discussed the configuration of two lenses required to project images and determined their magnification.37 Huygens aimed at providing a general and exact theory of the properties of lenses and their configurations. The generality of Huygens’ theory reached its high-point in a theorem that is inserted in part two of Tractatus as the sixth proposition. It may be of a later date, as the manuscript is on a different kind of paper and written with a different pen than the rest of this part.38 Nevertheless, the theorem states that the magnification of an arbitrary system of lenses remains the same when eye and object switch place.39 This theorem, so Huygens concluded his demonstration, would be useful in determining the magnification and distinctness of images.

Figure 12 Analysis of Keplerian telescope with erector lens. See also Figure 13.

Huygens applied the theorem in the third and fourth proposition included by the editors of Oeuvres Complètes in book three. The third proposition is certainly of a later date, as it analyses the eyepiece Huygens invented in 1662.40 The fourth proposition discusses a configuration of three convex lenses proposed by Kepler in 1611 (Figure 12).41 A telescope of two convex lenses ordinarily produces a reversed image, but a third lens inserted between the ocular and the objective may re-erect the image. Huygens explained that an upright and sharp image is attained as follows (Figure 13).42 AC is the focal distance of the objective lens YAB, and HF the focal distance of the ocular QHR. The third lens DET is identical with the ocular with a focal distance EL = HF. It is placed so that EC = 2EL and EH = 3EL. In this case, point C on the axis is the ‘punctum correspondens’ for rays through focus F of the ocular. Therefore a ray from S at a large distance is refracted by the lenses in such a way that it leaves the ocular parallel to the axis towards the eye PN. In order to determine the magnification by the 36

OC13, 244-247. OC13, 246-253. 38 Hug29, 151-167. 39 OC13, 198-199. 40 OC13, 252n1. See below, section 3.1.2. 41 Dating this theorem is difficult. It may have been written in 1653, as the configuration was well-known. Yet, Huygens also discussed the enlarged field of such a configuration, which may imply that it is of a later date. See note 20 on page 16 above. 42 OC13, 258-261. 37

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system, Huygens applied proposition six of book two. The eye is imagined at S and the object at PN. In this way the magnification is Figure 13 Diagram for the analysis in Figure 12. determined by the proportion YB : PN. It easily follows that this proportion is equal to AC : EL, the proportion of the focal distances of the objective and the ocular. Conclusion In Tractatus, Huygens addressed a specific question: how can the working of the telescope be understood mathematically? Regarding thin glass lenses his answers, as we shall see in the next section, were not that new. Yet, he had arrived at these answers by way of a rigorous mathematical analysis of the properties of lenses. With the sine law, Huygens derived general and exact theorems regarding the focal distances of thick lenses for both parallel and non-parallel rays, irrespective of the material lenses are made of. On the basis of this exact theory, he showed that these theorems reduce to the familiar, simpler ones when the thickness of the lens is ignored and a specific index of refraction is chosen. In the same way, he first established a general theorem regarding the magnification by a lens-system and then showed that, in the cases of actual telescopes, it reduced to the simple and familiar one. If the elaboration of the theory of Tractatus was markedly mathematical, its rationale was the telescope. Its goal was a ‘theory of the telescope’: an account of the working of the telescope on the basis of dioptrical theory. In this sense, the theory of the first two books was almost too elaborate. All in all, in his Tractatus, Huygens gave a rigorous answer to the question how the working of the telescope can be understood mathematically. Huygens was the first one to elaborate a theory of the teleoscope by means of the exact law of refraction. He knew that his treatise would fill gaps left by others, in particular Descartes, so we would expect him to publish it soon. However, as contrasted to other mathematical treatises he published in this period, he did not press ahead with Tractatus. He inquired with publishers and Van Schooten even proposed to append Huygens’ treatise to a Latin edition of Descartes’ Discours de la Methode, La Dioptrique and Les Météores, but nothing came of it.43 Despite repeated announcements between 1655 and 1665 that he was publishing Tractatus, Huygens never did.44 2.2 Dioptrics and the telescope The orientation on the telescope is essential to Tractatus. If Huygens was the first to apply the sine law to questions regarding the telescope, what had other students of dioptrics been doing? In this section, I sketch the 43 44

OC1, 280; 301-303; 321-322. Huygens did not pin much faith in Van Schooten’s proposal. I will say a bit more about his publishing pattern on page 174.

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development of seventeenth-century dioptrics, with a particular emphasis on the way questions regarding the telescope were addressed. The telescope was made public when in September 1608 a spectacle maker from Middelburg, Hans Lipperhey, came to The Hague to request a patent for a “… certain device by means of which all things at a very great distance can be seen as if they were nearby, …”45 It was a configuration of a convex and a concave lens fitted appropriately in a tube and turned out to magnify things seen through it. The patent was denied, as within a couple of week two other claimants turned up. It is doubtful whether Lipperhey had made the invention himself. He may have learned it from his neighbour Sacharias Janssen, who in his turn seems to have learned the secret of the device from an itinerant Italian.46 The history of the invention of the telescope is an intricate one, in which Jacob Metius of Alkmaar was the first to be publicly named its true inventor by Descartes. The first doubts were raised in the 1650s through the publication of Pierre Borel. Huygens himself was one of the first to perform some archival research on the matter, claiming that the credit should go to either Lipperhey or Janssen.47 The news of the device spread quickly through Europe and by the summer of 1609 simple telescopes were commonly for sale in the major cities of Europe.48 The news also reached the ears of scholars, who realized the device could be of use in astronomical observation. Most successful among them was Galileo in Venice, whose interest in the telescope was aroused in the spring of 1609. He figured out how to make one and how to improve it. Among the earliest telescopists, Galileo was the only one who not only knew how the telescope could be improved, but also had the means to do so. In August, he had made a telescope that magnified nine times, as opposed to the ordinary three-powered spyglasses. A couple of months later he had made telescopes that were even more powerful.49 In this way, Galileo turned the spyglass into a powerful instrument of astronomical observation.50 He observed the heavens and saw spectacular things: mountains on the Moon, satellites around Jupiter, and more. In March 1610, he published his observations in Sidereus nuncius. Galileo also sent a copy to the Prague court with a specific request for a comment by Kepler.51 In May, Kepler published his comment in Dissertatio cum nuncio sidereo. He primarily responded to Galileo’s observations, but he also said a few things about the instrument. In Sidereus nuncius, Galileo had explained its construction and use, but he had left out any mathematical account.52 In 45

Van Helden, Invention, 35-36; Galileo, Sidereus nuncius, 3-4 (Van Helden’s introduction). De Waard, Uitvinding, 105-225; Van Helden, Invention, 20-25. 47 OC13, 436-437. 48 Van Helden, Invention, 21, 36. 49 Van Helden, Invention, 26; Galileo, Sidereus nuncius, 6, 9 (Van Helden’s Introduction). 50 Van Helden, “Galileo and the telescope”, 153-157. 51 Galileo, Sidereus nuncius, 94 (Van Helden’s Conclusion). 52 Galileo, Sidereus nuncius, 37-39. 46

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reply, Kepler briefly explained how lenses refract rays of light so that they can produced magnified images.53 The explanation in Dissertatio was only a sketch, but the message was clear. The telescope was a remarkable invention, but its working needed mathematical clarification. A theory of the telescope was called for. Within a few months, Kepler developed one. In September 1610, he finished the manuscript of Dioptrice, published the next year. “Some have disputed over the priority of its invention, others rather applied themselves to the perfection of the instrument, as there chance mainly counted, here reason dominated. But Galileo scored the greatest triumph by exploring its use to disclose secrets, because zeal procured him with the design and fortune has not withheld him the success. I, driven by an honest emulation, have shown the mathematicians a new field to expose their acuteness, in which the causes and principles are retraced to the laws of geometry, the effects of which are so awaited with much impatience and are of such pleasing diversity.”54

The goal of Dioptrice was to provide a mathematical account of the working of the telescope. In Kepler’s view, the working of any instrument used in astronomy should be understood precisely. A decade earlier, he had approached the puzzling properties of the pinhole images used in the observation of solar eclipses. His answer had been a new theory of image formation, which he had published in Paralipomena (1604). In Dioptrice, Kepler applied this theory to lenses in order to determine the dioptrical properties of the telescope. Dioptrice had one substantial shortcoming: Kepler knew that he did not know the exact law of refraction. He used an approximate rule instead. 2.2.1 KEPLER AND THE MATHEMATICS OF LENSES Kepler’s concerns in Paralipomena were induced by a problem of astronomical observation. In 1598, Tycho Brahe had reported an anomalous observation of the apparent size of the moon during a solar eclipse.55 Brahe used a pinhole to project the image of the eclipsed sun. When he measured the diameter of the projection he realized that “the moon during a solar eclipse does not appear to be the same size as it appears at other times during full moons when it is equally far away”.56 He tried to produce consistent values by applying some ad hoc corrections to his measurements.57 Kepler took a different approach, analyzing mathematically the way the image was produced. He had known the anomaly of pinhole images for some time 53

Kepler, Conversation, [19-21]. Kepler, Dioptrice, dedication (KGW4, 331). “… circaque eam alij de palma primae inventionis certarent, alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est causarum lege geometrica demonstrandarum, quibus tam exoptati, tam jucunda varietate multiplices effectus inniterentur.” 55 Straker, “Kepler’s theory of pinhole images”, 276-278. 56 Cited and translated in: Straker, “Kepler’s theory of pinhole images”, 278. 57 Straker, “Kepler’s theory of pinhole images”, 275-276; 280-282. 54

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when in 1600 he set himself to see whether an ‘optical cause’ might account for it. The solution to the apparent anomaly of pinhole projections of solar eclipses would be the copestone of Paralipomena. I will only discuss Kepler’s theory of image formation and its application to the eye. The optical theory available around 1600 was the medieval tradition of perspectiva and it did not provide Kepler with an answer for the anomaly of pinhole observations. Perspectiva built on the great synthetic work from the eleventh century of the Arab mathematician Alhacen, when it was adopted by a line of thirteenth-century Christian mathematicians, Bacon, Witelo and Pecham. They elaborated a mathematical theory of optics, in addition to the natural philosophical and medical theories, in which vision was analyzed in terms of the behavior of light rays.58 The designation ‘perspectiva’ derives from the common title for their works and it constituted the canon of mathematical optics well into the seventeenth century. In the sixteenth century perspectiva texts had been published, with the 1672 edition by Friedrich Risner of Alhacen’s Optica and Witelo’s Perspectiva as the most important.59 The problem of pinhole images was well-known in perspectiva. It was known since Antiquity that the image of the sun, projected by a square aperture, can still be round. This seemed to contradict the basic principle of optics: the rectilinearity of light rays. The solutions given by perspectivist writers did not satisfy Kepler. Each had in the end sacrificed the principle of rectilinearity – the foundation of geometrical optics.60 Kepler had to resolve the problem by himself. His solution consisted of a new theory of the way rays form images of objects. This theory, in its turn, would be the foundation of his dioptrics as well as of seventeenth-century geometrical optics in general. Kepler approached the problem anew and did so by uncompromisingly applying the principle of rectilinearity. In Paralipomena, he describes how he replaced a ray of light by a thread. He took a book, attached a thread to one of its corner and guided it along the edges of a many-cornered aperture, thus tracing out the figure of the aperture. Repeating this for the other corners of the book, and many more points, he ended up with a multitude of Figure 14 Kepler’s solution to overlapping figures that formed an image of the pinhole problem 58

Further discussed in section 4.1.1. Dupré points out Risner’s programmatic discussion of the science of optics in the preface to the edition which constitute an important, yet still little studied, agenda for seventeenth-century optics. Dupré, Galileo, the Telescope, 54. 60 See Lindberg, “Laying the foundations”, 14-29. 59

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the book. In the same way, he argued, all the points of the sun project overlapping images of the aperture (Figure 14). The resulting image has the shape of the sun, albeit with a blurred edge. In the projection of an eclipse, the image of the shadow of the moon is partially overlapped by the image of the sun. Consequently, the diameter of the moon seems too small. In chapter two of Paralipomena, Kepler had solved the apparent anomaly of pinhole observations in principle, building on the previous chapter, he elaborated the exact solution in the eleventh and final chapter. Image formation Kepler came to the conclusion that there were more problems in perspectiva, in particular its core, the theory of vision. In chapter five of Paralipomena, he elaborated a new theory of vision on the basis of his newly gained understanding of image formation. In its fourth section, Kepler listed the defects of existing theories of vision, the most important being a wrong understanding of the anatomy of the eye and of the mathematics of image formation. Perspectivist theories considered the lens the sensitive organ of the eye, whereas recent anatomical investigations had demonstrated, convincingly according to Kepler, that the retina receives images from objects. He himself had shown the defects of the perspectivist understanding of image formation, calling Witelo by name, and he now went on to reconsider the optics of the eye. In perspectivist theory, each point of an object emits rays of light in each direction. This, however, raises the problem how a sharp image can be perceived, that is: how a one-to-one relationship between a point of the object and a point of the image in the eye is established. According to Alhacen there can be only one point in the eye where a ray from a point of the object can be perceived. He stated that this must be the one entering the eye perpendicularly (and thus perpendicular to the lens). He explained that the other rays are refracted by the eye, therefore weakened, and thus do not partake in the formation of the image.61 In medieval optics, images were therefore taken to be produced by single rays from each point of the object. Kepler saw no reason to differentiate between weak and strong rays. He did not see, for that matter, why refraction would weaken a ray. In his view, all rays emitted by a point should somehow partake in the formation of an image. In the case of pinholes this resulted in a fuzzy image, but what about the sharp images by which we generally see the world? Kepler’s answer was that the cone of rays coming from one point is somehow brought to focus on the retina. Following certain recent anatomical observations he considered the retina as the sensitive organ of the eye, in contrast to perspectivist theory that had assigned the power of visual perception to the crystalline humor. According to Kepler, the various humors of the eye can be regarded as one refracting sphere. In the fifth 61

Alhacen, Optics I, 68 (book 1, section 17) and 77 (book 1, section 46).

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chapter of Paralipomena, Kepler explained how images are formed on the retina. In order to account for spherical aberration, he argued that the pupil as well a the slightly a-spherical shape of the posterior side of the humors diminish the severest aberrations. Kepler’s analysis was based on his study of refraction in the fourth chapter of Paralipomena. In this chapter, he had tried unsuccessfully to find an exact law of refraction, but his understanding of refraction at plane surface sufficed for discussing the focusing properties of spheres at least qualitatively.62 With this Kepler completed his theory of image formation. It had originated in the solution of an anomalous astronomical observation and its ultimate rationale was astronomical observation. With his definition of optics and its indispensability to cosmology, Kepler fits in a Ramist trend in the sixteenth century that Dupré refers to with ‘the art of seeing well’ and to which Risner also belongs.63 The full title of Paralipomena starts with Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur, …. In his preface, Kepler proclaimed eclipses to be the most noble and ancient part of astronomy: “… these darknesses are the astronomers’ eyes, the defects are a cornucopia of theory, these blemishes illuminate the minds of mortals with the most precious pictures.”64 The eye being the fundamental instrument of observation, to Kepler a reliable theory of visual perception was indispensable for astronomers. His perspectivist forebears had not treated the matter satisfactorily and thus he had provided the necessary additions to Witelo. Revolutionary additions, to be sure. The eye perceives dots rather than things and in the analysis of vision “… we should not look to entire objects, but to individual points of objects, …”65 Kepler had made it clear that all rays from an object point partake in the formation of images, whose sharpness is not evident beforehand. Image formation was no longer a matter of tracing individual rays from object to image. The task of the optician now became to determine exactly how a bundle of rays is brought to focus again after it is emitted by a point of an object. Lenses Kepler approached the newly invented telescope in the same manner as the pinhole and the eye. The working of the telescope should be properly understood if it were to be used in astronomical observation. For Kepler, this meant that a mathematical theory was required, a mathematical theory of the telescope so to say. He had already treated lenses briefly in the final proposition of chapter five of Paralipomena. At that moment spectacle glasses were new topic in optical literature. Kepler expressed his amazement that no 62

Kepler’s efforts to find a law of refraction are discussed below, in section 4.1.2. Dupré, Galileo, the telescope, 31. 64 Kepler, Paralipomena, 4 (KGW2, 16). “… hae t e n e b r a e sint Astronomorum o c u l i , hi d e f e c t u s doctrinae sint a b u n d a n t i a , hi n a e v i mentes mortalium preciosissimis p i c t u r i s illustrent.” Translation Donahue, Optics, 16. 65 Kepler, Paralipomena, 201 (KGW2, 181). “Itaque non oportet nos ad res totas respicere, sed ad rerum singular puncta, …” Translation Donahue, Optics, 217. 63

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mathematical account of such an important and widespread device existed. We can understand his surprise, for spectacles had already been invented around 1300.66 A brief account by Francesco Maurolyco, that dated back to around 1521, was not to be published before 1611, in Diaphaneon seu transparentium libellus. Kepler would not have found much in it to his liking, for it was a qualitative theory based on a somewhat confusing variant of the perspectivist theory of vision and refraction.67 Kepler knew that Della Porta had written a study of refraction, but he had not been able to lay hands on De refractione (1593). He dismissed what Della Porta had written in Magia naturalis, namely that spectacles correct vision because they magnify images. Kepler elaborated his own account of lenses, dedicating it in Paralipomena to his patron Ludwig von Dietrichstein, whom he said had kept him busy for three years with the question of the secret of spectacles.68 Kepler explained the beneficial effects of spectacles as follows. Myopic and presbyotic vision occurs when rays are not brought to focus on the retina but in front of it or beyond. He gave a short, qualitative discussion of the effect of lenses on a bundle of parallel rays coming from a distant point. Convex and concave lenses – for myopics and presbyotics respectively – move the focus of rays to the retina. Some magnification may occur, but this is not the reason why spectacle lenses correct vision. With the introduction of the telescope in astronomy, the qualitative account of single lenses in Paralipomena did not suffice any more. In Dioptrice, Kepler extended his theory of image formation to a quantitative analysis of the properties of lenses and their configurations.69 As a matter of fact, he was the one to coin the term ‘dioptrics’.70 Compared to Huygens’ Tractatus, Kepler’s dioptrical theory was of more limited scope. His goal was to explain the formation of images by a telescope. He therefore restricted his theory to a few types of lenses and mainly confined himself to object points at infinite distance when incident rays are parallel. The basic concept was the focus of a lens, the point where parallel rays intersect after refraction. Kepler could not determine the focal distance with the exactness we have seen with Huygens. He could not, for example, determine the exact route of a ray through the refractions at both surfaces of a lens. The main obstacle in the way of a more extensive treatment was the fact that Kepler did not know the exact law of refraction. In Dioptrice, he used an approximation that was valid only for angles of incidence below 30º, and that, even so, applied solely to glass. According to this rule the angle of deviation is one third of the angle of

66

Rosen, “The invention of eyeglasses”, 13-46. Lindberg, “Optics in 16th century Italy”136-141. Maurolyco had preceded Kepler in his analysis of the pinhole image: Lindberg, “Optics in 16th century Italy”, 132-135; Lindberg, “Laying the foundations”. 68 Kepler, Paralipomena, 200-202 (KGW2, 181-183). 69 Malet, “Kepler and the telescope” offers a detailed discussion of Dioptrice, without however presenting it as a part of the ‘optical part of astronomy’. 70 Kepler, Dioptrice, dedication (KGW4, 331). 67

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incidence; the angle between the incident ray, produced beyond the refracting surface, and the refracted ray. Kepler began with a discussion of the focal distances of planoconvex lenses (Figure 15). A ray HG is incident on a convex surface with radius AC, the angle of incidence is GAC. As the angle of Figure 15 Focal distance of a plano-convex lens deviation is one third of this, HG will be refracted towards F, with AC : AF = 1 : 2.71 The focal distance is therefore approximately three times the radius of the convex face. Analogously, he argued that the focal distance of a plano-convex lens, the plane face turned towards the incident rays, is approximately twice the radius of curvature. For other cases Kepler established only rough estimations. If convergent rays are incident on the plane side of a plano-convex lens, the refracted rays intersect the axis within the focal distance. Combining these three theorems, Kepler showed that the focal distance of a bi-convex lens is both smaller than three times the radius of the anterior side and twice the radius of the posterior side. In the case of an equi-convex lens, this comes down to a focal distance approximately equal to the radius of its sides.72 Kepler did not determine the focal distance of a concave lens, he only showed that rays diverge after refraction.73 On this basis, the properties of images formed by lenses are easily found. The image DBF of an extended object CAE through a bi-convex lens GH is formed at focal distance (Figure 16). The picture is inversed as the rays from C are refracted towards D, etcetera. As the focal distance is roughly the radius of any side, the magnitudes of object and image will be in a proportion equal to their respective distances to the lens.74 In Dioptrice, Kepler briefly reiterated his theory of vision. On the one hand, so he said in the dedication, he did so for the sake of completeness, on the other hand because some readers had trouble understanding his account in Paralipomena.75 He explained that a perfectly focusing surface was not spherical, but should be hyperbolic, like the crystalline humor of the eye was.76 On the basis of his theory of the retinal image, he explained the effect of a lens placed before the eye once more. Depending upon the position of the eye with respect to the focal distance, the object will be perceived sharply.77 When the eye is placed not too far from the focus, a magnified image will be perceived.

71

Kepler, Dioptrice, 11 (KGW4, 363). Kepler, Dioptrice, 12-15 (KGW4, 363-367). 73 Kepler, Dioptrice, 45-49 (KGW4, 388-393). 74 Kepler, Dioptrice, 16-18 (KGW4, 367-369). 75 Kepler, Dioptrice, dedication (KGW4, 335). 76 Kepler, Dioptrice, 21-24 (KGW4, 371-372). 77 Kepler, Dioptrice, 35-42 (KGW4, 381-387). 72

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Kepler proceeded to discuss the combination of two convex lenses. He explained how these should be configured in order to perceive a sharp, magnified image.78 This is achieved when the foci of both lenses coincide. It is remarkable that that Kepler discussed the configuration of two convex lenses, because in 1611 only the combination of a convex objective and a concave ocular was known to produce a telescopic effect. Kepler probably arrived at this alternative configuration by theoretical considerations.79 He never manufactured this kind of telescope himself. The configuration has come to be known as a Keplerian or Astronomical telescope, as opposed to the Dutch or Galilean telescope with a concave ocular. Much later it became clear that the Keplerian type has the advantage of a larger field of view, but Kepler himself did not know this. He did realize that this configuration had a drawback, it produced inverted images. This could be corrected, he said, by inserting a third lens at an appropriate place between ocular and objective lens. Kepler then turned to an account of concave lenses and finally to a discussion of so-called Dutch Figure 16 Image formation by a lens telescopes. He explained the configuration of a convex objective and a concave ocular only in broad lines. As he did not speak of the focus of a concave lens, he could only roughly point out where the ocular should be placed with respect to the focus of the objective. His discussion of the configurations and the resulting properties of the images remained mainly qualitative. Kepler offered a wealth of practical guidelines as to the configurations of lenses and the way the best effects are achieved. Dioptrice does not consist of rigorously demonstrated theorems. Without an exact law of refraction, a quantitative and exact theory could hardly be attained. This was not necessarily Kepler’s intention. Rather, he intended to explain the working of the telescope mathematically. He did so by analyzing, on the basis of his theory of image formation, how it forms magnified images. All this may lead us to conclude that Huygens’ Tractatus can be seen as an up-to-date answer to the question Kepler had originally addressed in Dioptrice; updated in the sense that the analysis of lenses was based on the sine law. It established the dioptrical properties of spherical lenses and 78

Kepler, Dioptrice, 42-43 (KGW4, 387-388). A possible source of inspiration may have come from the analogous configuration of the eye and a convex spectacle glass, as the eye acts as a convex lens does. See also Malet, “Kepler and the telescope”, 119-120. 79

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focused on problems pertaining to their configurations in actual telescopes. Like Kepler, Huygens intended to found the dioptrical properties on a sound mathematical basis. Whether a continuation of Dioptrice was his actual goal, can only be surmised as he did not explicitly refer to it in such a programmatic sense. Huygens did know Dioptrice, it had been on the reading list of his mathematics tutor Stampioen and much later he commended it to his brother Constantijn as the best introduction to dioptrics.80 Huygens did not have much to offer that was not already known. Tractatus covered more types of lenses but the eventual results regarding the focal distances of lenses and the magnifying properties did not differ much from Dioptrice. The crucial difference is that Huygens founded his results on a general and exact theory of focal distances. It rigorously proved Kepler’s results. He had the exact law of refraction at his disposal and thus could be exact where Kepler necessarily had to leave his readers with approximate answers. Perspectiva and the telescope At the same time when Kepler wrote Dioptrice, two other scholars devised an account of the telescope. Della Porta’s ‘De telescopio’ remained unpublished, De Domini’s De Radiis Visus et Lucis was published in 1611. Both were based on perspectivist theory of image formation. Before I go on to discuss the impact of the sine law on dioptrics, I briefly discuss these in order to make it clear why that perspectivist theory was intrinsically inadequate to account fully for the effect of lenses. Shortly before his death, Della Porta extended his theory of lenses of De Refractione to telescopes in a manuscript ‘De telescopio’.81 It reveals the problems lenses posed for perspectivist theory of image formation. In order to determine the place where an object is seen, perspectiva used the cathetus rule. The cathetus is the line through the object point, perpendicular to the reflecting or refracting surface. The cathetus rule states that the image is the intersection of the ray entering the eye and the cathetus. Modern Keplerian theory shows that, although valid in many cases, this rule turns out to break down for curved surfaces in particular. To account for images of lenses another problem turns up. As a lens refracts a ray twice, this seems to imply that the rule has to be applied twice also. Della Porta avoided this problem by considering only one cathetus. Della Porta considered a lens in terms of refracting spheres, as he had done in De Refractione (Figure 17). The dotted lines indicate such spheres and the lens dcgf is formed by their overlap.82 The object ab is perceived as follows: a ray from point a is refracted along cd to the eye. Della Porta drew the cathetus ka of the lower surface of the lens, which also is its radius. When produced, the ray entering the eye intersects the cathetus in point h, 80

OC1, 6 (Stampioen’s list of recommended readings spans pages 5-10) and OC6, 215. Della Porta’s account of refraction by spheres and lenses in De refractione is discussed in Lindberg, “Optics in 16th century Italy”, 143-146. 82 Della Porta, De Telescopio, 113-114. 81

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where point a is seen. In the same manner point i is constructed and hi is the object as perceived through the lens. The question is why only the ray acd emanating from point a is singled out. Della Porta seemed to assume that this is the one that enters the eye perpendicularly. Yet, in the case of a distant object, he no longer chooses rays parallel to the axis of the system, but the crossing rays ad and bg (Figure 18). He probably did so to account for the reversing of the image, but he lacked a theoretical justification. Della Porta’s account of concave lenses was even more troublesome, as he ignored the implication of his reasoning that the eye cannot perceive the whole object at once. Moreover, he persistently has rays refracted from the perpendicular at the first surface (for example de in Figure 19).83 ‘De telescopio’ culminated in an account of a Galilean telescope (Figure 19). Della Porta traced the path of a ray emanation from point a of the object. He then chose the cathetus with respect to the upper surface of the concave lens and argued that qr is the image perceived.

Figure 17 Image of a near object

Figure 18 Image of distant object

Figure 19 Image by a telescope

All in all, from Kepler’s perspective Della Porta’s theory of lenses was fraught with difficulties and mathematically it was riddled with ambiguities. Part of these arose from his sloppiness and lack of understanding of certain problems. Part of the problem lies also with the perspectivist foundation of his account. How, for example, should the cathetus rule be applied to two or more refractions? More important, perspectivist theory offers no means of differentiating between sharp and fuzzy images, quite a relevant issue with respect to the telescope.84 Della Porta made no attempt to deal with it. Whether he chose ignore it or wass unaware of it is unclear. He was quite content with what he had written. As the inventor of the telescope – so he

83 84

Della Porta, De telescopio, 141-142. Compare Lindberg, “Optics in 16th century Italy”, 146-147.

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fancied – he regarded himself as the only authority in these matters.85 Shortly after he wrote ‘De telescopio’ he died, and the text remained unknown until 1940. De Radiis Visus et Lucis of De Dominis is well-known for its discussion of the rainbow, but it also contains an account of lenses and the telescope. Like Della Porta, De Dominis maintained perspectivist theory. His theory did not go beyond a brief, qualitative theory of the refraction of visual rays by lenses. In this way it avoided the problems revealed by Della Porta’s theory. It does not seem to have counted as a serious alternative to Dioptrice. Unlike Kepler, De Dominis was rarely referred to in matters dioptrical. In the widely read Rosa Ursina (1630), Scheiner adopted Kepler’s theory of image formation. He elaborately treated the construction and use of telescopes. Scheiner discussed the properties of lenses and their configurations, but he did not incorporate the quantitative part of Dioptrice – his account remained qualitative. Another authoritative book on geometrical optics at the time, Opticorum Libri Sex (1611) by the Antwerp mathematician Aguilón, did not discuss refraction or lenses at all. Dioptrice had been a reaction to Galileo’s neglect to explain the telescope dioptrically in Sidereus Nuncius. Although quite an able mathematician, Galileo never developed a theory of dioptrics. He applied himself to the improvement of the instrument by making better lenses and optimizing the quality of telescopic images. His friend Sagredo did take an interest in the dioptrics of lenses, but was not encouraged to pursue his study. Galileo wanted him to concentrate on matters of glass-making and lens-grinding.86 On Dioptrice Galileo kept silent altogether.87 Apparently, this self-styled mathematical philosopher was not interested in the mathematical properties of the instrument that had brought him fame. He did, however, have a clear understanding of the working of lenses and telescopes. Dupré has recently argued that Galileo relied on a tradition of practical knowledge, of mirrors in particular, that had developed in the sixteenth century next to the mathematical strand on which my account focuses.88 2.2.2 THE USE OF THE SINE LAW The exact law of refraction Kepler had to make do without, was soon found. More than that, it had been within his reach. The English astronomer Thomas Harriot had discovered it in 1601. After the publication of Paralipomena, he and Kepler had corresponded on optical matters. However, the correspondence broke off before Harriot had revealed his discovery.89 Long before that, but unknown until the late twentieth century, the tenth85

Ronchi, “Refractione au Telescopio”, 56 and 34. “They know nothing of perspective.” and “... and it pleases me that the idea of the telescope in a tube has been mine; ...” 86 Pedersen, “Sagredo’s optical researches”, 144-148. 87 KGW4, “Nachbericht”, 476. 88 Dupré, Galileo, the Telescope, chapters 4 to 6 in particular. 89 Harriot is discussed in section 4.1.2.

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century student of burning glasses Ibn Sahl had used a rule equivalent to the sine law.90 Around 1620, the Leiden professor of mathematics Willebrord Snel was next and in the late 1620s Descartes closed the ranks of discoverers of the law of refraction.91 He published it in La Dioptrique (1637), shortly after Pierre Hérigone had done so in the fifth volume of Cursus Mathematicus. Hérigone did not use it in his dioptrical account, which summarized Dioptrice. Harriot and Snel have left no trace of applying their find to lenses. Which leaves La Dioptrique for further inspection. Descartes and the ideal telescope La Dioptrique was the fruit of Descartes’ involvement in the activities of Parisian savants regarding (non-spherical) mirrors and lenses, which also places him in the sixteenth-century tradition of mirror-making.92 Descartes, however, added his natural philosophical leanings and Kepler’s optical teachings. In collaboration with the mathematician Mydorge and the artisan Ferrier, he allegedly managed to produce a hyperbolic lens and in the course of events he discovered the law of refraction. La Dioptrique had much influence on seventeenth-century optics, especially through its second discourse where Descartes derived the sine law.93 In the following discourses, Descartes first discussed the eye and vision – summarizing Kepler’s theory of the retinal image – and then went on to a consideration “Of the means of perfecting vision”.94 This seventh discourse anticipated his discussion of telescopes. He laid stress on the way spectacles enhance vision, instead of correcting it. The telescope itself was introduced in a peculiar way. Descartes explained how an elongated lens may further enhance vision. He then replaced the solid middle part by air, thus arriving at a telescope consisting of two lenses.95 The argument was clear, but the discussion of focal and magnifying properties of lenses was entirely qualitative and the sine law played no role in it. In the eighth discourse of La Dioptrique, Descartes applied the sine law to lenses under the title: “Of the figures transparent bodies must have to divert the rays by refraction in all manners that serve vision”.96 Its sole purpose was to show that lenses ought to have an elliptic or hyperbolic surface in order to bring rays to a perfect focus. Avoiding the subtleties of geometry he explained how these lines could be drawn by practical means and demonstrated the relevant properties of the ellipse and hyperbola. As regards the focal distances of lenses thus obtained with respect to configuration and 90

Rashed, “Pioneer”, 478-486. For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250. 92 Gaukroger, Descartes, 138-146. Dupré, Galileo, the Telescope, 53-54. 93 Discussed in section 4.1.3 94 Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.” 95 Descartes, AT6, 155-160. 96 Descartes, AT6, 165. “Des figures que doivent avoir les corps transparens pour detourner les rayons par refraction en toutes les façons qui servent a la veuë” 91

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magnification, the account remained qualitative. La Dioptrique was written, Descartes said in the opening discourse, for the benefit of craftsmen who would have to grind and apply his elliptic and hyperbolic lenses. Therefore the mathematical content was kept to a minimum.97 Apparently this implied that Descartes need not elaborate a theory of the dioptrical properties of lenses. Descartes adopted the term Kepler had coined for the mathematical study of lenses. He had not, however, adopted the spirit of Kepler’s study. Dioptrice and La Dioptrique approached the telescope from opposite directions. Kepler had discussed actual telescopes and drudged on properties of lenses that did not fit mathematics so neatly. Descartes prescribed what the telescope should be according to mathematical theory. The telescope, having been invented and thus far cultivated by experience and fortune, could now reach a state of perfection by explaining its difficulties.98 Huygens was harsh in his judgment of La Dioptrique. In 1693, he wrote: “Monsieur Descartes did not know what would be the effect of his hyperbolic telescopes, and assumed incomparably more about it than he should have. He did not understand sufficiently the theory of dioptrics, as his poor build-up demonstration of the telescope reveals.”99

We can say that Descartes, according to Huygens, had failed to develop a theory of the telescope. He had ignored the questions that really mattered according to Huygens: an exact theory of the dioptrical properties of lenses and their configurations. La Dioptrique glanced over a telescope that existed only in the ideal world of mathematics. Unfortunately for Descartes, no one during the following decades succeeded in actually grinding the a-spherical lenses of his design. Of mere anecdotal interest is the irony with which Huygens’ tutor Stampioen had in 1640 pointed out to Descartes’ yet unfulfilled promise of a perfect telescope: “… my servant Research will turn him a better spyglass without circles … But nevertheless, what this Mathematicien has promised to do for six years is still not satisfied.”100 But Stampioen was in the middle of a terrible dispute with Descartes at that moment. After Descartes Hobbes, Descartes’ most ardent rival in matters of mechanistic philosophy, developed an alternative derivation of the sine law too. In the elaboration of his dioptrical theory he also discussed spherical lenses. The unpublished “A 97

Descartes, AT6, 82-83. Ribe, “Cartesian optics” offers an enlightening account of the artisanal roots of La Dioptrique. 98 Descartes, AT6, 82. 99 OC10, 402-403. “Mr. des Cartes n’a connu quel seroit l’effet de ses Lunettes hyperboliques, et en a presumè incomparablement plus qu’il ne devoit. n’entendant pas assez cette Theorie de la dioptrique, ce qui paroit par sa demonstration très mal bastie des Telescopes.” 100 Stampioen, Wis-konstigh ende reden-maetigh bewys, 58. “… mijn Knecht Ondersoeck sal hem eens een beter Verre-kijcker sonder cirkeltjes daer toe weten te drayen : … Maer niettemin ’t geen dese Mathematicien al over 6 Iaren belooft heeft te doen, blijft nog on-vol-daen.”

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minute or First Draught of the Optiques” of 1646 (the most complete elaboration of his optics) included several chapters on lenses and telescopes.101 He did not, however, make the most of his knowledge of the sine law. The account consisted of qualitative theorems – without proof and often dubious – which applied mostly to single rays refracted by lenses. Despite the presence of an exact law of refraction, Hobbes’ account (if published) would have been no match for Dioptrice. With the exact law of refraction established and published, the road might seem open for a follow-up of Dioptrice in the form of an exact theory of the dioptrical properties of spherical lenses. It was not to be, for various reasons. First of all the sine law became generally known and accepted only around 1660.102 This delay may have been caused by a slow distribution of Descartes’ works – and this maybe partly because La Dioptrique was written in French – or the bad odor his ideas were in. As late as 1663, in Optica promota, Gregory showed that the ellipse and hyperbola are aplanatic without using the sine law. In 1647, Cavalieri extended the theory of Dioptrice to some more types of lenses, using Kepler’s original rule. As the title Exercitationes geometricae sex suggests, this was an exercise in mathematics not aimed at furthering the understanding of the telescope. In this regard, Cavalieri was not an exception. Further, and more importantly, mathematicians addressed questions raised in Kepler’s Paralipomena rather than in his Dioptrice, to wit abstract optical imagery pertaining to Kepler’s theory of image formation, and the ‘anaclastic’ problem that had been put in a different light by that theory. The anaclastic problem, or ‘Alhacen’s problem’, is closely related to the determination of aplanatic surfaces: to find the point of reflection or refraction of a ray passing from a given point to another.103 When all rays are considered, as is relevant in Kepler’s theory of image formation, to find these points means determining the aplanatic surface. In this theory images are formed by the focusing of bundles of rays, and in most cases of reflection and refraction the image of a point source will not be a point. The properties of these images became an important subject of study in seventeenth-century geometrical optics. In Optica promota, Gregory extended the theory of Paralipomena with his contributions to the theory of optical imagery and his determination of aplanatic surfaces. From this viewpoint, La Dioptrique embroidered on Paralipomena rather than Dioptrice. The seventeenth-century study of these topics reached its highpoint in the lectures Barrow and, later, Newton delivered at the university of Cambridge. Barrow’s lectures were published in 1669, those of Newton remained unpublished during his lifetime. With Huygens’ dioptrical work 101

Stroud, Minute, 20; Prins, “Hobbes on light and vision”, 129-132. On Hobbes’ derivation of the sine law, see section 5.2.1. 102 Lohne, ”Geschichte des Brechungsgesetzes”, 166. 103 Huygens worked on it in 1671-2, see page 160.

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remaining uncompleted and unpublished as well, Lectiones XVIII was the most advanced treatise on geometrical optics published until the end of the seventeenth century. The core of Lectiones XVIII consists of lectures IV through XIII, in which Barrow determined the image of a point source in any reflection or refraction in plane and spherical surfaces. Barrow developed a mathematical theory of imagery by analyzing the intersections the refractions of a bundle of rays. For example, a point A is seen by an eye off the axis AB, with COD being the pupil of the eye.104 (Figure 20). The pupil is perpendicular to the refracted ray NO, which passes through the center of the pupil. The extension KNO is called the principal ray. Now, draw the refracted rays MC and RD that pass through the edge of the pupil. Produced backwards, MC and RD will not intersect the principal ray NKO in one point, but in points X and V. Barrow demonstrated that point Z on the principal ray is the limit of these intersections. According to his definition of the image, Z is the place Figure 20 Barrow’s analysis of image of the image. Consequently, the formation in refraction. cathetus rule does not apply here, as it 105 would have point K as the image. Barrow applied this determination of the image point to various problems in refraction. In the case of spherical surfaces he derived expressions for the place of the image point for the eye being both on and off the axis of the surface. Barrow defined the image point in a similar way as Huygens defined the ‘punctum concursus’ and applied it with comparable rigor to study the refracting properties of spherical surfaces. Many of their results were equivalent. Yet, they had different goals. Huygens intended to explain the dioptrical properties of the telescope and therefore confined himself to paraxial rays, not discussing optical imagery. He ignored mathematically sophisticated problems that had no relevance to the telescope, like the focus of an oblique cone of rays. Barrow’s aim was to develop a general theory of optical imagery. He had no intention of explaining the telescope and many of the problems he treated had no relevance to it.106 Still, in lecture XIV, he also discussed spherical lenses. He gave, without proof, a series of equations for the focal and image points of all kinds of lenses by way of an example of the 104

Barrow, Lectiones, [82-83]. Compare Shapiro, “The Optical Lectures”, 130 & 133-134. 106 Compare Shapiro, “The Optical Lectures”, 150-151; and Malet, “Isaac Barrow”, 286. 105

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application of his preceding discussion of single spherical surfaces. It was an exact solution to the problem of Dioptrice, yet a complex and cumbersome one.107 Barrow had chosen the example “… with a view to common use, and particularly aimed at reducing the labour of anyone who comes across them.”108 Barrow’s exact theory of focal distances was the first in print, but it was no more than a theory of focal distances. He did not discuss magnification and configurations of lenses. Barrow’s footsteps were followed by Newton in his lectures. Their central subject was his mathematical theory of colors. In a section “On the Refractions of Curved Surfaces” he also treated some topics regarding monochromatic rays. Newton extended Barrow’s theory of image formation to three-dimensional bundles of rays. He demonstrated the existence of a second image point at the intersection of the axis and the principal ray. Newton had developed his own solution of the anaclastic problem – although in the lectures he abandoned it in favor of Barrow’s – and found a new way to derive Descartes’ ovals. The final goal of the lectures were, however, colors. So, when Newton, in his 31st proposition, determined the spherical aberration of a ray, he did so to compare it to chromatic aberration. The latter was larger and “Consequently, the heterogeneity of light and not the unsuitability of a spherical shape is the reason why we have not yet advanced telescopes to a greater degree of perfection.”109 In this way, Newton dismissed Descartes proposal as a dead-end. This was an important result for the understanding of the working of the telescope. In order to overcome the disturbing effects of aberration, Newton proposed to use mirrors instead of lenses. Newton’s theory of colors and his reflector are further discussed in section 3.2.3. Dioptrics as mathematics The discovery of an exact law of refraction had supplied geometrical optics with a foundation for the mathematical study of the behavior of refracted rays. This study consisted of deducing theorems from the postulates and definitions of dioptrics in a rigorous way aimed at generality. With Kepler’s new theory of image formation, a range of new problems were raised relating to the perfect and imperfect focusing of rays. To the ones already mentioned was added, at the end of the century, that of caustics; the locus of intersections of rays refracted by a curved surface.110 These problems were markedly theoretical, mathematical puzzles tackled without practical objectives. Halley, for example, in a paper of 1693 solved “the problem of finding the foci of optick glasses universally” by means of a single algebraic 107

Shapiro, “The Optical Lectures”, 149-150. Barrow, Lectiones, [168]. 109 Newton, Optical Papers 1, 427. 110 In a series of papers of the 1680s and 1690s Tschirnhaus, Jakob and Johann Bernoulli and Hermann attacked the problem. They were preceded by Huygens in 1677, but he did not publish his account until 1690, see section 5.1. Jakob Bernoulli published a general solution in Acta Eruditorum in 1693. In his Analyse des infiniments petits (1696), L’Hopital gave a definitive solution on basis of the differential calculus. 108

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equation.111 Despite his involvement in practical matters of telescopes Halley, like Barrow, did not further apply his finding to the effect of lenses. His principal goal seems to have been to supplement Molyneux’ theory of focal distances by means of giving “An instance of the excellence of modern algebra, …”112 All in all, the telescope rarely directed the dioptrical studies undertaken by mathematicians. Kepler is rightly regarded as the founder of seventeenth-century geometrical optics, yet it was Paralipomena rather than Dioptrice that constituted the starting-point for later studies. Similarly, Descartes’ La Géométrie was the starting-point for later studies of aplanatic surfaces rather than La Dioptrique. I find it remarkable that an instrument that had revolutionized astronomy was ignored by students of geometrical optics in the same way as spectacles had been previously. Kepler alone had, right upon its invention, insisted that a mathematical understanding of the telescope was needed for its use in observation, and Huygens was the only one to take the instruction to heart. His approach was that of a mathematician, yet he applied his mathematical abilities to a practical question: understanding the working of the telescope. In Tractatus, he used the sine law to derive an exact and general theory of the properties of spherical lenses and their configurations. It remains to be seen, however, whether such a mathematical theory of the telescope was really of any use. Tractatus remained unpublished, those interested had to do with Dioptrice. 2.2.3 THE NEED FOR THEORY Dioptrice had arisen from Kepler’s conviction that, in order to make reliable observations, and astronomical instrument should be understood precisely. The mathematicians I have discussed in the preceding section did not follow his lead. Even Descartes and Newton, who proposed innovations in telescope design, did not bother to elaborate theories of the way telescopes produce sharp, magnified images. Maybe this was so because they, like the others mathematicians that have been discussed, did were not much involved in telescopic observation. Could the case be different for the mathematicians who were, the observers? We have seen that Galileo, the most renowned telescopist, was not really interested in mathematical questions of dioptrics. He applied himself rather to practical matters of the manufacture and improvement of the telescope. It does not seem that Galileo had to invoke dioptrical arguments to defend the reality of telescopic observations, at least not arguments from the mathematical tradition of perspective and Kepler.113 To be sure, as a pioneer in astronomical telescopy Galileo was confronted with suspicions about the reality of heavenly things seen through the tube, but these soon wore off. Likewise, telescopists like Scheiner and Hevelius in 111

Halley, “Instance”, 960. See: Albury, “Halley, Huygens, and Newton”, 455-457. 113 Galileo, Sidereus nuncius, 112-113 and 92-93 (Van Helden’s conclusion). See Dupré, Galileo and the telescope, 175-178. 112

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their books on telescopic observation contented themselves with a cursory, qualitative account of the telescope, drawing on Kepler’s lessons. Dioptrice was the standard theory referred to well until the close of the century, but mostly as regards the basic theorems on focal lengths and configurations. Apparently this sufficed the needs of practical dioptrics leaving the mathematical details superfluous. Had Kepler made things more difficult than they really were? This theme may be illustrated with the example of Isaac Beeckman, a savant who combined an interest in practical affairs with a theoretical outlook. He was interested in many things, including optics in all its manifestations, and kept an elaborate diary of his ideas and observations. It enables us to get an idea what a knowledgeable man would do with the mathematics of dioptrics. The diary contains numerous notes on visual observation that show that he read the literature – Aguilón, Kepler – attentively. In addition, he was familiar with Descartes’ optical ideas and their development, being in close contact with him on and off since 1618.114 In the 1620s, Beeckman became interested in telescopes and he acquired some lenses and instruments and later, in the 1630s, he put much effort in grinding lenses and building telescopes.115 Working on them, he encountered the problem of spherical aberration (and later chromatic aberration) for which he considered several remedies. The notes concerned are interesting for they show a basic understanding of the working of lenses – as he would have acquired from Paralipomena and Dioptrice – but the actual problem, that the aberration is inherent to the spherical shape of a lens, seems to have eluded him. Besides the common use of diaphragms to decrease the disturbance, Beeckman thought up some sagacious ideas like combining lenses on a spherical surface in order to emulate one large lens or a lens built up in thin rings like a Fresnel lens.116 The first idea he tested, just to discover soon that it did not work and that he had overlooked a basic property of lenses.117 He was enough of an experimentalist not to trust ideas blindly. When Descartes informed him in 1629 of his project of a hyperbolic lens, Beeckman reacted skeptical.118 The micrometer and telescopic sights The principal reason why astronomers did not show much interest in dioptrics lies, I think, in the fact that the telescope was a qualitative instrument during the first decades after its introduction. It had revealed new, spectacular phenomena in the sky, but it had not been deployed in the

114

Schuster, “Descartes opticien” and Van Berkel, “Descartes’ debt”. Beeckman, Journal, II, 209-211; 294-296. For lens grinding see down, page 57. 116 For the second idea see Beeckman, Journal, II, 367-368. For a later consideration see for example: III, 296. 117 Beeckman, Journal, II, 296; 357. 118 Beeckman, Journal, III, 109-110. 115

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exact description of the universe.119 After all, the telescope was an artful means to reveal new things in the heavens, whereas astronomical measurement instruments aided the naked eye.120 The step to combine these two by aiding the artificial eyes with quadrants and the like, was not taken immediately. Around the middle of the century, astronomical measurements were still made by using the pre-telescopic methods and instruments developed by Tycho Brahe. Hevelius used telescopes extensively to study the surface of the Moon, but he turned to open-sight instruments when making measurements. Efforts had been made to use the telescope for measurements, but in vain. Until the 1670s, the accuracy of telescopic observations was determined by the acuity of the human eye. But then change set in. With the introduction of the micrometer, the telescope was transformed into an instrument of precision. Significantly, the men closely involved in that development were the ones to seek a more precise account of the working of the telescope. The configuration Kepler had thought up in 1611 had the drawback that it reversed the image. Given the quality of lenses made at that time, it was not advisable to add a third lens to re-erect the image. The two-lens Keplerian telescope was therefore used only to project images, whereas the Galilean type continued to be used for direct observation. In the course of time the first advantage of Kepler’s configuration was discovered: its wider field of view. When the length of a Galilean telescope is increased the field of view quickly diminishes, which makes it very difficult to use. Towards the 1640s, the Keplerian telescope was gaining ground, in particular through the good craftsmanship of telescope makers like Fontana in Naples and Wiesel in Augsburg.121 At some point in the early 1640s, the second advantage of this type was discovered by the Lancashire astronomer William Gascoigne. The Keplerian configuration has a positive focus inside the telescope; an object inserted into it will cast a sharp shadow over the object seen through the tube. Gascoigne relates that he discovered this by accident after a spider had spun its web in his telescope.122 Inserting some kind of ruler makes it possible to make measurements of telescopic images. He died in 1644, before he could publish his discovery and his measurements of the diameters of planets.123 Gascoigne’s accomplishments were made public in 1667 when Richard Towneley, backed by Christopher Wren and Robert Hooke, claimed British priority for the invention of the micrometer. This happened after a letter of 119

Van Helden, Measure, 118-119. Compare Dear, Discipline and Experience, 210-216. 121 Van Helden, “Astronomical telescope”, 26-32. See also below section 3.1.1. 122 Rigaud, Correspondence, 46: “This is that admirable secret, which, as all other things, appeared when it pleased the All Disposer, at whose direction a spider’s line drawn in an opened case could first give me by its perfect apparition, when I was with two convexes trying experiments about the sun, the unexpected knowledge.” 123 McKeon, “Les débuts I”, 258-266. 120

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Adrien Auzout was published in Philosophical Transactions, in which he described a method of determining the diameters of planets.124 He had devised – possibly with the help of Pierre Petit – and used – together with Jean Picard – a grate of thin wires and a moveable reference frame inserted in the focal plane of a telescope. In two letters, also published in Philosophical Transactions, Towneley argued that Gascoigne had made and used a micrometer much earlier. He described a pair of moveable fillets that could be inserted into the focal plane.125 He himself had used and improved the device – probably since late 1665 – to make accurate observations.126 The principle of the micrometer, however, had already been published in 1659; by Huygens in Systema Saturnium, his astronomical work in which he presented his discoveries regarding the ring and the satellites of Saturn. In its final section he explained the principle and described how to use it to make measurements. He inserted a ring in the focal plane and then measured the angular magnitude of the opening thus produced by timing the passage of a star. Next, he inserted a cuneiform strip through a hole in the tube until it just covered a planet. The angular diameter of the planet was determined by taking out the strip and comparing its width at the point found with the inner diameter of the ring.127 It was not a real micrometer, but Huygens’ rather cumbersome method did produce reliable, accurate data.128 It was a convenient method for measuring the size of planets, Huygens said, as one did not have to wait for a conjunction of the planet with the Moon or a star.129 Huygens had been acquainted with Auzout and Petit since 1660 and had come to Paris in 1666 to give leadership to the Académie. His explanation of the principle of the micrometer certainly inspired their work on the micrometer, but the precise nature of Huygens’ contribution is hard to determine.130 The principle of the micrometer had another important application: telescopic sights. By inserting crosshairs in the focal plane, a telescope could reliably be aligned on a measuring arc.131 With the telescopic sight the accuracy of Brahe’s measurements could finally be improved. Several programs of astronomical measurement now set off. In Paris, Picard and other members of the Académie – completed in 1669 by Cassini – put into use a new, well-equipped observatory.132 Picard’s work on cartographic 124

OldCorr3, 293: “… prendre les diametres du soleil, de la lune et des planetes par une methode que nous avons, Monsieur Picard et moy, que ie croy la meilleure de toutes celles que l’on a pratiquer Jusques a present, ...” 125 McKeon, “Les débuts I”, 266-269. 126 McKeon, “Les débuts I”, 286. In Micrographia (1665) Hooke had suggested that a scale may be inserted into the focal plane of telescopes. Hooke, Micrographia, 237. 127 OC21, 348-351. 128 Van Helden, Measure, 120-121. 129 OC21, 352-353. 130 McKeon, “Les débuts I”, 286; Van Helden, Measure, 118. 131 McKeon, “Renouvellement”, 122. 132 McKeon, “Renouvellement”, 126.

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measurements resulted in the determination of the arc of the meridian, published in Mesure de la Terre (1671). In London, Hooke and Wren devoted themselves to carrying out the idea of telescopic sights. In 1669, Hooke announced that he had established the motion of the earth by means of a mural quadrant thus equipped. His claim met with great skepticism. In 1675, Flamsteed was appointed Astronomer Royal at the London counterpart to the Paris observatory. At the Royal Observatory, he erected a wealth of precision instruments and set up a program of astronomical measurements, eventually resulting in Historia coelestis brittanica (1725). The usefulness of telescopic sights was not, however, beyond all doubt. Hevelius, the most renowned astronomer in those days, was suspicious. He believed that telescopic sights were unreliable and therefore preferred naked eye views.133 In 1672, a letter by Flamsteed was published in Philosophical Transactions in which he defended the use of telescopes for astronomical measurements.134 He praised Hevelius for having improved Brahe’s astronomical data, but doubted whether any further progress could be possible as long as the latter refrained from using ‘glasses’. Hevelius took offense at Flamsteed’s allegations, and responded in Machina coelestis pars prior (1673) and in a letter that was published in part in Philosophical Transactions of April 1674: “For it is not only a question of seeing the stars somewhat more distinctly (…) but whether the instruments point correctly in every direction, whether the telescopic sights of the instrument can be accurately directed many times to any observations, and can be reliably maintained; but I very much doubt whether this can be done with equal precision every time.”135

The argument went a bit out of hand when, later in 1674, Hooke interfered with a vehement attack on Hevelius in Animadversions on the first part of the machina coelestis. Deeply hurt, Hevelius sent Flamsteed a letter in which he once more explained his doubts about the reliability of telescopic sights. The dispute was settled only five years later after a visit to Gdansk by Halley. He reported that Hevelius’ naked eye observations were indeed incredibly accurate. Hevelius had fought a lost battle – so it can be said with hindsight – but he was right in his suspicions about the reliability of telescopic sights. He knew from experience how difficult it is to align instruments reliably. Already in 1668 – right after the announcement of the micrometer – he had written to Oldenburg: “For many things seem most certain in theory, which in practice often fall far enough from truth.”136 He was astonished that Hooke would claim great accuracy for his measurements on the basis of just single observations. Hevelius knew that accuracy was gained by hard and systematic work. Picard, Cassini and Flamsteed undertook such an arduous task, but 133

Flamsteed, Gresham lectures, 34-39 (Forbes’s introduction). OldCorr9, 326-327. 135 OldCorr10, 520. 136 OldCorr4, 448. 134

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were convinced that the new optical devices were useful. The telescopic sight and the micrometer, together with the pendulum clock, brought about a revolution in positional astronomy between 1665 and 1680.137 In dioptrics it raised the question of the exact properties of lenses anew. Understanding the telescope Apart from the practical problems of mounting and aligning, the theoretical problem of the working of the telescope now became a matter of sustained interest. As a result of his discussion with Hevelius over the reliability of telescopic sights, Flamsteed realized that a theoretical justification of his claims was also needed: “… to prove that optick glasses did not impose upon or senses. then to shew that they might be applyed to instruments & rectified as well as plaine sights.”138 His chance to elaborate a dioptrical account of the telescope came in the early 1680s, when, appointed Gresham professor of Astronomy, he could deliver a series of lectures on astronomy. In these lectures, he discussed instruments and their use at length and included an account of dioptrics. “Yet such has beene the fault of or time that hitherto very little materiall on this subject has been published in or language. [Tho severall learned persons have done well concerning opticks in ye latine Tongue. Yet how glasses may be applyed to instruments & how the faults commonly committed in theire applycation might be amended or rather shund & how all the difficultys suggested by ingenious persons who had not the good to understand them aright might be avoyded the best authors of Dioptricks have been hitherto silent. … I shall therefore make it my businesse in this & my following lectures of this terme fully to explain the Nature of telescopes the reason of their performances, how they may be applyed to Levells, Quadrants, & Sextants. & how the instruments furnished with them may be so rectified & adjusted that they may be free from all suspicion of errors]”139

Flamsteed began with a discussion of the focal distances of convex lenses. It has two notable features. First, he took the consequences of Newton’s theory of colors into account by pointing out the chromatic aberration of lenses. Second, the paucity of his demonstrations shows that he was not an outstanding geometer.140 By means of the sine law, he calculated the refraction of single rays numerically and then compared the result with the Keplerian rules for focal distances of a bundle of rays. By calculating spherical aberration he discovered – as Huygens had done earlier – that the aberration of a plano-convex lens varies considerably depending on which side is turned towards the incident rays. He gave only a qualitative account of chromatic abberation. On this basis he argued that only telescopes consisting of two convex lenses are useful in astronomy, because these admit the

137

Van Helden, “Huygens and the astronomers”, 156-157; Van Helden, Measure, 127-129. Flamsteed, Gresham lectures, 154. 139 Flamsteed, Gresham lectures, 119 & 132. Flamsteed later deleted the part between brackets. 140 Flamsteed, Gresham lectures, 120-127. 138

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insertion of a micrometer or crosshairs.141 He then went on to explain in detail how to mount a telescope on quadrants and other things.142 Flamsteed did not achieve the exactness and rigor of Huygens. His analysis of the properties of lenses consisted of numerical calculations rather than of general theorems. His account was larded with solutions to practical problems, and here indeed resided the eventual goal of giving a dioptrical account of the properties and effects of telescopes. Given the scarcity of suitable publications on these matters, Flamsteed did not have much to start from. He confessed that he had not taken the time to peruse Kepler’s Paralipomena and he claimed never to have read Dioptrice.143 He based himself instead on some letters in which Gascoigne discussed the foci of planoconvex and plano-concave lenses.144 He considered his own account of other lenses and of telescopes “… but a superstructure on yt foundation”.145 It sufficed to free the telescope of the imputation that “… all observations made with glasses [are] more doubtfull & uncerteine …”146 Flamsteed’s lectures attracted only a small audience and did not go to print until this century.147 Some of Flamsteed’s ideas were passed on by Molyneux. During the 1680s, the men had corresponded extensively on dioptrics, among other things. In 1692 Molyneux published Dioptrica Nova, an elaborate dioptrical account of the telescope. In its preface, he acknowledged his debt to Flamsteed: “… the Geometrical Method of calculating a Rays Progress, which in many particulars is so amply delivered hereafter, is wholly new, and never before publish’d. And for the first Intimation thereof, I must acknowledg my self obliged to my worthy Friend Mr. Flamsteed Astron. Reg. who had it from some unpublished Papers of Mr. Gascoignes.”148

Dioptrica Nova was a compilation of dioptrical works published during the seventeenth century.149 Molyneux’ own contribution consists of his particular presentation of the material, arranging theoretical knowledge in such a way that it was useful for understanding the working of telescopes. He gave his own demonstrations of many of its theorems, but he did not aim at mathematical rigor or completeness: “… [the Reader] is not to expect Geometrical Strictness in several Particulars of this Doctrine. … ; as being more desirous of shewing in gross the Properties of Glasses and

141

Flamsteed, Gresham lectures, 136. Flamsteed, Gresham lectures, 140-143. 143 Flamsteed, Gresham lectures, 40; 146n2 (Forbes’ introduction). 144 Flamsteed, Gresham lectures, 8-9; 40 (Forbes’ introduction). 145 Flamsteed, Gresham lectures, 39 (Forbes’ introduction). 146 Flamsteed, Gresham lectures, 149. 147 Flamsteed, Gresham lectures, 4-5 (Forbes’ introduction). 148 Molyneux, Dioptrica nova, (Admonition to the reader). 149 Molyneux mentioned Kepler, Cavalieri, Hérigone, Dechales, Fabri, Gregory and Barrow. 142

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their Effects in Telescopes, than of affecting a Nicety, which would be of little Use in Practice.”150

The limitations of Molyneux’ mathematics are easily noted. In proposition III, for example, he discussed refraction by a bi-convex lens of a ray parallel to the axis. 151 Taking into account both the distance of the ray from the axis and the thickness of the lens, he derived by means of the sine law the point where the refracted ray intersects the axis. In generalizing this to the focal distance of the lens, Molyneux was less exact: “If by this Method we calculate the Progress of a Ray through a Double Convex-Glass of equal Convexities; and the thickness of the Glass be little or nothing in comparison of the Radius of the Convexity; and the Distance of the Point of Incidence from the Axis be but small, we shall find the Point of Concourse to be distant from the Glass about the Radius of the Convexity nearly.”152

He then gave Kepler’s theorem and reproduced the latter’s proof. For a lens with unequal curvatures, he stated that the refracted ray could also be constructed exactly. He confined himself, however, to a “… Shorter Rule laid down by most Optick Writers”, which is identical with Huygens’ rule for a thin lens cited on page 19. 153 This pattern of exact constructions for single rays and questionable generalizations to bundles of rays recurs throughout Dioptrica Nova. His problem was that he somehow had to link Flamsteed’s discussion of single rays with the Keplerian rules of focal distances found in most published treatises. He was not able enough a mathematician to derive general theorems on focal distances by means of the sine law. The problem with Molyneux’ generalizations is that he thought that the intersection of single refracted ray with the axis was an approximation of the focal distance. He did not fully understand that the focus of a refracting body is the (limit) point of the refracted rays of a pencil of rays. His definition of ‘focus’ in terms of the intersection of a single ray with the axis makes this clear. 154 Taken literally, this would mean that a spherical surface has many foci for one point object. In a scholium following proposition III of Dioptrica Nova, he discussed spherical aberration. He began by reproducing Flamsteed’s calculations for single rays as well as his conclusions concerning the use of lenses. He then defined the distance between the ‘focus’ and the intersection of the refracted ray with the axis as the ‘depth of the focus’.155 Again, he mixed up the refraction of a single ray with the focusing of a pencil of rays. It is not difficult to point out flaws in Molyneux’ demonstrations, but we should bear in mind the practical aim of Dioptrica Nova. In his discussion of 150

Molyneux, Dioptrica nova, (Admonition to the reader). Molyneux, Dioptrica nova, 19-23. 152 Molyneux, Dioptrica nova, 20. 153 Molyneux, Dioptrica nova, 22. 154 Molyneux, Dioptrica nova, 9. 155 Molyneux, Dioptrica nova, 24. From the preceding it will be clear, that following Molyneux's line of thought this distance should be zero, for both points are by definition the same. 151

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images of extended objects, Molyneux displayed a better understanding of the focusing of rays. In a section on “… the Representation of outward Objects in a Dark Chamber; a Convex-Glass”, he described how the image is formed by the focusing of pencils of rays originating in the points of an object.156 He then remarked that “… tho all the Rays from each point are not united in an answerable Point in the Image, yet there are a sufficient quantity of them to render the Representation very perfect.”157 Rather than mathematically precise, this was a practical definition of focus. It explained why in practice images may appear sharp. Besides all the objections that can be raised against Molyneux’ theory from a mathematical point of view we should bear in mind that Dioptrica Nova was the first published dioptrical account of telescopes, since Dioptrice. It was up-to-date with developments in telescope making and was intended to be useful for practice. Before coming to a conclusion of this chapter, we go back in time and cross back over the Channel. Flamsteed’s ally in the debate over telescopic sights, Picard, also saw the importance of theory. In a letter to Hevelius, he had briefly explained the working of the telescopic sight in dioptrical terms.158 Somewhat earlier – probably in 1668 – he had pointed out the need for such an analysis: “[optical devices] can also be subject to certain refractions that should be known well.”159 In Mesure de la Terre, he had briefly discussed matters of aligning and rectifying telescopic sights in these terms. Picard was known for his interest and ability in matters dioptrical. At the Académie, he frequently discoursed of dioptrical theory.160 In this, the telescope stood central: “What we have just explained about the construction of telescopes, concerns only its use in instruments made for observation, …”161

Picard never published his dioptrics, but a collections of papers he had read at the Académie was published posthumously in 1693 under the title ‘Fragmens de Dioptrique’.162 Picard had a major advantage over Flamsteed. He was acquainted with one of the most knowledgeable men in dioptrics: Huygens. Besides his learning, in 1666 Huygens had brought a copy of the manuscript of Tractatus to Paris.163 At the Académie, Huygens had also discoursed on dioptrics. “Fragmens de Dioptrique” make it clear that Picard must have been among Huygens’ most attentive listeners. They are for the 156

Molyneux, Dioptrica nova, 36-38. Molyneux, Dioptrica nova, 38. 158 Picolet, “Correspondence”, 38-39. 159 “… peuuent aussi estre sujets a certaines refractions qu’il faut bien connoistre.” Quoted in McKeon, “Renouvellement”, 126-128. It is found in: A. Ac. Sc., Registres, t. 3, fol 156 ro - 164 vo spéc. 157 vo. 160 Blay, ”Travaux de Picard”, 329-332. Blay cites several references. 161 Blay, “Travaux de Picard” 343. “Ce que nous venons d’expliquer touchant la construction des lunettes d’approche, n’est que par rapport à l’usage que l’on en fait dans les instruments qui servent à l’observer, …” 162 Divers Ouvrages de Mathematique et de Physique, par Messieurs de l’Academie Royale des Sciences (1693), 375-412. 163 OC13, “Avertissement”, 7. 157

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most part derived from Huygens’ dioptrical theories, and I will not discuss them in further detail.164 Huygens’ position Picard’s dioptrical fragments bring us back to Huygens. What had he been doing in the meantime? In Systema saturnium he had alluded to an elaborate theory of dioptrics, which we know he possessed indeed. Yet, despite ongoing requests to publish it, he had kept it to himself. It may be clear by now that Tractatus is a unique work in the development of seventeenthcentury dioptrics. Huygens was the first and only man to follow the lead of Dioptrice. Like Kepler, he combined the two things necessary to develop a theory of the telescope: mathematical proficiency and an orientation on the instrument. Unlike Kepler, he had the exact law of refraction and thus he could rigorously develop an exact theory of the telescope. But did Huygens really follow Kepler? Did he want to understand the telescope in view of its use in astronomy? Tractatus came into being well before Huygens commenced his practical activities of telescope making and astronomical observation (discussed in the next chapter). Unlike Flamsteed and Picard, he did not seek answers to questions that had arisen in practice. Nevertheless, his orientation on the telescope is clear. He passed by all those sophisticated problems not relevant to the understanding of the telescope that preoccupied mathematicians like Barrow. However, nowhere does Huygens mention Kepler as an example. It looks as if developing a theory of the telescope on the basis of the sine law was to him an interesting mathematical puzzle, maybe just to correct Descartes’ useless approach to dioptrics. The problem had not yet been solved and Huygens only too gladly seized the opportunity. Which makes his exclusive orientation on the instrument all the more interesting. The transformation of the telescope into an instrument of precision brought back an interest in the dioptrical properties of the telescope. In this regard, one might say that Kepler had prematurely raised the question after a mathematical understanding of the telescope. In 1611, it was a qualitative instrument and remained so for another half century. To understand its working, a qualitative account of the effects of lenses therefore sufficed. Similarly, we can ask whether an exact theory like Huygens’ was really needed. It seems that Kepler’s or Keplerian theories satisfied the needs of men like Flamsteed and Molyneux pretty well. They lacked sufficient proficiency in mathematics to treat lenses in exact terms, but they may also have been perfectly satisfied with their approximate results. Huygens himself did not put much work in applying his theory to the questions that occupied Picard and Flamsteed. The principle of the micrometer may or may not have been the result of his theoretical understanding, in Systema saturnium he explained it only briefly. Huygens did 164

Blay, “Travaux de Picard”, 340.

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expect that theory could be useful. The discovery that a sphere is an aplanatic surface in some cases had given the initial impulse to his interest in dioptrics. In his letter to Van Schooten, he expressed the expectation that this theoretical insight would contribute to the improvement of the telescope. From the very start, Huygens saw a connection between the theory of dioptrics and the practice of telescope making. In the next chapter we shall see what he would make of it.

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Chapter 3 1655-1672 - 'De Aberratione' Huygens' practical optics and the aspirations of dioptrical theory

In the decade following Tractatus, Huygens was at home were his mathematical virtuosity grew to full stature. These are the years of his most renowned achievements: the invention – in 1656 – improvement and employment of the pendulum clock and the theory of pendulum that were the basis of his master piece Horologium Oscillatorium (1673); the discovery in 1655 of a satellite of Saturn and the identification of its ring. Through his correspondence and publications Huygens increasingly gained recognition among Europe’s scholars. He traveled abroad, first to Paris in 1655 to meet the leading French mathematicians, then to Paris and London in 1660-1, and again in 1663-4, the last time being elected fellow of the Royal Society. There were squabbles as well, in Italy in particular, over the priority of the pendulum clock with Florentine sympathizers of the late Galileo and with the Roman telescope maker Divini over the superiority of his telescopes. Probably as a result of the clock dispute, he did not obtain a position at the court of prince Leopold, but in 1666 Huygens realized his learned assets. At the instigation of Colbert he came to Paris to help organize an ‘académie des sciences’, thus confirming his status as Europe’s leading mathematician. Life in Paris, with it competitive milieu, was no unqualified pleasure. Huygens correspondence shows symptoms of homesickness, he particularly missed his brother Constantijn, and in 1670 he was was smitten with ‘melancholie’ for the first time. In these years he also experienced the first major setback in his science: a design for a perfect telescope proved useless. The design was the outcome of Huygens’ practical activities in telescopy of the late 1650s and his subsequent theoretical reflections thereupon of the 1660s. These are the subject of this chapter. When Huygens’ interest in dioptrics was sparked late 1652, it was both its theoretical and practical aspects. He immediately began inquiring about the art of lens making, but he engaged in practical dioptrics only after he put aside the manuscript of Tractatus. Around 1655, he and his brother Constantijn acquired the art of lens making and started building telescopes.1 The practice bore fruit almost immediately. In 1656, Christiaan published a pamphlet De saturni luna observatio nova on the discovery of a satellite around Saturn. It was the first new celestial body in the solar system to be

1

Editor’s comment, OC15, 10. See also Anne van Helden, “Lens production”, 70.

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discovered since Galileo.2 The tract ended with an anagram holding Huygens’ second discovery: the true nature of the inexplicable appearance of Saturn. Three years later, he elaborated his explanation in Systema saturnium. The strange attachments to the planet that disappeared from time to time were manifestations of a solid ring around the planet.3 He owed much to his instruments, Huygens wrote in Systema saturnium. He prided himself on his practical skills of telescope making and claimed that his success proved the unmatched quality of his telescopes. Van Helden explains that his discovery owed at least as much to his talents for geometrical and physical reasoning.4 Initially, Huygens had used a 12-foot telescope of their own make. After a trip to Paris, where he probably discussed his observations, the brothers built a new, 23-foot telescope which he started using in February 1656.5 He illustrated the difference between both pieces in Systema saturnium (Figure 21). Everyone could see for himself that Huygens could hold his own with the best of telescope makers. At least, that is how he saw it himself. His boasting offended Eustachio Divini in Rome, who saw his fame of being the best telescope maker in Europe challenged. In 1660 he published Brevis annotatio in systema saturnium, disputing the observational results as well as Huygens’ claims regarding his instruments.6 The tract was actually written by the Roman astronomer Fabri. In the ensuing dispute Divini/Fabri were no match for Huygens, at least not as regards the structure of the system of Saturn.7

Figure 21 Observations of Saturn with the 12- and a 23-foot telescope.

The dispute itself is less interesting than the fact that Huygens did not feel above at entering a dispute with a craftsman. It raises questions about the relationship between his theoretical and his practical pursuits, how he valued his mathematical expertise and his skilful craftsmanship. The last part of this 2

Huygens did not name it, he called it ‘saturni luna’ and sometimes ‘comes meus’. The name Titan was given by Herschel in 1847. 3 OC15, 296-299. 4 Van Helden, “Huygens and the astronomers”, 150-154. Van Helden, “Divini vs Huygens”, 48-50. 5 OC15, 177; 230. Huygens employed Rhineland feet (0,3139 meters) and inches (0,026 meters). 6 It is reprinted in OC15, 403-437. 7 Van Helden, “Divini vs Huygens”, 36-40.

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chapter examines these themes in a broader context of the scientific revolution, and forms a conclusion of this account of Huygens’ dioptrics prior to the metamorphosis of his optics discussed in the subsequent chapters. So much can be said that Huygens’ passion was with the instrument, not its employment. For Huygens telescopic astronomy was a pastime rather than a full-time job. Although he had solved the puzzle of Saturn’s bulges by systematic observation, this never became his vocation. His fascination was with its design and manufacture of telescopes.8 To this we may also count his interest in dioptrical theory, being a means of tinkering with the instrument and contemplating its workings. In the ten or so years after 1653 when the brothers engaged in practical pursuits, Huygens did not work on dioptrical theory (at least no traces ar left). During the 1660s he returned to theory and set out for what should have been the crowning glory of his dioptrical work: the design of a telescope in which spherical aberration was nullified. Not by means of imaginary lenses of the kind Descartes had thought up, but by means of actual spherical lenses. In the design came together Huygens’ theoretical understanding and practical experience with lenses and it brought him closer to bridging the gap between theory and practice than any other in the seventeenth century. Newton’s ‘New Theory’ of colors eventually shipwrecked the project. Newton’s approach of mathematical optics essentially differed from Huygens’. These differences shed light on the character of the Huygens’ dioptrics and may explain why Huygens did not manage to bridge the said gap completely. 3.1 The use of theory Around 1600, spectacle makers had advanced their art far enough to enable the discovery of the telescopic effect.9 Astronomers in their turn discovered the possibilities of this chance invention. Their pursuit demanded far greater power than the first spyglasses offered. They needed skillful hands: sometimes their own, but usually those of a craftsman. Galileo, not particularly all fingers and thumbs himself, had the advantage of living close to Venice, the center of European glass industry. After the success of Sidereus nuncius he established a workshop for telescopes. Simon Marius, in Germany, was less lucky: he had great trouble finding a good lens maker and could not put the new invention to fruitful use.10 During the first half of the seventeenth century, the manufacture of telescopes for astronomy developed into a small trade of specialized craftsmen. This section will not treat the history of lens and telescope manufacture, it focuses on the relationship between dioptrical theory and the art of telescope making. Central questions are: to what extent was theoretical knowledge used in practical dioptrics, if it

8

Van Helden, “Huygens and the astronomers” 148, 157-158. Van Helden, Invention, 16-20. 10 Van Helden, Invention, 26; 47-48. 9

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was useful at all; did the scholarly world contribute to the art of telescope making besides revenue, status, and stimuli for progress? René Descartes definitely believed art could learn from philosophy, and that it should. In La Dioptrique he intended to show the benefits of philosophy. The telescope, he said, was a product of practical wit and skills, but the explanation of its difficulties could bring it to a higher level of perfection.11 Just as the telescope surpassed the natural limitations of vision, so the scholar could teach the craftsman how to overcome his limitations. The sine law dictated that lenses should have a conical rather than a spherical shape, as we have seen in the previous chapter. Descartes had also considered the way his design could be put to practice. In the tenth and final discourse of La Dioptrique he described the way his lenses could be made. His account included an ingenious lathe to grind hyperbolic lenses. It reflected his efforts, during the late 1620s, to make a hyperbolic lens. Descartes was never lavish to point out his debt to others – to put it mildly – so he did not tell his readers that he owed much to his cooperation with the lens maker Jean Ferrier and the mathematician Claude Mydorge.12 Allegedly, the threesome succeeded in grinding a convex hyperbolic lens. “And it turned out perfectly well …”, Descartes wrote in 1635 to Huygens’ father Constantijn.13 It had been made by hand. The next step was to design a lathe. Towards the end of 1628 Descartes left for Holland. In the fall of 1629, he and Ferrier exchanged some letters in which an earlier design for a lathe was mentioned.14 They discussed a machine Descartes had contrived for making a cutting blade with a hyperbolic edge.15 Ferrier proposed several modifications and improvements that turn up in La Dioptrique.16 Throughout the seventeenth century, Descartes’ account gave rise to efforts to make elliptic and hyperbolic lenses. Around 1635, Constantijn Huygens arranged unsuccessful attempts to grind them.17 In 1643, Rheita claimed to have succeeded with tools he described in Novem stellae circum Iovem. Wren described a device to make a hyperboloid surface in an article published in Philosophical Transactions in 1669. During the 1670s, Huygens corresponded with Smethwick over another design.18 Much in these suggestions never went beyond the paper stage. To execute them skills and tools – as well as patience – were needed. To design a lens may have been a scholarly challenge, actually to make it required craftsmanship. And then it 11

Descartes, Dioptrique, 2-3 (AT6, 82-83). Shea, “Descartes and Ferrier”, 146-148. 13 AT1, 598-600. “Et il reussit parfaitement bien; …” It turned out that it was impossible to make a concave lens in the same way. 14 AT1, lts 8, 11, 12,13,22,21,27. Shea, “Descartes and Ferrier”. The letters not only reveal Ferrier’s mastery of the art but also his mathematical knowledge. 15 AT1, 33-35. 16 Descartes, Dioptrique, 141-150 (AT6, 215-224). 17 Ploeg, Constantijn Huygens, 34-38. 18 OC7, 111; 117; 487; 511-513. In 1654 Huygens described a mechanism to draw ellipses on the basis of a circle, apparently aimed at making elliptic lenses out of spherical ones; OC17, 287-292. 12

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remained to be seen whether it could be made to function properly. As for Descartes’ ideal lenses, theory had not advanced practice yet. 3.1.1 HUYGENS AND THE ART OF TELESCOPE MAKING In the middle the seventeenth century the art of lens making had progressed enormously, allowing telescopes to be made with more than two lenses and challenging the inventiveness of telescope makers. Skills, tools and materials were the principal necessities for the manufacture of ordinary spherical lenses. The manufacture was not fully under control: glass suffered from various flaws, the produced faces of lenses were spherical at best, and so on. Clever solutions were needed to obtain good telescopic images. The state of the art of lensmaking can be more or less reconstructed from the quality of surviving lenses, but this only provides indirect evidence of the art itself. Lensmaking practice in the first decades after the invention of the telescope is hardly documented. Some information can be distilled from Sirturus and Scheiner, but their accounts are quite succinct and rather superficial. A rare source of information is found in the diary of Isaac Beeckman, which meticulously records his trials and errors with using and manufacturing lenses.19 His interest in telescopes was excited in the early 1620s, but not until 1632 did he embark on serious lensmaking himself. In the meantime he recorded his dissatisfaction with the lenses he acquired with established lensmakers. The number of notes on grinding, polishing and the like, quickly grows in the early 1630s and in 1633 he acquired a grinding mould and commenced manufacturing his own lenses. Beeckman visited several artisans who taught him their art. The diary describes their techniques in much detail, particularly noting the differences.20 The attentive pupil was a quick learner. In the autumn of 1635, Beeckman compared one of his lenses with one from Johannes Sachariassen of Middelburg – one of his tutors and son of one of the claimants of the invention of the telescope – and found out it was better.21 Beeckman’s journal was a hidden treasure. He showed it only to Descartes, Mersenne and Hortensius and it remained unknown until Cornelis de Waard discovered it in 1905. An aspirant lens maker lacked published expositions to learn of the art. Rheita in 1645 was somewhat more elaborate, but when Huygens took on to lens making in the early 1650s, he had to consult, like Beeckman before him, experts and acquire the art by trial and error. The questions he fired at Gutschoven in 1652 display the diversity of the know-how involved in the process of cutting, grinding, and polishing to make a good lens out of a lump of glass.22 How to make grinding moulds? How can a perfect spherical figure

19

Next to numerous short entries, the main body is collected under the heading “Notes sur le rodage et le polisage des verres” in Beeckman, Journal, III, 371-431. 20 Beeckman, Journal, III, 69, 249, 308, 383. 21 Beeckman, Journal, III, 430. 22 OC1, 191. See also Anne van Helden, “Lens production”, 70-75.

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be created? What kind of sand is needed for grinding? Which glue is best to attach the handle? Et cetera, et cetera. Despite its orientation on the telescope, Huygens’ 1653 study of dioptrics did not aim at its improvement. Apparently, he had not found much use in the discovery of 1652. In the third part of Tractatus he had written down an the idea to add a little mirror to a Keplerian telescope to re-erect the image without the need to add an extra lens.23 This was not a new idea and it did not improve the defects of lenses directly. Having put aside the manuscript on the mathematical properties of lenses, Huygens turned to their material properties. In the practical work he pursued with his brother, he developed an artisanal understanding of lenses. The question is what such an understanding entailed and how it related to the theoretical understanding Huygens had developed in Tractatus. Huygens’ skills Some notes survive, in which Huygens described the process of grinding lenses.24 In 1658 he recorded how he had made a “good” 4½-foot bi-convex lens: “Always kept it fairly wet to preserve the dust. But not too much water in the beginning, or it will bump. Never forget to press evenly, and often lifted the hand and placed it evenly again. It is best to be alone. … At first I ground the other side wrongly: the reason for this was that I either took too much water in the beginning, or that I did not polish on the right spot. I first corrected somewhat by polishing at the right spot again; then with more polishing it got worse once again.”25

Making lenses was first of all a matter of ‘Fingerspitzengefühl’ acquired through much practice. Huygens and his brother did so and eventually became pretty good at it.26 One of the main problems of grinding and polishing was to secure an optimal shape of a lens. Both surfaces should be really spherical and the axes should coincide. As Huygens’ notes show this entailed a good deal of accuracy and patience. Proper tools did not only ease the laborious task but improve the control of the manufacture and thus the quality of the lens. In his notes, Huygens described a device (Figure 22) that relieved the hands and ensured a proper, even pressure on the glass.27 A similar device had been described by Beeckman, who added that it was a 23

OC1, 242. He distributed several telescopes of this design during the next decade. (OC1, 242; OC13, 264n3; OC4, 132-3; OC4, 224, 228-9) 24 OC17, 293-304. 25 OC17, 294. “altijdt redelijck nat gehouden om te beter de stof te bewaren. doch in ’t eerst niet al te veel waters, want anders stoot het aen. altijdt dencken om gelijck te drucken, en dickwils de hand af gelicht en weer gelijck aen geset. ’t is best alleen te sijn. … De andere sijde sleep ick eerst eens mis: daer de oorsaeck van was, of dat ick in ’t eerst te veel water nam of dat ick niet op de goeije plaets en polijsten. ick verbeterdense eerst wat met op de rechte plaets noch eens te polijsten; daer nae met noch meer polijsten wierd het weer erger.” 26 The earliest lenses that remain – one in the Utrecht University Museum and two at Boerhaave Museum in Leiden – are not very good. Their fame as lens makers stems from the 1680s. Anne van Helden, “Lens production”, 75-78; Anne van Helden, Collection, IV; 22. 27 OC17, 299.

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technique used by mirror-makers.28 It is not known where Huygens got the idea. Such tools for improving the grinding process had been thought up earlier, in particular by the most prestigious lens makers. In Galileo’s workshop – reigning until the 1640s – a lathe was introduced that permitted greater precision than was attained by ordinary spectacle makers.29 During the 1660s, the Campani brothers in Rome became the undisputed masters of the art. They used a range of machines of their own design, producing lenses unsurpassed until the eighteenth century. The Huygens brothers kept a Figure 22 Beam to facilitate lens grinding. close eye on developments like these and around 1665 references to a type of lathe designed by Campani turned up in their writings. The quality of lenses seems to have depended to some degree on the lens maker’s inventiveness to convert laborious and unsteady handiwork into reliable tools. Huygens had learned how to make lenses. He knew the limitations of the art and of its products. Even the best lenses might suffer from flaws like bubbles, irregular density, faults in shape, etcetera. In the end, the proof of the pudding was in the eating. The quality of telescopes was determined by trial, sometimes literally. Campani beat Divini early 1664 by a series of carefully organized ‘paragoni’: open contests in which printed sheets at a distance were read by means of the instruments of both competitors.30 Alternative configurations Besides the quality of lenses, telescopes could be improved by configuring lenses alternatively. Kepler could not have known that the configuration of two convex lenses he discussed in Dioptrice had several advantages over the Galilean one. The positive focus that made possible the micrometer has already been discussed in the previous chapter. Scheiner, who used a Keplerian telescope to project images of the sun, discovered by coincidence that it had a wider field of view. There are indications that Fontana was the first to put Kepler’s idea into practice, although Scheiner was the first to mention using it.31 Around 1640 Fontana was the first to challenge Galileo’s dominance in the trade and he did so with Keplerian telescopes. Around that time, the Galilean configuration was beginning to reach the limits to which its power could be increased without the field of view becoming too small. 28

Beeckman, Journal, III, 232. Bedini, “Makers”, 108-110; Bedini, “Lens making”, 688-691. 30 Bonelli, “Divini and Campani”, 21-25. 31 Van Helden, “Astronomical telescope”, 20-25. Compare Malet, “Kepler and the telescope”, 120. 29

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The inversion of the picture a Keplerian telescope produced could be overcome by adding a third lens. Given the quality of the earliest lenses it was not advisable to ‘multiply’ glasses, but by the 1640s multi-lens telescopes were beginning to become acceptable. With the increase of length and magnification, however, the field of a Keplerian telescope also became narrow. For example, the 23-foot telescope that Huygens used in his observations of Saturn had a field of only 17'. It could not display the entire Moon at once. The limited field of view could be overcome by adding a field lens. The Augsburg telescope maker Johann Wiesel was probably the first to make telescopes with such compound oculars.32 In a letter of 17 December 1649, Wiesel described a four-lens telescope. It had an eyepiece consisting of two plano-convex lenses fitted in a small tube. The eyepiece tube was inserted in a composition of tubes which held an objective lens at the far side and a plano-convex ocular which acted as a field lens. Wiesel added: “Sir you may bee assured this is y.e first starrie tubus wch I have made of this manner & so good yt it goes farre beyond all others wherof my selfe also doe not little rejoyce.”33

The fame of Wiesel’s telescopes spread quickly and throughout Europe telescope makers tried to equal them. On a visit to his relative Edelheer in Antwerpen on New Year’s eve 1652, Huygens saw a Wiesel telescope and was very impressed. It was a four-lens telescope, probably comparable to one Wiesel described in 1649. Towards the end of 1654 Huygens acquired two letters written by Wiesel - one to his cousin Vogelaer - describing the construction and use of several optical instruments.34 In the first letter a sixlens telescope was described, which could be used for both terrestrial and celestial purposes. Wiesel was an artisan, a very good one with an unmatched understanding of lenses and their configurations. His was another kind of understanding than the dioptrical theory Huygens developed in 1653. This kind of experiential knowledge Huygens acquired the following years in his practical dioptrics. Then, some ten years later, in an artisanal manner Huygens made his own compound eyepiece with excellent dioptrical properties. Experiential knowledge Telescope makers had a great deal of knowledge of dioptrics, as witnessed by the fruits of their labor that are preserved. Like the process of production, the thinking behind these products is more difficult to retrieve. It is barely documented as craftsmen in general were reluctant to reveal the secrets of their trade. There is reason to believe that their knowledge of dioptrics was of a different kind than that of mathematicians. That much we can infer from what little material that has been preserved. Lens makers knew very 32

Van Helden, “Compound”, 27-29; Keil, “Technology transfer”, 272-273. They are first mentioned in Rheita’s Oculus Enoch et Eliae (1645), who referred his readers to Wiesel. For the relationship between Rheita and Wiesel see Keill, Augustanus Opticus, 66-77. 33 Van Helden, “Compound eyepieces”, 34. The entire letter is reproduced on 34-35. 34 OC1, 308-311.

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well how to grind lenses to suitable proportions and configure appropriately. The nature of this knowledge and to what extent they understood the dioptrical properties of lenses will be discussed now. Note that this is a discussion of very limited scope, determined by the considerations and activities of Huygens, that passes over the a wealth of historical knowledge that can be, and has been, gathered regarding the telescope making trade.35 A booklet on spectacles written in 1623 gives an indication of the understanding their manufacturers had of glasses. Uso de los antojos was written by the Andalusian licentiate Daza de Valdez. It explained how to choose glasses of appropriate strength for a patient. Daza described a procedure to determine the ‘grado’ of a given lens (Figure 23).36 He drew two solid circles X and Q of unequal diameter on a sheet of paper as well as a specific scale at one of the circles. A glass was then positioned on the scale in such a way that both circles were seen equally large. The position of the lens on the scale gave its ‘grado’.37 Daza did not explain the method, he only described how it was Figure 23 Daza’s scale employed. It was a practical procedure that did not require any understanding of its effect on rays. A manuscript written around 1670 in Rome by a certain Giovanni Bolantio contains a similar kind of procedural knowledge. It discussed the manufacture of telescopes and probably recorded the daily routine in some workshop.38 It contains two tables listing the ocular and objective lenses needed to make a telescope of a specific strength, characterized by its length. The lenses are characterized by the doubled radius of the pattern in which they were ground.39 With these tables at hand, the workman could choose the patterns needed to make a telescope on order. Bolantio did not explain whether he had constructed the tables himself nor how they were made. Some dioptrical rules are implicit in them. For example, the length of a 35

See for example the recent, formidable study on Wiesel by Inge Keill which may serve as a guide to themes and literature: Keil, Augustanus Opticus. 36 Daza, Uso de los antojos, 137-140. It appears that this classification in terms of ‘degrees’ was, at that time, replacing an older one in terms of the common age of someone bearing spectacles of a particular strength. The ‘grados’ Daza employs seem to be identical with the ‘punti’ Garzoni mentions in his discussion of the craft in La piazza universale (1585). See also: Pflugk, “Beiträge”, 50-55. 37 Daza did not explain how the scale on the paper was established. Von Rohr has given an alluring suggestion as to how such a scale might be construed. Spectacle makers knew that multiple lenses of a given strength could be substituted by a stronger one to reach the same effect. Thus the first position on the scale was determined by a weak lens and the other positions determined by the amount of equal lenses which had to be put in those positions. Von Rohr, “Versuch”, 4. 38 Bedini & Bennet, “Treatise”. 39 Bedini & Bennet, “Treatise”, 120-121.

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Keplerian telescope is set equal to the doubled radius of its objective lens, which - correctly - implies that the focal distance of the objective is twice the radius of its convex side. Another table prescribed the size of the aperture th

for a given objective, in each case 801 of the tube’s length. Bolantio explained that the objective should be partially covered by a ring so that no light fell on the interior of the tube, which apparently implied the ratio used in the table.40 Whatever the dioptrical understanding implicit in Bolantio’s manuscript, it was presented in a procedural, how-to style that did not require further theoretical knowledge.41 Some telescope makers published their own observations, to promote their products.42 They did not publish the secrets of their art, as their revenues depended on them. Information on the manufacture of lenses and telescopes could be found in books that were mostly written by scholars. Examples are Telescopium (1618) by Girolamo Sirtori, Selenographia (1647) by Hevelius and La Dioptrique oculaire (1672) by Cherubin d’Orleans. In 1685, Huygens wrote a treatise on lens grinding in Dutch, Memorien aangaende het slijpen van glasen tot verrekijckers, published posthumously in Latin in the Opuscula posthuma (1703).43 Memorien was the elaboration of notes like the one cited above. Huygens described the process of lens making as a set of directives, procedures, tips and tricks. No attempt is made to explain why things work as they work: for example a geometrical account of the grinding device is absent. Memorien supplied the kind of experiential knowledge also found in Bolantio’s manuscript: a description of skills Huygens had acquired through long-time practice. To what extent a telescope maker like Campani understood the dioptrics implicit in tables like those in Bolantio’s manuscript cannot be determined. First rank, specialized telescope makers like Divini and Campani had received some formal education, so they may have been able to read and study a book like Dioptrice. It remains to be seen whether a question like this is relevant at all. I doubt whether dioptrical knowledge would have been of any use in the design and manufacture of telescopes. They knew very well the effect of diverse types of lenses, but this probably was experiential knowledge. Innovative craftsmen like Wiesel were able to find new configurations with improved properties. These are likely to have been the product of trial and error. It has been said that Kepler’s configuration was the only contribution from the theory of dioptrics to the improvement of the telescope.44 Still, its advantages had to be discovered in practice. The 40

Bedini & Bennet, “Treatise”, 117. Willach discusses dioptrical theory emerging from the correspondence of Rheita en Wiesel which suggests similar lines. Willach, “Development of telescope optics”, 390-394. 42 For example: Fontana’s Novae coelestium (1646) and Campani, Lettere di Giuseppe Campani intorne all'ombre delle Stelle Medicee (1665). 43 OC21, 252-290. 44 Van Helden, “The telescope in the 17th century”, 44-49. 41

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improvement of the telescope was the result of the artisanal process of trialand-error. Better configurations were designed by making them, not made by designing them. 3.1.2 INVENTIONS ON TELESCOPES BY HUYGENS After Tractatus followed a decade of practical dioptrics, that was crowned by the publication of Systema Saturnium. Together with his brother, Huygens had become a skilled telescope maker and could already pride himself on some innovations of the instrument. In the previous chapter, one of these innovations has been discussed: a device to make telescopic measurements. It is not known how Huygens discovered the principle of the micrometer. The discovery was probably related to an innovation of the telescope he had developed somewhat earlier: the diaphragm. The diaphragm improved the way images were enhanced by blocking part of the light entering the telescope. Early in 1610 Galileo discovered that telescopic images became more distinct when he covered the objective lens with a paper ring.45 He determined the optimal size and shape of the ring by means of trial-and-effort and did not try – at least not on paper – to explain the effect dioptrically. As contrasted to such an aperture stop, a diaphragm is inserted into the focal plane. It has the advantage of diminishing the effect we call chromatic aberration. In December 1659 Huygens first employed a diaphragm in his 23-foot telescope.46 As he related in 1684: “N.B. In 1659 in my system of Saturn, I have taught the use of placing a diaphragm, as it is called, in the focus of the ocular lens, without which those telescopes cannot be freed from the defects of colors.”47

Apparently, he recognized the combining a diaphragm with some measuring device a bit later.48 The fact that an object inserted in the focal plane casts a sharp shadow over things seen through the telescope seems a logical consequence of Huygens’ understanding of the dioptrics of a Keplerian configuration. Still, it took him some time to recognize its usefulness and this may well have been a chance discovery. The fact that a diaphragm reduces ‘the defects of colors’ did not follow from his dioptrical theory and had to be discovered in practice. Until the 1660s, Huygens’ approach to telescope making did not differ substantially from that of an ordinary craftsman. We have seen his unmatched understanding of dioptrical theory but it cannot be told what role it played in his practical pursuits. In Systema saturnium, he described his micrometer in a procedural way, without explaining it analytically in dioptrical terms. The book contained only one dioptrical passage. He wrote 45

Bedini, “The tube of long vision”, 157-159. OC15, 56. 47 OC13, 826. “N.B. me anno 1659 in Systemate Saturnio meo docuisse usum diaphragmatis quod vocant, in foco ocularis lentis ponendi, absque quo colorum vitio haec telescopia carere non poterant.” In 1694 he explicitly claimed that he was the first to use a diaphragm: OC13, 774. 48 McKeon, “Les débuts I”, 237. 46

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that the power of a telescope could better be determined by calculation than using the ordinary ways of comparison. He referred to a theorem in “Dioptricis nostris”: the magnification is equal to the proportion of the focal distances of objective and ocular.49

Figure 24 Huygens’ eyepiece. (see also the diagram in Figure 25)

The year 1662 marks a turn in Huygens’ dioptrics. He invented something new and then turned to dioptrical theory again. The invention was a particular configuration of three lenses in a compound ocular (Figure 24). Nowadays called ‘Huygens’ eyepiece’, it had considerable advantages over earlier solutions: it produced a large field of view and images that suffered relatively little from aberrations.50 Huygens had developed the eyepiece after his trip to Paris and London in 1660-1, where he had talked much on telescopes and related matters. In Paris he had seen the artisan Menard and the ingeneer Pierre Petit, who had the best collection of telescopes in Paris. In London he saw telescopes with compound eyepieces made by the telescope makers Paul Neile and Richard Reeve.51 In 1662, Huygens made his first telescopes with field lenses. Later that year, he found out what configuration of lenses produced bright images and a wide field. On 5 October he wrote to his brother Lodewijk in Paris: “As for oculars, you will see that I have found something new that causes that distinctness in daytime telescopes [i.e. terrestrial], and the same thing in the very long ones, while giving them at the same time a wide opening.”52

Huygens’ design quickly became known and was adopted widely. How Huygens had found the precise configuration is unknown, yet everything points at it being a matter of trial-and-error inspired by the examples he had seen.53 After the invention, however, Huygens did something others like Wiesel and Reeve did not do. He set out to understand how it worked by analyzing the dioptrical properties of his eyepiece. Huygens described its configuration in a proposition inserted in the third part of Tractatus.54

49

OC15, 230-233. Van Helden, “Compound eyepieces”, 33; Van Helden, “Huygens and the astronomers”, 158. 51 OC22, 568-576. 52 OC4, 242-3: “car pour les oculaires vous voyez bien que j’y ay trouvè quelque chose de nouveau, qui cause cette nettetè dans les lunettes du jour, et de mesme dans les plus longues, leur donnant en mesme temps une grande ouverture.” 53 Van Helden, “Compound eyepieces”, 33. 54 OC13, 252-259. The text in Oeuvres Complètes is probably from 1666. The notes contain some previous phrasing, probably from 1662. OC13, 252n1 50

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“Although lenses should not be multiplied without necessity, because much light is lost due to the thickness of the glass and the repeated reflections, experience has shown it is nevertheless useful to do so here.”55

When the single ocular lens is replaced by two lenses, so Huygens continued, the field of the telescope can be enlarged. Moreover, the images produced are less deformed and the irregularities of the lenses are less disturbing. The precise configuration of the eyepiece was as follows (Figure 25). AB is the objective lens, CD and EF form the eyepiece; the focal distances are LG, KT, and SH Figure 25 Diagram for the eyepiece, accompanying respectively. Now, KT is Figure 24. about four times SH, and the distance KS between the ocular lenses is about twice the focal distance SH of the outer one. Finally, the focus G of the objective lens AB should fall between the outer ocular EF and its focus H in such a way that H is the ‘punctum correspondens’ of point G with respect to lens CD. Rays coming from a distant point Q will therefore be parallel after refraction in the outer ocular lens EF. Having determined the position of the eye M and the magnification by the system, Huygens concluded by explaining that points P and Q of the object are seen sharp but reversed. Huygens had demonstrated that this configuration produced sharp, magnified images. This was a rather straightforward application of the theory he had developed in 1653. The text bears witness to the fact that Huygens had gained much experience with actual lenses since the days of Tractatus. At one point in the theorem, he indicated why images do not suffer much from the irregularities in the lenses. Because the eye is so close to the outer ocular, “the spots and tiny bubbles of air, that are always in the material of the glass, cannot be perceived in lens EF. But one does not see them in lens CD either, because the eye perceives objects placed there confusedly, but those that are located close to H distinctly.”56

Huygens did not, however, explicitly compare the field of his configuration to that of a telescope with a single ocular, nor did he explain why images were less deformed.57 His analysis offered a dioptrical understanding of his eyepiece but it did not improve it:

55

OC13, 252-253. “Quanquam lentes non frustra sint multiplicandae, quod et vitri crassitudine et iteratis reflexionibus non parum lucis depereat; hic tamen utiliter id fieri experientia docuit.” 56 OC13, 256-257. “Atque ex hac oculi propinquitate sit primum ut naevi, seu bullulae minutissimae, quibus vitri materia nunquam caret, in lente EF percipi non possint. Sed neque in lente CD; quoniam oculus confuse cernit quae hic objiciuntur, distincte vero quae ad H.” 57 He developed a systematic theory of the field of view of a telescope much later, after 1685: OC13, 450461, 468-73.

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“We give here, if not the best combination of all lenses, the investigation of which would take long and might be impossible, but one which experience has shown us to be useful.”58

The particular configuration of Huygens’ eyepiece was a product of trial-anderror, and theory could not, or not yet, add to that. Huygens the scholar had not yet been able to assist Huygens the craftsman. As contrasted to other telescope makers, however, Huygens was able to understand retrospectively and in mathematical terms, what he was doing when configuring lenses. That is to say, he understood the dioptrical properties of lenses and their configurations. He could explain whether and how a configuration of lenses produced sharp, magnified images. But he could not explain everything of the kind. In another proposition found in part III of Tractatus and apparently following the one discussed above, Huygens discussed a telescope with an erector-lens such as Kepler had proposed.59 He concluded with some remarks about the quality of images produced by various configurations. With a telescope consisting of a convex objective and a concave ocular – the Galilean configuration – images are more distinct “and defiled by no colored rims that can hardly be prevented in this composed of three lenses.”60 A well-chosen combination of lenses could counter these defects, but “different people combine ocular lenses differently with regard to each other, looking for the best combination with only the guide of experience. It would not be easy, to be sure, to teach something about this that is grounded in certainty, since the consideration of colors cannot be reduced to the laws of geometry, ….”61 [italics added]

In his practical work Huygens had found out that lenses suffered from all kinds of defects. Some of these eluded dioptrical analysis. But he had also found out that nuisances caused by fogs, bubbles and colors could be diminished. The diaphragm had already proven this. His eyepiece gave another means to improve the quality of images.62 He could not fully explain its advantages, nor could he improve it by means of dioptrical analysis. Still, the eyepiece had proven that a well-chosen configuration of lenses could be advantageous. And it made him realize that even better configurations could be found, even though he was as yet pessimistic about such an enterprise. If Huygens the scholar could gain a thorough understanding of the defects of 58

OC13, 252-253. “Dabimus autem in his, etsi non omnium optimam lentium compositionem, quam investigare longum esset ac forsan impossibile, at ejusmodi quam nobis experientia utilem esse ostendit.” 59 OC13, 258-265. Discussed above, section 2.1.2.. 60 OC13, 262-263. “… res visas, atque etiam distinctiores efficere, nullisque colorum pigmentis infectas quod in hic lentium trium compositione aegre vitari potest.” 61 OC13, 264-265. “Alij vero aliter lentes oculares in his inter se consociant, sola experientia duce quid optimum sit quaerentes. nec sane facile foret certa ratione aliquid circa haec praecipere, quum colorum consideratio ad geometriae leges revocari nequeat, …” 62 A way to reduce colors that was more commonly employed, was to make objective lenses with large focal distances. These, however, had the drawback that telescopes became very long and tubes too heavy to remain straight. In 1662, it occurred to Huygens that this could be circumvented by making a tubeless telescope. He realized it much later and published a little tract on it, Astroscopia Compendiaria (1684). OC21, 201-231.

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lenses, he might teach Huygens the craftsmen how to combine lenses in the best possible way. The next decade he actually set out to do this. 3.2 Dealing with aberrations According to Hugyens, not all defects of lenses could be explained dioptrically. One particular defect, however, was subject to the laws of geometry: spherical aberration. It could therefore be explained and, possibly, prevented. Huygens was not the first to design a solution to prevent the defects of lenses. Descartes had done so with his elliptic and hyperbolic lenses. Newton would built a mirror telescope in order to avoid the defects of (spherical) lenses. Huygens was the first to take spherical lenses as a starting point for a theoretical design, instead of ruling them out beforehand. In 1665, he began a study of spherical aberration with the intention to design a telescope consisting of spherical lenses such as to neutralize each others’ aberrations. The idea that the lenses of a telescope might cancel out their mutual aberrations had occurred to no-one yet: “Until this day it is believed that spherical surfaces are … less apt for this use [of making telescopes]. Nobody has suspected that the defects of convex lenses can be corrected by means of concave lenses.”63

The project added a new dimension to Huygens’ dioptrical studies. No longer did he just want to understand the telescope, but now he also wanted to improve it by means of dioptrical theory. In so doing, he followed Descartes’ ideal that the scholar could lead the craftsman, but it had taken on a different form. Huygens started out with what was practically feasible instead of what was theoretically desirable. Spherical lenses had been the focus of both his theoretical investigations and his practical activities. It looks like Huygens now wanted to combine these two sides of his involvement with telescopes. 3.2.1 PROPERTIES OF SPHERICAL ABERRATION In order to be able to determine an optimal configuration of lenses, Huygens first had to develop a theory of spherical aberration. The phenomenon had been known for a long time. In perspectivist theory it was known that a burning glass does not direct all sunrays to one point. No one, however, had gone beyond the mere recognition of the phenomenon, and its exact properties had not been studied. Kepler went farthest by pointing out the connection between a ray’s distance from the axis and its deviation from the focus, but this necessarily remained qualitative.64 Mathematicians like Descartes had focused on determining surfaces that did not suffer from such aberration. With his concept of ‘punctum concursus’ of Tractatus, Huygens had been the first to take spherical aberration into account in dioptrical theory, defining the focus as the limit point of intersecting rays. In 1665 he 63

OC13, 318-319. “creditum est hactenus … sphaericae superficies minus aptae essent his usibus, nemine suspicante vitium convexarum lentium lentibus cavis tolli posse.” 64 Kepler, Paralipomena, 185-186 (KGW2, 168-169). Kepler repeated his insights in Dioptrice.

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extended this by developing a theory of spherical aberration. He subjected, so to say, the ‘punctum concursus’ to a closer examination to see how exactly spherical aberration affected the imaging properties of lenses. Huygens’ study of spherical aberration had been preceded by a calculation of rays refracted by a plano-convex lens he had apparently carried out in 1653. In Tractatus, he remarked that rays “reunite somewhat better, i.e. that the points where they cut the axis are closer to one point, …, when the convex surface faces the incident rays, than when the plane surface faces them.”65 The 1653 calculation is lost, but was probably identical to later ones.66 The result implied that the orientation of a lens affected the degree of aberration. In 1665, Huygens went to see whether the amount of spherical aberration might deliberately be decreased by a proper configuration of lenses. He began a study of spherical aberration under the heading “Adversaria ad Dioptricen spectantia in quibus quæritur aberratio a foco”.67 After a decade of quiet, the ‘Adversaria’ was the next chapter of Huygens’ dioptrical studies. In ‘Adversaria’ Huygens derived expressions for the amount of spherical aberration as it depends upon the properties of a lens. The rigor familiar from Tractatus returns immediately. On the basis of the theorems of Tractatus, he took the refractions at both faces of the lens as well Figure 26 Spherical aberration of a as its thickness into account. In the first plano-convex lens. calculations Huygens returned to the claim of 1653. He derived an expression for the aberration of the extreme ray incident on a plano-convex lens GBC with focal distance GS (Figure 26). A parallel ray is refracted at the extreme point C of the lens towards T on the axis. The derivation of the aberration TS is straightforward and yields 65

OC13, 82-83. “…, accuratius aliquanto eos propiusque ad unum punctum convenire …, cum superficies convexa venientibus opposita est radijs, quam si plana ad illos convertatur.” Huygens had also written this to Gutschoven in his letter of 6 March 1653: OC1, 225. As we have seen above, Flamsteed carried out a numerical calculation and came to the same conclusion, which returned in Molyneux’ Dioptrica nova. Flamsteed, Gresham Lectures, 120-127. Molyneux, Dioptrica nova, 23-25. 66 OC13, LII (“Avertissement”), those later calculations are on pages 283-287. 67 OC13, 355-375.

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= 76 BG, where BG is the thickness of the lens.68 When the lens is reversed and rays are incident on the plane side, the aberration becomes 69 9 TS = 2 BG. The aberration is therefore considerably smaller – almost four times – when the convex side faces the incident rays. This time Huygens went further than the mere observation that the orientation of a lens affects the amount of aberration. The faces of a lens are surfaces with different radii – infinite in the case of a plane face. The proportion between these radii apparently determines how large the aberration is. Consequently, an ideal lens can be found by determining the optimal proportion of both radii. To do so, Huygens derived an expression for the aberration of a parallel ray HC incident on the extreme end of a lens IMCB (Figure 27). AB = a and NM = n are the radii of the anterior and posterior side and BG = b is the thickness of the anterior half of the lens. The thickness of the entire lens BM = q can be expressed as TS

q = b  ba . The anterior face refracts n

an extreme ray HC towards P, a little off its focus R. The posterior face, in its turn, refracts the extension CF of ray CP towards D, a little off the focus E of the lens. Huygens then expressed the spherical aberration DE of the extreme ray in terms of the radii of the faces and the thickness of half the lens: DE =

68 69

OC13, 357. OC13, 359.

7n 2 q  6anq  27a 2 q “… 6(a  n )2

Figure 27 Aberration of a bi-convex lens

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the space on the axis within which all parallel rays are brought together, which space DE is defined by this rule.”70 Or, the aberration DE is found by multiplying the thickness of the lens q by the expression

7n 2  6an  27a 2 , 6( a  n )2

which only depends on the radii of both faces. The shape of a lens that produces minimal aberration can be found by determining the minimum of this expression; this yields a : n = 1 : 6.71 In this case the aberration of the extreme ray DE = 15 q. Huygens found the same for a bi-concave lens, 14

whereas a converging meniscus lens yielded a meaningless outcome.72 Satisfied, he summarized the result: “In the optimal lens the radius of the convex objective side is to the radius of the convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.”73

The ‘Adversaria’ provided general expressions for spherical aberration in terms of the shape of a lens. It contained a set of derivations and calculations without explanation. He did not, for example, point at certain simplifications he had carried out. The results were not therefore fully exact, as will become clear later on. Still, it was the most advanced account of spherical aberration at the time. On the basis of his theory of spherical aberration he went on to design a configuration of lenses that minimized the ‘aberrations from the focus’. A note of clarification needs to be made. Huygens did not yet call the phenomenon he was investigating spherical aberration. Around 1665, Huygens referred to it in a general way: “aberration from the focus” and “Investigate which convex spherical lens brings parallel rays better together.”74 Only much later, when distinguishing the aberration caused by colors, did he explicitly called it “the aberrations of rays that arise from the spherical shape of the surfaces”.75 We should bear this in mind when interpreting Huygens’ study of aberrations and his designs for perfect telescopes. That is, we do not know for certain what exactly he thought his design would improve. Specilla circularia Before continuing with Huygens, mention has to be made of another study of spherical aberration. Not because it mattered much for the mathematical theory of spherical aberration – it did not – but because it approached the 70

OC13, 364. “DE spatium in axe intra quod radij omnes paralleli coguntur, quod spatium DE per regulam hanc definitur.” 71 OC13, 366-367. Modern methods yield the same result. 72 OC13, 375 and 370. In the latter case the solution yields a negative value for the radius of the posterior side. 73 OC13, 367. “Radius convexi objectivi ad radium convexi interioris in lente optima ut 1 ad 6. EUPHKA. 6 Aug. 1665.” 74 OC13, 280n2. “Quaenam lens sphaerica convexa melius radios parallelos coligat investigare.” 75 OC13, 280-281. “aberrationes radiorum quae ex figura superficierum sphaerica oriuntur”

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problem central to La Dioptrique in an original way. Moreover, it preceded Huygens’ study and he may have known it in some way. The study is found in two manuscript copies of Specilla circularia, a tract presumed to have been written in 1656 by Johannes Hudde, an acquaintance of Huygens.76 The fact that Hudde had written on dioptrics was known from his correspondence with Spinoza.77 Apparently Spinoza had a copy of Specilla circularia, as he referred to a ‘small dioptrica’ by Hudde and some of his own figures and calculations are clearly based on it. In addition, a tract called Specilla circularia turns up in Huygens’ correspondence in 1656. On 30 May 1656, Van Schooten wrote that he had recently bought an anonymously published treatise called Specilla circularia. He supposed it was written by Huygens “because of its accuracy”.78 Huygens replied that it was not and that he had never heard of it.79 He asked for a copy, but it is not clear whether he ever received one. Huygens corresponded with Hudde on mathematical topics, but they did not discuss dioptrics. Huygens visited Spinoza several times around 1665 and they discussed dioptrical matters extensively. Whether or not he knew Specilla circularia, it would not have added to Huygens’ understanding of spherical aberration. Probably he would not have accepted Hudde’s analysis and conclusions, either. The main goal of Specilla circularia was to show there was no point in striving after the manufacture of Descartes’ aspherical lenses. In practice one legitimately makes do with spherical lenses, because spherical aberrations are sufficiently In order to Figure 28 Hudde’s calculation of spherical aberration small.80 substantiate this claim, Hudde employed an original definition of the focus of a lens (Figure 28). AB is a ray parallel to the axis DNI at distance BF. It is refracted to BI by a convex surface with radius DN. Choosing DN = 1 and an index of refraction 20 : 13, Hudde calculated the length of NI for various values of BF, concluding that the smaller BF the larger NI (where I approaches K).81 Considering 76

The original tract is lost, but has been identified by Vermij with two manuscript copies discovered in London and Hannover. Both are dated 25 April 1656 and one gives the name of the author: “Huddenius consul Amstelodamensis”, which suggests the copy itself was made in or after 1672. Vermij, “Bijdrage”, 27; Vermij and Atzema, “Specilla circularia”, 104-107. 77 Spinoza, “Briefwisseling”, 251. Spinoza’s letters contain calculations that are similar to those in Specilla circularia. The letter can also be found in OC6, 36-39, where it is assumed to be addressed to Huygens. 78 OC1, 422. “propter accurationem” 79 OC1, 429. 80 Vermij and Atzema, “Specilla circularia”, 119. 81 Vermij and Atzema, “Specilla circularia”, 116: “Ex quibus patet, quanto x sive BF minor est, tanto etiam punctum I longius distare ab N;”

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numerically all rays between B and F, he calculated the proportion of BF to Im, through which all refracted rays pass. Seeing that IM is small compared to BF, Hudde concluded that K could be regarded as the focus.82 According to Hudde, the focus was not an exact, geometrical point, but a ‘mechanical point’, a point that cannot be divided mechanically or whose parts are not truly discernable.83 This practical outlook made him reject Descartes’ proposal as superfluous. Hudde’s study lacked Huygens’ rigor. From a mathematical point of view, he explained away spherical aberration. He attained ‘practical’ exactness, rather than mathematical, much in the same way as a Flamsteed or Molyneux. Hudde called in question whether spherical aberration was as relevant a problem as Descartes had claimed it to be. In Specilla circularia, he argued that in practice it was not. The spirit of Hudde’s study of dioptrics was similar to that of Huygens’: to see what mathematics could teach about the working of lenses in practice. Hudde’s conclusion was the opposite of Huygens’. In Huygens’ view, an exact understanding of the phenomenon might yield a telescope that actually smoothed aberrations away. Theory and its applications Sometime after writing the ‘Adversaria’, Huygens elaborated it into a rounded essay on spherical aberration. It contained his first solution to the problem his study was aimed at: a configuration of spherical lenses that neutralized spherical aberration. The essay is found in the Oeuvres Complètes under the title De Aberratione radiorum a foco. In De Aberratione Huygens worked up and extended his earlier notes. He set up his argument with a definition of the thickness of a lens and several auxiliary propositions.84 Besides the expressions he had given in the ‘Adversaria’ for the aberration of extreme rays, he established the relationship between the aberration of an arbitrary ray and its distance from the axis.85 In the fourth and fifth propositions of De Aberratione the results of the ‘Adversaria’ returned. Huygens now explicated the simplifications he had carried out earlier. He first derived a more exact expression – which I will not give – for the aberration of the extreme ray incident on the plane side of a plano-convex lens. When the radius of the convex side is 72 inches and the extreme ray is 1 inch from the axis, this expression yielded an aberration of 31253 inches. Huygens then stated – without proof – that the aberration 1000000 could be found more easily by multiplying the thickness of the lens by 29 , the rule found in the ‘Adversaria’.86 There is, he admitted, a slight difference 82

Vermij and Atzema, “Specilla circularia”, 117: “Unde constat, focum ipsum pro puncto mechanico tantum habendum esse.” 83 Vermij and Atzema, “Specilla circularia”, 114: “Punctum autem mechanicum appello, quod in mechanicis aut divisible non est, aut cujus partes hic non sunt considerata digna.” 84 OC13, 276-277. 85 OC13, 308-313. 86 OC13, 282-285. Each time he assumed an index of refraction 3 : 2.

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1 inches) but this was of no significance in actual telescopes.87 When ( 1000000 the convex side of the same lens faced the incident rays, the exact calculation 81021 yielded an aberration of 10000000 inches. In this case, the easier rule of

‘Adversaria’ – multiplying the thickness of the lens by

7 6

– gave

81022 10000000

,a

1 difference of only 1000000 inches. Again, the main goal of this exercise was to show that the aberration of a plano-convex lens is least when its convex side faces the incident rays. 88 Continuing with a bi-convex lens, Huygens sketched out how the aberration might be calculated exactly, but immediately moved on to an ‘abbreviated rule’ he had ‘found’.89 This was the expression of the ‘Adversaria’, found by “ignoring very little quantities, but judiciously so as needed.”90 The rule applied to convex as well as to concave lenses and yielded the optimal proportion of both radii of 1 : 6. The resulting bi-convex

lens produces an aberration of only 15 times its thickness.91 14

Surprisingly, these laborious derivations were not of great value for telescopes. After having explained the optimal proportions of bi-concave lenses, Huygens wrote that they were not useful as ocular lenses. In telescopes, he said, one should choose “… other, less perfect lenses, so that the defects of the convex lens are compensated and corrected by their defects.”92 Those less perfect lenses were diverging concavo-convex lenses. Huygens showed that these lenses always produce a larger aberration than biconvex or bi-concave lenses. As ocular lenses they could, however, be useful: “With concave and convex spherical lenses, to make telescopes that are better than the one made according to what we know now, and that emulate the perfection of those that are made with elliptic or hyperbolic lenses.”93

Here was what Huygens had been looking for: a configuration where lenses mutually cancel out their aberration. He had designed a telescope in which the ocular corrects for the aberration of the objective lens, thus equaling the effect of a-spherical lenses. The solution was as follows: given an objective lens and the required magnification of a telescope, determine the shape of the ocular lens (Figure 29). On the axis BDFE of lens ABCD, divide the focal distance DE by point F 87

OC13, 284-285. “Exigua quidem differentiola, sed quae in illa lentium latitudine quae telescopiorum usibus idonea est, nullius sit momenti.” 88 OC13, 284-287. 89 OC13, 290-291. “Et haec quidem methodus ad exactam supputationem adhibenda esset. Invenimus autem et hic Regulam compendiosam …” 90 OC13, 290-291. “Quae regula … inventa est neglectis minimis, sed necessario cum delectu.” 91 OC13, 290-291& 302-303. 92 OC13, 302-303. “…, sed aliae minus perfectae, quarum nempe vitijs compensantur ac corrigentur vitia lentis convexae, …” 93 OC13, 318-319. “Ex lentibus sphæricis cavis et convexis telesopia componere hactenus cognitis ejus generis meliora, perfectionemque eorum quæ ellipticis hyperbolicisve lentibus constant æmulantia.”

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according to the chosen magnification. The ocular GFH is to be placed in F. For example, DE : FE = 10 : 1 when the telescope should magnify ten times. Because the foci of both lenses should coincide, the focal distance of the ocular lens is given, namely FE. Due to its spherical aberration, the objective lens does not refract parallel rays KK, CC to E but to N and O along KLN and CHO. After refraction by the ocular lens rays LM, HI should all be parallel. This can be accomplished when the aberrations NE, OE are the same for the objective and ocular lenses with respect to the parallel rays KK, CC and LM, HI respectively. With the expression for the aberrations of both lenses, Huygens could determine the required radii of the ocular lens. The radius of 86 times its focal distance FE, the radius of the the convex side should be 100 86 concave side 272 times FE.94 Next, he proved that this ocular indeed canceled out the aberration of the objective.95 Ergo, the proposed configuration would produce almost perfect images.

Figure 29 Galilean configuration in which spherical aberration is neutralized.

Huygens supplied a table in which he listed telescopes with various magnifications against the ocular lenses required according to his analysis. These numerical examples were, so to say, the blueprint by means of which his design could be realized by any skilled worker. By way of conclusion, Huygens remarked that the advantages of his design could only be realized by lenses that were truly spherical. The manufacture of spherical lenses should therefore be resumed diligently. Huygens had made clear at the outset of his exposition that the usefulness of his design was limited. Only a concave ocular could correct for the aberration of the objective lens. The design was therefore useful only for Galilean telescopes. In astronomy, telescopes required a convex ocular: “However, it is certain that this mutual correction is not found in the composition of convex lenses. On the contrary, the defect of the exterior lens is always a bit augmented by the ocular lens and it cannot be remedied in any way.”96

Still, his theory was not entirely useless for the Keplerian telescopes required for astronomical observation. In the final propositions of De Aberratione, 94

OC13, 320-323. OC13, 324-327. For the rays KK and LM – that are not extreme rays – Huygens used the proposition on the linear proportion between aberration of a ray and the square of its distance to the axis. OC13, 308313. 96 OC13, 318-319. “Sed certum est in convexis inter se compositis emendationem illam mutuam non reperiri. Imo contra, vitium exterioris lentis a lente ocularis augetur semper nonnihil neque id ulla ratione impediri potest.” 95

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Huygens examined the means to enhance the quality of images produced by telescopes with a convex ocular. That is, he took a theoretical look at the matter. On the basis of this analysis, he could provide directions for optimizing the quality of telescopes with convex oculars. The magnifying power of a telescope depends upon the ratio of the focal distances of objective and ocular lenses, and can therefore be increased by reducing the focal distance of the ocular lens. This, however, simultaneously decreases the clarity and distinctness of images. To maintain clarity at the same time, the opening of the objective lens would have to be made larger.97 He began by considering a naked eye in front of which a telescope is placed. Assuming that an equal number of rays should enter the eye when a more powerful telescope is taken, Huygens argued that the opening should be kept proportional to the magnification. This implied that his 22-foot telescope would need an opening of 125 times the area of the pupil. In reality, he observed, a satisfactory telescope had a much smaller opening, only 15 times the area of the pupil. Evidently, in astronomical observation one could do with much smaller clarity. He therefore did not take the eye as starting-point, but a telescope with satisfactory quality. If the ocular is replaced by an ocular that magnifies twice as much, the clarity will be four times smaller. The opening of the objective should therefore be increased accordingly. Evidently, this cannot be done at will and one should “consider accurately which magnification the opening of the exterior lens can support”.98 Maintaining the clarity of images does not mean, however, that their quality is maintained. Increasing the opening of a lens renders images less distinct. Huygens made it clear that only experience could tell which configuration produced satisfactory images. Yet, when such a telescope is known, theory can explain how the quality of images is maintained when its strength increases. In his account, Huygens applied a new conception of spherical aberration that he had defined in an earlier proposition of De Aberratione. He called it the ‘circle of aberration’. As contrasted to the earlier conception, in which the aberration GD is measured along the axis, the circle of aberration is measured by the distance ED perpendicular to the axis (Figure 30). In other words, the circle of aberration is the spot produced by parallel rays coming from one point of a distant object. Consequently, the images produced by two Figure 30 ‘Circle’ lens systems are equally clear and equally distinct when the of aberration. respective circles of aberration are the same.99 97

OC13, 332-335. OC13, 336-337. “sed diligenter expendendum quale incrementum exterioris lentis apertura perferre valeat” 99 OC13, 340-343. 98

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Huygens supposed that the circle of aberration XV of a lens system is mainly produced by the objective lens AB (Figure 31). The ocular lens PO barely increases the diameter of the circle and could therefore be considered to have a perfect focus. He considered the opening BC of the objective lens required to maintain a constant circle of aberration when the focal distance CD of this lens is changed. He proved that the proportion CD3 : CB4 should remain constant.100 Finally, the quality of images will be maintained upon changing the ocular lens, when the proportion OD : 4 CD between the focal distances of both lenses is maintained.101 Again, Huygens converted these proportions into a table of numerical values, listing the optimal values of the focal distances of both lenses and the opening of the objective, as well as the resulting magnification of the system.102 This table concluded De Aberratione.

Figure 31 Aberration produced by a Keplerian configuration.

Huygens’ theoretical accomplishments in De Aberratione are beyond dispute. Like the theory of focal distances and magnification of Tractatus, his theory of spherical aberration was rigorous and general. And again his theoretical studies were aimed at understanding the telescope; in this case, at understanding how a system of lenses produces spherical aberration. Huygens could claim that he understood why an opening of such-and-such dimensions maintained the quality of images. His results were couched in two tables listing the required components to make these optimal systems, in a way quite comparable to the ones found in Bolantio’s manuscript. They prescribed how to assemble a telescope without presupposing theoretical knowledge of dioptrics. The difference is that Huygens’ tables were derived from his mathematical theory of lenses instead of a record of experiential knowledge. The table prescribing the aperture of telescopes was not gained by some implicit rule of thumb, but was based on an explicit theorem derived from dioptrical properties of lenses. Huygens could prove that the openings he prescribed were optimal. Whether this worked in practice remains to be seen. At least he could claim that he could calculate beforehand how to adjust the components of a telescope when its length was changed, thus avoiding a renewed process of trial-and-error. Huygens had realized the goal of De Aberratione. He had demonstrated that the aberrations of spherical lenses could be made to cancel out. 100

OC13, 342-345. OC13, 348-351. 102 OC13, 350-353. 101

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Moreover, he had employed theory to improve the telescope. The design was still a blueprint, and at this point his accomplishments were theoretical only. He had developed a further understanding of the properties of spherical lenses and found means to configure them optimally. He had not yet ‘tested’ his designs, nor had he verified his theory as a whole. For example, his concept of circle of aberration suggests a way to study the observational properties of spherical aberration, to see whether it correctly described the defects of lenses. Nowhere, however, did he refer to something of the kind. I will return to this point below. Huygens’ next step was an attempt to realize his design of a telescope in which ocular and objective lenses cancelled out their mutual aberrations. A test to the theory? 3.2.2 PUTTING THEORY TO PRACTICE Not until 1668 did Huygens set about realizing his design.103 By that time he lived in Paris, where he had arrived in the summer of 1666.104 In the meantime, telescopes had not been out of his mind, though. They were frequently discussed in his correspondence with Constantijn. He examined the quality of glass and lenses made by Parisian craftsmen, not being impressed.105 He was particularly dissatisfied by a telescope he had bought for his father – a campanine made by one Menard.106 He equipped a campanine of his own with lenses made by his brother and was pretty contented with it.107 In April 1668, he decided to have Constantijn make lenses for the design of De Aberratione.108 On 11 May 1668, Huygens gave his brother detailed instructions to grind a set of lenses. For the objectives Constantijn made – plano-convex lenses of 2 feet and 8 inches – a concavo-convex ocular was required. The radii ought to be 0,187 and 0,289 inches, respectively, and Huygens drew out the shapes in his letter.109 Combined, these lenses would perform like hyperbolic glasses, he said, “… because the concave lens corrects the defects arising from the spherical shape of the objective lens. therefore I cannot determine the opening of the objective that maybe might be 3 or 4 times larger than an ordinary one has, but if we can just double it much would be gained and the clarity will be sufficiently large for the magnification of 30.”110

103

OC13, 303n4; 331n4. OC5, 375; OC6, 23. 105 OC6, 151; 205; 207. 106 OC6, 86-87; 151; 205. 107 OC6, 207. 108 OC6, 209. 109 OC6, 214-215. 110 OC6, 214. “Ce composè, …, doibt faire autant que les verres hyperboliques, parce que le concave corrige les defauts de l’objectif qui vienent de la figure spherique. c’est pourquoy je ne puis pas determiner l’ouverture de l’objectif qui peut etre pourra estre 3 ou 4 fois plus grande qu’a l’ordinaire, mais si nous la pouvons seulement faire double ce sera beaucoup gaignè et la clartè sera assez grande pour la multiplication de 30.” 104

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Huygens did not explain the ‘secret’ of his new method to his brother. He urged him not to tell anyone about the plans. Constantijn responded quickly. On the first of June, Huygens answered two letters – now lost – his brother had sent on May 18 and 24.111 Constantijn had sent an ocular with only one side ground according to his instructions, the other being plane. Apparently, Constantijn had made some objections to his brother’s design. Huygens did not agree and urged his brother to make a lens exactly to his directives. Huygens did not await new lenses, but immediately tried the one Constantijn had sent him. A week later he reported on the disappointing results. When the objective lens was covered in ordinary fashion, the Figure 32 Rendering of Huygens’ sketch. system performed reasonably well. Yet, the system fell short of his expectations. According to his design, the quality of the image should be maintained when the whole objective lens was exposed to light. (Figure 32) “but uncovering the entire glass I see a bit of coloring which leads me to believe that there is an inconvenience therein, which results from the angle made by the two surfaces of the objective at the edges. This necessarily causes colors, in such a way that by making hyperbolic glasses one encounters the same things when making them very large.”112

Huygens here tentatively drew an important conclusion. That is, we recognize that he was on the right track by suspecting that those colors were inherent to the refraction of rays and could not be prevented by hyperbolic lenses. Moreover, his suggestion that the production of colors could be linked to the angle of the lens’ surfaces was promising in light of Newton’s later theory of colors. The remark may have been inspired by a measurement Huygens had performed in November 1665.113 Having read Hooke’s account, in Micrographia, of colors produced in thin films of transparent material, he set out to determine the thickness of the film, which Hooke had not been able to do. He pressed two lenses together to produce colored rings. The colors appear where the two lenses nearly meet, a situation comparable to the thin rim of a glass lens. Whether this measurement and the remark of 1668 are connected is, however, mere speculation. In Micrographia, he also would have found discussions of prism experiments, and the effect of a prism may also explain the emphasis on the angle between the faces of the lens at the edge. Whatever be the case, Huygens did not pursue this line of thinking. He suspected that the proportions of Constantijn’s lens were not the gist of the problem, “but before assuring that, I would be pleased to 111

OC6, 218-220. OC6, 220-221. “mais en decouvrant tout le verre je vois un peu de couleurs ce qui me fait croire qu’il y a un inconvenient de costè la, qui provient de l’angle que font les 2 surfaces de l’objectif vers les bords. qui cause necessairement des couleurs, de sorte qu’en faisant des verres hyperboliques l’on trouueroit la mesme chose en les faisant fort grands.” 113 OC17, 341. Huygens’ measurements, as well as the experiments Newton performed at the same time, are amply discussed in Westfall, “Rings”. 112

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carry out the plan with an entire glass, like I have asked you to make for me.”114 During the following months, Huygens kept reminding his brother that he was waiting for the proper lens.115 He even considered taking up his own grinding work and started looking around in Paris for able craftsmen.116 On November 30, he sent his brother additional directives for oculars.117 On 1 February 1669, Huygens brought his invention to Constantijn’s attention for the last time: “You don’t talk anymore about the oculars you have promised me.”118 This was the final, somewhat aggrieved sentence of a letter in which he informed his brother of another letter – one he had received from a certain baron de Nulandt, an acquaintance of Constantijn living in The Hague at that time.119 The baron was engaged in making telescopes and also had some ideas regarding dioptrical theory. On 20 December 1668, Nulandt had written to Huygens. In the letter of 1 February to his brother, Huygens wrote: “The worthy Baron de Nulandt begins to talk like a great savant, and lets me coolly know that he has found the same proportions of glasses to imitate the hyperbola of which I have talked to him in my letter, although I am sure that this is infinitely beyond his capacities. The calculations he sends me are far from the truth, and I will not refrain from showing him this.”120

Huygens had told Nulandt about his idea of nullifying spherical aberration by means of spherical lenses in a letter now lost. On 18 January, Nulandt had replied that he had also found that a concave meniscus lens could correct the aberration of the objective lens, but had not given any details.121 In that letter, Nulandt calculated the amount of aberration for two lenses and had drawn conclusions that were contrary to Huygens’ own. Huygens’ letters in reply are lost, but it is clear that he easily convinced Nulandt of his mistakes. In his next letter, Nulandt admitted that his configuration for nullifying spherical aberration was faulty, because he had calculated the aberration of lenses in a wrong way.122

114

OC6, 221. “mais devant que de l’assurer je serois bien aise de faire l’essay avec un verre entier, que je vous ay priè de me vouloir faire.” 115 OC6, 236; 266. He did not show consideration for the fact that Constantijn was getting ready for his marriage on 28 August 1668. 116 OC6, 266; 300. 117 OC6, 299-300. 118 OC6, 353. “Vous ne parlez plus des oculaires que vous m’avez promis.” 119 Little is known about him. He published an anti-Cartesian treatise Elementa physica in 1669 in which he included an extract of a letter written by Christiaan (OC6, 420-421). He first appears in a letter to Huygens of 20 December 1668, which suggests that they had met, probably in Paris. OC6, 304-305. 120 OC6, 353, “Le Seigneur Baron de Nulandt commence a parler en grand docteur, et me mande froidement, d’avoir trouvè les mesmes proportions de verres, pour imiter l’Hyperbole, dont je lui avois parlè dans ma lettre, quoique je sasche bien que cela passe infiniment sa capacitè. Les calcus qu’il m’envoye sont trop eloignez de la veritè, et je ne manqueray pas de le lui remontrer.” 121 OC6, 348-351; particularly 350. 122 OC6, 363-367; particularly 364.

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A new design We could have passed over this episode with Nulandt, if its conclusion had not coincided with the next phase in Huygens’ study of spherical aberration. On that same 1st of February, he gloriously wrote down “A composite lens emulating a hyperbolic lens. EUPHKA”123 He had found a new solution to the problem of neutralizing spherical aberration that made his earlier one superfluous. It consisted of a combination of two lenses that would replace one objective lens. This composite lens could therefore be used in telescopes for astronomical observation, whereas the earlier solution was useful for terrestrial telescopes only. On February 22, he asked his brother Lodewijk to tell Constantijn “… that I abandon the little ocular I had asked from him, because I have found something better and more substantial in these matters, that I would like to try out myself.”124

Huygens’ idea was as follows (Figure 33). The bi-concave lens VBC and the plano-convex lens KSTG have the same focus E, with respect to diverging rays MV coming from M, and parallel rays QK, respectively. In addition, the spherical aberration EN produced by each lens is the same for these rays. An arbitrary ray QK, parallel to the axis ASM, is refracted by lens KST towards point N, a little off its focus E. Lens VBC, in its turn, refracts a ray MV towards the same point N, at the same distance from its focus E. Ray CN is therefore refracted towards M. As a result, the composite lens brings all parallel rays QK to a perfect focus M and “… will emulate a hyperbolical or elliptical lens perfectly.”125 The system acts as a converging lens and can therefore replace the objective of any telescope. In his proof, Huygens worked

123

Figure 33 The invention of 1669

OC13, 408. “Lens composita hyperbolicae aemula. EUPHKA 1 Febr. 1669.” OC6, 377. “Vous pourrez luy dire que je le quite pour ce qui est du petit oculaire que je luy avois demandè, ayant trouvè quelque chose meilleur et de plus considerable en cette matiere, dont j’ay envie de faire moy mesme l’essay.” 125 OC13, 413. “…[lens] compositae ex duabus VBC, KST, quae Hyperbolicae aut Ellipticae perfectionem aemulabitur.” 124

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the other way around.126 The bi-concave lens VBC is given. M is the center of surface BV, so that rays from M are not refracted by it. Surface CB of this lens refracts a ray MC to KCN, intersecting the axis in N, where EN is the spherical aberration. The problem is to find a convex lens KST with the same focus E, which refracts a parallel ray QK, at distance KS to the axis, to the same point N. Huygens chose BE – nearly equal to GE – as the focal distance of this lens KST. Its spherical aberration EN is – by the rule from the ‘Adversaria’ – 76 times its thickness GS. This length EN is also the spherical aberration of surface BC of the bi-concave lens VBC. It can be expressed in terms of its radius AB, the length BG (proportional to the distance CG of the ray to the axis), and the length BM. Equating both expressions for EN, he found a proportionality between the radius of KST and BC. It is 100 to 254, or nearly 2 to 5. In addition MB, the radius of the other surface BV of the bi-concave lens, has to be twice that of BC or ten times that of KST. At the end of his calculations Huygens summarized the solution: “A lens composed of two emulates a hyperbolic lens, the one plano-convex the other concave on both sides. The radii of the surfaces are nearly two, five, ten.”127

Five days later, on 6 February, he sent a letter to Oldenburg to which he appended an anagram containing his ‘important invention’:128 a bc d e h i l m nop r s t u y 5 2 2 1 4 1 23 3 1 3 2 232 4 1 This second invention can be regarded as the final piece of Huygens’ project of canceling out spherical aberration by means of spherical lenses. He had shown that spherical lenses were indeed apt for telescopes by designing a configuration that produced an almost perfect focus. As contrasted to the earlier invention of 1665, this one could improve telescopes used for astronomical observation.129 What remained to be done, was to test the design. We should remember that it was not an ordinary project Huygens had embarked on. His theoretical investigations of spherical aberration served the practical goal of improving actual telescopes. With this he marked himself off from both theoreticians and practitioners. Unlike other telescope makers – as he manifested himself earlier – he had aimed at improving the telescope by means of theoretical study. The configuration in which aberration was to be neutralized was not the result of trial-and-error like his eyepiece, but of mathematical analysis of lenses and calculating the optimal 126

OC13, 411-413. OC13, 417n2. “Lens e duabus composita hyperbolicam aemulatur, altera planoconvexa altera cava utrimque. Semidiametri superficierum sunt proximè duo, quinque, decem.” 128 OC4, 354-355 and OC13, 417. The solution of the anagram is: “Lens e duabus composita hyperbolicam aemulatur”. 129 Huygens may have tested the idea to combine two lenses into an objective earlier, at the time of the invention of 1665. Hug29, 76v and 77r contain sketches reminiscent of the earlier invention as well as ones reminiscent of the 1669 invetion. The folios can date from any time between the two inventions, but appear to reflect some intermediate stage in his thinking. 127

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combination. He had made a blueprint, a design by which his perfect telescope should be made, instead of designing one by first making it. Unlike earlier theorists like Descartes, however, Huygens had not started from the ideal situation but from the actual materials available to a telescope maker. He worked halfway between the scholar and the craftsman in an unprecedented effort to combine their respective theoretical and practical goals. Apparently the earlier, unsuccessful test of his first invention had not shaken his confidence that spherical lenses could cancel out their mutual aberrations. He had not changed his theory of spherical aberration, including the values he used to approximate the amount of aberration produced by a lens. Did Huygens expect that the composite lens would not produce those disturbing colors? He may have thought that his new design was of a different kind. As contrasted to the earlier one, it was not the ocular lens that canceled out the aberration of the objective lens, but the aberration was neutralized within the composite objective. This raises the question how Huygens had hit upon the idea not to consider the configuration of a complete telescope, but of a single lens system. It may have dawned upon him when he was pointing out the flaws in Nulandt’s statements. It would indeed be ironical that Huygens would have drawn inspiration for this remarkable invention from a man he held so low. Despite the triumphant EUPHKA, little is heard of the invention after February 1669. On 26 June 1669 he wrote Oldenburg that he had been working on lenses for a couple of weeks. He pointed out difficulties of attaining truly spherical figures and of the glass available to him.130 It is not clear whether he was trying to execute his design or that he was working on the 60-foot lenses mentioned in several letters of this period. In the meantime appeals to publish his dioptrical studies were numerous. On 18 March, Oldenburg warned him not to wait too long: “Sir, allow me to urge you to be willing to finish your Dioptrique for fear that you will not be preceded in this by someone else.”131 At the end of October it was too late. Barrow published his Lectiones XVIII and Oldenburg sent Huygens a copy on November 21.132 With the publication of Lectiones XVIII, Huygens lost priority on a basic accomplishment of Tractatus: the application of the sine law to spherical lenses. Barrow’s lessons were, as we have seen, of a different nature than Tractatus. Barrow himself was aware of the differences. In a letter to Collins, written on Easter Eve 1669, he wrote: “… had I known M. Huygens had been printing his Optics, I should hardly have sent my book. He is one that hath had considerations a long time upon that subject, and is used to be very exact in what he does, and hath joined much experience with his 130

OC6, 460. In November the Royal Society decided to send Huygens a piece of the excellent glass made in England. OC6, 533 and note 5. 131 OC6, 389. “Monsieur permettez moy de vous presser de vouloir acheuer vostre Dioptrique de peur que vous n’y soyez prevenu de quelque autre.” He warned him again on April 8. OC6, 416. 132 OC6, 534.

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speculations. What I have done is only what, in a small time, my thoughts did suggest, and I never had opportunity of any experience.”133

Barrow was too humble about his mathematical abilities but he was right in observing that Huygens had more ‘experience’ in dioptrical matters. Huygens praised Barrow in a letter to Oldenburg of 22 January 1670, but added “… someday you will see that what I have written about it is completely different.”134 Yet, he did not hurry. The publication of Lectiones XVIII may have pushed his plans to the background. In February 1670 Huygens fell ill and he went to The Hague in September, with an explicit ban by his physician to engage in intellectual labor. In June 1671 he returned to Paris. Huygens’ dioptrics are not mentioned among the manuscripts he entrusted to Vernon in February 1670, when he feared the worst.135 In Holland, he was with Constantijn again and we may speculate that they also discussed dioptrical matters. In general, Huygens wrote little about dioptrics in these years. He exchanged letters with de Sluse on Alhacen’s problem, a mathematical problem regarding spherical mirrors.136 Much of his correspondence was taken up by a discussion about the laws of collision he had sent to Oldenburg. No trace is found that Huygens worked on executing the design of February 1669. Not long after his return to Paris in June 1671, Huygens received a letter that would eventually mean the end of his plans. 3.2.3 NEWTON’S OTHER LOOK AND HUYGENS’ RESPONSE The invention of February 1669 is found on two places in Huygens’ manuscripts. One is his notebook of that period, the other is in the folder also containing ‘Adversaria’ and seems to be the original calculation.137 Both contain the sketch of his invention and the ‘EUPHKA 1 feb. 1669’. In the last one, however, the EUPHKA is crossed out and a ‘P.S.’ is added: “This invention is useless as a result of the Newtonian aberration that produces colors.”138 Along with his invention, Huygens discarded all parts of De Aberratione dealing with the improvement of telescopic images, namely his earlier invention and his rules for the opening of keplerian telescopes. He tore them from his manuscript and put them in a cover which said: “Rejecta ex dioptricis nostris”.139 The P.S. is dated October 25, without a year, but it is likely to be 1672.140 Evidently, this drastic decision was occasioned by the preceding correspondence with Newton on colors. 133

Rigaud, Correspondence II, 70. OC7, 2-3. “… vous verrez quelque jour que ce que j’en ey escrit est encore tout different.” 135 OC7, 7-13; especially 10-11. 136 Discussed in: Bruins, “Problema Alhaseni”. 137 Hug2, 72r and Hug29, 87r respectively. 138 OC13, 409n2. “Hoc inutile est inventum propter Abberationem Niutoniana quae colores inducit.” 139 OC13, 314n1. 140 The editors of the Oeuvres Complètes date it 1673, but in a conversation Alan Shapiro and I came to the conclusion that it must have been 1672. I will return to this on page 92. 134

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Figure 34 The crossed out EUREKA.

In a letter of 11 January 1672, Oldenburg first made mention of Newton to Huygens.141 This ‘mathematics professor in Cambridge’ had invented a small telescope in which the objective lens was replaced by a mirror. According to Oldenburg it represented an object “without any color and very distinct in all its parts.”142 In his next letter of 25 January, Oldenburg sent him a drawing and a detailed description, and asked Huygens’ opinion.143 At the bottom of a relatively wide tube a concave mirror reflected rays to a plano-convex ocular lens via a small plane mirror. Huygens promptly sent Oldenburg his opinion on the device. In the 81st issue of Philosophical Transactions (15 March, O.S.), Oldenburg published Newton’s description of his reflector along with some of Huygens’ comments.144 In the meantime, Huygens had also sent a letter on Newton’s telescope to Gallois, the editor of the Journal des Sçavans, who published an extract of it in the issue of February 29.145 Huygens spoke in the highest terms of Newton’s telescope. He enumerated no less than four advantages over ordinary telescopes: a mirror suffers less from spherical aberration, it does not ‘impede rays at the edge of the glass due to the inclination of both surfaces’, there is no loss of light due to internal reflections, and inhomogeneities in the material which affect lenses play no part in mirrors.146 In short, the reflector was a promising device. The main obstacle for its success, already pointed out by Oldenburg, was to find a durable material for making reflecting surfaces which lent itself

141

N.S. All dates are New Style unless indicated otherwise. OC7, 124-125. “… qui envoye l’object à l’oeil, et l’y represente sans aucune couleur et fort distinctement en toutes ses parties.” 143 OC7, 129-131. 144 OC7, 131 Huygens’ note a; 140-143. 145 OC7, 134-136. 146 OC7, 134-136 (to Gallois); 140-141 (to Oldenburg). In a note added to the description of Newton’s reflector, Huygens calculated the difference of spherical aberration produced by a spherical lens and a spherical mirror. The aberrations produced by a lens and a mirror with the same focal distance and aperture are 28 to 3. Therefore, he concluded, the aperture of a mirror can be three times as large. OC7, 132. 142

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to good polishing.147 The second of the advantages Huygens listed is interesting. Although he did not mention colors, it is clear that he referred to the observation he had made in 1668. In his letter to Oldenburg he almost literally repeated it: “Besides, by [the mirror] he avoids an inconvenience, which is inseparable from convex Object-Glasses, which is the Obliquity of both their surfaces, which vitiateth the refraction of the rays that pass towards the sides of the glass, and does more hurt than men are aware of.”148

In his letter to Gallois, Huygens added that this defect could not be prevented by a-spherical lenses. Evidently, he still was aware that his earlier observation was of consequence to the use of lenses. Still, we have no idea how he thought it would affect his invention of 1669. Those disturbing colors would eventually induce Huygens to discard his invention, but not until Newton had convinced him about his own ideas on their cause. In his letter of 21 March, Oldenburg notified Huygens of a paper by Newton in the 80th issue of Philosophical Transactions (19 February, O.S.): “In this print you will find a new theory of Mr. Newton, (…) regarding light and colors: in which he maintains that light is not a similar thing, but a mixture of differently refrangible rays …”149

It was, of course, the famous paper in which Newton set forth his ‘New theory about Light and Colors’. According to Newton, rays of different colors have a different degree of refrangibility: to each color belongs one, immutable index of refraction. Moreover, he argued that white light is not homogeneous but a mixture of all colors. Colors therefore are produced when this mixture is separated, for example by refraction, into its components. In ‘New theory’, Newton described his experiments with prisms to substantiate his claim that color is an original and immutable property of light rays which depends solely upon a specific index of refraction. Newton also explained why he had developed his reflecting telescope. After he introduced his idea of different refrangibility, he wrote that it had made him realize that colors could not be prevented in any lens and that mirrors should be used instead. On the basis of the measurement of the spectrum produced by one of his prisms, he calculated that the difference between the refractions of the red and blue rays is about a 25th part of the mean refraction. Consequently chromatic aberration is about a 50th part of the opening of the lens and therefore considerable larger than the spherical aberration produced by the same lens.150 After this ‘digression’, Newton went 147

OC7, 134 (to Gallois); 141 (to Oldenburg). Oldenburg had pointed this out to Huygens in the letter accompanying the description of Newton’s reflector: OC7, 128. 148 Oldenburg’s translation of OC7, 140 in: OldCor8, 520. 149 OC7, 156. “Dans cet imprimé vous trouverez une theorie nouvelle de Monsieur Newton, (…) touchant la lumiere et les couleurs: ou il maintient, que la lumiere n’est pas une chose similaire, mais un meslange de rayons refrangibles differemment …” The paper was therefore published in the issue preceding the one containing the description of his reflector. 150 Newton, Correspondence I, 95.

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on to lay down his doctrine of the origin of colors in the form of 13 propositions substantiated by the experiments he had described.151 On 9 April, Huygens gave a first reaction to Newton’s theory: “… I see that he has noticed like me the defect of the refraction of convex objective glasses caused by the inclination of their surfaces. As regards his new Theory of colors, I consider it quite ingenious, but it will have to be seen whether it is compatible with all experiences.”152

Two things stand out in this comment. In the first place, Huygens was mainly interested in the significance of Newton’s findings for dioptrics. In the second place, he seemed to miss the point of the theory of different refrangibility.153 In his view, Newton had merely confirmed what he had observed earlier. Seemingly, he did not realize that Newton’s point was that chromatic aberration is a consequence of the constitution of light, rather than the shapes of lenses. In his next letter to Oldenburg, of July 1, Huygens went more deeply into the matter, though still along the same lines. After discussing Newton’s telescope a bit further, he wrote: “As regards his new hypothesis of colors of which you ask my opinion, I admit that it seems very plausible to me, and the experimentum crucis (if I understand it correctly, as it is described somewhat obscurely) confirms it very much. But I don’t agree with what he says about the aberration of rays through convex glasses. For while reading what he writes, I find that following his principles this aberration must be twice as large as he takes it, to wit

1 25

the opening of the glass, which experience however seems to

contradict. so that this aberration may not always be proportional to the angle of inclination of rays.”154

We see what kind of ‘experiences’ Huygens had in mind when he cast doubt on the validity of Newton’s theory: the colors he had seen in lenses. He did not believe that chromatic aberration was as large as Newton claimed. Consequently, Newton’s explanation of the aberration was questionable. But it does not appear that Huygens had considered Newton’s theory of colors in much detail. It seems that he had mainly read the part on lenses. He did not use the term or notion of different refrangibility and only talked in terms of aberrations.

151

Newton, Correspondence I, 96-100. OC7, 165. “… je vois qu’il a remarquè comme moy le defaut de la refraction des verres convexes objectifs a cause de l’inclination de leurs surfaces. Pour ce qui est de sa nouvelle Theorie des couleurs, elle me paroit fort ingenieuse, mais il faudra veoir si elle est compatible avec toutes les experiences.” 153 See also: Sabra, Theories of Light, 268-267. 154 OC7, 186. “Pour ce qui est de sa nouvelle hypothese des couleurs dont vous souhaittez scavoir mon sentiment, j’avoue que jusqu’icy elle me paroist tres vraysemblable, et l’experimentum crucis (si j’entens bien, car il est ecrit un peu obscurement) la confirme beaucoup. Mais sur ce qu’il dit de l’abberration des rayons a travers des verres convexes je ne suis pas de son avis. Car je trouvay en lisant son ecrit que cette 152

1

aberration suivant son principe devroit estre double de ce qu’il la fait, scavoir 25 de l’ouverture du verre, a quoy pourtant l’experience semble repugner. de sorte que peut estre cette aberration n’est pas tousjours proportionelle aux angles d’inclinaison des rayons.”

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Newton realized that Huygens did not grasp the full import of his theory. Reacting to Huygens’ first comment on his theory, he had written Oldenburg on 13 April (O.S.): “Monsieur Hugenius has very well observed the confusion of refractions neare the edges of a Lens where its two superficies are inclined much like the planes of a Prism whose refractions are in like manner confused. But it is not from ye inclination of those superficies so much as from ye heterogeneity of light that that confusion is caused.”155

This remark was not, however, communicated to Huygens. On July 8 (O.S.), Newton replied to Huygens’ second comment in a letter Oldenburg forwarded to Figure 35 Newton’s determination of chromatic aberration. Huygens on 28 July.156 He acknowledged that the presentation of his theory might have been obscure for reasons of brevity. Newton also realized that Huygens had misread his discussion of chromatic aberration. “But I see not,” he wrote, “why the Aberration of a Telescope should be more than about 1/50 of ye Glasses aperture”. He included a drawing of the way he had calculated the proportion (Figure 35): “Now, since by my principles ye difference of Refraction of ye most difforme rayes is about ye 24th or 25th part of their whole refraction, ye Angle GDH will be about a 25th part of ye angle MDH, and consequently the subtense GH (which is ye diameter of ye least space, in to which ye refracted rays converge) will be about a 25th of ye subtense MH, and therefore a 49th part of the whole line MN, ye diameter of ye Lens; or, in round numbers, about a fiftieth part, as I asserted.” 157

The same letter was accompanied by a copy of the 84th issue of Philosophical Transactions (17 June, O.S.). It contained a letter in which Pardies criticized Newton’s theory and a reply by the latter. Two weeks later, Oldenburg sent Huygens the next issue of Philosophical Transactions (15 July, O.S.) containing further correspondence of Pardies and Newton on the matter.158 Pardies, a Jesuit priest and a Parisian acquaintance of Huygens, also criticized Newton’s claims, but in a more searching manner and with a different line of approach. He questioned the core of Newton’s theory – different refrangibility – and raised several objections to his experiments and his interpretations thereof. For example, he initially doubted whether the oblong spectrum could not be explained by the accepted rules of refraction.159 He also questioned the very idea of different refrangibility, which in his view depended upon a corpuscular conception of light. In his view, colors could also be caused by a ‘diffusion’ of light, for example by a slight spreading of the waves he 155

Newton, Correspondence I, 137. Newton, Correspondence I, 212-213; OC7, 207-208. 157 OC7, 207-208. 158 OC7, 215. 159 Newton, Correspondence 1, 131-132. 156

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supposed light to consist of.160 In his two successive replies, Newton clarified his experiments and his claims. He fully convinced Pardies of his claims and the father ended the discussion by saying that he was ‘very satisfied’.161 One expects that by now, late July, it must have dawned upon Huygens what Newton’s new theory was about. Still, the letter he sent Oldenburg on 27 September does not give the impression that he really grasped the essence of different refrangibility.162 He regarded Newton’s replies as a further confirmation of the theory, but added that things could still be otherwise. It had, however, dawned upon him that Newton had also something to say about the nature of light, to wit the compound nature of white light. To this he raised objections of a different kind: “Besides, if it were true that the rays of light were, from their origin, some red, some blue etc., there would still remain the great difficulty of explaining by the physics, mechanics wherein this diversity of colors consists.”163

The remark was clearly inspired by the objections Pardies had made. It does not give the impression that Huygens had given the matter any further thought. Let it be noted that this was the first moment Huygens raised objections of a mechanistic nature against Newton, after their discussion had progressed in several letters, and that the objections are not brought out strongly. He still did not refer to the notion of different refrangibility. As a conclusion he admitted his misreading of Newton’s discussion of chromatic aberration. Huygens’ next letter to Oldenburg, four months later on 14 January 1673, displayed a drastic change in his attitude towards Newton’s theory. Not only did he show to have considered the claims about the nature of white light and colors, he also subjected them to a serious critique.164 In addition, the tone of his comments became sharper. In his view Newton unnecessarily complicated matters: “I also do not see why Monsieur Newton does not content himself with the two colors yellow and blue, because it will be much easier to find some hypothesis by motion that explains these two differences, than for so many diversities as there are of other colors. And until he has found this hypothesis he will not have taught us wherein the nature and diversity of colors consists but only this accident (which certainly is very considerable) of their different refrangibility.”165 160

Newton, Correspondence 1, 157. Newton, Correspondence 1, 205. “Je suis tres satisfait de la derniere réponse que M. Newton a bien voulu faire à mes instances.” 162 Sabra, Theories of Light, 270. 163 OC7, 228-229. “De plus quand il seroyt vray que les rayons de lumiere, des leur origine, fussent les uns rouges, les autres bleus &c. il resteroit encor la grande difficultè d’expliquer par la physique, mechanique en quoy consiste cette diversitè de couleurs.” 164 OC7, 242-244. 165 OC7, 243. “Je ne vois pas aussi pourquoy Monsieur Newton ne se contente pas des 2 couleurs jaune et bleu, car il sera bien plus aisè de trouver quelque hypothese par le mouvement qui explique ces deux differences que non pas pour tant de diversitez qu’il y a d’autres couleurs. Et jusqu’a ce qu’il ait trouvè cette hypothese il ne nous aura pas appris en quoy consiste la nature et difference des couleurs mais seulement cet accident (qui assurement est fort considerable) de leur differente refrangibilitè.” 161

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In his view, white light may also be produced by mixing yellow and blue alone. By maintaining that there are only two primary colors, Huygens drew upon a letter published in the 88th issue of Philosophical Transactions (18 November 1672, O.S.) in which Newton responded to comments by Hooke. Among other things, Hooke had claimed that two primary colors sufficed to explain the diversity of colors. Hooke’s comments had not been published and his name was not mentioned in Newton’s reply. Huygens referred to Hooke’s prism experiments in Micrographia (1665). He suggested an experiment to verify whether all colors are necessary to produce white light. Evidently, Huygens had problems with Newton’s claims about the nature of light. What these problems were exactly, why he would prefer just two colors, and what he meant by ‘explaining by physics, mechanics’ and ‘some hypothesis of motion’ is not explained in the letter. In chapter 6 we will be able to reconstruct, in retrospect, the background to Huygens’ remarks. He had a reasonably clear idea what mechanistic explanation ought to be, but it appears that by 1672 he had not yet elaborated in much detail his conception of the mechanistic nature of light. Besides raising objections to Newton’s ideas on the nature of light and colors, Huygens summed up his own treatment of chromatic aberration: “Apart from that, as regards the effect of the different refractions of rays in telescope glasses, it is certain that experience does not correspond with what Monsieur Newton finds, because by considering only the distinct picture that an objective of 12 feet makes in a dark room, one sees that it is too distinct and too sharp to be able to be produced by rays that disperse from the 50th part of the aperture so that, as I believe to have brought to your attention before, the difference of the refrangibility may not always have the same proportion in the large and small inclinations of the rays on the surfaces of the glass.”166

There is no reason to assume that Huygens had not actually performed this test.167 We may only wonder why he had not done so in 1665. Then again, the alleged observations remained qualitative. We may wonder what had prompted Huygens to consider the issue of chromatic aberration anew. Assuming that the full import of Newton’s theory had occurred to him as a result of Pardies’ comments, he may have realized at some time by late 1672 that chromatic aberration was a problem of refraction and thus inherent to lenses. As Newton emphasized in his reply to Hooke: “And for Dioptrique Telescopes I told you that the difficulty consisted not in the figure of the 166

OC7, 243-244. “Au reste pour ce qui est de l’effect des differentes refractions des rayons dans les verres de lunettes, il est certaine que l’experience ne s’accorde pas avec ce que trouve Monsieur Newton, car a considerer seulement la peinture distincte que fait un objectif de 12 pieds dans une chambre obscure, l’on voit qu’elle est trop distincte et trop bien terminée pour pouvoir estre produite par des rayons qui s’escarteroient de la 50me partie de l’ouverture de sorte que, comme je vous crois avoir mandè desia cy devant la difference de la refrangibilité ne suit pas peut estre tousjours de la mesme proportion dans les grandes et petites inclinations des rayons sur les surfaces du verre.” 167 Because he was a ‘devoted water-color painter’, Shapiro is puzzled about Huygens’ assertion that yellow and blue may produce white, “… because this is contrary to all beliefs about color mixing held in the seventeenth century.” Shapiro, “Evolving structure”, 223-224. We should bear in mind that Huygens was also an experienced employer of magic lanterns.

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glasse but in ye difformity of refractions.”168 Despite his doubts about the true properties of different refrangibility, Huygens now recognized that the disturbing colors in lenses are inherent to refraction. There is no word about spherical aberration in his letter, and he may indeed already have realized at this point that his project to design configurations to neutralize it had become useless. Would this help account for the sharpening in his tone? Newton partially granted Huygens both objections. He dropped the claim that all colors are necessary to compound white light restricting it now to sunlight.169 In his reply of April 3 (O.S.), he strongly objected to Huygens’ claim that two primary colors are more easily explained, but he explicitly refrained from proposing a ‘Mechanicall Hypothesis”.170 As regards the actual effect of chromatic aberration, he watered down his claim a bit. The rays that are dispersed mostly “… are but few in comparison to those, which are refracted more Justly; for, the rays which fall on the middle parts of the Glass, are refracted with sufficient exactness, as also are those that fall near the perimeter and have a mean degree of Refrangibility; So that there remain only the rays, wich fall near the perimeter and are most or least refrangible to cause any sensible confusion in the Picture. And these are yet so much further weaken’d by the greater space, through which they are scatter’d, that the Light which falls on the due point, is infinitely more dense than that which falls on any point about it. …”171

As a conclusion, Newton suggested a way to measure the chromatic aberration of the extreme rays to verify his claims. Huygens accepted Newton’s argument, but added that “… he must also acknowledge that this abstraction [dispersion] of rays does not therefore harm lenses as much as he seems to have wished to be believed, when he proposed concave mirrors as the only hope for perfecting telescopes.”172

Huygens was not, however, satisfied with Newton’s rebuttal concerning the nature of white light and colors: “… but seeing that he maintains his opinion with so much ardor, this deprives me of the appetite for disputing.”173 Two weeks later, he wrote to Oldenburg not to send Newton his last letter at all and to tell him only that he did not want to dispute anymore.174 Newton did receive the letter anyhow and replied on 23 June (O.S.) by a more precise reformulation of his theory, which was published in the 96th issue of

168

Newton, Correspondence I, 173. Shapiro, “Evolving structure”, 224-225. 170 OC7, 265-266 and Newton, Correspondence I, 264-265. 171 OC7, 267 and Newton, Correspondence I, 266. In Opticks, he elaborated this argument a bit further and mathematically, and reduced chromatic aberration to 1/250 of the aperture as contrasted to the original 1/50. Newton, Optical lectures, 429n15. 172 OC7, 302-303. “… mais aussi doit il avouer que cette abstraction des rayons ne nuit donc pas tant aux verres qu’il semble avoir voulu faire accroire, lors qu’il a proposè les mirroirs concaves comme la seule esperance de perfectionner les telescopes.” 173 OC7, 302. “…, mais voyant qu’il soustient son opinion avec tant de chaleur cela m’oste l’envie de disputer.” 174 OC7, 315. 169

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Philosophical Transactions (21 July O.S.).175 Newton invited him once again to compare by computation aberrations both of lenses and mirrors, but Huygens did not respond anymore. Thus came an end to a dispute that had run an odd course. In January 1672 Huygens had welcomed the newcomer on the scene of European scholarship as a kindred spirit in matters dioptrical; in June 1673 he refrained from discussing any further with someone who so obstinately clung to his claims. But most striking about the state of affairs I find the relative late moment at which Huygens recognized the purport of Newton’s paper. Until the letter of September 1672, the fact that Newton’s theory concerned the physical nature of light escaped him. And then again, he made only one – apparently non-committal – objection. Only in the letter of January 1673 did he engage in a dispute on Newton’s theory of colors, to break it off in the next letter. Until the letters of Pardies were published, Huygens only paid attention to what Newton had said about the aberrations of lenses. And even at this point, he failed to grasp Newton’s message. He only talked of chromatic aberration in the same terms as he had treated spherical aberration. One gets a strong impression that in 1672 Huygens lacked a certain sensibility for the kind of question Newton addressed, namely concerning the physical nature of light. This is all the more surprising since Huygens has become famous for a theory explaining the nature of light of his own. The preceding reconstruction sheds new light on this famous dispute. Huygens was not a Cartesian that a priori rejected Newton’s theory for reasons of its mechanistic inadequacy and untenability, like Hooke did and Pardies too initially, and like he is usually presented in historical literature.176 We should reconsider the his dispute from the perspective of Dioptrica. Huygens entered the dispute from his background in dioptrics. He was interested (and informed) in lenses and telescopes and he had something to loose. At first he did not look beyond issues directly pertaining to lenses and it took some time before he realized what Newton’s theory was about. He began to raise mechanistic doubts only during the final stages of the dispute, and probably when he realized the consequence for his project of nullifying spherical aberration. For in anything may explain Huygens relative reluctance in accepting Newton’s theory, it would be De Aberratione. Somewhere along the line, Huygens must have realized that Newton’s findings wrecked his project of perfecting the telescope. He crossed out the ‘Eureka’ of February 1669 and discarded a large part of his theory of spherical aberration. ‘Newtonian’ aberration had rendered his designs useless. Spherical aberration might be cancelled out by successive lenses, chromatic aberration could never be prevented. Despite the objections he raised in it, the letter of January 1673 reveals that Huygens had recognized 175 176

OC7, 328-333 and Newton, Correspondence I, 291-295. See also Shapiro, “Evolving structure”, 225-228. For example Sabra, Theories of Light, 268-272.

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different refrangibility. I find it reasonable to presume that this had happened at some time during the preceding months. He demonstrably had contacts with Pardies, who had accepted the crux of different refrangibility and might have pointed it out to Huygens. Moreover, at that time Mariotte carried out experiments that corroborated Newton’s results.177 I find it therefore likely that Huygens took the drastic decision to discard his project on 25 October 1672, rather than 1673 as the editors of the Oeuvres Complètes have it.178 I find it unlikely that Huygens would have taken the step when the whole event had long passed. The only fact supporting this interpretation is the publication of Newton’s and Huygens’ letters of 3 April and 10 June respectively in the issue of Philosophical Transactions of 6 October 1673 (O.S.). Although Huygens had long known their content, further reflection upon the dispute might conceivably have triggered the decision to discard the main part of De Aberratione. 3.3 Dioptrica in the context of Huygens’ mathematical science The drastic decision of October 1672 brought Huygens’ study of lenses to a temporary end. He was not to resume it, adjusting his theory of aberration by taking ‘Newtonian’ aberration into account, until his return to Holland in the 1680s.179 This would not change the character of his dioptrical studies as we have come to know it in these two chapters. Although he had lost the ambition to design a perfect telescope, the orientation on the telescope guided his dioptrical studies. He conducted his dioptrical studies in order to understand the instrument. With a Huygens one tends, however, to overlook the obvious. For him, understanding something meant mathematically understanding something. Together with his orientation on the instrument, his mathematical approach is the clue to Dioptrica. In addition to this scholarly contemplation, Huygens had applied himself to the craft of telescope making. De Aberratione can be seen as an effort to combine his dual capacities as a scholar and a craftsman. In this sense, it should have been the climax of his involvement with the telescope. It turned into an anti-climax. In this concluding section on Dioptrica, I first go through the nature of Huygens’ mathematical approach and its consequences for De Aberratione. Then I consider his orientation on the telescope in the broader context of our understanding of Huygens’ science. 3.3.1 THE MATHEMATICS OF THINGS Ignoring for a moment the particular aims of Huygens’ dioptrical studies, we may notice that De Aberratione has all the features of a geometrical treatise. It is structured as a set of propositions and definitions regarding spherical aberration. De Aberratione is not a geometrical treatise by appearance only. In 177

Shapiro, “Gradual acceptance”, 78-80. Additional evidence for this dating I find in the fact that Pardies and Huygens discussed Iceland Crystal in the summer of 1672. See further footnote 120 on page 140 below. 179 See section 6.1. 178

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its elaboration it consisted of a geometrical derivation of the properties of spherical aberration. Like Tractatus, it rested on little more than the sine law and a generous dose of Euclidean geometry. In the elaboration of the theory of spherical aberration, geometry had the upper hand. This stands out clearest in the simplifications Huygens employed. He used a simplified expression in order to determine the amount of aberration produced by a particular lens. He justified this by comparing the calculated differences between both expressions. What effects such differences would have in actual lenses, he did not tell. Nowhere in De Aberratione does Huygens give an indication that he had considered the question how the calculated properties of spherical aberration related to its observed properties. A modern reader would expect otherwise, but Huygens went about by geometrical deduction exclusively. This geometrical analysis resulted in a sophisticated theory of spherical aberration in which complex problems were solved of neutralizing it by configuring spherical lenses properly. But Huygens’ goal was not mere theory, he aimed at its practical application to real lenses and telescopes. This marked him off from his fellow dioptricians. Had he not applied his theory to design better telescopes and tested his design, he would not have been confronted with those disturbing colors. The fact that Huygens was taken by surprise by those disturbing colors need not surprise us. In his dioptrical study of lenses, Huygens confined himself to their mathematical properties and excluded the consideration of colors. Likewise, in his study of halos and parhelia, written around 1663, he confined himself to tracing the paths of rays of light through transparent particles in the atmosphere and left out any consideration of the colors of these phenomena.180 Colors eluded the laws of geometry, so he wrote there with even greater conviction than in Tractatus: “However, to investigate the cause of these colors further; to know why they are generated in a prism, I want to undertake by no means, I admit on the contrary not to know the cause at all, and I think that no one will comprehend their nature easily for as long as some major light will not have enlightened the science of natural things.”181

That major light had come, it was named Newton, and it had eclipsed Huygens’ grand project of perfecting telescopes. Huygens was well acquainted with the disturbing colors produced by lenses. Dealing with them was, in his view, a matter of trial-and-error configuring of lenses instead of purposive calculation. When colors came to disturb the predicted optimal working of his design, he did not do anything with them. Despite the importance of colors for his project, Huygens did not elaborate upon his observation that colors might be related to the angle of 180

With connected reproduced in OC17, 364-516. On the dating see OC17, 359. OC17, 373. “Doch de reden van dese couleuren verder te ondersoecken, te weten waerom die in een prisma gegenereert worden, wil ick geensins ondernemen, emo fateor rationem eorum me prorsus ignorare, neque facile quemquam ipsas perspecturum arbitror quandiu naturalium rerum scientiae major aliqua lux non affulserit.” 181

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incidence of a ray of light. He did not adjust his theory of spherical aberration, nor the way he intended to counter its effects in telescopes. Apparently, he saw no possibility to extend dioptrics to the properties of colors. Colors kept eluding his mathematical understanding. In other words, he did not take the step to leave the established domain of mathematical optics. This should not be strongly counted against him, for no-one in the seventeenth century did so. Except for Newton, who had an extraordinary scholarly disposition that combined a mathematical outlook with an interest in material things fostered by experimental philosophy and the new natural philosophies in general. Newton did see geometry in colors, but he looked at them from an entirely different perspective. His studies of prismatic colors had begun around 1665 with an experiment described by Boyle – with a thread that was half blue and half red and appeared broken when seen through a prism.182 Unlike Boyle, he interpreted this in terms of the refraction of rays of light. He realized that the rays coming from both parts of the thread are refracted at different angles. In other words, Newton interpreted the phenomenon in the geometrical terms of rays and angles. On this basis he began his prismatic experiments, deliberately studying the differences of the angles with which rays of various colors are refracted. Unlike Descartes, Boyle and Hooke before him, he tried to make the spectrum as large as possible, by projecting it as far as possible.183 He passed the beam of light at minimum deviation, so that the effect of the width of the beam was minimized. By turning the prism into a precision instrument, Newton discovered that it was the principles of geometrical optics that were violated by the spectrum. The solution of the anomaly consisted of linking ‘color’ with ‘refractive index’ and thus with the sine law of refraction. Different refrangibility reduced colors to the laws of geometry. It was not only the mere recognition of geometry that led to different refrangibility. In order to establish the laws to which colors were subject, Newton employed experiment in a new way. Combining mathematical thinking with a heuristic use of experiment, he developed the new methodological means of quantitative experiment. By measuring the phenomena produced in his prisms he was able to discover geometrical properties where previously there had been none. “But since I observe that geometers have hitherto erred with respect to a certain property of light concerning its refractions, while they implicitly assume in their demonstrations a certain not well established physical hypothesis, I judge it will not be unappreciated if I subject the principles of this science to a rather strict examination, adding what I have conceived concerning them and confirmed by numerous experiments to what my reverend predecessor last delivered in this place.”184

182

Newton, Certain philosophical questions, 467. Westfall, Never at rest, 163-164. 184 Newton, Optical papers 1, 47 & 281. 183

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Newton considered his discovery of different refrangibility an addition to geometrical optics. A necessary addition because it explained “… how much the perfection of dioptrics is impeded by this property and how that obstacle, insofar as its nature allows, may be avoided.”185 These lines could have been addressed to Huygens personally, had Newton known of De Aberratione. Newton was aware that he was breaking new ground. In the letter he sent to Oldenburg he wrote: “A naturalist would scearce expect to see ye science of [colours] become mathematicall, & yet I dare affirm that there is as much certainty in it as in any other part of Opticks.”186

These lines were, however, omitted when his ‘New theory’ appeared in Philosophical Transactions. With his theory Newton went beyond the recognized boundaries of geometrical optics by extending it to the study of colored rays. Huygens, on the other hand, stayed within the established domain of optical phenomena to be studied mathematically. He elaborated his dioptrical theories in the manner customary in geometrical optics. As a mathematical theory, the content of Dioptrica did not deviate in any principal way from the doctrines found in Aguilón or Barrow. In Paralipomena physical considerations – albeit within the traditional domain of mathematical optics – were much more integrated in mathematics, but Huygens did not follow this line of Kepler at this moment.187 As a topic of mixed mathematics, geometrical optics was principally a matter of geometrical deduction. The difference with ‘pure’ geometry was that lines and circles represented physical objects like rays, reflecting and refracting surfaces. Geometrical inference was preconditioned by a specific set of postulates: the laws of optics describing the behavior of unimpeded, reflected and refracted rays. Or, as Huygens would state it in Traité de la Lumière, optics is a science “where geometry is applied to matter.”188 Huygens ‘géomètre’ Thus Huygens treated spherical aberration as a geometrical problem which ought to be solved by mathematical analysis. Despite the vital importance of colors for his project, he did not go beyond the traditional boundaries of mixed mathematics in order to tackle the problem. He confined his investigation to effects known to be reducible to the laws of geometry. Geometrical optics did not provide the means to deal with colors, so he left them to the craftsman. In this sense Huygens’ Dioptrica fits in with his mathematical science in general. In his studies of circular motion and consonance he also focused on exploring their mathematical properties on the basis of established (mathematical) principles.

185

Newton, Optical papers 1, 49 & 283. Newton, Correspondence 1, 96. 187 See section 4.1.2. 188 Traité, 1. “… toutes les sciences où la Geometrie est appliquée à la matiere, …” 186

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During the final weeks of 1659, Huygens took up and solved a problem that Mersenne had discussed 12 years earlier in Reflexiones physico-mathematicae (1647). The problem was to determine the distance traversed by a body in its first second of free fall, which amounts to determining half the value of the constant of gravitational acceleration. After having tried Mersenne’s experimental approach, Huygens abandoned it in favor of a theoretical consideration of gravitational acceleration. He began a study of circular motion which in his view was closely connected to gravity: “The weight of a body is the same as the conatus of matter, equal to it and moved very swiftly, to recede from a center.”189 Circular motion had been discussed by both Descartes and Galileo, but only in qualitative and fairly rough terms.190 Huygens set out to analyze circular motion mathematically. He derived an expression for the tension on a chord exerted by a body moving in a circle, by equating it with the tension exerted by the weight of the body.191 He then considered the situation in which a body revolves on a chord in such a way that a stable situation is created and centrifugal and gravitational tension are counterbalanced. With the conical pendulum thus procured and reversing his calculations, Huygens found an improved value for gravitational acceleration and dismissed Mersenne’s original experiment.192 Analyzing the experiment mathematically and comparing the time of vertical fall to the time of fall along an arc, he derived a theory of pendulum motion eventually resulting in the discovery of the isochronity of the cycloid.193 The aim of Huygens’ studies of curvilinear fall and circular motion was to render these motions with the same exactness Galileo had achieved with free fall.194 In the case of curvilinear fall this meant to solve the tricky mathematical problem of relating the times with which curved and straight paths are traversed. In the case of circular motion, he quantitatively compared centrifugal and gravitational acceleration. Huygens’ success came from his proficiency in using infinitesimal analysis and his control of geometrical reasoning.195 He conceptualized the forces he was studying in a way that could be geometrically represented, which in his view meant to treat free fall and centrifugal force in terms of velocities.196 He considered, for example, gravity as mere weight, and acceleration as continuous alteration of inertial motion.197 In other words, rather than mathematizing these 189

Yoder, Unrolling time, 16-17. Yoder, Unrolling time, 33-34. 191 Yoder, Unrolling time, 19-23. This expression for centrifugal tendency amounts to the modern formula: F = mv2/r. 192 Yoder, Unrolling time, 27-32. 193 Yoder, Unrolling time, 48-59. 194 The first draft of De vi centrifuga opened with a quotation of Horace: “Freely I stepped into the void, the first”, above his discovery of the isochronicity of the cycloid he wrote: “Great matters not investigated by the men of genius among our forefathers; Yoder, Unrolling time, 42 and 61. 195 Yoder, Unrolling time, 62-64. 196 The same goes for his earlier study of impact, to be discussed in section 4.2.2. 197 Westfall, Force, 160-165. 190

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phenomena, he reduced them to concepts already mathematized. To be more specific: Huygens reduced these dynamical phenomena to the kinematic groundwork laid by Galileo. Yoder has pointed out Huygens’ talent for transferring physics to geometry.198 His proficiency in idealizing phenomena enabled him to mathematize not only the abstract objects of mechanics but also concrete bobs and cords. Once transformed into a geometrical picture, Huygens could apply his geometrical skills. Just as in his study of spherical aberration, the kind of experimentation by which Newton had mathematized colors was absent from Huygens’ studies of circular motion. He was surely a careful observer and capable of designing clever experiments as an independent means to test theoretical conclusions.199 Yet, the precision he achieved in measuring the constant of gravitational acceleration was made possible by his mathematical understanding of the matter. Exploring mathematical properties of a phenomenon empirically was not the way he approached his objects of study. On the contrary, he readily dismissed Mersenne’s experiment as indecisive, aware of the imprecision and bias of observation.200 He approached his subject first of all theoretically, interpreting concepts geometrically and analyzing phenomena by means of his mathematical mastery. In his dioptrical studies, Huygens had likewise relied on his geometrical proficiency. His theory of spherical aberration was the outcome of rigorous, sometimes clever deduction. At the point he could have broken really new ground – when colors emerged – Huygens halted. The process of geometrizing new phenomena that had proven to be so fruitful in his study of motion did not get going in dioptrics. Seemingly, he did not see possibilities to transform those disturbing colors into a geometrical picture, despite some promising observations he had made of them. However, we should bear in mind that motion, as contrasted to colors, had already been mathematized. In his geometrization of circular motion, Huygens could build on the groundwork laid by Galileo. Compared to his study of circular motion, De Aberratione was rather straightforward geometrical reasoning. In this regard, it comes closer to his study of consonance that occupied him, on and off, from 1661 onwards.201 The first problem Huygens attacked was the order of consonance, an issue that had arisen (anew) with the new theories of music of the sixteenth and seventeenth centuries. In the theory of consonance Huygens adopted, the coincidence theory of Mersenne and Galileo, the order of consonants was not evident. He derived a clever rule that only left one problem. His rule seemed to imply that 74 should be placed between the major third and the 198

Yoder, Unrolling time, 171-173. Yoder, Unrolling time, 31-32. 200 Yoder, Unrolling time, 170-171. 201 Cohen, Quantifying music, 209-230 and Cohen, “Huygens and consonance”, 271-301. 199

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fourth which implied that “… the number 7, …, is not incapable of producing consonance …”, a conclusion that ran in the face of all previous musical theory.202 At that time – around 1661 – Huygens decided not to accept the consonance of intervals with 7 because they had no regular place in the scale. Next, Huygens addressed a problem in tuning. When keyboards are tuned according to then customary mean tone temperament, the question was how the fifths employed ought to be adjusted with respect to pure fifths.203 In order to determine a mathematical solution, Huygens started by deriving the ratios of all twelve tones in terms of the string lengths of the octave and the fifth. In the course of his investigation, Huygens found a new property of mean tone temperament. It concerned the quantitative difference between the diatonic and the chromatic semitones.204 Calculating the ratio of both kinds of semitones, he concluded that C-D can be divided into 5 equal parts and, consequently, the octave into 31 equal parts. Thus Huygens arrived at the 31-tone division of the octave he had found discussed by Mersenne and Salinas. In a letter published 30 years later in Histoire des Ouvrages des Sçavans (October 1691), Huygens elaborately explained how he calculated the various string lengths and pointed out advantages of his 31-tone division.205 The paper did not contain a further consequence Huygens had drawn in his private notes: the consonance of intervals based on the number 7. Thus, Huygens’ ‘most original contribution to the science of music’ remained unknown to the world until this century.206 Huygens’ studies of consonance show, once more, his dexterity in exploring and elaborating the mathematics of a topic. He added rigor and precision to Mersenne’s science of music, using Galileo’s approach and

202 203

OC20, 37. Translation: Cohen, Quantifying music, 214. The tones of the octave are found using the consonances; this is called the division of the octave. The 3

seven tones of the diatonic scale are found by means of the fifth ( 2 ) and its complement, the fourth ( 74 ). Likewise the chromatic tones are found by addition of fifths. A problem arises, however, because a complete octave cannot be reached again by continuous addition of fifths. A small difference, called the Pythagorean comma, exists between 12 fifths

( 32 )12

and 7 octaves

( 12 ) 7 . As a result, the tones of the

octave ought to be tempered in musical practice, which means that the purity of some consonances is sacrificed. In mean tone temperament most major thirds are pure and the fifths are made a bit too large; in equal temperament all consonances save the octave are a bit impure. Huygens preferred the former, the latter has become standard tuning in Western music since the early nineteenth century. 204 The diatonic semitone is the difference between E and F, B and c, etc.; the chromatic semitone is the difference between, for example, C and C. The chromatically sharpened C and flattened D – C# and Db – differ, whereby C-Db and C#-D have the size of a diatonic semitone. The difference between C-C# and CDb is the difference between both kinds of semitones. 205 Most of Huygens’ musical studies is reproduced in OC20, 1-173. The French and Latin versions of the letter have been reprinted with Dutch and English translations by Rasch in: Huygens, Le cycle harmonique. 206 Cohen, Quantifying music, 225-226.

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extending it to problems the latter had ignored.207 As with his studies of dioptrics and circular motion, Huygens’ study of consonance did not develop in an empirical vacuum. He rejected Stevin’s theory, as purely mathematical and ignoring the demands of sense perception. But he also rejected systems that lacked theoretical foundation.208 His aim was to develop a sound mathematical theory that explained and founded his musical preferences. Mean tone temperament therefore was his natural starting point, and the 31tone division seems a natural outcome of his analysis of its mathematical properties as it conformed to both his theoretical and practical preferences. Like his studies of circular motion and consonance, Huygens’ study of spherical aberration, and this is almost a truism, was predominantly mathematical. Huygens fruitfully explored and rigorously examined mathematical theory. More revealing in the context of the present study is the relationship between mathematics and observation. Huygens was not blind for the empirical facts. On the contrary, they constituted the main directive of his investigation in such diverse ways as the measure of gravity, pleasing temperament and workable lens-shapes. Huygens knew how to check his theoretical conclusions empirically and he was not easily satisfied. Exploratory observation of phenomena was not the way Huygens approached a subject. In modern terms: he did not employ experiment heuristically. In the case of gravity, he had soon found out that mere observation did not yield reliable knowledge. The result proved him right: the analysis of the mathematical properties of circular motion gave him a better theory as well as a better means of measurement. Huygens successfully extended the Galilean, mathematical approach to gravity and circular motion. Newton likewise was a mathematician with a Galilean spirit, but in his study of colors he linked it with the experimental approach of Baconianism. Although he was favorably disposed to Bacon’s program for the organization of science (see below), Huygens did not regard the experimental collecting of data as a source for new theories, let alone a trustworthy basis for mathematical derivation. He explored the underlying mathematical structure of a phenomenon the results of which could be verified to see whether the supposed structure was real. In the case of consonance, the empirical foundation of the theory had already been established. In the case of spherical aberration, however, such preliminary work had not yet been done, unfortunately. It turned out that not all effects of lenses depended upon the known mathematical properties of lenses. Suppose he had pursued his idea that colors were related to the inclination of the sides of a lens. He might have taken some objective lenses, covered their center (instead of their circumference as was customary) and 207

Cohen, Quantifying music, 209. It should be noted that, unlike his predecessors, Huygens possessed 1

logarithms and was therefore readily able to calculate, for example, a 4 5 . 208

Cohen, “Huygens and consonance”, 293-294.

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see how the different inclinations affected the generation of colors. He might even have taken a prism to study the effect of twofold refraction on colors. He might even have measured the angles of the inclination and – even more speculative – made some measurements on the colors themselves. He did not, and left colors aside in De Aberratione. In short, he recognized the importance of the colors displayed by his lenses, but did not know what to do about them. Which amounts to saying that he did not know what to do about them mathematically. 3.3.2 HUYGENS THE SCHOLAR & HUYGENS THE CRAFTSMAN Which brings us back to what Huygens’ study of spherical aberration was all about: the improvement of telescopes. From the viewpoint of dioptrics, nothing was wrong with his theory of spherical aberration. It described the properties, derived from the principles of dioptrics, of light rays when refracted by spherical surfaces. From the viewpoint of Huygens’ project there was, however, a serious problem. He did not develop his theory in order merely to extend his dioptrical knowledge, but to find an improved configuration of lenses. Huygens’ theory of spherical aberration could not take colors into account – let alone explain how to minimize their disturbing effects. From the viewpoint of dioptrical theory, colors were a further effect yet to be understood; from the viewpoint of De Aberratione they were a fatal blow. Without the practical goal of De Aberratione, Huygens probably would never have run across the disturbing colors that spherical lenses also produced. I have amply argued that the orientation of Dioptrica on the telescope marked off Huygens’ dioptrical studies from those of most other seventeenth-century scholars. He was one of the very few who tried to acquire a theoretical understanding of the telescope and, in addition, he wanted to improve the instrument on this basis. That is not necessarily to say that this practical orientation is characteristic of Huygens’ science in general. Although applications of theory to instruments were never far from his mind, his studies of consonance and circular motion were not guided by an orientation on instruments as his studies of dioptrics were. The problem of tuning keyboard instruments was important for Huygens’ musical studies but their main goal was the mathematical theory of consonance. Having elaborated his 31-tone division, he readily saw the practical application in the guise of a suitable organ, on which one could switch easily between keys in mean tone temperament. Likewise, his study of circular motion was aimed at a physical problem (measuring gravity) and took the form of a thorough, mathematical analysis of circular motion in many of its manifestations. It was not a analysis of the clock he had invented earlier, nor did the question which pendulum would be isochronous guide it.209 Still, practical thinking of a kind was inherent in Huygens’ study of 209

Yoder, Unrolling time, 71-73.

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circular motion. He often couched his thoughts on circular motion in some mechanical form. And he designed several clocks that embodied his theoretical findings. As regards his original pendulum clock he reaped the rewards of his study by equipping it with cheeks that gave its bob an isochronous path. If instruments did not guide Huygens’ other studies the way they did in dioptrics, his approach to them was nevertheless similar. Horologium Oscillatorium of 1673 does not just describe the pendulum clock and the ideal cycloidal path, but also gives the mathematical theory of motion embodied in it. Going beyond the mere necessities of explaining its mechanical working – as in Dioptrica – he elaborated his theories of circular motion, evolutes and physical pendulums. Of the achievements of 1659, Horologium Oscillatorium included the study of curvilinear fall and cycloidal motion, transformed into a direct and refined derivation, but it listed only the resulting propositions of his study of circular motion and the conical clock. In addition, it contained a discussion of physical pendulums. Huygens imaginatively applied the insight that a system of bodies can be considered as a single body concentrated in the center of gravity, to a physical pendulum considered to be resolved into its constituent parts independently. With this he could express the motion of the pendulum by means of the accelerated motion of its parts. Next he compared the physical pendulum to an isochronous simple pendulum, deriving an expression for the length of the latter in terms of the length and the weights of the parts of the former.210 His organ likewise rested on an sound and even elegant theory of consonance. In this way he showed the solid theoretical basis on which his inventions rested, showing at the same time that he was not a mere empiricist but a learned inventor.211 De Aberratione stands out among Huygens’ studies in that he developed theory with the explicit aim of improving an instrument. Earlier, he had proven the working of his eyepiece on a mathematical basis, but he had not been able to demonstrate that it was the best configuration possible. In De Aberratione Huygens set out to design a configuration of lenses that he could prove mathematically was the best one possible. Huygens was not unique for trying to solve a practical problem by means of theory. Descartes’ a-spherical lenses were meant to serve as a solution to the same problem Huygens attacked. Descartes had tried to realize his design by thinking up a device fit for making those lenses. Examples from other fields can be found without much effort; the problem of finding longitude at sea is only the first to come to mind. The seventeenth century is pervaded by scholars who believed theory could or should be of practical use. The special thing about De Aberratione is the way Huygens set out to solve the problem of spherical aberration. His starting point consisted of the mathematical 210 211

Westfall, Force, 165-167. Cohen, Quantifying music, 224.

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theory of spherical lenses he had developed earlier. As contrasted to Descartes and others, his design for a better – or even perfect – telescope did not start out with the ideal lenses of geometry, but with the ‘poor’ lenses of actual telescopes. He did not avoid or explain away the defects of spherical lenses, like Descartes or Hudde. He analyzed these defects in order to take them into account and eventually correct them. Huygens’ design of a perfect telescope was not based on the theoretically desirable, but on the practically feasible. Although craftsmanship preconditioned De Aberratione, Huygens did not go the craftsman’s way as in his earlier inventions. He wanted to derive a blueprint for an improved configuration on the basis of his theoretical understanding of lenses. Instead of tinkering with lenses, he would be tinkering with mathematics. He replaced the trial-and-error configuring of lenses by mathematical design. Whether consciously or not, Huygens was trying to bridge the gap between craftsmanship and scholarship. It was an effort to make science useful for the solution of practical problems. An advanced one, as the limitations and possibilities of the actual art of telescope making were at the very heart of Huygens’ project. De Aberratione can be seen as an early effort to do science-based technology. How did Huygens set about it? He tried to understand mathematically the technical problem of imperfect focusing and to solve it by means of his theory. The configuration of lenses was the only part of the artisanal process of telescope making he replaced by theoretical investigation. Colors he left for crafty hands. Despite this close tie to practice, the subsequent elaboration of the project was a matter of plain mathematics. He reduced the problem of the imperfect focusing of spherical lenses to the mathematical problem of spherical aberration. He then designed a configuration of lenses that overcame the latter problem, assuming that it also solved the original practical problem. It did not, for the test of his design brought to light an additional technical problem that escaped his mathematical theory of lenses. In a way, it was not just a test of his design but of his theory of spherical aberration as well. The trial of 1668 can be seen as an empirical test of his theory of spherical aberration – the first one, as far as the sources reveal. Whether Huygens also saw it in this way may be doubted. His second design of 1669 was founded upon the same theory. We do not know whether he expected it to be free of colors, had it been realized. With hindsight, we can say that the failure of Huygens’ project is an example of the fact that technology goes beyond the mere application of science. Huygens had remarked earlier that colors were a technical problem. Minimizing their effects was a matter of craftsmanship and eluded mathematical understanding. Unlike Barrow, Huygens was not inexperienced with the craft of telescope making at all. With his diaphragm and his eyepiece he had shown that he was quite capable of handling such technical problems. The remark in his letter to Constantijn shows that he must have known, in a practical way, much more of the properties of those disturbing colors than

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his dioptrical writings reveal. Still, he did or could not integrate this knowledge into his theory of spherical aberration. Neither by adjusting it in some appropriate way, nor by extending it by a mathematical theory of colors. At the crucial point where colors thwarted his plans to design a perfect telescope, he did not know how to fit his experiential knowledge of lenses into his theoretical knowledge of them. Huygens did indeed appear as a scholar as well as a craftsman, but he did not weld both roles. Would it be reasonable to presume that Huygens’ project fell short of the kind of method Newton had successfully used to mathematize colors? If quantitative experimentation is the obvious way to get a mathematical grip on colors, one may say that it was in the wrong hands as far as the sciencebased improvement of the telescope is concerned. Newton’s methodological innovation stemmed from an entirely different context from Huygens’ dioptrics. Newton was after the physical nature of light and colors, a nature that in his view ought to be mathematically structured. His calculations of spherical aberration gave the same theoretical results as those of Huygens, but they were aimed at substantiating his claim that chromatic aberration was much larger.212 Newton saw the practical implications of his findings. He did not stick to his negative conclusion and set out to show how lenses could be replaced by mirrors.213 Yet, telescopes had not been the goal of Newton’s studies of light and colors. His original interest concerned their physical properties and the nature of matter. It may be questioned whether the kind of problem Huygens ran into – a technical problem that escaped his theory – would have given rise to a Newtonian quantification of those disturbing colors. The example of Hudde’s Specilla circularia makes it clear that a practical approach may also give cause for reasoning a problem away. Hudde explicitly distinguished mathematical and ‘mechanical’ exactness. In practice mechanical exactness would do, and Hudde accordingly simplified his mathematical analysis on the strength of explicitly practical considerations. It may have been precisely Huygens’ practical outlook that made him ignore colors in his theory of aberrations. He was studying the geometrical properties of lenses and those colors fell outside this domain. He knew that these could be dealt with by other means: the crafty tinkering with lenses he was also competent in. We can only speculate what form Huygens’ study would have taken, had he

212

In his lectures Newton derived a formula for spherical aberration. Newton, Optical papers 1, 405-411. His discovery of dispersion led him to conclude that no lens could ever prevent the disturbing effects of aberration and made him design his reflector. Shortly after he published his theory, he did consider the possibility that chromatic aberration could be prevented in lenses. In a letter to Hooke (Newton, Correspondence I, 172), he alluded to the possibility of constructing a compound lens that canceled out chromatic aberration. Pursuing an idea of Hooke’s, he considered the possibility of using a lens compounded of different refracting media in which chromatic aberration was cancelled in the course of consecutive refractions. (Newton, Mathematical Papers I, 575-576). In Opticks he ruled out this possibility, probably because it was at odds with the dispersion law he put forward in it. Shapiro, “Dispersion law”, 102-113; Bechler, “Disagreeable”, 107-119. 213

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pursued his thinking on the disturbing colors his configuration turned out to produce. The reality is that colors thwarted Huygens’ plan to design via theory a configuration of spherical lenses that minimized the effect of spherical aberration. It was not his own observation of those colors that made him drop the project. He needed a Newton to point out that those colors were inherent to lenses. And he only got the point when Newton made clear that it was an aberration; a mathematical property inherent to refraction. Huygens realized that his project was futile when he saw the ‘Abberationem Niutonianam’. Disappointed, he stroke out the larger part of what was one of the most advanced efforts in seventeenth-century science to do sciencebased technology. The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument Huygens’ orientation on the telescope may explain the form and content of Dioptrica, it does not explain it as such. Why did Huygens want to develop a theory of the telescope? Why did he want to prove mathematically that his eyepiece performed the way he knew by experience it did? Kepler’s motive for creating Dioptrice had been his conviction that an exact understanding of the telescope was needed for reliable observations. In the practice of midseventeenth-century telescopy this need did not turn out to be as pressing as Kepler had thought. Even when the telescope became an instrument of precision, astronomers could go about it with a rather superficial understanding of the dioptrics of the telescope. Kepler’s point of view does not seem to have been Huygens’ main motive to embark upon a study of the dioptrics of the telescope. In a preface he wrote in the 1680s for Dioptrica, he expressed his surprise that no one had explained the telescope theoretically. One would have expected this marvelous, revolutionary invention to have aroused the interest of scholars. “But it was far from that: the construction of this ingenious instrument was found by chance and the best learned men have not yet been able to give a satisfactory theory.” 214

In this preface, Huygens did not explain what further use such a theory would have. He wanted to explain the telescope and did not wonder whether others also found this important. To Huygens, I believe, the dioptrics of the telescope was a meaningful topic in its own right. Huygens’ practical activities strengthen the impression that he was fascinated by the instrument for its own sake. As we have seen, his interest in telescopes went far beyond mere dioptrical theory. He made telescopes and prided himself with the innovations he had made to the instrument as well as to the craft. Yet, making telescopes seems to have been a goal in itself.215 Despite his impressive discoveries around Saturn, Huygens never became a telescopist. He did not – and could not and need not – make some sort of a 214

OC13, 435. “Sed hoc tam longe abest, ut fortuito reperti artificij rationem non adhuc satis explicare potuerint viri doctissimi.” 215 Van Helden, “Huygens and the astronomers”, 148 & 158-159.

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living out of the manufacture of telescopes. He prided himself with making good instruments – the best, he claimed in Systema saturnium – but it seems making them was his ultimate goal and pleasure. To this desire to make his telescopes work, he added in Dioptrica the desire to figure out how they worked. Both intellectually and practically, Huygens was fascinated by the working of telescope in itself. During the 1660s he added an extra dimension to his zygomorphic interest in the telescope. In De Aberratione he aimed to put his theory to practice in order to design a perfect telescope. Deliberately or not, he tried to join his practical and theoretical activities regarding the telescope. In the wider context of seventeenth-century science, De Aberratione can be seen as an instance of the omnipresent utilitarian ideal. An ideal that took on various forms, ranging from invoking science to the general benefit of mankind to using it to understand and solve particular problems of river hydraulics or gunnery. Or, the other way around, Bacon’s call for an alliance between the sciences and the crafts would have scientists learn from and turn to the experiential knowledge acquired by craftsmen. At the academies in London and Paris programs were developed to take stock of the arts. Little came of those plans and the rare times scholars set out to offer their learning to practicians were even less successful.216 De Aberratione is an example of a specific application of science to a practical problem. Such instances were the exception in the seventeenth century, as utilitarianism often did not go beyond grand utopian schemes.217 Not accidentally, Huygens brought it about, as he combined the scholar and the craftsman in one person. He did so, however, without wasting his breath in Dioptrica on Baconian or otherwise inspired ideals. The only place where utilitarian ideas are explicit is the plan he wrote around 1663 for the Académie.218 Huygens’ own interests – dioptrics, harmonics, motion – were prominent in the plan, and he explicitly linked them to practical issues of astronomy, navigation and geodesy. In these plans, as in his own studies, it was a utilitarianism of sorts: centered around scientific instruments and thus focused on the advancement of science. He was no exception in this regard. Westfall has shown that almost all interrelations that were established during the seventeenth century between scholarship and craftsmanship concerned scientific instruments.219 It is a kind of utilitarianism that stays very close to science itself. In the plan for the Académie those instruments served as mediators for a selected set of issues for the common good. The general benefit of solving the problem of finding longitude may be clear, but for the rest the development of instruments was aimed at the advancement of science. This in its turn was apparently 216

Boas, “Oldenburg, the ‘Philosophical transactions’, and technology”, 27-35; Ochs, “Royal society” Another example is Castelli’s attempt to engineer river hydraulics, discussed in Maffioli, Out of Galileo. 218 OC4, 325-329. 219 Westfall, “Science and technology”, 72. 217

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considered a useful task in itself. In the preface cited above, Huygens sings the praise of the invention of the telescope. It had served the contemplation of the heavenly bodies tremendously and had revealed the constitution of the universe and our place in it. “What man, unless plain stupid, does not acknowledge the grandeur and importance of these discoveries?”220 Huygens’ interest in scientific instruments was not exceptional. The form it took was exceptional. Huygens gave a particular twist to the idea that theory could be used to improve the telescope. Instead of deriving an ideal solution to the problem of spherical aberration, he applied his mathematical understanding of real, spherical lenses. Gaining a theoretical understanding of the telescope was not that hard for a Huygens; applying it to solve practical problems proved a more tricky business. With his clocks he was more successful. His theoretical knowledge of circular motion enabled him to design an isochronous pendulum. Still, the usefulness of the cheeks was rather limited. He had to rack his brains considerably to find means to make his clocks seaworthy – with or without cheeks. Instruments may not have guided Huygens’ other pursuits as they did in dioptrics, they certainly were important to him. He published part of his studies of circular motion in the guise of a treatise describing and explaining his isochronous pendulum clock. One might say that Huygens used instruments to present himself and his scientific knowledge. This would ally with the way he emphasized, in Systema saturnium, the quality of his telescopes. It would also offer a (partial) explanation of the fact that he did not publish Dioptrica despite repeated requests. The book would lack a vital element: an impressing innovation of the telescope. The invention he had placed his hopes on – a configuration of spherical lenses neutralizing spherical aberration – had turned out to be worthless.

220

OC13, 439. “Quae magna et praeclara esse quis nisi plane stupidus non agnoscit?”

Chapter 4 The 'Projet' of 1672 The puzzle of strange refraction and causes in geometrical optics

Huygens was in Paris in the autumn of 1672. He was still a leading scholar, but some clouds had begun to appear in the sky. The discussions at the ‘Académie’ sometimes distressed him, in particular the interventions of Roberval.1 His status was challenged by aspiring newcomers. The previous chapter described how Newton with his new theory thwarted his plans to design a perfect telescope. And the successful entrée on the Parisian scene of Cassini put serious pressure on his position as 'primes' under Louis' savants. Cassini had arrived from Rome in 1669 and almost immediately had started to adorn his patron with a series of astronomical observations, where Huygens could set little against.2 With all that, sickness had begun to plague him. In February 1670 he had fallen ill and he went home to the ‘air natale’ of The Hague in September. June 1671 Huygens returned to Paris to resume his activities, but in December 1675 he would relapse into his ‘maladie’. Whether these illnesses were caused by his ‘professional’ troubles is hard to tell. Huygens biographer Cees Andriesse holds this view, developing a Freudian reading of Huygens’ personality, in which Christiaan identifies with his intellectual achievements to make up for the early loss of his mother.3 Still, going through his Paris letters to his brother Constantijn gives the impression that Christiaan was bothered by a good share of homesickness. And maybe the Paris environment just was not that good for Huygens’ constitution. Whether or not his failing health was related, it is certain that the move from the confines of his parental home to the competitive milieu of Paris had put his science under pressure in the early 1670s. Huygens did not stand by idly, however. In 1672, he was in the middle of preparing the description and explanation of his pendulum clock for publication. Horologium Oscillatorium, his masterpiece, appeared in 1673 dedicated to his patron Louis. And whoever might think that Huygens had given up on his dioptrics because of Newton's interference, was dead wrong. Huygens had discarded the results of his analysis of spherical aberration in October 1672. Around the same time, he drew up a plan for a publication 1

Gabbey, “Huygens and mechanics”, 174-175; Andriesse, Titan, 235-243. Van Helden, “Constrasting careers”, 97-101. 3 Andriesse, Titan, 244-247 and “The melancholic genius”, 8-11. I have discussed Andriesse’s account in Dutch in Dijksterhuis, “Titan en Christiaan”. 2

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on dioptrics. Under the heading ‘Projet du Contenu de la Dioptrique’, he first listed the topics he would discuss and then made an outline of the chapters.4 The treatise would contain a large part of the dioptrical theory he had developed since 1653. It would be a comprehensive account of the refraction of light rays in lenses and their configurations. With this, he finally prepared to give in to the persistent demands of his correspondents to publish his dioptrics. He would not be able to present an impressive innovation, like the cycloïdal pendulum of Horologium Oscillatorium. But also without the design of a flawless telescope, Huygens had something to offer. The theory of Tractatus was still worth publishing, despite the fact that Barrow had gotten ahead of him by publishing the derivation of the focal distances of spherical lenses from the sine law. A theory elaborating the dioptrical properties of telescopes was still not available. Huygens had enough material left to fill up a treatise on dioptrics. The ‘Projet’ – as I will refer to it – is a key text in the development of Huygens’ optics. On the one hand, it straightened out the remains of his previous studies of dioptrics. On the other hand, it pointed a new direction for his optics that would eventually lead to the Traité de la Lumière. This direction was sign-posted by two new topics the ‘Projet’ introduced to Huygens’ optics. First, the treatise would contain a chapter on the nature of light. Huygens planned to give an explanation of the sine law of refraction in terms of waves of light. Secondly, he would discuss an optical phenomenon recently discovered: the strange refraction of Iceland crystal. The topic bears no relevance whatsoever to the questions about telescopes that had occupied him in his previous dioptrical studies. The reason for treating strange refraction was that it posed a problem for the kind of explanation of the sine law he had in mind. All this is remarkable, for in his dioptrics Huygens had never before bothered about the nature of light or the cause of refraction. What is more, in his recent dispute with Newton, he appeared to have a blind spot for these very subjects. In this chapter, we follow Huygens’ switch from the mathematical analysis of the behavior of refracted rays to the consideration of its causes and the explanation of optical laws. The issue of causes became relevant for Huygens through the phenomenon of strange refraction. The first attack of the problem was inconclusive and, moreover, left the issue of the cause of refraction untouched. Together, this attack and the ‘Projet’ are illuminating, not only for the development of Huygens’ optics and his conception of mathematical science, but also for seventeenth-century optics in general. Optics was in the middle of a transition from medieval ‘perspectiva’ to new way of dealing mathematically with phenomena of light. This chapter focuses on the issue of causes and explanations in optics. Over the shoulder of Huygens we look back to the way Alhacen, Kepler, Descartes dealt with the 4

OC13, 738-745. I date the sketch in 1672, instead of 1673 as the editors of Oeuvres Complètes have it. See page 92 above and page 140 below.

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physical foundations of optical laws. Huygens’ opinions and conduct in 1672 turn out to be rather illustrative of the transition optics was going through. The problem of strange refraction would be solved five years later, but not without Huygens developing a different and innovative approach to the nature of light. That will be discussed in the next chapter. ‘Projet du Contenu de la Dioptrique’ The ‘Projet’ sketchily fills up the two sides of a manuscript page, with all kinds of additions and remarks inserted in and around a main line of contents.5 It begins with a short list of topics and continues with an outline of the chapters and their content. The planned treatise on dioptrics would, of course, be about the telescope: “my principal design is to show the reasons and measures of the effects of telescopes and microscopes.”6 The treatise would open with a historical chapter on the invention and advancement of the telescope and of telescopic discoveries, and was to include an account of the development of the mathematical understanding of lenses and related phenomena.7 In chapters four to seven Huygens would expound his own theory of dioptrics, the theory of the telescope of Tractatus. He would solely discuss spherical lenses – “the only ones useful until now” – and leave out the hyperbolic and elliptic lenses invented by Descartes. Huygens was clear about the principal defect of Descartes’ treatment of dioptrics: “What I have said about the necessity of the theory of spherical ones is so true that Descartes, for not having examined it, has not known to determine the most important thing in the effect of telescopes, which is the proportion of their magnification, for what he says about it means nothing; …”8

The final chapter of ‘Dioptrique’ would treat the structure and the working of the eye. The main part of these four chapters was ready and only needed some rewriting and restructuring. The second and the third are the chapters of the ‘Projet’ that interest us here. They introduced two topics new to Huygens’ dioptrics: the cause of refraction and the strange refraction of Iceland crystal. In chapter two, Huygens planned to treat the sine law and its causes. He would start with a historical account of the discovery of the sine law – in his view undeservedly attributed to Descartes – and discuss some features of refraction. Next, he would give an explanation of refraction and discuss the nature of light. Although sketchy, the gist of his plans is clear. He rejected the explanation of the sine law Descartes had given in La Dioptrique:

5

Hug2, 188r-188v. OC13, 740. “mon principal dessein est de faire voir les raisons et les mesures des effects des lunettes d’approche et des microscopes.” 7 OC13, 740. Huygens mentions Archimedes (things seen under water), Alhacen, Kepler and Galileo by name. He elaborated his historical account later during the 1680s. 8 OC13, 743. “Ce que j’ay dit de la necessitè de la theorie des spheriques est si vrai, que Descartes pour ne l’avoir point examinée n’a sceu determiner la chose la plus importante dans l’effect des lunnetes qui est la proportion de leur grossissement, car ce qu’il en dit ne signifie rien; …” 6

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“difficulties against Descartes. where would the acceleration come from. he makes light a tendency to move [conatus movendi], which makes it difficult to understand refraction as he explains it, at least in my view. … light extends circularly and not in an instant,…”9

The concluding words reveal Huygens’ own conception: the nature of light is to spread out circularly over time. In other words, light consists of waves. The notes also clarify where Huygens had got the idea to think of light as waves. “Refraction as explained by Pardies.”10 Ignace-Gaston Pardies was a Jesuit father with a keen interest in the mathematical sciences, who actively participated in the Parisian scientific life, and with whom Huygens maintained good relations. Pardies had proposed the idea that light consists of waves and had explained the sine law with it. Huygens listed some essentials of a wave theory: “transparency without penetration. bodies capable of this successive movement. Propagation perpendicular to circles.”11 In the margin he added “vid. micrograph. Hookij”, a reminder to check Hooke’s alternative wave theory of Micrographia.12 The original formulation of Pardies’ theory has been lost, so we cannot know what precisely Huygens knew of it. He had known of “… the hypothesis of father Pardies …” at least since August 1669, when he mentioned it in a discussion at the Académie.13 On 6 July 1672 Pardies sent him a treatise on refraction that probably revealed some more details. After Pardies died in 1673, his confrere Pierre Ango published his explanation of refraction – at least its main lines – in L’Optique divisée en trois livres (1682). Ango had taken ‘the best parts’ of Pardies’ theory and blended them with own ideas, but Huygens did not have a high opinion of Ango’s work.14 We do not know to what exact extent Huygens knew Pardies’ theory and derived his own understanding of the nature of light and refraction from it. We do know that they stood in close contact over these matters, that Huygens openly acknowledged the contributions of Pardies, and that the essentials of Pardies’ theory where central to Huygens’ subsequent attack of strange refraction. He explicitly recorded the main assumption of Pardies’ derivation of the sine law: “Propagation perpendicular to circles.” In other words, rays are always normal to waves.15

9

OC13, 742. “difficultez contre des Cartes. d’où viendrait l’acceleration. il fait la lumiere un conatus movendi, selon quoy il est malaisè d’entendre la refraction comme il l’explique, a mon avis au moins. … lumiere s’estend circulairement et non dans l’instant, …” 10 OC13, 742. “Refraction comment expliquee par Pardies.” 11 OC13, 742. “transparance sans penetration. corps capable de ce mouvement successif. Propagation perpendiculaire aux cercles.” 12 OC13, 742 note 1. 13 OC16, 184. “… l’hypothese du P. Pardies …” Pardies’ second letter to Newton in their dispute about colors suggests that Pardies’ wave conception of light was rooted in Grimaldi’s ideas. Shapiro, “Newton’s definition”, 197. 14 Shapiro, “Kinematic optics”, 209-210. OC10, 203-204. 15 This is discussed below, in section 4.2.2.

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In the ‘Projet’ the third chapter was only indicated by a title and a single remark: “Iceland crystal” and “difficulty of the crystal or talc of Iceland. its description. shape. properties.”16 Iceland crystal was a rarity from the barren nordic lands displaying remarkable properties. This had been known for ages, but a sample had recently been brought to Copenhagen to increase the collection of curiosities of the Danish king. Danmark’s leading mathematician, Erasmus Bartholinus, then made a study of the crystal and its phenomena and reported on its strange refraction properties in 1669 in a treatise called Experimenta crystalli islandici disdiaclastici (1669). The strange refraction of Iceland crystal contradicted the sine law. It refracts a perpendicularly incident ray, which is impossible according to the sine law. Still, Iceland crystal had no relevance whatsoever to telescopes. So why would Huygens include it in his ‘Dioptrique’? The reason is that strange refraction constituted a problem for Pardies’ explanation of refraction. The ‘difficulté’ of Iceland crystal was that the refraction of the perpendicularly incident ray could not be reconciled with the assumption that rays are normal to waves. Huygens did not say this explicitly, but the place where he indicated the ‘difficulté’ makes it clear that Iceland crystal was a problem for Pardies’ explanation of refraction. Moreover, in his first notes on the phenomenon of around the same time, Huygens explicitly phrased the problem this way.17 We now see why Huygens would want to include strange refraction in a treatise on the dioptrics of the telescope. If his explanation of refraction were to be acceptable, it should not be contradicted by this particular kind of refraction. But why would he care for the tenability of the explanation so much? Huygens had not bothered to explain refraction before. Part of the answer lies in the fact that it was customary to do so. Books on geometrical optics usually contained a preliminary account of the nature of light and the causes of the laws of optics. The explanation of the sine law was to complete Huygens’ dioptrics so that it could be published as a proper treatise in geometrical optics. It would also complete his critique of Descartes’ La Dioptrique. As his theory of spherical lenses corrected the latter’s failure to explain the telescope properly, the projected explanation of the sine law would correct the difficulties in Descartes’ explanation. Waves would do the job, assuming that the problem of strange refraction could be settled. But what job exactly would they do? Just before the sketch of his explanation of refraction Huygens added an epistemological remark. An utterance of this kind is rare with Huygens, and this one is particularly illuminating: “Although it suffices to pose these laws as principles of this doctrine, as they are very certain by experience, it will not be unbecoming to examine more profoundly the cause of the refraction in order to try to give also that satisfaction to the curiosity of the mind 16

OC13, 743: “Cristal d’Islande” and 739: “difficultè du cristal ou talc de Islande. sa description. figure. proprietez.” 17 See below at the beginning of section 4.2.

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that loves to know the reason of every thing. And to have at least the possible and probable causes instead of remaining in an entire ignorance.”18

Huygens here presents his causal account as only supplementary rather than foundational. This raises the question what status waves of light had. Apparently they were merely probable and did not convey some indisputable truth. Still, they ought to explain refraction and do so in a better way than Descartes’ ‘conatus’ had done. Moreover, the explanation of ordinary refraction must not be contradicted by another instance of refraction, exotic as it might be. Just like any mathematical theory, an explanatory theory ought to be consistent. Despite the limited importance of waves, Huygens took the problem that strange refraction posed for waves seriously. He went on to get it out of the way before publishing his ‘Dioptrique’. 4.1 The nature of light and the laws of optics The problem that Huygens recognized is historically significant. Optics was in the middle of a transformation initiated by Kepler and Descartes, in which the rising corpuscular view on essences was shifting the understanding of the nature of light as well as the relationship between causal explanations and mathematical descriptions of its properties. Whether he fully realized it or not, with the ‘difficulté’ of strange refraction Huygens found himself at the heart of this remapping of the scholarly treatment of light. Before turning to Huygens’ first efforts to reconcile strange refraction with light waves, I will sketch the historical context of the epistemological issues raised by the ‘Projet’, in particular the relationship between physics and mathematics of light. To this end, I sketch the way Huygens’ most significant precursors in optics treated the issue of causality with respect to reflection and refraction: Alhacen, Kepler, Descartes and Barrow. This will bring into perspective Huygens’ specific, and rather non-committal approach to the explanation of the law of refraction. In this regard, it should be noted that the phrase ‘law of refraction’ was rarely used in seventeenth-century optics.19 In the project Huygens spoke of ‘loix de refraction’, which included for example reciprocity as well, as he also would do in Traité de la Lumière. In Dioptrica he called the sine law the proportion or ratio of sines.20 The concept of a law of nature aroses in 18

OC13, 741. “Quoy qu’il suffise de poser ces loix pour principes de cette doctrine, comme estant tres certains par l’experience, il ne sera pas hors de propos de rechercher plus profondement la cause de la refraction pour tascher de donner encore cette satisfaction a la curiositè de l’esprit qui aime a scavoir raison de toute chose. Et d’avoir au moins les causes possibles et vraysemblables que de demeurer dans une entiere ignorance.” 19 Kepler used ‘mensura’ (Kepler, KGW2, 78; see below). Descartes spoke of the laws of motion but of ‘mesurer les refractions’ (Descartes, AT6, 102). In his optical lectures of 1670 Newton used ‘regula’ and ‘mensura’ (Newton, Optical papers 1, I, 168-171 & 310-311). In Opticks Newton, like Huygens in Dioptrica, used ‘proportion’ or ‘ratio’ of sines (Newton, Opticks, 5-6 & 79-82). 20 Huygens, OC13, 143-145. In Traité de la Lumière he simply called the sine law the ‘principale proprieté’ of refraction (others are its reciprocity and total reflection); Huygens, Traité de la Lumière, 32-33. In his notes he sometimes spoke of ‘laws’ or ‘principles’ (OC13, 741) as he did in the draft of Dioptrica prepared around 1666 (OC13, 2-9).

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seventeenth-century natural philosophy and entered the mathematical sciences only gradually. In the opening lines of the first chapter of Paralipomena, which treats the nature of light, Kepler points out a disciplinary division between physical and mathematical aspects of light. “Albeit that since, for the time being, we here verge away from Geometry to a physical consideration, our discussion will accordingly be somewhat freer, and not everywhere assisted by diagrams and letters or bound by the chains of proofs, but, looser in its conjectures, will pursue a certain freedom in philosophizing - despite this, I shall exert myself, if it can be done, to see that even this part be divided into propositions.”21

In the subsequent chapters Kepler naturally returned to the firm grounds of geometry, but not before he had pointed out an unfortunate side effect of this division. In the appendix to the chapter he complains that the insights mathematicians have acquired regarding light are neglected and undeservedly underrated by natural philosophers. Therefor, in this appendix, Kepler explains the common misunderstandings of them - notably the followers of Aristotle - although they could have corrected themselves had they taken notice of the writings of opticians.22 The gap between physical and mathematical accounts of the cosmos in pre-Keplerian astronomy is well-documented. It is tempting to take stock of Scholastic views on the nature and function of mathematical inquiry and generalize the status of mathematical astronomy towards that of the mathematical sciences as a whole. Smith argues that a historical link between classical astronomy and classical optics existed, consisting of shared conceptions, commitment and methodologies.23 Still, when considering the relationship between mathematical descriptions of light and its nature, caution should be taken. Compared to the other fields of mathematics, the development of geometrical optics followed a rather idiosyncratic course up to the early seventeenth century. Since Greek antiquity it was realized that the central object of study – the light ray – combines almost naturally physical and mathematical conceptualization. In this way geometrical optics had incorporated a realistic mode of geometrical reasoning since the very founding of the science by Euclid and Ptolemy. Through the influential work of Alhacen the onset of a physico-mathematical conception of optics was established at a much earlier time than would be the case in the other mathematical sciences. In its transmission through medieval perspectiva, Alhacen’s optics was the starting point for Kepler and Descartes and profoundly affected their innovations of the science. As a consequence, 21

Kepler, Paralipomena, 5 (KGW2, 18. Translation Donahue: Kepler, Optics, 17). “Caeterum cum hic à Geometria interdum in physicam contemplationem deflectamus: sermo quoque erit paulò liberior, non ubique literis et figuris accommodatus, aut demonstrationum vinculis astrictus, sed coniecturis dissolutior, libertatem aliquam philosophandi sectabitur: Dabo tamen operam, si fieri potest, ut in Propositiones et ipse dividâtur.” 22 Kepler, Paralipomena, 29 (KGW2, 38) 23 Smith, “Saving the appearances”, 73-91.

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methodological, epistemological and conceptual features of perspectivist optics are perceptible throughout seventeenth-century optics.24 4.1.1 ALHACEN ON THE CAUSE OF REFRACTION In the eleventh century, the Islamic scholar Ibn al-Haytham composed a voluminous work on optics, KitĆb al-ManĆzir. The Optics of Alhacen, as they are commonly referred to in the West, was intended to bring together mathematicians’ and physicists’ accounts of light and vision by giving a systematic treatment of optics that met the demands of both.25 This required the combination of the Aristotelian doctrine of forms received by the eye and Ptolemy’s ray-wise analysis of the perception of shape and position. According to Alhacen both these notions were partly true but incomplete. 26 The synthesis he had in mind - ‘tarkĩb’ - consisted of a theory in which the forms of light and color issue from every point of the object and extend rectilinearly in all directions.27 It met the demands of the Aristotelian doctrine by considering light rays as the direction in which light extended and those of Ptolemy by appointing rays as the ultimate tool of analysis. As contrasted to his mathematical precursors, Alhacen regarded a ray as a purely mathematical entity: “Thus the radial lines are imaginary lines that determine the direction in which the eye is affected by the form.”28 In a later work on optics, the Discourse on light, Alhacen expounded his conception of light and reflected on the character of the science of optics by discussing the distinction between mathematical and physical aspects of light. In his view, each provided answers to different kinds of questions: in physics one investigates the essence of light; in mathematics the radiation or spatial behavior of light. Physical theory classified the various kinds of bodies: luminous, shining, transparent, opaque. Mathematical theory described the perception of things by means of rays, rectilinear and inflected.29 In the Optics Alhacen adopted the Aristotelian concept of forms without further philosophical inquiry. His exposition on the nature of light in book 1 served as a physical foundation for the mathematical and experimental investigation of light and vision that constituted the heart of the Optics. Alhacen provided the basis for the flourishing of the study of light and vision in thirteenth-century Europe given shape to by Robert Grosseteste, Roger Bacon, John Pecham and Witelo.30 Alhacen’s work reached the west in 24

This theme is amplified by, among others, Schuster, Descartes, 332-334: Smith, Descartes’s Theory of Light and Refraction, 4-12. 25 Alhacen, Optics I, 3-6 (book 1). The content and scope of Alhacen’s optics is discussed in Sabra’s introduction to his translation of its first three books: Alhacen, Optics II, xix-lxiii. See further Lindberg, Theories, 85-86. 26 Alhacen, Optics I, 81 (book 1, section 61). 27 Alhacen’s account for the subsequent one-to-one correspondence between the points of the object and the image in the eye is discussed in section 2.2.1 above. 28 Alhacen, Optics I, 82 (book 1, section 62) 29 Alhacen, Optics I, li (Sabra’s introduction). See also Sabra, “Physical and mathematical”, 7-9. 30 Lindberg, Theories, 120-121 and 109-116.

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truncated form, for the Latin translation, in both manuscript and printed form, lacks his first three books. It was translated in the thirteenth (possibly twelfth) century and became known as Perspectiva (or De aspectibus).31 The Perspectiva communis (ca. 1279) of Pecham and the Perspectiva (ca. 1275) of Witelo are to be understood primarily as compendia of Alhacen’s optics. These works became textbooks of perspectiva - a common denominator of medieval optics. Friedrich Risner published (the remaining books of) Alhacen’s Optics together with Witelo’s Perspectiva in 1572, an edition that remained authoritative well into the seventeenth century. To provide for the now lacking physical foundation of Alhacen’s optics, perspectivist writers drew on the ideas of Grosseteste and Bacon. Through Bacon, Grosseteste’s theory of the multiplication of species was incorporated into perspectivist theories. Although it provided a mathematically structured account of the nature of light and its propagation, it served no function in perspectivist accounts of the behavior of light rays interacting with various media. The perspectivist writers reiterated Alhacen’s analysis of reflection and refraction, adding some clarifications on its basic assumptions.32 In his accounts of reflection and refraction Alhacen also discussed the causes of these phenomena. These appealed only to light qua radiation, however, not its physical essence. The core of Alhacen’s account consists of a mathematical analysis of rays in their components perpendicular and parallel to the reflecting or refracting surface. In reflection the parallel component remains unaltered whereas the perpendicular component is inverted, which readily yields the law of reflection.33 By differentiating the parallel and perpendicular components of a light ray he extended Ptolemy’s analysis, who had only considered the angles before and after reflection.34 In his account Alhacen appealed to an analogy between reflected rays and a rebounding ball: “We can see the same thing in natural and accidental motion, …”.35 He pictured a sphere attached to an arrow projected perpendicularly or obliquely to a mirror. This mechanical analogy applied to the mathematical analysis of the motion of light rays and did not appeal to the form-like nature of light. Light is reflected because its motion is fully or partially ‘terminated’ by an obstacle. Alhacen’s causal account of refraction proceeded along similar lines. Rays are refracted because their motion changes when they enter a medium of different density. In the case of refraction towards the normal – into a denser medium – he assumed that part of the parallel component was altered. He did so implicitly, in an comparison with a ball striking a thin slate. “For 31

Alhacen, Optics I, lxxiii-lxxix (Sabra’s introduction). Lindberg, “Cause”, 30-31. 33 Risner, Optica thesaurus, 112-113. Witelo relied the argument of the shortest path: Risner, Optica thesaurus, 198. 34 Lindberg, “Cause”, 25-29. Sabra, “Explanation”, 551-552. 35 Risner, Optica thesaurus, 112-113. “Huius aút rei simile in naturalibus motibus videre possumus, & etiá in accidentalibus.” Translation: Lindberg, in Grant, Source book, 418. 32

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things moved naturally in a straight line through some substance that will receive them, passage along the perpendicular to the surface of the body in which passage takes place is the easiest.”36 A couple of lines further, Alhacen continued: “Therefore, the motion [of the light] will be deviated toward a direction in which it is more easily moved than in its original direction. But the easier motion is along the perpendicular, and that motion which is closer to the perpendicular is easier than the more remote.”37 In the case of refraction away from the normal Alhacen abandoned the appeal to the easiest path. He considered the components of the ‘motion’ again and stated without argument that the parallel component is increased. Besides being inconsistent, Alhacen’s account of refraction remained qualitative, as he did not attempt to determine to what degree a refraction ray was bent towards the normal, nor to what proportion the parallel component was altered. Alhacen’s account of refraction primarily consists of an experimental analysis. In Risner’s edition it covers the first eleven or twelve propositions of the seventh book, which return in the second chapter of Witelo’s part. In the tenth chapter the latter added to the quantitative account of refraction by providing a table – supposedly observational – of angles of refraction for a set of incident rays. In Alhacen’s accounts of reflection and refraction two levels of inference can be distinguished. In the first place, the analysis of rays in their perpendicular and parallel components revealed some deeper lying mathematical structure of both phenomena. It unified his accounts to some extent, although he did not assume the parallel component unaltered in all cases like Descartes would later do. The second level involves mechanical analogies that illuminate rather than prove the mathematical analyses of reflected and refracted rays. The causal account provided additional support for the properties of reflection and refraction, but the ultimate justification was empirical.38 In this regard the analogies can be considered to serve didactical purposes. Alhacen’s analogies do not - and were not intended to - explain refraction and reflection by deriving their properties from an account of the nature of light. That is the way Huygens and his fellow seventeenth-century students of optics understood ‘explaining the properties of light’ and which his waves of light would have to bring about. Whereas the rectilinearity of rays followed rather naturally from Alhacen’s understanding of forms, reflection and refraction are discussed in terms of light rays instead of interactions between forms with reflecting and refracting substances. The ideals of 36

Risner, Optica thesaurus, 241. “Omnium autem moterum naturaliter, que recte moventur per aliquod corpus passibile, transitus super perpendicularem, que est in superficie corperis in quo est transitus, erit facilior.” Translation: Lindberg, “Cause”, 26. 37 Risner, Optica thesaurus, 241. “...: accidit ergo, ut declinetur ad partem motus, in quam facilius movebitur, quàm in partem, in quam movebatur : sed facilior motuum est super perpendicularem: & quod vicinius est perpendiculari, est facilius remotiore.” Translation (amended): Lindberg, “Cause”, 27. 38 Alhacen, Optics I, lxi (Sabra’s introduction); Risner, Optica thesaurus, XVII-XIX (Lindberg’s introduction).

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mechanical philosophy notwithstanding, this ‘physical’ understanding of rays and their behavior would crucially affect the investigations of Huygens and other seventeenth-century opticians. Kepler and Descartes set off where Alhacen and Witelo had left off. A law of reflection was known, as well as diverse mathematical properties of radiated light, but refraction remained to be understood only qualitatively. The thirteenth-century synthesis left perspectiva as a comprehensive body of knowledge – Alhacen’s theory of vision, solutions to various problems of reflection and so on – riddled with some persistent, well-known problems like the pinhole image.39 The sixteenth century witnessed major developments in optics, but mainly in its practical parts that bore on Galileo’s telescopic achievements rather than Kepler’s and Descartes’ theoretical pursuits.40 4.1.2 KEPLER ON THE MEASURE AND THE CAUSE OF REFRACTION The heritage of medieval perspectiva Kepler received, consisted of a welldefined set of aims and criteria for geometrical optics: mathematical analysis of the behavior of light rays.41 In Paralipomena he took up this heritage and transformed it radically. In chapter two we have seen how he created a new theory of image formation by rigorously applying the principle of rectilinear propagation of light rays. We now turn to his account of the causes underlying the behavior of light rays. Here the same approach is recognizable. In Kepler’s view, the mathematically established properties of things are real and should be directive in physical considerations. Kepler’s conception of the nature of light can be seen as a realist reading of perspectivist’s mathematical ideas which he then rigorously employed in the investigation of the behavior of light.42 At the start of this section I cited the opening lines of Paralipomena, where Kepler pointed out the relative freedom of reasoning he would employ in these matters pertaining to physics. In the first chapter, ‘De Natura Lucis’, he expounded the general concepts and principles pertaining to his account of optics. On the whole, his theory of light was the perspectivists’ theory of multiplication of species enriched with neoplatonist metaphysics.43 According to Kepler, light is an incorporeal substance which has two aspects, essence and quantity, by which it has two operations (‘energias’), illumination and local motion, respectively.44 Radiation is the form of propagation: light spreads in all directions and does so spherically. Light rays are the radii of this sphere and thus rectilinear. Light itself can be regarded as the twodimensional surface of an expanding sphere. The mathematical structure of 39

Lindberg, Theories, 122-132. Dupré, Galileo, 17-19. 41 Lindberg, “Roger Bacon”, 249-250. 42 The following discussion owes much to Buchdahl’s illuminating discussion of Kepler’s method: Buchdahl “Methodological aspects”. References are to the original text, corresponding pages in the Gesammelte Werke in parentheses. Except where noted, translations are by Donahue from Kepler, Optics. 43 Lindberg, “Incorporeality”, 240-243. 44 Kepler, Paralipomena, 13 (KGW2, 24) 40

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this theory was clearly perspectivist in origin, but to Kepler it represented the physical nature of light, not only its mathematical behavior. On the basis of this theory of light, Kepler discussed the behavior of propagated light. The rectilinearity of light rays is a direct outcome of Kepler’s conviction that light ‘strives to attain the configuration of the spherical’.45 Where light is deflected from its straight path this must be the effect of the interaction of light and matter. As light is a two-dimensional surface this interaction can only occur with the surface of reflecting and refracting media. Kepler attributed a form of density to surfaces and argued that light is hindered in its passage through the surface of a body proportionally to its density.46 In the case of reflection the density of the surface is so high that light falling upon it “... is made to rebound in the direction opposite to that whence it approached.”47 Kepler specified that this applied to the perpendicular component of a ray - i.e. the part of the motion towards the surface. The law of reflection now followed naturally, thus clearing the way for an exact analysis of the properties of reflection in chapter 3 of Paralipomena. In this chapter Kepler took up the classical topic of perspectiva to determine mathematically the location where a reflected image is perceived.48 The measure of refraction For refraction things were more complicated. The causal account in chapter 1 did not yield an exact law, so in the fourth chapter of Paralipomena Kepler could not readily embark on an analysis of refractional phenomena. Instead, he first had to find such a ‘measure’ of refraction. The course of the chapter reveals Kepler’s conception of the distinction between causes and measures in optics. Initially he stuck to the epistemic organization of his treatise by analyzing refraction in term of rays, the components of their motion, and regularities in the various angles at which they are refracted. However, when all this yielded no satisfactory results he took the nature of light into consideration to see where these ‘proper’ causes could lead him to find the measure of refraction. In proposition XX of chapter 1 Kepler derived from his suppositions about the interactions of light (surfaces) with (the surface of) a dense medium that a ray is refracted towards the perpendicular.49 His argument comes down to the idea that the surface impedes the spreading of the sphere of light. Kepler explained that this understanding is based on the fact that motion belongs to light and that said interaction is general for moving 45

Kepler, Paralipomena, 8 (KGW2, 20). “Nam diximus affectari à luce figurationem Sphaerici.” Propositions XII-XIV: Kepler, Paralipomena, 10-11 (KGW2, 22-23) 47 Kepler, Paralipomena, 13 (KGW2, 25). “Lux in superficiem illapsa repercutitur in plagam oppositam, unde advenit.” 48 An important part of this was Kepler’s negation of the generality of the cathetus rule and the introduction of his new theory of image formation, which have been discussed above in section 2.2.1. 49 Proposition XX: Kepler, Paralipomena, 15-21 (KGW2, 26-31). 46

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matter. In this way he followed up on the mechanical analogies employed in perspectivist causal accounts, but he did not do so without appropriating that line of reasoning to his own means. “For it may be permissible here for me to use the words of the optical writers in a sense contrary to their own opinion, and carry them over into a better one.”50 Kepler went on to develop the analysis of a ball spun into water by distinguishing between the dynamics of the parallel and perpendicular components of its motion, whereby light is rarified in the former direction and merely transported in the latter direction. He then proceeded with a short discussion of the underlying physics, to wit the statics of a balance. In this way Kepler transformed the mechanical analogies employed by his perspectivist forebears to illuminate the mathematics of refraction into a physical foundation of the analysis of refraction. The account in chapter 1 only yielded a qualitative understanding of refraction, and only partial for that matter, for Kepler did not discuss the passage of light into a rarer medium. At the opening of chapter 4, ‘De Refractionum Mensura’, Kepler still lacked an exact law of refraction. He needed this ‘measure’ in the first place for his account of the dioptrics of the eye in the next chapter (see above section 2.1.1.), but in the end principally for his account of atmospheric refraction later in Paralipomena. After all, it was a treatise in the optical part of astronomy for which the laws of optics were instrumental. Nevertheless my discussion will be confined to the optics per se: Kepler’s tour the force to tackle the mathematics of refraction. Kepler began with a review of the received opinions regarding the measure of refraction. In this section, he tied in with the traditional approach of considering the physical properties of light rays and their components. After negating several opinions, Kepler laid down the - in his view - generally established understanding: first, that the density of the refracting medium is the cause of refraction and, second, the angle of incidence contributes to its cause. The question therefor was how these two aspects are connected. Kepler ran through several options as they had been set forth, rejecting each as insufficient. Next, he contemplated how the two said aspects could correctly be combined.51 Kepler proceeded to represent these conditions geometrically (Figure 36). BC is the refracting surface of a medium BCED and AB, AG, AF are incident rays. Kepler now extended the medium to DEKL, thus representing the greater density of its surface. He then constructed a refracted ray FQ by drawing HN perpendicular to the lower surface and joining N at the imaginary bottom with F.52 Comparing the results of this method with Witelo’s table, Kepler simply concluded that it was refuted by

50

Kepler, Paralipomena, 16 (KGW2, 27). “Liceat enim hîc mihi verba Opticorum contra mentem ipsorum usurpare, et in meliorem sensum traducere.” 51 Kepler, Paralipomena, 85-87 (KGW2, 85-86) 52 This is equivalent with sini : tanr = constant. Lohne, “Kepler und Harriot”, 197. Compare Buchdahl, “Methodological aspects”, 283.

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experience.53 He tried some more ideas flowing from this geometry, including some ways of evaluating the ‘refractaria’ - the locus of images where the points of a line are percieved, D for point L, I for point M, etcetera.54 All ideas were refuted by experience and Kepler abandoned his attempt of finding a measure of refraction on the basis of an analysis of the physics of light rays. In the next three sections, Kepler temporarily ignored the causes of refraction and focused on finding mathematical regularities in the given angles of incidence and refraction. Building on the known properties of reflection, he tried certain analogies between reflection and refraction. Kepler argued that in the case of refraction in a medium with infinite density, all rays must be refracted into the perpendicular. He then correlated this case to reflection by a parabolic mirror with rays coming from its focus. This led him to consider the Figure 36 The first stage of attack of relationship of conic sections with refraction. He Kepler’s refraction. constructed a diagram of angles of incidence and refraction and considered the intersection of the accompanying rays. The resulting curve is similar to a hyperbola, but points from where the rays come are not the matching foci, so Kepler dismissed this attempt as well. This and other trials with conic sections – including the effort to construct an anaclastic curve – still did not give Kepler a correct ‘measure of refractions’ and he abandoned this line of thought as well. Finally, Kepler returned to his causal analysis of refraction of chapter 1 to query whether - “may God look kindly upon us” - this would yield the measure of refraction.55 As contrasted to the ray analysis of the first stage, he now considered the interaction of the surface of light with the surface of the refracting medium.56 Kepler warned beforehand he would perhaps stray somewhat from his goal of finding the measure of refractions in its causes, and halfway through his exercise he would acknowledge “In demonstrating the true cause of this directly and a priori, I am stuck.”57 He did not formally deduce a ‘measure’ from the causes of refraction, but rather had employed (in Buchdahl’s words) “physical considerations to guide the intuitive search for responsible factors relevant to the result.”58 53

Kepler, Paralipomena, 86 (KGW2, 86). “Hic modus refutatur experientiâ: ...”. Kepler, Paralipomena, 88-89 (KGW2, 87-88). 55 Kepler, Paralipomena, 110 (KGW2, 104). “quod Deus benè vertat” 56 Kepler, Paralipomena, 110-114 (KGW2, 104-108). 57 Kepler, Paralipomena, 110 and 113 (KGW2, 104 and 107). “Etsi enim à scopo forsan etiamnum nonnihil aberrabimus: ...” and “In genuina huius rei causa directè et à priori demonstranda haereo.” 58 Buchdahl, “Methodological aspects”, 291. 54

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Kepler began with the understanding of the nature of light as the surface of an expanding sphere laid down in the opening chapter of Paralipomena (Figure 37). ABMK is the section of a physical ray obliquely incident on the surface BC and refracted towards QBMR. According to Figure 37 The final stage of Kepler’s Kepler the angle of deviation must be analysis of refraction proportional to the angle of incidence. This condition is met when only the (surface)density of the refracting medium is assumed to be effective. With increasing obliquity, BM increases and therefore the resistance met by the light increases. Now “… there is more density in BM than in LM ...” so that the proportion LM to BM must be added as a factor of refraction onto the proportionality of angles of incidence and deviation.59 However, the proportion LM to BM – or sec i – implied a paradox. Horizontal rays would be refracted at an infinitely large angle. Kepler therefore changed his perspective and now considered BR of the refracted ray. He concluded that the secans of the angle at the upper surface of the denser medium ‘plays a part’ in refraction.60 Refraction was thus a composite of two factors: the proportionality of i-r to i and the proportionality of i-r to sec r – in other words: i-r = c· i· sec r, where c is some constant. In proposition 8, Kepler gave instructions how to apply this analysis to calculate angles of refraction. It is in the form of a ‘problem’, a procedural statement of the sort the later Dioptrice was composed of, as we saw above in section 2.2.1. After all, Kepler’s struggle had not yielded a general ‘measure of refraction’ independent of specific media and transcending measurements. First, both factors are determined for the medium by means of one known pair of incident and refracted rays. Then the angle of refraction for any other angle of incidence is computed. By means of an example, Kepler calculated a table for refraction from air into water. The values differed somewhat from Witelo’s data which Kepler had plied so rigorously in the previous sections. This time he was more tolerant: “This tiny discrepancy should not move you; believe me: below such a degree of precision, experience does not go in this not very well-fitted business.”61 Moreover, he (correctly) suspected that Witelo had modified his table on the basis of Ptolemy’s false supposition that the secondary differences of the angles are constant. “Therefore, the fault lies in Witelo’s refractions”, and Kepler proceeded to use his own result to consider atmospheric refraction.62 Although the empirical correctness was 59

Kepler, Paralipomena, 111 (KGW2, 105). “Plùs igitur densitatis est in BM, quàm in LM.” Kepler, Paralipomena, 113 (KGW2, 107). “..., sciendum igitur, eorum angulorum incidentiae secantes concurrere ad mensuram refractionum, qui constituuntur ad superficiem in medio densiori.” 61 Kepler, Paralipomena, 116 (KGW2, 109). “Neque te moveat tantilla discrepantia, credas mihi, infra tantam subtilitatem, experientiam in hac minus apt materia non descendere.” 62 Kepler, Paralipomena, 116 (KGW2, 109). “Ergò in Vitellionis refractionibus culpa haeret.” 60

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not beyond doubt, Kepler preferred his own data over Witelo’s because it was based on “regularity and order”. In the final propositions of this section and the remaining sections of the chapter, Kepler was now able to dealt with proper subject of the chapter: the quantitative treatment of atmospheric refraction. Kepler’s search for a ‘measure’ refraction clearly reveals the idiosyncrasies of his thinking. He laboriously reported on his persistent efforts to find a satisfactory law, and although – so we can see with hindsight – he came tantalizingly close he did not succeed. The successive stages of his attack display his ever inventive mathematical reasoning, mixed with those typical Renaissance conceptions of his that make it hard for a modern reader to distinguish mathematics and physical ideas. In the light of ensuing developments in seventeenth-century optics, the final stage of Kepler’s struggle with refraction is the most interesting. Here he took his conception of the nature of light into account in order to find a law of refraction. In a kind of microphysical, though far from corpuscular, analysis he considered the interaction of a surface of light and the refracting medium. At this stage he move farthest away from traditional approaches. Although the resulting ‘rule’ was phrased in terms of rays, he had taken the true nature of light into account while analyzing the interaction of rays and (refracting) media. As I see it, this was possible because of his realist view of mathematical description. With Kepler, the mathematics of light propagation necessarily reflected the nature of light. One may argue that mathematics took the lead in his thinking. Kepler more or less reduced light to a mathematical entity, a two-dimensional surface. The geometry of refraction was rather autonomous in his final attempt to derive a law.63 Yet, pure formalisms would have been meaningless for him. Kepler maintained geometrical optics as a mathematical theory explaining the behavior of light rays. He adopted many concepts of perspectivist theories of light and refraction, but he applied them in a radical and sometimes radically different way. On the level of methodology, all relevant components – physics, mathematics, observation – had been present in perspectivist optics, but Kepler sought a closer connection between them and often used these means in a much stricter way. He repeatedly allowed Witelo’s data to refute the outcome of his trials. Kepler’s wanted to establish a closer tie between the nature of light and the laws of optics and derive ‘measure’ from ‘cause’. He openly acknowledged that he could not realize this ideal. He resorted to a freer mode of reasoning because, as I see it, he was far too creative a thinker to stick too rigidly to his ideals.

63

See for example: Buchdahl, “Methodological aspects”, 291.

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True measures Even without a true measure of refraction, an inventive mathematician like Kepler could solve problems in the behavior of refracted rays. In Dioptrice, he determined properties of spherical lenses in a less rigorous way, pragmatically applying a rule that had only limited validity. Likewise, his predecessors had used their limited knowledge to discuss isolated problems regarding refraction. In order to turn ‘dioptrics’ into a genuine part of the mathematical science of optics, a true measure of refraction was still needed. However impressive Kepler’s persistence to find a true measure of refraction, his efforts will always have a tragic side. Around the same time he was struggling with the phenomenon, across the Channel the exact law had already been found by the very man Kepler had been corresponding with: Thomas Harriot. Harriot had done so by traditional means that were accessible to Kepler too: analysis of the observed propagation of light rays. The difference was that Harriot made new observations and had a lucky hand in this. Harriot’s success shows that, in the case of refraction, traditional methods could yield the required result. Around 1597, Harriot had begun looking for a law of refraction. Initially, he also tried to find a law on the basis of Witelo’s tables. As these efforts were unsuccessful, he decided that Witelo was unreliable and started to measure angles of refractions anew.64 After some fruitless attempts, he chose a way of measurement that proved very lucky. In 1601, he measured refraction by means of an astrolabe suspended in water (Figure 38). Viewing along the center R of the astrolabe, he determined the positions O where a point was seen when moved along the lower edge of the astrolabe. Then he determined the image points B. The cathetus rule (see page 33) said that the image point is the intersection of the normal to the refracting surface and the incident ray. All image points were on a circle. Figure 38 Harriot’s measurements (Lohne). This meant that RO and RB were in constant proportion, and likewise were the sines of i and r. In a table Harriot compared angles of deviation as he had measured them with calculated ones, but he did not reveal how he had used the figure with two concentric circles – which he called ‘Regium’ – for his calculations. The calculated values give reason to believe that it was the sine relation he used.65

64 65

Lohne, “Geschichte des Brechungsgesetzes”, 159-160. Lohne, “Kepler und Harriot”, 202-203.

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Harriot had reconsidered and reapplied traditional methods anew and found – what might be called – an empirical law of refraction. As contrasted to Kepler, he had turned to the measurement of refraction, instead of the theoretical trench-plowing of his hapless correspondent. Harriot does not seem to have considered the ‘proper cause’ of refraction with which his law may have been understood. His accomplishments were known only to a small circle of acquaintances. It is possible that they spread through correspondence, but he became known as a discoverer of the law of refraction only in the twentieth century.66 Around 1620, Willebrord Snel was the next to discover the exact measure of refraction - again by means readily available to Kepler. He did not publish his discovery, but it became generally known in the 1660s. How he discovered the law will remain a matter of conjecture. Snel’s papers on optics are lost, except for the notes he made in Risner’s Opticae libri quatuor (1606) and an outline of a treatise on optics discovered in the 1930s.67 Hentschel has been the first to make a thorough attempt at reconstruction. In his view, Snel was inspired by an ‘experimentum elegans’ described by Alhacen and copied by Witelo that involved a segmented disc lowered into water. This led him to study the refractaria and, facilitated by his geodetic expertise, to the law of refraction in secans form.68 I do not fully agree with Hentschel’s analysis, for I think that the idea of a contraction of the unrefracted perpendicular ray may have opened to Snel a more direct route to his discovery. Whichever interpretation is preferable, the main point is that Snel employed means readily available to Kepler. What is more, his approach of rational analysis of mathematical regularities in a set of refracted rays was precisely how Kepler set about initially. He even analyzed the refractaria from various perspectives, which makes it all the more surprising that Snel was seemingly unfamiliar with Paralipomena.69 It remains to be seen why Snel was successful - or: why he was satisfied with what he found, as contrasted to Kepler’s fruitless struggle. Maybe he was less strict in empirical matters or he was - like Harriot - just lucky with looking at the issue from the right perspective. Paralipomena and the seventeenth-century reconfiguration of optics The central concept of perspectiva was the visual ray, which established the visual relation between objects and observer.70 In seventeenth-century optics the concept of ray underwent two substantial changes, both anticipated by Kepler: the subordination of vision to light and the physicalization of the ray. 66

Lohne, “Geschichte des Brechungsgesetzes”, 160-161. Harriot corresponded with Kepler after the publication of Paralipomena. The correspondence broke off, however, before Harriot could reveal his findings. KGW2, 425. 67 The notes are in Vollgraff, Risneri Opticam. The outline was discovered by Cornelis de Waard, who transcribed and translated it in Waard, “Le manuscript perdu de Snellius”. A German translation is given in Hentschel, “Das Brechungsgesetz”, 313-319. 68 Hentschel, “Das Brechungsgesetz”, 302-308. 69 Hentschel, “Das Brechungsgesetz”, 334 note 22. 70 Smith, “Saving the appearances”, 86-89.

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The most important change in the mathematical study of light was the abandonment of questions of cognition. Perspectivist theory not only consisted of a theory of perception but also seized epistemological and psychological problems of visual cognition.71 The eye was crucial in that the behavior of rays was understood on the basis of an understanding of visual cognition. In the seventeenth-century optics the eye became a subordinate topic in the mathematical study of optics and questions of cognition were abandoned altogether. Kepler’s theory of image formation was a theory of rays painting pictures on a dead, passive surface. His theory of the retinal image was a theory only of ray tracing and he passed over physiological and psychological issues. Only in the fifth chapter of Paralipomena did he explain how the eye paints pictures on the retina, after he had explained image formation, reflection and refraction. The mathematical analysis of the behavior of light rays was turned into the study of the paths of light rays without an eye necessarily being present. Instead of the foundation of geometrical optics, vision became an application of it. The ray became a light ray instead of a visual ray. Kepler’s theory was readily assimilated in the first decades of the seventeenth century. The subordination of vision to the theory of image formation is clear in most seventeenth-century works on geometrical optics. This includes Huygens, who deferred his discussion of the eye to the last chapter of his projected ‘Dioptrique’. Shapiro has pointed out that Barrow’s thinking in terms of images as the eye perceives them was crucial to his extension of Kepler’s theory of image formation, as had been the case with Gregory.72 Yet, they too confined themselves to retinal imagery and adopted the Keplerian understanding of the eye as an optical instrument that painted images on the retina. Closely connected with the changing role of the eye and vision in the mathematical study of light is the changing meaning of the optician’s elementary tool: the ray. Whereas the mathematical line used in optical analysis in perspectiva represented a real line in space, it came to represent an imaginary line in time in the course of the seventeenth century.73 Instead of constituting light itself, a ray of light became – in various ways – the path traced out by some substance that constituted light. The corpuscular conceptions in the new philosophies of the seventeenth century transformed the light ray into an effect of some material action. Kepler’s conception of light and his analysis of reflection and refraction anticipated this, but with him light remained expressly incorporeal. One may say that the combined subordination of questions of vision and ‘physicalization’ of light constitutes the transition from medieval perspectiva to seventeenth-century geometrical optics. 71

Smith, “Big picture”, 587-589. Shapiro, “The Optical lectures”, 137. 73 Smith, “Ptolemy’s search”, 239-240. 72

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It is beyond dispute that Kepler was crucial to the development of seventeenth-century optics. With his seminal work, he gave the study of optics a new start at the beginning of the seventeenth century. What his influence was exactly is harder to determine. As a result of the advent of corpuscular conceptions of nature, his explanation of the nature of light was outdated almost immediately. On the level of theories and mathematical concepts his influence is clear: his theory of image formation and of vision were the starting-point of all subsequent studies. However, his contribution was largely obscured by the uncredited adoption of his ideas by Descartes most notably. On the level of methodology the matter is less clear. Descartes called Kepler his “first teacher in Optics”, but what he had been taught he did not say.74 He did not, for one thing, adopt Kepler’s candor as regards the way he discovered things. Seventeenth-century savants found Kepler’s Renaissance conceptions hard to take and the odor of mysticism that surrounded him seems to have been responsible for the fact that few referred to Kepler directly. As regards the way mathematical reasoning could be applied to understand natural phenomena, he was quickly overshadowed by Descartes and Galileo. Huygens, in particular, was silent on Kepler as regards his approach to optics. 4.1.3 THE LAWS OF OPTICS IN CORPUSCULAR THINKING The new philosophies of the seventeenth century came to see light as an effect of some material action. As a consequence, the mechanical analogies used in perspectivist accounts of reflection and refraction were put in a different light. Discussions of motions and impact regarding the causes of reflection and refraction were now connected directly with the essence of light. Yet, accounting for the nature of light was not integrated with mathematical analysis of the behavior of light rays at one go. This is evident in Descartes’ account of refraction in La Dioptrique, a peculiar amalgam of perspectivist and mechanistic reasoning. In La Dioptrique Descartes made public the sine law, which he had discovered in Paris in the late 1620s during his collaborative efforts to realize non-spherical lenses (see section 3.1). How exactly he arrived at the sine law remains a subject for debate, but it is certain Descartes did not discover it along the lines of his account in La Dioptrique. Descartes’ account of refraction is difficult to comprehend in twentiethcentury parlance. A quick detour via the correspondence of Claude Mydorge, one of his Parisian collaborators, will be enlightening for modern readers. In a letter to Mersenne from around 1627, Mydorge used a rule to calculate angles of refraction, given the angles of one pair of incident and refracted rays (Figure 39). If FE-GE is the given pair, the refraction EN of HE is found in the following way. Draw a semicircle around E that cuts EF in F. Draw IF parallel to AB, and from I drop IG parallel to CE, cutting EG in G. Draw a second semicircle around E through G. Now draw HM, cutting the first 74

AT 2, 86 (to Mersenne, 31 March 1638).

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semicircle in M, and drop MN, cutting the second semicircle in N. EN is the required refracted ray. The rule comes down to a cosecant ‘law’: cosec i : cosec r = FE : EG. Later in the letter, Mydorge applied this rule to lenses and transformed it into sine form.75 Mydorge’s rule embodies the two assumptions that formed the core of Descartes’ derivation of the sine law in La Dioptrique. First, a constant ratio between the incident and refracted rays, represented by the constant ratio Figure 39 Mydorge's rule of the radii of the two semi-circles. Second, the constant length of the parallel components FO and OI before and after refraction. In the diagram of Mydorge these assumptions are represented directly by the lengths of the respective lines. However, instead of a distance diagram, in La Dioptrique Descartes used a time diagram where the lengths of lines represent duration (Figure 40). Instead of the two semicircles representing the constant ratio of the effect of the media, it shows a single circle. As a consequence the constancy of the parallel component was represented by lines of differing length (AH and HF). I have begun with Mydorge’s rule because it somewhat bridges the gap between Cartesian conceptualization of refraction and our understanding of the sine law. It gives the modern reader a clear idea of the assumptions fundamental to Descartes’ derivation as well as the way he adapted it to his own line of thinking. As I will argue below, the diagrams Descartes used fitted Figure 40 Descartes’ analysis of refraction perspectivist analysis of refraction rather than his own account, and he chose them deliberately. The account of La Dioptrique, with all its complicating facets, was how seventeenth-century readers got to know the sine law and the mechanistic interpretation of refraction. It formed the starting-point of all subsequent accounts of the causes of refraction, although few adopted Descartes’ conceptual and methodological notions in full. 75

Mersenne, Correspondence I, 404-415. This letter and its import for Descartes’ optics is discussed thoroughly in Schuster, “Descartes opticien”, 272-277 and Schuster, Descartes and the Scientific Revolution, 304308.

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Refraction in La Dioptrique Descartes began La Dioptrique with an explication of the way rays of light enter the eye and are deflected on their way to it. He did not intend to explain the true nature of light, he said, as the essay ought to be intelligible to the common reader. He took the liberty, he said, to employ a threesome of comparisons between the behavior of light and everyday phenomena: “…; imitating in this the Astronomers, who, although their assumptions are almost all false or uncertain, nevertheless, because these assumptions refer to different observations they have made, do not fail to draw many true and well-assured conclusions from them.”76

First, light acts like the white stick that enables a blind man to sense objects; it is an action instantaneously propagated through a medium without matter being transported. Second, this action is like the tendency of a portion of wine in a barrel of half-pressed grapes to move to a hole in the bottom. It works along straight lines that can cross each other without hindrance. In other words, light is not a motion but a tendency to motion: “And in the same way, considering that it is not so much the movement as the action of luminous bodies that must be taken for their light, you must judge that the rays of this light are nothing else but the lines along which this action tends.”77

Although essentially light is a tendency to movement rather than actual motion, with respect to the deflections from its straight path rays of light follow the laws of motion, Descartes maintained. So, in the third comparison, the way light interacts with mediums of different nature is compared to the deflections of a moving ball encountering hard or liquid bodies. Thus the three comparisons of the first discourse of La Dioptrique established a qualitative basis for the mathematical account of refraction in the next. The second discourse ‘Of refraction’ opens with an account of reflection providing the conceptual basis for Descartes’ explanation of the ‘way in which refractions ought to be measured’.78 It introduces a crucial distinction with regard to the powers governing the motion of an object: one that works to continue the ball’s motion and one that determines the particular direction in which the ball moves.79 Instead of the more accurate ‘absolute quantity of force of motion’ and ‘directional quantity of force of motion’, for sake of convenience I will speak of ‘quantity’ and ‘direction’ both of which may 76

Descartes, AT6, 83. “imitant en cecy les Astronomes, qui, bien que leurs suppositions soyent presque toutes fausses ou incertaines, toutefois, a cause qu’elles se rapportent a diverses observations qu’ils ont faites, ne laissent pas d’en tirer plusieurs consequences tres vrayes & tres assurées.” (Translation based on Olscamp) 77 Descartes, AT6, 88. “& ainsy, pensant que ce n’est pas tant le mouvement, comme l’action des cors lumineus qu’il faut prendre pour leur lumiere, vous devés iuger que les rayons de cete lumiere ne sont autre chose, que les lignes suivant lesquelles tend cete action.” (Translation based on Olscamp) 78 “… en quelle sorte se doivent mesurer les refractions”, AT6, 101-102. 79 “Seulement faut il remarquer, que la puissance, telle qu’elle soit, qui fait continuer le mouvement de cete balle, est differente de celle que la determine a se mouvoir plustost vers un costé que vers un autre, …” AT6, 94.

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apply to Cartesian motion proper as well as to tendency to movement. When a ball rebounds from the surface of an impenetrable body the following happens. The quantity of its motion is unaffected because it remains moving through the same medium - the air surrounding the body - and only the direction changes. Regarding the parallel and perpendicular components of the direction, Descartes noted that the body offers resistance only in the direction perpendicular to its surface. Thus the parallel component is unaltered. To determine the path of the ball after the impact, Descartes switched to a derivation in which he graphically mathematized the assumptions just established (Figure 41). In circle AFD radius AB represents the path along which the ball approaches the surface where it rebounds from B in some direction. As the quantity of motion is constant, the ball must traverse the same distance Figure 41 Descartes’ analysis of reflection after reflection. It thus reaches the circumference of the circle somewhere. Since the parallel component of its direction is also constant, it follows that the horizontal distance traversed after reflection must be equal too. Therefore, BE is equal to BC. Under these conditions the ball can either arrive at point D or point F on the circle. It cannot penetrate the body below GE and so F is the only option left. “And thus you will easily see how reflection occurs, namely according to an angle always equal to the one that is called angle of incidence”, Descartes concluded without much further ado.80 Like reflection, refraction is understood as the combined effect on the quantity and the direction of motion. The only difference is that in refraction the ball penetrates the medium. In other words, it enters a medium of different density. Therefore the quantity of motion changes. It does so at the passing of the surface separating both mediums. This can be compared to smashing a ball through a thin cloth. It loses part of its speed, say half. Again only the perpendicular component of the direction of the motion is affected and the parallel component remains unaltered. As in the case of reflection, Descartes switched to a mathematical derivation in the form of a diagram to determine the exact path of the ball after impact (Figure 40). As a result of the loss of speed, it takes the ball twice as long to reach the circumference of the circle after impact at B. However, as its determination to advance parallel to the surface is unchanged, it moves twice as far to the right in this time. 80

“Et ainsy vous voyés facilement comment se fait la reflexion, a sçavoir selon un angle tousiours esgal a celuy qu’on nomme l’angle d’incidence.” AT6, 96.

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Therefore the distance between lines FE and HB must be twice a large as that between AC and HB. As a result, the ball reaches point I on the circle. The same is the case when instead of a cloth the ball hits the surface of a body of water. For the water does not alter the motion of the ball any further after it has passed the surface, according to Descartes. When the ball passes a boundary where in some way or another its quantity of motion is augmented, it reaches the circumference of the circle earlier and is deflected towards the normal of the surface. Note that Descartes did not specify the change of the perpendicular component, a point that is often overlooked. He did not know that amount and he did not need to, for the two assumptions he used suffice for the derivation of the sine law.81 As Descartes took the motions of the ball to reflect the deflections of light, he could now draw his main conclusion. Rays of light are deflected in exact proportion to the ease with which a transparent medium receives them compared to the medium from which they come. The only remaining difference between the motion of a ball and the action of light is that a denser medium like water allows rays of light to pass more easily. The deflection caused by the passage from one medium into another ought to be measured, not by the angles made with the refracting surface, but by the lines CB and BE. Unlike the proportion between the angles of incidence and refraction, the proportion between these sines remains the same for any refraction caused by a pair of mediums, irrespective of the angle of incidence. Et voilà, the law of sines. Epistemic aspects of Descartes’ account in historical context Both historically and intrinsically, Descartes’ account of refraction is a key text in the transition from medieval perspectiva to seventeenth-century optics. Yet, the line of inference is subtle and, at many points, implicitly pursued. I will have to enlarge in some detail on its epistemic aspects in their historical context. At least three levels of inference can be distinguished in Descartes’ account. In the first place the level of mathematics. This holds the derivation of the sine law from the two assumptions conveyed in the diagrams accompanying his discourse. First, the passage to another medium alters in a fixed ratio the quantity of motion. This ratio is represented by the radius of the circle. Second, the parallel component of the direction of motion is unaffected. This is represented by drawing horizontal lines in proportion to the successive times to travel to and from the center of the circle. The mathematical inference of Descartes’ account constituted a successful culmination of perspectivist optics, in that Descartes was the first to derive a law of refraction on the analytical groundwork laid by Kepler and his forebears. He brought consistency to the analysis of reflection and refraction by having the parallel component constant in all cases. More important, in 81

When both aspects of the motion are interpreted as speeds the assumptions can be written as: vr = nvi and vi sini = vr sinr, which directly yield sini = n sinr. See Sabra, Theories of Light, 111.

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the first assumption, he stated an exact relationship between the medium and the length of a ray. Combined with the second assumption – which was not new – the sine law could be derived. Mathematically speaking, the proof – as Newton later phrased it – was not inelegant. It was fairly undisputed in the seventeenth century and the starting point for much optical investigations. 82 Descartes’ first assumption was more than a purely mathematical assumption, which brings us to the second level of inference that holds the physical properties of rays. The physics of rays had been central in perspectivist optics, but the content of Descartes’ assumptions was innovative. According to Sabra and Schuster, stating a positive dependence of the motion of light on the density of the medium, irrespective of the direction of propagation, made up the decisive break with tradition.83 Descartes may have drawn inspiration for this from his reading of Paralipomena (which he did not acknowledge at all in La Dioptrique). In proposition XX of chapter 1 and the sequel section of chapter 4, Kepler also associated the propagation of a ray with the medium. Descartes may have read Kepler’s diagrams physically, so that the length of the rays represent the action of light as affected by the media.84 Descartes’ diagram represented the actions involved when a ray enters a refracting medium and served to justify his assumptions. He did so by drawing an analogy between a refracted ray and a tennis ball struck through a frail canvas by the man in the diagram (Figure 40). As we have seen, these mechanical analogies had a long history in optics with a direct line from Alhacen to Kepler and, now, Descartes. The mechanical analogies had a different meaning for Descartes than for his perspectivist forebears. To an Alhacen the motions of bodies compared to light only with respect to its propagation, not its essence. According to Descartes light was essentially corpuscular. He made clear that they went further than a mere analogy: “… when [rays] meet certain other bodies they are liable to be deflected by them, or weakened, in the same way as the movement of a ball or a rock thrown in the air is deflected by those bodies it encounters. For it is quite easy to believe that the action or the inclination to move which I have said must be taken for light, must follow in this the same laws as does movement.” 85

However, Descartes took care not to transgress the conceptual and methodological boundaries of perspectiva openly. He presented his account 82

Huygens’ case is discussed below in section 4.2.1., Newton in section 5.2.2. of the next chapter. This theme is leading in Dijksterhuis, “Once Snel breaks down”. Newton’s view is cited below on page 133, footnote 98. 83 Sabra, Theories, 97-107; Schuster, Descartes, 333-334. 84 Schuster, “Descartes opticien”, 279-285; Schuster, Descartes, 334-336. 85 Descartes, AT6, 88-89. “mais, lors qu’ils rencontrent quelques autres cors, ils sont sujets a estre détournés par eux, ou amortis, en mesme façon que l’est le mouvement d’une balle, ou d’une pierre iettée dans l’air, par ceux qu’elle rencontre. Car il est bien aysé a croire que l’action ou inclination a se mouvoir, que j’ay dit devoir estre prise pour la lumiere, doit suivre en cecy les mesmes loys que le mouvement.” (Translation based on Olscamp)

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in terms of analogies and explicitly said these did not reflect the true nature of light. Restricting in this way his account to the behavior of rays, he methodologically tied in with tradition. Still, mechanistic thinking was at the heart of La Dioptrique. Assuming a proportionality between density and motion is almost unthinkable outside a corpuscular framework. Indeed, at the close of the second discourse Descartes showed his hand. The comparisons had a much higher content of realism than suggested by his circumspect introduction of them. “For finally I dare to say that the three comparisons which I have just used are so correct, that all the particularities that that can be noted in them correspond to certain others which are found to be very similar in light; …”86

If the mechanisms Descartes employed in the analogies and to which he ascribed a fair degree of realism do little to persuade our post-Galilean minds, one ought to remember that they were modeled on an understanding of motion that was rooted in a hydrostatics of pressures rather than a kinematics of velocities. Probably this was also one of the reasons the analogies did not convince his seventeenth-century readers either.87 Descartes’ intricate employment of mechanical analogies brings us to the third level of inference in his account of refraction, where the physical nature is involved in the analysis. Although he did not elaborate his theory of light and circumspectly presented the mechanics of deflected motion as analogy, Descartes’ line of reasoning strongly suggests that the laws of optics to be derived from his mechanistic understanding of light. In Sabra’s words: “As repeatedly asserted by Descartes, the ‘suppositions’ at the beginning of the Dioptric belong to this [domain of a priori truth]”.88 This is substantiated by the fact that Descartes deviated from perspectivist tradition in a second important respect as well. In La Dioptrique he did not explicitly call for an empirical foundation of the sine law. In this way, Descartes’ derivation of the sine law was intended as a derivation from the true nature of light. Historian’s assessment of Descartes’ optics The question whether or not Descartes actually succeeded in deriving the sine law from his mechanistic theory of light has been a matter of incessant debate among historians of science. Although few seventeenth-century students of optics were convinced by Descartes argument, I think it appropriate to digress somewhat to contemporary evaluations because these are illuminating as regard the exact purport of his account. Many have argued that Descartes’ claim, that a tendency to move is subject to the same laws as motion itself, was mere rhetoric. Schuster, on the other hand, argues that Descartes’ theory of light did provide the basis of the 86

“Car enfin j’ose dire que les trois comparaisons, dont je viens de me servir, sont si propres, que toutes les particularités que s’y peuvent remarquer, se raportent a quelques autres qui se trouvent toutes semblables en la lumiere; …” AT6, 104. 87 Except Clerselier who expressly defended Descartes’ mechanistic models; Sabra, Theories, 116-135. 88 Sabra, Theories, 44.

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analogies, despite the fact that it hardly appears in La Dioptrique.89 Drawing on the work of Mahoney, he says that the analogies provided a ‘heuristic model’ that legitimately compared the action of light with the motion of a ball. By leaving specific material factors in the motion of the ball aside, Descartes could single out the ball’s tendency to move rather than its motion. He then was ready to consider this tendency and distinguish between “… the power, …, which causes the movement of this ball to continue …” and “… that which determines it to move in one direction rather than in another, …”90 According to Schuster this does not refer to a distinction between force of motion and direction of motion, but to a distinction between quantity of force of motion and directional magnitude of force of motion. The two assumptions of Descartes’ derivation are based on this distinction: the quantity depended on the medium and the parallel component of directional magnitude was constant. In La Dioptrique Descartes labeled the directional magnitude with the term ‘determination’ in order to analyze the components of the action without implicating the notion of velocity.91 With this interpretation of the analogies, Descartes’ analysis is not directly at odds with the system he expounded in Principia Philosophiae and Le Monde. There he had made the same distinction between quantity and directional magnitude. The first law of nature states that the quantity of force of motion is constant when a body is in uniform rectilinear motion; the third law states that a force of motion is conserved in a unique direction (tangent to the path of motion).92 According to Schuster, the tension between the analogies and the tendency theory can be resolved when Descartes’ heuristic use of the analogies is interpreted in the terms of his theory of motion.93 In the light of the Galilean conception of motion Huygens and Newton employed (as do we), Descartes’ claim that he derived the laws of optics from his mechanistic principles was untenable. Sabra has sufficiently pointed this out.94 Yet, this was not so much because his system was incoherent or inconsistent as, rather, because the interpretation of the underlying principles had changed. Descartes usually considered motion at an instant of impact and discussed it in terms of the body’s force to move. In the light of this science of motion, the mathematical derivation of the sine law can indeed be physically interpreted in a plausible manner. Yet, through his crude presentation in La Dioptrique Descartes made little effort to prevent misunderstandings and misinterpretations.

89

Schuster, “Descartes opticien”, 261-272 Schuster, Descartes, 273; Mahoney, Fermat, 387-393; Sabra, Theories, 78-89. 90 Descartes, AT6, 94-95. 91 Schuster, “Descartes opticien”, 258-261; Schuster, Descartes, 293 92 Schuster, Descartes, 288. 93 Schuster, “Descartes opticien” 261-265. 94 Sabra, Theories, 112-116.

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Schuster has proposed a possible route along which Descartes’ discovery of the sine law may have taken place.95 In it, his collaboration with Mydorge plays an important role - the cosecant rule being a crucial step towards the law of sines. According to Schuster, the actual discovery was independent of Descartes’ mechanistic ‘predilections’; rather the other way around: the latter were triggered by the former.96 Shea has argued for a different route to the discovery, via measurements of angles of refraction by means of a prism.97 In this variant too, the discovery was the result of an analysis of the behavior of rays. Descartes developed his mechanistic interpretation of his analysis of refraction after the discovery. With the presentation in La Dioptrique, he then obscured his analysis and explanation considerably. He adopted the use of analogies and adapted his derivation of the sine law to the perspectivist analysis of refraction. It appears as if Descartes tried to make his theory look as traditional as possible. Yet, he deviated from tradition by reversing the way in which he justified the law. Descartes suggested that the laws of optics ought to be based on prior principles regarding the nature of light. And despite the circumspection of his presentation, it was clear that he regarded the mechanistic causes of refraction an important, if not crucial, matter. In this way, he shifted the focus of the mathematical study of light towards the nature of light and the causes of the laws of optics. La Dioptrique is hard to characterize in terms of seventeenth-century geometrical optics. On the one hand, it was obviously a treatise on geometrical optics. It discussed the behavior of light in terms of the mathematical laws of the propagation of rays, in particular as they are refracted by lenses. Still, it did not offer quite as thorough an account of lenses as one would expect from a mathematical treatise. As we have seen in chapter two, La Dioptrique did not elaborate the mathematical theory of refraction in a way modeled after Kepler’s Dioptrice. Its main goal was to establish the law of refraction and explain its main consequences for the working of the telescope. Reception of Descartes’ account of refraction In Descartes’ system of natural philosophy, natural phenomena were explained from mechanistic principles. His optics was the most elaborate example of this project. Even if this elaboration was not fully unproblematic, it made clear what a new, mechanistic science of optics should be about. It did not halt at the mathematical description of natural phenomena, nor at depicting micro-mechanisms to explain them, but sought to explain the mathematical laws of nature by its mechanistic nature. Notwithstanding recent pleas by historians of science for Descartes’ integrity, few contemporaries accepted his explanation of refraction. “The author would have demonstrated not inelegantly the truth of this, if only he had not left 95

Schuster, Descartes, 321-326. See also: Costabel, “Refraction et La Dioptrique”. Schuster, Descartes, 343-346. 97 Shea, Magic, 156-157. 96

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room for doubt concerning the physical causes he assumed”, Newton wrote 30 years later.98 In view of Galileo’s science of motion it is doubtful whether the motion of a ball struck through a frail canvas is subject to the assumptions Descartes made. His explanation of refraction into a denser medium – towards the normal – was regarded most problematic. In order to account for the necessary increase of motion, he introduced the rather ad hoc assumption that the ball was struck again at the refracting surface. Besides rejecting Descartes’ theory of light on the conviction that the speed of light is finite, in the ‘Projet’ Huygens explicitly mentioned this extra assumption as one of the difficulties in Descartes’ derivation.99 Newton and Huygens wrote at a time when the law of sines as such had been generally accepted. This had taken some twenty years, during which it only slowly became widely known. Cavalieri in 1647 did not employ the law of sines and Gregory seems to have been ignorant of it as late as 1663. As we have seen in the previous chapters, Huygens was one of the very few to pursue the study of dioptrics in this period. Compared to the preceding decades, the 1660s witnessed a true upsurge of the study of light. The investigations of Grimaldi, Boyle, Hooke, Newton, Bartholinus, brought to light a collection of new properties shaking the foundations of optics. Remarkably, the final acceptance of the law of sines coincided with accusations of plagiarism directed at Descartes. In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snel’s papers and concocted his own proof. We now know this charge to be undeserved but it has been adopted by many since. Descartes may have heard of Snel’s achievement through his contacts with the circle that included Golius (Snel’s successor) and Constantijn Huygens sr. around 1632, but he had found the law much earlier. Christiaan Huygens started to display doubts regarding Descartes’ originality since the early 1660s. Probably spurred by Vossius’ claims, he traced and examined Snel’s papers. As late as 1693 he voiced his opinion as follows: “It is true that from all appearances these laws of refraction aren’t the invention of Mr. des Cartes, because it is certain that he has seen the manuscript book of Snel, which I also have seen.” 100 Most remarkable about this is that Huygens could have known, through his father, much earlier about Snel’s achievement. Constantijn sr. had heard of it through a letter from Golius of 1 November 1632. Apparently the topic had never entered their conversation. The slow adoption of the sine law may have been brought about by the bad odor of Descartes’ philosophy, or simply the slow diffusion of his works. Fermat was convinced of the sine law’s validity only after he found 98

Newton, Optical lectures, 170-171 & 310-313. A similar conclusion can be drawn with respect to Descartes’ theory of hydrostatics on which his concept of ‘conatus’ was based. Shapiro, “Light, pressure”, 260-266. 100 “Il est vray que ces loix de la refraction ne sont pas l’invention de Mr. des Cartes selon toutes les apparences, car il est certain qu’il a vu le livre manuscrit de Snellius, que j’ay vu aussi; ...” OC10, 405-6. See also OC13, 9 note 1. 99

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his own demonstration. Right after the publication of La Dioptrique he had severely criticized Descartes’ derivation, and maintained his objections when supporters of Descartes reopened the debate in 1657.101 Employing the principle of natural economy, previously used by Hero and Witelo for reflection, Fermat deduced the law of sines in 1662, thus strengthening his conviction that Descartes’ mechanistic line of reasoning had been false. To know that the law was independent of Descartes’ mechanistic reasoning may have facilitated its acceptance, although it may well be that the ostensible non-acceptance was simply a matter of inactivity on the front of optics during the 1640s and 1650s. The reception of La Dioptrique makes clear that the treatise is hard to situate in the development of seventeenth-century optics. It formed the starting-point of most subsequent investigations in optics, and has therefore been the focus of many historical studies.102 La Dioptrique showed how the properties of light could be discussed in corpuscular terms and its readers got this message. Although few agreed with the details of Descartes’ derivation of the sine law, nor with his system of mechanistic philosophy in full, he set the idiom for the all-prevailing thinking on light in corpuscular terms. As a consequence, the traditional analogies between light and motion implied a potential claim about the true nature of light and could not be used as informally as before. Descartes had intended to found the laws of optics in the mechanistic nature of light, but his derivation was not free from ambiguities and obscurities. A mathematician like Barrow adopted the corpuscular understanding of nature but not Descartes’ approach to explanation. We now turn to him, to see how he dealt with questions regarding the status of the corpuscular nature of light and how it ought to explain the laws of optics. Barrow’s causal account of refraction Barrow was a mathematician with a clear awareness of the epistemological intricacies of mathematics and its applications to nature. The lectures on mathematics which he delivered at Cambridge between 1664 and 1666 dealt at great length with the status of mathematical concepts and methods and their relevance for the study of nature.103 His subsequent lectures on optics are likewise riddled with epistemic statements. Lectiones XVIII of 1669 is illuminating with respect to Huygens’ ‘Projet’ as it assigns a similar role to explanations of the causes of the laws of optics. The subject of the lectures was ‘Optics’, one of the fields that are “… bright with the flowers of Physics and sown with the harvest of Mechanics,…”104 The core of this science 101

The debate is listed in Smith, Descartes’s theory of light and refraction, 81-82 and discussed in detail in Sabra, Theories of Light, 116-135. 102 Sabra, Theories, 12. 103 Published in 1666 as Lectiones mathematicae XXIII. They were translated by John Kirkby and published in 1734 under the title The usefulness of mathematical learning etc. It is cited in Shapiro, Fits providing improved translations. 104 Barrow, Lectiones, [10].

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consisted of six generally accepted principles required to elaborate mathematical theory. In a way reminiscent of the ‘Projet’, Barrow said that these hypotheses, as he called them, were empirically founded but also needed some sort of explanation: “The hypotheses agree with observation, but we must also fortify them with some support of reason, by treading on the foundations and suppositions laid down.”105

In his first lecture Barrow discussed these foundations and suppositions, although he mainly defined terms like ‘light’ (in relation to illumination, images, ‘phasmata’ and the like), ‘refraction’ and ‘opaque’. He then proposed a theory of light that is a hybrid of viewing light as a pulse and as a pressure propagated simultaneously in the first and second matter of the Cartesian scheme.106 Whatever he meant precisely, Barrow did not lend much weight to this theory. “Still, since it is desirable for me to lay some preliminary foundations about the nature of light, to agree with my explanation of hypotheses which I shall later offer, I conceive the facts to be these, or something like them: …”107

These preliminary foundations merely needed to be consistent with the ensuing explanations of the laws of optics and Barrow expressly did not claim any authority in these matters.108 In what followed, Barrow’s theory of light came down to considering a ray to be the path traced out by a pulse-like entity, “… two-dimensional and like a sort of rectangular parallelogram lying in a plane at right angles to the surface of the inflecting medium, …”109 This conception of a physical ray traced out by a line of light emitted by a shining object went back to Hobbes’ theory of light. Barrow’s derivation of the law of sines can likewise be traced back to Hobbes.110 With this definition of a ray, Barrow now could make ‘some attempt to explain’ the laws of optics, stressing once more that they were empirically founded: “… I need practically nothing else to explain the hypotheses which all opticians in common with each other assume and which must necessarily be laid down as a foundation for building up this science. I shall make no effort to prove what I have said, since … it seems clearer than light itself that such proofs cannot be given, although a number of experiments show that they are given in actuality.”111

Besides accounting for the rectilinearity of light rays, he discussed some basic assumptions of geometrical optics, like the fact that ‘inflections’ take place in a plane perpendicular to the surface of the ‘inflecting’ medium. Then, in the second lecture, he moved on to these ‘inflections’ proper, reflection and

105

Barrow, Lectiones, [26]. Barrow, Lectiones, [15-16]. 107 Barrow, Lectiones, [15]; (emphasis in original). 108 Barrow, Lectiones, [8, 15]. 109 Barrow, Lectiones, [26]. 110 Shapiro, “Kinematic optics”, 177-181. Hobbes’ optics is discussed in the next chapter, section 5.2.1. 111 Barrow, Lectiones, [17] 106

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refraction.112 For reflection, he considered BD – a line of light in the most realist sense of the word – colliding obliquely with a reflecting surface EF (Figure 42). He argued that, after B hits the surface, this end of the line of light rebounds while end D continues its way, resulting in the rotation of BD around its center Z. Figure 42 Barrow’s explanation of reflection. This rotation lasts until D hits the surface and the line of light is in position ƢƤ. The line of light then continues towards ơƪ. From the symmetry of the situation the equivalence of the angles of incidence and reflection follows directly. To substantiate his claim, Barrow invoked a general ‘law’ of motion: “… that it is constantly found in nature, when a straight movement degenerates into a circular one, that it is the extreme parts of the moving objects that direct and control all motion.”113

He applied the same law to derive the sine law (Figure 43). On entering the more resisting medium below EF, point B of the line of light BD will be slowed down while D continues with the original speed. As a consequence DB will be rotated around a point Z until D also reaches the ‘denser’ medium. Then the line of light ƢƤ will continue along a straight path. Now, the proportion between ZD and ZB is constant for any angle of incidence and depends upon the particular difference of the densities. From this it easily Figure 43 Explanation of refraction. follows that for i =‘GBM and 114 r =‘NƤƪ, sin i : sin r = ZD : ZB. After thus explaining refraction into a rarer medium and total reflection, Barrow was ready to elaborate the ‘Optic Science’ of his lectures in the common manner: “…considering rays as one-dimensional (seeing that the other dimensions, in which physicists delight, have no importance for the calculations here undertaken).”115

112

Like Maignan in his Perspectiva Horaria (1648), Barrow added a derivation of the law of reflection which Hobbes had not provided. See Shapiro, “Kinematic optics”, 175-178. 113 Barrow, Lectiones, [28]. 114 Barrow, Lectiones, [29-31] 115 Barrow, Lectiones, [39-41]

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Lectiones XVIII treated optics as the mathematical science aimed at the analysis of the behavior of light rays. Priority was with the laws of optics, being laws of rays that were justified empirically and generally accepted. In this sense the lectures stood with both feet in traditional geometrical optics. Yet, Barrow was too conscious of epistemic issues regarding mathematics and of the new developments in natural philosophy to treat optics in the outright traditional manner of other contemporary works. A good example is the Opera mathematica, a mathematics textbook from 1669 by the Flemish Jesuit Andreas Tacquet, a correspondent of Huygens. In its catoptrical chapters, Tacquet makes room for a noncommittal survey of explanations of reflection: some give natural economy as the ‘ratio’ of reflection, others maintain that the perpendicular component of a ray’s motion is inverted, and so on.116 Even Descartes is reviewed, stripped of all corpuscular trimmings to be sure. Tacquet did not show preference for any of the alternatives, he only explained the various ways in which the law of reflection could be deduced. The business of a mathematical student of light was to establish those properties of rays interacting with varying mediums so that the laws describing its behavior could be derived logically. For Barrow mixed mathematics - where natural things are considered in their quantitative aspects - was a genuine part of mathematics. In his lectures on mathematics, Barrow effectively discarded the distinction between sensible and intelligible matter, so that a science like optics could approach the certainty of geometry. The certainty of inferences only depended on the certainty of the presuppositions - axioms, postulates, principles.117 Barrow presented his explanations as a non-committal elucidation of empirically founded laws, similar to the mechanical analogies of perspectivist theory. The new mode of thought regarding the nature of things had changed the understanding of the nature of light and the causes of reflection and refraction. Yet, compared to these, corpuscular accounts of the causes of reflection and refraction obtained a different meaning, as it implied a potential claim about the true nature of light. This, combined with his epistemic awareness, may explain Barrow’s reluctance to make strong claims about his explanations. In his comments, Barrow considerably qualified the status of his theory of light and his causal accounts. His focus was on the laws and he did not elaborate his account of the mechanistic nature of light in any detail or explore its consequences. He was rather vague about the necessity and role of such an account. The laws of optics should ‘not be repugnant to reason’ and be given ‘some support of reason’. He invoked a law of motion, but did not intend to prove the laws like Descartes, by deriving them from his theory of light. He offered a physical rationale for the laws, without making clear the exact purpose of his explanations. As a consequence, he parried the 116 117

Tacquet, Opera mathematica (Antwerp, 1669), Catoptricae libri tres, 217-218 Shapiro, Fits, 31-36.

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question raised by the new philosophies of what status the corpuscular nature of light should have and how it ought to explain the laws of optics. 4.2 The mathematics of strange refraction Kepler and Descartes had drawn attention to the problem of the relationship between a theory expounding the true nature of light and the mathematical behavior of light rays. It remains to be seen how Huygens considered this issue. What exactly did he mean by explaining refraction with waves? What were those waves and how would he proceed from there to the sine law? The statements in the ‘Projet’ suggest that his opinion about causal accounts was similar to Barrow’s. Explaining refraction was a rather non-committal affair to satisfy the minds of the particularly curious. Still, he wanted to solve the problem strange refraction posed for Pardies’ explanation of ordinary refraction. Apparently, the nature of light was serious enough a matter for Huygens first to wish to get this inconsistency out of the way.118 Given the definition of the problem, the line of his first attack of strange refraction is rather surprizing. Huygens’ first attempt at understanding strange refraction is found on some ten pages in his notebook.119 In my view, it must have taken place around the same time he noted down the ‘Projet’, somewhere during the second half of 1672.120 On the first pages Huygens jotted down some sketches characterizing the phenomenon. The first shows five pairs of incident and refracted rays (Figure 44). One of each pair, indicated by the letter r is refracted regularly (‘regelmatig’) according to the sine law, the other one indicated by the letter o is refracted irregularly (‘onregelmatig’).121 Below, Huygens wrote what is irregular about it: “The perpendicularly incident [ray] is refracted It does not make a double reflection.”122

118

Ziggelaar correctly points out that the problem of strange refraction was a reason Huygens did not directly elaborate ‘Projet’ (which he sees as a new plan for a treatise on dioptrics), but he does not discuss his first attempt to solve it beyond a single, and incorrect, characterization. Ziggelaar, “How”, 181-182. See also page 162. 119 Hug2, 173v-178v. It consists of seven pages numbered by Huygens (175r-178r), preceded by two and a half pages with some notes and followed by a page containing a further note plus the record of an experiment performed in 1679 (discussed in section 5.3.1) Parts of their content are reproduced in OC19, 407-415. 120 I disagree with the editors of the Oeuvres Complètes regarding the dating of the papers. I think this first study took place around the time of Pardies’ letter, much earlier than they presume. On 4 September 1672, hardly a month later, Huygens wrote to his brother Constantijn, saying he was not yet going to publish “what I have observed of the crystal or talc of Iceland” (OC7, 219. “…ce que j’ay observè du Chrystal ou Talc d’Islande; …”). I think this remark refers to his discovery of another peculiar phenomenon displayed by Iceland crystal – polarization – recorded on the final pages of his investigation. The discovery is in OC19, 412-414. The editors date these between December 1672 and June 1673, but it is possible that they – or similar notes now lost – were written at the same, earlier date. 121 Hug2, 173v. One half of Hug2, 174r is torn away; the page contains a remark that seems of a later date. 122 Hug2, 173v; OC19, 407. “Perpendiculariter incidens refringitur Non facit duplicem reflexionem.” The editors combine this with a remark written on Hug2, 175v.

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Figure 44 Sketch of refracted rays in Iceland crystal: r (‘regelmatig’) for ordinary refraction; o (‘onregelmatig’) for strange refraction.

On 8 July 1672, Pardies wrote Huygens about strange refraction. He had visited Picard and taken a look at a piece of Iceland crystal brought from Denmark. Pardies did not believe the phenomenon contradicted the sine law, as he thought it could be explained from the composition of the crystal. “… it seems to me that it is not as troublesome as I had imagined to explain this effect. … I am very much mistaken if one cannot demonstrate that, if one were to cut various pieces of glass in rhomboid shape and simply put one on the other to make a total rhomboid out of them, two refractions would present themselves.”123

Some sketches Huygens made in his notebook around the same time are reminiscent of Pardies’ view (Figure 45). They seem to explore how the composition of the crystal may explain strange refraction. The surface is drawn indented, so that part of the perpendicularly incident light actually falls on an oblique surface. A perpendicular ray falls upon the indented surface so that part of the Figure 45 A refracted wave is divided into many small wavelets, that perpendicular caused by the composition of the crystal. proceed obliquely to the surface.124 Apparently, Huygens did not accept Pardies’ idea, for he did not elaborate it beyond these sketches. Moreover, he ended his first study with the conclusion that the refracted perpendicular contradicted the wave explanation of refraction.125 A sketch on the next page of his notebook makes 123

OC7, 193. “… il me semble qu’il n’est pas si malaisé que je m’estois imaginé, d’expliquer cét effet. Je suis fort trompé si l’on ne peut démonstrer que si l’on taillait plusiers pieces de verre en rhomboide et qu’on les mit simplement l’une sur l’autre pour en faire un rhomboide total, il s’y feroit deux refractions.” 124 Hug2, 178v; OC19, 415. 125 See below page 151 footnote 148.

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it clear what kind of problem strange refraction constituted for the wave theory (Figure 46). It shows the strange refraction of a perpendicular ray along with, what seems to be, the propagation of waves.126 After having passed the refracting surface, the waves proceed obliquely to their direction of propagation, which contradicts the assumptions of Pardies' theory. Thus this tiny sketch illustrates what Huygens called the ‘difficulté’ of strange refraction. Strange refraction posed a problem for the explanation of ordinary refraction that Huygens intended to adopt and he first wanted to solve it. His first attempt is recorded in those Figure 46 ten notebook pages. The notes are revealing. Despite the fact Waves through that strange refraction was a problem of waves, this tiny the crystal. sketch is the only place where waves enter his investigation. Instead, Huygens approached strange refraction in a rather traditional way. He tried to find out what mathematical regularities rays refracted in Iceland crystal might display. This was the same way Bartholinus had approached the phenomenon, namely by trying to find a law describing the behavior of strangely refracted rays. 4.2.1 BARTHOLINUS AND HUYGENS ON ICELAND CRYSTAL Huygens began by recording the main characteristics of the crystal (Figure 47).127 With explicit reference, he reproduced the crystallographic data of Bartholinus. The crystal has the form of a parallelepiped, of which the obtuse angles of each parallelogram like ACB are 101q. Consequently, the angle AXB between faces GOCA and FOCB is 103q40' and those between lines OC and CI (bisecting angle BCA) is 72q34'. A ray of light falling on a Figure 47 Shape and main angles of the crystal. face of the crystal is double refracted. One of the refractions conforms to the sine law, whereby the index of refraction is approximately 5 to 3, a value Bartholinus had determined empirically. The other refraction does not follow the sine law and is therefor called extraordinary or strange. Huygens observed some physical characteristics of the crystal as well, in particular the fact that the 126

I experienced some difficulty seeing this sketch as a two-dimensional section, as most historians have done. I once thought it was meant to be drawn in perspective, a ray refracted out of the paper towards the reader. Despite this ambivalence, I think after all that the two-dimensional interpretation is correct. 127 Hug2, 173v and 175r. OC19, 407-408; Bartholinus, Experimenta, 8-11 and 40.

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crystal is easily cleft but only along surfaces parallel to its faces. He noted that other ways of cutting it had not been successful yet. He evidently alluded to Bartholinus’ discussion of refraction in planes that are not parallel to the natural faces of the crystal. These data were written mainly in French. The investigation continues with an analysis, written in Latin, of the way rays are refracted by the crystal. Huygens did not adopt the conclusion Bartholinus had drawn from his experiments with the crystal, in the form of a ‘law’ of strange refraction. Huygens proposed an alternative ‘law’. Bartholinus’ experimenta To see where he took off in his study of strange refraction and to compare Huygens’ analysis with Bartholinus’, we first turn to the main argument of Experimenta crystalli islandici disdiaclastici. As regards its methodological structure, Experimenta is an interesting work in the history of optics. Bartholinus explicitly discussed how the main principle for the mathematical analysis of the phenomenon under consideration - his law of strange refraction - was derived from and subsequently founded upon empirical investigation. He did not report literally on his experiments, but stylized and ordered them into a mathematical argument.128 This way of integrating experimental inquiry into mathematical inference is akin to the approach of men like Pascal and Mariotte, and seems to have been a common strategy in

Figure 48 Double refraction according to Bartholinus.

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that period of (continental) mathematicians to appropriate the new philosophies.129 Experimenta is divided in two parts: seventeen ‘experimenta’ and ten ‘propositiones’. These parts pivot around a section ‘Observationes ad demonstrationem præcedentium’, in which he put forward his law of strange refraction. The ‘experimenta’ describe Bartholinus’ observations of the crystal and the behavior of rays refracted by it. From these experiments he inferred his law. The content of the ‘propositiones’ is identical with the findings of the ‘experimenta’ in the first part. The difference is that in these propositions Bartholinus used his law of strange refraction to derive the results of his observations. In this way, Bartholinus demonstrated that the law agreed with observation. In the twelfth experiment, Bartholinus introduced the line ED which bisects the obtuse angle of the upper face (Figure 48).130 This line has particular properties. In the first place, when the eye is in a plane perpendicular to the face of the crystal and through ED, both images of an object A will appear on this line. In the second place, the separation of the images is maximal on this line. In the following, Bartholinus confined his account to refractions of rays in the plane perpendicular to the upper face of the crystal through ED and the edge EM. Huygens later introduced the term principal section for this plane. In the next, thirteenth experiment, Bartholinus explained what is strange about strange refraction. When the crystal is rotated on the table, the image C of object A is fixed, as it ought to. The second image B, however, moves around. Therefore, B must be produced by some strange, extraordinary refraction.131 Throughout Experimenta, Bartholinus distinguished between ordinary and strange refraction by referring to the ‘fixed’ and the ‘mobile’ images.132 Only in the sixteenth experiment, after he had ruled out reflection as a cause for the second image, did he introduce the epithet ‘extraordinary refraction’.133 Huygens did not speak of a mobile image, but only talked of strange or extraordinary refraction.

128

Lohne has published and translated an earlier draft of Experimenta, which shows that Bartholinus reordered the propositions for the final version and apparently rewrote the context of discovery. Lohne, “Nova experimenta”, 106-107. 129 See Dear, Discipline and experience, 201-209. 130 Bartholinus, Experimenta, 19-20. The figure is erroneous, as Bartholinus pointed out too, for ED bisects the acute angle of the upper surface. 131 Bartholinus, Experimenta, 20-22. In the text the thirteenth experiment is also numbered XII. 132 In his fourteenth experiment, he explained how the mobile image might be rendered fixed – and viceversa – by considering alternative surfaces for observing the images. (Bartholinus, Experimenta, 22) It is questionable whether Bartholinus had actually made the observations he described here. Lohne has pointed out that, had it been carried out, it would have appeared to yield trivial or erroneous results. Lohne, “Nova Experimenta”, 135 note 29. Buchwald and Pedersen point out, however, that these observations are quite difficult to perform. Bartholinus, Experiments, 19-20 (Introduction). 133 Bartholinus, Experimenta, 29-30. Bartholinus attributed his discovery of strange refraction - that the duplicate image is caused by refraction instead of reflection - to this experiment, which however the earlier draft of his treatise does not include. Lohne, “Nova experimenta”, 106-107.

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Figure 49 Refraction in two positions of the crystal.

The mobile image contradicts the ordinary laws of refraction, but it does not “seem to vary with uncertain laws”, so Bartholinus began his fifteenth experiment.134 To deduce the ‘laws’ governing strange refraction, he recorded some empirical properties. The mobile image rotates around the fixed image and does not describe a perfect circle. When the eye is in O, the separations DC and CB of the images for two positions of the crystal are unequal. When the eye is in N, on the other hand, the separations QC and CP are the same. The properties Bartholinus employed to define strange refraction consisted of qualitative observations where the behavior of strangely refracted rays is linked to the crystallographic data. This also applies to the observation crucial to his law of strange refraction: a ray parallel to the edge of the crystal is not refracted. This probably was a naked eye observation and, as we will see in the next chapter, it was inaccurate. He did not perform direct optical measurements of strange refraction. From these rather meager data, Bartholinus concluded that the mobile image does describe a perfect circle when the eye is in N. After seventeen experiments, Bartholinus was ready to formulate a law of strange refraction: a mathematical construction explaining how to construct the strange refraction of a ray in the principal section. Bartholinus’ solution is ingenious; we may surmise his line of thought like the following. As the sine law does not apply to strange refraction, refractions cannot be measured with reference to the normal of the refracting plane. Some other instance of reference should be found, and this was the oblique ray that passed the surface unrefracted: “… the extraordinary refraction took for its normal a parallel to the sides of the birefringent crystal, while the ordinary refraction is

134

Bartholinus, Experimenta, 24. Translation by Archibald.

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directed to the perpendicular to Bartholinus the surface.”135 reproduced the Cartesian diagram of the sine law and added the lines governing strange refraction (Figure 50). QGS is the face of the crystal, DN the normal governing ordinary refraction. FGL is an ordinarily refracted ray, so FK : LN is constant. Bartholinus had determined empirically the index of (ordinary) refraction for the crysal FK : LN = 5 : 3. CP is the unrefracted oblique ray parallel to Figure 50 Bartholinus’ law of strange refraction. the edge of the crystal. It governs strange refraction in the same way as DN does its ordinary counterpart. Consequently, when FGM is an extraordinarily refracted ray, the sines FI and PM are in constant ratio, namely 5 : 3.136 It is clear that Bartholinus’ law of strange refraction was an extension of Descartes’ law of ordinary refraction. According to Pedersen and Buchwald, the leading idea behind Bartholinus’ law of strange refraction was to preserve Descartes’ sine law of refraction, changing only its frame of reference. Strange refraction is strange because its ‘normal’ is oblique to the refracting surface rather than perpendicular. In one sentence, Bartholinus suggested a physical explanation of the law, which resembled Descartes’ explanation of ordinary refraction: “For it appears that this birefringent crystal has pores running along the faces and parallel to them, since we may observe that the fracture and separation of fragments follows this disposition of the sides; and [further] one image, namely the mobile one, passes through these same [pores].”137

He seems to have adopted Descartes’ theory of light, but he did not elaborate a causal analysis of strange refraction. Buchwald and Pedersen point out that Bartholinus expressly distinguished the mathematical law and the physical structure regarding strange refraction.138 His sole objective was to establish the law governing the behavior of strangely refracted rays. In his view he had succeeded in formulating a law from which its observed properties could be derived. He had also suggested an experiment further to substantiate it. In the fourteenth experiment, he discussed refraction in planes that are not parallel to the natural faces of the crystal. He claimed that

135

Bartholinus, Experimenta, 32. Translation by Archibald. Bartholinus, Experimenta, 46-48. Modern notation: sin(i – 17):sin(r – 17) = 5:3; Lohne, “Nova experimenta”, 142. 137 Bartholinus, Experimenta, 54. Translation by Archibald. 138 Bartholinus, Experiments, 18-19 (Introduction). 136

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the fixed and the mobile image would swap place, but had not been able to substantiate this, as he could not cut the crystal appropriately.139 Huygens’ alternatives Huygens followed Bartholinus’ approach to consider only the observed properties of strangely refracted rays. He adopted the Dane’s data and he even seems to follow him in his line of thinking: to extend Descartes’ account of ordinary refraction. Nevertheless, Huygens’ analysis differs in two respects. In the first place he changed perspective by focusing on the refracted perpendicular ray instead of the unrefracted oblique ray. Which is not unexpected, for the refraction of the perpendicular ray formed the heart of the ‘difficulté’ of strange refraction. Accordingly, the one original datum Huygens supplied was the angle of the refracted perpendicular: slightly smaller than 7o.140 Secondly, he went beyond Bartholinus by considering rays outside the principal section. The outcome was a new law of strange refraction. Having described the crystal, Huygens began his analysis by drawing the principal section and some rays (Figure 51). This plane is formed by the edge of the crystal and the line AB that bisects the obtuse angle of the upper face of the crystal. The perpendicular ray GH is refracted to HE. This meant, according to Huygens, that each ray in plane GH – the plane through GH, perpendicular to the paper – is refracted into the plane HE. KLE is the unrefracted oblique ray (parallel to the edge of the crystal through B). Now, Huygens writes, the Figure 51 Rays in the principal section. refraction of rays in plane KL (the plane through KL, perpendicular to the paper) that are not parallel to ray KL, do not lie in plane LE. These rays outside the principal section are refracted towards the perpendicular, and the more oblique they are to KL, the closer their refractions are to the plane through LS, the refracted perpendicular.141 By considering rays outside the principal section, Huygens surpassed Bartholinus’ account. The ‘oblique’ sine law applied only to rays in the principal section and therefore was not a general law. By considering rays 139

Bartholinus, Experimenta, 22-24. Hug2, 175v; OC19, 410 “Angulus FBC refractionis radii perpendicularis est paulo minor 7 grad. cum ad solis radios inquiritur.” The reference is to Figure 51. 141 Hug2, 175v; OC19, 410 “… introrsum versus perpendicularem refringuntur ut in LS, idque tanto magis quanto erunt ad KL radium obliquiores.” 140

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Figure 52 Construction for strangely refracted rays in the principal section

outside the principal section the refracted perpendicular NLS came into view as an important line of reference. Bartholinus’ analysis had been fully based on the unrefracted oblique ray. He had not assigned special significance to the refracted perpendicular. The refracted perpendicular promptly suggested an alternative law, which Huygens elaborated on the next, third page of his investigation. Huygens began with a new drawing of the principal section (Figure 52). In DBCF are given the normal to the refracting surface ABF and the refracted perpendicular ABC, with FBC slightly smaller than 7q. TTC is the unrefracted oblique ray, parallel to the edge of the crystal. In the case of ordinary refraction, Huygens applied the sine law as he was used to do in his dioptrics. To find the refraction DD of a ray DF, draw S on BF so that DS is to DF as the ratio of sines 5 to 3 (or slightly smaller than 8 to 5). Next, he simply stated the following: in strange Figure 53 Main lines of Figure 52 refraction rays DD are refracted towards C, where the refracted perpendicular reaches the bottom of the crystal.142 In other words, strange refraction adds the line FC to the ordinarily refracted ray. So, to find the strange refraction DC of a ray DD, draw its ordinary refraction DF and add the line FC. The fact that a parallel component is added precisely marks the strangeness of strange refraction: the line FC is the component Iceland crystal ‘strangely’ adds to the perpendicular ray. 142

Hug2, 176r; OC19, 411.

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Figure 54 Huygens’ alternative for Bartholinus’ law.

The previous construction applied to rays in the principal section only, but Huygens went on to extend it to arbitrary rays (Figure 54). PD is an arbitrary ray in the plane PDE perpendicular to the surface of the crystal. At the point of incidence D, the perpendicular ray is refracted towards DG. Now draw through DB and DG plane DGCB, in which the refracted ray will lie. In order to find the strange refraction DC of PD, determine its ordinary refraction DF and then add the ‘strange component’ KG, by drawing FC equal and parallel to KG. “So that the motion of the refracted ray within the crystal is composed, as it were, of the motion it would regularly have, and of a lateral motion whose quantity in the whole descent through the crystal is equal to the line FC.”143

This, then, was Huygens’ alternative to Bartholinus. He gave no justification for his construction. He did not reveal whether it accorded with the observed properties of strange refraction, nor why it was better. Huygens evidently did not accept Bartholinus’ law, but he nowhere says so explicitly, nor does he give reasons for his non-acceptance. To be sure, Huygens’ alternative was more general, not being confined to rays in the principal section. But it was likewise a mathematical construction applicable to rays. It is comparable to Bartholinus’ law in another respect as well, as it is also based on Descartes’ account of refraction. Huygens’ construction was a rather straightforward extension of Descartes’ derivation. It added a ‘strange parallel component’ to the components of a regularly refracted ray, equal to the one added by strange refraction to the perpendicular ray. Huygens seized

143

Hug2, 176v; OC19, 412. “Adeo ut motus radij refracti intra crystallum sit veluti compositus ex motu quem regulariter haberet, et ex motu laterali cujus quantitas in toto descensu per crystallum est æqualis rectæ FC.”

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the refracted perpendicular in order to understand the behavior of strangely refracted rays, whereas Bartholinus started from the unrefracted oblique ray. Huygens was not done yet. He directed a ray AB through two pieces of crystal GKVH and LNM, aligned with all their faces parallel (Figure 55).144 By strange refraction one ray BC continues unrefracted while the other BD is refracted ordinarily. Upon leaving the crystal the rays CE and DF become parallel, as expected. Yet, a curious thing happened as they entered the second crystal. Ray CE was not split up into EO and EP, nor was DF split up into FP and FQ, as ought to be CE continued expected. Instead, (unrefracted) along EO and OS, and DF was refracted (ordinarily) to FQ and QR. When, on the other hand, AB was not parallel to the edge of the crystal, or when both pieces were not parallel to each other, the rays were split up by the second piece. A sketch on the preceding page of his notes shows how the phenomenon puzzled Huygens.145 Even if a pair of rays EP-FP would join, three instead of two Figure 55 Description of polarization. images should have been visible. Huygens’ notes do not reveal whether he just happened to align two pieces of crystal in this way. The experiment may also have been induced by his alternative law of strange refraction. The sketches make it clear that he was considering the unrefracted oblique ray and he may have wondered what happened if the strange component added to it was equal to the distance EF between two rays. Huygens may also have been considering the explanation of strange refraction Pardies had given. If, as Pardies would have it, the crystal was composed of tiny congruent pieces of crystal, some idea of the effects might be got by aligning two pieces of the crystal. In that case the observation would confirm Pardies’ idea as the incident ray would not be infinitely split up. Whatever induced Huygens to perform the experiment, in all probability he was the first to observe the phenomenon, polarization as it is called nowadays. It puzzled him. In an effort to ‘make sense’ of it, waves returned: 144 145

Hug2, 177v; OC19, 412-413. Hug2, 177r. Not reproduced in the Oeuvres Complètes.

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“I have imagined that in the crystal there are two different matters, and that there are likewise two different ones in the air or ether where the motion happens that we call light. And that the two motions of undulation of these two matters of the ether have power to move each its analogous matter of the two that compose the crystal, and reciprocally, that these different matters of the crystal being stirred, these would be able to impress this motion of light only upon its analogous matter of the ether.”146

In this case ray AB contains both motions, whereas CE and DF contain only the motions belonging to strange and ordinary refraction respectively. But then, Huygens said, it remains to be seen why CE and DF are split up for other positions of the crystals. “Which is very difficult, because for this it is required, that these rays CE, DF, although not composed when hitting the surface LN of the crystal in some direction, are able to move the two different matters that compose it, and in other directions not.”147

Besides this puzzling phenomenon, the actual problem of strange refraction was not really near its solution. Huygens concluded his first study of strange refraction with a formulation of the true problem with the anomalous refraction of the perpendicular, accompanied by a sketch of a Pardies-like explanation (Figure 45 on page 142): “How can the perpendicular ray become oblique by the refraction, for it happens that the waves will not be at right angles to the line of their extension or emanation, contrary to what our hypothesis of light demands.”148

Huygens had now found a means to construct a strangely refracted ray, which he apparently preferred over Bartholinus’ law. It was more general but Huygens still did not have a clue towards an explanation. The core of his alternative ‘law’ was the refracted perpendicular, which indeed was the core of the problem of strange refraction. Maybe Huygens thought that explaining strange refraction might benefit from the mathematical regularity expressed by his law, or the other way around. As of yet, he had no idea what might happen to waves so as to explain strange refraction, nor do his notes suggest that he had considered the matter in any detail. He had extended ordinary refraction by adding a strange component. What this component – or any of the components in Descartes’ derivation – might mean in terms of waves he had not considered. At this point Huygens may have brought some clarity to the behavior of strangely refracted rays, but the

146

Hug2, 178r; OC19, 413-414. “Pour rendre raison du phenomene de la page precedente, je me suis imaginè que dans ce crystal il y a deux matieres differentes, et qu’il y en a pareillement deux differentes en l’air ou ether dont le mouvement fait ce que nous appellons lumiere. Et que les deux mouvements d’undulation de ces deux matieres de l’ether ont pouvoir d’emouvoir chacun sa matiere analogue des deux qui composent le crystal, et que reciproquement, ces matieres differentes du crystal estant esbranlees, ne sçavroient imprimer ce mouvement de lumiere qu’a leur matiere analogue de l’ether.” 147 Hug2, 178r; OC19, 414. “Ce qui est tres difficile, car il faut pour cela, que ces rayons CE, DF quoyque non composez en frappant en certain sens la surface du crystal LN, puissent esbransler les 2 differentes matieres qui le composent, et en d’autres sens point.” 148 Hug2, 178v; OC19, 414-415. “Comment le rayon perpendiculaire peut il devenir oblique par la refraction, car il arrivera que les ondes ne seront pas a angles droits a la ligne de leur extension ou emanation, contre ce que demande notre hypothese de la lumiere.”

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behavior of strangely propagated waves was just as obscure as when he began his investigation. 4.2.2 RAYS VERSUS WAVES: THE MATHEMATICS OF THINGS REVISITED In the view of the way he had formulated the problem of strange refraction, Huygens’ first investigation of strange refraction may strike us as odd. He clearly considered it to be a problem of waves, but these do not enter his analysis. He approached strange refraction in a rather traditional way as a problem regarding the behavior of rays. Moreover, the fact that it was an extension of Descartes’ derivation of the sine law may strike us as odd. Such a line of thought seems contradictory to his adoption of a Pardies-like explanation of refraction, and to his express rejection of Descartes’ explanation. Of course, Huygens may just have been trying to see how far he could get. We need not make too much of the alternative Huygens devised in 1672, for we never hear of it again. Still, I will briefly discuss the possible conflict between a Cartesian analysis of strange refraction and a Pardies conception of ordinary refraction, as this will be illuminating for our understanding of the physical conceptualization of light involved. The ‘Projet’ reveals that Huygens rejected Descartes’ view of light as a tendency that propagates instantaneously. In addition, he noticed difficulties in Descartes’ derivation of the sine law, particularly the assumed increase when a ray is refracted towards the normal. Does this rule out an analysis in terms of components of motion as Huygens employed in his ‘law’ of strange refraction? I believe it does; Descartes’ derivation of the sine law conflicts with Pardies’. Yet, Huygens need not have been aware of such a conflict. He might have accepted the mathematical structure of Descartes’ derivation in a general sense – refraction adds a component to the motion of light – regardless of the question how the motion of rays and its components ought to be interpreted physically. He may have assumed – deliberately or not – that an interpretation of the derivation in terms of waves would also be possible. It was not, I will argue, as the assumptions of Descartes’ derivation are meaningless with a wave conception of light. In order to substantiate the last point, I now sketch Ango’s derivation of the sine law, assuming it to be similar to Pardies’ account.149 The derivation depends on two premises: rays are always normal to waves and waves propagate with a definite speed in different media.150 With these premises a ray refracted according to the sine law can be constructed. Consider spherical waves passing the surface BED between two media (Figure 56). Ray ccC is the direction of propagation of the wave and normal to the tangent Cm. In order to construct the propagated wave in the second medium, its tangent Cn is constructed. Draw an arbitrary circle that cuts BED in C and K and Cm in m. Draw Kn so that Km : Kn = v1 : v2 , where v1 and v2 are the speeds of propagation in the respective media. Now Cn must be tangent to the 149 150

Ango, Optique, 61-66. Shapiro, “Kinematic optics”, 209-218.

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refracted wave and thus perpendicular to its direction of propagation Cee. Thus Cee is the refracted ray for incident ray ccC. It is easily shown that the sine law holds. Now compare Descartes’ derivation and Huygens’ extension of it to strange refraction. Descartes assumed that the parallel component of the ray was conserved. He did not say anything about the perpendicular component. This accords with Huygens’ Figure 56 Ango’s explanation of refraction. construction, which adds a ‘lateral’ component to an ordinarily refracted ray. The second assumption of Descartes’ derivation was a constant proportion of the motions of the ray before and after refraction. Pardies also assumed such a constant proportion, but exactly the other way around. Waves move faster in air than in glass, whereas in Descartes’ derivation rays necessarily move fastest in glass. Consequently, a Cartesian derivation contradicts a Pardies-like explanation of refraction. Moreover, in Pardies’ derivation of the sine law, both components of the ray have changed after refraction, rendering the Cartesian analysis meaningless.151 How could so gifted a man as Huygens overlook such an obvious inconsistency? We should bear in mind that, in Dioptrica, Huygens never used the Cartesian circle diagram (Figure 40 on page 127). He always visualized the sine law as a ‘cathetus’ construction, where CG and CD are constructed according to the ratio of sines (Figure 57). A similar approach is also suggested by DS and DF of the ordinarily refracted ray (Figure 54 on page 149). The Figure 57 The sine law in Tractatus. details of Descartes’ derivation need not have been on top of his head when Huygens added his strange component. Although he stayed closer to the drift of Descartes’ derivation as compared to Bartholinus – who merely extended the circle diagram – he nevertheless 151

In Ango’s diagram the proportion of speeds is directly represented by the ratio Kn : Km and subsequently by the distances cc and ee traversed by the waves of light. These distances can, in its turn, be analyzed in parallel and perpendicular components, but both change according to the figure. The assumption visini = vrsinr is meaningless.

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left aside the possible physical implications of his Cartesian analysis of strange refraction. It remains to be seen how seriously he took his proposal. In view of his commitment to a wave conception one might say that Huygens was just taking considerable liberty of reasoning in order to see how far he could get at fathoming the oddities of strange refraction. Huygens also afforded himself liberty in another respect. In his notes, he did not explain his motives for proposing an alternative to Bartholinus’ law. Like Bartholinus, he extended the mathematical structure of the sine law through rational analysis. He did not question Bartholinus’ data and confined his study to mathematical analysis.152 Also, irrespective of the virtues of Bartholinus’ verification, Huygens made no effort to justify his conclusions empirically. If his alternative was more general, he did not check whether it was anywhere near the truth. Huygens was familiar with such an approach of mathematical reasoning and it had been successful several times. Although in his optical studies in De Aberratione it had led him somewhat astray, in his studies of motion this strategy had been very rewarding. As mentioned in chapter three, an empirical study of gravitational acceleration had got him nowhere. The breakthrough had been effected by mathematical analysis of circular motion. In his correction of Descartes’ rules of impact, Huygens likewise relied on rational analysis. He built upon the established, Galilean laws of motion to find the true laws of impact by means of rational analysis, a strategy he also chose in his analysis of strange refraction where he built upon the established law of ordinary refraction. Huygens carried out his study of impact between 1652 and 1656. It has been discussed in full detail by Westfall.153 He had found out that Descartes’ rules of impact - a crucial topic in mechanistic philosophy - where wrong save for the first. In particular in the case of unequal bodies or speeds, the rules proved inconsistent and failed to obey Galileo’s principle of relativity. Huygens’ solution lay in rigorously applying this principle, in combination with the principle of inertia. In so doing, he converted the study of impact into an extension of Galileo’s theory of uniform motion, namely, the inertial motion of the center of gravity of two colliding bodies.154 As Westfall shows elaborately, the main thread in Huygens’ study of impact was an increasing desire to treat impact in terms of velocities instead of forces, which in his view defied mathematical clarity. As we shall see in the next chapter, the concept of velocity would be crucial to Huygens’ understanding of the mechanistic causes of natural phenomena. In his study of impact Huygens repeatedly solved problems by transforming them into problems subject to known principles. The principle of relativity made possible the treatment of all collisions of equal bodies. Collisions of 152

There is no way Ziggelaar’s observation that “Huygens repeats carefully the experiments of Bartholin on the crystal, measures more exactly, …” can be substantiated. Ziggelaar, “How”, 182. 153 Westfall, Force, 149-155. 154 Westfall, Force, 152-153.

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unequal bodies could be solved with the principle of inertia combined with the hypothesis stating that one body conserves its original motion if the other does so. Huygens approached strange refraction in a way similar to his analysis of impact: by trying to reduce it to established principles, in this case Descartes’ derivation of the sine law. His study of strange refraction consisted of a mathematical analysis of the behavior of strangely refracted rays, trying to transform the new phenomenon into a phenomenon whose mathematical properties were already known. Measuring or experimenting was not the way in which Huygens explored strange refraction. He first tried to fathom the mathematical structure of the phenomenon. As we have seen in the case of chromatic aberration, he ruled out the possibility that such understanding could be attained beforehand, consequently he did not set up an empirical investigation of it. The precision he achieved in measuring the constant of gravitational acceleration was made possible by his mathematical understanding of the matter. Huygens approached his subject first of all theoretically, interpreting concepts geometrically and analyzing phenomena by means of his mathematical mastery. The particular problem of strange refraction: waves versus masses Huygens’ initial approach to strange refraction does not suggest that his account would stand out much as compared to Bartholinus’. It does not give reason to expect that it would lead to a memorable contribution to the history of optics. Had Huygens realized his ‘Projet’, this study of mine would probably not have been written. Huygens’ ‘Dioptrique’ would have been a treatise in geometrical optics, an analysis of the behavior of rays where the nature of light played an elucidative role at the most. Its most distinguishing feature would have been the way it combined rigorous analysis with a focus on the telescope. The dioptrical theory would have been preceded by some mechanistic justification of the sine law – one that would probably be counted nowadays as a repetition or a variant of Pardies. Maybe it would also have contained a discussion of strange refraction in terms of waves. What this would have looked like, we cannot tell. Huygens had tried to make sense of strange refraction by reducing it to a strange component added to ordinary refraction. From this point of view, his first investigation seems successful. He had found a law that was more general than Bartholinus’ law, and was also more in line with Descartes’ analysis of refraction. Yet, his alternative law did not solve the actual problem of strange refraction, which was a problem of waves. Whatever the virtues of his law, it did not explain how waves could become oblique to rays. And this is the most remarkable thing of all about Huygens’ study. Strange refraction was a problem of waves but they did not enter his analysis. He was thinking in terms of rays. He tried to establish a general law by analyzing the mathematical regularities of rays refracted in Iceland crystal. In this sense his approach was traditional. His ideas about the nature of light

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and his analysis of strangely refracted rays remained separate. As yet, Huygens had not considered the interaction of waves with Iceland crystal beyond the observation that it constituted a problem. This, then, is where the strategy that had been so successful with motion broke down. Whereas motion only involves observable entities like balls and pendulums, refraction also entails a consideration of unobservable entities. As contrasted to the laws of motion, mechanistic causes are involved in the laws of optics. Huygens believed that a Pardies-like theory of light was a plausible way to explain refraction, but he had not figured out how it related to strange refraction. Still, he had committed himself to this theory, and this was why his earliest study of strange refraction ended inconclusively. In view of geometrical optics, it may appear strange that Huygens would reject his law on the basis of mechanistic considerations. Why did he not drop Pardies’ explanation? Looking at his writings prior to 1672, one would not expect Huygens to feel so strongly about a mechanistic theory. Prior to the ‘Projet’, Huygens never considered the nature of refraction more than incidentally. During his trip to Paris, on 3 January 1661 he discussed refractions – “contre des Cartes” – with Clerselier and others.155 In a letter to Moray of 9 June 1662 he ridiculed Vossius’ ideas on light and said that he had a totally different “… doctrine concerning refraction …”156 What it was he did not say. The same year his brother Lodewijk copied him a letter containing Fermat’s derivation of the sine law, based on the assumption that light follows the quickest path. On 8 March 1662 he answered that he admired the ingenuity of Fermat’s proof, but he found his fundamental assumption “pitoyable” and considered his doctrine unsatisfactory. According to him, neither Descartes nor Fermat were capable of proving the law of refraction, “the fundamental theorem of refractions” – and only experience renders it certain.157 Apparently Huygens saw no point in going more deeply into the matter of the cause of refraction. Despite his deep involvement in dioptrics, Huygens had never elaborated or adopted a theory of light or refraction. Whether he was simply not really interested in the matter or had ideas of his own on the back of his head, is hard to tell.158 In general, his attitude towards Cartesians and mechanistic philosophizing had been ambivalent during the previous years. In Paris, as he had already 155

OC22, 544. OC4, 149. “… doctrine touchant la refraction …”. 157 OC4, 71. “Pour faire donc l’accord entre luy [Fermat] et Monsieur des Cartes je dirois que ny l’un ny l’autre a prouvè la theorem fondamental des refractions, et qu’il n’y a que la seule experience qui nous en rende certains.” 158 There is one reference to the nature of light in a planned introduction, of 1656, to his treatise on impact, which appears, however, to be in contradiction with his later views, as Huygens seems to adopt instantaneous propagation of light: “… if nature as a whole consists of certain corpuscules from the motion of which every diversity arises, and from the fastest impulse of which light is propagated in a moment of time and flows throughout the immense expanse of the sky, …” (OC16, 150; translation: Gabbey, “Huygens and mechanics”, 189) 156

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experienced during his earlier travels, savants displayed much more concern for philosophical and metaphysical topics than he did. During his trip to Paris and London in 1660-1661, he expressed his appreciation for the downto-earth attitude of the Londoners as compared to the more esoteric bent of the Parisians. The particular group of Parisians he started to cultivate during that visit consisted of like-minded ‘mathématiciens’ like Auzout and Petit.159 In this light, his commitment to a mechanistic theory of light seems paradoxical. There is reason to believe that Huygens’ move to Paris brought about a change in his interests. After 1666, at the Académie, he was confronted with many discussions about natural philosophical topics. Huygens took part, in particular, in a discussion on gravity in August 1669.160 Van Berkel has suggested that Huygens began to emphasize mechanistic explanations because he was dissatisfied with the many theories put forward at the Académie that were not (properly) mechanistic.161 His opponents in the discussion on gravity assumed, for example, attractive forces. Huygens’ paper on gravity of 1669 may have had the effect that he saw the value of discussing natural philosophical questions. In the discussion about gravity, allusions to the nature of light also come out for the first time. In the notes he took during this discussion the question is asked how light can be understood when perfectly hard bodies do not rebound. And he added that if the corpuscles explaining light were elastic and composed this would accord with “…l’hypothese du P. Pardies …”.162 It is reasonable to suggest that the Parisian scene compelled Huygens to think more and more actively on questions of mechanistic philosophy than he had done in The Hague. Pardies himself may have been a decisive factor in this regard. It is not inconceivable that Pardies’ wave theory showed Huygens that it was possible to pursue mechanistic philosophizing in a satisfactory manner. Although we do not know in what manner Huygens intended to treat the mechanistic causes of refraction, the Pardies-like theory may have offered the kind of middle course between a non-committal, Barrovian account and a Cartesian derivation that suited him. Statements in the ‘Projet’ seem to rule out a Cartesian view, whereas his investigation of strange refraction suggests that Huygens took questions regarding the nature of light more seriously than Barrow did. In the next chapter, I discuss what may have attracted Huygens in Pardies’ theory. I think the example set by Pardies made Huygens realize that it was possible to treat the nature of light in a ‘comprehensible’ way.163 He saw no alternative for Pardies’ waves and, strange as it may seem in view

159

Hahn, ”Huygens and France”, 58-59. See below, section 6.3.1. 161 Van Berkel, “Legacy”, 55-59. 162 OC16, 184. 163 I owe this suggestion to Alan Shapiro. It is elaborated in section 5.2.2. 160

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of his earlier lack of interest for mechanistic explanation, he did not want to drop they idea of giving an explanation of refraction. I suspect few contemporaries would have objected if Huygens had passed over strange refraction in a treatise on dioptrics. Few would realize that it contradicted his explanation of refraction; even Pardies himself thought that strange refraction could easily be resolved with his theory. It looks like Huygens made things difficult for himself by choosing to bring up the ‘difficulté’ of strange refraction. Not only did he decide to include an account of strange refraction, but he also wanted to reconcile this with his mechanistic conception of ordinary refraction. According to Huygens, the causes of the various properties of light could only be plausible when they were consistent with one another. Therefore, his study of strange refraction had not yet come to an end. The question Huygens had posed in the ‘Projet’ – the ‘difficulté’ of strange refraction – carried the seed of a turn towards a new way to consider the problem of finding a new law of optics. His attitude towards the justification of established laws and the way he had searched for a law of strange refraction had been traditional. Had he been satisfied with the results, ‘Dioptrique’ would have been a traditional treatise in geometrical optics, unaffected by the changes initiated by Kepler and Descartes. The problem of strange refraction turned out to be not traditional, as it was a problem of waves instead of rays. In the next chapter we shall see how Huygens took the remaining step, that of analyzing strange refraction on the level of waves of light.

Chapter 5 1677-1679 - Waves of Light The road to the wave theory and the transformation of geometrical optics

At the end of his first attack on strange refraction in 1672, Huygens was literally back at his original question: how can strange refraction be reconciled with Pardies-like waves? The refracted perpendicular contradicted the principal assumption of wave propagation, that rays are normal to waves. He left the problem unsolved and left his ‘Projet’ for what is was, a project. Five years later, he returned to the problem. This time he went straight to the heart of the matter: what happens to waves when they enter Iceland crystal? This was the question he had left aside in 1672. Now, in the summer of 1677, he found an answer that solved the problem of strange refraction in one stroke. “EUPHKA”, he exclaimed on 6 August 1677. This solution was preceded by a study of some topics of refraction in which Huygens was reconsidering, so we can say with hindsight, the question of how exactly waves propagate. In the course of these investigations, he implicitly formulated his principle of wave propagation and subsequently applied it to waves in Iceland crystal. He did not elaborate the principle until he presented his theory of light at the Académie in the summer of 1679. This theory makes it clear that something special had happened in the summer of 1677. He had developed a rigorously mathematized definition of light waves, that is: of the mechanistic nature of light. Huygens’ principle of wave propagation enabled him to derive the laws of optics from a theory explicating the mechanistic nature of light. It enabled him to give mutually consistent explanations of the rectilinearity of rays, of reflection and refraction, and of strange refraction. It enabled him, in other words, to elaborate the ‘Projet’. But then the question is whether his explanations still fitted the methodological scheme laid out in the ‘Projet’. It did not, as Huygens was really doing a new kind of optics that went beyond traditional geometrical optics. In his new theory the mechanistic nature of light was no longer an additional elucidation of the laws of optics but the very heart of the theory. The question is to what extent Huygens realized that he was breaking new ground, beyond the mere solving of the puzzle of strange refraction. The text of the eventual Traité de la Lumière presents the epistemic novelties of his theory of light in a rather matter-of-fact way. Yet, to objections raised at the Académie in 1679 to his explanation of strange refraction, Huygens had replied in a way that brought its innovative character into sharp relief. His new theory did not allow of direct empirical proof, but required an

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indirect empirical confirmation by means of an experimental test of an hypothesis derived from it. The test succeeded, giving rise to a second “EUPHKA” on 6 August 1679. This event underscores the remarkable fact that until this late point, Huygens had largely proceeded by rational analysis without much thought for the empirical foundation of his ideas. This chapter discusses the development of Huygens’ wave theory of light from 1677 until the events of 1679. It is marked by a twofold “EUPHKA” written out in Huygens’ notebook, and that may be said to signify the context of discovery and of justification respectively. Unlike the “EUPHKA” that had hailed Huygens design for a perfect telescope in 1669, these two would stand the test of time. Traité de la Lumière of 1690 contains the larger part of the text he read at the Académie in 1679. It remained unchanged until 1689, when Huygens made some corrections and additions as he finally prepared his theory for publication.1 Times were still eventful for Huygens. After his return, upon his recovery, to Paris in June 1671, years of productivity followed. His encounter with Newton and his reflective telescope has been discussed in chapter 3, the ‘Projet’ and the first attack on strange refraction in the previous chapter. A discussion of Alhacen’s problem with de Sluse had started early 1671 and reached its peak late 1672.2 On 9 August 1673 he wrote a letter for Colbert, in which he summarized the state of the art in contemporary dioptrics. He explained the value of dioptrical theory: with “… the rules of refraction …, one could predict in advance the effect of telescopes”.3 He described the most powerful telescopes available and the problems with grinding good lenses, especially in Paris. A design for grinding non-spherical lenses by Smethwick had caught his attention in 1671. In 1675, he discussed it, without becoming fully convinced of the validity of the method.4 In addition, he was engaged in various activities, like trials of his pendulum clock at sea and the invention of the spring balance and its subsequent priority disputes, early 1675, with Hooke and Hautefeuille. Several papers in Journal des Sçavans and Philosophical Transactions appeared and in 1673 Huygens published his master piece Horologium Oscillatorium. He dedicated it to Louis XIV, in spite of the fact that his patron had invaded his fatherland the previous year and occasioning the Republic’s ‘disaster year’ and the lynching of his mathematics soul mate Johan de Witt . 1

The text in which these changes are made is preserved in two manuscript copies. OC19, “Avertissement”, 383. 2 The problem is, to find the point on a spherical mirror were a light ray is reflected when the position of the light source and the eye of the observer are given. In June 1669, Huygens sent his initial solution to Oldenburg, who began sending de Sluse’s work to Huygens in August 1670. Extracts of ensuing letters were printed in Philosophical Transactions of October and November 1673 after the discussion had ended in January. It is primarily a mathematical problem and less relevant for my account of Huygens’ optics, so I will not discuss it. For a detailed account of the problem and its solution: Bruins, “Problema”. 3 OC7, 350-351. “… les regles de refraction …, l’on pouvoit predire par avance l’effect des lunettes d’approche” It is not clear on what occasion he wrote this. 4 OC7, 111 (October 1671); 117 (November 1671); 487 (July 1675); 511-513 (October 1675)

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Amidst these various activities there is no sign of further work on the issues raised in the ‘Projet’. After the inconclusive end of his investigation of strange refraction, Huygens seems to have let the matter rest. Then, in the winter of 1675/6 his ‘melancholie’ reared its head again. Huygens went home to The Hague the following summer, returning to Paris two years later in June 1678. But he came back with valuable stuff. He had discovered Van Leeuwenhoek and his microscopes, adding several innovations as well as a new topic for his dioptrics. And he had a new insight in the nature of light and the solution to the puzzle of 1672: how can Iceland crystal refract a perpendicular ray? 5.1 A new theory of waves On 15 September 1676, Constantijn Sr. wrote to Oldenburg that ‘his Archimedes’ had brought a piece of that remarkable Iceland crystal with him.5 In the Hague, sometime during the next year, Christiaan returned to the problem of strange refraction. On 14 October 1677 he wrote to Colbert that he had recently demonstrated the properties of Iceland crystal “…, which is not a small wonder of nature, nor easy to fathom”.6 The solution is found in Huygens’ notebook, right after an investigation of caustics in which he first formulated his principle of wave propagation. This principle then turned out to provide the basis for solving the problem strange refraction posed for waves. The study of caustics and the solution of strange refraction together take up 11 pages of the notebook, which in my view form one continuous whole. However, in their customary manner, the editors of the Oeuvres Complètes have split up the contents into five paragraphs of the section ‘La Lumière’ in OC19.7 They blended material from different pages into what they considered coherent issues, to the point of inserting material that dates from years later.8 This not just disturbs chronology but even Huygens’ actual line of thinking. I shall now offer my reconstruction of how his conception of the propagation of waves developed hand in hand with the study of 5

OC8, 19. OC8, 36-37. “… demontrè … depuis peu celle [les proprietez] du Cristal d’Islande, qui n’est pas une petite merveille de la nature, ni aisée a aprofondir.” 7 Hug9, 38r-48v; OC19, 416-431. With considerable effort, the original order may be reproduced on the basis of information given in the editors’ annotations. In order to give an idea of the way the manuscript material has been mixed up in the Oeuvres Complètes, I will list the way the order in which the illustrations are given (page number in Hug9, number of the illustration in OC19 - section number in OC19. ‘nu’: an illustration not used) 38r, 137-3, nu (shortest path); 38v, 148-6 (ovals); 39r, 149-6, nu (ovals); 39v, nu nu nu (ovals); 40r, nu nu (ovals); 40v, 138-3, 139-3, nu nu (shortest path); 41r, 141-4, 144-5, 140-4, nu (caustics); 41v, 145-5, nu (caustics); 42r, 146-5, nu (caustics); 42v, 150-6 (caustics, wave propagation); 43r, 142-5, 143-5, nu nu (wave propagation, principle); 43v, calculations; 44r, nu nu (telescope and a wavefront); 44v, nu nu nu (idea of spheroidal wave?); 45r, nu nu nu (idem); 45v, nu (spheroidal waves, sketch related to Eureka); 46r, ...; 46v, calculations; 47r, 151-7, 152-7, 156-7 (eureka); 47v, nu nu (waves?); 48r, 154-7, 153-7 (shape of crystal); 48v, 147-5, nu nu (athm refraction, (faulty?) propagation (?) of spheroidal waves) 8 §4 on OC19, 430-431 is of a much later date than the insertion in the section on the explanation of August 1677 suggests. With respect to content and place in the notebook it must be closer to the experiment of August 1679. 6

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Cartesian ovals and caustics also conducted on these pages. In my view, the solution to the problem of strange refraction came directly out of this line of thought, although it may be argued that at some point some break occurred.9 The editors failed to reproduce the material on the five pages preceding the EUPHKA. It is indisputably related to the solution and, although somewhat obscure, it clarifies the way the solution may have taken shape in Huygens’ mind. Both Huygens’ account of caustics and his explanation of strange refraction depend upon Huygens new conception of wave propagation. Yet, it remains implicit throughout the notes involved. Only one or two tiny sketches reveal that an adjustment of Pardies’ wave theory had taken shape in his mind. The crucial insight is first found in the sketch reproduced in Figure 58. All points on a wave are centers of a multitude of wavelets spreading in all directions; the tangent to these wavelets is the propagated wave. Only in Traité de la Lumière did Huygens elaborate his principle of wave propagation and Figure 58 Huygens’ principle. its application to the behavior of light rays. To my knowledge, only two previous historians have consulted parts of the manuscripts reproduced in volume 19 of the Oeuvres Complètes. They were puzzled. According to Shapiro, they contain “the most subtle refinement of Huygens’ optics” which cannot have been its starting point.10 Ziggelaar has suggested that they reflect the “flash of genius” in which Huygens found his principle, that he then applied to the sophisticated problem of caustics.11 In my view, rather than a matter of application, the principle gradually emerged from the analysis of caustics. By following the manuscript material, we may find out what the “flash of genius” was – if it was one at all – and what sparked it. 5.1.1 A FIRST EUPHKA The pages begin with a drawing of a ray refracted by a plane surface, accompanied by a formulation of Fermat’s principle of least time: a ray is refracted in such a way that light travels between two points in different media in minimal time.12 On the next pages Huygens used the ensuing equation for the lengths of the two parts of the refracted ray to construct a Cartesian oval, the curve that refracts rays from one point to exactly a

9

Hug9, 42v is clearly written later, as it is dated 24 March 1678. The following three pages contain some scattered sketches and calculations. 10 Shapiro, “Kinematic optics”, 241. 11 Ziggelaar, “How”, 187. 12 Hug9, 39r; §1on OC19, 416.

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second point.13 (In a way this brought him back to the very beginning of his dioptrical studies.) Then he returned to the principle of least time, now to derive it from the sine law (Fermat had worked the other way around).14 All of this dealt with rays refracted by plane and curved surfaces.

Figure 59 Two rays refracted by a plane surface. Lettering added.

Figure 60 Wave VK refracted by a plane surface VM forming a caustic VHN.

On the next page, Huygens moved on to the case when rays do not intersect in one point after refraction.15 In such cases, the intersections of refracted rays form a caustic. Huygens first considered two rays refracted from glass (top) to air (bottom) at a plane surface (Figure 59). Before refraction, the rays AP and DP intersect in P. AP : AE = DP : DF = 3 : 2, according to the sine law. The refracted rays AE and DF intersect in H. Huygens continued: “the difference of the two PA, PD must be 23 of the difference of the two EA, FD.”16 These differences are AB and AC respectively; AC equals 23 AB. They indicate the paths traversed by rays in air and glass in equal times: in the time light covers the distance AC in air, its covers AB in glass. With this, Huygens derived an expression for the position of H on AE in terms of the position of S (OS perpendicular to AE).

13

Hug9, 38v-40r; partly reproduced in §1and §2 on OC19, 424-425. Hug9, 40v; §2 on OC19, 416-417. 15 Hug9, 41r. This page (reproduced in OC19, 419 §2) begins with an analysis, invoking his theory of spherical aberration, of the point of intersection of two near parallel rays refracted by a sphere. 16 OC19, 418. “differentia duarum PA, PD debet esse 3/2 differentiae duarum EA, DF.” 14

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In a drawing plus text right above this, the same case is considered, but now in terms of a wave.17 (Figure 60) In other words, all rays intersecting in P and all the points H of intersection of refracted rays are considered. A wave VK propagates from the glass above VM into the air below it. It propagates in such a way that all incident rays VP, CP and KP – like the rays AP and DP above – intersect in P. These rays are refracted towards VM, CG and MP.18 The intersections of the refracted rays – like point H above – form a curve VHN tangent to all refracted rays. Huygens now wanted to prove VHN = NM + 23 MK. That is, the time for light to cover VHN in air, is equal to the time required to cover NM in air and MK in glass. He explained the meaning of this statement by considering the moment when point K of the wave has reached the refracting surface. In the time K moves to M through glass, point V moves to Q through the air (VQ = 23 KM). At this moment, Huygens said without explanation, a wave is formed consisting of parts RM and RQ, which are the involutes of parts VR and NR of curve VHN. Ergo, NR + RV equals NM + VQ = NM + 23 MK. It is still not clear what Huygens exactly was after. He was thinking in terms of rays being paths covered by light in a certain time, but the point of considering the unfolding wave QR-RM after refraction is unclear.

Figure 61 A wave refracted at the plane surface of a glass medium. (Letters ABCDE added by editors, Xxxxx by me) 17

OC19, 421 §2. KP is perpendicular and not refracted; VP is incident at about 48o, the critical angle, and is refracted to the parallel VM. It is an odd case, a wave propagating to a single point instead of away from it. It might be connected to the preceding discussion of ovals, in that Huygens is considering what happens when the wave has passed the aplanatic surface and crosses a plane surface. 18

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On the next two pages, the issue at stake does become clear.19 Here he considered the same wave, but now propagating from air to the glass between E and D. The incident rays intersect in C (Figure 61). When the whole wave has passed the refracting surface, the wave XxxxxE is formed in the glass. The accompanying text reads: “The common tangent curve of all the particular waves will be the propagation of the principal wave in glass. Therefore, the straight lines which cut this tangent curve at right angles will be the refracted rays. These, however, are given otherwise. Therefore these will cut that curve at right angles. Therefore the curve is the involute of the other curve which is the common tangent of these rays. It is sufficient to know that the waves are propagated along these straight lines. But since the lines must cut the waves at right angles, it can appear surprising how the lines, not tending to one center, can always cut the waves at right angles. But this is now explained by the involute.”20

This text makes two things clear. In the first place, Huygens finally says what problem had been involved in the preceding exercises. In both cases discussed, rays do not intersect in one point after refraction. The question therefore is how the accompanying wave ought to be imagined. Apparently, as we shall soon see, caustics (or aberration in general) also raised questions with Pardies’ theory, as it is not immediately clear whether or how rays are normal to waves. Huygens had settled the matter by means of involutes. The refracted rays form a curve AB, like the curve VHN in Figure 60. The wave XxxxxE is the involute of this curve. In the second place, Huygens was applying a new conception of wave propagation. The particular waves he talks about are not drawn, but are thought to be the various spherical waves spreading in all directions through the glass around the points of incidence. At the time the wave in air reaches E, these wavelets have covered the distance to points x. Their tangent is XxxxxE, the propagation of the principal wave. Huygens leaves out this step and immediately goes on to draw the ‘straight lines’ along which it is propagated, the rays that is. He can do so because these are ‘given otherwise’, namely by the sine law. So, instead of determining the refracted rays by constructing the propagated wave, he determines the propagated wave by constructing the refracted rays. As I see it, instead of applying his principle to construct the propagated ray, Huygens was using it to justify the construction by means of refracted rays. The insight underlying this justification does not go beyond a small sketch one page further down (Figure 58 on page 162).21 We cannot be certain to what extent the insight was already in his mind. I believe it was beginning to take shape when he was analyzing caustics in figures 59 to 61. I find the preceding study of Fermat’s principle and of aplanatic surfaces telling. Huygens was beginning to consider rays in terms of a path covered in 19

Hug9, 41v and 42r. OC19, 421 §3 and 422 respectively. I only discuss 42r. OC19, 422; translation from Shapiro, “Kinematic optics”, 236. 21 Hug9, 43r. On the intermediate page 42v he applies it to the propagation of a wave crossing an aplanatic surface (OC19, 425-426 §2), but this apparently is much later as Huygens dated it 24 March 1678. 20

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a certain amount of time, that is, as optical paths. He then applied this to caustics. In this case he considered a number of rays. He measured out equal times covered along different rays and recognized the relationship between caustics and involutes, curves he was fully acquainted with. Somewhere along the line, Huygens realized that in constructing waves this way he was assuming that rays are always normal to waves. If that is so, the notion of wavelets spreading in all directions from the points of incidence and subsequently forming a propagated wave by their enveloping curve must have come up as a justification.22 Huygens’ account of caustics resolved two ambiguities in Pardies’ derivation of refraction. Pardies had determined the direction into the refracted wave propagates by constructing the refracted ray (Figure 56 on page 153). The line sections Ce, ee, etc. on the refracted rays are regarded as the intervals by which the wave proceeds in a given time. The resulting wave is not spherical anymore, but this is not accounted for any further. The meaning of the curve in terms of light waves thus remains vague. Furthermore, as Shapiro has pointed out: “Pardies has not explained why at the point of refraction the wave should be refracted in one direction, … He has simply assumed that refraction occurs. Therefore, he has demonstrated only that if the wave is refracted, then it must be propagated in the new medium in a direction such that the rays are always normal to the wave fronts.”23 Huygens now stated that light, in the form of ‘particular waves’, spreads in all directions from the points of refraction. The propagated wave can be constructed by drawing the envelope of the wavelets, even if the wave is not spherical. Implicitly, Huygens stated that the wave is a surface of constant phase, i.e. the locus of points where wavelets unite. It remains to be seen whether Huygens was explicitly reconsidering Pardies’ theory of light at this moment. It is unclear, for example, whether the ‘surprising’ observation that ‘rays not tending to one center, can always cut the waves at right angles’ had given rise to the study of ovals and caustics.24 Irrespective of the question of whether he was explicitly tackling problems with Pardies’ theory, Huygens had begun to focus on distances covered by light in a specific time. And irrespective of the question of whether, and if so how, his principle of wave propagation arose from the ensuing analysis of caustics, he realized that both waves and rays are 22

In other words, I tend to disagree with Shapiro’s view that this ‘most subtle refinement of Huygens’ optics’ cannot have been the starting point for the formulation of Huygens’ principle. (241) I do not consider the equality of optical paths to have been derived from his theory of light (232), but rather to be Huygens’ starting point in these studies, which he subsequently, and rather implicitly, justified by a vague notion of his principle. I suspect that Shapiro has been misled by following the text of Traité de la Lumière, that is, the analysis of the enveloping wave refracted by a spherical surface, which is indeed the most subtle refinement and application of Huygens’ theory of light. It must, however, be of a much later date as this case is not found in the 1667 notes. Shapiro, “Kinematic optics”, 231-236. 23 Shapiro, “Kinematic optics”, 215-217. Emphasis in the original. 24 Ziggelaar states, without argument, that caustics posed a crucial objection to Pardies’ theory and that this induced Huygens to formulate his own theory: Ziggelaar, “How”, 186-187.

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subordinate to the speed of propagation of light. What distinguishes the notion implicit in this analysis from Pardies is the idea that light spreads in all directions. The flash of genius would then consist in the insight that Pardies’ premise, rays are normal to waves, still holds as it directly follows from the fact that the wave is tangent to all wavelets. What Huygens had done was to reduce waves to their speed of propagation. Another question is what stimulated Huygens to resume the riddles of light waves. It had, after all, been 5 years since he put aside the unsolved questions raised by strange refraction. May an article in the Philosophical Transactions of 25 June 1677 (6 July N.S.) have been the occasion for Huygens’ study? The article, by Ole Rømer, contained a proof of the finitude of the speed of light.25 It was based on observed irregularities in the motions of Jupiter's satellites. The idea had already been proposed by Cassini in 1675, but he had withdrawn it shortly afterwards.26 Cassini would remain one of the main opponents of Rømer’s assertions.27 Understandably, Huygens was to welcome this observational confirmation of the main premise of his understanding of light. In Traité de la Lumière he would repeat “the ingenious demonstration of Mr. Romer.”28 In his view, the wave theory and the finite speed of light were necessary linked.29 He wrote Rømer on 16 September 1677 to express his gratitude.30 In the following months they exchanged various letters on the subject. Rømer’s article would not have affected Huygens’ conviction that the speed of light is finite, but it is likely that reading it induced him to look at the propagation of light anew. The centrality of the speed of propagation in the notes we have just examined – from Fermat’s principle to the analysis of caustics – makes this plausible. This would mean that this study took place between the middle of July and 6 August 1677.31 It would imply that he moved on to the problem of strange refraction almost at once upon solving the problem of caustics. In view of the content and appearance of the manuscript material I find such a short span of time quite plausible.

25

OC8, 30n1. It was the translation of an article that had been published in French on 7 December 1676: “A demonstration concerning the Motion of Light, communicated from Paris, in the Journal des Sçavans, and here made English.” Apparently, Huygens had not seen the issue of Journal des Sçavans. 26 Sabra, Theories of Light, 205. Not all historians agree on this point; compare: Cohen, “First determination”, 345-346; Van Helden, “Roemer’s speed of light”, 140n1 and Wroblewski, “De Mora Luminis”, 629. 27 Other opponents to the view that the speed of light is finite were Hooke and Fontenelle. Wroblewski argues that the controversy came to an end in 1729. 28 Traité, 7. “l’ingenieuse demonstration de Mr Romer.” 29 In the ‘Projet’ he had made one reservation: “Light extends circularly and not in an instant, at least in the bodies down here, because for the light of stars it is not without difficulty to say that it would not be instantaneous.” Such a reservation was now needless, for Rømer’s argument was based on astronomical observations. OC19, 742. “lumiere s’estend circulairement et non dans l’instant, au moins dans les corps icy bas, car pour la lumiere des astres il n’est pas sans difficulté de dire qu’elle ne seroit pas instantanee.” 30 OC8, 30-31. 31 The editors of Oeuvres Complètes think that the study of caustics may have taken place as early as 1676.

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The solution of the ‘difficulté’ of Iceland Crystal In August 1677, what we see as Huygens’ principle was nothing but an implicit application in the analysis of caustics and a tiny sketch reflecting this idea. Huygens had not yet elaborated his principle nor explained how it ought to be applied. This did not refrain him from passing on to the puzzle that still remained: what happens to waves when they enter Iceland crystal? In his analysis of caustics, Huygens had confirmed Pardies’ premise that waves are normal to rays. Yet, it had become secondary to his understanding of wave propagation. But strange refraction still contradicted it. As we cannot tell with certainty to what extent Huygens was tackling problems in Pardies’ theory with his study of caustics, it is not clear in what way the problem of strange refraction was on his mind. In his notebook the study of caustics is preceded by a single leaf filled with sketches relating to strange refraction.32 All relevant matters are reviewed, Bartholinus’ law and the supposed pores of the crystal, Pardies’ explanation and the propagation of a ray through successive layers of crystal. No progress, in comparison with the 1672 investigation, is apparent. Dating the page is hazardous. On the basis of its place in the notebook it can be prior to December 1674, but it might as well have been a vacant page Huygens scribbled on later during his stay in The Hague. The next allusion to strange refraction follows almost immediately upon the analysis of caustics and the sketch of his principle. After two pages with calculations, a diagram of a telescope and another sketch (apparently) of wavelets, a page follows with intriguing content.33 From left to right we see three sketches: a horizontal ellipse with two rays parallel to its axis refracted to its focus; an ellipse plus axis, yet drawn obliquely; rays refracted with some wavelets indicated. The last sketch is remarkable, as the refracted ray seems to be drawn obliquely to its accompanying wave. The problem of strange refraction had returned, and the following pages make it clear that Huygens had found the solution: an oblique ellipse. On the mirror page the ellipse returns, now with some small wavelets and – this was the solution – a horizontal tangent.34 The line connecting the point of tangency and the center of the ellipse makes an angle with the tangent. In other words: ray and wave intersect obliquely. The speed of propagation of light in Iceland crystal is not equal in all directions, light spreads spheroidally instead of spherically. Consequently, the wave is not normal to its direction of propagation. We are able to understand these sketches in this way, as we have the hindsight knowledge of Huygens’ elaborating in Traité de la Lumière. In his notebook he did not explain what precisely the oblique ellipse was and how it explained strange refraction. On the next three pages, he only offered

32

Hug9, 7r. None of it is reproduced in the Oeuvres Complètes. Hug9, 44v. Not reproduced in the Oeuvres Complètes. 34 Hug9, 45r. Not reproduced in the Oeuvres Complètes. 33

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some more sketches and numbers calculations. Then a glorious EUPHKA follows: “EUPHKA. 6 August 1677. The cause of strange refraction in Iceland crystal.”35

The page contains a large drawing of this ‘cause’, surrounded by explanatory texts and additional drawings and calculations. In the central figure the oblique ellipse returns, now abundantly rigged up with geometry. The editors of the Oeuvres Complètes have proposed a plausible order in which the texts around it have been written. This means that Huygens started with the figure reproduced in Figure 62, then wrote the ‘Eureka’ at the top-right corner and started the explanation of the drawing, continuing in the top-left corner with a smaller sketch and further notes.

Figure 62 “Causam mirae refractionis in Crystallo Islandica”.

The explanation begins with a description of the figure. The crystal has a principal section for each dimension and AS, the axis of the obtuse solid angle of the crystal, is the intersection of these (from the center diagonally down to the right). The plane of the paper is one of the principal sections. The upper face of the crystal is KA, the lower face DƦ, and AC (from the center down second to the left) is parallel to the edge of the crystal. AC is also the unrefracted oblique ray (i.e. parallel to the edge of the crystal). Also drawn is AB, the refracted perpendicular (from the center down first to the left). BPSHN is an ellipse, with a circle drawn inside it with center A and radius AS. The ellipse and circle therefore touch in S. The axis of the ellipse AP (from the center diagonally down to the left) is normal to AS. Huygens showed that DƦ is tangent to the ellipse in point B, its intersection with the refracted perpendicular AB. The smaller ellipse ưƧƪ is constructed by drawing the quadrangle µƪLA (ƪµ parallel to the tangent of 35

Hug9, 47r. OC19, 427, it is preceded by a facsimile of this manuscript page. “EUPHKA. 6 Aug 1677. Causam miræ refractionis in Crystallo Islandica.”

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the large ellipse in C, µL parallel to AC and normal to LA). Huygens then explained the meaning of all this: “In the time light forms a sphere with radius µL in the air, it forms a spheroid ưƧƪ, congruent with PCH, within the crystal.”36

The ellipse signifies a wave of light and presumably this explains strange refraction. How it did Huygens did not explain in detail, but – again with hindsight – we can tell. The horizontal line BD is tangent to the wave at the point where the refracted perpendicular AB cuts it. In other words, the wave is oblique to its direction of propagation.

Figure 63 Strange refraction of an arbitrary ray.

Huygens went on to explicate how the ellipse can be used to determine the strange refraction of an arbitrary ray. In a sketch at the top-left of the page he discusses the incident ray MA (Figure 63). Again, Cµ is tangent to the ellipse, µL parallel to CA (the unrefracted oblique ray) and perpendicular to LA. Now draw AV perpendicular to MA, VX perpendicular to AV (where VX = µL), and XF tangent to the ellipse. Then AF is the refracted ray. Huygens only proved that XF is the tangent. In terms of waves, the following happens. A plane wave AV propagates along MA reaching the surface of the crystal obliquely in A. In the time light covers the distance VX in the air, a spheroidal wave HFBC is formed inside the crystal. The tangent XF is the refracted wave propagating along AF, which therefore is the refracted ray. Again, the meaning of the construction in terms of waves remained largely implicit in Huygens’ proof. He had defined a construction to determine the refraction of an arbitrary ray. In other words, he had formulated a new law of strange refraction, the kind of thing he had been looking for in 1672. The new law differed considerably from the proposal of 1672, however. The latter depended upon the refracted perpendicular – as contrasted to Bartholinus’ law that depended upon the unrefracted oblique ray. The refracted perpendicular offered a straightforward extension of Descartes’ derivation of the sine law in terms of components added to rays. The new law depended upon both the refracted perpendicular and the unrefracted oblique ray: AB is the basic parameter of the ellipse and AC is used to draw VX (via µL), the propagation of light in air proportional to the propagation 36

OC19, 427. “Quo tempore lux in aere facit sphaeram cujus radius µL, eadem intra crystallum facit spheroides ưƪƧ simile PCH.”

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of the ellipse in the crystal. Yet, the most important difference between both laws is their nature. Whereas the proposal of 1672 was fully phrased in terms of rays and their components, the new law utilizes waves, unobservable and hypothetical entities expressing the mechanistic nature of light. The refracted ray is constructed by means of the tangent to the ellipse, by means of waves and their properties. The unrefracted oblique ray and the refracted perpendicular have become secondary to the speeds with which light propagates through the crystal. The core of the new law was the strange mode of wave propagation in Iceland crystal. With this Huygens had fully departed from both his own line of thinking of 1672 and from Bartholinus’. He had returned to his ideas regarding the nature of light to understand the peculiar phenomena displayed by Iceland crystal. The EUPHKA of 6 August 1677 signaled the solution to what Huygens saw as the problem of strange refraction. As with his proposal of 1672 – and also with Bartholinus’ law – no empirical confirmation is given. Huygens had, by the way, improved his empirical data. He introduced an accurate method of determining crystallographic angles that required only one measurement.37 Yet, he had not improved or added optical data, nor did he explicitly verify his new law empirically. Huygens seems to have been fully convinced that the ‘cause’ he had found was valid. What was the ‘cause of strange refraction’ Huygens had found? The EUPHKA did not hail the discovery of spheroidal waves. Instead, it hailed the invention of the way a strangely refracted ray could be constructed by means of the ellipse. Although Huygens said what the ellipse was – which in retrospect was sufficient – he was relatively silent on the question how the construction should be interpreted in terms of waves. Drawing on a distinction made by Shapiro, we can say that Huygens was more concerned with the question how an spheroidal wave could explain strange refraction than with the question whether it could.38 He did not explain the idea that light produces a spheroid wave in the crystal. Only implicitly did he assume that the speed of propagation or the action of the crystal differs in each direction. As with his principle of wave propagation, the physical concepts underlying the mathematical construction were at the back of his head, but he did not take the trouble to elaborate them. The proof of the pudding was in the eating: before spelling out the idea of his principle of wave propagation, Huygens first saw to it that it could be successfully applied. He found out that, with his new idea, he could understand both caustics and strange refraction in terms of waves. He focused on the new mathematical structure of wave propagation, and even this he did not elaborate in detail. In his notes, he did not explain if and how the construction conformed to his ideas on the propagation of light waves. He only brooded a little on the composition of the crystal. Scattered around 37 38

Discussed in Buchwald, “Experimental investigations”, 313-314.. Shapiro, “Kinematic optics”, 238-239.

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the central figure are sketches of piles of round and elliptical balls. Apparently, he was figuring out how these could make up a rhomboid and maybe also how elliptical particles could pass on a movement asymmetrically. This was a question that still stood open regarding his theory of light in general. He had thought out a principle describing its wavelike propagation, but he had not answered the question what light is and why these waves propagate this way. For answers to all these questions we have to turn to the theory as he presented it at the Académie in 1679. 5.1.2 UNDULATORY THEORY Announced directly after his return to Paris in the summer of 1678, the reading of “the treatise of Mr. Huygens on dioptrics” began on 13 May 1679.39 Huygens expounded his ideas on the nature of light and its wavelike propagation, concluding with an explanation of the rectilinearity of light rays on this basis. This account is found in chapter one of Traité de la Lumière. Chapters two to six deal successively with reflection, refraction, atmospheric refraction, strange refraction, and finally aplanatic surfaces and caustics. Hereafter, I follow the text of Traité de la Lumière except where otherwise indicated. Huygens began with an exposition of the mechanistic nature of light. The whole argument is aimed at establishing the one defining characteristic of waves: their finite and constant speed of propagation. Besides reproducing Rømer’s proof (“But what I used only as a hypothesis has recently received every appearance of a definite truth, by the ingenious demonstration of Mr. Rømer …”40) he explained how an action is propagated, with finite speed, through imperceptibly small particles. Light, according to Huygens, originates from the agitation of particles in luminous objects, colliding with the particles of the surrounding, all-pervading ether. These collisions are communicated in all directions through the ether, without particles being displaced. The waves thus produced constitute light. On the basis of the mechanical properties of hard, elastic balls, Huygens argued that this impact spreads in all directions with a finite and uniform velocity. The velocity depends only on the degree of elasticity of the particles and is independent of the strength of the impact. Moreover, any particle can communicate different impulses simultaneously. At the bottom, Huygens’ account of the mechanisms explaining the propagation of light waves had only one purpose: to show that the ether consists of elastic particles. If this be the case, the basic premise of his theory is valid: the collisions propagate with a finite and uniform velocity that

39

OC19, 441. (The dates when the reading was continued are given on 441-443) “… le traitté de Mr. Hugens de la Dioptrique.” 40 Traité de la Lumière, 7. “Mais ce que je n’employois que comme une hypothese, a recue depuis peu grande apparence d’une verité constante, par l’ingenieuse demonstration de Mr. Romer …”

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depends only on the medium.41 In other words, Huygens mechanistically justified his reduction of light propagation to velocity. Consequently, light consists of waves: “If light thus takes time for its passage (which we now will examine) it will follow that this movement impressed on matter is successive; and consequently it spreads, …, by spherical surfaces and waves: …”42

After Huygens had shown how light can be thought to consist of spherical waves propagating with a considerable yet finite velocity, he moved on to consider their propagation in more detail.43 This is where he finally elaborated his principle of wave propagation. First of all, each point of a luminous source produces spherical waves (Figure 64). The circles represent the propagation of a single wave, so he added, and should not suggest any regular succession of particular waves. Although Huygens’ theory is in fact a pulse theory, for sake of convenience I will speak of waves. While a wave moves away from its origin, its speed is maintained although it gradually loses its strength. In the long Figure 64 Waves around a run, the waves will become imperceptible to our source of light eyes. Still, light produced by such small actions can be perceptible over long distances, because innumerable waves “… unite in such a way that to the senses they make up only one single wave, that consequently must have enough force to make itself felt.”44

In a ‘Particular remark on the extension of light’, Huygens went on to describe this uniting more precisely. It was an elaboration of the premise he had formulated earlier in his discussion of caustics: “The common tangent curve of all the particular waves will be the propagation of the principal wave …”. Although he introduced this principle of wave propagation as a ‘remark’, it was the core of Huygens’ theory. Waves of light were not his idea; his contribution consisted of this principle: “This is what was not known to those who previously began to consider the waves of light, among whom are Mr. Hooke in his Micrographia, and father Pardies, who in a treatise of which he has shown me a part, …, undertook to prove by these waves the

41

The notes on caustics already reveal that the speed of propagation had become central for Huygens. Implicitly, he constructed the refracted wave by considering rays as isochronous paths. He explicitly considered the caustic as a path travelled in a specific time, when he showed that its length is equal to a rectilinear distance traveled by light. 42 Traité, 4. “Que si avec cela la lumiere employe du temps à son passage; ce que nous allons examiner maintenant; il s’ensuivra que ce mouvement imprimé à la matiere est successif, & que par consequent il s’etend, … , par des surfaces & des ondes spheriques: …” 43 Traité, 15-17. 44 Traité, 17. “… s’unissent en sorte que sensiblement elles ne composent qu’une onde seule, qui par consequent doit avoir assez de force pour se faire sentir.”

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effects of reflection and refraction. But the principal foundation, that consists of the remark I have made, was lacking in his demonstrations, …”45

This principal foundation – Huygens’ principle – explained how a propagated wave can be constructed mathematically: “There is the further consideration of these waves, that each particle of matter in which a wave spreads, ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point, but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement. So it arises that around each particle there is made a wave of which that particle is the center. Thus (Figure 65) if DCF is a wave emanating from the luminous Figure 65 Huygens’ principle. point A, which is its center, the particle B, one of those comprised within the sphere DVF, will have made its particular wave KCL, which will touch the wave DCF at C at the same moment that the principal wave emanating from the point A has arrived at DCF; and it is clear that it will be only the region C of the wave KCL which will touch the wave DCF, to wit, that which is in the straight line drawn from AB. Similarly the other particles of the sphere DCF, such as bb, dd, etc., will each make its own wave. But each of these waves can be infinitely feeble only as compared with the wave DCF, to the composition of which all others contribute by the part of their surface which is most distant from the center A. One sees, in addition, that the wave DCF is determined by the distance attained in a certain space of time by the movement which started from the point A; there being no movement beyond this wave, though there will be in the space which it encloses, namely in parts of the particular waves, those parts which do not touch the sphere DCF. And all this ought not to seem fraught with too much minuteness or subtlety, since we shall see in the sequel that all the properties of Light, and everything pertaining to its reflection and its refraction, can be explained principally by this means.”46

Huygens had drawn the ultimate consequence of the notion that in a fluid medium an action must spread in all directions: each particle in the medium is the source of a new wave. These wavelets are too feeble to be perceptible, but when they unite at certain loci they form a principal wave. The action of this principal wave constitutes visible light. Why this laborious exposition of waves producing wavelets forming a new wave? Because the laws of optics can be explained properly only with this principle. Huygens explicitly warned 45

Traité, 18. “C’est ce qui n’a point esté connu à ceux qui cy-devant ont commencé à considerer les ondes de lumiere, parmy lesquels sont Mr. Hook dans sa Micrographie, & le P. Pardies. qui dans un traitté dont il me fit voir une partie, …, avoit entrepris de prouver par ces ondes les effets de la reflexion & de la refraction. Mais le principal fondement, qui consiste dans la remarque que je viens de faire, manquoit à ses demonstrations, …” 46 Traité, 17-18; translation: Shapiro, “Kinematic Optics”, 222-223.

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his readers not to judge this principle foundation at face value, but to wait and see how the behavior of rays was derived from it. The subsequent explanation of the laws of optics is indeed the key to Huygens’ wave theory. The ‘remark’ was not a mere explication of his idea how feeble actions may produce visible light over tremendous distances. It was the mathematical representation of his understanding of wave propagation. Waves are represented by circles and defined as the “distance attained in a certain space of time by the movement which started from point A”. These express his premise that waves propagate with a uniform and finite velocity. Huygens’ principle explains how to construct a principal wave that has traveled for a certain time: construct the secondary waves and draw their tangent. It did so, as we shall see, for all situations where the speed of propagation changes when the medium changes. The only assumption to be made is the distance covered by waves in a certain time. The first thing to be explained was the rectilinearity of light rays. Light propagates rectilinearly when the medium is homogeneous and the speed of propagation does not change. In his explanation, Huygens stated the mutual relationship between the original wavelets and the principal wave more explicitly. Huygens argued that a wave spreads in such a way (in Figure 65, from BG on to CE) that it is always between the same straight lines (ABC and AGE) drawn from the luminous points. “… For although the particular waves produced by the particles comprised within the space CAE spread also outside this space, they yet do not concur at the same time instant to compose a wave which terminates the movement, as they do precisely at the circumference CE, which is their common tangent.”47

The wavelets outside the region CAE are “… too feeble to produce light there.” Because this applies to any portion of the principal wave, the opening BG can be made arbitrary small. “Thus then we may take the rays of light as if they were straight lines.”48 Refraction now is explained by the change of the speed of waves propagating from one medium to another. During the time a wave covers LL in the medium above the refracting surface AB, it will cover a smaller distance OO in the medium below (Figure 66). The points AHHHC reach the Figure 66 Huygens’ explanation of refraction. 47

Traité, 19; translation: Shapiro, “Kinematic optics”, 223. Traité, 19-20; translation: Shapiro, “Kinematic optics”, 223. For an extensive discussion of the validity of his argument, see Shapiro, “Kinematic optics”, 225-227; De Lang, “Originator”.

48

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surface in points AKKKB successively, producing wavelets spreading in all directions through the refracting medium around these points. When the whole wave has reached the surface – when C arrives in B – around A a wavelet will have propagated over the distance AN. The common tangent NB of all wavelets around the points of incidence is the propagated principal wave. The sine law of refraction easily follows. CB represents the speed of light in the upper medium, but also the sine of the angle BAC, equal to the angle of incidence DAE. Likewise, AN is the speed in the lower medium and the sine of angle ABN, equal to the angle of refraction FAN. The assumption that CB and AN are in constant proportion directly yields the sine law. In the same way, Huygens could derive the law of reflection by considering only the propagation of waves. Assuming that the motion of light rebounds at a reflecting surface, the tangent of wavelets spreading around the points of reflection is constructed and the equality of the angles of incidence and reflection readily follows. Huygens’ theory was simpler and contained less ambiguities than Pardies’. He had reduced waves to a single property of light, its speed of propagation. Waves are the effect of an action spreading with a certain speed. In his derivation of the sine law, Huygens did not have to presume that a wave refracts. He only had to consider the consequence of an alteration in the speed of propagation. The curve (or line) resulting from his construction has an unambiguous meaning, established by his principle of wave propagation. He preserved the premise that rays are normal to waves, at least in the case of spherical waves. As a ray is the path traveled by a point of a wave in a specific time, it is a direct consequence of the fact that the principal wave is the tangent of secondary waves. Explaining strange refraction Precisely by this reduction of waves to speed of propagation, the puzzle of strange refraction had been solved. In Iceland crystal light propagates with differing speeds in differing directions. In his notes, Huygens had not explained why spheroidal waves account for the refracted perpendicular, nor why light propagates spheroidally in Iceland crystal in the first place. In the fifth chapter of Traité de la Lumière, Huygens elaborated his discovery of 6 August 1677. He began with a description of the crystal and its peculiar properties. It displays double refraction, so supposedly light propagates through the crystal in two different ways. The first one was regular and produced ordinary refraction in agreement with the sine law. The other one was irregular, as a perpendicular ray was refracted. To account for this strange phenomenon, Huygens “wanted to try what elliptical, or better speaking spheroidal, waves would do”.49 In other words: try and see what would happen when light propagates in this direction faster than that. 49

Traité, 58. “Quant à l’autre émanation qui devoit produire la refraction irreguliere, je voulus essaier ce que feroient des ondes Elliptiques, ou pour mieux dire spheroïdes; …”

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He began with a qualitative account of spheroidal waves produced by a perpendicularly incident wave (Figure 67).50 RC is part of a plane wave incident perpendicularly on the surface AB of the crystal, so that all points RHhhC arrive in AKkkB at the same time. Suppose spheroidal wavelets SVT spread around these points, as the speed of propagation in the direction AV is larger than in the direction AZ. According to Huygens’ principle Figure 67 Refraction of the perpendicular. the common tangent of these spheroidal waves is the refracted wave. “And it is thus that I have comprehended, what had seemed to me very difficult, how a ray perpendicular to a surface could suffer refraction on entering the transparent body; seeing that the wave RC, having come to the aperture AB, continued forward thence, extending between the parallels AN, BQ yet itself remaining always parallel to AB, such that here the light does not extend along lines perpendicular to its waves, as in ordinary refraction, but these lines cut the waves obliquely.”51

Thus the application of Huygens’ principle applied to spheroidal waves showed that these could explain the refracted perpendicular. Huygens had solved the original problem of strange refraction, as the refracted perpendicular implied that waves would not be at right angles to the line of their extension. Strange refraction was strange precisely because of this: the propagation of light in Iceland crystal is extraordinary and produces waves that are not normal to rays. At this point in the Traité de la Lumière the problem was solved, but only in principle. For Huygens the most important question still had to be answered: how spheroidal waves could explain strange refraction. The answer, the exact properties of the spheroid, was that of 6 August 1677. In Traité de la Lumière, he began with observing that in the principal sections of each face of the crystal (the dotted lines in Figure 68) strange refraction behaves the same. Consequently, the spheroid must have the same section in all three planes. This is so when the axis of the spheroid is also the axis of the obtuse solid angle C of the crystal. Choosing plane GCF, Huygens drew the ellipse PSG, the cross-section of the spheroid around center C (Figure 69). CS is the axis of the obtuse solid angle C as well as the axis of 50

Traité, 60-62. Traité, 61-62. “Et c’est ainsi que j’ay compris, ce qui m’avoit paru fort difficile, comment un rayon perpendiculaire à une surface pouvoit souffrir refraction en entrant dans le corps transparent; voyant que l’onde RC, estant venue à l’ouverture AB, continuoit de là en avant à s’étendre entre les paralleles AN, BQ demeurant pourtant elle mesme tousiours parallele à AB, de sorte qu’icy la lumiere ne s’étend pas par des lignes perpendiculaires à ses ondes, comme dans la refraction ordinaire, mais ces lignes coupent les ondes obliquement.” 51

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Figure 68 Orientation of spheroid in the crystal.

Figure 69 Shape of the spheroidal wave.

revolution of the spheroid, with angle GCS = 45º20´. CM is the refracted perpendicular and thus – on account of the preceding – the ellipse must be tangent in M to the lower surface FH of the crystal, with angle MCL = 6º40´. Choosing CM = 100,000 yields CP = 105,032, CS = 93410 and CG = 98779.52 Except for differences in the specific values (due to later measurements) this what his new method of August 1677 gave.53 In Traité de la Lumière Huygens went on to explain how the strange refraction of an arbitrary ray can be found. He applied his principle of wave propagation to spheroidal waves, which had been implicit in the notes of August 1677 (Figure 70): “Coming now to a search for the refractions that the obliquely incident rays must make, following the hypothesis of these spheroidal waves, I saw that these refractions depended upon the proportion of the speed that is between the movement of the light outside the crystal in the ether, and the movement inside the same. For supposing for example that this proportion was such that, while the light in the crystal makes the spheroid GSP, as I have just said, outside it makes a sphere of which the semi-diameter is equal to the line N, which will be determined further down; then this is the manner to find the refraction of the incident rays.”54

Except for the line N, the construction is the same as on 6 August 1677: RC is incident on surface kCK of the crystal. To find the (strangely) refracted ray CI, draw CO normal to RC and OK normal to CO, with KO = N. Drawing the tangent through K to the ellipse GSP yields point I, and CI is the refracted ray. The proof, using spheroidal wavelets, proceeds along the same lines as in the case of ordinary refraction. By the time O arrives in K, points H have arrived at the points x on the surface and spheroidal wavelets have spread in the

52

Traité, 62-63. CM = 100,000 yields CP = 105,022, CS = 93095 and CG = 98473. In 1677 angle FCL = 70º57´. As a result of the measurement of August 1679 (see section 5.3.1) FCL = 73º20´ in Traité de la Lumière. 54 Traité, 63-64. “Or passant à la recherche des refractions que les rayons incidens obliques devoient faire, suivant l’hypothese de ces ondes spheroides, je vis que ces refractions dependoient de la proportion de la vitesse qui est entre le mouvement de la lumiere hors du cristal dans l’éther, & le mouvement au dedans du mesme. Car supposant par exemple que cette proportion fût telle que, pendant que la lumiere dans le cristal fait le spheroide GSP, tel que je viens de dire, elle fasse au dehors une sphere dont le demidiametre soit égal à la ligne N, laquelle sera determinée cy apres; voicy la maniere de trouver la refraction des rayons incidens.” 53

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Figure 70 Construction of the refraction of an arbitrary ray in Traité de la Lumière.

crystal. IK is the common tangent of these wavelets and therefore IK is the propagated wave and IC the refracted ray.55 In this construction the line N replaces the unrefracted oblique ray in the construction of August 1677. It represents the proportion of the speeds of propagation in the air and in the crystal. As Buchwald explains, it provided an absolute parameter for the construction.56 Originally, this proportion followed from the unrefracted oblique ray. In 1679 he found out, as we will see in section 5.3.1, that this ray was not parallel to the edge of the crystal. Although this did not change his construction, he could not use it as a parameter anymore. Instead he used the line N, introduced as an observational value: “To find the length of the line N, proportional to CP, CS, CG, it is through the observations of the irregular refraction that occurs in this section of the crystal, that it must be determined; and I find in this way that the ratio of N to GC is a little less than 8 to 5.”57

With the ellipse construction thus quantitatively determined, Huygens derived several properties of the strange refraction. He showed which ray passed without refraction; considered rays outside the principal section, and discussed the apparent position of images. Finally, Huygens could conclude: 55

Traité, 65. Buchwald, Rise, 315-316. However, Buchwald only discusses the final text of Traité de la Lumière and therefor does not take into account the historical background of this choice of parameters. On page 317 he raises the possibility that Huygens did not determine the value of N directly - as the next quote suggests - but deduced it from the angle of the unrefracted oblique ray and subsequently reversed the calculation to confirm is theory. In the manuscripts he could have caught Huygens more of less in flagrante delicto. 57 Traité, 66. “Pour trouver la longueur de la ligne N, à proportion des CP, CS, CG, c’est par les observations de la refraction irreguliere qui se fait dans cette section du cristal, qu’elle se doit determiner; & je trouve par là que la raison de N à GC est tant soit peu moindre que de 8 à 5.” 56

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“In this way, I have searched in every detail the properties of the irregular refraction of this crystal, to see whether each phenomenon that is derived from our hypothesis agrees with what is actually observed. This being so, it is not a light proof of the truth of our suppositions and principles.”58

In glaring contrast with the acuity with which he thus derived the behavior of strangely refracted rays from his hypothesis stands the vagueness with which Huygens dealt with the question why light produces spheroidal waves in Iceland crystal. When he proposed the idea, he said that he only needed to assume that the speed of light differed for various directions of the crystal: “As for the other emanation that must produce the irregular refraction, I wanted to try what elliptical or, speaking better, spheroidal waves would do; and these I supposed would spread indifferently both in the ethereal matter diffused throughout the crystal and in the particles of which it is composed; …It seemed to me that the disposition, or regular arrangement, of these particles could contribute to forming the spheroidal waves (nothing more being required for this than that the successive movement of light should spread a little more quickly in one direction than in the other) and I hardly doubted that there is in this crystal such an arrangement of equal and similar particles, due to its shape and its angles of definite and invariable measure.”59

The crucial assumption that light propagates somewhat faster in one direction of the crystal is being introduced here rather incidentally, in parentheses. Huygens only vaguely suggested how this assumption in its turn could be explained mechanistically: by the disposition of the particles of the crystal. That was about all Huygens said about it, and it was quite meagre compared to the work he put in accounting for the finite and uniform speed of light. At the end of the chapter, Huygens discussed the composition of the crystal in some detail. He pictured a pile of balls and explained how it would produce a body with a specific shape. According to this line of reasoning Iceland crystal would be composed of spheroidally shaped particles. “… these little spheroids might very well contribute to forming the spheroids of the light waves assumed above; both being situated the same, and with their axis parallel.”60

Except for this suggestion, Huygens said nothing about the mechanistic explanation of his hypothesis. I figure it is quite difficult indeed to explain

58

Traité, 85. “J’ay recherché ainsi par le menu les proprietez de la refraction irreguliere de ce Cristal, pour voir si chaque phenomene, que se deduit de nostre hypothese, conviendroit avec ce qui s’observe en effet. Ce qui estant ainsi, ce n’est pas une legere preuve de la verité de nos suppositions & principes.” 59 Traité, 58. “Quant à l’autre émanation qui devoit produire la refraction irreguliere, je voulus essaier ce que feroient des ondes Elliptiques, ou pour mieux dire spheroïdes; lesquelles je supposay qu’elles s’estendoient indifferement, tant dans la matiere étherée repandue dans le cristal, que dans les particules dont il est composé; … Il me sembloit que la disposition, ou arrangement regulier de ces particles, pouvoit contribuer à former les ondes spheroïdes, (n’estant requis pour cela si non que le mouvement successif de la lumiere s’étendit un peu plus viste en un sens qu’en l’autre,) & je ne doutay presque point qu’il n’y eust dans ce cristal un tel arrangement de particules égales & semblables, à cause de sa figure & ses angles d’une mesure certaine & invariable.” 60 Traité, 96. “J’ajouteray seulement que ces petits spheroides pourroient bien contribuer à former les spheroides des ondes de lumiere, cy dessus supposez; les uns & les autres estant situez de mesme, & avec leur axes paralleles.”

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differing speeds of propagation in terms of successive impact. At any rate, in Traité de la Lumière Huygens simply avoided the problem. 5.1.3 TRAITÉ DE LA LUMIÈRE AND THE ‘PROJET’ With the solution of the problem of strange refraction the last obstacle for elaborating the ‘Projet’ was out of the way. Huygens had begun reading his treatise on dioptrics in May 1679. At the end of June 1679 the reading began of “the first part of his treatise that contains the physical causes of refraction and the phenomena of Iceland crystal.”61 Compared to the ‘Projet’ he had altered the organization of its contents. The chapters on the causes of refraction and on strange refraction had become a separate part. However, as the opening lines of Traité de la Lumière reveal, his views on the place and function of an explanatory theory had not altered: “The demonstrations that concern optics, as is the case in all sciences where geometry is applied to matter, are founded upon truths derived from experience; such as that the rays of light extend in right lines; that the angles of reflection and incidence are equal; and that in refraction the ray is broken according to the rule of sines, nowadays so wellknown, and no less certain than the preceding ones. The majority of those who have written concerning the different part of optics have contented themselves with presupposing these truths. But some more curious have wanted to investigate their origin and causes, considering them as admirable effects of nature themselves. Having advanced ingenious things in this, but not to the extent that the most intelligent would not want explications that satisfy them better, I want to propose here what I have considered on this subject, to contribute as much as I can to the clarification of that part of natural science that is not without reason reputed to be one of the most difficult.”62

Although Huygens now spoke less in disparaging fashion about those curious minds that want to know the reason of everything, he had not changed his mind about the necessity of explanations. The laws of optics were empirical laws, whose causes could be investigated, in a supplementary way, as effects of nature. However plausible, explanation does not add to their validity. In wording akin to the ‘Projet’, he continued: “In this book I will therefore try, by the principles accepted in the philosophy of today, to give clearer and more probable reasons, firstly of these properties of light directly extended; secondly of that which is reflected by the encounter with other bodies.

61

OC19, 440. “… la premiere partie de son traitté qui contient les raisons physiques de la réfraction et des phenomenes du cristal d’Islande” 62 Traité, 1-2. “Les demonstrations qui concernent l’Optique, ainsi qu’il arrive dans toutes les sciences où la Geometrie est appliquée à la matiere, sont fondées sur des veritez tirées de l’experience; telles que sont que les rayons de lumiere s’etendent en droite lignes; que les angles de reflexion & d’incidence sont egaux; & que dans les refractions le rayon est rompu suivant la regle des Sinus, desormais si connue, & qui n’est pas moins certaine que les precedentes. La pluspart de ceux qui ont écrit touchant les differentes parties de l’Optique se sont contentés de presupposer ces veritez. Mais quelques uns plus curieux en ont voulu rechercher l’origine, & les causes, les considerant elles mesmes comme des effets admirables de la Nature. En quoy ayant avancé des chose ingenieuses, mais non pas telles pourtant que les plus intelligens ne souhaittent des explications qui leur satisfassent d’avantage; je veux proposer icy ce que j’ay medité sur ce sujet, pour contribuer autant que je puis à l’éclaircissement de cette partie de la Science naturelle, qui non sans raison en est reputée une des plus difficiles.”

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Further I will explicate the symptoms of rays that, are said to, suffer refraction when passing through transparent bodies of a different kind. … Then I will examine the causes of the strange refraction of a certain crystal that is brought from Iceland.”63

The reconciliation of strange refraction with his wave theory had cleared the way to propose waves as a plausible cause of refraction. Yet, how did strange refraction fit in this scheme of experiential truths additionally explained by the principles of accepted philosophy? Huygens did not mention the properties of strange refraction among the common ‘truths derived from experience’. He did not explicate what law strange refraction was subject to. He would examine its causes – spheroidal waves as we know – but what were these to explain? In view of Huygens’ epistemological statement, strange refraction would be properly accounted for if he could give a ‘law’ of strange refraction that was empirically valid. Similar to the sine law, such a law should prescribe in exact fashion how rays are refracted in Iceland crystal. This is what Huygens had been looking for in 1672. Rejecting Bartholinus’ law, he searched for a ‘law’ of strange refraction that was general. The proposal of 1672 provided a general construction for the strangely refracted ray, but did not solve the underlying problem that strange refraction could not be reconciled with the wave explanation of ordinary refraction. In August 1677 Huygens found what he had been looking for: a general construction that could be accounted for in terms of waves. But was this construction a law of optics in the same sense as the sine law? In chapter five of Traité de la Lumière, Huygens presented an account of strange refraction that, at first sight, had been structured like the previous chapters on rectilinear propagation, reflection and refraction. In the case of refraction, he first laid down ‘the principal properties of refraction’: the sine law and the reciprocity of refraction.64 Then he went on to explain these by means of his wave theory.65 In the case of strange refraction, he also began with a description of the (mathematical) properties of strange refraction. Huygens carefully described the observable properties of strange refraction – the perpendicular is refracted, strange refraction contradicts the sine law, the unrefracted oblique ray is not parallel to the edge of the crystal. He even 63

Traité de la Lumière, 2. “J’essaieray donc dans ce livre, par des principes recues dans la Philosophie d’aujourd’huy, de donner des raisons plus claires & plus vraysemblables, premierement de ces proprietés de la lumiere directement etenduë; secondement de celle qui se reflechit par la rencontre d’autres corps. Puis j’expliqueray les symptomes des rayons qui sont dits souffrir refraction en passant par des corps diaphanes de differente espece: … Ensuite j’examineray les causes de l’étrange refraction de certain Cristal qu’on apporte d’Islande.” 64 Note by the way as was explained above on page 112 that Huygens seldom spoke of ‘laws’ of optics (and few did in the seventeenth century). In Traité de la Lumière he called the sine law the principle property of refraction, elsewhere he spoke of ‘ratio of sines’. Sometimes he used ‘laws’ to indicate the various properties of refraction. He neither called his construction for strange refraction a ‘law’. I will use the modern terminology of ‘law of refraction’ and in analogy call his construction a law. Later readers conceived of it as such. 65 Traité, 32-33.

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formulated a ‘remarkable rule’: if two rays from opposing directions are incident with equal angles, the distance between the refracted rays and the refracted perpendicular was also equal.66 This was, however, not a general law prescribing how an arbitrary ray was refracted by the crystal. Besides, it probably was a residue from the line of thinking guiding his 1672 attack on strange refraction. Unlike with the common properties of light, in the case of strange refraction he did not give an experiential law before turning to its causes. Instead he first proposed the hypothesis of spheroidal waves, and then showed how all observable properties could be derived from it. The hypothesis did yield a law of strange refraction, a general procedure to construct strangely refracted rays. It was, however, a law of a different kind than the laws of reflection and refraction. The law of Traité de la Lumière was a law of waves. His proposal of 1672 had been a law that described strange refraction in terms of rays and their components and therefore in principle an experiential law. The ellipse construction also described the behavior of strangely refracted rays, but it was based on an analysis of the propagation of waves. Moreover in its implementation it employed spheroidal waves, that is, unobservable, hypothetical entities. This ‘manner of finding the refraction of incident rays’ at the same time incorporated its probable cause. In the ellipse construction waves and rays were inseparably tied, being explanans and explanandum at the same time. In this sense, strange refraction did not fit Huygens’ scheme of providing explanations for empirically established optical laws. Huygens had found his law of strange refraction by exploring the propagation of light waves. In a way similar to Kepler’s analysis of ordinary refraction, he had figured out the mathematical properties of strangely refracted rays by liberally applying the idea of spheroidal waves. It was not a particularly new thing to speculate upon causes in optics, not even to use this in order to find new laws. Yet, by presupposing them in his eventual construction this law was not a traditional law of optics, mathematically describing the behavior of rays. By deriving the observed properties of strange refraction from it, he gave ample proof for the empirical soundness of his assumptions. Still, the ellipse construction was not a ‘truth drawn from experience’ like the sine law. It was an application of Huygens’ principle, which expressed his conception of the propagation of waves. Huygens could derive the properties of reflected and refracted rays from his principle, by assuming that the speed of propagation of these waves depended upon the medium traversed. Unlike the laws of reflection and ordinary refraction, in his ‘law’ of strange refraction he could not separate the properties of rays from the properties of waves after his derivation. Huygens did offer an abridged version of the ellipse construction, similar to the circle diagram for the sine law.67 A refracted ray was constructed by 66 67

Traité, 57. Buchwald, Rise, 316 calls it the law of proportions.

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means of a single ellipse, without secondary waves being used. The ellipse can be seen as a mere geometrical tool without a meaning in terms of waves, but Huygens called it an abridged repetition of the preceding ‘maniere’. One can read the abridged version of the ellipse construction as a purely mathematical construction in which the ellipse lacks physical meaning. Yet, the order of his presentation contradicts this interpretation. He put forward the abridged version only after his explanation of strange refraction. He did not present it as an empirically founded law applying solely to rays. The observed properties of strange refraction could be derived from the ellipse construction, but an empirically founded law of strange refraction was not among them. The solution of the problem of strange refraction thus produced an unanticipated result. The goal of his study had been to attain consistency in the causes of the various forms of refractions. The original problem of strange refraction had been the refracted perpendicular. This did not just contradict the sine law, it constituted a problem for his Pardies-like explanation of refraction. It was a problem of waves. In order for waves to be a plausible cause of ordinary refraction, the explanation ought not be contradicted by strange refraction. After his initial attempt to solve it in terms of rays, the solution had come from a reconsideration of the microphysics involved. The opening lines of chapter five of Traité de la Lumière, then, sum up what had constituted the problem of strange refraction and wherein consisted its solution: “From Iceland, …, is brought a kind of crystal, or transparent stone, very remarkable for its shape, and other qualities, but above all for its strange refractions. The causes of which seemed to me all the more worthy to be investigated curiously, as among diaphanous bodies only this one, in respect of the rays of light, does not follow the ordinary rules. I even had some necessity to make this investigation, because the refractions of this crystal seemed to overthrow our preceding explication of regular refraction; which, on the contrary, it will be seen they confirm a good deal, upon being reduced to the same principle.”68

The problem had turned into its opposite. Whereas strange refraction had first constituted a problem for his explanation of ordinary refraction, it now confirmed it. The solution did not, however, fit the original scheme anymore. There was no empirical law of strange refraction that could be reduced to the principle of wave propagation. The law that described the behavior of strangely refracted rays was a law of waves. Solving the problem of strange refraction had yielded a novelty in the mathematical study of optics. And Huygens’ principle was the key. 68

Traité, 48-49. “L’on apporte d’Islande, …, une espece de Cristal, ou pierre transparente, fort remarquable par sa figure, & autre qualitez, mais sur tout par celle de ses estranges refractions. Dont les causes m’ont semblé d’autant plus dignes d’estre curieusement recherchées, que parmy les corps diaphanes celuy cy seul, à l’egard des rayons de la lumiere, ne suit pas les regles ordinaires. J’ay mesme eu quelque necessité de faire cette recherche, parce que les refractions de ce Cristal sembloient renverser nostre explication precedente de la refraction reguliere; laquelle, au contraire, l’on verra qu’elles confirment beaucoup, apres reduites au mesme principe.”

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The ellipse construction broadened the traditional meaning of what constituted a law of optics. As contrasted to the sine law, Huygens’ law of strange refraction was not independent of the underlying conception of the nature of light. It was derived from his wave theory, as an application of his principle of wave propagation, in the same way as he had managed to derive the laws of rectilinear propagation, of reflection and refraction from his wave theory. The novelty of Traité de la Lumière went deeper than this peculiar law of strange refraction. Huygens’ principle of wave propagation constituted the means to reduce the laws of optics to one and the same ‘principle foundation’. 5.2 Comprehensible explanations In Traité de la Lumière, Huygens ‘wanted to propose’ what he had considered on the subject of the origin and causes of the laws of optics. After the passage quoted at the beginning of section 5.1.3, he continued: “I acknowledge to be much indebted to the first ones who have commenced to dispel the strange obscurity in which these matters were shrouded, and to give hope that they could be explicated by intelligible reasoning. But on the other hand, I am also amazed how the same have quite often wanted to make pass little evident arguments for very certain and demonstrative: not finding anyone who has yet explicated in a probable way these first, notable phenomena of light, namely, why it extends only along right lines, and how the visual rays, coming from an infinity of diverse places, cross without impeding each other in any way.”69

Huygens would do a better job. He would give ‘clearer and more probable reasons’ of the laws of optics than his predecessors had. By means of the ‘principles accepted in the philosophy of today’, he added. One page further down he explained what these principles were. Light consists of the movement of a certain matter. Both the origin of light – flames and such – as well as its effects – heat and burning – indicate motion, “… at least in the true philosophy, in which one comprehends the cause of all natural effects by reasons of mechanics. That is what must be done in my view, or give up all hope ever to comprehend anything in physics.”70

Huygens was going to give better explanations by means of ‘raisons de mechanique’.71 Although these words about the proper conduct in physical explanation are clear, Huygens put his opinion in a fairly general way. What did he 69

Traité, 1-2. “Je reconnois estre beaucoup redevable à ceux qui ont commencé les premiers à dissiper l’obscurité estrange ou ces choses estoient enveloppées, & à donner esperance qu’elles se pouvoient expliquer par des raisons intelligibles. Mais je m’étonne aussi d’un autre costé comment ceux là mesme, bien souvent ont voulu faire passer des raisonnements peu evidents, comme tres certains & demonstratifs: ne trouvant pas que personne ait encore expliqué probablement ces premiers, & notables phenomenes de la lumiere, sçavoir pourquoy elle ne s’étend que suivant des lignes droites, & comment les rayons visuels, venant d’une infinité de divers endroits, se croisent sans s’empêcher en rien les uns et les autres.” 70 Traité, 3. “… la vraye Philosophie, dans laquelle on conçoit la cause de tous les effets naturels par des raisons de mechanique. Ce qu’il faut faire à mon avis, ou bien renoncer à toute esperance de jamais rien comprendre dans la Physique.” 71 Since I do not want to prejudge my ensuing discussion of what ‘raisons de mechanique’ are meant by Huygens to be I here leave the phrase untranslated.

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understood by ‘raisons de mechanique’ and what did it entail to comprehend phenomena by them? In Traité de la Lumière he mentioned some others, Descartes, Hooke and Pardies, but criticized their theories in only general terms.72 The best way to get answers to questions like these is to compare Huygens approach with that of others. After all, he did not operate in an intellectual vacuum but against the rise of mechanistic philosophy in which Descartes’ trailblazing work in optics was critically assessed. This section discusses relevant texts and see how Huygens predecessors and contemporaries realized and employed the so-called true philosophy in optics, that is: how they gave shape to the mechanistic explanation of the properties of light. In this way I intend to expose the contours of Huygens’ implementation of mechanistic philosophy. The discussion of the texts of others is instrumental to this, they are judge from the point of view of Traité de la Lumière. Huygens had a very clear goal, I will argue, to acquire in mechanistic philosophy the same level of comprehensibility as in mathematical science. He did so by rigorously treating the matter the world was thought to be made of in the same way as the observable phenomena arising from it. Others were to be judged by this same standard. 5.2.1 MECHANISMS OF LIGHT Descartes was, of course, the one who had set the tone in the ‘true philosophy’. For Huygens too, he was a major point of reference, but it was not all euphony he heard. In the previous chapter we saw the difficulties with Descartes’ optics revealed in the ‘Projet’. In Traité de la Lumière, Huygens expressed his critique in no uncertain terms: “Because it has always seemed to me, and to many others with me, that even Mr. Descartes, who had the goal of treating intelligibly of all subjects of physics, and who certainly succeeded in this much better than any person before him, has said nothing that was not full of difficulties, or even inconceivable, regarding light and its properties.”73

Like the remarks in the ‘Projet’, these lines were probably aimed at the account of refraction in La Dioptrique. In the previous chapter, we have seen that as regards its mechanistic underpinnings, this account left open many questions. Descartes had put the nature of light between brackets and the physical foundations of the proposed mechanisms were only intimated. To consider the details of the ‘raisons de mechanique’ of light propagation we have to turn to the natural philosophical works in which Descartes elaborated them. 72

That is: with respect to the nature of light. He mentions Bartholinus of course with respect to his (faulty) ideas about strange refraction. In addition he discusses Rømer’s proof of the speed of light and Fermat’s principle of least time. Leibniz (aplanatic surfaces), Barrow (caustics), Newton (aplanatic surfaces and dispersion) he only mentions in the passing. 73 Traité, 6-7. “Car il m’a tousjours semblé, & à beaucoup d’autres avec moy, que mesme Mr. Des Cartes, qui a eu pour but de traitter intelligiblement de tous les sujets de Physique, & qui assurément y a beaucoup mieux reussi que personne devant luy, n’a rien dit qui ne soit plein de difficultez, ou mesme inconcevable, en ce qui est de la Lumiere & de ses proprietez.”

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Descartes presented his tendency theory of light in Le Monde (written 1630-1632, published posthumously in 1664) and Principia Philosophiae (written 1641-1644, published 1644). Light was central in his system of natural philosophy, the nature of light being ultimately connected with the essence of the cosmos. The full title of Le Monde was Le Monde ou Traité de la Lumière. Descartes envisaged a system of natural philosophy founded solely on mathematical principles.74 Quantity was the only thing to be investigated about material substance and it was subject to the laws of motion. According to Descartes, the cosmos is completely filled with matter, which is manifest in three kinds or elements. The third element, composed of the bulkiest parts, constitutes the visible objects around us, like the earth, the planets and comets. The first and finest element makes the Sun and the stars; the second consists of spherical particles and makes the heavens.75 All three elements have their share in the explanation of light: the bodies of the first element produce light, the second element makes up the medium that propagates it, and reflecting and refracting bodies are made of the third element. Descartes argued that the Sun and Stars, in rotating about their axes, exert a radial tendency (‘conatus’) upon the Heavens which instantaneously spreads outward along straight lines. This tendency is light, and Descartes explained its properties in a discussion of circular motion.76 In Traité de la Lumière Huygens did not waste his breath on the tendency theory. He only mentions it once, to reject it on the basis of just one argument: “… [Descartes] has light consist in a continual pressure, that only tends to movement. As this pressure cannot act at once from two opposing sides, against bodies that have no inclination whatsoever to approach, it is impossible to comprehend what I have just said of two persons who mutually see each others’ eyes, nor how two flambeaus can illuminate each other.”77

Huygens did not bother to criticize the mechanistic underpinnings of the tendency theory in detail. Although we can figure that Huygens would reject the ‘raisons de mechanique’ employed, he regarded instantaneous propagation as the decisive problem of the theory. It returns unremittingly whenever mention is made of Descartes. To Huygens the speed of light was necessarily finite.

74

Descartes, Principles, [76]. Descartes, Principles, [110]. 76 Descartes, Principles, [111-118]. For a detailed discussion see: Shapiro, “Light, pressure”, 243-266. 77 Traité, 20. “… Descartes, qui fait consister la lumiere dans une pression continuelle, qui ne fait que tendre au mouvement. Car cette pression ne pouvant agir tout à la fois des deux costez opposez, contre des corps qui n’ont aucune inclination à s’approcher; il est impossible de comprendre ce que je viens de dire de deux personnes qui se voyent les yeux mutuellement, ni comment deux flambeaux se puissent éclairer l’un l’autre.” 75

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“I have therefore not had any difficulty, …, in supposing that the emanation of light occurs with time, seeing that therewith all its phenomena could be explained, and that following the contrary opinion all was incomprehensible.”78

An instantaneously propagated action flew in the face of everything his Galilean understanding of motion entailed. This understanding was geometrical at the same time. As in his studies of impact and circular motion, Huygens reduced light to velocity, its speed of propagation. It is a concept that lends itself to geometrical representation in an obvious way, by means of a line segment. Instantaneity, on the other hand, is hard to picture. In addition to the incomprehensibility - in Huygens’ view - of its ‘raisons de mechanique’, there was another problem with Descartes’ optics. Where Huygens said the main objective for treating the nature of light is to explain its properties - the laws of optics - Descartes never thoroughly connected these two parts. Le Monde and Principia did not give an explanation of the laws of optics, they only treated the mechanistic nature of light. Instead, Descartes referred his readers to La Dioptrique.79 There the nature of light remained below the surface. The only more or less explicit connection between the nature and the properties of light consisted of his claim that tendency obeys the same laws as motion proper. Descartes’ derivation of the sine law was problematic, to say the least. The analogies implied that light was of a different nature than he himself proclaimed. The claim that moving balls and tendencies were compatible may have been acceptable to himself, but he did not – whether in Le Monde or in La Dioptrique – explain why and how a tendency refracted. Huygens wanted to explain the laws of optics. Le Monde and Principia were probably hardly relevant to Huygens. Descartes derived the laws of optics in La Dioptrique, and this is where Huygens found his difficulties. In my view, the argument about the unimpeded crossing of rays should be read with the sticks and wine barrels of La Dioptrique in mind, rather than the various kinds of elements of Le Monde. Although Huygens may initially have considered the mathematics of the derivation a useful way to look at strange refraction, he never accepted Descartes’ physical interpretation of it. He could not see how diagrams invoking distances could be understood in terms of an instantaneously propagating action. From 1672 on he remained completely silent on Descartes’ derivation. Whatever the merits of Descartes’ theory of light, we can fairly say that Huygens did not accept the way it should have explained the laws of optics. Although Huygens unrelentingly rejected Descartes’ theory of light, he openly acknowledged his indebtedness in a more general sense. Descartes had been the first to show how physics can be treated in a comprehensible way. That is: ‘par des raisons de mechanique’. In optics Descartes’ project 78

Traité, 6. “Je n’ay donc pas fait difficulté, …, de supposer que l’emanation de la lumiere se faisoit avec le temps, voyant que par là tous ses phenomenes se pouvoient expliquer, & qu’en suivant l’opinion contraire tout estoit incomprehensible.” 79 Descartes, Principles, [159-164].

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may not have materialized satisfactory, but Huygens would show how it ought to be done. Despite all the critique, Huygens’ theory of light shared one fundamental element with Descartes’: the idea that light must be an action propagated without transport of matter. In Descartes’ case this conception was dictated by his conception of the cosmos as a plenum. In Huygens’ case it was the only viable alternative. Emission conceptions of light - where light is thought to consist of corpuscles emitted by its source – he rejected on basic grounds: it conflicted with the unimpeded crossing of light rays.80 He did not waste more words on the matter. Wave theories - or more generally: the idea of successive motion without transport of matter - formed Huygens’ frame of reference. He knew and credited two predecessors: Hooke and Pardies. The progenitor of wave theories was Thomas Hobbes, who formulated a wave theory in direct response to La Dioptrique. It displays a sustained effort to mathematize the mechanistic nature of light. His ideas were passed on by Maignan and Barrow, to whom they are often attributed.81 Huygens was familiar with Barrow’s Lectiones - discussed in the previous chapter - but he nowhere referred to its physical parts. Although Huygens reacted to none of these wave theorists, a discussion of Hobbes’ theory is illuminating with respect to the pitfalls of corpuscular reasoning. He did respond to Hooke’s rendering of the wave conception, which enlarged the deficiencies in question. Hobbes, Hooke and the pitfalls of mechanistic philosophy: rigid waves Descartes had provided the most elaborated attempt to explain the nature and properties of light in mechanistic terms.82 Despite all its flaws, Descartes’ optics was of overriding importance for the development of seventeenthcentury thinking on light. Hobbes, a fierce defender of mechanistic thinking, was the first of many to react to the derivation of the sine law. In the course of a dispute over La Dioptrique he devised an alternative derivation of the sine law.83 It was published by Mersenne in 1644 – probably against Hobbes’ intentions – under the title of Tractatus opticus as the seventh book of Universae geometriae mixtaque mathematicae synopsis. It is rather sketchy and confusing. In the unpublished Tractatus Opticus II (1640) and in ‘A Minute or First Draught of the Optiques’ (around 1646) the theory is elaborated in more detail.84 According to Hobbes, a source of light dilates and contracts like a heart, thus producing an action that is propagated as pulses in the surrounding 80

Traité, 3. Shapiro, “Kinematic optics”, 143-145. Hobbes is regarded here as the onset of seventeenth-century continuum theories. As the paper focusses on the so-called kinematic tradition, Shapiro does not include Descartes as he did not contribute to the the concepts of waves and rays. 82 Gaukroger, Descartes, 269. 83 Prins, “Hobbes on light and refraction”, 132. 84 Stroud, First draught, 18-20. The dating of Tractus Opticus II at 1640 is derived from Horstmann, “Hobbes und das Sinusgesetz”, who also argues that the published Tractatus Opticus, commonly referred to as ‘I’, is in fact of later date. 81

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medium. He called such a pulse a ‘line of light’. A ‘line of light’ traces out a rectilinear path, and the resulting parallelogram is a ray of light. In Hobbes’ view the rectilinearity of a ray followed from the nature of this action (Figure 71): “… the luminous object acts with its entire force; for if the sides of the ray would leave obliquely from line of light AB, as AE and BF, the luminous object would not act with its entire force, but diminished in the ratio of AG to 85 AB.”

Hobbes justified this statement by drawing a comparison with a cylinder pushed with equal force at each end.86 In the ‘First draught’, this argument is Figure 71 Hobbes’ rays. used to explain that a perpendicular incident ray is not refracted. When both ends of the cylinder are not pushed with equal force, its motion can be compared to that of a cone with bases AE and BF tracing out a curved path AH, BR (Figure 72). If combined with Hobbes’ assumption that rarer media like air are less resistant to motion than denser media like glass and water, the sine law can be derived simply from this cone model. Barrow was to employ this Figure 72 Refraction. reasoning (see above, page 138). Hobbes used this comparison only to discuss refraction in qualitative terms. Hobbes’ derivation of the sine law was a formal proof based on two assumptions: a line of light has a constant width and there is a constant ratio with which a ray submerges into a refracting medium. Hobbes demonstrated that CD follows the curved path to GH when D submerges into the denser medium but C has not yet reached its surface ED (Figure 73). His demonstration rests upon the constancy of the proportion between LF (the ‘quantity of submergence’) and CQ. G is an arbitrary point on ED, and GH is found by GH = AB and MH = LF.87 Hobbes did not discuss the mechanics of a line of light passing the boundary of two media of differing resistance. In the published Tractatus opticus, Hobbes did not prove the equivalence of this derivation with his the cone model.88 A probable reason why Hobbes left out this explicit linking of his derivation and his physical model is that he claimed to be considering infinitesimally narrow rays of light. The very reason G can be drawn arbitrarily on ED lies in the fact that “… all of ED, just as AB or GH, must be 85

Shapiro, “Kinematic optics”, 151. Shapiro, “Kinematic optics”, 152; Stroud, First draught, 122-125. 87 For a detailed discussion of the demonstration see Shapiro, “Kinematic optics”, 261-262. 88 Shapiro, “Kinematic optics”, 260n410; Stroud, First draught, 126a-126m. This equivalence rests, Shapiro explains, on the equality of LF and MH. In the unpublished Tractatus Opticus II, Hobbes proved this by considering the curved path CG, DH the line of light traces in refraction. The extension of lines CD and EF intersect in N and CN : DN = CE : DF, the proportionality of the resistance of the media. Now MH = DP can be constructed and by the proportionality CG : DH = CN : DN it follows that CQ : MH = CE : FD. 86

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Figure 73 Hobbes’ derivation of the sine law.

understood as insensible.”89 With this assumed insensibility the equivalence of Hobbes’ physical rays and mathematical rays was secured. In his physical model such a transition to infinitesimal lines was problematic because the width of a line of light was crucial in this account. Despite its flaws, Hobbes’ account of refraction was an advance over Descartes in one respect. It directly invoked his understanding of the nature of light by phrasing his derivation in terms of the behavior of lines of light. Yet, a more fundamental problem emerges in Hobbes’ understanding of the nature of light. By his ‘lines of light’ he explained light and refraction in terms of bodies in motion, rather than of particles in motion. He implicitly regarded a line of light as a coherent entity. Refraction therefore involved the mechanics of a rigid body, rather than the quasi point-masses of Descartes. The problem is that no laws governing the motion of such rigid bodies were known. Although they soon became superseded, Descartes had at least worked out a set of laws to which the matter constituting light in his view was subject. Hobbes had not, he simply presumed that a rigid rod behaved like he claimed. The mechanisms he employed to explain the phenomena are speculative and qualitative because he did not provide a theoretical or empirical justification of the supposed motion. He did not come out with observational evidence in terms of macroscopic rods, nor did he produce some theory of motion to substantiate it. We see here a fundamental problem of corpuscular reasoning and quite common in the work of seventeenth-century savants. 89

Shapiro, “Kinematic optics”, 260.

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In Maignan’s and Barrow’s elaborations of the sketchy theory of Tractatus opticus, this problem only got worse. Both transformed Hobbes’ theory into an emission theory by interpreting a line of light explicitly as a moving body. I have discussed Barrow’s explanations of reflection and refraction in section 4.1.3. He did invoke a general law of motion to justify his claim that a rod ‘gyrates’ in the way he claimed, but the status of this law was unclear, to say the least. As a matter of fact, around 1680 the physical pendulum was the only rigid body whose motion was understood mathematically.90 The difference with Huygens’ wave theory is clear. His waves were not some coherent body but the effect of motions of ethereal particles.91 In the guise of a macroscopic model he advanced empirical evidence for his basic claim – that this action propagates with finite speed and does not displace the ethereal particles. Moreover, he knew the laws governing impact.92 We may infer that this is what Huygens meant with ‘raisons de mechanique’. The mechanisms assumed to be at work on the microscopic level ought to be understood on the macroscopic level. Imperceptible matter should be subject to the laws of an established science of motion. Huygens did not mention Hobbes, Barrow, or Maignan, but it is not difficult to see why he would not accept their theories. They employed the method of transduction, which extends the properties of macroscopic bodies to the unobservable motion of corpuscules, in a deficient way.93 Deficient, to wit, from the perspective of Traité de la Lumiére. On a qualitative level of everyday observation they may have thought rigid bodies to behave the way they claimed, as no laws describing these motion were available, mathematically the extension was incomplete. By reducing the propagation of light to the one property of velocity, Huygens steered clear of this pitfall. He possessed a theory of motion and impact that covered his claims about the waves propagated in ether, given that ether corpuscles were indeed hard. In his view he thus had succeeded in applying his beloved rigor of mathematics to the mechanistic nature of things. These pitfalls of corpuscular reasoning are accentuated in the theory of waves Hooke included in his Micrographia (1665). Hooke was one of the precursors Huygens mentioned in Traité de la Lumière and elsewhere he severely criticized his theory. From the point of view of the mathematician Huygens this is to be expected, for exactness was precisely the weakness in Hooke’s account. Hooke did, however, provide the most detailed and complete account of colors, experimental and theoretical, of the time, the very subject Huygens had avoided and would remain silent on. As a theoretical exposition instead of description of microscopic observations, 90

Shapiro, “Kinematic optics”, 177. Newton also understood that a wave cannot be conceived as some kind of coherent entity without smuggling in unproven assumptions. Shapiro, “Definition”, 195-196. 92 Although gaps in the transition from the behavior of single particles to waves in the sea of ether may be pointed out, as Burch has done. Burch, “Huygens’ pulse models”, 56-60. 93 For an exposition of this concept and references to the literature see Shapiro, Fits, 40-48. 91

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Hooke’s causal account of colors is somewhat the odd man out of Micrographia. The book is a sustained effort to show that the world is made up of seemingly imperceptible particles and structures and that the newly invented microscope has enabled manhood to open up these uncharted world plus ultra the visible surface of things. Hooke’s theory of waves is found in ‘Observation XI’ of Micrographia (1665). This section contains an experimental investigation of the colors produced in thin films of transparent material, like the lamina of ‘Muscovy glass’, or the space between two lenses pressed together, or soap-bubbles. According to Hooke, these colors meant a refutation of the explanation of prismatic colors Descartes had given in Les Météores (1637). The latter had argued that no colors are produced when there is no net refraction. Hooke argued that in thin films there is no net refraction, so according to Descartes’ theory no colors should be produced. Yet, observation shows they are.94 In addition he argued that according to Descartes’ own theory no colors would be produced in rain-drops.95 In both cases, Hooke gave an alternative explanation, thus demonstrating the superiority of his theory of colors over Descartes’. These explanations were based on his own pulse theory of light, according to which the short, vibrating motions of luminous objects produce pulses that propagate rectilinearly through a transparent, homogenous medium.96 When such a light pulse falls obliquely on the surface of a denser medium, the following happens.97 Adopting Descartes’ viewpoint, Hooke assumed that the pulses propagate faster in the denser medium.98 The end of the pulse that first reaches the surface will therefore come to move ahead of the other end. As a result, the pulse will become oblique to its direction of propagation, the refracted ray as found by means of the sine law. According to Hooke the preceding end of the pulse is resisted most by the medium and thus becomes weaker than its other end, whose passage has been prepared by the first. This difference accounts for the primary colors red and blue. With this ‘hypothesis’ Hooke explains the production of colors when light passes a drop of water in a succession of refraction, reflection and another refraction.99 It may be clear that in this account a pulse is necessarily a coherent whole, otherwise no difference can be made between its acute and obtuse ends. Moreover, Hooke made no effort to mathematize the mechanism presumed in the passage of the pulse to the denser medium, nor did he suggest a macroscopic phenomenon comparable to it.

94

Hooke, Micrographia, 54. Hooke, Micrographia, 59. 96 Hooke, Micrographia, 54-56. 97 Hooke, Micrographia, 56-59. 98 Compare Shapiro, “Kinematic optics”, 194-196. 99 Hooke, Micrographia, 61-62. 95

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Things became even more problematic when Hooke turned to the colors produced by thin films.100 On the basis of several experiments, he came to the conclusion that the various colors depended upon the thickness of the film.101 In his view, part of the incident light is reflected at the upper surface of the film, part at the lower surface. Consequently, two pulses following shortly upon each other are produced of which the second, having traveled a longer distance, is weaker. The amount of retardation depends upon the thickness of the layer, thus explaining the variety of colors. In this way, Hooke had formulated two different, even inconsistent theories of color. This did not keep him from a generalization: “That Blue is an impression on the Retina of an oblique and confus’d pulse of light, whose weakest part precedes, and whose strongest follows. And, that Red is an impression on the Retina of an oblique and confus’d pulse of light, whose strongest part precedes, and whose weakest follows.”102

This explanation ought to encompass both the oblique pulse (with the acute end being the weakest) and the retarded pulse (with the pulse reflected at the lower surface of the film being the weakest). It is, however, stated in most general terms and Hooke did not explicitly consider the question whether his two mechanisms explaining colors could be made consistent. The formulation rules out any possibility of mathematization the two individual theories may have had. Several other inconsistencies and obscurities in Hooke’s theory can be pointed out.103 Huygens’ verdict was merciless. The annotations in his copy of Micrographia make it clear that he did not think much of Hooke. “This must not be presumed but ought to be demonstrated, …” Huygens wrote in the margin of the page where Hooke introduced the sine law.104 He was quick to point out the sloppiness of Hooke’s reasoning, in particular his use of ‘pulses’ and ‘rays’.105 In Traité de la Lumière, he mentioned Hooke without comment, but elsewhere he made no secret of his dissatisfaction. In optics Hooke had only made ‘shameful blunders’, he wrote to Leibniz in 1694.106 If Micrographia did not come up to the standards of Traité de la Lumière, these were not, after all, Hooke’s standards. His goal was not to elaborate a mathematical, but an experimental theory of colors derived from and founded on exhaustive empirical evidence. He accepted the sine law as an empirically founded truth that needed no further mathematical or other

100

Hooke, Micrographia, 65-67. Hooke, Micrographia, 50. 102 Hooke, Micrographia, 64. 103 In his subsequent analyses of refracted pulses, of the refraction of a beam (by a drop of water), his interpretation of ‘ray’ and ‘pulse’ continuously changed, switching without notice from a microscopic point of view to a macroscopic and back. Huygens noted several gaps, and some vagueness as well: Barth, “Huygens at work”, 612-613. See also: Shapiro, “Kinematic optics”, 198-199. 104 Barth, “Huygens at work”, 612. 105 Barth, “Huygens at work”, 612 (in particular 57 II & III). 106 OC10, 612. “… bevues honteuses …” 101

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proof.107 He set great store by his experimental refutation of Descartes’ theory. As an experimental philosopher he proceeded by minutely recording observations and experiments of colors in order to infer their causes. His observations were largely qualitative. As regards the colors in thin films, Hooke admitted that he had not been able to “… determine the greatest or least thickness requisite for these effects, …” 108 He suggested that the colors in thin films were periodical in some way, without attempting to state this in more exact terms. Upon reading Micrographia both Huygens and Newton readily determined the thickness of the film and the kind of periodicity involved.109 In this way, Micrographia is typical of the experimental philosophy in which observations and explanations were qualitative and theories never rose to an exact level. 5.2.2 ‘RAISONS DE MECHANIQUE’ From the perspective of Traité de la Lumière, Pardies’ theory of waves met the standards of proper ‘raisons de mechanique’, even if it could be improved somewhat. In the first place, it employed mechanistic concepts Huygens accepted. It was based on the idea that light consists of motion without transport of matter, and crystallized in a conception of waves produced by successive collisions of ethereal particles. The basic corpuscular entity was the particle and the basic motion was impact, instead of some kind of body whose exact motions were obscure. The combination of these constituted an action governed by established mathematical laws. Waves produced by impacts of ethereal particles were, in other words, proper ‘raisons de mechanique’. Secondly, the form of Pardies’ theory met Huygens’ demands. It was cast in the form of a geometrical construction. In this way, the mechanistic consideration of refracting waves reduced to geometrical manipulation on the basis of some mathematical premises. As we have seen in section 4.2.2, this was not wholly unproblematic in Pardies’ explanation of refraction. The curve resulting from the construction had an ambiguous meaning in terms of waves. In Ango’s rendition Pardies’ waves remain entities whose existence is presupposed in the derivation of the sine law. In his own theory, Huygens rigorously defined waves as an effect of colliding ethereal particles. At all times, a wave is the resultant of the way this action has spread indifferently through a sea of disconnected particles. Waves are reduced to the one property that was central to his understanding of motion: velocity. The idea that each part of a wave is the source of a new wave, combined with the assumption that visible light is produced only where secondary waves coincide, enabled Huygens to consider speeds of propagation only. In his ‘principal foundation’ this was cast in mathematical form, thus reducing the consideration of wave propagation to geometrical 107

Hooke, Micrographia, 57. Hooke, Micrographia, 67. 109 Westfall, “Rings”, 64-65. 108

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construction.110 It improved the mechanistic conceptualization and its mathematical representation of Pardies’ theory. These two aspects of Huygens’ wave theory are two sides of the same coin: comprehensibility. To Huygens, sound mechanistic concepts were those that could be represented mathematically. In Huygens’ principle the mechanics of wave propagation was absorbed into geometry.111 However, in Traité de la Lumière Huygens did not explicate what precisely he meant with ‘raisons de mechanique’. The strict definition of a mathematized model of actions that are governed by established laws of motion remains implicit in his treatment. Against the background of the problems in Descartes’ optics of not completely integrating mathematical science and natural philosophy, Huygens got further than his contemporaries in turning the principles of mechanistic philosophy in true ‘raisons de mechanique’. In this sense I tend to disagree when Buchwald says Huygens’ principle of wave propagation does not explain the rectilinear propagation of rays or the sine law any better than Pardies/Ango. Conceptually it was clearer and it was a more successful exercise in transduction. Yet, these are subtle differences and it is indeed little surprising that the fact that only Huygens’ theory could account for strange refraction was not enough to gain him many adherents.112 Newton’s speculations on the nature of light Given Huygens’ (implicit) conception of ‘raisons de mechanique’, an emission conception of light may seem a viable alternative but this he had ruled out in advance. Only one seventeenth-century adherent took the trouble of mathematically elaborating an emission conception of light: Newton. In Principia and Opticks he presented an analysis of the dynamics of a particle passing into a denser medium. Newton understood ‘raisons de mechanique’ in a way similar to Huygens: the established laws of motion applied to unobservable particles. Before continuing my discussion of the achievements of Traité de la Lumière, I discuss Newton’s analysis of refraction in some detail. Although Newton had a similar idea of the proper ‘raisons de mechanique’, he dealt with them quite differently. Only with the greatest circumspection did he reveal his ideas on the nature of light in public. He had adopted the idea that light consists of moving particles since his earliest studies of prismatic colors.113 Yet, according to Newton this was irrelevant for his theory of colors. An experimentally founded theory explicating the behavior and properties of rays could and should be separated from speculative ideas regarding their causes and the nature of light. Different 110

Compare Shapiro, “Kinematic optics”, 208. Shapiro (“Kinematic optics”, 244) puts it as follows: “The key to Huygens’ success in optics was his continual ability to rise above mechanism and to treat the continuum theory of light purely kinematically and, thereby, mathematically.” 112 Buchwald, Rise, 5. 113 Hall, Unpublished, 403; Newton, Certain, 432-435; Westfall, Never at rest, 159-163, 170-172. 111

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refrangibility was an observational property derived from experiment. In his paper on colors of 1672 he remained silent on the mechanistic nature of light and colors, and in his lectures on optics he likewise kept his mouth shut. In a paper Newton sent Oldenburg a few years later, on 7 December 1675, he disclosed his views in public for the first time. “An Hypothesis explaining the Properties of Light discoursed of in my severall Papers” it was called. It gave qualitative explanations for reflection, refraction and the diversity of colors. Twelve years later, in Principia, Newton mathematized the explanation of the sine law sketched in the ‘Hypothesis’. However, he did not present it explicitly as a derivation of the sine law. Section XIV of Book I discussed “The motion of minimally small bodies that are acted on by centripetal forces tending towards each of the individual parts of some great body”.114 Only in a scholium after the exposition did Newton point out the resemblance of the results here obtained with the behavior of light rays. He added cautiously: “Therefor because of the analogy that exists between the propagation of rays of light and the motion of bodies, I have decided to subjoin the following propositions for optical uses, meanwhile not arguing at all about the nature of rays (that is, whether they are bodies or not), but only determining the trajectories of bodies, which are very similar to the trajectories of rays.”115

In proposition 94, Newton pictured (Figure 74) a space AabB between two similar media, bounded by parallel planes, through which a body passes that is attracted or impelled perpendicularly towards either of those media and showed “… that the sine of the angle of incidence onto either plane will be to the sine of the angle of emergence from the other plane in a given ratio.”116 A body moves along GH and is attracted upwards between Aa and Bb. During its passage it follows the curve HI, then leaves the layer along IK. The curve is constructed by producing GH to M and IK to L, drawing IM perpendicular to Bb, and a semi-circle PNIQ with center L and radius LI. Newton then showed that when the attraction is uniform, HI will be part of a Figure 74 Refraction in Principia. parabola with the following 114

Newton, Principia, 622. Newton, Principia, 622- 626. The following propositions were on anaclastics. 116 Newton, Principia, 622. 115

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properties: HM2 = MI· p (where p is the given latus rectum of the parabola), HM bisected in L and IR = MN. Consequently “… the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given.”117 Next, he showed that if AabB is divided into several parallel planes the same holds for the angle of incidence on the first plane and the angle of emergence from the last plane. Newton did not explain the physical meaning of parameter p. Bechler has shown that it depends upon the properties of the material in the layer AabB and is constant for all angles of incidence. Furthermore, he argues convincingly that this leads to seeing the ratio of sines – that is, the index of refraction – as dependent upon the initial velocity of the body. So for the sine law to hold, all incident rays must have the same velocity.118 It may be clear that this derivation of the sine law employed proper ‘raisons de mechanique’ as Huygens understood them implicitly. In the preceding sections of Principia Newton had established his theory of accelerated motion which he now applied to ‘very small bodies’. It goes without saying that Huygens’ mechanistic vocabulary lacked the forces employed (see chapter six). In Book 1, Part 1 of Opticks, Newton presented an adjusted and abbreviated version of the demonstration in Principia. He carefully eliminated all traces of unobservable particles and couched the derivation solely in terms of rays and their properties. He did not explicitly say that the ray is curved at the refracting surface and referred to an ordinary ray diagram, considering the parallel and perpendicular components of the ray’s motion. Although he said the sine law to be adequately founded empirically – which had been sufficient in his optical lectures of 1670 – he now added a demonstration in order to show that it is ‘accurately true’.119 He was, after all, a mathematician rather than a mere experimentalist. He did so by the supposition “That Bodies refract Light by acting upon its Rays in Lines perpendicular to their Surfaces.”120 As Sabra has shown, the supposition comes down to saying that only the perpendicular component of the rays is affected and its parallel component remains constant.121 This means that Newton adopted the mathematical assumptions of Descartes’ demonstration of the sine law, while giving them a new physical interpretation. Newton did not, like Descartes, simply assume the second assumption – the velocities in the respective media are in constant proportion, and larger in denser media. He derived it by stating, without proof, a consequence of the demonstration in Principia: the perpendicular velocity at emergence is equal to the square root of the sum of the square of the perpendicular 117

Newton, Principia, 623. Bechler, “Newton’s search”, 16-17. 119 Newton, Opticks, 79. Newton, Optical papers 1, 169-171; 311-313. 120 Newton, Opticks, 79. 121 Sabra, Theories, 300-301. 118

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velocity at incidence and of the square of the perpendicular velocity at emergence of a ray at grazing incidence (Figure 75).122 The following relationship holds between rays MCN and ACE: CF = ¥(DC2 + CG2), where CG is a constant determined by the medium. Combined with AD = DH, as the diagram should be read, the proposition yields the sine law of refraction.

199

Figure 75 The sine law in Opticks.

“And this Demonstration being general, without determining what Light is, or by what kind of Force it is refracted, or assuming any thing farther than that the refracting Body acts upon Rays in Lines perpendicular to its Surface; I take it to be a very convincing Argument of the full truth of this Proposition.”123

Nevertheless, assumptions on the level of unobservable entities are implicit: AC and MC have equal velocity and the constancy of the index of refraction depends upon the velocity of the incident ray.124 Bechler argues that Newton chose to use the confusing one-circle diagram – instead of a diagram representing the relative ‘velocities’ in both media – precisely to obscure the meaning and physical implications of the fact that MC : NG is given. In this way, Newton could refer to the components of the rays without explicitly referring to the velocities. There may have been an extra reason for Newton to obscure the central role velocity played in his derivation of the sine law, besides his effort to convey the demonstration in terms of experimentally founded entities. The derivation had unfortunate consequences for his understanding of colors, the very ‘raison d’être’ of Newton’s optics. The derivation implied that the index of refraction depends upon the velocity of rays. In this way, different refrangibility might be identified with the different velocities of rays of various colors. This in turn might lead to the derivation of a law of dispersion correlating the refractive indices of various colors. In fact, Newton had already formulated the law that would result from such an extension of the sine law. In his optical lectures of the 1670s he had laid down a law of dispersion without proof – be it experimental or

122

Newton, Opticks, 79-80. Newton, Opticks, 81-82. 124 Bechler, “Newton’s search”, 28-31. 123

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mathematical.125 It assumed a constant difference between the parallel components of the colored rays. The law can be seen as a natural extension to dispersion of Descartes’ derivation of the sine law, identifying the variety of colors with various sizes of the parallel component.126 In Opticks this ‘Cartesian’ dispersion law has disappeared. Supposedly, Newton had realized that his dispersion law implied that color depended upon velocity. This he could not accept, as the immutability of colors was the core of his theory. As velocity changes in even the most elementary mechanisms, it seemed a unlikely candidate for an original and conservable property of light rays.127 Therefore, the ‘Cartesian’ dispersion law of the optical lectures was unacceptable.128 This left a model for dispersion based on size or mass, but Newton never articulated a mechanism through which different refrangibility might be explained this way. What is more, the only mechanism he elaborated for refraction – based on perpendicular forces acting upon particles – was at odds with such a model, for acceleration is independent of mass. In Opticks, Newton put forward an alternative law of dispersion with dubious empirical evidence and whose mechanistic causes he had never elaborated – not even in private.129 The status of ‘raisons de mechanique’ In the seclusion of his private quarters, Newton allowed himself a far greater liberty of reasoning than in his publications. From the very start, his experimental inquiries had been accompanied by speculations on the corpuscular nature of light and colors. Shapiro has analyzed the way in which Newton employed his vibration model to develop his theory of periodicity of colors.130 The derivation of the sine law shows that Newton was on a par with Huygens as regards the mathematization of mechanistic causes. He employed the method of transduction with a comparable meticulousness mathematizing the physics of unobservable particles by founding them upon the established laws of motion. Unfortunately, Newton’s consideration of ‘raison de mechanique’ turned problematic because it produced discrepancies that left considerable gaps in his mathematical science of colors. It did not affect his experimental theory of color, though. Although he did not have an exact law of dispersion, the experimentally disclosed and secured theory of different refrangibility stood unshaken. 125

Newton, Optical papers 1, 199 & 335-337. See Shapiro, “Dispersion law”, 99-104 & 126-127; Bechler, “Newton’s search”, 4-5. I discuss this matter in more detail in my “Once Snel breaks down”. 127 Bechler, “Newton’s search”, 32-33. 128 In 1691, Newton figured out a test for the assumption that color differs with velocity: when a moon of Jupiter disappears behind the planet the slowest color – red – should be seen last. In February 1692, Flamsteed reported that such a difference could not be observed. This empirical evidence definitely ruled out velocity. Shapiro, Fits, 144-146. 129 Newton, Opticks, 128-130. Shapiro, “Dispersion law”, 97-99; 126-127. Shapiro suggests that it might be based upon a small angle approximation of refracting angles. 130 Shapiro, Fits, 200-201. 126

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If Newton and Huygens had comparable opinions about the nature of mechanistic principles, the took opposite positions regarding their status and their relationship to the laws of optics. The way Huygens derived and presented his law of strange refraction would have been unacceptable for Newton, who had gone to such great lengths to erase all traces of speculation from his experimentally established theory of light and colors. Huygens deliberately used hypotheses to explain the observed properties of light. True, this implied speculation, but to do so was inevitable when it came to explanations. “One will see here demonstrations of the kind as not to produce a certainty as great as those of geometry, and that even differ much from it, for whereas the geometers prove their propositions by secure and incontestable principles, here the principles are verified by the conclusions drawn from them; as the nature of these matters does not allow that this is done otherwise.”131

The wave theory belonged to ‘la Physique’, offering a plausible explanation of the empirical truths on which ‘l’Optique’ was founded. Yet, explanation was not arbitrary. In order to be plausible an explanation ought to employ proper ‘raisons de mechanique’. His principle of wave propagation satisfied this demand. But still, it was probable at best. It derived no proof value from being plausible. Huygens did not claim a priori truths. Consequently, the burden of proof for the wave theory did not rest with its merits as a mechanistic theory. The raison d’être of Huygens’ principle was that ‘all properties of light, and everything pertaining to its reflexion and its refraction, can be explained principally by this means’.132 His ‘principal foundation’ should not be judged at face value but on its adequacy to derive the laws of optics. Pardies’ theory had the same focus on explaining the laws of optics. Huygens pointed out the advantage of his principle of wave propagation precisely in this context. It filled a gap in Pardies’ demonstrations of the laws of reflection and refraction. What this gap was, and why his own demonstrations were better, Huygens did not explicate. He had found a better ‘foundation’ for deriving the laws of optics. Huygens was rather awkward in spelling out the mechanics of wave propagation in full and did not always – as in the case of strange refraction – elaborate it in full detail. His focus was on his principle of wave propagation. First of all, he assured himself that all properties of reflection and refraction could be reduced to it. His wave theory had developed accordingly. In his notes on caustics and the explanation of strange refraction, Huygens satisfied himself that he could construct a propagated wave in the troublesome situation of caustics, and a refracted ray in the equally troublesome situation of strange refraction. 131

Traité, “Preface”, [2-3]. “On y verra de ces sortes de demonstrations, qui ne produisent pas une certitude aussi grande que celles de Geometrie, & qui mesme en different beaucoup, puisque au lieu que les Geometres prouvent leurs Propositions par des Principes certains & incontestables, icy les Principes se verifient par les conclusions qu’on en tire; la nature de ces choses ne souffrant pas que cela se fasse autrement.” 132 See the passage quoted above on page 174.

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With the full elaboration of his theory in Traité de la Lumière, Huygens showed that with his principle propagated waves could be constructed in any situation. Only the speed of propagation, as it depended upon the medium traversed, needed to be varied. In this way he derived all observable properties of light rays from one and the same principle in a mutually consistent way. This reduction was what he understood by explanation. Reducing the properties of light rays to Huygens’ principle was explaining these properties mechanistically, because the principle explicated the essentials of successive impact in ethereal particles. The validity of this ‘principal foundation’ rested upon the fact that the laws of optics could be reduced to it. In other words, it did not rest upon the appropriateness of ‘raisons de mechanique’, but on the plausibility of mathematical inference. In Huygens’ wave theory three levels can be distinguished: a mechanistic model of colliding particles, the laws of optics and – in between – Huygens’ principle. As the mathematical representation of the mechanistic nature of wave propagation, Huygens’ principle serves as a intermediary of a special kind between the nature of light and the laws of optics. It was the indispensable link between Huygens’ mechanistic picture of collisions of ethereal particles and the mathematical laws of light rays. In the light of seventeenth-century geometrical optics, where the laws of optics functioned as the postulates or principles of mathematical science, Huygens’ principle of wave propagation can be called a law of optics. Not in the modern sense of a law of nature in physical science, but in a then traditional sense. Remember that the sine law and the like were rarely called laws then, but rules, measures or properties. In the mathematical science of optics Huygens had disclosed a new law, a more fundamental one to which the various properties of light propagation were subordinated to. However, this new ‘law’ was of an entirely different nature than the traditional principles of optics. Huygens’ principle did not describe the behavior of rays but the behavior of waves; it was a mathematical law describing the behavior of unobservable entities. Comprehended in this way, Huygens’ principle was a novel element in the mathematical science of optics. Huygens’ principle not only unified ordinary and strange refraction, it unified all properties of light rays. It was a more general law and a law of different character at the same time, describing the behavior of unobservable waves mathematically. One might say that Huygens had brought geometrical optics to a new level, that of microphysics. He focused on the geometrical constructions with his principle and did not spell out its mechanistic underpinning. In Traité de la Lumière, waves have taken the place of rays. Waves are entities with well-defined mathematical properties, the causes of which are explained rather informally, like in Barrow’s elucidations. Huygens switched to the mathematical consideration of waves in a matter-of-course way. He applied geometry to these unobservable entities with the same ease as he applied it to observable balls and pendulums. In his wave theory he extended Galileo’s mathematical physics of observables to that of unobservables. As one

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applied geometry to matter in optics or any other branch of mathematics, one could apply it to the ether. To Huygens, it went without saying that the ‘vraye Philosophie’ meant applying the ‘raisons de mechanique’ to unobservables just as one applied them to pendulums. He gave no sign of an awareness of the possibility that mathematizing the ether might raise philosophical or ontological problems. The only difference between applying geometry to observable matter and applying it to ethereal particles was the level of certainty that could be attained. The nature of these things does not allow other than speculations, but – so the text quoted on page 201 continued: “It is still possible in this to arrive at a degree of probability that quite often yields hardly to full evidence. This is so when the things that are demonstrated by the supposed principles correspond perfectly with the phenomena that experience has drawn attention to; particularly so when there is a great number of them, and moreover principally when one conceives and foresees new phenomena that must follow from the hypotheses one employs, and when one finds that in this the effect answers our expectation. When all these proofs of probability converge in what I have designed to treat of, as it seems to me they are, this must be quite a great confirmation of the success of my research, and it is only with difficulty possible that things would not be more or less as I represent them.”133.

The wave theory occupied the twilight zone of probability that lies between the truth of the laws of optics and the arbitrariness of mere speculations. This was an area where Huygens, unlike Newton, dared to tread because in his own view he could still cling here to the reliability of mathematical inference. The reduction of the laws of optics to the principle of wave propagation did not add to their validity, it only proved the probability of the explanation. These laws were empirically founded, as contrasted to the principle of wave propagation that described unobservables. His principle of wave propagation was probable at most, precisely because it could not be demonstrated by direct observation. By means of sound and mutually consistent derivations, Huygens demonstrated that his principle was more plausible and more probable than other explanatory theories. This is, of course, what we call hypothetico-deductive inference. The explanation of strange refraction had been decisive. It singled out Huygens’ theory because it was the only one that accounted for the phenomenon in a way that could be reconciled with the other properties of light rays. In Huygens’ view, this matter had been settled. But it would not be that easy. This had everything to do with the particular nature of the ellipse construction. As we have seen in the previous section, it differed essentially 133

Traité, “Preface”,[3]. “Il est possible toutefois d’y arriver à un degré de vraisemblance, qui bien souvent ne cede guere à une evidence entiere. Sçavoir lors que les choses, qu’on a demontrées par ces Principes supposez, se raportent parfaitement aux phenomenes que l’experience a fait remarquer; sur tout quand il y en a grand nombre, & encore principalement quand on se forme & prevoit des phenomenes nouveaux, qui doivent suivre des hypotheses qu’on employe, & qu’on trouve qu’en cela l’effet repond à nostre attente. Que si toutes ces preuves de la vraisemblance se rencontrent dans ce que je me suis proposé de traiter, comme il me semble qu’elles sont, ce doit estre une bien grande confirmation de succês de ma recherche, & il se peut malaisement que les choses ne soient à peu pres comme je les represente.”

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from the established laws of optics. It was no ‘truth drawn from experience’, for it mixed the properties of rays with the properties of waves. The ellipse construction was inextricably bound up with the principle of wave propagation. 5.3 A second EUPHKA With his theory explaining the established laws of refraction as well as the strange refraction of Iceland crystal, such as we have set it forth in the preceding section, Huygens addressed the Académie in the summer of 1679, nearly two years after the EUPHKA of 6 August 1677. We can imagine his expectations. He would present to his colleagues a truly mechanistic explanation of the properties of light, firmly founded upon the laws of motion. Only with his principle could the laws of optics be derived in a sound and coherent way. In addition, he would present a wonderful confirmation by explaining the baffling phenomenon of strange refraction with it. Things could hardly be otherwise. His theory of waves was the only comprehensible explanation conceivable. It would show what rigorous thinking could yield. Thinking that was not easily satisfied, but aimed at rendering matters intelligible without compromise. At least one member of the Académie was not convinced immediately. It was Rømer, the same who in 1677 had provided Huygens with observational proof of the finite speed of light. Rømer’s intervention forced Huygens out of the safe domain of rational analysis, where the properties of light are derived from clear and distinct concepts by means of rigorous deduction, to the empirical domain of tinkering with the unpolished reality of measurement and experimentation. Huygens managed to counter Rømer’s objections by measurements acquired by a precise and powerful observing technique and a ingenious experiment that reveals a remarkable command. Remarkably, as up to this point Huygens had repeatedly steered clear of empirical grounds. These measurements of 1679 provided the data of the eventual Traité de la Lumière which, in other words, date from Huygens’ third go at strange refraction. In a letter of 11 November 1677 Huygens had informed Rømer of a letter he had written to Colbert on October 14.134 He had praised Rømer’s “belle invention”, and now added that he had always assumed the same in order to explain the properties of light.135 He added further that his hypothesis to explain strange refraction was so simple and so accurate and agreed so well with observation that he did not doubt that everyone would accept it.136 Replying on December 3, Rømer expressed the opinion that optical principles that could not account for strange refraction were useless.137 He was curious after Huygens’ ideas and added that he himself had also done 134

OC8, 41. OC8, 36-37. This letter is quoted on page 161. 136 OC8, 41. 137 OC8, 45. 135

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some thinking on refraction, in particular on Descartes’ account of it. He threw doubts of the validity of Descartes’ proof. It was unclear to him what kind of impact Descartes had in mind when comparing light rays with balls struck into water. And, according to him, contrary conclusions could be derived from Descartes’ assumptions. Rømer himself had read a paper to the Académie in which he rejected the assumption that the speed of light is larger in denser mediums and derived the sine law in a way similar to Fermat’s derivation.138 Huygens’ letter had made him consider strange refraction and he had promising ideas, he said. But without a piece of crystal and precise data he could not pursue his thinking in a satisfactory manner.139 5.3.1 DANISH OBJECTIONS Rømer had to wait for one and a half year before hearing the details of Huygens’ explanation. And when the time came, in the summer of 1679, he raised serious objections. Huygens was to recall what happened when he sent him a copy of Traité de la Lumière in 1690.140 From this letter it is also clear that some notes Huygens wrote in July and August 1679 were directed at Rømer’s objections. From all this we can infer that Rømer had advanced the theory of Bartholinus – his father in law – as a viable alternative to Huygens’ ellipse construction.141 Bartholinus had argued that strange refraction is governed by an ‘oblique perpendicular’ that is parallel to the edge of the crystal. Evidence for this he found in the fact that the unrefracted oblique ray is parallel to the edge of the crystal. Therefore, so Bartholinus had concluded, the unrefracted oblique ray must have a function similar to the perpendicular ray in ordinary refraction. According to Bartholinus’ ‘oblique’ sine law, the sines of incident and strangely refracted rays are in constant proportion when measured with respect to the unrefracted oblique ray. Bartholinus had suggested that pores in the crystal could explain the unrefracted passage of the ray parallel to the edge of the crystal. According to the registers of the Académie, Huygens read from his ‘Dioptrique’ on 1 July 1679.142 This may well have been the session at which Rømer pointed out that Bartholinus’ explanation was equally plausible and had not been refuted by Huygens’ explanation. On 3 July, Huygens in his turn could refute a central assumption of Bartholinus’ explanation: “Observation made on 3 July 1679. which proves manifestly that it is not the ray parallel to the sides of the crystal that passes without refraction as I thought until now.”143

138

Cohen, “Roemer”, 344. OC8, 45-46. 140 OC9, 489. No direct evidence from the late 1670s of Rømer’s objections is available. 141 As Ziggelaar also assumes; “How”, 185. 142 OC19, 440n2. 143 OC19, 440. “Observation faite le 3 juillet 1679. qui prouve manifestement que ce n’est pas le rayon parallele aux costez du cristal qui passe sans refraction comme j’avois creu jusqu’icy.” 139

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Huygens described a simple but precise method for comparing the angle of the edge of the crystal and the angle of the unrefracted oblique ray (Figure 76).144 The observation technique is reminiscent of the one employed by Bartholinus to determine the index of (ordinary) refraction.145 The manuscripts suggest that Huygens went back to his earliest notes on strange refraction for Figure 76 The new measurement. the observations are recorded on a remaining part of the last page of the 1672 investigation.146 On the upper surface of the crystal he marked a point D and determined the point B on the opposite surface so that BD is parallel to the edges AH and KL. Then he positioned a ruler EF under the crystal, perpendicular to the diameter AC and through point B. On the diameter AC he then marked the place were B is seen and found that this was not D. The angle between the edge of the crystal and the refracted oblique ray was about 2½ degrees. On the next session of the Académie – 8, 15 and 22 July – Huygens continued to read his ‘Dioptrique’ and presumably presented this new result. Huygens’ observation still did not suffice to counter the Dane’s objections. Rømer was not convinced that the ‘oblique’ sine law had been refuted. Apparently, he had referred to Bartholinus’ suggestions about refraction in surfaces of non-natural sections of the crystal. According to Bartholinus, strange refraction was related to the orientation of the refracting surface with respect to the crystal. He had predicted that ordinary and strange refraction would swap place – so that the ‘fixed’ image becomes mobile and vice versa – in alternative, non-natural sections of the crystal. On 6 August 1679, Huygens found out that this was not the case. He figured the crystal cut along plane MN, which makes an angle of 45º20' with the axis of the spheroid governing strange refraction (Figure 77). According to the ellipse construction, this plane should produce the same refractions as the natural plane gG. The verification of this Figure 77 The EUPHKA of August 1679. assumption was troublesome because it was not easy to cut and polish the crystal, but Huygens claimed to 144

See the assessment of Buchwald, Rise, 312-313. Bartholinus, Experimenta, 34-41 (experimentum XVII). 146 Hug2, 178r. 145

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have succeeded: “In this way I have made section MNO, and I have found that the surfaces it makes have the same refractions as the surface gG, …”147 After considering various ways of cutting the crystal, he concluded: “It appears that it is not the disposition of the layers of the crystal that contributes to the irregular refraction.”148 Bartholin’s explanation and Rømer’s objections thus lacked a foundation. A second EUPHKA followed: “EUPHKA. The confirmation of my theory of light and of refractions.”149

Huygens had proven that his was the only acceptable explanation of strange refraction. Unexpectedly, Rømer had made it clear that Bartholinus’ law could not be dismissed forthwith and that Huygens should produce decisive evidence against it. In doing so, Huygens showed that his was the only theory that could also explain strange refraction. Ergo, to take Rømer by his own words, his principle of wave propagation was the only useful principle in optics. On 12 August he continued the reading of his ‘Dioptrique’ at the Académie. Forced innovation In the summer of 1679 Huygens showed himself an able measurer and inventive experimenter. In a two-stage reaction to Rømer’s objections he refuted Bartholinus’ law and confirmed his own ellipse construction. The measurement and the experiment added a new, empirical element to his study of strange refraction. It is remarkable that Huygens had not questioned Bartholinus’ data previously. In 1672 he had improved Bartholinus’ measurements of the angles of the crystal by means of a more reliable technique.150 Yet, apart from the angle of the refracted perpendicular ray – which Bartholinus had not provided – he had not measured any angle of refraction. He had never measured the unrefracted oblique ray or any other rays. That he developed the technique to measure the refraction of a ray only in 1679 appears from the fact that this section of Traité de la Lumière was inserted into the original manuscript.151 Until that time his theory had developed in an empirical void. He discovered the ‘law’ of strange refraction by mathematical reasoning, not from precise observations as Buchwald concluded from his study of Traité de la Lumière.152 The empirical solidity of the finalized theory was acquired only at the third stage of Huygens’ studies of strange refraction, when he was forced by Rømer’s objections to take a closer look. 147

OC19, 442. “De cette maniere j’ay fait la section MNO, et j’ay trouvè que les surfaces qu’elle a faites avoient les mesmes refractions que la surface gG, …” On 3 November 1679 he wrote his brother: “I have found means to grind and polish this crystal which was thought impossible, …” OC8, 241. “J’ay trouvè moyen de tailler et de polir ce cristal ce qu’on croioit impossible,…” 148 OC19, 443. “Il paroit que ce n’est point la disposition des feuilles du cristal qui contribue a la refraction irreguliere.” 149 OC19, 441. “EUPHKA. La confirmation de ma theorie de la lumiere et des refractions.” 150 Buchwald, “Experimental investigations”, 313-314. 151 OC19, “Avertissement”, 385. 152 Buchwald, “Experimental investigations”, 313 & 316-317 and Buchwald, Rise, 313.

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The Eureka of 6 August 1679 provided an additional confirmation of Huygens’ explanation of strange refraction. It was presented in Traité de la Lumière accordingly. Having derived in detail the properties of strange refraction from his ellipse construction, he concluded: “This being so, it is not a light proof of the truth of our suppositions and principles. But what I am going to add here confirms them marvelously once more. These are the different cuts of this crystal, of which the surfaces produced by them bring about refractions precisely such as they must be and as I have foreseen them, following the preceding theory.”153

Huygens inserted the results of his measurements and the experiment in the original manuscript. He had to adjust his earlier ellipse construction, as it had originally employed Bartholinus’ observation that the unrefracted oblique ray runs parallel to the edge of the crystal. This did not, as we have seen, affect the ellipse construction as such. At the beginning of the chapter, when he gave the angle of the unrefracted oblique ray, he had warned his readers: “This is to be noted, so that one does not search in vain the cause of the singular property of this ray, in its parallelism to said sides.”154 He did not bother to tell his readers that he had long assumed the same. Rømer’s objections had not shaken Huygens’ confidence in the validity of the ellipse construction and in his wave theory as a whole. He could reduce all properties of light to a single principle by assuming only that the speed of propagation varied with a specific medium. The explanation of strange refraction singled out his theory, as it was the only one that could account for the phenomenon in a consistent way. He had dismissed Bartholinus’ law without further notice right in 1672 and never mentioned it. Only when Rømer pointed out that it was a viable alternative did he put some work into refuting it. He decided upon a remarkable way to counter Rømer’s objections. What makes it remarkable is that his study of strange refraction had consisted of rational analysis. He could have argued that Bartholinus’ law was not general because it applied only to rays in the principal section. He could have questioned the effect of the suggested pores – why do they affect rays only partially? Huygens did not take this line of approach. Instead, he called upon nature to refute Bartholinus’ law. He did so in a particular way. The improved measurement of the unrefracted oblique ray undermined the logic of Bartholinus’ law. If the unrefracted ray was not parallel to the edge of the crystal, its connection with the pores of the crystal became dubious. The result did not take away Rømer’s objections, though. The ‘oblique perpendicular’ governing strange refraction need not be parallel to the edge of the crystal for an ‘oblique’ sine law to be plausible. It might well be that in strange refraction the sines 153

Traité, 85. “Ce qui estant ainsi, ce n’est pas une legere preuve de la verité de nos suppositions & principes. Mais ce que je vais adjouter icy les confirme encore merveilleusement. Ce sont les coupes differentes de ce Cristal, dont les surfaces, qu’elles produisent, font naistre des refractions precisement telles qu’elles doivent estre, & que je les avois prevuës, suivant la Theorie precedente.” 154 Traité, 57. “Ce qui est à noter, afin qu’on ne cherche pas en vain la cause de le proprieté singuliere de ce rayon, dans son parallelisme ausdits costez.”

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should be measured with respect to an arbitrary line, empirically determined. So, even after the measurement, Bartolinus’ law remained a viable alternative to Huygens’ ellipse construction. It was an exact law, that could account for the basic observable properties of strange refraction. Huygens had to find a more substantial refutation. He succeeded, as the experiment of 6 August 1679 went right to the heart of Bartholinus’ law. He could have challenged its empirical accuracy directly, by offering a set of measurements that refuted the constancy of sines. Instead, he devised an experiment of a special kind. He set Bartholinus’ law and his ellipse construction side by side and compared them on their merits as exact laws. He thought up a situation in which the two constructions yielded opposite results and let nature decide. Compared to the way Huygens rejected other theories the refutation of Bartholinus’ law takes a special place in Traité de la Lumière. He refuted Descartes’ theory and emission conceptions by means of straightforward, qualitative observations. Bartholinus’ law was now refuted by means of a crucial experiment of the sort Hooke had employed to refute Descartes’ theory of colors. The colors of Muscovy glass, however, were a qualitative observation and in this sense they decided between Descartes’ and Hooke’s theories. Huygens’ experiment, on the other hand, produced a situation in which exact predictions of two mathematical laws were compared. This was a new kind of experiment, a ‘mathematized’ crucial experiment and Huygens had been forced into this innovation by Rømer’s unanticipated objections. Huygens’ experiment undeniably refuted Bartholinus’ law in favor of the ellipse construction. That law was inherently connected with the particular shape of the crystal and the measurement and the experiment were aimed at undermining this connection. First, by casting doubts on the relationship between the angles of the crystal and the line governing strange refraction. Then, by questioning the relationship between strange refraction and the orientation of the refracting surface. Huygens, too, explained strange refraction by properties of the crystal. In his case, however, it was a property of the material the crystal was made of instead of its (macroscopic) shape. Strange refraction was caused by a property of the medium (affecting the speed of propagation) just as any optical phenomenon was caused by various properties of the media. The experiment did not so much confirm the ellipse construction, it confirmed Huygens’ explanation of strange refraction by spheroidal waves and thereby his wave theory as a whole. 5.3.2 HYPOTHESES AND DEDUCTIONS The experiment of 1679 made the hypothetico-deductive structure of Huygens’ wave theory manifest. Prior to it, the evidence for Huygens’ principle consisted of the reduction of the common properties of light to one and the same principle. In addition, the successful explanation of strange refraction by the same means reinforced its probability. It singled out his principle of wave propagation as the only useful – as Rømer would put it –

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principle in optics. Still, Huygens’ waves were hypothetical entities. Unlike Descartes, Huygens did not intend to prove the laws of optics by means of his theory of waves. It was the other way around: the verities of optics proved the probability of the way he imagined waves to propagate. The laws of optics were the ultimate foundation of Huygens’ theory of light. However, the ‘law’ of strange refraction did not really fit this scheme as it mixed up waves and rays and was not an empirical truth. Indirectly, via the successful derivation of the ellipse construction, Huygens’ principle was founded upon singular – but important – observations of strange refraction. Rømer’s objections made it clear that the ellipse construction was not the only law that was consistent with those observations. In order to counter these objections, Huygens chose to employ the keystone of hypothetico-deductive inference: experimental verification. Although he could have refuted Bartholinus’ law otherwise, Huygens went to the heart of the matter. He devised an experiment with the suggested relationship between the crystal and strange refraction in mind. An unnatural section of the crystal would reveal whether a law of strange refraction should be related to the shape and structure of the crystal or to its material. He put the very foundation of his wave theory at stake: waves are defined by their speed of propagation, which depends solely on the medium traversed. This could not be verified directly, but only by comparing consequences drawn from the alternatives. The crux of Huygens’ employment of hypotheticodeductive inference was that he had the laws predict what would happen. He derived exact predictions to be put to the test. The drawing accompanying the experiment can be regarded as the essence of Traité de la Lumière: a wave with respect to an unnatural section. The mathematical representation of the mechanistic nature of light is here being experimentally verified. The Eureka of 6 August 1679 was the ultimate consequence of Huygens’ mathematico-mechanistic thinking. Unexpectedly drawn into the problem of the nature of light, our dioptrical geometer had set up a search for the mechanistic causes of the properties of light. He had found waves caused by collisions of ethereal particles and fitted out with mathematically defined properties. Huygens’ principle was the plausible cause he needed, a law of waves. It was a new kind of law, unifying the observable properties of light rays by reference to unobservable waves. It also was a hypothetical law, as it was not drawn from experience. The ellipse construction derived from it was likewise hypothetical, although less explicitly so. It described strangely refracted rays while presupposing spheroidal waves. When forced to test it, Huygens chose to put to test this assumption of a medium-dependent propagation of waves. The experiment was not a necessary step, but it was the obvious choice. Waves were not just a plausible cause of the properties of light, ultimately they were their true cause. Things could not reasonably be otherwise than Huygens imagined. Therefore one could deduce phenomena from this hypothesis, which experiment should show to be real.

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Huygens’ waves, while hypothetical and probable, were nonetheless thought to be real things. This marked his explanations off from the analogies invoked to elucidate the laws of optics. The balls and swords of Alhacen clarified the assumptions of his mathematical account of reflection and refraction, but did not prove them. Light was compared to cleaving swords and bouncing balls only with respect to its ability to be deflected, not because its nature was sword- or ball-like. Descartes’ moving balls were likewise analogies not meant to represent the true nature of light. Only Kepler had tried to derive the measure of refraction from its proper cause, although he did not have a corpuscular conception of light. Barrow’s pulses may have given a reasonable idea of the nature of light, they did no more than that. Barrow did not explore this idea, to see what new consequences it might reveal. The waves of Huygens were meant to reflect the true nature of light, so that its properties could be derived from it. Huygens did, and thus set the wheel of hypothetico-deductive inference in motion. One might say that he did what Galileo had done in Discorsi, except that Huygens applied mathematics to the motions of unobservable objects. Between 1672 and 1679, a new way of doing the mathematical science of optics developed. The wave theory contains three elements typical of early modern science: mathematical description, mechanistic explanation, experiment. By mathematically formulating his principle of wave propagation, Huygens brought these three elements to a fruitful synthesis that made possible the discovery and establishment of the nature of strange refraction. In the attack upon strange refraction, a methodical process had got going that (as Hakfoort tentatively concluded earlier) is typical of modern mathematical physics, in which theory is extended by mathematically derived consequences that are experimentally verified.155 Within the limited scope of reflected and refracted rays, Traité de la Lumière constitutes the birth of physical optics. Not the whole of it, but at least it contained a new, mathematical science of optics in which the nature of light and its observed behavior was fruitfully integrated for the first time.156

155

Hakfoort, Optics in the age of Euler, 183-184. In his theory of colors Newton had invented another kind of physical optics, in which experiment was used as a heuristic tool for finding new, mathematical properties of light. Newton at the same time refrained from integrating explanatory hypotheses into his mathematico-experimental theories. I return to this in section 6.2. 156

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Chapter 6 1690 - Traité de la Lumière Retrospection upon the coming about of the wave theory in the context of Huygens’ oeuvre and the mathematical sciences in the seventeenth century

Huygens’ new science of optics developed in a markedly contingent way. If he had not conceived of a plan for the elusive publication of his dioptrics; if he had not fallen in with the custom of providing some explanation for refraction; if he had not recognized the problem strange refraction posed for his Pardies-like explanation of refraction; if he had not decided to include it in his treatise; if he had not pressed ahead after his investigations of strange refraction of 1672; and if Rømer had not compelled him to devise the special experiment of 1679. If all this had been otherwise, then his celebrated wave theory had not come about. In that case, some time, some kind of ‘Dioptrique’ may have been published. As a treatise in geometrical optics it would hardly have marked itself off from – say – Barrow’s lectures, except for its practical outlook. But Traité de la Lumière was something different. What had begun as a fairly conventional, natural philosophical introduction to a treatise on dioptrics had become a new way of treating light mathematically that went beyond traditional geometrical optics. We are now ready to look back and ask how the wave theory related to the seventeenthcentury development of optics and of mathematical science in general and, second, what light it sheds on Huygens’ oeuvre. Traité de la Lumière was not presented by its author as a revolutionary new way of doing optics. Hypotheses were simply inevitable in these matters, Huygens said as a matter of fact, and he did not draw attention in any way to the special character of his principle of wave propagation and his account of strange refraction. Did he realize he was breaking new ground? We cannot read his mind, of course, but there is reason to think that he did not value his findings in the same vein as we do, as some kind of methodological innovation, that is. He never abandoned the original plan of 1672, in which his theory of light would be a preparatory part to his dioptrics. Only at the very last moment did he abandon his plan to publish a ‘dioptrica’ and created a ‘traité de la lumière’. He published his wave theory in 1690 as Traité de la Lumière, a title he had chosen at the very last moment. He did so after many hesitations over the best way to present his wave theory, which suggests that Huygens himself was also not sure about its exact status. The publication of the wave theory took no less than twelve years. The years after its presentation at the Académie witnessed Huygens’ step by step

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departure from Paris. In 1681, he fell ill again and he returned to The Hague in September. In Paris, the climate for Protestants was growing less tolerant in Paris, and when in 1683 Colbert died, Huygens decided to remain in Holland. He spent some happy years enjoying the reunion with his brother until duty in the form of Stadtholder William III (= King William) called upon Constantijn to go to London in 1688. Their correspondence in these final years of separation gives once again proof of their intimate comradeship. In 1687, at the ripe old age of ninety, their father Constantijn sr. passed away. As the second son, Huygens inherited Hofwijck, the county house in Voorburg, and the title ‘Lord of Zeelhem’, an estate of the family in what is now Belgian Limburg.1 The last years of his life he spent much time at the seclusion of Hofwijck, where his science experienced somewhat of a prime with, among other things, contributions to the recent developments in mathematics. This chapter begins with the publication history of Traité de la Lumière and a short outline of his later dioptrics. It continues with a review of seventeenth-century optics from the perspective of Triaté de la Lumière. The development of Huygens’ wave theory has revealed some themes that in my view are important for our understanding of the development of seventeenth-century optics. I will not offer a worked-out history but rather sketch the lines of a re-interpretation. In the final part of this chapter, I turn to Huygens’ science as a whole, in particular his alleged Cartesianism. Read as a textbook example of Cartesian science, Traité de la Lumière is often seen as exemplary for Huygens’ science. The eventual Traité de la Lumière should not, however, be taken at face value. When assessing Huygens’ scholarly goals and conceptions the winding road of its creation needs to be taken into account. And it particularly casts doubts on its reputed cartesianist essence. 6.1 Creating Traité de la Lumière With the solution of the problem of strange refraction, nothing stood in the way of elaborating the ‘Projet’ of 1672. Yet, it lasted more than ten years before Huygens put his wave theory to print. In the preface of the eventual Traité de la Lumière, he mentioned three reasons for the delay: “One may ask why I have tarried so much with publishing this work. The reason is that I had written it rather negligently in the language in which one sees it, with the intention to translate it into Latin, doing so in order to have more attention to things. Upon which I planned to give it together with another treatise on dioptrics, where I explain the effects of telescopes, and the other things that also belong to that science. But the pleasure of the novelty being gone, I have gone on postponing the execution of this

1

Father Constantijn had given the estate Zeelhem - and probably the title too - to his son Constantijn in 1651. Christiaan did not have a title, but he bore ‘Lord of Zuylichem’, for example on the title page of Horologium Oscillatorium. After their father’s death, Constantijn inherited the house at the ‘Plein’ in The Hague and the title ‘Lord of Zuylichem’, while Christiaan now became a ‘real’ lord, of Zeelhem. Keesing, “Wanneer”, 63 and Keesing, Constantijn en Christiaan, 112-113.

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plan, and I do not know when I would have been able to put this to order, being often diverted either by things to do or by some new study. ”2

It was true, many things had distracted him since his ‘Eureka’s’. In the summer of 1678, Hartsoeker and Leeuwenhoek had kindled his interest in microscopes and microscopical observation. He devised several Leeuwenhoek-style simple microscopes himself and made some technical improvements and additions.3 In 1680, he published a design for a telescopeenhanced level, followed by disputes which lasted for years.4 Around the same time he studied the properties and nature of magnetism. Back in Holland, Huygens designed his planetarium and continued working on his pendulum clock and its application at sea. With Constantijn he recommenced the grinding of lenses and manufacture of telescopes, earning fame for their skills and creating demand for their products.5 They built a grinding lathe and Huygens published the description of an aerial telescope of his design, Astroscopia Compendaria (1684). Around the same time the brothers were also discussing a treatise on the grinding of lenses.6 So, even if the list is confined to optical matters there was enough to divert his attention from his treatise in dioptrics. Moreover, Huygens somewhat lost interest in his theory, witness his response to Leibniz’ inquiries after his wave theory. During the 1680s Leibniz repeatedly asked him his opinion on the nature of light and refraction, but Huygens proved rather reluctant to discuss the details of his wave theory. He replied Leibniz’ questions were often much later and when he did he was rather succinct and shallow.7 The delay was not only effected because other things caught Huygens’ eye. His indecisiveness as regards the final format of his optics, to which he alluded in the first sentences quoted above, also played a part; a substantial part in my view. Although the preface does not say so, Traité de la Lumière had been intended as part of a treatise on dioptrics, rather than an accompanying discours. Until Traité de la Lumière went into print, Huygens had maintained his original plan of a ‘Dioptrique’ of which his theory of light was an integral part. When he presented it to the Académie, he had upgraded his wave theory to form a separate part, but still the ‘first part’ of his ‘Dioptrique’. Now, in 1690, Huygens mentioned ‘Dioptrique’ as a separate 2

Traité, ‘Preface’, [1]. “On pourra demander pourquoy j’ay tant tardé à mettre au jour cet Ouvrage. La raison est que je l’avois escrit assez negligement en la Langue où on le voit, avec intention de le traduire en Latin, faisant ainsi pour avoir plus d’attention aux choses. Apres quoy je me proposois de le donner ensemble avec un autre Traité de Dioptrique, ou j’explique les effets des Telescopes, & ce qui apartient de plus à cette Science. Mais le plaisir de la nouveauté ayant cessé, j’ay differé de temps à autre d’executer ce dessein, & je ne sçay pas quand j’aurois encore pû en venir à bout, estant souvent diverti, ou par des affaires, ou par quelque nouvelle étude.” 3 For example: OC8, 112-113. For details on his observations: Fournier, “Huygens’ observations”. 4 OC8, 263-266; 273-276, and further. 5 The quality and distribution of their lenses is recorded in Van Helden & van Gent, The Huygenscollection and Van Helden and van Gent, “Lens production”. 6 For example: OC8, 432-435; OC9, 8; 25. 7 See OC8, 244-245; 250-251; 256-257; 267; OC9, 259.

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work not inherently connected to Traité de la Lumière. Still, the two parts of his ‘Dioptrique’ had only gradually drifted apart between 1672 and 1690.8 Below the surface the two parts were of a fundamentally different nature. The innovative character of his eventual theory of waves had broken it loose from the geometrical optics of lenses and Huygens only gradually came aware of the gulf that had formed between both parts. His allusion to a Latin translation of Traité de la Lumière underscores the close tie he remained to see between the two parts of his ‘Dioptrique’. 6.1.1 COMPLETING ‘DIOPTRIQUE’ The text of the eventual Traité de la Lumière had been ready for by far the greater part in 1678. The events of August 1679 necessitated some corrections and additions, but these could be inserted into the existing text, as indeed they were. Neither the theory as such, nor the main line of his argument was affected by these changes. What, then, needed in view of the ‘Projet’ to be done about the second part of ‘Dioptrique’? Huygens decided to consider the content of the dioptrical part anew. The content of Tractatus still had to be rearranged to begin with and a final text elaborated. On and off, Huygens worked on this during the 1680s, making some tables of content, writing some prefaces, and composing parts of the theory. Huygens’ dioptrics in the 1680s In the meantime, Huygens added new dioptrical studies. The recent developments, and his own involvement, in microscopy made Huygens decide to include a discussion of the dioptrical properties of these instruments, too. To this particular topic he had assigned only historical importance in 1672.9 The application of his theory to microscopes is similar to his treatment of telescopes, so I will not discuss much detail.10 The most interesting, and somewhat unexpected, dioptrical addition was his investigation of lens aberrations. Although it had been discarded with the ‘Projet’, Huygens made a fresh start with his ill-fated theory of spherical aberration and combined it with a mathematical analysis of chromatic aberration. Due to the 1672 debacle, Huygens now set the aims of his dioptrical theory considerably lower. He had given up hope of neutralizing aberrations. He set full focus on the mathematical understanding of the quality of images and on deriving guidelines for improving it. This was a continuation of the table that had concluded De Aberratione, in which he listed the optimal configurations of Keplerian telescopes. Probably by early 1684 Huygens began to examine these anew. He remarked that his earlier studies were useless as they presupposed that spherical aberration was the key factor in 8

Yoder has suggested that, even after the publication of Traité, Huygens intended to attach it to the second part of ‘Dioptrique’. Yoder, “Archives”, 106.. 9 OC13, 748-749. 10 Most of it can be found in OC13, 512-585.

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the limited quality of telescope images. He now realized that the effect of spherical aberration was small as compared to the aberration “… that arises from the Newtonian dispersion of rays.”11 He set out to make a new table of optimal configurations, which would also take the ‘Newtonian’ aberration into account.12 Huygens explained the difficulty with telescopes in the same way he had done in 1665: increasing the magnification renders images fuzzy and obscure.13 The problem was how to increase the power of a telescope whilst maintaining the clarity and distinctness of images. This came down to determining the aperture of the objective lens in proportion to the aperture of a given telescope of good optical qualities.14 Huygens first determined the amount of chromatic aberration of a lens relative to its focal distance. He more or less repeated what he had written to Newton in 1672. Huygens had argued that the ratio between the aberration and the focal distance was 1 : 25, whereas Newton had used 1 : 50.15 This meant that chromatic aberration exceeded spherical aberration 39 times and would imply that it was superfluous to take spherical aberration into account when dealing with the quality of images.16 Huygens explained that in reality things were not as bad as these figures suggested. Repeating further arguments from his dispute with Newton, he said that many of the dispersed rays were not perceptible. Therefore lenses did produce images that were sufficiently distinct, although they might be surrounded by a faint ‘nebula’.17 First, Huygens considered the chromatic aberration NM, produced on the retina by a telescope consisting of two convex lenses AC and DP (Figure 78). The axis of the system is TPC, F is the focus of the red rays refracted by the objective lens AC and B the focus of the violet rays. In this type of telescope the foci of objective and ocular lens should coincide, but the focal distances of the various colors differ. Huygens assumed the foci of the red rays to coincide. F is also the focus of the red rays for the ocular PD, G is the focus of the violet rays. Consequently, red rays will be refracted along AFO, towards LK parallel to the axis, and point N, where the axis intersects with the retina. Next, Huygens considered the path of the violet rays. These are refracted by the objective lens towards ABD. As G is the focus of the violet rays for the ocular, a ray GD will be refracted towards DE and N on the retina. Ray ABD is not refracted towards DE, however, but towards DK and thus reaches the retina in M. Consequently NM is the aberration produced by the system. Because – by a small angle approximation – angle NKM is equal 11

OC13, 621. “… ex dispersione radij Newtoniana.” OC13, 496-499. 13 OC13, 480. 14 OC13, 482. Compare section 3.2.1 on De aberratione below. 15 OC13, 484-487 and 485 note 8. The manuscript is confusing as Huygens first derived his own figure of 1 : 25 but used Newton’s figure of 1 : 50 when he later inserted the numbers into the text. 16 According to his own figure of 1 : 25 this should be 79. 17 OC13, 486-487. 12

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to angle DKL and EDK equal to BDG, it follows that BDG fixes the aberration at the back of the eye.18 Next, Huygens queried how the length of the telescope can be increased while maintaining the degree of aberration, that is: keeping the angle BDG constant. A second telescope consists of objective lens ac, ocular pd and focus f. A straightforward derivation yields the conditions for the aberration to be equal in both telescopes. The apertures need to be in proportion to the square roots of the focal distance of the objective: ac : AC cf : CF . Consequently, the focal distances of the ocular are in proportion to the apertures: fp : FP = ac : AC.19 From this he derived a rule to determine the aperture and the ocular when the objective lens is given. Given a satisfactory 30-foot telescope the rule is as follows. Multiply the focal distance of the objective lens (measured in inches) by 3000; the square root of this figure gives the diameter of the aperture (in hundredths of inches); adding one tenth yields the focal distance of the ocular (in hundredths of inches).20 It follows that the apparent magnification by the system is in proportion to the aperture. The table listing the appropriate configurations found in the manuscript was, however, calculated by a simpler rule where the focal distance is equal to the diameter of the aperture.21 Thus Huygens revised and updated De 18

Figure 78 Chromatic aberration

OC13, 488-491. OC13, 492-493. 20 OC13, 494-495. 21 It was this simpler rule he communicated to his brother on 23 April 1685. OC9, 6-7. 19

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Aberratione, now taking into account the ‘newtonian’ aberration as well. Huygens applied his account of chromatic aberration to microscopes. In the first place he considered simple microscopes in which only the aperture could be adjusted but loss of clarity could not be prevented.22 With compound microscopes the application of the theorem had the surprising outcome that the aperture could be increased without loss of clarity and distinctness.23 This compelled Huygens to take spherical aberration into account as well. Calculating the degree of both aberrations in a specific microscope, he came to the conclusion that in this case the spherical aberration exceeds the chromatic.24 This being established, he could apply his theory of spherical aberration in order to prescribe the optimal configurations in a compound microscope.25 Although of limited use for telescopes, Huygens’ theory of spherical aberration was still of some avail for the improvement of microscopes. It does not seem that he put his findings to practice. His own microscopic observations date back to the late 1670s, when he had used simple microscopes like Leeuwenhoek’s. Without professing to do full justice to Huygens’ dioptrical studies of the 1680s, I leave it at this. It suffices to make clear that they remained in line with his work prior to 1672, both qua content and qua character. Huygens maintained his focus on instruments, which now included microscopes as well. Huygens seems to have learned from the debacle of De Aberratione. In his discussion of the effect of aberrations on the quality of images he was more perceptive of the relative nature of mathematical results. Having demonstrated how clarity of images relates to the dimensions of telescopes, he remarked that his results did not fully agree with experience, in particular when it came to observing faint objects.26 He then came to a discussion of dimensions actually employed, elaborating on their justification in a fairly qualitative way and appealing to his personal experience.27 6.1.2 FROM ‘DIOPTRIQUE’ TO TRAITÉ DE LA LUMIÈRE From time to time during the 1680s, Huygens made sketches for a revised ‘Dioptrique’. The earliest can be dated in or after 1682.28 It contains, among other things, two sketches of a transitional paragraph between part one “… on the physical causes of the rules that light observes, …” and part two containing “… the explanation of the effects of glass lenses …”29 The historical sketch originally planned as the first chapter in the ‘Projet’ would 22

OC13, 530-535. OC13, 542-543. 24 OC13, 564-565. 25 OC13, 576-585. 26 OC13, 502-503. 27 OC13, 502-509. 28 OC13, 745n11. 29 OC13, 747. “… des causes physiques des regles qu’observe la lumiere, …”; “… l’explication des effects des lentilles de verre …” 23

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now open the second part of ‘Dioptrique’. In the adjusted plan, this part would be a trimmed down version of Tractatus. Huygens remarked that he had written his theory of lenses a long time ago and that some of its content had been dealt with since by others. He probably had Barrow’s lectures in mind. Matters like these he would sketch only briefly and then present the most important theorems of Tractatus. The principal topic of the second part of ‘Dioptrique’ was magnification, a topic that according to Huygens still passed without proper treatment in existing literature.30 Huygens’ dioptrical studies of the 1680s have been gathered by the editors of the Oeuvres Complètes in a separate section in OC13, called “Dioptrica. Pars Tertia”. For the large part it consists of an essay labeled ‘De Telescopiis’. Dating it is hazardous, but it appears that the latest version is of 1692. Material with varying dates – ranging from the 1660s to the 1690s – is collected and alternative versions of several parts can be found in appendices. Huygens probably started assembling and elaborating this in 1684 or 1685.31 In or after 1684, he made another outline for the second part of ‘Dioptrique’, listing the order of subjects treated.32 This outline corresponds for the most part with ‘De Telescopiis’. The essay cannot have been intended as a definite second part of ‘Dioptrique’, though. At several places Huygens referred to theorems of Tractatus. If ‘De Telescopiis’ was eventually intended to replace Tractatus entirely, Huygens still would have to find a way to integrate these theorems. At any rate, Huygens did not round off his new plans. In 1687 he considered the state of his ‘Dioptrique’ once again. He still considered his theory of waves and his theory of dioptrics as two parts of a single work. He doubted, however, whether ‘Dioptrique’ was still an appropriate title. A note gives a new title: ‘Optique. I partie.’33 In addition, the problem of languages remained. He made a start with a Latin translation of the first part under the title ‘Versio Diatribæ de Luce’ but got no further than some 10 pages.34 At the beginning of 1690, Traité de la Lumière was published as a autonomous treatise. It did not, as we have seen, reveal that it had been intended as the first part of a larger work. Still, it was not fully separated from the ‘Dioptrique’. At the time the Traité de la Lumière was being printed, Huygens began a French translation of his dioptrics: “Beginning of the treatise on my dioptrics in French that I planned to join with the treatise on light, …”35 The decision to treat them as separate treatises had been made just before. The opening had at first read: “Beginning of my second part of the ‘Dioptrique’

30

OC13, 746-748. OC13, 434-511. See notes 1 and 2 of pages 434-435 on the dating. 32 OC13, 750-752. 33 OC13, 754. 34 OC19, 458-470. He probably did not start translation before May 1687: OC9, 133. 35 OC13, 754; 755-770. “Commencement du Traitè de ma Dioptrique en François que j’avois dessein de joindre au Traitè de la Lumière, …” 31

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in French to join it with the first which is in this same language. This plan has changed for it will remain in Latin.”36 After the publication of Traité de la Lumière, Huygens continued to work on the dioptrical treatise. In 1692, his plans changed once again after he had read Molyneux’ Dioptrica Nova. Huygens’ notes on Dioptrica Nova survive.37 Molyneux had treated telescopes better than anyone before, but had “little of the things my treatise contains on this matter”38 Huygens knew that he was a better mathematician than the Irishman, and realized that he had still something valuable to present in dioptrics. Some time after his reading of Dioptrica Nova, he made a new (also the last) outline for his dioptrics: “De Ordine in Dioptricis nostris servando”39 He would leave out what Molyneux had treated and emphasize his own strong points: the theories of spherical aberration and magnification. He gave particular attention to his theorem on the magnification produced by a given system of two lenses, the lens-formula as it is called nowadays.40 Huygens would prove it (instead of just stating it, as Molyneux had done) and extend it to more complex systems. ‘De Ordine’ brought together scattered material ranging from the 1650s to the early 1690s. In accordance with this scheme, Huygens ordered his manuscripts and numbered the pages in red.41 Roughly the set runs as follows: the first part of Tractatus, with De Aberratione (without the rejected parts) inserted after proposition twenty, propositions one and two of part three of Tractatus, part two of Tractatus, various fragments of ‘De Telescopiis’. It makes it clear that in the course of 40 years Huygens’ views on what dioptrics was about had not changed: it was about telescopes. It had to account for the working and improvement of telescopes mathematically. Huygens did not live to see his dioptrical treatise through the press. In his will he instructed De Volder and Fullenius to look over his “mathematical writings” and “to edit as best they can whatever in it might be fit to publish”42 He explicitly named the ‘Dioptrika’ and three other treatises. De Volder and Fullenius followed Huygens’ ordering pretty closely, so that the 1704 edition gives a good indication of his final idea of Dioptrica.43

36

OC13, 754n4. “Commencement de ma seconde partie de la Dioptrique en francois pour la joindre a la Première qui est en cette mesme langue. Ce dessein est changè car elle demeurera en Latin.” 37 OC13, 826-844. 38 OC10, 279. “… peu de ce que contient mon Traitè sur cette matiere.” 39 OC13, 770-778. 40 OC13, 773-774. The original version was: OC13, 186-197 and is treated in section 0. 41 Hug29, 101bis-205. 42 OC22, 775-776. “schriften van Mathematique” and “… ‘tgeen daerin soude mogen weesen bequaem om gepubliceert te werden, hetselve willen besorgen ten besten sij sullen connen, …” 43 The editors of the Oeuvres Complètes chose not to follow the final ordering, but have attempted to make a chronological reconstruction of Huygens’ dioptrical papers.

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The publication of Traité de la Lumière Why did Huygens not publish his dioptrics himself, given the quite publishable state he left it in?44 After the unceasing postponements of the previous decades, with none of his plans – beginning with that of 1652 – completely executed, it does not really come as a surprize he did not finish ‘De Ordine’. Still, this leaves the question why he never went public with his dear dioptrics. Huygens’ general tardiness of publishing is often pointed out, suggesting some psychological factor for failing to publis his important discoveries and inventions. However, there may well be cultural factors in play as well. I will not elaborate this, except for advancing some questions. Most important, in my view, is reversing the question that opened this paragraph. Rather than asking why Huygens did not publish what from our modern point of view was well worth publishing, we should ask why he published what he did? What credit could a savant like Huygens have gained by publishing? To answer questions like this, his social position needs to be taken into account. Was an ‘aristocrat’ like Huygens in the position to publish whatever he liked at the moment that suited him, or did he need to observe specific forms? Were books and articles the obvious means for disseminate one’s ideas, or would someone like him prefer letters to wellchosen peers? In order to explain Huygens’ tardiness in publishing Dioptrica (and other texts), his publication pattern ought to be surveyed. This would account for the swiftness he published his early mathematical works with around 1650, as well as the haste with which he usually applied for patents. So, we let the question why Huygens did not publish his dioptrics rest and turn to the more important question why, in the end, he did publish the first part of his ‘Dioptrique’, the wave theory and his explanation of strange refraction. A direct incentive to publish Traité de la Lumière may have been plans at the Académie to publish papers of its (former) members.45 On 8 September 1686 Huygens received a letter from De la Hire asking his permission to publish some of his manuscripts kept in Paris.46 Huygens did not hesitate to list some interesting treatises, but at first no mention was made of his theory of light.47 In his letter of 20 April 1687 De la Hire started to inquire about the state of affairs concerning Huygens’ treatise of dioptrics.48 Huygens answered that it was almost ready; at least the part on “… physics, Iceland crystal etc.”.49 The following letter makes it clear that Huygens’ explanation of strange refraction was well remembered in Paris but also that it was not fully

44

Yoder, “Archives”, 91-92. Divers Ouvrages de Mathématique et de Physique. Par Messieurs de l’Academie Royale des Sciences was published in 1693, containing eight papers by Huygens. 46 OC9, 91. 47 OC9, 95-95. 48 OC9, 129. 49 OC9, 133. “… la Physique, le Cristal d’Islande &c.” 45

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understood.50 This interest displayed in his theory, combined with an apparent ignorance of its contents, may have influenced the decision to publish Traité de la Lumière. In his preface to Traité de la Lumière, Huygens alluded to another reason. During the 1680s, Leibniz and Newton had published on anaclastic curves. Huygens had first heard of Newton’s derivation of anaclastic surfaces from Fatio de Duillier in June 1687.51 Fatio, who had visited Huygens at the end of the preceding year, wrote him about the excitement among the members of the Royal Society over Newton’s forthcoming Principia. According to Fatio, Newton’s method for finding anaclastic surfaces accorded with Huygens’, in that he assumed that each ray travels in the same time from one focus to the other, although he employed a different principle.52 Huygens, in a letter of 11 July 1687, responded that he did not see how the same conclusion could be reached from a different assumption.53 He did not return to the matter in his correspondence. When Principia was published, he would discover that Newton assumed bodies instead of waves to travel in equal times. Leibniz, in the meantime, had also heard of the Principia (he claimed only to have read the review in Acta eruditorum at that point) and acted appropriately. For reasons of priority, he sent three articles to the Acta, including one on anaclastic surfaces. In chapter 6 of Traité de la Lumière, Huygens presented his own derivation on the basis of his wave theory. In addition, this chapter contained his determination of caustics by means of his wave theory. It was an elaboration of the notes of 1677 discussed in section 5.1.1. Some hold that Newton’s Principia was the main incentive for Huygens to publish Traité de la Lumière together with his theory of gravity, Discours de la Cause de la Pesanteur. After he had read the Principia, Huygens wrote his brother Constantijn in London that he was impressed and would like to come to Cambridge only to meet Newton.54 In the summer of 1689, he went to England and met Newton at the Royal Society. Huygens spoke about his theories of light and gravity; Newton is reported to have discussed, out of all possible subjects, strange refraction.55 Unfortunately, no records remain but it is likely that the two men did not fully agree. In Principia, Newton had asserted that a wave-like motion diverts after passing through an aperture. This implied that waves could not explain the rectilinearity of rays.56 In Discours de la Cause de la Pesanteur – a treatise on gravity published together with Traité de la Lumière – Huygens retarded that waves do spread around

50

OC9, 164. OC9, 167-171. 52 Newton, Principia, 626-628 (Propositions 97 and 98 of section XIV of book I). I do not precisely know what and how Fatio knew of Huygens’ ideas. 53 OC9, 190. 54 OC9, 305. 55 Westfall, Never at rest, 488; Shapiro, “Pursuing and eschewing hypotheses”, 223. 56 Newton, Principia, 762-767. For an extensive discussion see: Shapiro, “Light, pressure”, 284-291. 51

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corners but that these dispersing waves are too weak to produce light.57 This argument was, of course, elaborated in the first chapter of Traité de la Lumière.58 At the beginning of 1690, Traité de la Lumière was printed and he sent some copies to England.59 Huygens wrote Fatio on 7 February 1690 that he was anxious to know what Newton thought of his “… explanations of refraction and of the phenomena of Iceland crystal, but I am not quite sure whether he understands French, …”60 He would not live to see Newton entirely reject the idea of light waves, and ignore his explanation of strange refraction, in the queries appended to Opticks. With Traité de la Lumière now published, we may ask what Huygens thought he had published. The hesitations about titles and languages suggests that Huygens was not sure anymore of the status of his wave theory. Did it still belong to his ‘Dioptrique’? He had written it in French, the language used at the Académie. But French was also the language of the particular topics discoursed of there, issues in physics and so on. Latin, on the other hand, was the language he used for mathematical topics.61 His doubts about translating it might also indicate that he was doubting whether it still belonged to the mathematical science of optics. At the very last moment Huygens decided to publish it as an independent treatise in French, under the title Traité de la Lumière. In so doing, he cut through the umbilical cord of his wave theory, thus withdrawing it from the realm of mathematics. Huygens had put, so to speak, his theory of light in the milieu of the Paris Académie and the post-Cartesian debates proliferating there. Through the decoupling of Traité de la Lumière from ‘Dioptrique’ he focused attention on the wave theory, that is, on the explanation of the laws of optics instead of their application to the behavior of rays. The title he chose, Traité de la Lumière instead of Dioptrique, suggests that the treatise had switched from a correction of La Dioptrique to a rebuttal of Le Monde. However, its stated aim was much more modest: 57

OC21, 475. Cohen argues that one of the reasons Huygens was urged to publish Traité de la Lumière was to guarantee his priority regarding the theory of wave propagation. Cohen, “Missing author”, 32. 59 Cohen explains that a series of Traité de la Lumière exist that was issued by the publisher Vander Aa with the author’s name spelled out on the title page: “Par Monsieur Christian Huygens, seigneur de Zeelhem.” The other copies of Traité de la Lumière, including the large-size edition Huygens also printed to distribute among acquaintances, only have the author’s initials on the title page: “Par C.H.D.Z.”. Cohen assumes that the title page was altered just after printing had begun. The reason may have been that Huygens had used the title ‘Lord of Zuylichem’ in earlier publications and wanted to prevent confusion by used a neutral ‘D.Z.’. Since the death of their father, Constantijn formally was ‘Lord of Zuylichem’. Cohen, “Missing author”, 33-35. 60 OC9, 358. “… Explications de la Refraction et des phenomènes du cristal d’Islande, mais je ne suis pas bien assurè s’il entend le François, …” 61 After his move to Paris in 1666 Huygens began writing more and more in French, with the exception however of his works on mathematics. Illustrative are his 1672 notes on strange refraction, where – as we have seen above – he switched between both languages accordingly. Huygens’ use of languages corresponds with what seems to be a general pattern. Halfway the seventeenth century, the vernacular had begun to replace Latin in scholarly writings, especially in France and England. The notable exception are mathematical texts, which remained to be written in Latin well into the nineteenth century. A systematic study of the use of languages in scholarly writings would be well worth pursuing. 58

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“Where are explained the causes of what happens to it in reflection and in refraction. And particularly in the strange refraction of Iceland crystal.”62

Huygens did not intend to expound the principles of natural philosophy nor the methodology of mathematical science. The subtitle of Traité was not something like ‘Treatise of light, where the nature and all properties of light are wonderfully explained in a clear and most probable way according to the true philosophy’. He expounded natural philosophical principles insofar as these want to explain the mathematical laws of optics. He considered the principle of wave propagation – this ‘principal foundation’ – his main achievement. The validity of his principle was based upon, and implicitly confined to, the successful derivation of those laws. Traité de la Lumière offered an example of the proper use of mechanistic philosophy. Traité de la Lumière did indeed offer better explanation – more plausible, more probable – but we value it for its epistemic innovations. The wave theory had originally been planned as a non-committal, explanatory introduction to his mathematical theory of dioptrics. The eventual outcome really exceeded geometrical optics. The few methodological issues Huygens raised, were passed over as a matter of course. In other words, he does not seem to have been aware of the epistemically innovative character of the wave theory. Too modestly, from our point of view, Huygens presented his wave theory as a better explanation of the laws of optics, instead of a new way of doing the mathematical science of optics. 6.2 Traité de la Lumière and the advent of physical optics With Traité de la Lumière, Huygens created a new kind of optics, an instance of what we would call physical optics. I have argued that his actual interest in optics was the dioptrics of telescopes but that the phenomenon of strange refraction rather coincidentally directed him to questions pertaining to the mechanistic nature of light, which he subsequently subjected to the rigorous mathematical treatment of his dioptrics. This account of its historical development sheds new light upon our understanding of Huygens’ science, as I will argue in the next section. This section deals with lines of interpretation Huygens’ case suggests for the history of seventeenth-century optics. I will therefore broaden the outlook of my discussion and see how the themes in my account of Huygens’ optics may be generalized. I do not profess to offer a new history of seventeenth-century, rather I want to suggest some lines of interpretation that I consider important for our historical understanding of the origin physical optics. A major point of reference will be, of course, Newton, who created his own particular instance of a physical optics. Despite fundamental differences in their outlook, intentions and activities, there are important parallels between the optics of Huygens and Newton.

62

Traité de la Lumière, title-page. “Où sont expliquées les causes de ce qui luy arrive dans la reflexion, & dans la refraction. Et particulierement dans l’etrange refraction du cristal d’Islande.”

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Historical studies of seventeenth-century optics tend to focus on the development of theories of light. They do so with good reason, for it is in the changing thinking over the nature of light and its properties that the development of optics exemplifies the transformation of science known as the scientific revolution. However, such a focus presupposes some kind of coherence and continuity in the pursuits of thinkers on matters optical. In my view the study of phenomena of light in the seventeenth century was a rather heteroneous affair. More specifically, I think that historians have tended to overlook the fact that something akin to physical optics did not yet exist before Newton’s and Huygens’s work.63 The part of exact science that is organized around the question ‘what is light and how does this explain its properties?’ did not come about until they created and established it. The one synthetic study of seventeenth-century optics, Sabra's incomparable Theories of Light from Descartes to Newton, is telling in this respect.64 It is structured around the conceptualizations of the physical nature of light and the way seventeenth-century students of optics employed these to explain the properties of light. Likewise, most monographic studies treat seventeenth-century optics qua physical optics. Yet, the issues basic to modern physical optics did not guide the development of seventeenthcentury optics but were gradually recognized as fundamental to the science of optics through the pursuits of Newton and Huygens. For a historical understanding of their optics, and of the development of seventeenth-century optics in general, its origins need to be taken into account. The historian should turn his eyes away from their future achievements and look back to the mathematical science of optics as they had encountered it. In this regard, traditional geometrical optics is historically significant. It formed a prominent disciplinary, conceptual, and methodological context for the pursuits of seventeenth-century students of optics, and Huygens and Newton in particular. In contrast to later physical optics, this science was organized around the questions ‘what are the properties of rays in their interaction with various media and how can their behavior be deduced geometrically?’, questions that were directional in seventeenth-century optics and, as such, have left their mark on Huygens’ and Newton’s pursuits as well. That does not mean that geometrical optics was the sole root of early modern optics. The emerging new philosophies of nature were equally significant, interacting with the established mathematical sciences and regauging their basic principles. Yet, existing literature considers seventeenth-century optics primarily within the context of these philosophies 63

Cantor points this out but does not elaborate this theme. Cantor, “Physical optics”, 627-628 in particular. Cohen does point out the novelty of the content of Traité de la Lumière and Opticks compared to traditional geometrical optics, but he neither elaborates it further; Cohen, “Missing author”, 30-32. 64 A.I. Sabra, Theories of Light, 11-15 in particular. I call it incomparable in the way Leibniz called Huygens an ‘incomparable man’ at the news of his death (Acta eruditorum August 1695, 369): although he regretted Huygens’ lack of interest in metaphysical issues, he greatly admired his scientific abilities.

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and underrate, in my view, the historical significance of geometrical optics. Shapiro’ studies of the development of Newton’s optics offer a favorable exception. He has analysed Newton’s theory of colors as a failed effort to establish a mathematical science of colors after the model of geometrical optics.65 His work has been very inspiring and instructive in developing my ideas on the historical significance of geometrical optics. Taking into account the influence of geometrical optics does not, to be sure, yield a radically new historical picture, but I do think that it can further illuminate the developments of seventeenth-century theories of light. Mathematization by extending mathematics Unlike Huygens, Newton was explicit about the new ground he was breaking. The lectures on optics he delivered from 1670 on as newly appointed professor at Cambridge, were the first occasion where he presented his new ideas in optics. The core of it is formed by the theory of different refrangibility which is integrated into an elaborate discourse in geometrical optics. Laboriously, he accounted for his digression into the origins of colors, a topic that traditionally belonged to philosophy rather than mathematics: “But lest I seem to have exceeded the bounds of my position while I undertake to treat the nature of colors, which are thought not to pertain to mathematics, it will not be useless if I again recall the reason for this pursuit. The relation between the properties of refractions and those of colors is certainly so great that they cannot be explained separately. Whoever wishes to investigate either one properly must necessarily investigate the other. Moreover, if I were not discussing refractions, my investigation of them would not then be responsible for my undertaking to explain colors; nevertheless the generation of colors includes so much geometry, and the understanding of colors is supported by so much evidence, that for their sake I can thus attempt to extend the bounds of mathematics somewhat, just as astronomy, geography, navigation, optics, and mechanics are truly considered mathematical sciences even if they deal with physical things: the heavens, earth, seas, light, and local motion. Thus although colors may belong to physics, the science of them must nevertheless be considered mathematical, insofar as they are treated by mathematical reasoning.”66

Newton here calls his mathematization of prismatic colors an extension of the bounds of mathematics. He justifies it by comparing it to the fields of mixed mathematics where physical things were traditionally treated mathematically. From a historical point of view these are telling words. They provide contemporary reflections on what it meant to subdue new domains of natural inquiry to mathematical treatment in early modern science. Newton’s words indicate that mathematization is not a mere application of cut-anddried concepts and methods plucked from some mathematical air, but that existing fields of mathematical sciences provided the context for it and formed a steppingstone to treat new domains mathematically. The explicit methodological awareness Newton displays here, is vainly searched for in 65 66

In particular his “Experiment and mathematics” and “Dispersion law” Newton, Optical papers 1, 88-87 & 438-439.

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Traité de la Lumière. For Huygens it went without saying to apply mathematics to matter in ‘Physique’ just like it was done ‘Optique’, in spite of the very different nature of this was matter. Besides the difference in tone in the way Huygens and Newton made public their achievements, there is, of course, a big difference in the character of their pursuits. Each created a form of physical optics by mathematizing new domains of light, but of an essentially different nature. Newton mathematized a new range of optical phenomena (which Huygens did not); Huygens extended mathematics into the new kind of realm of the unobservable, hypothetical nature of light (what Newton could do equally well but only did so privately). In each case, traditional geometrical optics was a major starting point, yet embedded in different natural philosophical contexts and problem definition. Huygens primarily responded to the issues raised by mechanistic philosophy, continuing along the lines of mathematical optics that run from Alhacen over Kepler to Descartes. The roots of Newton’s optics were more diverse. As much, if not more, as his optics was guided by his proficiency in and commitment to mathematical science, it was informed by his quest for the true nature of matter. In addition to mathematical optics, it built on the teachings of experimental philosophy and questions of matter theory articulated by Aristotle, Descartes and Gassendi, and Boyle. Despite these differences, the development of their pursuits show conspicuous similarities. Huygens and Newton came to their novel ways of doing optics only after they had moved beyond the confines of traditional geometrical optics. At an early stage, each worked much closer to the tenets of traditional geometrical optics than there final theories suggest. In section 4.2, I have shown that, in first attack on strange refraction in 1672, Huygens approached the phenomenon in a traditional way aimed at establishing the properties of rays interacting with Iceland crystal. Only at a later stage did he focus his attention on the mechanics of waves involved. Newton likewise formulated a law of dispersion in his Optical Lectures that defined additional properties of rays to mathematically account for the amount of dispersion. In these lectures he also allowed himself an epistemological freedom of solely providing a rational foundation that can only be understood in the context of geometrical optics.67 In addition, both employed a similar strategy in their early efforts to fathom the mathematical regularities of the two phenomena of strange refraction and color dispersion that challenged the universality of the newly discovered law of refraction. Both extended on Descartes’ analysis of ordinary refraction adding some extra component to the (now irregularly) refracted ray. These examples, which I have elaborated in detail elsewhere, serve to show that mathematization also involves transferring to new domains ideas and strategies from established fields of

67

See Shapiro, Fits, 24-26.

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mathematics.68 However, by that time establishing the properties of rays was not a straightforward a matter as it had used to be in traditional geometrical optics. With the rise of corpuscular thinking the ray no longer was a selfevident physical concept. Properties of rays now needed further justification, beyond the realm of visible phenomena or everyday experience. Huygens and Newton recognized the full import of these new questions and were the first to directly face them. The matter of rays Kepler can be said to have sharpened the question after the nature of rays and their properties. In perspectivist accounts these were answered only loosely, by an appeal to analogies between the motion of rays and that of bodies. With his rigorously realist reading of mathematical description, Kepler thought that the causes of rectilinearity, reflection and refraction ought to be contained in their measure. In his theory, this implied considering the interaction of incorporeal surfaces with the surfaces of diverse media. In the case of refraction this indeed led to a quasi-physical analysis of refraction on a microscopic level, as we have seen in section 4.1.2. In Paralipomena, Kepler explicitly distinguished the mathematical ray from the physical ray and, in a note on what he calls the ‘fourth kind of light’ meaning light communicated by the interaction with bodies, he can be said to have put the question after the physical nature of light propagation on the agenda.69 He did so in the first place, however, by the general reorientation of perspectiva into optics: from a theory of vision to a theory of the behavior and properties of light. As contrasted to his achievements in geometrical optics proper, however, Kepler’s ideas on the physics of light were little referred to later. Besides the Renaissance idiom of his thinking, the conduct of Descartes seems to have blocked the view on Kepler. Not only did he conceal the inspiration he had drawn from him, more importantly, he gave a radical twist to the perspectivist-cum-keplerian understanding of the behavior of light rays. Descartes’ mechanistic interpretation of perspectivist causal analyses of the laws of optics, turned these into material interactions. By the same token a good deal of traditional conceptualization was channeled into seventeenthcentury theories of light. Of old, geometrical optics had been geometry applied to matter, the matter of light rays. Descartes now raised the question of what matter these rays were and how this could explain their behavior. Still, the question of the nature of the light ray no longer was a simple one. Hobbes’ concept of a line of light indicates that the once natural identification with a geometrical line no longer was valid. Questions arose concerning the relationship between light and the geometrical line, whether it somehow expressed the nature of light, or whether it was the route of the its propagation, or merely an abstraction of some kind. Depending on one’s 68 69

Dijksterhuis, “Once Snel breaks down”. Kepler, Paralipomena, 37 note (KGW2, 46) in particular; Kepler, Paralipomena, 35 and note (KGW2, 31)

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specific conception of the corpuscular nature of light, questions like these could be answered in a more or in a less straight-forward manner. Broadly speaking, corpuscular conceptions of light can be divided in two: emission and medium conceptions. That is, light is either propagated matter or an action propagated through matter. The first maintained the primacy of the light ray in a rather natural way. The rectilinearity of light rays seems to follow directly from the view of a moving particle. The most prominent exponent of the emission conception was Newton, who considered its purport more thoroughly than anybody else. Apart from his explicit refutation of medium conceptions - in particular waves - he carefully considered his own understanding of light. At least in his early years he thought of light in terms of atoms, but soon developed a precise definition of a light ray that covered his emission conception without being dependent on it, as well as carefully determining the relationship between a geometrical ray and a physical ray, where a physical ray not necessarily is the mathematical line of geometrical optics.70 When he later reconsidered the papers and letters he had published in 1672 in the Philosophical Transactions, he added a footnote where he once again went into the details of the question whether light is a “body” or “the action of a body”.71 Medium conceptions of light marked a more decisive break with traditional geometrical optics. A light ray came to be seen as the effect of the propagation of light, not as its essence. This implied abandoning the idea that a ray has much intrinsic physical significance, as Buchwald explains, and he adds that few at that time were willing to do so.72 In a medium conception rectilinearity requires explanation. Significantly, Descartes, who originally formulated the idea that light consists of an action propagated without transport of matter, evaded the question.73 The same can be said for Hooke and Barrow who turned Hobbes’ pulse theory into what in essence was an emission conception of moving rods. Huygens, who adopted his medium conception in an almost a priori manner, took the problem seriously. In the first chapter of Traité de la Lumière, he put great stock in demonstrating that, and how, waves are capable of producing rectilinear light rays. The choice for an emission or a medium conception determined the way in which refraction could be conceptualized. In an emission conception, one must account for the fact that a transition to a different medium results in an instantaneous change of direction. The conception of refraction as a surface phenomenon rooted in perspectivist analyses. Kepler (himself neither a medium nor an emission theorist, to be sure) had explicated this by his notion of surface density. In his final analysis of refraction, he tried to analyze the interaction between the two-dimensional surfaces of light and 70

Shapiro, “Definition”, 206-208. Cohen, “Missing author”, 23-26. 72 Buchwald, Rise, 5. 73 See also: Shapiro, “Light, pressure”, 254-260. 71

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medium. The conceptualization of refraction as a surface phenomenon was to culminate in proposition 14 of Principia, where it is articulated as an event occurring at the boundary layer between two media. It subsequently to be reformulated entirely in terms of rays and their properties in the proof in Opticks. In medium conceptions, refraction could be explained in a much more straightforward way. The explanation reduced to accounting for the fact that a change of velocity results in a change in the direction of propagation. In his explanation of refraction, Descartes inserted this notion into a perspectivist analysis of refraction. The result was a rather ambiguous account, as he blended his medium conception of light with a surface conception of refraction. He was the first to state the propagation of light in terms of properties of the refracting medium. In the first assumption of his derivation, he mathematized this insight. Yet, in his second assumption he maintained the conception of refraction as a surface phenomenon by attributing the constancy of action to the surface of the refracting medium. In terms of the corpuscular nature of light, Descartes’ derivation thus raised more problems than it solved, which his successors did not refrain from pointing out. The mathematics of Descartes’ derivation, however, made an indelible impression on seventeenth-century savants. By stating the interaction between light and media in terms of rays and their actions, the derivation gained a significance that went beyond refraction per se. It provided a promising clue to seize all phenomena in which refraction was involved. Extending Descartes’ diagrams is a strategy that recurred many times in course of the seventeenth century. Bartholinus took this lead to fathom the behavior of strangely refracted rays, and Huygens initially did so, too. Newton in particular would always remain impressed with the cogency of Descartes’ elegant proof.74 In Opticks, he preserved it while putting it on a firmer (emission) foundation than La Dioptrique had done. In his search for a law of dispersion Newton’s first proposal was an extension of Descartes’ derivation. Naturally, so one would say, as this would preserve the harmony with monochromatic refraction and preserve the analysis of the phenomenon in terms of rays. Likewise, when Newton turned his mind to strange refraction in Opticks, he proposed – without justification – a construction that added the irregular component of the refracted perpendicular to the ordinary refraction of each ray.75 Indeed, the same construction Huygens had proposed in 1672. So, even after he had dismissed his ‘Cartesian’ dispersion law (see above 5.2.2), Newton remained confident that a Cartesian analysis had broader significance for phenomena of refraction.

74 75

Shapiro, “Light, pressure”, 239-241. Newton, Opticks, 356-357.

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The mathematics of light In their early optics, both Newton and Huygens employed strategies and methods common in traditional geometrical optics, i.e. determining properties of rays in order to account for phenomena of light mathematically and thus establishing principles for the mathematical science of optics. Both realized, however, that the new philosophies of nature had by then imposed limitations to such an affair, that the ray was not a natural physical entity and its properties needed supplemental accounting for. Besides the question what precisely was the (corpuscular) nature of light – discussed in the previous section – epistemic issues had arisen: how can the corpuscular nature of light be understood mathematically, how do conceptions regarding the nature of light relate to the laws of optics, what part do they have in the mathematical science of optics? Such questions were already shining through in Kepler’s account, but Descartes made them manifest without, however, fully answering them. The status of his models was ambiguous and the explanation problematic. Moreover, La Dioptrique lacked an explicitly empirical foundation of the sine law, and the derivation suggested that it was founded upon a priori mechanistic principles. In this sense he put the mathematical science of optics upside-down. Later students of geometrical optics were quick to restore the primacy of empirically founded laws, among which the sine law could now be counted as well. The validity of the laws of optics was independent of the mechanisms underlying them. Few directly addressed such questions regarding the physical nature of rays and light, be it in terms of empirical validation or corpuscular explanations. The explanations of a mathematician like Barrow were non-committal elucidations reminiscent of the way analogies had functioned in perspectivist theory. In their further optical studies, Huygens and Newton were the first to fully confront the issues involved. The content of their pursuits differed greatly, but in each case can be interpreted as dealing with the question what a new physical basis of the mathematical science of optics ought to look like. In his study of strange refraction Huygens moved, more or less unseen, from a consideration of the properties of rays to the actual problem of the properties of waves. In this domain of the hypothetical corpuscles and their motions, his basic problem was to ascertain the mechanistic causes of the laws of optics in a properly mathematical way. Newton expressly rejected this kind of ‘Hypothetical Philosophy’. His goal was to establish a mathematical science of colors on the precepts of ‘Experimental Philosophy’, to propose and prove the properties of light by reason and experiments.76 Newton’s use of the two pairs of ‘proposing and proving’ and ‘reason and experiments’ show the consciousness with which he pursued his 76

Newton used the terms ‘hypothetical’ and ‘experimental philosophy’ in a letter to Cotes in 1713, cited in Shapiro, Fits, 16. The phrase ‘propose and prove the properties of light by reason and experiment’ is derived from the opening lines of Opticks, Book I: Newton, Opticks, 1.

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project. They express his unique combining of mathematical science and experimental philosophy. In mathematical science, including Galileo’s ‘nuova scienza’, experiment was used as a tool of verification of hypotheses and theories. From the experimental philosophers, Newton adopted the heuristic use of the experiment, to discover and explore new phenomena and properties. Yet Newton looked upon his experiments with the eye of a mathematician. He saw rays and he looked for laws and did so by measuring and analyzing mathematically the outcomes of his experiments. In so doing he extended geometrical optics to new properties of rays: colors. In the Optical Lectures, Newton still confidently proposed properties of light by reason, but he soon qualified his statements. In particular after the disputes over the ‘New Theory’, he distinguished the certainty of mathematical demonstration from the conditional certainty of experimental conclusions.77 Explaining his view on the certainty of mathematical science to Hooke in 1672, he wrote “… the absolute certainty of a Science cannot exceed the certainty of its Principles”. And in optics these principles were physical.78 In his view different refrangibility was an experimentally proven property of rays and the true cause of the appearance of colors, and he was trying to convince Hooke of it. Newton maintained the conceptual primacy of the light ray in optics, thus ensuring the connection of his theory of colors with the mathematical science of optics. He did not take the physical significance for granted, like traditional geometrical optics, but carefully defined its mathematical and physical meaning respectively, and carefully determined its properties experimentally. The theory of fits of Book 2 of Opticks, in which he attributed periodicity to rays and that should account for the colors in thin films, can be seen as the culmination of this project of establishing a mathematical science of colors. As such, the project had foundered, however. Newton had not been able to establish the theory of unequal refrangibility as a mathematical science of colors. The first step in finding the laws of colored rays was to establish a one-to-one correspondence between the color of a ray and its index of refraction. The next, crucial step for the science of colors to become mathematical was to determine the regularity of the various indices by means of a law of dispersion. Having dropped the ‘Cartesian’ dispersion law of the Optical Lectures, in Opticks he resorted to a law whose validity, both by reason and experiment, was unclear. This is only a symptom of the fact that Newton never succeeded in elaborating the mathematical science of colors projected in the lectures.79 The first book of Opticks is the direct descendant of the Optical Lectures, but Newton had transformed his theory of unequal refrangibility from a mathematical deduction into an experimental exposition. In this he consolidated the presentation of the ‘New theory’. 77

Shapiro, Fits, 12-14. Newton, Corrspondence 1, 187-8. Cited and discussed in Shapiro, Fits, 36-38. 79 See the quote above on page 227 78

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Books 2 and 3 concern his later experimental investigations of the colors in thin films and of diffraction. Although Newton’s mathematical perspective is unmistakable in the concepts employed and the quantification effected, the mathematical reasoning at the heart of his understanding of colors is implicit. Opticks presented the new science of colors as an experimental theory. The core of Newton’s optical investigations consisted of the search of phenomenological laws and properties of light. He did speculate on the corpuscular nature of light and color and their properties, but from the onset he barred these from his established theories. In his view, experimental philosophy should not be contaminated by hypotheses or other ill-founded assumptions, as Descartes had done. This level of causation, distinguished from the level of experimentally demonstrated properties, implied so-called hypothetical philosophizing which Newton pursued publicly only once, in his 1675 paper for the Royal Society ‘Hypothesis explaining the properties of light’. Speculations on unobservable matter in motion, did play a part in Newton’s optics however. One of the reasons he dropped the ‘Cartesian’ dispersion law seems to have been that it conflicted with his most private thought on the corpuscular nature of colors. Yet, this remained concealed from its readers. To stress the lucidity of mathematics against the obscurity of natural philosophy was rather a ‘topos’ in the seventeenth-century. Like Newton, Huygens thought that Descartes had gone astray in presuming that the laws of optics could be derived from a priori truths. A mechanistic hypothesis could not by itself prove anything. Unlike Newton however, Huygens did not banish hypotheses from his optics. On the contrary, the question how to establish proper ‘raisons de mechanique’ and built a mathematical science of optics from them, was his main concern. According to Huygens, the causes of the behavior of light were ultimately hypothetical. His problem was how to find the right hypotheses. In the first place, this meant to establish veritably mechanistic causes of the properties of light. As contrasted to the speculations of Hooke and Descartes, Huygens wanted his mechanistic explanations to be comprehensible. That is, a hypothetical mechanism had to be exact and to conform to the established laws of motion. Huygens’ principle defined ethereal waves mathematically and prescribed how a propagated wave could be constructed geometrically. In this way it explained the laws of optics accurately, by means of mathematical derivation. From the perspective of seventeenth-century geometrical optics, Huygens’ principle can be seen as a law. But it was a new kind of law: a law of unobservable waves instead of rays. Light consisted of waves and these could be treated in the same manner as the rays of traditional optics. He defined the properties of these waves in the same law-like manner. By mathematizing the mechanistic causes of the laws of optics, Huygens extended geometrical optics into the realm of the unobservable. For methodological reasons Newton would not allow such a thing, although he was quite capable of mathematizing mechanistic causes. As compared to the

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derivation in Principia, Huygens’ model of colliding particles was fairly economic. Huygens’ focus was on the construction and its applications to variously deflected rays not on the subtleties of ethereal collisions. Causes in optics ought to be comprehensible, that is, be mathematical in the first place. But what about their status? Of course waves were real, they must be, but were they true? By admitting hypotheses Huygens sacrificed the full, indisputable certainty Newton wanted to preserve at all costs. According to him, one could and should conjure up a picture of light propagation, as long as one showed that the established rules of motion did not leave room for alternatives. Such a plausible cause could be used subsequently to derive a possible law of strange refraction. Huygens conjured up a principle to which all laws of optics could be reduced. The principle was demonstrated by confirming experimentally conclusions drawn from it. Such a proof was necessarily indirect and less than fully conclusive. Huygens’ ultimate goal was not the mechanistic theory per se, but a theory that properly explained the laws of optics. These conditioned his explanatory theory and were its ultimate foundation. Generalizing my findings regarding Huygens’ optics, I have tried to show how the traditional mathematical science of optics offers useful clues for the historical understanding for the development of physical optics in the seventeenth century. Geometrical optics is not the only root, in particular not in the case of Newton, but I think it is an important one that has been relatively neglected. To interpret Huygens’ and Newton’s pursuits in optics as extensions of the mathematical science of optics, reveals some noticeable similarities that tend to be overshadowed by the vast differences between them. Let me conclude by setting their eventual publications side by side. There are many resemblances, as Cohen has amply shown. In particular the novelty of the subject combined with the imperfectness of both works, has led him to suggest that Newton may have used Traité de la Lumière as a model for Opticks.80 For my argument, the most conspicuous parallel between Opticks and Traité de la Lumière is the way their mathematical roots are obscured. Although Opticks preserved the deductive structure of definitions, axioms, and propositions, the line of inference was clearly experimental. The deductive structure of Traité de la Lumière is not visualized as such. Moreover, both publications only established the principles of their new mathematical sciences of optics. Traité de la Lumière had been intended as the prelude for a dioptrics in which the mathematical theory would be elaborated by applying the principles to lenses and their configurations. Newton applied his principles to a few problems like the rainbow, but did not elaborate his theory of colors in the way he had done in his lectures. In this way, Traité de la Lumière and Opticks can be said to spotlight their new ways of doing the mathematical science of optics.

80

Cohen, “Missing author”, 30-33.

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6.3 Traité de la Lumière and Huygens’ oeuvre Traité de la Lumière looks like a complete and purposively elaborated whole, an exemplar of a seventeenth-century mathematical physics in which the principles of optics are derived from a mathematized theory of the corpuscular nature of light. Traité de la Lumière is often regarded as exemplary for Huygens’ science as well. In particular, historians has regarded it as a proof of the fundamental role mechanistic philosophy played in his science. Yet, it is risky to base an interpretation on the eventual text, as it barely hints at the winding road towards the final result. In my view, the historicization of Traité de la Lumière of the preceding chapters sheds new light on Huygens’ science in general. In the historical literature, Traité de la Lumière is often seen as a direct response to Descartes’ optics. Sabra first characterizes Huygens’ Cartesianism and then shows how it produced the wave theory.81 Along similar lines, E.J. Dijksterhuis calls Traité de la Lumière the high-point of seventeenth-century mechanistic science and its author the first perfect Cartesian.82 In Traité de la Lumière the mechanistic conception of nature was indeed perfected, but mechanistic science had not given the momentum to its materialization. Seeing it as a response to Descartes fails to account for the fact that mechanistic thinking is virtually absent in his optics prior to the 1670s. At a relatively late stage, Huygens began considering the causes of the laws of optics and only while developing the wave theory did Huygens become a ‘mechanistic thinker’. The new form Traité de la Lumière gave to mechanistic explanation was the outcome of questions pertaining to geometrical optics, rather than some preconceived plan or mechanistic programme. I think that Huygens’ commitment to mechanistic philosophy was not as decisive as is often assumed. His ideas on the nature of light were, of course, based on prevailing mechanistic conceptions, and Huygens was well aware of the problems in Descartes’ optics. Yet, questions of mechanistic theory were not the impetus or drive of his consideration of causes in optics. The problem of strange refraction set it going and its persistence gave it an unexpected twist, eventually resulting in the wave theory as we know it. Huygens’ mechanistic conceptions regarding the nature of light only played a limited role in the development and establishment of the wave theory. He displayed a particular lack of interest in elaborating the mechanistic finesses of his theory, for example by avoiding the question why waves propagate asymmetrically in Iceland crystal. The strict definition of what counts as a ‘raison de mechanique’ is largely my interpretation, as Huygens himself did not explicate it in Traité de la Lumière.

81

“There is in fact evidence to show that Huygens first arrived at his views regarding the nature of light and the mode of its propagation through an examination of Descartes’ ideas.” Sabra, Theories of Light, 198. 82 Dijksterhuis, Mechanization, 503-507.

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The question now is, to what extent this interpretation of Traité de la Lumière stands up with regard to his science in general. Historians’ view of Huygens as a Cartesian at heart has been fostered by his confrontations with Newton over colors and gravity. They have created the impression that Huygens refused to accept Newton’s discoveries because they did not fit his mechanistic conception of nature. I believe that this view of Huygens’ Cartesianism can, and must, be qualified by drawing upon my interpretation of the development of Traité de la Lumière. 6.3.1 HUYGENS’ CARTESIANISM Huygens himself has given ample reason for seeing him as a Cartesian. In 1693 wrote a commentary of Baillet’s biography of Descartes which developed into a reflection upon the virtues of his teachings. He went back to his earliest encounter with Cartesian philosophy: “When I read this book of Principles for the first time it seemed to me that everything in the world went as well as it could, and I believed that, when I found some difficulty in it, it was my fault for not grasping his thought well enough. I was only 15 or 16 years old. But having since then discovered in it from time to time things visibly false, and others very little probable I have well returned from the preoccupation where I had been, and right now I find almost nothing that I can approve of as true in his entire physics, nor in his metaphysics, nor in his meteors. What was very pleasant in the beginning when this philosophy began to appear, is that one understood what Mr. Descartes said, instead of the other philosophers who gave us words that made nothing comprehensible, such as those qualities, substantial forms, intentional species, etc. He rejected more universally than anyone before this impertinent ragbag. But what above all recommended his philosophy, is that he did not confine himself to instilling distaste for what is old, but that he dared to substitute for it causes which one can comprehend of all there is in nature.”83

Concluding his comment with: “Notwithstanding this little amount of truth I find in the book of Principles of Mr. des Cartes, I do not deny that he displayed quite a good deal of wit in fabricating, the way he did, this whole new system, and in giving it such a twist of truth-likeneness as to make infinitely many people satisfied with it and pleased with it. One may also say that by presenting those dogmas with much assurance, and becoming a very celebrated

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OC10, 403. “Il me sembloit lorsque je lus ce livre des Principes la premiere fois que tout alloit le mieux du monde, et je croiois, quand j’y trouvois quelque difficultè, que c’etoit ma faute de ne pas bien comprendre sa pensée. Je n’avois que 15 à 16 ans. Mais y ayant du depuis decouvert de temps en temps des choses visiblement fausses, et d’autres tres peu vraisemblables je fuis fort revenu de la preoccupation ou j’avois estè, et à l’heure qu’il est je ne trouve presque rien que je puisse approuver comme vray dans toute la physique ni metaphysique, ni meteores. Ce qui a fort plu dans le commencement quand cette philosophie à commencè de paroitre, c’est qu’on entendoit que disoit M. des Cartes, au lieu que les autres philosophes nous donnoient des paroles que ne faisoient rien comprendre, comme ces qualitez, formes substantielles, especes intentionnelles, etc. Il a rejettè plus universellement que personne auparavant cet impertinent fatras. Mais ce qui a surtout recommandé sa philosophie, c’est qu’il n’est pas demeurè à donner du degout pour l’ancienne, mais qu’il a osè substituer des causes qu’on peut comprendre du tout ce qu’il y a dans la nature.” Also quoted and translated in Westman, “Problem” 95-96 and 99.

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author, he has excited all the more those who wrote after him to resume it and to try to find something better.”84

This can hardly be read otherwise than as the outline of a program of Cartesian physics. On the basis of the clear and comprehensible philosophical foundation laid by Descartes’ followers would erect the mechanistic science he himself failed to realize. At this moment, Huygens does not tell that not long after his introduction to Cartesian philosophy another protagonist of the new sciences would make an even deeper an more decisive impression on him: Galileo. But first his reception Descartes’ teachings. Undeniably, Huygens’ ideas about the ultimate nature of things have always been uncompromisingly mechanistic. The question is, however, what relevance these philosophical ideas for his science and when, and how, this mechanistic framework was mobilized in his actual investigations. At what moments did Huygens become a mechanistic philosopher, philosophizing about the mechanistic causes of the phenomena? In Traité de la Lumière his mechanistic thinking had a designate yet limited role of providing plausibility of his hypotheses. His focus was on the derivation of the laws of optics. This aspect of Huygens’ optics developed late in his career, probably not before the late 1669s during his sojourn in Paris. In Huygens’ optics in general, La Dioptrique formed the main point of reference, first as regards the dioptrics of telescopes, then the mathematics and mechanics of refraction. Responding to Descartes does indeed form a thread in Huygens’ optics, but one that needs qualification. In the early decades of his career he primarily responded to Descartes’ ideas and achievements insofar as they pertained to the mathematical sciences. ‘Physique’, the consideration of topics in mechanistic philosophy and the response to Descartes’ natural philosophical conceptions, enters his oeuvre at a later stage, in the context of his Académie activities. These qualifications of Huygens’ Cartesianism put his famous confrontations with Newton in 1672 and 1690 in a different perspective. The first, in 1672, concerned Newton’s theory of colors and has been discussed in chapter 3. The second, around 1690, was Huygens’ reaction to Newton’s theory of universal gravity, and I will now discuss it briefly. To do so, we have to go back to the late 1660s, when Huygens laid the foundation of his ideas about the mechanistic cause of gravity. The subtle matter of 1669 On one page in his notebook, probably between September 1667 and February 1668, Huygens listed a series of statements about “… a matter very

84

OC10, 406. “Nonobstant ce peu de veritè que je trouve dans le livre des Principes de Mr. des Cartes, je ne disconviens pas qu’il ait fait paroitre bien de l’esprit à fabriquer, comme il a fait, tout ce systeme nouveau, et a luy donner ce tour de vraisemblance qu’une infinitè de gens s’en contentent et s’y plaisent. On peut encore dire qu’en donnant ces dogmes avec beaucoup d’assurance, et estant devenu autheur tres celebre, il a excitè d’autant plus ceux qui escrivoient apres luy a le reprendre et tacher de trouver quelque chose meilleur.”

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subtle and lithe and that is agitated by an extremely swift movement.”85 The behavior of this subtle matter may explain gravity, the effect of gunpowder, flames, magnetism, elasticity of solid bodies and of air. Huygens did not make clear what united these phenomena in his view. Was it because they all arise from vortices, streams of subtle matter as Descartes had proposed them? This might explain why light was not mentioned among the phenomena subtle matter would explain, as according to his (later) views it was not a vortex but an action propagated through the ether. Or did the phenomena Huygens mentioned constitute a different class? The ensuing discussion at the Académie might suggest that they did, namely, a group of hidden forces in nature that called for mechanistic explanation. It might also suggest that his recent move to Paris somehow occasioned Huygens to make these notes on a subject he had not considered before. He may have been reacting – as was often the case with him – to some discussions going on at the Académie. For the cases of gravity and magnetism Huygens elaborated the idea of vortices in papers he read before the Académie in 1669 and 1680 respectively. The “Traité de l’aimant” of 1680 is mainly an effort to determine the exact way in which turbulences of subtle matter stream on the basis of observations of patterns of filings of iron.86 I will not discuss this paper. Huygens considered his ideas on gravity further in some notes he made probably between February and May 1668.87 Besides remarks about various materials like iron, lead and water, he laid down what he considered its main properties. Gravity works towards the center of the earth; the subtle matter causing it easily penetrates all bodies; weight is proportional to the quantity of matter in a body. In these notes he discussed some of Descartes’ claims, including an experiment to simulate the effect of vortices. It described a vessel containing rotating water in which pieces of lead pushed pieces of wood towards the center. Because lead is heavier than wood, Huygens thought that “this experiment does not serve to show the cause of gravity, …”.88 Gravity ought to be explained by movement alone so there should not be a difference in weight between the various materials in the vessel. In the paper he read at the Académie on 29 August 1669, Huygens described an alternative experiment. In a vessel filled with water, two strings allow a small sphere to move along the diameter. Turning the vessel around its axis will make the water rotate. If it is suddenly made to stop, the water

85

OC19, 553. “Qu’il y a une matiere tres subtile et deliée et qui est agitée d’un mouvement extremement viste.” 86 OC19, 575-581. Huygens appears to have modified his interpretations afterwards, laying particular stress on the pores of the magnet and the direction of the streams: OC19, 591-603. 87 OC19, 625-637. See also: Westfall, Force, 185-186. 88 OC19, 626. “Cette experience ne sert point a faire voir la cause de la pesanteur, …” Descartes had described the experiment in a letter to Mersenne of 16 October 1629, AT2, 593-594.

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will continue to rotate, thereby pushing the sphere towards the center.89 What did this experiment demonstrate? “Well then, having found in nature an effect equal to that of gravity and of which the cause is known it remains to be seen whether one can suppose that something similar happens as regards the earth, namely some movement of matter that constrains bodies to tend towards the center and that matches at the same time with all the other phenomena of gravity.”90

The emphasis should be on the phrase ‘of which the cause is known’. In Huygens’ view the experiment showed that vortices are a plausible cause of gravity because a comparable motion exists in nature. As in Traité de la Lumière, in his 1669 paper on gravity Huygens intended to derive the observable properties of the phenomena with a hypothesis employing proper ‘raisons de mechanique’. There are, however, marked differences. In Traité de la Lumière, Huygens founded his hypothesis on a discussion of the law-like behavior of single particles. The action causing gravity he could not specify in such terms. He did not explain the circularity of vortices by picturing the motions of single ethereal particles. The cause of the push exerted by rotating water was known to Huygens only on the basis of the observation of a macroscopic instance of such a motion, not in terms of established laws describing it. Moreover, he did not mathematize the picture. In other words, his theory of gravity could hardly count as providing proper ‘raisons de mechanique’. Consequently, the explanation of the properties of gravity was not a geometrical derivation as in Traité de la Lumière, but rather an account of the behavior of vortices with respect to heavy bodies. These properties, in their turn, lacked mathematical specification. To the ones mentioned above, Huygens had added Galileo’s law of fall: bodies are continually accelerated proportional to the time.91 It was the only mathematical one, but he explained it in just one paragraph. The first and the third can be said to be potentially mathematical, but in his explanation he did not go beyond a broad, qualitative formulation.92 Huygens’ improvements and adjustments of Descartes’ explanation were rather marginal. With his alternative experiment, he could argue more convincingly that a vortex-like action exists in nature. He brought in some quantity by calculating the speed of the subtle matter at the surface of the earth at 17 times the speed of rotation of the earth. He defined more precisely what needed to be explained (whereby the proportionality of weight and quantity of matter was a new and important insight). All in all, the 1669 89

OC19, 633. OC19, 634. “Or ayant trouvé dans la nature un effect semblable a celuy de la pesanteur et dont la cause est connue il reste a voir si l’ont peut supposer qu’il arrive quelque chose de pareil à l’esgard de la terre, sçavoir quelque mouvement de matiere qui contraigne les corps a tendre au centre et qui convienne en mesme temps a tous les autres phoenomenes de la pesanteur.” 91 OC19, 640. 92 Westfall, Force, 186-187 discusses some of these problems. 90

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paper contained a mechanistic theory that remained essentially qualitative. In the light of the precepts of Traité de la Lumière it could not be called an instance of ‘true philosophy’. The message of the paper could not, however, be mistaken. It made clear what, in Huygens’ view, mechanistic explanation ought to be about. His was the fourth and last of a series of papers on gravity read at the Académie in August 1669. On the 7th, Roberval had opened the debate with a paper that seems to be an express denial of everything Huygens stood for.93 He chose to account for gravity by proposing a mutual attraction between bodies of the same kind. He did specify how such an attraction explained the properties of gravity. He explicitly rejected efforts to explain it by the movement of a subtle matter because he had never seen anything that was not problematic. In his reaction to Huygens’ paper, four weeks later, Roberval said: “… he excludes from nature without proof attractive and expulsive qualities and he wants to introduce without foundation solely sizes, shapes and movement.”94

In reply, Huygens said that he excluded those qualities because “… I search for an intelligible cause of gravity, as it seems to me that it would be saying as much as nothing when attributing the cause why heavy bodies descend to the earth to some attractive quality of the earth or of these bodies themselves, but for the movement, the shape and the sizes of bodies I do not see how one can say that I introduce them without foundation since the senses make use know that these things are in nature.”95

This, then, was the raison d’être of the paper on gravity. In Roberval’s use of attractive qualities he saw a relapse into the incomprehensible thinking Descartes had dispensed with. He would not accept active principles in nature that were not intelligible in terms of matter in motion. But did Huygens regard attractive qualities as incomprehensible only because of his Cartesian leanings? I think not. The 1669 paper on gravity had been preceded by years of mathematical study of motions. In these he had persistently reduced all phenomena of impact and acceleration to the Galilean science of motion. Westfall describes in full detail his “… constant effort to eliminate dynamic concepts and to treat mechanics as kinematics, …”96 Huygens found that whatever could not be expressed in terms of velocities escapes mathematical treatment. In my view, Huygens rejected attractive forces not only because his mechanistic convictions forbade them, but also because such concepts were problematic from a mathematical point 93

OC19, 628-630. Frenicle and Buot followed on 14 and 21 August respectively, summarised in OC19, 630-631. Huygens read his paper – the longest – on 28 August. On 4 September, Roberval and Mariotte gave their comments to which Huygens replied on 23 October; OC19, 640-644. 94 OC19, 640. “… il exclud de la nature sans preuve les qualitez attractives et expulsives et il veut introduire sans fondement les seules grandeurs, les figures et le mouvement.” 95 OC19, 642. “… parce que je cherche une cause intelligible de la pesanteur, car il me semble que ce seroit dire autant que rien que d’attribuer la cause pourquoy les corps pesants descendent vers la terre, a quelque qualité attractive de la terre ou des corps mesmes, mais pour le mouvement la figure et les grandeurs des corps je ne vois pas comment on peut dire que je les introduicts sans fondement puisque les sens nous font connoistre que ces choses sont dans la nature.” 96 Westfall, Force, 177.

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of view. This does not mean that mechanistic philosophy did not influence Huygens’ studies of motion. Mechanistic views were a source of inspiration, for example when he equated gravity and circular motion in 1659.97 Yet, their role was limited and they were virtually absent in his subsequent analysis of circular motion.98 Forces were gradually pushed back and velocity became Huygens’ ultimate concept of motion. Not just because of the dictates of mechanistic philosophy, it was the only way he could deal with motion mathematically. Huygens lack of interest in the mechanistic nature of light, at least the absence of any recorded consideration up to the late 1660s, was not singular. In this sense, his optics is exemplary for his science in general. Before his move to Paris in 1666, philosophizing about the mechanistic nature of things is virtually absent in his writings. As I see it, with his brief note on subtle matter Huygens returned for the first time to the realm of thought that had made such an impression on his youthful mind. By that time his thinking on matter and motion had ripened into a thoroughly Galilean conception that he subsequently injected into mechanistic philosophy. He understood mechanistic philosophy as an ultimately mathematical idiom. Not the ontological idiom of Principia Philosophiae, but the mathematical idiom of the laws of motion of the Discorsi. The 1669 paper on gravity makes clear that the conception of mechanistic philosophy that underlies Traité de la Lumière had already taken shape. In Huygens’ view ‘raisons de mechanique’ invoked matter moving according to established rules and they were intended to formulate hypotheses that explained the observable properties of phenomena. As regards gravity the doctrine had not been realized in full, and in optics it had not yet taken form. The mechanistic nature of light first enters Huygens’ writings with his approving reference to Pardies’ theory in course of the 1669 debate on gravity. Huygens versus Newton In his dispute with Newton on colors Huygens took a more strict position regarding the need for mechanistic explanation than he did in the ‘Projet’ jotted down at the same time. When in September 1672 Huygens finally realized the full import of the new theory of light and colors he (grudgingly) accepted different refrangibility but he did not accept the compound nature of white light.99 According to him, Newton first had to solve the difficulty of explaining it mechanistically. Apparently, Huygens thought that sometimes mechanistic explanation was more than just an optional subject to satisfy ‘the mind that loves to know the reason of everything’ as he phrased it in the ‘Projet’.

97

See above, page 96. Yoder, Unrolling time, 17-19. 99 See above, page 88. 98

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What difficulties Huygens was thinking of, is not clear. I suspect the letters of Pardies had given him the idea that Newton’s theory presupposed an emission conception of light. This he could not accept, as it conflicted with his basic notions about the nature of light. Did he fear a diversity of ‘lights’ would undermine his understanding of the nature of light? His (later) view of light being an action propagated with a velocity depending only upon the nature of the medium, cannot be reconciled with such a diversity. On the other hand, around the same time he had suggested the idea that double refraction, including its odd absence in a second crystal, is caused by two undulations linked to two kinds of particles in Iceland crystal. I think there was a good deal of rhetoric in Huygens’ remark, and I do not believe he had considered the issue in any great detail at this point. It appears that he considered the mere semblance of being unclear in terms of the mechanistic nature of light sufficient to request clarification on this point. As long as Newton had not done so Huygens could not accept his conclusions about the nature of light on the basis of the ‘accident’ of different refrangibility. The dispute, 15 years later, over Principia followed a similar course. Huygens accepted the ‘accident’ of the inverse-square law, but rejected Newton’s conclusions regarding universal gravity. Principia did what Huygens had not been able to do: to unify all forms of accelerated motion.100 He did so by means of a new, mathematized concept of force. Newton considered circular motion in terms of a force that seeks the center and coined the term centripetal force. With this force he could treat all frictionless motions of point masses. Book one of Prinicipia laid down ‘the science of motions that result from any forces whatever’ as they can be investigated from the phenomena.101 The aim of Principia went beyond a mere science of motion. The laws and conditions of motions and forces established in book one were the principles of a philosophy from which the phenomena of nature were to be derived. “For many things lead me to have a suspicion that all phenomena may depend on certain forces by which the particles of bodies, by causes not yet known, either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede.”102

In book three, Newton gave an example of this by unfolding a system of the world founded upon the force of gravity. He showed that the force that holds satellites in their orbit is the same as the force that causes an apple to fall on earth. The centripetal force established mathematically in book one formed the basis. He correlated the centripetal acceleration of the moon and the acceleration of gravity and showed that both are instances of a force that varies inversely as the square of the distance. 100

Westfall (Force, 178-179) discusses a paper from about 1675 (OC18, 496-498) that was the start of a generalised theory of accelerated motion, but in which Huygens failed to see the dynamical equivalence of change of direction and change of linear velocity. 101 Newton, Principia, 382. 102 Newton, Principia, 382-383.

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From the viewpoint of mechanistic orthodoxy the introduction of attractive forces posed, to say it mildly, a problem. In being causes external to matter they seemed to attribute an active quality to it. Newton admitted that he had not yet been able to deduce the cause of gravity from the phenomena, but this did not refrain him from setting forth its properties. “And it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.”103

In definition four, Newton defined ‘impressed force’ descriptively: “Impressed force is the action exerted on a body to change its state either of resting or of moving uniformly straightforward.”104 Forces could thus be measured by changes in the motion of a body, as expressed in the second law: “A change of motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.” 105 The mathematical correlation of the change of motion of a satellite and of an apple convinced Newton that the inverse square law defined gravity as a really existing force of which the cause was yet to be discovered. When Huygens heard of the forthcoming Principia, his response was as might be expected. Fatio informed him in 1687 of the forthcoming publication of a book that would change all of physics. Huygens replied that he was curious to see the demonstrations that Fatio had sketched. “I do not mind that he is no Cartesian as long as he does not give us suppositions like that of attraction.”106 Seemingly, a new Roberval had stood up across the Channel. Upon reading Principia Huygens must have realized, however, that this one was of different stature. Newton did derive the properties of gravity. What is more, he derived a whole science of motion from the concept of gravitational attraction. Here was an able mathematician treading on ground that Huygens himself had explored earlier and on which he had acquired fame. What was Huygens’ reaction upon reading Principia? “Vortices destroyed by Newton. Vortices of spherical movement in their place.”107 In 1690, he published – together with Traité de la Lumière – Discours de la Cause de la Pesanteur, his 1669 paper on gravity extended with a critique of Principia. However, in the second book Newton had demolished Descartes’ notion of vortices. In a discussion of the motion of bodies through fluids, he had demonstrated that a system of vortices could not explain Kepler’s laws of planetary motion and, moreover, could not exist without some external agent. How could Huygens believe that Principia did not refute his explanation of gravity? 103

Newton, Principia, 943. Newton, Principia, 405. 105 Newton, Principia, 416. 106 OC9, 190. “Je veux bien qu’il ne soit pas Cartesien pourveu qu’il ne nous fasse pas des suppositions comme celle de l’attraction.” 107 OC21, 437. “Tourbillons destruits par Newton. Tourbillons de mouvement spherique a la place.” 104

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First of all because he did not believe that Newton intended to explain gravity by attraction: “It would be a different matter if one supposes that gravity is a quality inherent to corporeal matter. But that is what I do not believe Mr. Newton agrees with, because such an hypothesis would lead us far from mathematical or mechanical principles”.108

Secondly, Huygens had reason to believe that Newton’s concept of gravity was not universal. His calculations yielded a degree of oblateness of the earth that stood in sharp contrast to the value Newton had derived on the basis of the supposition of gravitational attraction between its parts.109 In addition, these calculations agreed with the data gathered on the most recent test of his clocks at sea.110 Huygens had not examined Newton’s derivation in detail “… because I likewise do not agree with a principle he supposes in that calculation and elsewhere; which is that all the little parts, …, mutually attract or tend to approach one another. That I would not know how to admit, because I believe to see clearly that the cause of such an attraction is not explicable by any principle of mechanics, nor by rules of motion.”111

Huygens had no objections to Newton’s mathematical demonstrations regarding the inverse-square law per se: “Thus I have nothing against the Vis Centripeta, as Mr. Newton calls it, by which he makes the planets be heavy towards the sun, and the moon towards the earth, but I rather remain in agreement with that without difficulty: … I had not thought either of that regular diminution of gravity, namely, that it is in reciprocal ratio to the squares of the distances from the center: which is a new and very remarkable property of gravity, of which it is well worth to seek the cause.”112

In explicitly refraining from explaining the properties of gravity mechanistically, Newton did not offer an alternative for spherical vortices. Huygens would be happy to fill this gap by means of his alternative to Descartes’ theory. Only, what did vortices explain? He could argue that his spherical vortices were consistent with an inverse-square law. But could he derive the properties of gravitational acceleration in the way he had derived the properties of light rays? No, for he did not show that this kind of movement follows from the behavior of single particles as he did for waves in Traité de la Lumière. Moreover, in the light of book two of Principia,

108

OC21, 474 (Discours, 163). “Ce seroit autre chose si on supposoit que la pesanteur fust une qualité inherente de la matiere corporelle. Mais c’est à quoy je ne crois pas que Mr. Newton consente, parce qu’un telle hypothese nous eloignoit fort des principes Mathematiques ou Mechaniques.” 109 Smith, “Huygens’ empirical challenge”, 2-3. 110 Mahoney, “Determination”, 258-260. 111 OC21, 471 (Discours, 159). “… parce qu’aussi bien je ne suis pas d’accord d’un Principe qu’il suppose dans ce calcul & ailleurs; qui est, que toutes les petites parties, …, s’attirent ou tendent à s’approcher mutuellement. Ce que je ne sçavrois admettre, par ce que je crois voir clairement, que la cause d’une telle attraction n’est point explicable par aucun principe de Mechanique, ni des regles du mouvement.” 112 OC21, 472 (Discours, 160). “Je n’ay donc rien contre la Vis Centripeta, comme Mr. Newton l’appele, par la quelle il fait peser les Planetes vers le Soleil, & la Lune vers la Terre, mais j’en demeure d’accord sans difficulté: … Je n’avois pensé non plus à cette diminution reglée de la pesanteur, sçavoir qu’elle estoit en raison reciproque des quarrez des distances du centre: qui est une nouvelle & fort remarquable proprieté de la pesanteur, dont il vaut bien la peine de chercher la raison.”

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Huygens should have been more cautious regarding claims about such motions. Why did Huygens publish Discours? Huygens may have felt that Newton challenged him on his own territory, having seen things – the inverse square law – he had admittedly missed. With Principia Newton threatened to eclipse the stature of Horologium Oscillatorium and of its author as one of Europe’s most eminent scholars. Fortunately, Huygens still had this old paper lying around that treated of just the thing Newton refused to do: the mechanistic nature of gravity. So, by publishing Discours, Huygens could show that he could not yet be written off and, in one stroke, how such occult qualities ought to be properly dealt with. Besides, the 1669 paper on gravity was already going to be published anyway. It had been among the papers he had, in 1686, allowed De la Hire to publish.113 Publishing an extended version offered Huygens the opportunity to make some corrections and additions with a view to Principia.114 The confrontations between Huygens and Newton ran a most unfortunate course. Their philosophical differences prevented a fruitful exchange of ideas on the mathematical ground they shared. Discours offered the very kind of explanation Newton refused to give. Not only did it employ hypothetical objects like the particles and the streams of subtle matter, but it also assumed laws applicable to them that lacked foundation. It was the kind of reasoning Newton would not even do in private and one we would not expect from the author of Traité de la Lumière. For Newton this made no difference, Huygens’ waves were as speculative as his vortices and neither had a place in good science. Still, Huygens and Newton ultimately wanted the same: to do better where Descartes had been led astray. This meant that the ‘raisons de mechanique’ ought to be based on the established laws of motion and were exempt from a priori truth. Descartes, Huygens observed in his comment on Baillet’s biography, “… should have proposed to us his system of physics as an essay of what one can say with probability in this science whilst admitting nothing but principles of mechanics…”115

Newton would agree, but he gave a different interpretation. To the ‘mechanical principles’ he added a new, mathematized concept of force. This was something Huygens could not do. In the end, he remained faithful to mechanistic philosophy by restricting its principles to their mathematical core: Galileo’s science of motion. At the same time, however, in his wave theory he was the first to realize Descartes’ ideal of a mathematical physics. He freely applied the Galilean laws of motion to the imperceptible particles 113

See above, page 222. Mahoney argues that Discours offered Huygens an opportunity to publish corrected values for the Earth’s rotation, resulting from the V.O.C. trials of his clock. Mahoney, “Determination”, 259. 115 OC10, 405. “Il devoit nous proposer son systeme de physique comme un essay de ce qu’on pouvoit dire de vraisemblable dans cette science en n’admettant que les principes de mechanique …” 114

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of the ether. Thus he mathematized the mechanistic foundations of natural science, which Newton had not dared to. As regards the essential probability of mechanistic explanations, Newton had drawn a conclusion opposite to Huygens’. If speculations lead you astray, then stop speculating (at least in public), Newton said. If they do so, speculate better, Huygens would say. Huygens’ self-image We have seen, in his comments on Baillet, how Huygens saw the merits and faults of Descartes’ science. According to Huygens, Descartes had shown what the principles of natural philosophy ought to be like, but at the same time he had failed to elaborate them properly in explaining particular phenomena. Still, if he had been too confident of the truth of his ideas, this may have been a stimulus for so many to do better. The picture returns in Discours: “I confess that his essays, and his views, although false, have served to open to me the road to what I have found on this very subject.”116

Yet the clear and distinct ideas that had made such an impression on the youthful Huygens were activated only occasionally during his scholarly career. His science consisted primarily of mathematical inquiry, sometimes inspired by his mechanistic thinking. He turned mechanistic philosopher only rarely. He did so, in my view, mainly when he saw ghosts of the occult thinking he thought Descartes had disposed of.117 And he did so in Traité de la Lumière, establishing the ‘raisons de mechanique’ of the laws of optics. Or should we say that, with his waves, he went beyond mechanistic philosophy by applying Galileo’s science of motion to the imperceptible particles of the ether? In his comments on Baillet’s biography, Huygens moved from a reflection upon Descartes’ merits to a discussion of the virtues of other natural inquirers. This is illuminating regarding the way he saw himself. The ancient atomists had had the right categories, but could not explain natural phenomena. Moderns like Telesio, Campanella and Gilbert were no better than Aristotelians in that they maintained occult qualities. Gassendi and Bacon exposed the inadequacies of Peripatetic philosophy but, like so many moderns, lacked understanding in mathematics. Only one man could stand the test of Huygens’ critique: “For wit and knowledge of Mathematics Galileo had all that is necessary to make progress in physics, and one must acknowledge that he has been the first to make beautiful discoveries concerning the nature of motion although he has left very considerable ones still to be made. He had neither the audacity nor the conceit to want to take it upon himself to explain all natural causes, nor the vanity to wish to be head of a sect. He was modest and loved the truth too much; he believed moreover to have

116

OC21, 446 (Discours, 127). “Et cependant j’avoue que ses essais, & ses vuës, quoyque fausses, ont servi à m’ouvrir le chemin à ce que j’ay trouvé sur ce mesme sujet.” 117 Probably, in the eyes of new age thinkers like Lewis Mumford, this would turn Huygens into a criminal of the stature of Galileo and Descartes. See Vanheste, Copernicus is ziek, 40-45.

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acquired sufficient reputation – one which would endure forever by his new discoveries. But Mr. Descartes, who seems to me to have been very jealous of Galileo’s renown, [and who] had this great urge to pass for the author of a new philosophy, as appears from his efforts and his hopes to have it taught at the academies instead of Aristotle’s, or from his wish that the Society of Jesus embrace it: but in the end because he stuck at all costs to things once he had put them forward, even though often very wrong ones.” 118

It is not difficult to understand where Huygens placed himself: as the heir of the unpretentious mathematician rather than the cocky philosopher. Westman has pointed out he was actually describing himself in his picture of Galileo as a humble, moderate lover of the truth.119 With mathematics rather than philosophy he had made progress in physics. In Traité de la Lumière he had succeeded to extend it to the causes of natural phenomena. In it, Huygens took the modest stance he ascribed to Galileo. His theory was not all-compassing, as he had nothing to say about colors.120 After his explanation of strange refraction he described a new phenomenon (polarization) he had discovered adding that his theory could not explain it: “For although until now I have not been able to find its cause, I do not want to refrain from indicating it, in order to give occasion to others to seek it. It seems that still other suppositions would have to be made besides the ones I have made; which nevertheless will not fail to preserve all their probability, after having been confirmed by so many proofs.”121

The limited reach of his theory did not alter the fact that only waves could explain the reflections and refractions of light rays properly. The focus and the foundation of Traité de la Lumière were on the mathematical science of optics. In its opening lines Huygens was explicit about his conception of optics. Optics was one of the sciences where geometry is applied to matter. He did not explicitly say that he extended this to the unobservable matter of mechanistic philosophy. He applied geometry 118

OC10, 404. “Galilee avoit du costè de l’esprit, et de la connoissance des Mathematiques tout ce qu’il faut pour faire des progres dans la Physique, et il faut avouer qu’il a estè le premier à faire de belles decouvertes touchant la nature du mouvement, quoy qu’il en ait laissè de tres considerables à faire. Il n’a pas eu tant de hardiesse ni de presomption que de vouloir entrepretendre d’expliquer toutes les causes naturelles, ni la vanitè de vouloir estre chef de secte. Il estoit modeste et aimoit trop la veritè; il croioit d’ailleurs avoir acquis assez de reputation et qui devoit durer à jamais par ses nouvelles decouvertes. Mais M. des Cartes qui me paroit avoir estè fort jaloux de la renommee de Galilee avoit cette grande envie de passer pour autheur d’une nouvelle philosophie. Ce qui paroit par ses efforts et ses esperances de la faire enseigner aux academies à la place de celle d’Aristote; de ce qu’il souhaitoit que la societè des Jesuites l’embrassast: et en fin parce qu’il soutenoit a tort et a travers les choses qu’il avoit une fois avancees, quoyque souvent tres fausses.” 119 Westman, “Problem”, 97. Yet, in what seems to be a slip of the eye, Westman misses the phrase: “… ni de presomption que de vouloir entrepretendre d’expliquer toutes les causes naturelles …” 120 Traité, “Preface”. 121 Traité, 88-89. “Car bien que je n’en aie pas pû trouver jusqu’icy la cause, je ne veux pas laisser pour cela de l’indiquer, afin de donner occasion à d’autres de la chercher. Il semble qu’il faudroit faire encore d’autres suppositions outre celles que j’ay faites; qui ne laisseront pas pour cela de garder toute leur vraisemblance, apres avoir esté confirmées par tant de preuves.”

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to matter, observable and unobservable alike, without racking his brain over philosophical issues involved. Mechanistic explanations raised a methodological problem – as compared to the ordinary laws of mathematical science – and as a matter of course Huygens explained that these required a different mode of inference. Huygens had a clear conception of proper method, without putting this into a prescriptive doctrine. This does not mean that he did not know methodological constraints. He relied on mathematics as he had learned to pursue it and submitted to it unconditionally. In this way he had discovered a new property of light, namely the exact way in which waves of light propagate. Huygens did not realize that his principle of wave propagation and its application to strange refraction went beyond traditional ways of studying light mathematically. He looked upon himself as a mathematician who had advanced seventeenth-century science by new discoveries and better theories. He perfected the science of motions and the measure of time, the telescope and telescopic observation, and so on. He believed that he had discovered the true causes of refractions and that he had perfected the study of mechanistic causes in optics. He did not realize that, with the latter, he had pursued optics in a new, unwittingly and inadvertently revolutionary kind of way. 6.3.2 THE RECEPTION OF HUYGENS Contemporaries looked upon Huygens as a prominent mathematician. He made his name with his astronomical and mechanical inventions and discoveries. In optics he had raised high expectations since the 1660s. During the late 1670s, many thought he would finally publish his long awaited dioptrics. Huygens’ father had begun to spread word that his son’s treatise was ready for publication.122 On 15 may 1679, Leeuwenhoek wrote “Sir, your father writes me …, that the main part of your dioptrica is almost ready to be printed from a good copy, …”123 Because Hooke had begun publishing papers on similar subjects, Leeuwenhoek urged him to publish his own.124 On 8 September 1679, Leibniz wrote to Huygens to breathe new life into their relation. He told him about his recent work in mathematics and said he had heard “… from Mr. de Mariotte that you will soon give the Dioptrique so long desired.”125 Still, he would have to wait another decade. When finally published, Traité was praised by several men. Leibniz was impressed. He was surprised by the ease with which the properties of light could be explained by waves, but when he proceeded to the explanation of 122

For example the letter of Susanna of 1 February 1680: OC8, 272. OC8, 166-167. “U.Edele hr. vader schrijft mij …, dat het voornaemste deel van U.Edele dioptica (sic) bij na in staet is, om uijt een goede copie gedruct te connen werden, …” 124 OC8, 166-167. Early 1678, Hooke had commenced publishing Lectures and collections made by Robert Hooke which were continued in 1679 under the title Philosophical Collections (the successor of Philosophical Transactions). 125 OC8, 214. “… de Mr. de Mariotte que vous donnerés bien tost la Dioptrique si longtemps souhaittée.” 123

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strange refraction, his esteem turned into admiration.126 He valued Huygens’ theory more highly than Pardies’, let alone Ango’s, but he still wanted to hear Huygens’ opinion on colors and on diffraction. Papin’s response was similar, but he had doubts whether the hypothesis explaining strange refraction could be true.127 From Paris, too, Huygens received appreciation.128 From London, Fatio called it a pleasure to read it but also had some comments.129 Amidst profuse apologies, he confessed he did not quite understand Huygens’ explanation of strange refraction. How could the wave strike the eye along rays not perpendicular to it? Huygens’ reply is lost, but Fatio withdrew his doubts completely and apologized for making objections without having studied Huygens’ explanation in a satisfactory manner.130 The correspondence with Fatio is revealing in another regard. The main part of Fatio’s comments did not concern Traité de la Lumière but Discours de la Cause de la Pesanteur. In this Fatio was no exception: Principia dominated even Huygens’ own correspondence. It therefor does not come as a surprise that the reactions to his own publication were dominated by Discours, his response to Principia. In this sense, Traité de la Lumière fell between two stools of gravity. After Huygens had gathered some compliments, it more or less disappeared from his correspondence. Huygens took no trouble to change this; apparently he had lost interest again. Early 1690, unbeknown to Huygens, Traité de la Lumière was discussed at the Royal Society. Hooke raised objections, mainly pointing out its failure to account for colors. Halley responded with a paper in which he discussed the virtues of wave theories, preferring Huygens’ over Hooke’s.131 Newton, who owned and dog-eared two copies of Traité de la Lumière, first referred to Huygens in the second English edition of Opticks in 1717.132 The subject returned once more in a letter Leibniz wrote on 26 April 1694. Huygens’ theory, he reported, had been expounded by Martin Knorre at the University of Wittenberg.133 Along with his own letter, Leibniz sent a copy of a letter he had received from Fatio, who reported on his and Newton’s opinions concerning the nature of light and gravity. Fatio and Newton, Leibniz wrote, still upheld an emission conception of light and explained different refrangibility with it. Leibniz still had problems with such 126

OC9, 522. OC9, 559-560. It seemed to imply that Iceland crystal was not a homogeneous substance, which Huygens presupposed in order to explain refraction. Huygens responded that this was a question concerning the structure of the crystal, on which he had only some speculations: OC10, 177-179. 128 For example from La Hire and Huet: OC10, 5-6; 53. 129 OC9, 381. 130 OC9, 410. On Fatio’s letter, he made a note: “IC is the light ray, but it affects the eye as if coming along the perpendicular of the wave IK”: OC9, 388. “IC est le rayon de lumiere, mais il agira sur l’oeil comme venant suivant la perpendiculaire de l’onde IK.” (See Figure 70) 131 Albury, “Halley and Traité de la Lumière”, 449-454. Albury seems to miss the point that Halley was discussing wave theories only, by claiming that the paper displays his rejection of Newton’s optics. 132 Cohen, “Missing author”, 32. 133 OC10, 601. 127

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a conception, and noted that Mariotte had not been able to verify the invariability of colored rays. He thought it was difficult to explain refraction with light conceived as particles. He still preferred Huygens’ hypothesis but also wanted his opinion on the matters discussed by Fatio.134 In reply to Leibniz, Huygens wrote he was glad that his theory was being approved of, although he was not pleased to see it equated with those of Hooke and Pardies as the Wittenberg professor did.135 For one thing, Pardies and Hooke had not been able to explain the “…bizarreries du cristal d’Islande, …” The explanation of strange refraction was the ‘Experimentum Crucis’ of his theory and as long as they could not explain ordinary refraction satisfactorily – let alone strange refraction – their views lacked a solid foundation. As regards Newton’s nice and interesting observations on different refrangibility, Huygens was of the same opinion as of universal gravitation: “… he does not explain what color in those rays is, and it is because of this that I, too, have not been fully satisfied until now.”136 What Huygens’ own thoughts on colors were, he once again did not tell. Leibniz went so far as to place Traité de la Lumière at the same level as Principia: these were in his view the two most important works in contemporary science.137 In this, Leibniz was the exception. Huygens believed that he had surpassed all his predecessors in establishing a plausible cause for the laws of optics. It was not hailed by his contemporaries as the success he saw in it. Newton, for one, was not convinced by its argument as he rejected a wave conception of light altogether. And he probably was not impressed by Huygens’ arguments against an emission conception. But he did take the trouble to discuss the difficulties of a wave conception in detail. In addition, he gave his own account of strange refraction, apparently in order to undo the uniqueness of Huygens’ explanation. After all, one strength of Huygens’ wave theory was that it was the only one that could explain this phenomenon. As Fatio put it: “You, Monsieur, always have the advantage, that one cannot claim to have something better until one has explained the phenomena of Iceland crystal so successfully …”138

This was true for the few – Leibniz, Papin, de la Hire – who accepted Huygens’ theory, but not for the majority in the eighteenth century who did not.139 Even ’s Gravesande, who published Opera Varia and Opera Reliqua of Huygens, in optics followed Newton in his widespread textbook Physices elementa mathematica of 1720. 134

OC10, 602. According to Fatio, Newton’s view that space is empty posed a serious but not insurmountable problem for Huygens’ theory: OC10, 606. 135 OC10, 611-612. 136 OC10, 613. “… il n’explique pas ce que c’est que la couleur dans ces raions, et c’est en quoy je ne me suis pas pleinement satisfait non plus jusqu’à present.” 137 Heinekamp, “Huygens vu par Leibniz”, 108. 138 OC9, 381; translation: Shapiro, “Kinematic optics”, 244. 139 Shapiro, “Kinematic optics”, 245. Shapiro offers a concise discussion of the way several scholars understood and reacted upon the physical concepts of Huygens’ wave theory: 245-252.

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A combination of several factors caused the almost complete rejection of Huygens’ theory in the eighteenth century.140 His account of rectilinear propagation was generally thought to be inadequate, even by those who did not follow Newton’s emission conception of light. The scope of the theory was limited. In particular Huygens’ omission of colors was a drawback in comparison with Opticks. In addition, the phenomenon on which Huygens’ theory was founded – strange refraction – was largely ignored during the eighteenth century. Students of crystallography consulted Traité de la Lumière, but only for his description of the crystal.141 Only with the studies of Haüy, Malus, and Wollaston the optical theory of strange refraction was rediscovered. In some German textbooks Huygens’ theory was appealed to, but adopted only in broad outline.142 Huygens’ explanations of specific phenomena such as rectilinear propagation were passed over. Hakfoort explains this by the mathematical content of Huygens’ theory, that exceeded the goals of books on natural philosophy, in which the nature of light was customarily being discussed. Hakfoort broadens his explanation of the neglect of Traité de la Lumière by pointing out a disciplinary factor. Huygens’ mathematical treatment of mechanistic causes eluded the capacities and interests of scholars dealing with such explanatory theories. As had been the case previous to Huygens, they resorted to qualitative explanations. Mathematicians, on the other hand, continued to confine themselves to the behavior of light rays irrespective of its underlying causes. According to Hakfoort, the eighteenth-century disciplinary barriers between mathematics and natural philosophy, and the lack of savants who successfully overcame them, caused Traité de la Lumière to fall in neglect.143 Opticks, on the other hand, had the advantage that it could be read as an experimental theory of colors. Newton’s queries offered a qualitative account of the nature of light and were adopted accordingly. This would mean that the true ‘raisons de mechanique’ Huygens prided himself to have established were an important factor in the neglect of his theory. What Huygens considered as a comprehensible explanation eluded the savants of the eighteenth century. If Traité de la Lumière fell into oblivion, such was not the fate of Huygens’ oeuvre as a whole. Smith’s A Compleat System of Opticks is exemplary in this regard. Whereas he ignored Traité de la Lumière – even for strange refraction he adopted Newton’s account – his praise for Huygens’ accomplishments in both practical and theoretical dioptrics was high. If Huygens was forgotten as a mechanistic philosopher, he remained renowned as a mathematician. In particular Horologium Oscillatorium earned him fame. In perfecting Galileo’s science of motion, he was perceived as preparing the ground for Newton. 140

Hakfoort, Euler, 53. Shapiro, “Kinematic optics”, 257. 142 Hakfoort, Euler, 119-126. 143 Hakfoort, Euler, 183-185. 141

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Some ambivalence resounds in the judgements of men like Laplace and Lagrange. Huygens brought the science of motion to a new level, but did not develop this into a new science like Newton.144 By the end of the eighteenth century Huygens’ investigation of strange refraction underwent a more positive valuation. However, his natural philosophical ideas met with doubts. In his Exposition du système du Monde of 1796 Laplace observed: “ … [the insufficient explanation of spheroidal waves and polarization] combined with the difficulties the theory of luminous waves presents is the cause why Newton and the majority of geometers who have followed him failed to appreciate with justice the law Huygens attached to it.”145

Malus praised Huygens for finding an accurate law of strange refraction, but lamented his troublesome ‘system of undulations’: “That law, considered in itself and cleared of the explanation to which Huygens had attached it is one of the finest discoveries of that celebrated geometer.”146

He did not realize that this troublesome system was essential to the discovery of the law and was inherently connected to it.147 He did not recognize the new way of studying light mathematically that was being pursued in Traité de la Lumière. His physical optics still had to be rediscovered. Nineteenthcentury students like Fresnel developed a Huygens-like way of doing optics in which microphysical hypotheses were the starting point of the investigation – a way Huygens himself had barely recognized as a kind of ‘Optique’ that went beyond geometrical optics.

144

Bachelard, “Influence”, 244-247. Laplace, Oeuvres Complètes 6, 353-354. “… joint aux difficultés que présente la théorie des ondes lumineuses est la cause pour laquelle Newton et la plupart des géomètres qui l’ont suivi n’ont pas justement apprécié la loi qu’Huygens y avait attachée.” 146 Malus, Theorie de la double réfraction…, 289-290. “Cette loi, considérée en elle-même et débarrassée de l’explication à laquelle Huygens l’avait attachée, est une des plus belles découvertes de ce célèbre géomètre.” 147 The eighteenth-century development of mechanics, in which mathematical science and natural philosophy tacitly drifted apart, is illuminating in this regard. See Boudri, Het mechanische van de mechanica, in particular 257-265. 145

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Chapter 7 Conclusion: Lenses & Waves A sketch of Huygens in the light of his optics

This study has been aimed at finding out the coming into being of Traité de la Lumière. How did Huygens’ work in optics develop into the wave theory of light, a new way of doing optics in which the laws of optics are derived from an experimentally confirmed, mathematized theory of the mechanistic nature of light? Huygens’ work in optics comprises in the first place his dioptrical studies, but these have hardly been taken into account by historians. Dioptrica has very little been studied historically and its potential relevance for understanding Traité de la Lumière has not been considered previously. Huygens’ optics tends to be identified with his wave theory and the mechanistic reasoning in it is often taken as a natural part of his science. Yet, little in Huygens’ work in optics prior to 1672 gives reason to suspect that he would have given a new form to mechanistic science by 1679. Taking Dioptrica into account while discussing the development of Huygens’ optics raises a historical problem. The kind of theorizing pursued in Traité de la Lumière is completely absent from Dioptrica. Generally speaking, Huygens does not appear to have had a particular interest in mechanistic topics prior to the 1670s. The Huygens of Dioptrica was a seventeenth-century mathematician who does not at all resemble the alleged, ‘first thoroughgoing Cartesian’ of Traité de la Lumière. What I wish to do now, is to forget about Huygens’ Cartesianism for a while and focus on the Huygens of Dioptrica. By comparing his pursuits in dioptrics to those of his precursors and contemporaries a picture has arisen of a mathematician with an idiosyncratic approach to questions of mathematical theory. I shall generalize this picture to include his pursuits in other fields and make a sketch of his scientific persona. Only then shall I ask how Traité de la Lumière may fit in and how we should assess his alleged Cartesianism. A seventeenth-century Archimedes The Huygens who went to Paris in 1666 pursued the various branches of the mathematical sciences: geometry, arithmetic, statics, optics, harmonics, some astronomy, and the study of motions. He pursued these brilliantly, and marked himself off by a particular sense of practical possibilities. Huygens’ orientation on instruments in Dioptrica was unique for its day, having his theoretical investigations guided by questions of practical relevance.

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Huygens was the first (and for a long time the only one) to pursue the question raised by Kepler right after the invention of the telescope: how can we understand its working in a mathematical way? Students of dioptrics like Descartes, Barrow, and Newton focused on solving sophisticated mathematical problems like determining aplanatic surfaces and analyzing optical imagery. They did not elaborate their findings to explain the dioptrical properties of ordinary lenses and their configurations. Astronomical observers, starting with Galileo, did not elaborate a dioptrical theory of telescopes either. Only when the telescope was turned, towards 1670, into an instrument of precision did its users like Flamsteed and Picard begin to bother about questions of dioptrics. Without Huygens’ mathematical proficiency they could not, however, obtain the rigor and generality of Tractatus and De Aberratione. But Huygens was not of help, he never came to publish his dioptrics. Despite the fact that he always had an open eye for practical implications, the orientation on instruments characteristic of Dioptrica cannot be directly generalized. The organ did not direct his studies of consonance, as clocks did not direct his studies of motion.1 Still, in a broader sense Dioptrica does reveal a particular feature of Huygens’ science. Whereas Descartes contented himself with establishing the principles of refraction and perfect vision, Huygens applied the sine law to establish the properties of actual lenses and telescopes. Whereas others analyzed the mathematics of lenses in order to find perfectly focusing surfaces, he did so in order to understand the properties of real lenses and fathom their imperfections mathematically. What Huygens did in Dioptrica – apart from the practical relevance of his pursuits – was elaborating mathematical theory by applying general principles to specific problems of real objects. Huygens ‘applied geometry to matter’, to use his phrase in Traité de la Lumière. Real, rather than ideal matter. Application in the sense of elaborating established mathematical theory for particular cases, rather than mathematization of new phenomena in the sense Newton did with colors. Even in his theories of impact and circular motion he substantially built upon mathematical foundations already laid by Galileo. As contrasted to Newton, he mathematized no phenomena that had not already latently been mathematized. Rather than establishing an investigation into the physics of consonance, he elaborated the mathematics of the coincidence theory. In dioptrics, he confined himself to the analysis of the properties of refracted rays and left colors for what they were. Brilliantly pursuing mathematical reasoning, he rarely went beyond the established boundaries of the mathematical sciences. One cannot escape the impression that the elaboration of mathematical theory for particular problems interested him more than laying new 1

Although these have never, to my knowledge, been studied from the viewpoint of the relationship between theory and practice.

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foundations. Bos’ aptly wrote: “Huygens as a mathematician was not a man of abstract theories and methods, his preference lay towards the use of these to solve problems, preferably problems in physics.”2 In analytical geometry he did not develop new methods but applied existing methods to new curves.3 Having established a theory of circular motion, Huygens quickly moved on to apply it to a physical pendulum. In the broad sense of a propensity to application, his dioptrics seems to me typical of Huygens’ mathematics in general. Such elaborating of theories gives rise to a different kind of science than the development of new principles. Lowbrow science so to say, that distinguishes itself by the complexity of problems and the elegance of solutions. It is mathematization in the reverse direction of the process we usually associate with the term; it is exploring the richness of theory by deriving specific theorems from its primary principles, rather than unfolding the mathematical nature of new phenomena. In the hands of Huygens, elaborating mathematical theory took on a special form. His theorems where not solutions to problems merely arising from theory, but applications to concrete, physical objects like lenses and pendulums. As contrasted to the mathematical mirrors of perspectiva, he applied the principles of optics to real lenses and telescopes. This idiosyncrasy of Huygens stands out clearest in comparison with Barrow, an equally gifted mathematician who stuck to the abstractions of optical theory. Huygens’ mathematics was applied in a more modern sense of the word, to real objects rather than abstract puzzles. With this, Huygens stood out among his contemporaries, dealing with problems others left aside, took for granted, or simply did not notice. Huygens applied geometry to things, real things, we can paraphrase the line from Traité de la Lumière. The application of mathematical theory to real (rather than ideal) objects was not very common in the seventeenth century, all the more so because for Huygns it seems to have been an end in itself. Newton was capable of this kind of elaboration, too, and he did so in his application of his theory of universal gravity to the system of the world. Yet, this exercise had the higher goal of giving an experimental proof of its principles.4 Similar epistemological aims are absent with Huygens. In his case application seems to be a natural part of mathematics, a challenge in its own right. This challenge then consists of tackling problems with ever greater complexity. Getting a grip on an increasing number of parameters, from the foci of spherical lenses to their aberrations; from ideal pendulums to physical pendulums. It is as if for Huygens the fun only started when the principles of a phenomenon had already been established. Unlike Newton or Galileo, he did not bother too much about epistemological, methodological, or ontological difficulties the application of mathematics to nature might give 2

Bos, “Huygens and mathematics”, 126. Bos, “Huygens and mathematics”, 143-144. 4 Cohen, Newtonian revolution, 62-64 and 100. 3

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rise to. In this sense Huygens’ mathematics is indeed lowbrow. He was more fascinated by the frayed fringes of mathematics when it came down to penetrating the behavior of concrete objects like brass pendulums and glass lenses, than in developing indisputable foundations or in gaining access to the truth about nature. However tempting, it would be too easy to explain the factual character of his mathematics by some sort of artisanal attitude. True, one can discern interesting parallels between artisanal practice and Huygens’ mathematics. The visual reasoning by which his mathematics has been characterized has also been pointed out to typify the way of thinking of the craftsman and the engineer.5 Like a craftsman wants things to work, Huygens first of all wanted to get the mathematics right. By his propensity for application he had to reckon with parameters eluding mathematical theory, as a craftsman has to cope with the imperfections of concrete materials standing in the way of a perfectly functioning apparatus. In either case this yields a different view of knowledge as never final and always open to improvement. As said, an explanation of the character of Huygens’ mathematics along such lines would be too easy. De Aberratione suggests that he did not fully grasp the mismatch between mathematical and artisanal knowledge. Yet, he probably had a clearer few of the gap between science and technology than anybody else in the seventeenth century, and he came closest to bridging it. In the end, however, Huygens could not integrate his theoretical and practical abilities. Huygens always had a keen eye for useful applications, but these did not drive his theoretical studies. Moreover, his interest in mathematical theory went beyond mere instrumentalism, in view of the way he sought to establish general theories that made possible a rigorous mathematical analysis of the subject at hand. Even in dioptrics, where the tie between theory and practice was closest, his practical approach cannot be said to originate in practice. Tractatus was written prior to his work on lens grinding and his telescopic observations. At the end of chapter three I have pointed out that, however strange it may seem, his orientation on telescopes emanated from itself. The same applies, I believe, to his approach in general. The Huygens I see was fascinated by figuring out the mathematics of real, tangible things and this concrete puzzling was to him of intrinsic value.6 All in all, I see quite a lot of coherence in Huygens’ activities before his move to Paris. First of all, most formed a part of the mathematical sciences.7 Secondly, he displayed a marked predilection for the elaboration of mathematical theory, including its application to concrete objects and 5

Bos, “Huygens and mathematics”, 132 and Ferguson, Engineering and the mind’s eye, 1-12. 6 In a way, his tutor Henricus Bruno foresaw this with the fourteen-year old Christiaan, when he wrote Constantijn sr. that they would have to fear that he might turn into an engineer, given the his fascination and skill with taking apart clocks. OC1, 552. 7 His work on pumps in 1661 being the most notable exception, but in this his principal interest was in apparatus rather than vacuum. For an overview see Sparnaay, Adventures in vacuums.

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phenomena. The coherence I see is therefore not one of content of theories, but one of a common approach and of disciplinary-connected fields of study. He believed in the power and fertility of rigorous mathematical reasoning, as opposed to the mere empiricism of ordinary craftsmen. Our pre-Parisian Huygens was a mathematician, a seventeenth-century mathematician with an idiosyncratic approach. A new Archimedes, Mersenne concluded when he was confronted with the youthful Huygens.8 From mathematics to mechanisms What happens, we may now ask, when a mathematician of this Archemidean inclination meddles with questions of the mechanistic nature of light? Nothing special needs to happen and nothing did at first. The nature of light became of interest to Huygens when he needed a preparatory chapter for his ‘Dioptrique’ that would explain the laws of optics. No problem, Pardies had shown how refraction could be explained by waves of light. Only an exotic phenomenon displayed by Iceland crystal posed a bit of a problem. A refracted perpendicular ray negated the perpendicularity of rays and waves that was crucial to Pardies’ explanation. Although the problem of strange refraction thus pertained to the wavelike nature of light, Huygens first approached the phenomenon in the meanwhile traditional, mathematician’s way. He sought a law of strange refraction in terms of the properties of rays. Not surprisingly, the law he found did not solve the problem of strange refraction. Five years later, Huygens returned to the problem. And this time something special did happen. Following on his analysis of waves refracted by curved surfaces, he considered the question what happened to waves when they traverse Iceland crystal. The special thing is that he now took the propagation of waves mathematically. He defined a wave as the result of a disturbance propagated with a specific velocity in all directions, which he could then apply by geometrical construction only. At the background was the conviction that the mechanics of wave propagation ought to follow from the laws of motion. But the mechanistic picture was explicated – and maybe also recognized – only afterwards. In the notes of 1677 we see Huygens less concerned about the broad ideas of his principle and of spheroidal waves than about their mathematical elaboration. The same line of reasoning that explained refraction should also explain the other properties of light rays, including strange refraction. The result was a law of wave propagation yielding an indissoluble tie between the mechanistic nature of light and the laws governing the behavior of rays. What Huygens did not realize, was that he had not just solved another problem in optics. It was a problem regarding the physical foundations of geometrical optics, but Huygens had phrased it in a particular way. Reconciling strange refraction with waves was a problem of reconciling 8

Yoder, Unrolling time, 179. OC1, 47. “Je ne croy pas s’il continue, qu’il ne surpasse quelque jour Archimede.”

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mathematical description and physical explanation. That had not been Bartholinus’ problem, who sought only a correct mathematical description of the phenomenon. As a matter of fact, few students of optics made a problem of the coherence of physical explanation and mathematical laws. Kepler, Descartes, Newton, and possibly Pardies were likely to recognize the problem. The first two, who did not know strange refraction, did not solve the problem of ordinary refraction in a satisfactorily coherent manner. Pardies saw strange refraction as a problem of the crystal, not as a problem of waves. And Newton … With respect to strange refraction he avoided the problem by proposing a law of strangely refracted rays without even suggesting, whether in print or in private, a possible explanation in terms of light particles. This law happened to be identical with the first stab Huygens had made at the problem of strange refraction but had failed to solve it. With a surefootedness that can only be called astonishing Huygens had taken precisely this problem of reconciling mathematical description and physical explanation seriously and brought it to a fortunate conclusion. Posing the problem in this way was, however, not a matter of course, nor was the eventual solution. Huygens did not realize that he was doing something new in a broader sense. In his view, he had merely solved the problem of strange refraction. And in a sense he was right. He had set his teeth in another challenging mathematical problem: reconciling waves and strange refraction. Just as he did not content himself with rough answers about pendulums and lenses, he wanted to get the mathematics of wave propagation right. Given the conscientiousness with which he handled all problems, it is quite natural that Huygens ended up with a coherent and thoroughly mathematical answer. In his attack on the problem we see the same versatility in applying his mathematical skills to concrete objects. In this case, however, these concrete objects were invisible waves of light. As if it went without saying, he had approached unobservable particles in the same Archimedean way he approached the tangible objects of his earlier mathematical studies. He had mathematized the mechanistic causes of the behavior of light rays. Huygens was therefore wrong as well. He had not just solved the problem of strange refraction. In effect he had invented a new way of doing optics. Within the limited scope of reflected and refracted rays, Huygens had invented that part of physical optics in which mathematics fruitfully integrated the nature of light and its observed behavior. Kepler had realized that the mathematical description of light rays also ought to reflect its physical nature, but had not succeeded in deriving the ‘measure’ of refraction from its ‘cause’. Descartes had proclaimed the mechanistic nature of light but, by seeking mathematics in the ontology of matter rather than its motions, had not succeeded in mathematizing his picture. Newton could mathematize the motions of particles of light, but he would not allow this to be integrated with his experimentally established theory of the mathematical behavior of colored rays. Parallel to Huygens, Newton had developed that

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other part of physical optics in which experiment was used as a heuristic tool for exploring new phenomena of light and establishing their mathematical properties. In this way he had extended the mathematical science of optics to the quality of color. Huygens had extended it to the mechanistic causes of the laws of optics. By applying Galileo’s science of motion to the motions of ethereal particles, he had invented the most complete form of mathematical physics in the seventeenth century. Huygens and Descartes Traité de la Lumière gave a new form to mechanistic science, the first ‘thoroughly Cartesian’ theory of light. Yet – and this is the gist of my argument – it was not the outcome of some program in mechanistic, or even Cartesian, science. A careful reconstruction of what exactly were the leading questions for Huygens, juxtaposed with comparable pursuits of other protagonists of seventeenth-century optics, reveals that Huygens’ wave theory was the outcome of his typically rigorous and tenacious approach to a problem raised in the context of geometrical optics. As was his wont, he first of all wanted to get the mathematics of his solution right. He wanted the explanations of the various laws to be mathematical derivations that were mutually consistent. As a result of the particular character of the ‘matter’ of geometrical optics – light rays, which had come to be seen as being of a mechanistic nature – he got involved in mechanistic questions. He did so in a deliberately mathematical way, intending to stick to the rigor of mathematics he missed in the reasonings of his fellows at the Académie. He believed in the power of mathematical reasoning and did not content himself with illdefined mechanisms. This reaction to the Parisian Cartesians can be seen as a continuation of what I regard as Huygens’ lifelong reaction to Descartes. Much of his oeuvre was a direct response to what Descartes had said on impact, circular motion, curves, lenses, light, halos, etc. He did so in a clearly mechanistic context, accepting fundamental concepts and drawing inspiration from some of Descartes’ ideas. In his theories of gravity and light he also considered the conceptualization of the mechanistic nature of things. Discours was induced by the – in his view – obscurities vented on the Parisian scene. Pardies may have inspired his thinking on the nature of light and the intellectual climate at the Académie, but strange refraction – together with the problem of caustics – may well have been the sole occasion for Huygens’ consideration of the mechanics of light propagation. As an adolescent, Huygens had soaked up Principia Philosophiae and its clarity of reasoning had made an indelible impression on him. The idea that nature ultimately consists of passive matter in motion was always at the back of his mind. But this does not turn Huygens into a Cartesian. Mechanistic philosophy was merely a tacitly assumed background of his thinking. Huygens quite consistently confined himself to the mathematics of these

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matters – or, more properly speaking, to Descartes’ dealings with mathematics. Descartes set Huygens’ agenda as he did for seventeenth-century science in general, but in this case it was a mathematical agenda instead of a natural philosophical one. He was no builder of a system of natural philosophy, neither in the Cartesian sense, nor in the way Newton was. Leibniz criticized him for failing to draw philosophical conclusions from his laws of impact. “He had no taste for metaphysics”.9 Huygens did not pursue questions raised by Descartes’ natural philosophical program, he responded to his contributions to the various branches of mathematics. He firmly criticized these and even his whole approach. In Huygens’ view, Descartes had corrupted mathematical science and he would do better. The point I want to make here is that the historical significance of Traité de la Lumière has blown up Huygens’ alleged Cartesianism and distorted our view of the whole of his optics – and his science in general. Sabra offers an example of the pitfall created by presuming Huygens to be pursuing Cartesian science. He discusses the wave theory prior to the dispute on colors with Newton, which is historically incorrect, thus making him more mechanistic than he actually was at that time.10 Huygens did not have some kind of research program aimed at unraveling the mechanistic nature of things, not even a program aimed at establishing the mathematical nature of things. The small Archimedes For Huygens, applying mathematics to real things (large and small) went without saying. He investigated the mathematical aspects of phenomena and one suspects that he did not have explicit ideas about the ultimate mathematical nature of nature like Galileo had. Huygens seems to have lacked “…a personal conviction about access to deep secrets of nature.”11 I regard his revolutionary conception of the probable nature of explanatory knowledge not as an outcome of some philosophical or epistemological conviction, but rather as a reflection of the cumulative character of his Archimedean mathematics. I do not think that the preface of Traité de la Lumière reflects some scepticist attitude. Colors or polarization were simply additional parameters not yet fathomed, but for Huygens this did not detract from the validity of his wave theory. Huygens has been called a problem solver, and this he was, marking himself off by solving problems his contemporaries passed over. He was perfectly happy with brilliantly solving sophisticated problems of 9

Heinekamp, “Huygens vu par Leibniz”, 106. Leibniz, Philosophische Schriften III, 611. “Il n’avoit point de goust pour la Metaphysique.” 10 Sabra, Theories of Light: chapter VI “Huygens’ Cartesianism and his theory of conjectural explanation”, chapter VIII “Huygens’ wave theory”, chapter X “Three critics of Newton’s theory: Hooke, Pardies, Huygens” 11 Hall, “Summary”, 307.

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mathematical physics without bothering too much about their mutual connections or their theoretical and philosophical background. As a result of Huygens’ pragmatism or eclecticism, his oeuvre may sometimes seem like a mishmash of isolated problems, brilliantly yet pragmatically solved. If we expect a brilliant savant to be searching for new or better foundations, Huygens may indeed pose a problem. Elaborating and applying theories does not yield new foundations. This makes it difficult to situate him among the Galileos and Newtons. Yet, Huygens’ pre-1670s oeuvre has historical significance in several other respects. It reveals another aspect of modern science, the application of mathematics to concrete things. Huygens’ mathematics was the kind of the rational mechanics that was to develop in the course of the eighteenth century.12 I have the impression that this kind of science tends to be overlooked by historians of science. When it comes to the scientific revolution, they primarily look at the development of new conceptual and methodological foundations.13 As regards seventeenth-century optics this is evident: whereas the discovery of the sine law and of dispersion as well as the mathematization of the corpuscular nature of light have amply been studied, the development of geometrical optics as cultivated by Huygens in Dioptrica has received hardly any attention. Historians seem to feel a bit awkward about the fact that someone of the stature of Huygens was a mere problem solver. Some have tried to distill some kind of underlying philosophical scheme from his activities.14 I do not expect that seeing him as some kind of neo-Cartesian will shed more light on the character of his oeuvre.15 I am more taken by the fact that a seventeenthcentury savant of his stature was so unprogrammatic and displayed such a lack of interest in epistemological, methodological and philosophical issues. Those historians tend to overlook that this problem solving for the pleasure of problem solving has historical significance in its own right. It made Huygens into one of the first – perhaps the very first – of a new kind of scientist, an investigator of nature desiring to do things better than others,

12

Mulder, “Pure, mixed and applied mathematics”, 37-39. This has also been suggested by Gabbey, at the 1979 symposium, in trying to understand why Huygens’ mechanics has received relatively little attention from historians: “…, I would suggest that historians have clustered around Galileo, Descartes, or Newton because one of their central aims was to describe the fundamental nature and workings of the physical world, and since this is the primordial purpose of physical science, the concomitant difficulties and inconsistencies, the associated philosophical and mathematical problems, the false starts, the anomalies, the blind allies, all the unfinished business, are irresistible to the historian. By contrast, Huygens has been visited relatively infrequently by historians because he solves problems, and does so magnificently, by an appeal to principles and hypotheses his intuition and empirical sense tell him are right, rather than erect an explanatory system of the world that has its roots in an original analysis of the nature of things.” Gabbey, “Huygens and mechanics” 175-176. Fortunately, since 1979 the situation around Huygens’ mechanics has changed to the better with, first of all, Yoder’s Unrolling time and, more recently, with Mormino’s ‘Penetralia motus’ . 14 Elzinga does so on the basis of Traité de la Lumière. Elzinga, On a research program. 15 Hall, “Summary”, 309-310. 13

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rather than seeking to fathom the deep secrets of nature. A ‘contemplator of nature’ he did not become until he composed Kosmotheoros.16 Huygens has obtained his historic stature as a pioneer in mathematical physics because he solved a special problem: the problem strange refraction appeared to him to pose to waves. In Traité de la Lumière, the small Archimedes – as his father used to call him – made new science. Galileo had mathematized motion, Newton mathematized qualities, Descartes had ontologized mathematics. Huygens mathematized the invisible nature of things. He had become the Archimedes of the small.

16

See Harting, Christiaan Huygens, 45-50.

List of figures Figure 1 Huygens: sketch of 6 August 1679 Figure 2 Spherical aberration Figure 3 Cartesian oval. Figure 4 Huygens: focal distance of a bi-convex lens Figure 5 Huygens: punctum concursus Figure 6 Huygens: refraction at the anterior side of a bi-convex lens Figure 7 Huygens: refraction at the posterior side of a bi-convex lens. Figure 8 Huygens: focal distance of a bi-convex lens Figure 9 Huygens: extended image. Figure 10 Huygens: magnification by a convex lens. Figure 11 Huygens: four of the cases of magnification by telescopes. Figure 12 Huygens: analysis of Keplerian telescope with erector lens. Figure 13 Diagram for Keplerian telescope with erector lens. Figure 14 Kepler’s solution to the pinhole problem Figure 15 Kepler: focal distance of a plano-convex lens Figure 16 Kepler: image formation by a lens Figure 17 Della Porta: image of a near object Figure 18 Della Porta: image of distant object Figure 19 Della Porta: image by a telescope Figure 20 Barrow’s analysis of image formation in refraction. Figure 21 Huygens: observations of Saturn with the 12- and a 23-foot telescope. Figure 22 Huygens: beam to facilitate lens grinding. Figure 23 Daza’s scale Figure 24 Huygens’ eyepiece. Figure 25 Diagram for Huygens’ eyepiece. Figure 26 Huygens: spherical aberration of a plano-convex lens. Figure 27 Huygens: spherical aberration of a bi-convex lens Figure 28 Hudde’s calculation of spherical aberration Figure 29 Huygens: Galilean configuration in which spherical aberration is neutralized. Figure 30 Huygens: ‘Circle’ of aberration. Figure 31 Huygens: Aberration produced by a Keplerian configuration. Figure 32 Rendering of Huygens’ sketch of chromatic aberration. Figure 33 Huygens’ invention of 1669 Figure 34 Huygens’ crossed out EUREKA. Figure 35 Newton’s determination of chromatic aberration. Figure 36 The first stage of Kepler’s attack of refraction. Figure 37 The final stage of Kepler’s analysis of refraction Figure 38 Harriot’s measurements. Figure 39 Mydorge’s rule Figure 40 Descartes’ analysis of refraction Figure 41 Descartes’ analysis of reflection Figure 42 Barrow’s explanation of reflection. Figure 43 Barrow’s explanation of refraction. Figure 44 Huygens: sketch of refracted rays in Iceland crystal. Figure 45 Huygens: a refracted perpendicular caused by the composition of the crystal. Figure 46 Huygens: waves through Iceland crystal. Figure 47 Huygens: shape and main angles of the crystal. Figure 48 Bartholinus: double refraction. Figure 49 Bartholinus: refraction in two positions of the crystal. Figure 50 Bartholinus’ law of strange refraction. Figure 51 Huygens: rays in the principal section. Figure 52 Huygens: construction for strangely refracted rays in the principal section

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Figure 53 Diagram of Huygens’ construction for strange refraction. Figure 54 Huygens’ alternative for Bartholinus’ law. Figure 55 Huygens: description of polarization. Figure 56 Ango’s explanation of refraction. Figure 57 The sine law in Tractatus. Figure 58 Huygens’ principle. Figure 59 Huygens: two rays refracted by a plane surface. Figure 60 Huygens: wave refracted by a plane surface forming a caustic. Figure 61 Huygens: wave refracted at the plane surface of a glass medium. Figure 62 Huygens: “Causam mirae refractionis in Crystallo Islandica”. Figure 63 Huygens: strange refraction of an arbitrary ray. Figure 64 Huygens: waves around a source of light Figure 65 Huygens’ principle. Figure 66 Huygens’ explanation of refraction. Figure 67 Huygens: refraction of the perpendicular. Figure 68 Huygens: orientation of spheroid in the crystal. Figure 69 Huygens: shape of the spheroidal wave. Figure 70 Construction of the refraction of an arbitrary ray in Traité de la Lumière. Figure 71 Hobbes’ rays. Figure 72 Hobbes: refraction. Figure 73 Hobbes’ derivation of the sine law. Figure 74 Refraction in Principia. Figure 75 The sine law in Opticks. Figure 76 Huygens: new measurement of strange refraction. Figure 77 Huygens’ EUPHKA of August 1679. Figure 78 Huygens: chromatic aberration of lenses.

Bibliography References are made by author and short title. References to the twenty-two volumes of the Oeuvres Complètes are given by OC followed by the volume and page numbers, as in OC10, 153. References to the manuscripts, all of which belong to the collection Codices Hugeniorum housed at the University of Leiden, are given by manuscript number and folio, as in Hug2, 45v. Huygens is left out as author. I do not explicitly refer to all seventeenth-century works listed in the bibliography, but I have drawn upon these in general conclusions regarding optics in the seventeenth century. Acloque, Paul. ‘‘L’oculaire de Huygens, son invention et sa place dans l’instrumentation’’ In Huygens et la France, Pais, 27-29 mars 1979: table ronde du Centre national de la recherche scientifique, pp. 177-186. Paris: Vrin, 1981 Aguilón, Francois de. Opticorum libri sex juxta ac mathematica utiles. Antwerp, 1613 Albury, William R. ‘‘Halley and the ‘Traité de la lumière’ of Huygens: new light on Halley’s relationship with Newton’’ Isis 62 (1971): 445-468 Allen, Ph. ‘‘Problems connexted with the development of the telescope (1609-1687)’’ Isis 34 (1942): 302-311 Andriesse, C.D. Titan kan niet slapen. Een biografie van Christiaan Huygens, Amsterdam: Contact, 1993 Andriesse, C.D. “The melancholic genius” De Zeventiende Eeuw 12-1 (1996): 3-13 Ango, Pierre. L’Optique divisée en trois livres. Paris, 1682 Ariotti, Piero E. ‘‘Bonaventura Cavalieri, Marin Mersenne, and the reflecting telescope’’ Isis 66 (1975): 303-321 Aris, Daniel. ‘‘La découverte de la France par Christiaan Huygens (1655)’’ In Les récits de la voyage, edited by Jean-Marc Pastré, pp. 58-72. Paris: Nizet, 1986 Baarmann, J. ‘‘Abhandlung über das Licht von Ibn al-Haitam’’ Zeitschrift der Deutschen Morgenländischen Gesellschaft 36 (1882): 145-237 Bachelard, Suzanne. ‘‘L’influence de Huygens au XVIIIe et au XIXe siècles’’ In Huygens et la France, Pais, 27-29 mars 1979: table ronde du Centre national de la recherche scientifique, pp. 241-258. Paris: Vrin, 1981 Barchillon, Jacques. ‘‘Les frères Perrault a travers la correspondance de Christian Huygens’’ XVIIe siècle 56 (1962): 19-36 Barrow, Isaac. The usefulness of mathematical learning explained and demonstrated: being mathematical lectures read at the publick schools at the University of Cambridge. Translated by John Kirkby. (Reprint. London, 1970) London, 1734 Barrow, Isaac. Isaac Barrow's Optical Lectures (Lectiones XVIII). Translated by H.C. Fay. Edited by A.G. Bennet, London: the Worshipful Company of Spectacle Makers, 1987 Barth, Michael. ‘‘Huygens at work: annotations in his rediscovered personal copy of Hooke’s ‘Micrographia’’’ Annals of Science 52 (1995): 601-613 Bartholinus, Erasmus. Experiments on birefringent Icelandic crystal. Translated by Th. Archibald. Introduction by J. Buchwald and K. Pedersen, Copenhagen: The Danish Library of Science and Medicine, 1991 Beale, John. “Perspective Tubes & Telescopes”, copied by Nathaniel Higmore, British Museum Manuscript: Sloane 548 Bechler, Zev. ‘‘Newton’s search for a mechanistic model of colour dispersion: a suggested interpretation’’ Archive for History of Exact Sciences 11 (1973): 1-37 Bechler, Zev. ‘‘Newton’s 1672 optical controversies: a study in the gramma of scientific dissent’’ In The interaction between science and philosophy, edited by Y. Elkana, pp. 115-142. Atlantic Highlands: Humanities Press, 1974

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Index Académie Royale des Sciences 44, 49, 53, 105, 107, 110, 157, 159, 160, 172, 204, 205, 206, 207, 213, 215, 222, 224, 238, 239, 241, 261 Aguilón, François d’ (1567-1617) 35, 42, 95 Alhacen, or Ibn al-Haytham (ca.9651039) 27-28, 38, 83, 108, 112-116, 124, 131, 160, 211, 228 Andriesse, C............................................ 107 Ango, Pierre (1640-1694) 110, 152, 153, 195-196, 250 Auzout, Adrien (1622-1691) .......... 44, 157 Bacon, Francis (1561-1626) ...99, 105, 247 Bacon, Roger (1214-1292).............. 27, 114 Baillet, Adrien (1649-1706) . 237, 246, 247 Barrow, Isaac (1630-1677) 38-41, 50, 95, 102, 108, 112, 125, 136, 138-140, 157, 189, 190, 192, 213, 220, 230, 232, 256, 257 cause of refraction ........137-138, 202, 211 image formation......................................39 Lectiones XVIII (1669) 39, 82, 83, 136, 139 Bartholinus, Erasmus (1625-1692)..... 111, 135, 142-147, 149, 151, 153, 154, 155, 168, 170, 171, 182, 205-210, 231, 260 Beeckman, Isaac (1588-1637) ....42, 57, 58 Berkel, K. van......................................... 157 Bolantio, Giovanni Christophoro () ... 6162, 76 Bos, H.J. .......................................................3 Boyle, Robert (1627-1691) .....94, 135, 228 Brahe, Tycho (1546-1601)... 26, 43, 44, 45 Buchdahl, G............................................ 120 Buchwald, J.Z....... 146, 179, 196, 207, 230 Campani, Guiseppe (1635-1715)..... 59, 62 Cassini, Gian Domenico (1625-1712) . 4445, 107, 167 Cavalieri, Bonaventura (ca.1598-1647) .38, 135 Cherubin d’Orleans (1613-1697)............62 Colbert, Jean-Baptiste (1619-1683) .......53, 160-161, 204, 214 Daza de Valdez, Benito (1591-1634) .....61 Descartes, René (1596-1650) 2, 5-9, 15, 25, 36, 37, 40-42, 50, 55-57, 67, 71, 72, 82, 94, 96, 101, 108-109, 112-113, 116-117, 126-127, 129-136, 139-140, 146-147, 149, 151-156, 158, 170, 186189, 191, 193, 195-196, 198, 200, 205,

209-211, 228-232, 234, 236-241, 244247, 256, 260-262, 264 cause of refraction .................126-130, 187 Discours de la Methode (1637) ..............24 La Dioptrique (1637) 8, 11, 13-14, 16, 24, 36, 37, 38, 41, 49, 56, 62, 71, 82, 109, 111, 125-128, 131-134, 136, 155, 186, 188-189, 206, 213, 224, 231-232, 238 La Géométrie (1637)...........11, 13-14, 41 Le Monde, ou Traité de la Lumière (1664) ...................133, 187, 188, 224, 253 lenses ............................................... 36-37 Les Météores (1637)...............13, 24, 193 Principia Philosophiae (1644) 133, 187, 242, 261 Dijksterhuis, E.J............................. 2, 6, 236 Divini, Eustachio (1610-1685) 53-54, 59, 62 with H. Fabri, Brevis Annotatio in Systema Saturnium (1660).............54 Domini, Marko Antonij (1560-1624) ....33 Dupré, S. ............................................. 29, 35 Fatio de Duillier, Nicolas (1664-1753) ...........................223-224, 244, 250-251 Fermat, Pierre de (1601-1665) 135, 156, 162, 165, 167, 205 Ferrier, Jean (fl. 1620-1640) ............. 36, 56 Flamsteed, John (1646-1719) 45-50, 72, 256 Fontana, Francesco (ca.1585-1656) 43, 59 Fresnel, Augustin (1788-1827)....... 42, 253 Fullenius, Bernardus (1640-1707) ........221 Galileï, Galileo (1564-1642) 9, 41, 53-55, 59, 63, 96, 97, 98, 117, 126, 135, 154, 202, 211, 233, 238, 240, 246-248, 252, 256, 257, 261-262, 264 Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze (1638) ............................................ 211, 242 Sidereus Nuncius (1610) ..........25, 35, 55 Gascoigne, William (ca.1610-1644). 43, 47 Gassendi, Pierre (1592-1655)....... 228, 247 Gravesande, Willem Jacob ‘s (1688-1742) ............................................................251 Gregory, James (1638-1675) ..38, 125, 135 Grimaldi, Francesco Maria (1618-1663) ............................................................135 Grosseteste, Robert (ca.1170-1253).....114 Gutschoven, Gerard van (1615-1668) 1418, 57 Hakfoort, Casper .......................9, 211, 252

286 Hall, A.R. .....................................................8 Halley, Edmond (1656-1743)...40, 45, 250 Harriot, Thomas (ca.1560-1621) 35, 123124 Harting, Pieter (1812-1885).......................7 Hartsoeker, Nicolaas (1656-1725)....... 215 Hérigone, Pierre (-ca.1643) .....................36 Hevelius, Johannes (1611-1689) 41, 43, 45-46, 49, 62 Hire, Philippe de la (1640-1718) ...... 222, 246, 251 Hobbes, Thomas (1588-1679) 37, 137, 189-192, 229, 230 cause of refraction .........................190-191 Hooke, Robert (1635-1703) 43, 45, 78, 89, 91, 94, 110, 135, 160, 186, 189, 192-194, 209, 230, 233-234, 249-251 Hudde, Johannes (1628-1704) 11, 71-72, 102-103 Huygens, Christiaan (1629-1695) 1, 2, 53, 107, 135 ‘Adversaria ad Dioptricen’ (1665) 68, 70, 72, 81, 83 ‘De Aberratione Radiorum a Foco’ (1666) 5, 72, 74-77, 83, 91-93, 95, 97, 100-102, 105, 154, 216, 219, 221, 256, 258 ‘De Ordine in Dioptricis nostris servando’ (ca. 1692).............................221-222 ‘De Telescopiis’ (1680s)...............220-221 ‘mon Archimède’.......... 10, 161, 259, 264 ‘Projet du Contenu de la Dioptrique’ (1672) 108-109, 111-112, 135-136, 140, 152, 155-161, 181, 186, 214, 216, 219, 242 ‘Tractatus de refractione et telescopiis’ (1653) 4, 12, 15-20, 22-24, 30, 32, 41, 49-50, 53, 58, 63-68, 76, 82, 93, 108-109, 153, 216, 220-221, 256, 258 accelerated motion ........................... 95-100 Astroscopia Compendaria (1684) ....... 215 De Saturni Luna Observatio Nova (1656) .......................................................53 Dioptrica 6, 8, 91-91, 95, 100-101, 104105, 112, 153, 220-221, 255-256, 263 Discours de la Cause de la Pesanteur (1690)........................ 223, 244, 250 Horologium Oscillatorium (1673) 53, 101, 107-108, 160, 246, 252 impact................................................. 154 Memorien aangaande het slijpen van glasen tot verrekijckers (1703)...................62 music ............................................. 97-100

INDEX

nature of gravity ...................239, 244-245 Systema Saturnium (1659) 44, 50, 54, 63, 105-106 Traité de la Lumière (1690) 2-4, 6, 9, 95, 108, 112, 159, 160-162, 167-168, 172, 176-179, 181-187, 192, 194196, 202, 204-205, 207-211, 213216, 220-225, 228, 230, 235-238, 240-242, 246-257, 261-262, 264 Huygens, Constantijn jr. (1628-1697)...11, 33, 53, 58, 63, 64, 77-80, 107, 214, 223 Huygens, Constantijn sr. (1596-1687) ..11, 56, 77, 88, 135, 214, 249, 264 Huygens, Lodewijk (1631-1699) 64, 80, 156 Kepler, Johannes (1571-1630)4-7, 12, 23, 27, 34, 36, 38, 40-43, 48, 50, 66-67, 104, 108, 112-113, 117, 120-125, 130131, 134, 140, 158, 183, 228-230, 232, 244, 256, 260 Ad Vitellionem Paralipomena (1604) ..2631, 35, 38, 41, 47, 95, 113, 117119, 121, 124-125, 131, 229, 276 cause of refraction 118-122, 126, 131, 211 Dioptrice (1611) 8, 15, 26, 30-33, 35-38, 40-41, 47, 49-50, 59, 62, 104, 121, 123, 134 Dissertatio cum Nuncio Sidereo (1610)..25 image formation................................ 28-29 lenses ............................................... 29-32 Knorre, Martin (-1699) ..........................250 Laplace, Pierre-Simon (1749-1827)......253 Leeuwenhoek, Antony van (1632-1723) .................................. 161, 215, 219, 249 Leibniz, Gottfried Wilhelm (1646-1716) ....... 9, 10, 194, 215, 223, 249-251, 262 Lipperhey, Hans (-1619)..........................25 Maignan, Emanuel (1601-1676) .. 189, 192 Malus, Etienne Louis (1775-1812) .... 252253 Mariotte, Edme (ca.1620-1684) 92, 143, 249, 251 Marius, Simon (1573-1624) .....................55 Maurolyco, Francesco (1494-1575)........30 Mersenne, Marin (1588-1648) 10, 57, 9698, 126, 189, 259 Molyneux, William (1656-1698) 22, 41, 47-48, 50, 72, 221 Mydorge, Claude (1585-1647) 36, 56, 126-127, 134 Newton, Isaac (1642-1727) 2, 5-7, 9, 38, 41, 46, 55, 67, 78, 83-91, 93-95, 97, 99, 103-104, 107-108, 131, 133, 135,

INDEX

160, 195, 197-201, 203, 217, 223, 225228, 230-235, 237-238, 242-246, 250253, 256-257, 260, 262, 264 ‘New Theory about Light and Colors’ (1672)..............85-87, 89, 91, 94-95 cause of refraction ................ 196, 198-199 image formation......................................40 Lectiones Opticae (1670-1672) 40, 197200, 227-228, 233, 235 Opticks (1704) 196, 198, 200, 224, 231, 233, 235, 250, 252 Philosophiae Naturalis Principia Mathematica (1687) 196-198, 223, 231, 243-244, 246, 250 Nulandt, Francois Guillaume Baron de () ..................................................79-80, 82 Oldenburg, Henry (ca.1618-1677) 45, 8188, 90, 95, 161, 197 Papin, Denis (1647-ca.1712) .........250-251 Pardies, Ignace-Gaston (1636-1673).... 8789, 91-92, 110-111, 140-141, 150-153, 155-159, 162, 165-166, 168, 176, 184, 186, 189, 195-196, 201, 213, 242-243, 250-251, 259-261 Pecham, John (ca.1240-1292) ........ 27, 114 Pedersen, K.M........................................ 146 Petit, Pierre (-1677) ...................44, 64, 157 Picard, Jean (1620-1682) 44-45, 49-50, 141, 256 ‘Fragmens de Dioptrique’ (1693) ...........49 Porta, Giambattista della (1535-1615) ..30, 33-35 lenses......................................................33 Reeve, Richard (-1666).............................64 Rheita, Anton Maria Schyrlaeus (15971660) ..............................................56-57 Risner, Friedrich (-ca.1580) 27, 29, 115116, 124

287 Roberval, Gilles Personne de (1602-1675) ...........................................107, 241, 244 Rømer, Ole Christensen (1644-1710).167, 172, 204-209, 213 Sabra, A.I. 7, 131-133, 198, 226, 236, 262 Sagredo, Giovanfrancesco (1571-1620) 35 Sahl, ... Ibn (fl. 970-990) ..........................36 Scheiner, Christoph (1573-1650) 35, 41, 57, 59 Schooten, Frans van, jr. (1615-1660)... 1115, 24, 51, 71 Geometria à Renato Des Cartes (1649 and 1659-1661) ............................. 13-14 Schuster, J.A. ...................................131-134 Shapiro, A.E. 2, 7, 125, 162, 166, 171, 200, 227 Sirtori, Girolamo (-1631).........................62 Sluse, René-François de (1622-1685)....83, 160 Smith, A.M. .............................................113 Smith, Robert (1689-1768)....................252 Snel, Willebrord (1580-1626) .36, 124, 135 Spinoza, Baruch (1632-1677)..................71 Stampioen, Jan Jansz. the younger (1610after 1689).................................... 33, 37 Stevin, Simon (1548-1620) ......................99 Tacquet, André (1612-1660) .....15-18, 139 Volder, Burchardus de (1643-1709).....221 Vossius, Isaac (1618-1689)........... 135, 156 Waard, C. de..............................................57 Westfall, R.S. ..........................105, 154, 241 Wiesel, Johann (1583-1662) 15, 43, 60, 62, 64 Witelo (ca.1230-ca.1280) 27-29, 114, 116117, 119, 121-124, 136 Witt, Johan de (1625-1672) ............ 11, 160 Wren, Christopher (1632-1723).43, 45, 56 Yoder, J.G..................................................97 Ziggelaar, A. ............................................162