# Lenses and Waves

Lenses and Waves

Lenses and Waves

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Archimedes<br />

Volume 9

Archimedes<br />

NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF<br />

SCIENCE AND TECHNOLOGY<br />

VOLUME 9<br />

EDITOR<br />

JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology,<br />

Pasadena, CA, USA.<br />

ADVISORY BOARD<br />

HENK BOS, University of Utrecht<br />

MORDECHAI FEINGOLD, Virginia Polytechnic Institute<br />

ALLAN D. FRANKLIN, University of Colorado at Boulder<br />

KOSTAS GAVROGLU, National Technical University of Athens<br />

ANTHONY GRAFTON, Princeton University<br />

FREDERIC L. HOLMES, Yale University<br />

PAUL HOYNINGEN-HUENE, University of Hannover<br />

EVELYN FOX KELLER, MIT<br />

TREVOR LEVERE, University of Toronto<br />

JESPER LÜTZEN, Copenhagen University<br />

WILLIAM NEWMAN, Harvard University<br />

JÜRGEN RENN, Max-Planck-Institut für Wissenschaftsgeschichte<br />

ALEX ROLAND, Duke University<br />

ALAN SHAPIRO, University of Minnesota<br />

NANCY SIRAISI, Hunter College of the City University of New York<br />

NOEL SWERDLOW, University of Chicago<br />

Archimedes has three fundamental goals; to further the integration of the histories of<br />

science <strong>and</strong> technology with one another: to investigate the technical, social <strong>and</strong> practical<br />

histories of specific developments in science <strong>and</strong> technology; <strong>and</strong> finally, where<br />

possible <strong>and</strong> desirable, to bring the histories of science <strong>and</strong> technology into closer contact<br />

with the philosophy of science. To these ends, each volume will have its own<br />

theme <strong>and</strong> title <strong>and</strong> will be planned by one or more members of the Advisory Board in<br />

consultation with the editor. Although the volumes have specific themes, the series itself<br />

will not be limited to one or even to a few particular areas. Its subjects include any<br />

of the sciences, ranging from biology through physics, all aspects of technology, broadly<br />

construed, as well as historically-engaged philosophy of science or technology.<br />

Taken as a whole, Archimedes will be of interest to historians, philosophers, <strong>and</strong> scientists,<br />

as well as to those in business <strong>and</strong> industry who seek to underst<strong>and</strong> how science<br />

<strong>and</strong> industry have come to be so strongly linked.

<strong>Lenses</strong> <strong>and</strong> <strong>Waves</strong><br />

Christiaan Huygens <strong>and</strong> the Mathematical Science<br />

of Optics in the Seventeenth Century<br />

by<br />

FOKKO JAN DIJKSTERHUIS<br />

University of Twente,<br />

Enschede, The Netherl<strong>and</strong>s<br />

KLUWER ACADEMIC PUBLISHERS<br />

NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 1-4020-2698-8<br />

Print ISBN: 1-4020-2697-8<br />

©2005 Springer Science + Business Media, Inc.<br />

Print ©2004 Kluwer Academic Publishers<br />

Dordrecht<br />

All rights reserved<br />

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,<br />

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Created in the United States of America<br />

Visit Springer's eBookstore at: http://ebooks.springerlink.com<br />

<strong>and</strong> the Springer Global Website Online at: http://www.springeronline.com

Contents<br />

CHAPTER 1 INTRODUCTION – ‘THE PERFECT CARTESIAN’ 1<br />

A history of Traité de la Lumière 2<br />

Huygens’ optics 4<br />

New light on Huygens 8<br />

CHAPTER 2 1653 - 'TRACTATUS' 11<br />

2.1 The Tractatus of 1653 12<br />

2.1.1 Ovals to lenses 13<br />

2.1.2 A theory of the telescope 16<br />

The focal distance of a bi-convex lens 17<br />

Images 20<br />

Conclusion 24<br />

2.2 Dioptrics <strong>and</strong> the telescope 24<br />

2.2.1 Kepler <strong>and</strong> the mathematics of lenses 26<br />

Image formation 28<br />

<strong>Lenses</strong> 29<br />

Perspectiva <strong>and</strong> the telescope 33<br />

2.2.2 The use of the sine law 35<br />

Descartes <strong>and</strong> the ideal telescope 36<br />

After Descartes 37<br />

Dioptrics as mathematics 40<br />

2.2.3 The need for theory 41<br />

The micrometer <strong>and</strong> telescopic sights 42<br />

Underst<strong>and</strong>ing the telescope 46<br />

Huygens’ position 50<br />

CHAPTER 3 1655-1672 - 'DE ABERRATIONE' 53<br />

3.1 The use of theory 55<br />

3.1.1 Huygens <strong>and</strong> the art of telescope making 57<br />

Huygens’ skills 58<br />

Alternative configurations 59<br />

Experiential knowledge 60<br />

3.1.2 Inventions on telescopes by Huygens 63<br />

3.2 Dealing with aberrations 67<br />

3.2.1 Properties of spherical aberration 67<br />

Specilla circularia 70<br />

Theory <strong>and</strong> its applications 72

vi CONTENTS<br />

3.2.2 Putting theory to practice 77<br />

A new design 80<br />

3.2.3 Newton’s other look <strong>and</strong> Huygens’ response 83<br />

3.3 Dioptrica in the context of Huygens’ mathematical science 92<br />

3.3.1 The mathematics of things 92<br />

Huygens ‘géomètre’ 95<br />

3.3.2 Huygens the scholar & Huygens the craftsman 100<br />

The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument 104<br />

CHAPTER 4 THE 'PROJET' OF 1672 107<br />

‘Projet du Contenu de la Dioptrique’ 109<br />

4.1 The nature of light <strong>and</strong> the laws of optics 112<br />

4.1.1 Alhacen on the cause of refraction 114<br />

4.1.2 Kepler on the measure <strong>and</strong> the cause of refraction 117<br />

The measure of refraction 118<br />

True measures 123<br />

Paralipomena <strong>and</strong> the seventeenth-century reconfiguration of optics 124<br />

4.1.3 The laws of optics in corpuscular thinking 126<br />

Refraction in La Dioptrique 128<br />

Epistemic aspects of Descartes’ account in historical context 130<br />

Historian’s assessment of Descartes’ optics 132<br />

Reception of Descartes’ account of refraction 134<br />

Barrow’s causal account of refraction 136<br />

4.2 The mathematics of strange refraction 140<br />

4.2.1 Bartholinus <strong>and</strong> Huygens on Icel<strong>and</strong> Crystal 142<br />

Bartholinus’ experimenta 143<br />

Huygens’ alternatives 147<br />

4.2.2 Rays versus waves: the mathematics of things revisited 152<br />

The particular problem of strange refraction: waves versus masses 155<br />

CHAPTER 5 1677-1679 - WAVES OF LIGHT 159<br />

5.1 A new theory of waves 161<br />

5.1.1 A first EUPHKA 162<br />

The solution of the ‘difficulté’ of Icel<strong>and</strong> Crystal 168<br />

5.1.2 Undulatory theory 172<br />

Explaining strange refraction 176<br />

5.1.3 Traité de la Lumière <strong>and</strong> the ‘Projet’ 181<br />

5.2 Comprehensible explanations 185<br />

5.2.1 Mechanisms of light 186

CONTENTS vii<br />

Hobbes, Hooke <strong>and</strong> the pitfalls of mechanistic philosophy:<br />

rigid waves 189<br />

5.2.2 ‘Raisons de mechanique’ 195<br />

Newton’s speculations on the nature of light 196<br />

The status of ‘raisons de mechanique’ 200<br />

5.3 A second EUPHKA 204<br />

5.3.1 Danish objections 205<br />

Forced innovation 207<br />

5.3.2 Hypotheses <strong>and</strong> deductions 209<br />

CHAPTER 6 1690 - TRAITÉ DE LA LUMIÈRE 213<br />

6.1 Creating Traité de la Lumière 214<br />

6.1.1 Completing ‘Dioptrique’ 216<br />

Huygens’ dioptrics in the 1680s 216<br />

6.1.2 From ‘Dioptrique’ to Traité de la Lumière 219<br />

The publication of Traité de la Lumière 222<br />

6.2 Traité de la Lumière <strong>and</strong> the advent of physical optics 225<br />

Mathematization by extending mathematics 227<br />

The matter of rays 229<br />

The mathematics of light 232<br />

6.3 Traité de la Lumière <strong>and</strong> Huygens’ oeuvre 236<br />

6.3.1 Huygens’ Cartesianism 237<br />

The subtle matter of 1669 238<br />

Huygens versus Newton 242<br />

Huygens’ self-image 247<br />

6.3.2 The reception of Huygens 249<br />

CHAPTER 7 CONCLUSION: LENSES & WAVES 255<br />

A seventeenth-century Archimedes 255<br />

From mathematics to mechanisms 259<br />

Huygens <strong>and</strong> Descartes 261<br />

The small Archimedes 262<br />

LIST OF FIGURES 265<br />

BIBLIOGRAPHY 267<br />

INDEX 285

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Preface<br />

“Le doute fait peine a l’esprit.<br />

C’est pourquoy tout le monde<br />

se range volontiers a l’opinion de ceux<br />

qui pretendent avoir trouvè la certitude.” 1<br />

This book evolved out of the dissertation that I defended on April 1, 1999 at<br />

the University of Twente. At the successive stages of its development critical<br />

readers have cast doubts on my argument. It has not troubled my mind; on<br />

the contrary, they enabled me to improve my argument in ways I could not<br />

have managed on my own. So, I want to thank Floris Cohen, Alan Shapiro,<br />

Jed Buchwald, Joella Yoder, <strong>and</strong> many others. Most of all, however, I have to<br />

thank Casper Hakfoort, who saw the final text of my dissertation but did not<br />

live to witness my defence <strong>and</strong> the further development of this study of<br />

optics in the seventeenth century.<br />

This book would not have been possible without NWO (Netherl<strong>and</strong>s<br />

Organisation for Scientific Research) <strong>and</strong> NACEE (Netherl<strong>and</strong>s American<br />

Commission for Educational Exchange) who supplied me with a travel grant<br />

<strong>and</strong> a Fulbright grant, respectively, to work with Alan Shapiro in<br />

Minneapolis. The book would also not have been possible without the<br />

willingness of Kluwer Publishers <strong>and</strong> Jed Buchwald to include it in the<br />

Archimedes Series, <strong>and</strong> the unrelenting efforts of Charles Erkelens to see it<br />

through.<br />

During the years this text accompanied my professional <strong>and</strong> personal doings,<br />

numerous people have helped me grow professionally <strong>and</strong> personally. I want<br />

to thank Peter-Paul Verbeek, John Heymans, Petra Bruulsema, Kai Barth,<br />

Albert van Helden, Rienk Vermij, Paul Lauxtermann, Lissa Roberts, <strong>and</strong><br />

many, many others.<br />

Still, the idea to study Huygens <strong>and</strong> his optics would not have even<br />

germinated – let alone that this book would have matured – without my life<br />

companion, Anne, with whom I now share a much more valuable creation.<br />

Thank you.<br />

Fokko Jan Dijksterhuis<br />

Calgary, June 2004<br />

This book is dedicated to Casper Hakfoort<br />

In memory of Lies Dijksterhuis<br />

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

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Chapter 1<br />

Introduction – ‘the perfect Cartesian’<br />

Christiaan Huygens, optics & the scientific revolution<br />

“EYPHKA. The confirmation of my theory of light <strong>and</strong> refractions”,<br />

proclaimed Christiaan Huygens on 6 August 1679. The line is accompanied<br />

by a small sketch, consisting of a parallelogram, an ellipse (though barely<br />

recognizable as such) <strong>and</strong> two pairs of perpendicular lines (Figure 1). The<br />

composition of geometrical figures does not immediately divulge its<br />

meaning. Yet, it conveys a pivotal event in the development of seventeenthcentury<br />

science.<br />

What is it? The parallelogram is a<br />

section – the principal section – of a piece<br />

of Icel<strong>and</strong> crystal, which is a transparent<br />

form of calcite with extraordinary optical<br />

properties. It refracts rays of light in a<br />

strange way that does not conform to the<br />

established laws of refraction. The ellipse<br />

represents the propagation of a wave of<br />

light in this crystal. It is not an ordinary,<br />

spherical wave, as waves of light are by<br />

Figure 1 The sketch of 6 August 1679<br />

nature, but that is precisely because the elliptical shape explains the strange<br />

refraction of light rays in Icel<strong>and</strong> crystal. The two pairs of lines denote the<br />

occasion for Huygens’ joy. They are unnatural sections of the crystal, which<br />

he had managed to produce by cutting <strong>and</strong> polishing the crystal. They<br />

produced refractions exactly as his theory, by means of those elliptical waves,<br />

had predicted.<br />

The elliptical waves were derived from the wave theory he had developed<br />

two years earlier, with the formulation of a principle of wave propagation.<br />

Like ordinary spherical waves, these elliptical waves were hypothetical<br />

entities defining the mechanistic nature of light. Now, seventeenth-century<br />

science was full of hypotheses regarding the corpuscular nature of things.<br />

But Huygens’ wave theory was not just another corpuscular theory. His<br />

principle defined the behavior of waves in a mathematical way, based on a<br />

theory describing the mechanics of light propagation in the form of<br />

collisions between ether particles. The mathematical character of Huygens’<br />

wave theory is historically significant. Huygens was the first in the<br />

seventeenth century to fully mathematize a mechanistic explanation of the<br />

properties of light. As contrasted to the qualitative pictures of his<br />

contemporaries, he could derive the exact properties of rays refracted by

2 CHAPTER 1<br />

Icel<strong>and</strong> crystal, including refractions that could only be observed by cutting<br />

the crystal along unnatural sections. The sketch records the experimental<br />

verification of Huygens’ elliptical waves <strong>and</strong>, with it, the confirmation of his<br />

theory of light <strong>and</strong> refractions.<br />

This brief synopsis explains what ‘actually’ happened on that 6 th of<br />

August in 1679. The various terms <strong>and</strong> concepts will be explicated later on in<br />

this book. For this moment, it suffices to make clear the core of Huygens’<br />

wave theory <strong>and</strong> its historical significance. For Huygens the successful<br />

experiment meant the confirmation of his explanation of strange refraction<br />

<strong>and</strong> his wave theory in general. In the context of the history of seventeenthcentury<br />

optics, <strong>and</strong> of the mathematical sciences in general, the importance<br />

of the event lies in the twofold particular nature of Huygens’ theory: a<br />

mathematized model of the mechanistic nature of light considered as a<br />

hypothesis validated by experimental confirmation. With the mathematical<br />

form of his theory, Huygens can be said to have restored the problematic<br />

legacy of Descartes’ natural philosophy, by defining mathematical principles<br />

for the mechanistic explanation of the physical nature of light. The<br />

hypothetical-deductive structure of his theory implied the ab<strong>and</strong>onment of<br />

the quest for certainty of that same Cartesian legacy <strong>and</strong> of seventeenthcentury<br />

science in general. Huygens presented waves of light, the inextricable<br />

core of his account of optical phenomena, explicitly as hypothetical entities<br />

whose certainty is inherently relative. In so doing, he set off from Descartes<br />

in a direction diametrically opposite to Newton, the principal other restorer<br />

of mechanistic science.<br />

About a decade after the EUREKA of 6 August 1679, Huygens published<br />

his wave theory of light <strong>and</strong> his explanations of ordinary <strong>and</strong> strange<br />

refraction in Traité de la Lumière (1690). This book established his fame as a<br />

pioneer of mathematical physics as evidenced by the fact his principle of<br />

wave propagation is still known <strong>and</strong> used in various fields of modern physics<br />

under the name ‘Huygens’ principle’. Its historical importance is also<br />

generally acknowledged. According to Shapiro, Huygens stood out for his<br />

“…continual ability to rise above mechanism <strong>and</strong> to treat the continuum<br />

theory of light mathematically.” 1 E.J. Dijksterhuis calls it the high point of<br />

mechanistic science <strong>and</strong> its creator the first ‘perfect Cartesian’: “In Huygens<br />

does Cartesian physics for the first time take the shape its creator had in<br />

mind.” 2 How did this come about? How did Christiaan Huygens came to<br />

realize this historical l<strong>and</strong>mark? Or more specifically, how did he arrive at his<br />

wave theory of light? That is the central question of this study.<br />

A history of Traité de la Lumière<br />

Unlike its eventual formulation in Traité de la Lumière, the development of<br />

Huygens’ wave theory has hardly been subject to historical investigation. The<br />

1 Shapiro, “Kinematic optics”, 244. (For referencing see page 267)<br />

2 Dijksterhuis, Mechanization, IV: 212 & 283 (references to this book will be made by section numbers). It<br />

should be noted that Dijksterhuis mainly focuses on the mathematical model of wave propagation.

‘THE PERFECT CARTESIAN’ 3<br />

first step for such a study is to go into the papers documenting Huygens’<br />

optics. The historian who does so on the basis of existing literature, awaits a<br />

surprise. There is much more to Huygens’ optics than waves. He elaborated<br />

a comprehensive theory of the dioptrical properties of lenses <strong>and</strong> their<br />

configurations in telescopes, that goes by the title of Dioptrica. A second<br />

surprise is in store when one takes a closer looks to these papers on<br />

geometrical optics. The papers on dioptrics cover the Huygens’ complete<br />

scientific career <strong>and</strong> form the exclusive content of the first two decades of<br />

his optical studies. The wave theory <strong>and</strong> related subjects are fully absent; not<br />

before 1672 do they turn up. In other words, the optics that brought<br />

Huygens future fame dates from a considerably late stage in the development<br />

of Huygens’ optics.<br />

In this way new <strong>and</strong> more specific questions arise regarding the question<br />

‘how did Huygens arrive at his wave theory’? What exactly was his optics?<br />

How did he move from Dioptrica to Traité de la Lumière? And what does this<br />

teach us about the historical significance of his wave theory, Huygens’<br />

creation of a physical optics, <strong>and</strong> the character of his science? The point is<br />

that Huygens’ dioptrics turns his seemingly self-explanatory wave theory into<br />

a historical problem. It did not develop from some innate cartesianism, for<br />

he was no born Cartesian, certainly not in optics at least. Fully absent from<br />

Dioptrica is the central question of Traité de la Lumière: what is the nature of<br />

light <strong>and</strong> how can it explain the laws of optics. Huygens first raised this<br />

question in 1672 – five years before he found his definite answer (which he<br />

confirmed another two years later in 1679). His previous twenty years of<br />

extensive dioptrical studies give scarcely any occasion to expect that this man<br />

was to give the mechanistic explanation of light <strong>and</strong> its properties a wholly<br />

new direction. In view of Dioptrica, the question is not only how Huygens<br />

came to treat the mechanistic nature of light in his particular way, but even<br />

how he came to consider the mechanistic nature of light in the first place.<br />

In the literature on Huygens’ optics, mechanistic philosophy has been<br />

customarily considered a natural part of his thinking. Only Bos, in his entry<br />

in the Dictionary of Scientific Biography, points at the relatively minor role<br />

mechanistic philosophizing played in his science before his move to Paris in<br />

the late 1660s. 3 The question of how the wave theory took shape in Huygens’<br />

mind thus becomes all the more intriguing. What caused Huygens to tackle<br />

this subject he had consistently ignored throughout his earlier work on<br />

optics? How do Dioptrica <strong>and</strong> Traité de la Lumière relate <strong>and</strong> what light does<br />

the former shed on the latter? Part of the answer is given by the fact that<br />

only at the very last moment, short before its publication, Huygens decided<br />

to change the title of his treatise on the wave theory from ‘dioptrics’ to<br />

‘treatise on light’. In his mind the two were closely connected, questions now<br />

3 Bos, “Huygens”, 609. Van Berkel further alludes to the influence of Parisian circles on the prominence<br />

of mechanistic philosophy in Huygens’ oeuvre: Van Berkel, “Legacy”, 55-59.

4 CHAPTER 1<br />

are: how exactly <strong>and</strong> what does this mean for our underst<strong>and</strong>ing of Huygens’<br />

optics?<br />

In addition to the question of where in Huygens’ oeuvre Traité de la<br />

Lumière properly belongs, a more general question may be asked: where in<br />

seventeenth-century science may this kind of science be taken to belong?<br />

Were the questions Huygens addressed in Traité de la Lumière part of any<br />

particular scientific discipline or coherent field of study? In the course of my<br />

investigation, it has became increasingly clear to me that the term ‘optics’ is<br />

rather problematic regarding the study of light in the seventeenth century,<br />

just like the term ‘science’ in general. Our modern underst<strong>and</strong>ing of ‘optics’<br />

implies an investigation of phenomena of light much like Traité de la Lumière:<br />

a mathematically formulated theory of the physical nature of light explaining<br />

the mathematical regularities of those phenomena. Yet, optics in this sense<br />

was only just beginning to develop during the seventeenth century. The term<br />

‘optics’ in the seventeenth century denoted the mathematical study of the<br />

behavior of light rays that we are used to identify with geometrical optics.<br />

This is what Huygens pursued in Dioptrica, prior to developing his wave<br />

theory. A general question regarding the history of optics raised by Huygens’<br />

Traité de la Lumière is how a new kind of optics, a physical optics, came into<br />

being in the seventeenth century <strong>and</strong> how this related to the older science of<br />

geometrical optics. This transformation of the mathematical science of optics<br />

is manifest in the title Huygens eventually chose for his treatise.<br />

Huygens’ optics<br />

This book offers in the first place an account of the development of<br />

Huygens’ optics, from the first steps of Dioptrica in 1652 to the eventual<br />

Traité de la Lumière of 1690. The following chapters take a chronological<br />

course through his engagements with the study of light, whereby the<br />

historical connection of its various parts sets the perspective. Terms like<br />

‘optics’ <strong>and</strong> ‘science’ are problematic historically. Nevertheless, for sake of<br />

convenience, I will freely use them to denote the study of light in general <strong>and</strong><br />

natural inquiry, except when this would give rise to (historical)<br />

misunderst<strong>and</strong>ings. When discussing their historical character <strong>and</strong><br />

development specifically, I will use appropriately historicizing phrases.<br />

Chapter two discusses Tractatus – the unfinished treatise of 1653 on<br />

dioptrics that marks the beginning of Huygens’ engagement with optics.<br />

Tractatus contained an comprehensive <strong>and</strong> rigorous theory of the telescope<br />

<strong>and</strong> I will argue that this makes it unique in seventeenth-century<br />

mathematical optics. Huygens was one of the few to raise theoretical<br />

questions regarding the properties <strong>and</strong> working of the telescope, <strong>and</strong> almost<br />

the only one to direct his mathematical proficiency towards the actual<br />

instruments used in astronomy. Kepler had preceded him, but he had not<br />

known the law of refraction <strong>and</strong> therefore could not derive but an<br />

approximate theory of lenses <strong>and</strong> their configurations. Some four decades<br />

afterwards, <strong>and</strong> two decades after the publication of the sine law, Huygens

‘THE PERFECT CARTESIAN’ 5<br />

was the first to apply it to spherical lenses <strong>and</strong> remained so for almost two<br />

decades more. Chapter three discusses his practical pursuits in dioptrics<br />

leading into his subsequent treatise on dioptrics, De Aberratione of 1665.<br />

Huygens made a unique effort to employ dioptrical theory to improve the<br />

telescope. The contrast with Descartes is particularly conspicuous, for<br />

Huygens did not fit the telescope into the ideal mold prescribed by theory<br />

but directed his theory towards the instruments that were practically feasible.<br />

The effort was unsuccessful, for with his new theory of light <strong>and</strong> colors<br />

Newton made him realize the futility of his design. Taking into account<br />

Huygens’ background in dioptrics sheds, I will argue, new light on the<br />

famous dispute with Newton in 1672.<br />

These chapters are confined to what we would call geometrical optics <strong>and</strong><br />

to its relationship to practical matters of telescopy. 4 I try to explain what this<br />

science was about <strong>and</strong> what was particular about the way Huygens pursued<br />

it. These chapters offer a fairly detailed account of Dioptrica within the<br />

context of seventeenth-century geometrical optics, <strong>and</strong> as such open fresh<br />

ground in the history of science. At the turn of the century, Kepler had laid a<br />

new foundation for geometrical optics. Image formation now became a<br />

matter of determining where <strong>and</strong> how a bundle of diverging rays from each<br />

point of the object is brought into focus again (or not) instead of tracing<br />

single rays from object point to image point. Of old, the ray was the bearer<br />

of the physical properties of light, but in the course of the seventeenth<br />

century this began to be qualified <strong>and</strong> questioned. In the wake of Kepler <strong>and</strong><br />

Descartes the mathematical science of optics gradually transformed into new<br />

ways of doing optics. The traditional, geometrical way of doing optics did<br />

not vanish, though. It was ray optics in which the question of the nature of<br />

light need not penetrate further than determining the physical properties of<br />

rays in their interaction with opaque <strong>and</strong> transparent materials. This is the<br />

mathematical optics Huygens grew up with <strong>and</strong> that set the tone in his<br />

earliest dealings with the physics of light propagation. Only on second<br />

thought did he focus on the new question what is light <strong>and</strong> how can this<br />

explain its properties. This transformation is the subject of the next chapters.<br />

In chapters four <strong>and</strong> five my focus shifts, along with Huygens’, to the<br />

mechanistic nature of light. In 1672 a particular problem drew his attention<br />

to the question what light is <strong>and</strong> how its properties can be explained: the<br />

strange refraction in Icel<strong>and</strong> Crystal which created a puzzle regarding the<br />

physics of refraction that Huygens wanted to solve. These two chapters<br />

discuss the three stages of his investigation, his first analysis of the<br />

mathematics of strange refraction in 1672 <strong>and</strong> his eventual solution by means<br />

of elliptical waves in 1677 <strong>and</strong> its confirmation in 1679. Huygens’ first attack<br />

on the problem of strange refraction is historically significant because he<br />

approached it along traditional lines of geometrical optics. Only in second<br />

4<br />

Preliminary results are published in: Dijksterhuis, “Huygens’ Dioptrica” <strong>and</strong> Dijksterhuis, “Huygens’s<br />

efforts”.

6 CHAPTER 1<br />

instance did he turn to the actual question underlying the problem: how<br />

exactly do waves of light propagate. And only in third instance, <strong>and</strong> forced<br />

by critical reactions, did Huygens seek for experimental validation of the<br />

theory he initially had developed primarily rationally. These chapters offer a<br />

new, detailed reconstruction of the origin of the wave theory on the basis of<br />

manuscript material that has not been taken into account earlier. It is also a<br />

reconstruction of how Huygens got from Dioptrica to Traité de la Lumière, in<br />

which I compare his approach to questions pertaining to the physical nature<br />

of light in the mathematical science of optics to his predecessors <strong>and</strong><br />

contemporaries. Central themes are the way the nature of light was<br />

accounted for in the mathematical study of light in the seventeenth century<br />

<strong>and</strong> to relationships between explanatory theories of light <strong>and</strong> the laws of<br />

optics. I will argue that Huygens was the first to successfully mathematize a<br />

mechanistic conception of light. He was rivaled only by Newton, but for<br />

epistemological reasons he kept his hypotheses private.<br />

Chapter six reviews the development of Traité de la Lumière, its<br />

significance for the history of seventeenth-century optics, <strong>and</strong> for our<br />

underst<strong>and</strong>ing of Huygens’ science. After discussing the publication history<br />

of Traité de la Lumière, which reveals that Huygens disconnected it from<br />

Dioptrica only at the very last moment, I sketch some lines for a new<br />

perspective of the history of seventeenth-century optics in which traditional<br />

geometrical optics is taken into account as an important root. The<br />

mathematico-physical consideration of light of Traité de la Lumière was a<br />

particular answer to a new kind of question. A kind of question also<br />

addressed by such diverse scholars as Kepler, Descartes <strong>and</strong> Newton. In this<br />

sense, my study of the development of Traité de la Lumière, in particular in<br />

relationship with Dioptrica, is also a study of the origins of a new science of<br />

optics, nowadays denoted by the term ‘physical optics’. Some instances of<br />

physical optics developed in the seventeenth-century, most notably by<br />

Huygens <strong>and</strong> Newton. But the primacy of the question ‘what is the physical<br />

nature of light <strong>and</strong> how may this explain its properties? first had to be<br />

discovered <strong>and</strong> this only gradually came about in the pursuit of the<br />

mathematical science of optics. In the case of Huygens this emergence was<br />

particularly quiet. While solving the intriguing puzzle of strange refraction,<br />

he developed a new way of doing mathematical optics but he seems to have<br />

been hardly aware of the new ground he was breaking. At the close of this<br />

chapter, I discuss his alleged Cartesianism <strong>and</strong> I will argue that Huygens<br />

stumbled into becoming a ‘perfect Cartesian’ rather than determinedly <strong>and</strong><br />

systematically create it.<br />

This study is based on the optical papers in the Oeuvres Complètes <strong>and</strong><br />

additional manuscript material. A large part of these have as yet not been<br />

studied. The Oeuvres Complètes split up Huygens’ optics in two parts – volume<br />

13 for Dioptrica <strong>and</strong> volume 19 for Traité de la Lumière. This subdivision along<br />

modern disciplinary lines resounds in the historical literature. E.J.<br />

Dijksterhuis, for example, separates explicitly ‘geometrical optics’ – where he

‘THE PERFECT CARTESIAN’ 7<br />

merely mentions Huygens – <strong>and</strong> physical theories of optics. 5 No doubt all<br />

this has contributed its share to the fact that the relationship between Traité<br />

de la Lumière <strong>and</strong> Dioptrica – historical, conceptual as well as epistemical – has<br />

gone unexamined so far. 6 As for Traité de la Lumière, most historical<br />

interpretations are based on <strong>and</strong> confined to the published text. The<br />

additional manuscript material published in OC19 has hardly been taken into<br />

account <strong>and</strong> no-one to my knowledge has used the original manuscript<br />

material in the Codices Hugeniorum in the Leiden university library. 7 Huygens’<br />

wave theory has been the subject of several historical studies. Each in their<br />

own way has been valuable for this study. E.J. Dijksterhuis gives an<br />

illuminating analysis of the merits of Traité de la Lumière as a pioneering<br />

instance of mathematical physics considered in the light of Descartes’<br />

mechanistic program. Sabra includes an account of Huygens’ wave theory in<br />

his study of the historical development of the interplay of theory <strong>and</strong><br />

observation in seventeenth-century optics. Shapiro offers a searching analysis<br />

of the historical development of the physical concepts underlying Huygens’<br />

wave theory. I intend to add to our growing historical underst<strong>and</strong>ing of Traité<br />

de la Lumière by reconstructing its origin <strong>and</strong> development in the context of<br />

his optical studies as a whole <strong>and</strong> of that of seventeenth-century optics in<br />

general.<br />

Huygens’ lifelong engagement with dioptrics as such has hardly been<br />

studied. 8 Even Harting, the microscopist who by mid-nineteenth century<br />

gives Huygens’ telescopic work a central place in his biographical sketch,<br />

mentions dioptrical theory only in passing. 9 The editorial remarks in the<br />

‘Avertissement’ of OC13 form the main exception <strong>and</strong> are one of the few<br />

sources of information on the history of seventeenth-century geometrical<br />

optics in general. Some topics pertaining to seventeenth-century geometrical<br />

optics have been studied in considerable detail, but for the most part in the<br />

context of the seventeenth-century development of physical science. These<br />

are Kepler’s theory of image formation, the discovery of the sine law <strong>and</strong><br />

Newton’s mathematical theory of colors <strong>and</strong> they are integrated in my<br />

5<br />

Dijksterhuis, Mechanisering, IV: 168-171, 284-287.<br />

6<br />

Hashimoto hardly goes beyond noting that “… two works were closely related in Huygens’s mind.”:<br />

Hashimoto, “Huygens”, 87-88.<br />

7<br />

Dijksterhuis, Mechanization, IV: 284-287 <strong>and</strong> Sabra, Theories, 159-230 are confined to Traité de la Lumière.<br />

Shapiro uses some of the manuscripts published in Oeuvres Complètes. Ziggelaar, “How”, draws mainly on<br />

OC19. Yoder has pointed out that the wave theory is no exception to the rule that in general, studies of<br />

Huygens’ work tend to focus on his published works.<br />

8<br />

Hashimoto has published a not too satisfactory article in which he discusses Huygens’ dioptrics in<br />

general terms. Apart from some substantial flaws in his analyses <strong>and</strong> argument, Hashimoto fails to<br />

substantiate some of his main claims regarding Huygens’ ‘Baconianism’. Hashimoto, “Huygens”, 75-76;<br />

86-87; 89-90. For example, he reads back into Tractatus the utilitarian goal of De aberratione (60, compare<br />

my section 3.3.2), he thinks Huygens determined the configuration of his eyepiece theoretically (75,<br />

compare my section 3.1.2), maintains that Systema saturnium grew out of his study of dioptrics (89,<br />

compare my section 3.1.2) <strong>and</strong> that Huygens ‘went into the speculation about the cause of colors’ after his<br />

study of spherical aberration (89, compare my section 3.2.3)<br />

9<br />

Harting, Christiaan Huygens, 13-14. Harting based himself on manuscript material disclosed in<br />

Uylenbroek’s oration on the dioptrical work by the brothers Huygens: Uylenbroek, Oratio.

8 CHAPTER 1<br />

accounts of, respectively, seventeenth-century dioptrics in chapter 2, the<br />

epistemic role <strong>and</strong> status of explanations in optics in chapter 4, <strong>and</strong> Huygens’<br />

own dealings with colors in chapter 3 as well as his specific approach to<br />

mechanistic reasoning in chapter 5. Little literature on the history of the field<br />

of geometrical optics <strong>and</strong> its context exist. 10<br />

A substantial part of my argument is based upon comparisons with the<br />

pursuits of other seventeenth-century students of optics. In order to come to<br />

a historically sound underst<strong>and</strong>ing of what Huygens was doing, I find it<br />

necessary to find out how his optics relates to the pursuits of his<br />

predecessors <strong>and</strong> contemporaries. What questions did they ask (<strong>and</strong> what<br />

not) <strong>and</strong> how did they answer them? Why did they ask these questions <strong>and</strong><br />

what answers did they find satisfactory? For example, in chapter 2 the<br />

earliest part of Dioptrica is compared with, among other works, Kepler’s<br />

Dioptrice <strong>and</strong> Descartes’ La Dioptrique. All bear the same title, yet the<br />

differences are considerable. Descartes discussed ideal lenses <strong>and</strong> did so in<br />

general terms only, rather than explaining their focusing <strong>and</strong> magnifying<br />

properties as Kepler had done. Huygens, in his habitual search for practical<br />

application, expressly focused on analyzing the dioptrical properties of real,<br />

spherical lenses <strong>and</strong> their configurations, thus developing a rigorous <strong>and</strong><br />

general mathematical theory of the telescope. By means of such comparisons<br />

it is possible to determine in what way Huygens marked himself off as a<br />

seventeenth-century student of optics, or did not. These comparisons are<br />

focused on Huygens’ optics, so I confine my discussions of seventeenthcentury<br />

of optics to the mathematical aspects of dioptrics <strong>and</strong> physical<br />

optics. Other themes like practical dioptrics <strong>and</strong> natural philosophy in<br />

general will be treated only in relation to Huygens.<br />

This is an intellectual history of Huygens’ optics <strong>and</strong> of seventeenthcentury<br />

optics in general. The nature of the available sources – as well that of<br />

the man – are not suitable for some kind of social or cultural history. He<br />

operated rather autonomously, mainly because he was in the position to do<br />

so, <strong>and</strong> he was no gatherer of allies <strong>and</strong> did not try very hard to propagate his<br />

ideas about science <strong>and</strong> gain a following.<br />

New light on Huygens<br />

This study offers, in the first place, a concise history of Huygens’ optics. Yet<br />

it is not a mere discussion of Huygens’ contributions to various parts of<br />

optics. I also intend to shed more light on the character of Huygens’<br />

scientific personality. The issue of getting a clear picture of his scientific<br />

activity <strong>and</strong> its defining features is an acknowledged problem. In summing<br />

up a 1979 symposium on the life <strong>and</strong> work of Huygens, Rupert Hall<br />

10 For example, the precise application of the sine law to dioptrical problems, for example, has hardly been<br />

studied. Shapiro, “The Optical Lectures” is a valuable exception, discussing Barrow’s lectures <strong>and</strong> their<br />

historical context. The relationship between the development of the telescope <strong>and</strong> of dioptrical theory –<br />

essential to my account of Dioptrica – has never been investigated in any detail. Van Helden has pointed<br />

out the weak connection between both in general terms: Van Helden, “The telescope in the 17 th century”,<br />

45-49; Van Helden, “Birth”, 63-68.

‘THE PERFECT CARTESIAN’ 9<br />

concluded that “… it isn’t at all easy to underst<strong>and</strong> how all the multifarious<br />

activities of this man’s life fit together.” 11 Huygens has been called the true<br />

heir of Galileo, the perfect Cartesian, <strong>and</strong> also a man deftly steering a middle<br />

course between Baconian empiricism <strong>and</strong> Cartesian rationalism. 12 Huygens<br />

himself has not been much of a help in this. He always was particularly<br />

reticent about his own motives. He was an intermediate figure between<br />

Galileo <strong>and</strong> Descartes on the one h<strong>and</strong> <strong>and</strong> Newton <strong>and</strong> Leibniz on the<br />

other but, lacking as he did a pronounced conception of the aims <strong>and</strong><br />

methods of his science, he is difficult to situate among the protagonists of<br />

the scientific revolution.<br />

In 1979, the most apt characterization of Huygens seemed to be that of<br />

an eclectic, who took up loose issues <strong>and</strong> solved them with the means he<br />

considered appropriate without some sort of central direction becoming<br />

apparent. 13 The original idea behind this study was that in Traité de la Lumière<br />

this eclecticism grew into a fruitful synthesis of mathematical, mechanistic<br />

<strong>and</strong> experimental approaches. This idea originates from the work my advisor,<br />

the Casper Hakfoort. In his study of eighteenth-century optics he formulated<br />

the idea when he pointed out the significance of natural philosophy for the<br />

development of optics, which in his view was hithertho neglected. 14 I<br />

consider it an honor to have been able to pursue this idea <strong>and</strong> to have had it<br />

bear unanticipated fruit. By trying to underst<strong>and</strong> how the multifarious<br />

aspects of his optics fit together, I hope to be able to shed light on the<br />

character of his science in general <strong>and</strong> on his place in seventeenth-century<br />

science. Dioptrica, while adding to the st<strong>and</strong>ing impression of the great<br />

versatility of Huygens’ oeuvre, has not changed my expectation that an<br />

underst<strong>and</strong>ing of the way the possible coherence of these aspects evolved<br />

may contribute to a better characterization of Huygens’ science <strong>and</strong> of his<br />

place in seventeenth-century science as a whole.<br />

In the final chapter of this book, I review my account of the development<br />

of Huygens’ optics to see what light this may shed on his scientific<br />

personality. By way of conclusion it offers a sketch of his science, based on<br />

the previous chapters that go into the details – often technical – of his optics<br />

<strong>and</strong> its development. This chapter can be read independently as an essay on<br />

Huygens.<br />

Huygens was a puzzle solver indeed, an avid seeker of rigorous, exact<br />

solutions to intricate mathematical puzzles. But these puzzles do have<br />

coherence, they all concerned questions regarding concrete, almost tangible<br />

subjects in the various fields of seventeenth-century mathematics. He was an<br />

eclectic, but only in comparison with the chief protagonists of the scientific<br />

11<br />

Hall, “Summary”, 311.<br />

12<br />

Westfall, Construction, 132-154; Dijksterhuis, Mechanization, 212; Elzinga, Research program <strong>and</strong> Westman,<br />

“Problem”, 100-101.<br />

13<br />

Hall, “Summary”, 305-306. As regards his studies of motion, Yoder has further specified this<br />

characterization; Yoder, Unrollling time, 169-179.<br />

14<br />

Hakfoort, Optics in the age of Euler, 183-184.

10 CHAPTER 1<br />

revolution, that set up schemes to lay new foundations for natural inquiry.<br />

Huygens did not have such a program <strong>and</strong>, as a result, his science seems to<br />

lack coherence, unless a coherence is looked for on a different level of<br />

seventeenth-century science. The essay therefore first forgets about<br />

Huygens’ alleged Cartesianism to sketch the mathematician <strong>and</strong> his<br />

idiosyncratic focus on instruments. He was not a half-baked philosopher but<br />

a typical mathematician. A new Archimedes, as Mersenne foretold in 1647.<br />

The incomparable Huygens, as Leibniz said in 1695 upon the news of his<br />

death. 15 Then I ask anew how his Cartesianism fits into the picture. We may<br />

have trouble getting a balanced idea of what he was doing, but it appears that<br />

he was hardly aware of the size of the new ground he had been breaking. He<br />

had, in fact, developed a new mathematical science of optics.<br />

15 OC1, 47 <strong>and</strong> OC10, 721.

Chapter 2<br />

1653 - 'Tractatus'<br />

The mathematical underst<strong>and</strong>ing of telescopes<br />

“Now, however, I am completely into dioptrics”, Huygens wrote on 29<br />

October 1652 in a letter to his former teacher in mathematics, Frans van<br />

Schooten, Jr. 1 His enthusiasm had been induced by a discovery in dioptrical<br />

theory he had recently made. It was an addition to Descartes’ account of the<br />

refracting properties of curves in La Géométrie, that promised a useful<br />

extension of the plan for telescopes with perfect focusing properties<br />

Descartes had set out in La Dioptrique. During his study at Leiden University<br />

in 1645-6, Huygens had studied Descartes’ mathematical works, La Géométrie<br />

in particular, intensively with Van Schooten. Although Van Schooten was<br />

professor of ‘Duytsche Mathematique’ at the Engineering school, appointed<br />

to teach practical mathematics in the vernacular to surveyors <strong>and</strong> the like.<br />

Huygens was not the only patrician son he introduced to the new<br />

mathematics: the future Pensionary Johan de Witt <strong>and</strong> the future Amsterdam<br />

mayor Johannes Hudde.<br />

From 1647 Huygens <strong>and</strong> Van Schooten had to resort to corresponding<br />

over mathematics, when Huygens had to go to Breda to the newly<br />

established ‘Collegium Auriacum’, the college of the Oranges to which the<br />

Huygens family was closely connected politically. In 1649 Huygens had<br />

returned home to The Hague <strong>and</strong> now, in 1652, he was ‘private citizen’. He<br />

did not feel like pursuing the career in diplomacy his father had planned for<br />

him <strong>and</strong>, with the Oranges out of power since 1650, not many duties were<br />

left to call on him. Huygens could, in other words, freely pursue his one<br />

interest, the study of the mathematical sciences. An appointment at a<br />

university was out of order for someone of his st<strong>and</strong>ing <strong>and</strong>, as Holl<strong>and</strong><br />

lacked a centrally organized church <strong>and</strong> a gr<strong>and</strong> court, interesting options for<br />

patronage were not directly available. 2 So, with a room in his parental home<br />

at the ‘Plein’ in The Hague <strong>and</strong> a modest allowance from his father, he could<br />

live the honorable life of an ‘amateur des sciences’. He enjoyed to company<br />

of his older brother Constantijn, who joined him in his work in practical<br />

dioptrics (see next chapter), <strong>and</strong> dedicated himself to mathematics.<br />

Geometry <strong>and</strong> mechanics were his main focus in these years, with his<br />

theories of impact <strong>and</strong> of pendulum motion <strong>and</strong> his invention of the<br />

1<br />

OC 1, 215. “Nunc autem in dioptricis totus sum ...”<br />

2<br />

Berkel, “Illusies”, 83-84. In the 1660s Huygens would start to seek patronage abroad, first in Florence<br />

<strong>and</strong> then, successfully in Paris.

12 CHAPTER 2<br />

pendulum clock as the most renowned achievements. Yet, the discovery<br />

made late 1652 had sparked his interest in dioptrics, which largely dominated<br />

his scholarly activities the next two years. The letter to Van Schooten was the<br />

onset to a lifelong engagement with dioptrics, which nevertheless has little<br />

been studied historically. 3<br />

In the months following the letter to Van Schooten, Huygens elaborated<br />

a treatise that contained a mathematical theory of the dioptrical properties of<br />

lenses <strong>and</strong> telescopes. I will refer to this treatise as Tractatus <strong>and</strong> it is the<br />

subject of this chapter. 4 In Tractatus Huygens treated a specific set of<br />

dioptrical questions, directed at underst<strong>and</strong>ing the working of the telescope.<br />

In the first section of this chapter the content <strong>and</strong> character of the treatise<br />

are discussed. In the second section Huygens’ approach to dioptrics is<br />

compared with that of contemporaries, by examining how other<br />

mathematicians dealt with the questions that stood central in Tractatus. In this<br />

discussion of seventeenth-century dioptrics the relationship between<br />

dioptrical theory <strong>and</strong> the development of the telescope is the central topic. I<br />

will argue that Huygens in his mathematical theory stood out for his focus<br />

on questions that were relevant to actual telescopes. In this he was the first<br />

to follow Kepler’s lead; other theorists were absorbed by abstract questions<br />

emerging from mathematical theory for which men of practice, in their turn,<br />

did not care. In the next chapter Huygens’ own telescopic practices are<br />

discussed. Now first the theoretical considerations of Tractatus.<br />

2.1 The Tractatus of 1653<br />

The background to Huygens’ letter to Van Schooten was a problem with the<br />

lenses used in the telescopes of those days. <strong>Lenses</strong> were spherical, i.e. their<br />

cross section is circular. As a result they do not focus parallel rays perfectly.<br />

Rays from a distant point source that are<br />

refracted by a spherical surface do not<br />

intersect in a single point, rays close to<br />

the axis are refracted to a more distant<br />

point on the axis than rays farther from<br />

the axis (Figure 2). This is called spherical<br />

aberration <strong>and</strong> results in slightly blurred Figure 2 Spherical aberration<br />

3 The most thorough-going account still are the ‘avertissements’ by the editors of the Oeuvres Complètes.<br />

Southall, “Some of Huygens’ contributions” reported on Huygens’ dioptrics after the publication of<br />

volume 13. Harting, Christiaan Huygens had earlier discussed it briefly. In relationship with his astronomical<br />

work <strong>and</strong> his practical dioptrics, Albert van Helden, “Development” <strong>and</strong> Anne van Helden/Van Gent,<br />

The Huygens collection <strong>and</strong> “Lens production” discuss some topics. In the context of the history of<br />

seventeenth-century geometrical optics – which in its own right has little been studied – Shapiro,<br />

“’Optical Lectures’” mention Huygens’ contributions. They are remarkably absent from the Malet, “Isaac<br />

Barrow” <strong>and</strong> “Kepler <strong>and</strong> the telescope”. Hashimoto, “Huygens, dioptrics” is the only effort to discuss<br />

Huygens’ dioptrics in the context of his broader oeuvre.<br />

4 OC13, 1-271. The editors of the Oeuvres Complètes have labeled it Dioptrica, Pars I. Tractatus de refractione et<br />

telescopiis. Its content stems from the 1650s. The original version of Tractatus does not exist anymore. A<br />

copy was made in Paris by Niquet – probably in 1666 or 1667, at the beginning of Huygens’ stay in Paris<br />

– on which the text of the Oeuvres Complètes is based. The editors assume Niquet’s copy of Tractatus is<br />

largely identical with the original 1653 manuscript; “Avertissement”, XXX.

1653 - TRACTATUS 13<br />

images. In La Dioptrique (1637), Descartes had explained that surfaces whose<br />

section is an ellipse or a hyperbola do not suffer this impediment. They are<br />

called aplanatic surfaces. Descartes could demonstrate this by means of the<br />

sine law, the exact law of refraction he had discovered some 10 years earlier.<br />

According to the sine law, the sines of incident <strong>and</strong> of refracted rays are in<br />

constant proportion. This ratio of sines is nowadays called index of<br />

refraction, it depends upon the refracting medium.<br />

The discovery Huygens made in late 1652 sprang from Descartes’ La<br />

Géométrie. Together with La Dioptrique <strong>and</strong> Les Météores, this essay was<br />

appended to Discours de la methode (1637). In La Géométrie, Descartes had<br />

introduced his new analytic geometry. In La Géométrie mathematical proof<br />

was given of the claim of La Dioptrique that the ellipse <strong>and</strong> hyperbola are<br />

aplanatic curves. In his letter to Van Schooten, Huygens wrote that he had<br />

discovered that under certain conditions circles also are aplanatic. This<br />

discovery implied that spherical lenses could focus perfectly in particular<br />

cases. Consequently, Huygens considered it of considerable importance for<br />

the improvement of telescopes.<br />

Huygens’ expectation that his discovery would be useful in practice, was<br />

fostered by the fact that Descartes’ claims had turned out not to be<br />

practically feasable. Around 1650, no one had succeeded in actually grinding<br />

the lenses prescribed in La Dioptrique. 5 Apart from that, the treatise did not<br />

discuss the spherical lenses actually employed in telescopes. Descartes had<br />

applied his exact law only to theoretical lenses. When he made his discovery,<br />

Huygens must have realized that no-one had applied the sine law to spherical<br />

lenses yet. In the aftermath of his discovery, Huygens set out to correct this<br />

<strong>and</strong> develop a dioptrical theory of real lenses.<br />

2.1.1 OVALS TO LENSES<br />

In his letter to Van Schooten, Huygens did not explain the details of his<br />

discovery. He did so much later, in an appendix to a letter of 29 October<br />

1654 that contained comments upon Van Schooten’s first Latin edition of<br />

La Géométrie: Geometria à Renato Des Cartes (1649). 6 In book two, Descartes<br />

had introduced a range of special curves, ovals as he called them. This was<br />

not a mere abstract exercise, he said, for these curves were useful in optics:<br />

“For the rest, so that you know that the consideration of the curved lines here<br />

proposed is not without use, <strong>and</strong> that they have diverse properties that do not yield at<br />

all to those of conic sections, I here want to add further the explanation of certain<br />

ovals, that you will see to be very useful for the theory of catoptrics <strong>and</strong> of dioptrics.” 7<br />

By means of the sine law, Descartes derived four classes of ovals that are<br />

aplanatic curves. If such a curve is the section of a refracting surface, rays<br />

5<br />

With the possible exception of Descartes himself. See below, section 3.1<br />

6<br />

OC1, 305-305.<br />

7<br />

Descartes, Geometrie, 352 (AT6, 424). “Au reste affin que vous sçachiées que la consideration des lignes<br />

courbes icy proposée n’est pas sans usage, & qu’elles ont diverses proprietés, qui ne cedent en rien a celles<br />

des sections coniques, ie veux encore adiouster icy l’explication de certaines Ovales, que vous verrés estres<br />

tres utiles pour la Theorie de la Catoptrique, & de la Dioptrique.”

14 CHAPTER 2<br />

coming from a single point are refracted towards another single point. In<br />

certain cases, the ovals reduce to the ellipses <strong>and</strong> hyperbolas of La Dioptrique.<br />

Huygens in his turn discovered that a particular class of these ovals may also<br />

reduce to a circle.<br />

In La Géométrie Descartes introduced the<br />

said class of ovals as follows (Figure 3). The<br />

dotted line is an oval of this class. If the<br />

right part 2X2 of the oval is the right<br />

boundary of a refracting medium, rays<br />

intersecting in point F are refracted to point<br />

G. 8 The oval is constructed as follows. Lines<br />

FA <strong>and</strong> AS intersect in A at an arbitrary<br />

angle, F is an arbitrary point on FA. Draw a<br />

circle with center F <strong>and</strong> radius F5. Line 56 is<br />

drawn, so that A5 is to A6 as the ratio of<br />

sines of the refracting medium. G is an arbitrary point between A <strong>and</strong> 5, S is<br />

on A6 with AS = AG. A circle with center G <strong>and</strong> radius S6 cuts the first circle<br />

in the points 2, 2. These are the first two points of the oval. This procedure<br />

is repeated with points 7 <strong>and</strong> 8, et cetera until the oval 22X22 is completed. 9<br />

Huygens’ discovered that the oval reduces to a circle when the ratio of AF to<br />

AG is equal to the ratio of A5 to A6, the ratio of sines. 10 This means that with<br />

respect to rays tending to F, a spherical surface 2X2 will focus them exactly in<br />

G. Van Schooten was a bit skeptical about Huygens’ claim. Could such a<br />

simple fact have escaped Descartes? Nevertheless, he included it in the<br />

second edition of Geometria à Renatio Des Cartes (1659). 11<br />

Discovering that a spherical surface is aplanatic in certain cases is one<br />

thing, applying it to lenses in practice is another. It remained to be seen what<br />

shape the second surface should have <strong>and</strong> how it might be employed in<br />

telescopes. For one thing, it does not seem useful for objective lenses, the<br />

front <strong>and</strong> most important lens of a telescope that receives parallel rays. It<br />

appears the usefulness of the discovery was limited, for Huygens never<br />

returned to it in his dioptrical studies. 12<br />

The historical importance of the discovery lies in the fact that it aroused<br />

Huygens’ interest in dioptrics. He did not exaggerate when he said he was<br />

engrossed in dioptrics. Not only its theory, practice too. Five days after his<br />

letter to Van Schooten, on 4 November, he wrote to Gerard Gutschoven, an<br />

acquainted mathematician in Antwerp. 13 After some introductory remarks,<br />

8 Descartes, Geometrie, 358-359 (AT6, 430-431). The left part 2A2 is a mirror that reflects rays intersecting<br />

in G so that they (virtually) intersect in F, provided that it diminishes the ‘tendency’ of the rays to a given<br />

degree.<br />

9 Descartes, Geometrie, 353-354 (AT6, 424-426). The curve satisfies the equation F2 – FA = n(G2 – GA).<br />

10 OC1, 305. See note 9: the equation becomes AF = nAG.<br />

11 Reproduced in OC14, 419.<br />

12 In Tractatus, he merely mentioned that a spherical surface is aplanatic for certain points: OC13, 64-67.<br />

13 OC1, 190-192.<br />

Figure 3 Cartesian oval.

1653 - TRACTATUS 15<br />

Huygens launched a series of questions on the art of making lenses. What<br />

material are grinding moulds made of, how is the spherical figure of a lens<br />

checked, what glue is used to attach the lenses to a grip, et cetera. Only after<br />

these questions did he explain to Gutschoven that he wanted to know all<br />

these things because he had discovered something that would greatly<br />

improve telescopes.<br />

Figure 4 Focal distance of a bi-convex lens<br />

The letter reveals that Huygens had already begun to investigate the<br />

dioptrical properties of spherical lenses. It contained a theorem on the focal<br />

distance of parallel rays refracted by a bi-convex lens CD (Figure 4). 14 AC <strong>and</strong><br />

DB are the radii of the anterior <strong>and</strong> posterior side of the lens. L <strong>and</strong> E are<br />

determined by DL : LB =CE : EA = n, the index of refraction. O is found by<br />

EL : LB = ED : EO. Rays parallel to the axis EL come from the direction of L.<br />

Without proof Huygens said that O is the focus of the refracted rays. He said<br />

that he could prove this <strong>and</strong> that he had found many more theorems. A<br />

month later, in a letter of 10 December to André Tacquet, a Jesuit<br />

mathematician in Louvain, he added an important insight. 15 As a result of<br />

spherical aberration point O is not the exact focus. Nevertheless, it may be<br />

taken as the focus: “… since beyond point O no converging rays intersect<br />

with the axis.” 16 In later letters to Tacquet <strong>and</strong> Gutschoven he called this<br />

point the ‘punctum concursus’. 17 It is the where rays closest to the axis are<br />

refracted to. This definition would be fundamental to the theory of Tractatus,<br />

which apparently was well under way. Huygens told Tacquet that he had<br />

already written two books of a treatise on dioptrics: one on focal distances,<br />

another one on magnification. A third one on telescopes was in preparation.<br />

Within a month or two after his letter to van Schooten, Huygens’<br />

underst<strong>and</strong>ing of dioptrics was rapidly developing. It was also developing in<br />

a particular direction. Huygens was studying the dioptrical properties of<br />

spherical lenses. He must have found out that little had been published on<br />

the subject. The only mathematical theory of spherical lenses was Kepler’s<br />

Dioptrice (1611), but it lacked an exact law of refraction. Only Descartes had<br />

applied the sine law to lenses, but he had ignored spherical lenses. Huygens<br />

had begun to develop an exact theory of spherical lenses by himself. He<br />

combined this theoretical interest with an interest in practical matters of<br />

telescope making. He reported to have seen a telescope made by the famous<br />

craftsman Johann Wiesel of Augsburg. He was impressed <strong>and</strong> regretted that<br />

14<br />

OC1, 192.<br />

15<br />

OC1, 201-205.<br />

16<br />

OC1, 204. “… adeo ut nullius radij concursus cum axe contingat ultra punctum O.”<br />

17<br />

OC1, 224-226.

16 CHAPTER 2<br />

Holl<strong>and</strong> did not have such excellent craftsmen. 18 On 10 February, 1653,<br />

Gutschoven finally informed him on the art of lens making. 19 Huygens did<br />

not put the information to practice right-away, he first elaborated his<br />

dioptrical theory.<br />

2.1.2 A THEORY OF THE TELESCOPE<br />

Huygens had written to Tacquet that his treatise would consist of three parts:<br />

a theory of focal distances of lenses, a theory of the magnification produced<br />

by configurations of lenses, <strong>and</strong> an account of the dioptrical properties of<br />

telescopes based on the theory of the two preceding parts. The third part<br />

was not yet finished when Huygens wrote Tacquet, in fact he never<br />

elaborated in the form originally conceived. The third part of Tractatus as it is<br />

found in the Oeuvres Complètes is a collection of dispersed propositions<br />

collected by the editors. Only the first two seem to be from the 1650s. 20 In<br />

the arrangements of manuscripts Huygens made in the late 1680s, part one<br />

of Tractatus appears for the large part as it is found in the Oeuvres Complètes. 21<br />

Judging from the various page numberings, Huygens has not edited it very<br />

much, except that he inserted – probably in the late 1660s – parts of his<br />

study of spherical aberration after the twentieth proposition. Part two of<br />

Tractatus has been reshuffled somewhat more, but the main line appears to<br />

be sufficiently original. In the following discussion of Tractatus, I follow the<br />

text of the Oeuvres Complètes in so far as it appears to reflect the original<br />

treatise.<br />

Huygens coupled his orientation on the telescope with the mathematical<br />

rigor typical of him. Although he singled out dioptrical problems that were<br />

relevant to the telescope, he treated these with a generality <strong>and</strong> completeness<br />

that often exceeded the direct needs of explaining the working of the<br />

telescope. Huygens’ rigorous approach is clear from the very start of<br />

Tractatus. Basic for his treatment of focal distances was the realization that<br />

spherical surfaces do not focus exactly. This had been noticed earlier <strong>and</strong> had<br />

been the rationale behind La Dioptrique. Nobody, however, had gone beyond<br />

the mere observation of spherical aberration. Huygens got a firmer<br />

mathematical grip on the imperfect focusing of lenses by defining which<br />

point on the axis may count as the focus. Although he only discussed focal<br />

points, Huygens took spherical aberration into account by consistently<br />

determining the focus as the ‘punctum concursus’.<br />

18 OC1, 215.<br />

19 OC1, 219-223.<br />

20 The decisions the editors made for the remaining propositions are sometimes somewhat mysterious.<br />

For example, the fourth proposition has been assembled of fragments from various folios. And from<br />

folio Hug29, 177 they put a diagram in part three of Tractatus, but they transferred the main contents to<br />

‘De telescopiis’ (see section 6.1.2).<br />

21 On this arrangement see page 221. By the way, the two first propositions of part three are inserted after<br />

part one.

1653 - TRACTATUS 17<br />

In part one of Tractatus he defined ‘punctum<br />

concursus’ as follows. In the third proposition, he<br />

defined the focus as the limit point of the intersections of<br />

refracted rays with the axis (Figure 5). 22 ABC is a planoconvex<br />

lens <strong>and</strong> parallel rays are incident from the<br />

direction of D. Consequently, they are only refracted by<br />

the spherical surface. Huygens showed that the closer<br />

rays are to the axis DE, the closer to E they reach it.<br />

Beyond E no refracted rays crosses the axis. This limit<br />

point E he defined as the ‘punctum concursus’ of the<br />

spherical surface ABC. If the surface is concave, rays do<br />

not intersect at all after refraction, they diverge. In this<br />

case, the ‘punctum concursus’ is the virtual focus, the<br />

limit point of the intersections with the axis of the<br />

backwards extended refracted rays.<br />

In the first part of Tractatus, Huygens derived focal<br />

distances of all types of spherical lenses by determining<br />

exactly the ‘punctum concursus’ in each case. Refraction of parallel rays by a<br />

lens consists in most cases of the two successive refractions by each side of<br />

the lens. Determining the focal distance thus consists of three problems.<br />

First, the refraction of parallel rays from air to glass by a spherical surface.<br />

Second, that of the refraction of converging or diverging rays from glass to<br />

air. Finally, combining both. Huygens built up his theory accordingly. He<br />

first derived theorems expressing focal distance of spherical surfaces for<br />

parallel rays in terms of their radii. Secondly, he derived theorems expressing<br />

the focal distance for non-parallel rays in terms of the radius <strong>and</strong> the focal<br />

distance for parallel rays. Finally, he expressed the focal distance of the<br />

various kinds of lenses in terms of the radii of their sides. In each case he<br />

took the thickness of the lens into account. Only afterwards did he derive<br />

simplified theorems for thin lenses, in which their thickness is ignored. I now<br />

sketch the typical case of a bi-convex lens, the theorem that Huygens<br />

included without proof in his letters to Gutschoven <strong>and</strong> Tacquet. The<br />

determination of the focal distances of other lenses – plano-convex, biconcave,<br />

etc. – went along similar lines.<br />

The focal distance of a bi-convex lens<br />

The eighth proposition of Tractatus dealt with parallel rays refracted at the<br />

convex surface of a denser medium (Figure 6). AC is the radius of ABP; Q is a<br />

point on the axis AC so that AQ : QC = n, where n is the index of refraction.<br />

Huygens demonstrated that Q is the ‘punctum concursus’ of parallel rays OB,<br />

NP. A refracted ray BL intersects axis AC in a point L between A <strong>and</strong> Q. With<br />

the sine law BL : LC = AQ : QC = n. For any ray OB, BL is smaller than AL <strong>and</strong><br />

AL is smaller than AQ. Therefore no refracted rays intersect the axis beyond<br />

22 OC13, 16-19<br />

Figure 5 Punctum<br />

concursus

18 CHAPTER 2<br />

Q. In order to prove that Q is the ‘punctum concursus’ of ABP, consider ray<br />

NP <strong>and</strong> its refraction PK. PK is found with the sine law <strong>and</strong> KQ is therefore a<br />

given interval. On KQ choose L <strong>and</strong> draw T, close to A, so that LT : CL =<br />

AQ : CQ = n. Now PL : LC < PK : KC = n. PL is smaller than TL, which in its<br />

turn is smaller AL. A circle with center L <strong>and</strong> radius TL intersects the<br />

refracting surface ABP between A <strong>and</strong> P in a point B. Draw BL <strong>and</strong> BC <strong>and</strong> it<br />

follows that BL : LC = TL : LC = n. Therefore BO is refracted to L. So, the<br />

closer a paraxial ray is to the axis, the closer to Q the refracted ray will<br />

intersect with the axis. Q is the limit point of these intersections <strong>and</strong><br />

therefore the ‘punctum concursus’. When the index of refraction<br />

AQ : QC = 3 : 2 – the approximate value for glass – AQ is exactly three times<br />

the radius AC.<br />

The refraction at the<br />

posterior side of the lens is dealt<br />

with in the twelfth proposition.<br />

This case is more complex as<br />

the incident rays are converging<br />

due to the refraction at the<br />

anterior side. Huygens dealt<br />

with eight cases of non-parallel<br />

rays. 23 For all cases, he<br />

expressed the focal distance of<br />

the non-parallel rays in terms of<br />

the focal distance of the surface<br />

for paraxial rays. The case at<br />

h<strong>and</strong> is the fourth part of the<br />

proposition (Figure 7). 24 Rays<br />

converge towards a point S,<br />

outside the dense medium<br />

bounded by a spherical surface<br />

AB with radius AC. Q is the ‘punctum concursus’ of paraxial rays coming<br />

from R. With SQ : SA = SC : SD, the ‘punctum concursus’ D of the converging<br />

rays LB is found. Huygens’ proof consisted of a reversal of the first case<br />

treated in this proposition: rays diverging from D are refracted so that they<br />

(virtually) intersect in S. 25 This proof is similar to the one above.<br />

Finally, in the sixteenth proposition of Tractatus, Huygens determined the<br />

focal distance of a convex lens by combining the preceding results. It was<br />

equal to the theorem he put forward in his letters to Tacquet <strong>and</strong><br />

Gutschoven. CD is a bi-convex lens with radii of curvature AC <strong>and</strong> BD<br />

(Figure 8). The foci for paraxial rays are respectively E <strong>and</strong> L. According to<br />

the eighth proposition CE : EA = DL : LB = n. With the twelfth proposition,<br />

23 OC13, 40-79.<br />

24 OC13, 70-73.<br />

25 OC13, 42-47.<br />

Figure 6 Refraction at<br />

the anterior side of a<br />

bi-convex lens<br />

Figure 7 Refraction<br />

at the posterior side<br />

of a bi-convex lens.

1653 - TRACTATUS 19<br />

the ‘punctum concursus’ N for parallel rays from the direction of L is found<br />

with EL : ED = EB : EN. After the refraction at the surface C, the rays<br />

converge towards E; they are then refracted at the surface D towards N. In<br />

modern notation:<br />

nAC BD<br />

BC CD<br />

DN <br />

n 1<br />

n(<br />

AC BD)<br />

(<br />

n 1)<br />

CD<br />

, where DN is the focal distance<br />

measured from the anterior face of the lens. For rays coming from the other<br />

direction O is the ‘punctum concursus’.<br />

The case of a bi-convex lens was only one out of many cases Huygens<br />

treated in the fourteenth to seventeenth proposition of Tractatus. Taking both<br />

spherical aberration <strong>and</strong> the thickness of the lens into account, he derived<br />

exact theorems for the focal distance of each type of lens. In each case, he<br />

also showed how to simplify the theorem when the thickness of the lens is<br />

not taken into account. In the case of a bi-convex lens, he started by<br />

comparing the focal distances CO <strong>and</strong> DN when the radii of both sides of the<br />

lens are not equal. Their difference vanishes when the thickness of the lens<br />

CD is ignored <strong>and</strong> both refractions are assumed to take place simultaneously.<br />

The focal distance N is then easily found by first determining point L with<br />

AC BD<br />

DL : LB = n <strong>and</strong> then AB : AD = DE : EA, or DN <br />

AC BD<br />

2<br />

. 26 In the case of<br />

a glass lens (n = 3 : 2) LB is twice BD <strong>and</strong> (AC + BD) : AC = 2BD : DN. It<br />

follows directly that the focal distance is equal to the radius in the case of an<br />

equi-convex lens. In the twentieth proposition of Tractatus, Huygens<br />

extended the results for thin lenses to non-parallel rays. In this case rays<br />

diverge from a point on the axis relatively close to the lens <strong>and</strong> are refracted<br />

towards a point P found by DO · DP = DC2 (DO is the focal distance for<br />

parallel rays coming from the opposite direction). Huygens had to treat all<br />

cases of positive <strong>and</strong> negative lens sides separately, but the result comes<br />

down to the modern formula 1 1 1 . 27<br />

<br />

p p'<br />

f<br />

In the remainder of the first book of Tractutus, Huygens completed his<br />

theory of focal distances by determining the image of an extended object,<br />

rounded off in the twenty-fourth proposition (Figure 9). The diameter of the<br />

image IG is to the diameter of the object KF as the distance HL of the image<br />

26 OC13, 88-89. Equivalent to the modern formula 1<br />

f<br />

27 OC13, 98-109.<br />

Figure 8 Focal distance DN of a bi-convex lens<br />

= (n -1)( + )<br />

1 1<br />

R1<br />

R 2<br />

.

20 CHAPTER 2<br />

to the lens is to the distance EL of the object to the lens. 28<br />

The point L has a special property that Huygens had<br />

established in the preceding proposition. An arbitrary ray<br />

that passes through this point leaves the lens parallel to the<br />

incident ray. 29 In the twenty-second proposition, Huygens<br />

had demonstrated that the focal distance LG of rays from a<br />

point K of the axis is more or less equal to that of a point E<br />

on the axis. 30 The triangles KLF <strong>and</strong> GLI are therefore<br />

similar, which proves the theorem. 31<br />

Images<br />

The theory of focal distances formed the basis of Huygens’ discussion of the<br />

properties of images formed by lenses <strong>and</strong> lens-systems in the second book<br />

of Tractatus. Huygens’ theory of images is once again both rigorous <strong>and</strong><br />

general. The central questions in book two were how to determine the<br />

orientation of the image <strong>and</strong> the degree of magnification. For the time being,<br />

Huygens ignored the question whether an image is in focus. In this way he<br />

could derive general theorems on the relationship between the shape of<br />

lenses <strong>and</strong> their magnifying properties. He then showed how these reduced<br />

to simpler theorems in particular cases, for example for a distant object. In<br />

the third book he showed what configurations produced focused images.<br />

Figure 10 Magnification by a convex lens.<br />

Figure 9 Extended<br />

image.<br />

In the second <strong>and</strong> third propositions of book two, Huygens discussed a<br />

convex lens. His aim was to determine the magnification of the image for the<br />

various positions of eye D, lens ACB <strong>and</strong> object MEN (Figure 10). In order to<br />

distinguish between upright <strong>and</strong> reversed images, Huygens defined the<br />

‘punctum correspondens’ (later called ‘punctum dirigens’). 32 This is the focus<br />

of rays emanating from the point where the eye is situated <strong>and</strong> is thus found<br />

by means of the theory of the first book. First, the eye D is between a convex<br />

28<br />

OC13, 122-125.<br />

29<br />

OC13, 118-123. In modern terms, L is the optical center.<br />

30<br />

OC13, 114-119.<br />

31<br />

Huygens added that when the thickness of the less is taken into account, point V in the lens instead of<br />

L should be taken as the vertex of the triangle.<br />

32<br />

OC13, 176n1.

1653 - TRACTATUS 21<br />

lens ACB <strong>and</strong> its focus O. In this case, the object MEN is seen upright <strong>and</strong><br />

magnified. The lens refracts a ray NBP to BD so that point N of the object is<br />

seen in B, whereas it would be seen in C without the lens. AB is larger than<br />

AC <strong>and</strong> on the same side of the axis. Huygens then showed that<br />

AB : AC = (AO : OD)·(ED : EP), which in the case of a distant object reduces<br />

to AB : AC = AO : OD. 33 If, on the other h<strong>and</strong>, the eye is placed in the focus<br />

(so that AD = AO) <strong>and</strong> NB is taken parallel to the axis, AB : AC = EO : AO<br />

which becomes infinitely large when the object is placed at large distance. In<br />

the next proposition, Huygens considered the cases where the eye is placed<br />

beyond the focus O. In this case the ‘punctum correspondens’ P is on the<br />

other side of the lens <strong>and</strong> the image will be reversed when the object is<br />

placed beyond it.<br />

With the same degree of generality, in the fifth proposition Huygens<br />

discussed the images produced by a configuration of two lenses. 34 He figured<br />

(Figure 11, most left one) two lenses A <strong>and</strong> B with focal distances GA <strong>and</strong> HB,<br />

the eye C <strong>and</strong> the object DEF, all arbitrarily positioned on a common axis. He<br />

then constructed point K on the axis, the ‘punctum correspondens’ of the<br />

eye with respect to lens B, the ocular lens. Next, he constructed point L on<br />

the axis, the ‘punctum correspondens’ of point K with respect to lens A, the<br />

objective lens. In this way, a ray LD will be refracted by the two lenses to the<br />

eye via points M on lens A <strong>and</strong> N on lens B. Without lenses, the eye sees<br />

point F of the object – where DE = DF – along line COF. The degree of<br />

magnification is therefore determined by the proportion BN : BO. In this<br />

general case, the magnification follows from<br />

BN : BO = (HB : HC)·(AG : GK)·(EC : EL). Huygens derived this proportion for<br />

the case of a concave ocular <strong>and</strong> a convex objective, but the same applied to<br />

a system of two convex lenses.<br />

In the adjoining drawings, Huygens sketched various positions of eye,<br />

lenses <strong>and</strong> object (Figure 11 gives four cases). These showed whether the<br />

image was upright or reversed. In addition, he showed how the general<br />

theorem reduced to a simpler one in particular cases. For example, when the<br />

‘punctum correspondens’ of the ocular K <strong>and</strong> the focus of the objective G<br />

coincide, it reduces to (HB : HC)·(EC : AK). Likewise, the configurations used<br />

in practice were only a special case that Huygens discussed as he went along.<br />

If a concave ocular <strong>and</strong> a convex objective are positioned in such a way that<br />

BG = BH, where the ocular is between the objective <strong>and</strong> its focus, the<br />

magnification of a distant object is AG : BH. The same applies to two convex<br />

lenses that are positioned with their foci coinciding in between. In other<br />

words, the magnification is equal to the quotient of the focal distances of<br />

both lenses. In this roundabout way, Huygens proved what had been, <strong>and</strong><br />

33 OC13, 174-179.<br />

34 OC13, 186-197.

22 CHAPTER 2<br />

continued to be, assumed for quite some time, as Molyneux was to remark in<br />

1690. 35<br />

Figure 11 Four of the cases discussed (additional lettering<br />

added).<br />

With the magnifying properties of lens-systems thus established in a most<br />

general way in parts one <strong>and</strong> two of Tractatus, Huygens’ subsequent account<br />

of actual telescopes came down to a rather straightforward application to a<br />

few specific cases. The state in which he left the third part of Tractatus in<br />

1653 is hard to determine. It probably consisted of only two or three<br />

theorems. Huygens did not discuss optimal configurations of lenses in<br />

telescopes systematically, but only explained under what conditions ocular<br />

<strong>and</strong> objective produced sharp images. The solution was simple, as he stated<br />

in the first proposition. In order to see a sharp image, the rays from the<br />

35 “This is the great Proposition asserted by most Dioptrick Writers, but hitherto proved by none (for as much<br />

as I know) …” Molyneux, Dioptrica nova, 161.

1653 - TRACTATUS 23<br />

object should leave the ocular parallel to the axis. 36 The foci of the lenses<br />

should therefore coincide. For myopic people <strong>and</strong> those using a telescope to<br />

project images things are different. In these cases the rays should be brought<br />

to focus after they have passed the ocular <strong>and</strong> the foci of the lenses should<br />

not coincide. In the second proposition, Huygens discussed the<br />

configuration of two lenses required to project images <strong>and</strong> determined their<br />

magnification. 37<br />

Huygens aimed at providing a general <strong>and</strong> exact theory of the properties<br />

of lenses <strong>and</strong> their configurations. The generality of Huygens’ theory reached<br />

its high-point in a theorem that is inserted in part two of Tractatus as the sixth<br />

proposition. It may be of a later date, as the manuscript is on a different kind<br />

of paper <strong>and</strong> written with a different pen than the rest of this part. 38<br />

Nevertheless, the theorem states that the magnification of an arbitrary<br />

system of lenses remains the same when eye <strong>and</strong> object switch place. 39 This<br />

theorem, so Huygens concluded his demonstration, would be useful in<br />

determining the magnification <strong>and</strong> distinctness of images.<br />

Figure 12 Analysis of Keplerian telescope with erector lens. See also Figure 13.<br />

Huygens applied the theorem in the third <strong>and</strong> fourth proposition included by<br />

the editors of Oeuvres Complètes in book three. The third proposition is<br />

certainly of a later date, as it analyses the eyepiece Huygens invented in<br />

1662. 40 The fourth proposition discusses a configuration of three convex<br />

lenses proposed by Kepler in 1611 (Figure 12). 41 A telescope of two convex<br />

lenses ordinarily produces a reversed image, but a third lens inserted between<br />

the ocular <strong>and</strong> the objective may re-erect the image.<br />

Huygens explained that an upright <strong>and</strong> sharp image is attained as follows<br />

(Figure 13). 42 AC is the focal distance of the objective lens YAB, <strong>and</strong> HF the<br />

focal distance of the ocular QHR. The third lens DET is identical with the<br />

ocular with a focal distance EL = HF. It is placed so that EC = 2EL <strong>and</strong> EH =<br />

3EL. In this case, point C on the axis is the ‘punctum correspondens’ for rays<br />

through focus F of the ocular. Therefore a ray from S at a large distance is<br />

refracted by the lenses in such a way that it leaves the ocular parallel to the<br />

axis towards the eye PN. In order to determine the magnification by the<br />

36 OC13, 244-247.<br />

37 OC13, 246-253.<br />

38 Hug29, 151-167.<br />

39 OC13, 198-199.<br />

40 OC13, 252n1. See below, section 3.1.2.<br />

41 Dating this theorem is difficult. It may have been written in 1653, as the configuration was well-known.<br />

Yet, Huygens also discussed the enlarged field of such a configuration, which may imply that it is of a<br />

later date. See note 20 on page 16 above.<br />

42 OC13, 258-261.

24 CHAPTER 2<br />

system, Huygens applied<br />

proposition six of book<br />

two. The eye is imagined<br />

at S <strong>and</strong> the object at PN.<br />

In this way the<br />

magnification is<br />

determined by the Figure 13 Diagram for the analysis in Figure 12.<br />

proportion YB : PN. It easily follows that this proportion is equal to AC : EL,<br />

the proportion of the focal distances of the objective <strong>and</strong> the ocular.<br />

Conclusion<br />

In Tractatus, Huygens addressed a specific question: how can the working of<br />

the telescope be understood mathematically? Regarding thin glass lenses his<br />

answers, as we shall see in the next section, were not that new. Yet, he had<br />

arrived at these answers by way of a rigorous mathematical analysis of the<br />

properties of lenses. With the sine law, Huygens derived general <strong>and</strong> exact<br />

theorems regarding the focal distances of thick lenses for both parallel <strong>and</strong><br />

non-parallel rays, irrespective of the material lenses are made of. On the basis<br />

of this exact theory, he showed that these theorems reduce to the familiar,<br />

simpler ones when the thickness of the lens is ignored <strong>and</strong> a specific index of<br />

refraction is chosen. In the same way, he first established a general theorem<br />

regarding the magnification by a lens-system <strong>and</strong> then showed that, in the<br />

cases of actual telescopes, it reduced to the simple <strong>and</strong> familiar one. If the<br />

elaboration of the theory of Tractatus was markedly mathematical, its<br />

rationale was the telescope. Its goal was a ‘theory of the telescope’: an<br />

account of the working of the telescope on the basis of dioptrical theory. In<br />

this sense, the theory of the first two books was almost too elaborate. All in<br />

all, in his Tractatus, Huygens gave a rigorous answer to the question how the<br />

working of the telescope can be understood mathematically.<br />

Huygens was the first one to elaborate a theory of the teleoscope by<br />

means of the exact law of refraction. He knew that his treatise would fill gaps<br />

left by others, in particular Descartes, so we would expect him to publish it<br />

soon. However, as contrasted to other mathematical treatises he published in<br />

this period, he did not press ahead with Tractatus. He inquired with<br />

publishers <strong>and</strong> Van Schooten even proposed to append Huygens’ treatise to<br />

a Latin edition of Descartes’ Discours de la Methode, La Dioptrique <strong>and</strong> Les<br />

Météores, but nothing came of it. 43 Despite repeated announcements between<br />

1655 <strong>and</strong> 1665 that he was publishing Tractatus, Huygens never did. 44<br />

2.2 Dioptrics <strong>and</strong> the telescope<br />

The orientation on the telescope is essential to Tractatus. If Huygens was the<br />

first to apply the sine law to questions regarding the telescope, what had<br />

other students of dioptrics been doing? In this section, I sketch the<br />

43<br />

OC1, 280; 301-303; 321-322. Huygens did not pin much faith in Van Schooten’s proposal.<br />

44<br />

I will say a bit more about his publishing pattern on page 174.

1653 - TRACTATUS 25<br />

development of seventeenth-century dioptrics, with a particular emphasis on<br />

the way questions regarding the telescope were addressed.<br />

The telescope was made public when in September 1608 a spectacle<br />

maker from Middelburg, Hans Lipperhey, came to The Hague to request a<br />

patent for a “… certain device by means of which all things at a very great<br />

distance can be seen as if they were nearby, …” 45 It was a configuration of a<br />

convex <strong>and</strong> a concave lens fitted appropriately in a tube <strong>and</strong> turned out to<br />

magnify things seen through it. The patent was denied, as within a couple of<br />

week two other claimants turned up. It is doubtful whether Lipperhey had<br />

made the invention himself. He may have learned it from his neighbour<br />

Sacharias Janssen, who in his turn seems to have learned the secret of the<br />

device from an itinerant Italian. 46 The history of the invention of the<br />

telescope is an intricate one, in which Jacob Metius of Alkmaar was the first<br />

to be publicly named its true inventor by Descartes. The first doubts were<br />

raised in the 1650s through the publication of Pierre Borel. Huygens himself<br />

was one of the first to perform some archival research on the matter,<br />

claiming that the credit should go to either Lipperhey or Janssen. 47 The news<br />

of the device spread quickly through Europe <strong>and</strong> by the summer of 1609<br />

simple telescopes were commonly for sale in the major cities of Europe. 48<br />

The news also reached the ears of scholars, who realized the device could<br />

be of use in astronomical observation. Most successful among them was<br />

Galileo in Venice, whose interest in the telescope was aroused in the spring<br />

of 1609. He figured out how to make one <strong>and</strong> how to improve it. Among the<br />

earliest telescopists, Galileo was the only one who not only knew how the<br />

telescope could be improved, but also had the means to do so. In August, he<br />

had made a telescope that magnified nine times, as opposed to the ordinary<br />

three-powered spyglasses. A couple of months later he had made telescopes<br />

that were even more powerful. 49 In this way, Galileo turned the spyglass into<br />

a powerful instrument of astronomical observation. 50 He observed the<br />

heavens <strong>and</strong> saw spectacular things: mountains on the Moon, satellites<br />

around Jupiter, <strong>and</strong> more. In March 1610, he published his observations in<br />

Sidereus nuncius. Galileo also sent a copy to the Prague court with a specific<br />

request for a comment by Kepler. 51<br />

In May, Kepler published his comment in Dissertatio cum nuncio sidereo. He<br />

primarily responded to Galileo’s observations, but he also said a few things<br />

about the instrument. In Sidereus nuncius, Galileo had explained its<br />

construction <strong>and</strong> use, but he had left out any mathematical account. 52 In<br />

45<br />

Van Helden, Invention, 35-36; Galileo, Sidereus nuncius, 3-4 (Van Helden’s introduction).<br />

46<br />

De Waard, Uitvinding, 105-225; Van Helden, Invention, 20-25.<br />

47<br />

OC13, 436-437.<br />

48<br />

Van Helden, Invention, 21, 36.<br />

49<br />

Van Helden, Invention, 26; Galileo, Sidereus nuncius, 6, 9 (Van Helden’s Introduction).<br />

50<br />

Van Helden, “Galileo <strong>and</strong> the telescope”, 153-157.<br />

51<br />

Galileo, Sidereus nuncius, 94 (Van Helden’s Conclusion).<br />

52<br />

Galileo, Sidereus nuncius, 37-39.

26 CHAPTER 2<br />

reply, Kepler briefly explained how lenses refract rays of light so that they<br />

can produced magnified images. 53 The explanation in Dissertatio was only a<br />

sketch, but the message was clear. The telescope was a remarkable invention,<br />

but its working needed mathematical clarification. A theory of the telescope<br />

was called for. Within a few months, Kepler developed one. In September<br />

1610, he finished the manuscript of Dioptrice, published the next year.<br />

“Some have disputed over the priority of its invention, others rather applied themselves<br />

to the perfection of the instrument, as there chance mainly counted, here reason<br />

dominated. But Galileo scored the greatest triumph by exploring its use to disclose<br />

secrets, because zeal procured him with the design <strong>and</strong> fortune has not withheld him<br />

the success. I, driven by an honest emulation, have shown the mathematicians a new<br />

field to expose their acuteness, in which the causes <strong>and</strong> principles are retraced to the<br />

laws of geometry, the effects of which are so awaited with much impatience <strong>and</strong> are of<br />

such pleasing diversity.” 54<br />

The goal of Dioptrice was to provide a mathematical account of the working<br />

of the telescope. In Kepler’s view, the working of any instrument used in<br />

astronomy should be understood precisely. A decade earlier, he had<br />

approached the puzzling properties of the pinhole images used in the<br />

observation of solar eclipses. His answer had been a new theory of image<br />

formation, which he had published in Paralipomena (1604). In Dioptrice, Kepler<br />

applied this theory to lenses in order to determine the dioptrical properties<br />

of the telescope. Dioptrice had one substantial shortcoming: Kepler knew that<br />

he did not know the exact law of refraction. He used an approximate rule<br />

instead.<br />

2.2.1 KEPLER AND THE MATHEMATICS OF LENSES<br />

Kepler’s concerns in Paralipomena were induced by a problem of astronomical<br />

observation. In 1598, Tycho Brahe had reported an anomalous observation<br />

of the apparent size of the moon during a solar eclipse. 55 Brahe used a<br />

pinhole to project the image of the eclipsed sun. When he measured the<br />

diameter of the projection he realized that “the moon during a solar eclipse<br />

does not appear to be the same size as it appears at other times during full<br />

moons when it is equally far away”. 56 He tried to produce consistent values<br />

by applying some ad hoc corrections to his measurements. 57 Kepler took a<br />

different approach, analyzing mathematically the way the image was<br />

produced. He had known the anomaly of pinhole images for some time<br />

53<br />

Kepler, Conversation, [19-21].<br />

54<br />

Kepler, Dioptrice, dedication (KGW4, 331). “… circaque eam alij de palma primae inventionis certarent,<br />

alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio<br />

dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum<br />

triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego<br />

doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est<br />

causarum lege geometrica demonstr<strong>and</strong>arum, quibus tam exoptati, tam jucunda varietate multiplices<br />

effectus inniterentur.”<br />

55<br />

Straker, “Kepler’s theory of pinhole images”, 276-278.<br />

56<br />

Cited <strong>and</strong> translated in: Straker, “Kepler’s theory of pinhole images”, 278.<br />

57<br />

Straker, “Kepler’s theory of pinhole images”, 275-276; 280-282.

1653 - TRACTATUS 27<br />

when in 1600 he set himself to see whether an ‘optical cause’ might account<br />

for it. The solution to the apparent anomaly of pinhole projections of solar<br />

eclipses would be the copestone of Paralipomena. I will only discuss Kepler’s<br />

theory of image formation <strong>and</strong> its application to the eye.<br />

The optical theory available around 1600 was the medieval tradition of<br />

perspectiva <strong>and</strong> it did not provide Kepler with an answer for the anomaly of<br />

pinhole observations. Perspectiva built on the great synthetic work from the<br />

eleventh century of the Arab mathematician Alhacen, when it was adopted<br />

by a line of thirteenth-century Christian mathematicians, Bacon, Witelo <strong>and</strong><br />

Pecham. They elaborated a mathematical theory of optics, in addition to the<br />

natural philosophical <strong>and</strong> medical theories, in which vision was analyzed in<br />

terms of the behavior of light rays. 58 The designation ‘perspectiva’ derives<br />

from the common title for their works <strong>and</strong> it constituted the canon of<br />

mathematical optics well into the seventeenth century. In the sixteenth<br />

century perspectiva texts had been published, with the 1672 edition by<br />

Friedrich Risner of Alhacen’s Optica <strong>and</strong> Witelo’s Perspectiva as the most<br />

important. 59<br />

The problem of pinhole images was well-known in perspectiva. It was<br />

known since Antiquity that the image of the sun, projected by a square<br />

aperture, can still be round. This seemed to contradict the basic principle of<br />

optics: the rectilinearity of light rays. The solutions given by perspectivist<br />

writers did not satisfy Kepler. Each had in the end sacrificed the principle of<br />

rectilinearity – the foundation of geometrical<br />

optics. 60 Kepler had to resolve the problem by<br />

himself. His solution consisted of a new theory<br />

of the way rays form images of objects. This<br />

theory, in its turn, would be the foundation of<br />

his dioptrics as well as of seventeenth-century<br />

geometrical optics in general.<br />

Kepler approached the problem anew <strong>and</strong><br />

did so by uncompromisingly applying the<br />

principle of rectilinearity. In Paralipomena, he<br />

describes how he replaced a ray of light by a<br />

thread. He took a book, attached a thread to<br />

one of its corner <strong>and</strong> guided it along the edges<br />

of a many-cornered aperture, thus tracing out<br />

the figure of the aperture. Repeating this for<br />

the other corners of the book, <strong>and</strong> many more<br />

points, he ended up with a multitude of<br />

overlapping figures that formed an image of<br />

Figure 14 Kepler’s solution to<br />

the pinhole problem<br />

58 Further discussed in section 4.1.1.<br />

59 Dupré points out Risner’s programmatic discussion of the science of optics in the preface to the edition<br />

which constitute an important, yet still little studied, agenda for seventeenth-century optics. Dupré,<br />

Galileo, the Telescope, 54.<br />

60 See Lindberg, “Laying the foundations”, 14-29.

28 CHAPTER 2<br />

the book. In the same way, he argued, all the points of the sun project<br />

overlapping images of the aperture (Figure 14). The resulting image has the<br />

shape of the sun, albeit with a blurred edge. In the projection of an eclipse,<br />

the image of the shadow of the moon is partially overlapped by the image of<br />

the sun. Consequently, the diameter of the moon seems too small. In chapter<br />

two of Paralipomena, Kepler had solved the apparent anomaly of pinhole<br />

observations in principle, building on the previous chapter, he elaborated the<br />

exact solution in the eleventh <strong>and</strong> final chapter.<br />

Image formation<br />

Kepler came to the conclusion that there were more problems in<br />

perspectiva, in particular its core, the theory of vision. In chapter five of<br />

Paralipomena, he elaborated a new theory of vision on the basis of his newly<br />

gained underst<strong>and</strong>ing of image formation. In its fourth section, Kepler listed<br />

the defects of existing theories of vision, the most important being a wrong<br />

underst<strong>and</strong>ing of the anatomy of the eye <strong>and</strong> of the mathematics of image<br />

formation. Perspectivist theories considered the lens the sensitive organ of<br />

the eye, whereas recent anatomical investigations had demonstrated,<br />

convincingly according to Kepler, that the retina receives images from<br />

objects. He himself had shown the defects of the perspectivist underst<strong>and</strong>ing<br />

of image formation, calling Witelo by name, <strong>and</strong> he now went on to<br />

reconsider the optics of the eye.<br />

In perspectivist theory, each point of an object emits rays of light in each<br />

direction. This, however, raises the problem how a sharp image can be<br />

perceived, that is: how a one-to-one relationship between a point of the<br />

object <strong>and</strong> a point of the image in the eye is established. According to<br />

Alhacen there can be only one point in the eye where a ray from a point of<br />

the object can be perceived. He stated that this must be the one entering the<br />

eye perpendicularly (<strong>and</strong> thus perpendicular to the lens). He explained that<br />

the other rays are refracted by the eye, therefore weakened, <strong>and</strong> thus do not<br />

partake in the formation of the image. 61 In medieval optics, images were<br />

therefore taken to be produced by single rays from each point of the object.<br />

Kepler saw no reason to differentiate between weak <strong>and</strong> strong rays. He did<br />

not see, for that matter, why refraction would weaken a ray. In his view, all<br />

rays emitted by a point should somehow partake in the formation of an<br />

image. In the case of pinholes this resulted in a fuzzy image, but what about<br />

the sharp images by which we generally see the world?<br />

Kepler’s answer was that the cone of rays coming from one point is<br />

somehow brought to focus on the retina. Following certain recent<br />

anatomical observations he considered the retina as the sensitive organ of<br />

the eye, in contrast to perspectivist theory that had assigned the power of<br />

visual perception to the crystalline humor. According to Kepler, the various<br />

humors of the eye can be regarded as one refracting sphere. In the fifth<br />

61 Alhacen, Optics I, 68 (book 1, section 17) <strong>and</strong> 77 (book 1, section 46).

1653 - TRACTATUS 29<br />

chapter of Paralipomena, Kepler explained how images are formed on the<br />

retina. In order to account for spherical aberration, he argued that the pupil<br />

as well a the slightly a-spherical shape of the posterior side of the humors<br />

diminish the severest aberrations. Kepler’s analysis was based on his study of<br />

refraction in the fourth chapter of Paralipomena. In this chapter, he had tried<br />

unsuccessfully to find an exact law of refraction, but his underst<strong>and</strong>ing of<br />

refraction at plane surface sufficed for discussing the focusing properties of<br />

spheres at least qualitatively. 62<br />

With this Kepler completed his theory of image formation. It had<br />

originated in the solution of an anomalous astronomical observation <strong>and</strong> its<br />

ultimate rationale was astronomical observation. With his definition of optics<br />

<strong>and</strong> its indispensability to cosmology, Kepler fits in a Ramist trend in the<br />

sixteenth century that Dupré refers to with ‘the art of seeing well’ <strong>and</strong> to<br />

which Risner also belongs. 63 The full title of Paralipomena starts with Ad<br />

Vitellionem paralipomena, quibus astronomiae pars optica traditur, …. In his preface,<br />

Kepler proclaimed eclipses to be the most noble <strong>and</strong> ancient part of<br />

astronomy: “… these darknesses are the astronomers’ eyes, the defects are a<br />

cornucopia of theory, these blemishes illuminate the minds of mortals with the<br />

most precious pictures.” 64 The eye being the fundamental instrument of<br />

observation, to Kepler a reliable theory of visual perception was<br />

indispensable for astronomers. His perspectivist forebears had not treated<br />

the matter satisfactorily <strong>and</strong> thus he had provided the necessary additions to<br />

Witelo. Revolutionary additions, to be sure. The eye perceives dots rather<br />

than things <strong>and</strong> in the analysis of vision “… we should not look to entire<br />

objects, but to individual points of objects, …” 65 Kepler had made it clear<br />

that all rays from an object point partake in the formation of images, whose<br />

sharpness is not evident beforeh<strong>and</strong>. Image formation was no longer a<br />

matter of tracing individual rays from object to image. The task of the<br />

optician now became to determine exactly how a bundle of rays is brought to<br />

focus again after it is emitted by a point of an object.<br />

<strong>Lenses</strong><br />

Kepler approached the newly invented telescope in the same manner as the<br />

pinhole <strong>and</strong> the eye. The working of the telescope should be properly<br />

understood if it were to be used in astronomical observation. For Kepler,<br />

this meant that a mathematical theory was required, a mathematical theory of<br />

the telescope so to say. He had already treated lenses briefly in the final<br />

proposition of chapter five of Paralipomena. At that moment spectacle glasses<br />

were new topic in optical literature. Kepler expressed his amazement that no<br />

62<br />

Kepler’s efforts to find a law of refraction are discussed below, in section 4.1.2.<br />

63<br />

Dupré, Galileo, the telescope, 31.<br />

64<br />

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<br />

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.”<br />

Translation Donahue, Optics, 16.<br />

65<br />

Kepler, Paralipomena, 201 (KGW2, 181). “Itaque non oportet nos ad res totas respicere, sed ad rerum<br />

singular puncta, …” Translation Donahue, Optics, 217.

30 CHAPTER 2<br />

mathematical account of such an important <strong>and</strong> widespread device existed.<br />

We can underst<strong>and</strong> his surprise, for spectacles had already been invented<br />

around 1300. 66 A brief account by Francesco Maurolyco, that dated back to<br />

around 1521, was not to be published before 1611, in Diaphaneon seu<br />

transparentium libellus. Kepler would not have found much in it to his liking,<br />

for it was a qualitative theory based on a somewhat confusing variant of the<br />

perspectivist theory of vision <strong>and</strong> refraction. 67 Kepler knew that Della Porta<br />

had written a study of refraction, but he had not been able to lay h<strong>and</strong>s on<br />

De refractione (1593). He dismissed what Della Porta had written in Magia<br />

naturalis, namely that spectacles correct vision because they magnify images.<br />

Kepler elaborated his own account of lenses, dedicating it in Paralipomena to<br />

his patron Ludwig von Dietrichstein, whom he said had kept him busy for<br />

three years with the question of the secret of spectacles. 68 Kepler explained<br />

the beneficial effects of spectacles as follows. Myopic <strong>and</strong> presbyotic vision<br />

occurs when rays are not brought to focus on the retina but in front of it or<br />

beyond. He gave a short, qualitative discussion of the effect of lenses on a<br />

bundle of parallel rays coming from a distant point. Convex <strong>and</strong> concave<br />

lenses – for myopics <strong>and</strong> presbyotics respectively – move the focus of rays to<br />

the retina. Some magnification may occur, but this is not the reason why<br />

spectacle lenses correct vision.<br />

With the introduction of the telescope in astronomy, the qualitative<br />

account of single lenses in Paralipomena did not suffice any more. In Dioptrice,<br />

Kepler extended his theory of image formation to a quantitative analysis of<br />

the properties of lenses <strong>and</strong> their configurations. 69 As a matter of fact, he was<br />

the one to coin the term ‘dioptrics’. 70 Compared to Huygens’ Tractatus,<br />

Kepler’s dioptrical theory was of more limited scope. His goal was to explain<br />

the formation of images by a telescope. He therefore restricted his theory to<br />

a few types of lenses <strong>and</strong> mainly confined himself to object points at infinite<br />

distance when incident rays are parallel. The basic concept was the focus of a<br />

lens, the point where parallel rays intersect after refraction. Kepler could not<br />

determine the focal distance with the exactness we have seen with Huygens.<br />

He could not, for example, determine the exact route of a ray through the<br />

refractions at both surfaces of a lens. The main obstacle in the way of a more<br />

extensive treatment was the fact that Kepler did not know the exact law of<br />

refraction. In Dioptrice, he used an approximation that was valid only for<br />

angles of incidence below 30º, <strong>and</strong> that, even so, applied solely to glass.<br />

According to this rule the angle of deviation is one third of the angle of<br />

66<br />

Rosen, “The invention of eyeglasses”, 13-46.<br />

67<br />

Lindberg, “Optics in 16th century Italy”136-141. Maurolyco had preceded Kepler in his analysis of the<br />

pinhole image: Lindberg, “Optics in 16th century Italy”, 132-135; Lindberg, “Laying the foundations”.<br />

68<br />

Kepler, Paralipomena, 200-202 (KGW2, 181-183).<br />

69<br />

Malet, “Kepler <strong>and</strong> the telescope” offers a detailed discussion of Dioptrice, without however presenting it<br />

as a part of the ‘optical part of astronomy’.<br />

70<br />

Kepler, Dioptrice, dedication (KGW4, 331).

1653 - TRACTATUS 31<br />

incidence; the angle between the incident ray, produced beyond the<br />

refracting surface, <strong>and</strong> the refracted ray.<br />

Kepler began with a discussion<br />

of the focal distances of planoconvex<br />

lenses (Figure 15). A ray<br />

HG is incident on a convex surface<br />

with radius AC, the angle of<br />

incidence is GAC. As the angle of<br />

deviation is one third of this, HG<br />

will be refracted towards F, with AC : AF = 1 : 2. 71 The focal distance is<br />

therefore approximately three times the radius of the convex face.<br />

Analogously, he argued that the focal distance of a plano-convex lens, the<br />

plane face turned towards the incident rays, is approximately twice the radius<br />

of curvature. For other cases Kepler established only rough estimations. If<br />

convergent rays are incident on the plane side of a plano-convex lens, the<br />

refracted rays intersect the axis within the focal distance. Combining these<br />

three theorems, Kepler showed that the focal distance of a bi-convex lens is<br />

both smaller than three times the radius of the anterior side <strong>and</strong> twice the<br />

radius of the posterior side. In the case of an equi-convex lens, this comes<br />

down to a focal distance approximately equal to the radius of its sides. 72<br />

Kepler did not determine the focal distance of a concave lens, he only<br />

showed that rays diverge after refraction. 73<br />

On this basis, the properties of images formed by lenses are easily found.<br />

The image DBF of an extended object CAE through a bi-convex lens GH is<br />

formed at focal distance (Figure 16). The picture is inversed as the rays from<br />

C are refracted towards D, etcetera. As the focal distance is roughly the radius<br />

of any side, the magnitudes of object <strong>and</strong> image will be in a proportion equal<br />

to their respective distances to the lens. 74 In Dioptrice, Kepler briefly reiterated<br />

his theory of vision. On the one h<strong>and</strong>, so he said in the dedication, he did so<br />

for the sake of completeness, on the other h<strong>and</strong> because some readers had<br />

trouble underst<strong>and</strong>ing his account in Paralipomena. 75 He explained that a<br />

perfectly focusing surface was not spherical, but should be hyperbolic, like<br />

the crystalline humor of the eye was. 76 On the basis of his theory of the<br />

retinal image, he explained the effect of a lens placed before the eye once<br />

more. Depending upon the position of the eye with respect to the focal<br />

distance, the object will be perceived sharply. 77 When the eye is placed not<br />

too far from the focus, a magnified image will be perceived.<br />

71 Kepler, Dioptrice, 11 (KGW4, 363).<br />

72 Kepler, Dioptrice, 12-15 (KGW4, 363-367).<br />

73 Kepler, Dioptrice, 45-49 (KGW4, 388-393).<br />

74 Kepler, Dioptrice, 16-18 (KGW4, 367-369).<br />

75 Kepler, Dioptrice, dedication (KGW4, 335).<br />

76 Kepler, Dioptrice, 21-24 (KGW4, 371-372).<br />

77 Kepler, Dioptrice, 35-42 (KGW4, 381-387).<br />

Figure 15 Focal distance of a plano-convex lens

32 CHAPTER 2<br />

Kepler proceeded to discuss the combination of<br />

two convex lenses. He explained how these should<br />

be configured in order to perceive a sharp,<br />

magnified image. 78 This is achieved when the foci of<br />

both lenses coincide. It is remarkable that that<br />

Kepler discussed the configuration of two convex<br />

lenses, because in 1611 only the combination of a<br />

convex objective <strong>and</strong> a concave ocular was known<br />

to produce a telescopic effect. Kepler probably<br />

arrived at this alternative configuration by<br />

theoretical considerations. 79 He never manufactured<br />

this kind of telescope himself. The configuration has<br />

come to be known as a Keplerian or Astronomical<br />

telescope, as opposed to the Dutch or Galilean<br />

telescope with a concave ocular. Much later it<br />

became clear that the Keplerian type has the<br />

advantage of a larger field of view, but Kepler<br />

himself did not know this. He did realize that this<br />

configuration had a drawback, it produced inverted<br />

images. This could be corrected, he said, by inserting<br />

a third lens at an appropriate place between ocular<br />

<strong>and</strong> objective lens.<br />

Kepler then turned to an account of concave<br />

lenses <strong>and</strong> finally to a discussion of so-called Dutch<br />

telescopes. He explained the configuration of a<br />

Figure 16 Image formation<br />

by a lens<br />

convex objective <strong>and</strong> a concave ocular only in broad lines. As he did not<br />

speak of the focus of a concave lens, he could only roughly point out where<br />

the ocular should be placed with respect to the focus of the objective. His<br />

discussion of the configurations <strong>and</strong> the resulting properties of the images<br />

remained mainly qualitative. Kepler offered a wealth of practical guidelines as<br />

to the configurations of lenses <strong>and</strong> the way the best effects are achieved.<br />

Dioptrice does not consist of rigorously demonstrated theorems. Without an<br />

exact law of refraction, a quantitative <strong>and</strong> exact theory could hardly be<br />

attained. This was not necessarily Kepler’s intention. Rather, he intended to<br />

explain the working of the telescope mathematically. He did so by analyzing,<br />

on the basis of his theory of image formation, how it forms magnified<br />

images.<br />

All this may lead us to conclude that Huygens’ Tractatus can be seen as an<br />

up-to-date answer to the question Kepler had originally addressed in<br />

Dioptrice; updated in the sense that the analysis of lenses was based on the<br />

sine law. It established the dioptrical properties of spherical lenses <strong>and</strong><br />

78 Kepler, Dioptrice, 42-43 (KGW4, 387-388).<br />

79 A possible source of inspiration may have come from the analogous configuration of the eye <strong>and</strong> a<br />

convex spectacle glass, as the eye acts as a convex lens does. See also Malet, “Kepler <strong>and</strong> the telescope”,<br />

119-120.

1653 - TRACTATUS 33<br />

focused on problems pertaining to their configurations in actual telescopes.<br />

Like Kepler, Huygens intended to found the dioptrical properties on a sound<br />

mathematical basis. Whether a continuation of Dioptrice was his actual goal,<br />

can only be surmised as he did not explicitly refer to it in such a<br />

programmatic sense. Huygens did know Dioptrice, it had been on the reading<br />

list of his mathematics tutor Stampioen <strong>and</strong> much later he commended it to<br />

his brother Constantijn as the best introduction to dioptrics. 80 Huygens did<br />

not have much to offer that was not already known. Tractatus covered more<br />

types of lenses but the eventual results regarding the focal distances of lenses<br />

<strong>and</strong> the magnifying properties did not differ much from Dioptrice. The crucial<br />

difference is that Huygens founded his results on a general <strong>and</strong> exact theory<br />

of focal distances. It rigorously proved Kepler’s results. He had the exact law<br />

of refraction at his disposal <strong>and</strong> thus could be exact where Kepler necessarily<br />

had to leave his readers with approximate answers.<br />

Perspectiva <strong>and</strong> the telescope<br />

At the same time when Kepler wrote Dioptrice, two other scholars devised an<br />

account of the telescope. Della Porta’s ‘De telescopio’ remained<br />

unpublished, De Domini’s De Radiis Visus et Lucis was published in 1611.<br />

Both were based on perspectivist theory of image formation. Before I go on<br />

to discuss the impact of the sine law on dioptrics, I briefly discuss these in<br />

order to make it clear why that perspectivist theory was intrinsically<br />

inadequate to account fully for the effect of lenses.<br />

Shortly before his death, Della Porta extended his theory of lenses of De<br />

Refractione to telescopes in a manuscript ‘De telescopio’. 81 It reveals the<br />

problems lenses posed for perspectivist theory of image formation. In order<br />

to determine the place where an object is seen, perspectiva used the cathetus<br />

rule. The cathetus is the line through the object point, perpendicular to the<br />

reflecting or refracting surface. The cathetus rule states that the image is the<br />

intersection of the ray entering the eye <strong>and</strong> the cathetus. Modern Keplerian<br />

theory shows that, although valid in many cases, this rule turns out to break<br />

down for curved surfaces in particular. To account for images of lenses<br />

another problem turns up. As a lens refracts a ray twice, this seems to imply<br />

that the rule has to be applied twice also. Della Porta avoided this problem<br />

by considering only one cathetus.<br />

Della Porta considered a lens in terms of refracting spheres, as he had<br />

done in De Refractione (Figure 17). The dotted lines indicate such spheres <strong>and</strong><br />

the lens dcgf is formed by their overlap. 82 The object ab is perceived as<br />

follows: a ray from point a is refracted along cd to the eye. Della Porta drew<br />

the cathetus ka of the lower surface of the lens, which also is its radius.<br />

When produced, the ray entering the eye intersects the cathetus in point h,<br />

80<br />

OC1, 6 (Stampioen’s list of recommended readings spans pages 5-10) <strong>and</strong> OC6, 215.<br />

81<br />

Della Porta’s account of refraction by spheres <strong>and</strong> lenses in De refractione is discussed in Lindberg,<br />

“Optics in 16th century Italy”, 143-146.<br />

82<br />

Della Porta, De Telescopio, 113-114.

34 CHAPTER 2<br />

where point a is seen. In the same manner point i is constructed <strong>and</strong> hi is the<br />

object as perceived through the lens. The question is why only the ray acd<br />

emanating from point a is singled out. Della Porta seemed to assume that<br />

this is the one that enters the eye perpendicularly. Yet, in the case of a distant<br />

object, he no longer chooses rays parallel to the axis of the system, but the<br />

crossing rays ad <strong>and</strong> bg (Figure 18). He probably did so to account for the<br />

reversing of the image, but he lacked a theoretical justification. Della Porta’s<br />

account of concave lenses was even more troublesome, as he ignored the<br />

implication of his reasoning that the eye cannot perceive the whole object at<br />

once. Moreover, he persistently has rays refracted from the perpendicular at<br />

the first surface (for example de in Figure 19). 83 ‘De telescopio’ culminated in<br />

an account of a Galilean telescope (Figure 19). Della Porta traced the path of<br />

a ray emanation from point a of the object. He then chose the cathetus with<br />

respect to the upper surface of the concave lens <strong>and</strong> argued that qr is the<br />

image perceived.<br />

Figure 17 Image of a near object<br />

Figure 18 Image of<br />

distant object<br />

All in all, from Kepler’s perspective Della Porta’s theory of lenses was<br />

fraught with difficulties <strong>and</strong> mathematically it was riddled with ambiguities.<br />

Part of these arose from his sloppiness <strong>and</strong> lack of underst<strong>and</strong>ing of certain<br />

problems. Part of the problem lies also with the perspectivist foundation of<br />

his account. How, for example, should the cathetus rule be applied to two or<br />

more refractions? More important, perspectivist theory offers no means of<br />

differentiating between sharp <strong>and</strong> fuzzy images, quite a relevant issue with<br />

respect to the telescope. 84 Della Porta made no attempt to deal with it.<br />

Whether he chose ignore it or wass unaware of it is unclear. He was quite<br />

content with what he had written. As the inventor of the telescope – so he<br />

83 Della Porta, De telescopio, 141-142.<br />

84 Compare Lindberg, “Optics in 16 th century Italy”, 146-147.<br />

Figure 19 Image by a<br />

telescope

1653 - TRACTATUS 35<br />

fancied – he regarded himself as the only authority in these matters. 85 Shortly<br />

after he wrote ‘De telescopio’ he died, <strong>and</strong> the text remained unknown until<br />

1940.<br />

De Radiis Visus et Lucis of De Dominis is well-known for its discussion of<br />

the rainbow, but it also contains an account of lenses <strong>and</strong> the telescope. Like<br />

Della Porta, De Dominis maintained perspectivist theory. His theory did not<br />

go beyond a brief, qualitative theory of the refraction of visual rays by lenses.<br />

In this way it avoided the problems revealed by Della Porta’s theory. It does<br />

not seem to have counted as a serious alternative to Dioptrice. Unlike Kepler,<br />

De Dominis was rarely referred to in matters dioptrical. In the widely read<br />

Rosa Ursina (1630), Scheiner adopted Kepler’s theory of image formation. He<br />

elaborately treated the construction <strong>and</strong> use of telescopes. Scheiner discussed<br />

the properties of lenses <strong>and</strong> their configurations, but he did not incorporate<br />

the quantitative part of Dioptrice – his account remained qualitative. Another<br />

authoritative book on geometrical optics at the time, Opticorum Libri Sex<br />

(1611) by the Antwerp mathematician Aguilón, did not discuss refraction or<br />

lenses at all.<br />

Dioptrice had been a reaction to Galileo’s neglect to explain the telescope<br />

dioptrically in Sidereus Nuncius. Although quite an able mathematician, Galileo<br />

never developed a theory of dioptrics. He applied himself to the<br />

improvement of the instrument by making better lenses <strong>and</strong> optimizing the<br />

quality of telescopic images. His friend Sagredo did take an interest in the<br />

dioptrics of lenses, but was not encouraged to pursue his study. Galileo<br />

wanted him to concentrate on matters of glass-making <strong>and</strong> lens-grinding. 86<br />

On Dioptrice Galileo kept silent altogether. 87 Apparently, this self-styled<br />

mathematical philosopher was not interested in the mathematical properties<br />

of the instrument that had brought him fame. He did, however, have a clear<br />

underst<strong>and</strong>ing of the working of lenses <strong>and</strong> telescopes. Dupré has recently<br />

argued that Galileo relied on a tradition of practical knowledge, of mirrors in<br />

particular, that had developed in the sixteenth century next to the<br />

mathematical str<strong>and</strong> on which my account focuses. 88<br />

2.2.2 THE USE OF THE SINE LAW<br />

The exact law of refraction Kepler had to make do without, was soon found.<br />

More than that, it had been within his reach. The English astronomer<br />

Thomas Harriot had discovered it in 1601. After the publication of<br />

Paralipomena, he <strong>and</strong> Kepler had corresponded on optical matters. However,<br />

the correspondence broke off before Harriot had revealed his discovery. 89<br />

Long before that, but unknown until the late twentieth century, the tenth-<br />

85<br />

Ronchi, “Refractione au Telescopio”, 56 <strong>and</strong> 34. “They know nothing of perspective.” <strong>and</strong> “... <strong>and</strong> it<br />

pleases me that the idea of the telescope in a tube has been mine; ...”<br />

86<br />

Pedersen, “Sagredo’s optical researches”, 144-148.<br />

87<br />

KGW4, “Nachbericht”, 476.<br />

88<br />

Dupré, Galileo, the Telescope, chapters 4 to 6 in particular.<br />

89<br />

Harriot is discussed in section 4.1.2.

36 CHAPTER 2<br />

century student of burning glasses Ibn Sahl had used a rule equivalent to the<br />

sine law. 90 Around 1620, the Leiden professor of mathematics Willebrord<br />

Snel was next <strong>and</strong> in the late 1620s Descartes closed the ranks of discoverers<br />

of the law of refraction. 91 He published it in La Dioptrique (1637), shortly after<br />

Pierre Hérigone had done so in the fifth volume of Cursus Mathematicus.<br />

Hérigone did not use it in his dioptrical account, which summarized Dioptrice.<br />

Harriot <strong>and</strong> Snel have left no trace of applying their find to lenses. Which<br />

leaves La Dioptrique for further inspection.<br />

Descartes <strong>and</strong> the ideal telescope<br />

La Dioptrique was the fruit of Descartes’ involvement in the activities of<br />

Parisian savants regarding (non-spherical) mirrors <strong>and</strong> lenses, which also<br />

places him in the sixteenth-century tradition of mirror-making. 92 Descartes,<br />

however, added his natural philosophical leanings <strong>and</strong> Kepler’s optical<br />

teachings. In collaboration with the mathematician Mydorge <strong>and</strong> the artisan<br />

Ferrier, he allegedly managed to produce a hyperbolic lens <strong>and</strong> in the course<br />

of events he discovered the law of refraction. La Dioptrique had much<br />

influence on seventeenth-century optics, especially through its second<br />

discourse where Descartes derived the sine law. 93 In the following discourses,<br />

Descartes first discussed the eye <strong>and</strong> vision – summarizing Kepler’s theory<br />

of the retinal image – <strong>and</strong> then went on to a consideration “Of the means of<br />

perfecting vision”. 94 This seventh discourse anticipated his discussion of<br />

telescopes. He laid stress on the way spectacles enhance vision, instead of<br />

correcting it. The telescope itself was introduced in a peculiar way. Descartes<br />

explained how an elongated lens may further enhance vision. He then<br />

replaced the solid middle part by air, thus arriving at a telescope consisting of<br />

two lenses. 95 The argument was clear, but the discussion of focal <strong>and</strong><br />

magnifying properties of lenses was entirely qualitative <strong>and</strong> the sine law<br />

played no role in it.<br />

In the eighth discourse of La Dioptrique, Descartes applied the sine law to<br />

lenses under the title: “Of the figures transparent bodies must have to divert<br />

the rays by refraction in all manners that serve vision”. 96 Its sole purpose was<br />

to show that lenses ought to have an elliptic or hyperbolic surface in order to<br />

bring rays to a perfect focus. Avoiding the subtleties of geometry he<br />

explained how these lines could be drawn by practical means <strong>and</strong><br />

demonstrated the relevant properties of the ellipse <strong>and</strong> hyperbola. As regards<br />

the focal distances of lenses thus obtained with respect to configuration <strong>and</strong><br />

90<br />

Rashed, “Pioneer”, 478-486.<br />

91<br />

For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat<br />

later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250.<br />

92<br />

Gaukroger, Descartes, 138-146. Dupré, Galileo, the Telescope, 53-54.<br />

93<br />

Discussed in section 4.1.3<br />

94<br />

Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.”<br />

95<br />

Descartes, AT6, 155-160.<br />

96<br />

Descartes, AT6, 165. “Des figures que doivent avoir les corps transparens pour detourner les rayons par<br />

refraction en toutes les façons qui servent a la veuë”

1653 - TRACTATUS 37<br />

magnification, the account remained qualitative. La Dioptrique was written,<br />

Descartes said in the opening discourse, for the benefit of craftsmen who<br />

would have to grind <strong>and</strong> apply his elliptic <strong>and</strong> hyperbolic lenses. Therefore<br />

the mathematical content was kept to a minimum. 97 Apparently this implied<br />

that Descartes need not elaborate a theory of the dioptrical properties of<br />

lenses.<br />

Descartes adopted the term Kepler had coined for the mathematical<br />

study of lenses. He had not, however, adopted the spirit of Kepler’s study.<br />

Dioptrice <strong>and</strong> La Dioptrique approached the telescope from opposite<br />

directions. Kepler had discussed actual telescopes <strong>and</strong> drudged on properties<br />

of lenses that did not fit mathematics so neatly. Descartes prescribed what<br />

the telescope should be according to mathematical theory. The telescope,<br />

having been invented <strong>and</strong> thus far cultivated by experience <strong>and</strong> fortune,<br />

could now reach a state of perfection by explaining its difficulties. 98 Huygens<br />

was harsh in his judgment of La Dioptrique. In 1693, he wrote:<br />

“Monsieur Descartes did not know what would be the effect of his hyperbolic<br />

telescopes, <strong>and</strong> assumed incomparably more about it than he should have. He did not<br />

underst<strong>and</strong> sufficiently the theory of dioptrics, as his poor build-up demonstration of<br />

the telescope reveals.” 99<br />

We can say that Descartes, according to Huygens, had failed to develop a<br />

theory of the telescope. He had ignored the questions that really mattered<br />

according to Huygens: an exact theory of the dioptrical properties of lenses<br />

<strong>and</strong> their configurations. La Dioptrique glanced over a telescope that existed<br />

only in the ideal world of mathematics.<br />

Unfortunately for Descartes, no one during the following decades<br />

succeeded in actually grinding the a-spherical lenses of his design. Of mere<br />

anecdotal interest is the irony with which Huygens’ tutor Stampioen had in<br />

1640 pointed out to Descartes’ yet unfulfilled promise of a perfect telescope:<br />

“… my servant Research will turn him a better spyglass without circles …<br />

But nevertheless, what this Mathematicien has promised to do for six years<br />

is still not satisfied.” 100 But Stampioen was in the middle of a terrible dispute<br />

with Descartes at that moment.<br />

After Descartes<br />

Hobbes, Descartes’ most ardent rival in matters of mechanistic philosophy,<br />

developed an alternative derivation of the sine law too. In the elaboration of<br />

his dioptrical theory he also discussed spherical lenses. The unpublished “A<br />

97<br />

Descartes, AT6, 82-83. Ribe, “Cartesian optics” offers an enlightening account of the artisanal roots of<br />

La Dioptrique.<br />

98<br />

Descartes, AT6, 82.<br />

99<br />

OC10, 402-403. “Mr. des Cartes n’a connu quel seroit l’effet de ses Lunettes hyperboliques, et en a<br />

presumè incomparablement plus qu’il ne devoit. n’entendant pas assez cette Theorie de la dioptrique, ce<br />

qui paroit par sa demonstration très mal bastie des Telescopes.”<br />

100<br />

Stampioen, Wis-konstigh ende reden-maetigh bewys, 58. “… mijn Knecht Ondersoeck sal hem eens een beter<br />

Verre-kijcker sonder cirkeltjes daer toe weten te drayen : … Maer niettemin ’t geen dese Mathematicien al<br />

over 6 Iaren belooft heeft te doen, blijft nog on-vol-daen.”

38 CHAPTER 2<br />

minute or First Draught of the Optiques” of 1646 (the most complete<br />

elaboration of his optics) included several chapters on lenses <strong>and</strong><br />

telescopes. 101 He did not, however, make the most of his knowledge of the<br />

sine law. The account consisted of qualitative theorems – without proof <strong>and</strong><br />

often dubious – which applied mostly to single rays refracted by lenses.<br />

Despite the presence of an exact law of refraction, Hobbes’ account (if<br />

published) would have been no match for Dioptrice.<br />

With the exact law of refraction established <strong>and</strong> published, the road<br />

might seem open for a follow-up of Dioptrice in the form of an exact theory<br />

of the dioptrical properties of spherical lenses. It was not to be, for various<br />

reasons. First of all the sine law became generally known <strong>and</strong> accepted only<br />

around 1660. 102 This delay may have been caused by a slow distribution of<br />

Descartes’ works – <strong>and</strong> this maybe partly because La Dioptrique was written<br />

in French – or the bad odor his ideas were in. As late as 1663, in Optica<br />

promota, Gregory showed that the ellipse <strong>and</strong> hyperbola are aplanatic without<br />

using the sine law. In 1647, Cavalieri extended the theory of Dioptrice to some<br />

more types of lenses, using Kepler’s original rule. As the title Exercitationes<br />

geometricae sex suggests, this was an exercise in mathematics not aimed at<br />

furthering the underst<strong>and</strong>ing of the telescope. In this regard, Cavalieri was<br />

not an exception.<br />

Further, <strong>and</strong> more importantly, mathematicians addressed questions<br />

raised in Kepler’s Paralipomena rather than in his Dioptrice, to wit abstract<br />

optical imagery pertaining to Kepler’s theory of image formation, <strong>and</strong> the<br />

‘anaclastic’ problem that had been put in a different light by that theory. The<br />

anaclastic problem, or ‘Alhacen’s problem’, is closely related to the<br />

determination of aplanatic surfaces: to find the point of reflection or<br />

refraction of a ray passing from a given point to another. 103 When all rays are<br />

considered, as is relevant in Kepler’s theory of image formation, to find these<br />

points means determining the aplanatic surface. In this theory images are<br />

formed by the focusing of bundles of rays, <strong>and</strong> in most cases of reflection<br />

<strong>and</strong> refraction the image of a point source will not be a point. The properties<br />

of these images became an important subject of study in seventeenth-century<br />

geometrical optics. In Optica promota, Gregory extended the theory of<br />

Paralipomena with his contributions to the theory of optical imagery <strong>and</strong> his<br />

determination of aplanatic surfaces. From this viewpoint, La Dioptrique<br />

embroidered on Paralipomena rather than Dioptrice.<br />

The seventeenth-century study of these topics reached its highpoint in<br />

the lectures Barrow <strong>and</strong>, later, Newton delivered at the university of<br />

Cambridge. Barrow’s lectures were published in 1669, those of Newton<br />

remained unpublished during his lifetime. With Huygens’ dioptrical work<br />

101<br />

Stroud, Minute, 20; Prins, “Hobbes on light <strong>and</strong> vision”, 129-132. On Hobbes’ derivation of the sine<br />

law, see section 5.2.1.<br />

102<br />

Lohne, ”Geschichte des Brechungsgesetzes”, 166.<br />

103<br />

Huygens worked on it in 1671-2, see page 160.

1653 - TRACTATUS 39<br />

remaining uncompleted <strong>and</strong> unpublished as well, Lectiones XVIII was the most<br />

advanced treatise on geometrical optics published until the end of the<br />

seventeenth century. The core of Lectiones XVIII consists of lectures IV<br />

through XIII, in which Barrow determined the image of a point source in any<br />

reflection or refraction in plane <strong>and</strong> spherical surfaces. Barrow developed a<br />

mathematical theory of imagery by analyzing the intersections the refractions<br />

of a bundle of rays.<br />

For example, a point A is seen by<br />

an eye off the axis AB, with COD being<br />

the pupil of the eye. 104 (Figure 20). The<br />

pupil is perpendicular to the refracted<br />

ray NO, which passes through the<br />

center of the pupil. The extension<br />

KNO is called the principal ray. Now,<br />

draw the refracted rays MC <strong>and</strong> RD<br />

that pass through the edge of the<br />

pupil. Produced backwards, MC <strong>and</strong><br />

RD will not intersect the principal ray<br />

NKO in one point, but in points X <strong>and</strong><br />

V. Barrow demonstrated that point Z<br />

on the principal ray is the limit of<br />

these intersections. According to his<br />

definition of the image, Z is the place<br />

of the image. Consequently, the<br />

cathetus rule does not apply here, as it<br />

Figure 20 Barrow’s analysis of image<br />

formation in refraction.<br />

would have point K as the image. 105 Barrow applied this determination of the<br />

image point to various problems in refraction. In the case of spherical<br />

surfaces he derived expressions for the place of the image point for the eye<br />

being both on <strong>and</strong> off the axis of the surface.<br />

Barrow defined the image point in a similar way as Huygens defined the<br />

‘punctum concursus’ <strong>and</strong> applied it with comparable rigor to study the<br />

refracting properties of spherical surfaces. Many of their results were<br />

equivalent. Yet, they had different goals. Huygens intended to explain the<br />

dioptrical properties of the telescope <strong>and</strong> therefore confined himself to<br />

paraxial rays, not discussing optical imagery. He ignored mathematically<br />

sophisticated problems that had no relevance to the telescope, like the focus<br />

of an oblique cone of rays. Barrow’s aim was to develop a general theory of<br />

optical imagery. He had no intention of explaining the telescope <strong>and</strong> many of<br />

the problems he treated had no relevance to it. 106 Still, in lecture XIV, he also<br />

discussed spherical lenses. He gave, without proof, a series of equations for<br />

the focal <strong>and</strong> image points of all kinds of lenses by way of an example of the<br />

104 Barrow, Lectiones, [82-83].<br />

105 Compare Shapiro, “The Optical Lectures”, 130 & 133-134.<br />

106 Compare Shapiro, “The Optical Lectures”, 150-151; <strong>and</strong> Malet, “Isaac Barrow”, 286.

40 CHAPTER 2<br />

application of his preceding discussion of single spherical surfaces. It was an<br />

exact solution to the problem of Dioptrice, yet a complex <strong>and</strong> cumbersome<br />

one. 107 Barrow had chosen the example “… with a view to common use, <strong>and</strong><br />

particularly aimed at reducing the labour of anyone who comes across<br />

them.” 108 Barrow’s exact theory of focal distances was the first in print, but it<br />

was no more than a theory of focal distances. He did not discuss<br />

magnification <strong>and</strong> configurations of lenses.<br />

Barrow’s footsteps were followed by Newton in his lectures. Their<br />

central subject was his mathematical theory of colors. In a section “On the<br />

Refractions of Curved Surfaces” he also treated some topics regarding<br />

monochromatic rays. Newton extended Barrow’s theory of image formation<br />

to three-dimensional bundles of rays. He demonstrated the existence of a<br />

second image point at the intersection of the axis <strong>and</strong> the principal ray.<br />

Newton had developed his own solution of the anaclastic problem –<br />

although in the lectures he ab<strong>and</strong>oned it in favor of Barrow’s – <strong>and</strong> found a<br />

new way to derive Descartes’ ovals. The final goal of the lectures were,<br />

however, colors. So, when Newton, in his 31 st proposition, determined the<br />

spherical aberration of a ray, he did so to compare it to chromatic aberration.<br />

The latter was larger <strong>and</strong> “Consequently, the heterogeneity of light <strong>and</strong> not<br />

the unsuitability of a spherical shape is the reason why we have not yet<br />

advanced telescopes to a greater degree of perfection.” 109 In this way,<br />

Newton dismissed Descartes proposal as a dead-end. This was an important<br />

result for the underst<strong>and</strong>ing of the working of the telescope. In order to<br />

overcome the disturbing effects of aberration, Newton proposed to use<br />

mirrors instead of lenses. Newton’s theory of colors <strong>and</strong> his reflector are<br />

further discussed in section 3.2.3.<br />

Dioptrics as mathematics<br />

The discovery of an exact law of refraction had supplied geometrical optics<br />

with a foundation for the mathematical study of the behavior of refracted<br />

rays. This study consisted of deducing theorems from the postulates <strong>and</strong><br />

definitions of dioptrics in a rigorous way aimed at generality. With Kepler’s<br />

new theory of image formation, a range of new problems were raised relating<br />

to the perfect <strong>and</strong> imperfect focusing of rays. To the ones already mentioned<br />

was added, at the end of the century, that of caustics; the locus of<br />

intersections of rays refracted by a curved surface. 110 These problems were<br />

markedly theoretical, mathematical puzzles tackled without practical<br />

objectives. Halley, for example, in a paper of 1693 solved “the problem of<br />

finding the foci of optick glasses universally” by means of a single algebraic<br />

107<br />

Shapiro, “The Optical Lectures”, 149-150.<br />

108<br />

Barrow, Lectiones, [168].<br />

109<br />

Newton, Optical Papers 1, 427.<br />

110<br />

In a series of papers of the 1680s <strong>and</strong> 1690s Tschirnhaus, Jakob <strong>and</strong> Johann Bernoulli <strong>and</strong> Hermann<br />

attacked the problem. They were preceded by Huygens in 1677, but he did not publish his account until<br />

1690, see section 5.1. Jakob Bernoulli published a general solution in Acta Eruditorum in 1693. In his<br />

Analyse des infiniments petits (1696), L’Hopital gave a definitive solution on basis of the differential calculus.

1653 - TRACTATUS 41<br />

equation. 111 Despite his involvement in practical matters of telescopes Halley,<br />

like Barrow, did not further apply his finding to the effect of lenses. His<br />

principal goal seems to have been to supplement Molyneux’ theory of focal<br />

distances by means of giving “An instance of the excellence of modern<br />

algebra, …” 112 All in all, the telescope rarely directed the dioptrical studies<br />

undertaken by mathematicians.<br />

Kepler is rightly regarded as the founder of seventeenth-century<br />

geometrical optics, yet it was Paralipomena rather than Dioptrice that<br />

constituted the starting-point for later studies. Similarly, Descartes’ La<br />

Géométrie was the starting-point for later studies of aplanatic surfaces rather<br />

than La Dioptrique. I find it remarkable that an instrument that had<br />

revolutionized astronomy was ignored by students of geometrical optics in<br />

the same way as spectacles had been previously. Kepler alone had, right<br />

upon its invention, insisted that a mathematical underst<strong>and</strong>ing of the<br />

telescope was needed for its use in observation, <strong>and</strong> Huygens was the only<br />

one to take the instruction to heart. His approach was that of a<br />

mathematician, yet he applied his mathematical abilities to a practical<br />

question: underst<strong>and</strong>ing the working of the telescope. In Tractatus, he used<br />

the sine law to derive an exact <strong>and</strong> general theory of the properties of<br />

spherical lenses <strong>and</strong> their configurations. It remains to be seen, however,<br />

whether such a mathematical theory of the telescope was really of any use.<br />

Tractatus remained unpublished, those interested had to do with Dioptrice.<br />

2.2.3 THE NEED FOR THEORY<br />

Dioptrice had arisen from Kepler’s conviction that, in order to make reliable<br />

observations, <strong>and</strong> astronomical instrument should be understood precisely.<br />

The mathematicians I have discussed in the preceding section did not follow<br />

his lead. Even Descartes <strong>and</strong> Newton, who proposed innovations in<br />

telescope design, did not bother to elaborate theories of the way telescopes<br />

produce sharp, magnified images. Maybe this was so because they, like the<br />

others mathematicians that have been discussed, did were not much involved<br />

in telescopic observation. Could the case be different for the mathematicians<br />

who were, the observers? We have seen that Galileo, the most renowned<br />

telescopist, was not really interested in mathematical questions of dioptrics.<br />

He applied himself rather to practical matters of the manufacture <strong>and</strong><br />

improvement of the telescope. It does not seem that Galileo had to invoke<br />

dioptrical arguments to defend the reality of telescopic observations, at least<br />

not arguments from the mathematical tradition of perspective <strong>and</strong> Kepler. 113<br />

To be sure, as a pioneer in astronomical telescopy Galileo was confronted<br />

with suspicions about the reality of heavenly things seen through the tube,<br />

but these soon wore off. Likewise, telescopists like Scheiner <strong>and</strong> Hevelius in<br />

111<br />

Halley, “Instance”, 960.<br />

112<br />

See: Albury, “Halley, Huygens, <strong>and</strong> Newton”, 455-457.<br />

113<br />

Galileo, Sidereus nuncius, 112-113 <strong>and</strong> 92-93 (Van Helden’s conclusion). See Dupré, Galileo <strong>and</strong> the<br />

telescope, 175-178.

42 CHAPTER 2<br />

their books on telescopic observation contented themselves with a cursory,<br />

qualitative account of the telescope, drawing on Kepler’s lessons. Dioptrice<br />

was the st<strong>and</strong>ard theory referred to well until the close of the century, but<br />

mostly as regards the basic theorems on focal lengths <strong>and</strong> configurations.<br />

Apparently this sufficed the needs of practical dioptrics leaving the<br />

mathematical details superfluous. Had Kepler made things more difficult<br />

than they really were?<br />

This theme may be illustrated with the example of Isaac Beeckman, a<br />

savant who combined an interest in practical affairs with a theoretical<br />

outlook. He was interested in many things, including optics in all its<br />

manifestations, <strong>and</strong> kept an elaborate diary of his ideas <strong>and</strong> observations. It<br />

enables us to get an idea what a knowledgeable man would do with the<br />

mathematics of dioptrics. The diary contains numerous notes on visual<br />

observation that show that he read the literature – Aguilón, Kepler –<br />

attentively. In addition, he was familiar with Descartes’ optical ideas <strong>and</strong> their<br />

development, being in close contact with him on <strong>and</strong> off since 1618. 114 In the<br />

1620s, Beeckman became interested in telescopes <strong>and</strong> he acquired some<br />

lenses <strong>and</strong> instruments <strong>and</strong> later, in the 1630s, he put much effort in grinding<br />

lenses <strong>and</strong> building telescopes. 115 Working on them, he encountered the<br />

problem of spherical aberration (<strong>and</strong> later chromatic aberration) for which<br />

he considered several remedies. The notes concerned are interesting for they<br />

show a basic underst<strong>and</strong>ing of the working of lenses – as he would have<br />

acquired from Paralipomena <strong>and</strong> Dioptrice – but the actual problem, that the<br />

aberration is inherent to the spherical shape of a lens, seems to have eluded<br />

him. Besides the common use of diaphragms to decrease the disturbance,<br />

Beeckman thought up some sagacious ideas like combining lenses on a<br />

spherical surface in order to emulate one large lens or a lens built up in thin<br />

rings like a Fresnel lens. 116 The first idea he tested, just to discover soon that<br />

it did not work <strong>and</strong> that he had overlooked a basic property of lenses. 117 He<br />

was enough of an experimentalist not to trust ideas blindly. When Descartes<br />

informed him in 1629 of his project of a hyperbolic lens, Beeckman reacted<br />

skeptical. 118<br />

The micrometer <strong>and</strong> telescopic sights<br />

The principal reason why astronomers did not show much interest in<br />

dioptrics lies, I think, in the fact that the telescope was a qualitative<br />

instrument during the first decades after its introduction. It had revealed<br />

new, spectacular phenomena in the sky, but it had not been deployed in the<br />

114<br />

Schuster, “Descartes opticien” <strong>and</strong> Van Berkel, “Descartes’ debt”.<br />

115<br />

Beeckman, Journal, II, 209-211; 294-296. For lens grinding see down, page 57.<br />

116<br />

For the second idea see Beeckman, Journal, II, 367-368. For a later consideration see for example: III,<br />

296.<br />

117<br />

Beeckman, Journal, II, 296; 357.<br />

118<br />

Beeckman, Journal, III, 109-110.

1653 - TRACTATUS 43<br />

exact description of the universe. 119 After all, the telescope was an artful<br />

means to reveal new things in the heavens, whereas astronomical<br />

measurement instruments aided the naked eye. 120 The step to combine these<br />

two by aiding the artificial eyes with quadrants <strong>and</strong> the like, was not taken<br />

immediately. Around the middle of the century, astronomical measurements<br />

were still made by using the pre-telescopic methods <strong>and</strong> instruments<br />

developed by Tycho Brahe. Hevelius used telescopes extensively to study the<br />

surface of the Moon, but he turned to open-sight instruments when making<br />

measurements. Efforts had been made to use the telescope for<br />

measurements, but in vain. Until the 1670s, the accuracy of telescopic<br />

observations was determined by the acuity of the human eye. But then<br />

change set in. With the introduction of the micrometer, the telescope was<br />

transformed into an instrument of precision. Significantly, the men closely<br />

involved in that development were the ones to seek a more precise account<br />

of the working of the telescope.<br />

The configuration Kepler had thought up in 1611 had the drawback that<br />

it reversed the image. Given the quality of lenses made at that time, it was<br />

not advisable to add a third lens to re-erect the image. The two-lens<br />

Keplerian telescope was therefore used only to project images, whereas the<br />

Galilean type continued to be used for direct observation. In the course of<br />

time the first advantage of Kepler’s configuration was discovered: its wider<br />

field of view. When the length of a Galilean telescope is increased the field<br />

of view quickly diminishes, which makes it very difficult to use. Towards the<br />

1640s, the Keplerian telescope was gaining ground, in particular through the<br />

good craftsmanship of telescope makers like Fontana in Naples <strong>and</strong> Wiesel<br />

in Augsburg. 121 At some point in the early 1640s, the second advantage of<br />

this type was discovered by the Lancashire astronomer William Gascoigne.<br />

The Keplerian configuration has a positive focus inside the telescope; an<br />

object inserted into it will cast a sharp shadow over the object seen through<br />

the tube. Gascoigne relates that he discovered this by accident after a spider<br />

had spun its web in his telescope. 122 Inserting some kind of ruler makes it<br />

possible to make measurements of telescopic images. He died in 1644,<br />

before he could publish his discovery <strong>and</strong> his measurements of the diameters<br />

of planets. 123<br />

Gascoigne’s accomplishments were made public in 1667 when Richard<br />

Towneley, backed by Christopher Wren <strong>and</strong> Robert Hooke, claimed British<br />

priority for the invention of the micrometer. This happened after a letter of<br />

119 Van Helden, Measure, 118-119.<br />

120 Compare Dear, Discipline <strong>and</strong> Experience, 210-216.<br />

121 Van Helden, “Astronomical telescope”, 26-32. See also below section 3.1.1.<br />

122 Rigaud, Correspondence, 46: “This is that admirable secret, which, as all other things, appeared when it<br />

pleased the All Disposer, at whose direction a spider’s line drawn in an opened case could first give me by<br />

its perfect apparition, when I was with two convexes trying experiments about the sun, the unexpected<br />

knowledge.”<br />

123 McKeon, “Les débuts I”, 258-266.

44 CHAPTER 2<br />

Adrien Auzout was published in Philosophical Transactions, in which he<br />

described a method of determining the diameters of planets. 124 He had<br />

devised – possibly with the help of Pierre Petit – <strong>and</strong> used – together with<br />

Jean Picard – a grate of thin wires <strong>and</strong> a moveable reference frame inserted<br />

in the focal plane of a telescope. In two letters, also published in Philosophical<br />

Transactions, Towneley argued that Gascoigne had made <strong>and</strong> used a<br />

micrometer much earlier. He described a pair of moveable fillets that could<br />

be inserted into the focal plane. 125 He himself had used <strong>and</strong> improved the<br />

device – probably since late 1665 – to make accurate observations. 126<br />

The principle of the micrometer, however, had already been published in<br />

1659; by Huygens in Systema Saturnium, his astronomical work in which he<br />

presented his discoveries regarding the ring <strong>and</strong> the satellites of Saturn. In its<br />

final section he explained the principle <strong>and</strong> described how to use it to make<br />

measurements. He inserted a ring in the focal plane <strong>and</strong> then measured the<br />

angular magnitude of the opening thus produced by timing the passage of a<br />

star. Next, he inserted a cuneiform strip through a hole in the tube until it<br />

just covered a planet. The angular diameter of the planet was determined by<br />

taking out the strip <strong>and</strong> comparing its width at the point found with the<br />

inner diameter of the ring. 127 It was not a real micrometer, but Huygens’<br />

rather cumbersome method did produce reliable, accurate data. 128 It was a<br />

convenient method for measuring the size of planets, Huygens said, as one<br />

did not have to wait for a conjunction of the planet with the Moon or a<br />

star. 129 Huygens had been acquainted with Auzout <strong>and</strong> Petit since 1660 <strong>and</strong><br />

had come to Paris in 1666 to give leadership to the Académie. His<br />

explanation of the principle of the micrometer certainly inspired their work<br />

on the micrometer, but the precise nature of Huygens’ contribution is hard<br />

to determine. 130<br />

The principle of the micrometer had another important application:<br />

telescopic sights. By inserting crosshairs in the focal plane, a telescope could<br />

reliably be aligned on a measuring arc. 131 With the telescopic sight the<br />

accuracy of Brahe’s measurements could finally be improved. Several<br />

programs of astronomical measurement now set off. In Paris, Picard <strong>and</strong><br />

other members of the Académie – completed in 1669 by Cassini – put into<br />

use a new, well-equipped observatory. 132 Picard’s work on cartographic<br />

124<br />

OldCorr3, 293: “… prendre les diametres du soleil, de la lune et des planetes par une methode que nous<br />

avons, Monsieur Picard et moy, que ie croy la meilleure de toutes celles que l’on a pratiquer Jusques a<br />

present, ...”<br />

125<br />

McKeon, “Les débuts I”, 266-269.<br />

126<br />

McKeon, “Les débuts I”, 286. In Micrographia (1665) Hooke had suggested that a scale may be inserted<br />

into the focal plane of telescopes. Hooke, Micrographia, 237.<br />

127<br />

OC21, 348-351.<br />

128<br />

Van Helden, Measure, 120-121.<br />

129<br />

OC21, 352-353.<br />

130<br />

McKeon, “Les débuts I”, 286; Van Helden, Measure, 118.<br />

131<br />

McKeon, “Renouvellement”, 122.<br />

132<br />

McKeon, “Renouvellement”, 126.

1653 - TRACTATUS 45<br />

measurements resulted in the determination of the arc of the meridian,<br />

published in Mesure de la Terre (1671). In London, Hooke <strong>and</strong> Wren devoted<br />

themselves to carrying out the idea of telescopic sights. In 1669, Hooke<br />

announced that he had established the motion of the earth by means of a<br />

mural quadrant thus equipped. His claim met with great skepticism. In 1675,<br />

Flamsteed was appointed Astronomer Royal at the London counterpart to<br />

the Paris observatory. At the Royal Observatory, he erected a wealth of<br />

precision instruments <strong>and</strong> set up a program of astronomical measurements,<br />

eventually resulting in Historia coelestis brittanica (1725).<br />

The usefulness of telescopic sights was not, however, beyond all doubt.<br />

Hevelius, the most renowned astronomer in those days, was suspicious. He<br />

believed that telescopic sights were unreliable <strong>and</strong> therefore preferred naked<br />

eye views. 133 In 1672, a letter by Flamsteed was published in Philosophical<br />

Transactions in which he defended the use of telescopes for astronomical<br />

measurements. 134 He praised Hevelius for having improved Brahe’s<br />

astronomical data, but doubted whether any further progress could be<br />

possible as long as the latter refrained from using ‘glasses’. Hevelius took<br />

offense at Flamsteed’s allegations, <strong>and</strong> responded in Machina coelestis pars prior<br />

(1673) <strong>and</strong> in a letter that was published in part in Philosophical Transactions of<br />

April 1674:<br />

“For it is not only a question of seeing the stars somewhat more distinctly (…) but<br />

whether the instruments point correctly in every direction, whether the telescopic sights<br />

of the instrument can be accurately directed many times to any observations, <strong>and</strong> can<br />

be reliably maintained; but I very much doubt whether this can be done with equal<br />

precision every time.” 135<br />

The argument went a bit out of h<strong>and</strong> when, later in 1674, Hooke interfered<br />

with a vehement attack on Hevelius in Animadversions on the first part of the<br />

machina coelestis. Deeply hurt, Hevelius sent Flamsteed a letter in which he<br />

once more explained his doubts about the reliability of telescopic sights. The<br />

dispute was settled only five years later after a visit to Gdansk by Halley. He<br />

reported that Hevelius’ naked eye observations were indeed incredibly<br />

accurate.<br />

Hevelius had fought a lost battle – so it can be said with hindsight – but<br />

he was right in his suspicions about the reliability of telescopic sights. He<br />

knew from experience how difficult it is to align instruments reliably. Already<br />

in 1668 – right after the announcement of the micrometer – he had written<br />

to Oldenburg: “For many things seem most certain in theory, which in<br />

practice often fall far enough from truth.” 136 He was astonished that Hooke<br />

would claim great accuracy for his measurements on the basis of just single<br />

observations. Hevelius knew that accuracy was gained by hard <strong>and</strong> systematic<br />

work. Picard, Cassini <strong>and</strong> Flamsteed undertook such an arduous task, but<br />

133<br />

Flamsteed, Gresham lectures, 34-39 (Forbes’s introduction).<br />

134<br />

OldCorr9, 326-327.<br />

135<br />

OldCorr10, 520.<br />

136<br />

OldCorr4, 448.

46 CHAPTER 2<br />

were convinced that the new optical devices were useful. The telescopic sight<br />

<strong>and</strong> the micrometer, together with the pendulum clock, brought about a<br />

revolution in positional astronomy between 1665 <strong>and</strong> 1680. 137 In dioptrics it<br />

raised the question of the exact properties of lenses anew.<br />

Underst<strong>and</strong>ing the telescope<br />

Apart from the practical problems of mounting <strong>and</strong> aligning, the theoretical<br />

problem of the working of the telescope now became a matter of sustained<br />

interest. As a result of his discussion with Hevelius over the reliability of<br />

telescopic sights, Flamsteed realized that a theoretical justification of his<br />

claims was also needed: “… to prove that optick glasses did not impose<br />

upon or senses. then to shew that they might be applyed to instruments &<br />

rectified as well as plaine sights.” 138 His chance to elaborate a dioptrical<br />

account of the telescope came in the early 1680s, when, appointed Gresham<br />

professor of Astronomy, he could deliver a series of lectures on astronomy.<br />

In these lectures, he discussed instruments <strong>and</strong> their use at length <strong>and</strong><br />

included an account of dioptrics.<br />

“Yet such has beene the fault of or time that hitherto very little materiall on this subject<br />

has been published in or language. [Tho severall learned persons have done well<br />

concerning opticks in ye latine Tongue. Yet how glasses may be applyed to instruments<br />

& how the faults commonly committed in theire applycation might be amended or<br />

rather shund & how all the difficultys suggested by ingenious persons who had not the<br />

good to underst<strong>and</strong> them aright might be avoyded the best authors of Dioptricks have<br />

been hitherto silent. … I shall therefore make it my businesse in this & my following<br />

lectures of this terme fully to explain the Nature of telescopes the reason of their<br />

performances, how they may be applyed to Levells, Quadrants, & Sextants. & how the<br />

instruments furnished with them may be so rectified & adjusted that they may be free<br />

from all suspicion of errors]” 139<br />

Flamsteed began with a discussion of the focal distances of convex lenses. It<br />

has two notable features. First, he took the consequences of Newton’s<br />

theory of colors into account by pointing out the chromatic aberration of<br />

lenses. Second, the paucity of his demonstrations shows that he was not an<br />

outst<strong>and</strong>ing geometer. 140 By means of the sine law, he calculated the<br />

refraction of single rays numerically <strong>and</strong> then compared the result with the<br />

Keplerian rules for focal distances of a bundle of rays. By calculating<br />

spherical aberration he discovered – as Huygens had done earlier – that the<br />

aberration of a plano-convex lens varies considerably depending on which<br />

side is turned towards the incident rays. He gave only a qualitative account of<br />

chromatic abberation. On this basis he argued that only telescopes consisting<br />

of two convex lenses are useful in astronomy, because these admit the<br />

137<br />

Van Helden, “Huygens <strong>and</strong> the astronomers”, 156-157; Van Helden, Measure, 127-129.<br />

138<br />

Flamsteed, Gresham lectures, 154.<br />

139<br />

Flamsteed, Gresham lectures, 119 & 132. Flamsteed later deleted the part between brackets.<br />

140 Flamsteed, Gresham lectures, 120-127.

1653 - TRACTATUS 47<br />

insertion of a micrometer or crosshairs. 141 He then went on to explain in<br />

detail how to mount a telescope on quadrants <strong>and</strong> other things. 142<br />

Flamsteed did not achieve the exactness <strong>and</strong> rigor of Huygens. His<br />

analysis of the properties of lenses consisted of numerical calculations rather<br />

than of general theorems. His account was larded with solutions to practical<br />

problems, <strong>and</strong> here indeed resided the eventual goal of giving a dioptrical<br />

account of the properties <strong>and</strong> effects of telescopes. Given the scarcity of<br />

suitable publications on these matters, Flamsteed did not have much to start<br />

from. He confessed that he had not taken the time to peruse Kepler’s<br />

Paralipomena <strong>and</strong> he claimed never to have read Dioptrice. 143 He based himself<br />

instead on some letters in which Gascoigne discussed the foci of planoconvex<br />

<strong>and</strong> plano-concave lenses. 144 He considered his own account of other<br />

lenses <strong>and</strong> of telescopes “… but a superstructure on yt foundation”. 145 It<br />

sufficed to free the telescope of the imputation that “… all observations<br />

made with glasses [are] more doubtfull & uncerteine …” 146 Flamsteed’s<br />

lectures attracted only a small audience <strong>and</strong> did not go to print until this<br />

century. 147<br />

Some of Flamsteed’s ideas were passed on by Molyneux. During the<br />

1680s, the men had corresponded extensively on dioptrics, among other<br />

things. In 1692 Molyneux published Dioptrica Nova, an elaborate dioptrical<br />

account of the telescope. In its preface, he acknowledged his debt to<br />

Flamsteed:<br />

“… the Geometrical Method of calculating a Rays Progress, which in many particulars is so<br />

amply delivered hereafter, is wholly new, <strong>and</strong> never before publish’d. And for the first<br />

Intimation thereof, I must acknowledg my self obliged to my worthy Friend Mr.<br />

Flamsteed Astron. Reg. who had it from some unpublished Papers of Mr. Gascoignes.” 148<br />

Dioptrica Nova was a compilation of dioptrical works published during the<br />

seventeenth century. 149 Molyneux’ own contribution consists of his particular<br />

presentation of the material, arranging theoretical knowledge in such a way<br />

that it was useful for underst<strong>and</strong>ing the working of telescopes. He gave his<br />

own demonstrations of many of its theorems, but he did not aim at<br />

mathematical rigor or completeness:<br />

“… [the Reader] is not to expect Geometrical Strictness in several Particulars of this<br />

Doctrine. … ; as being more desirous of shewing in gross the Properties of Glasses <strong>and</strong><br />

141<br />

Flamsteed, Gresham lectures, 136.<br />

142<br />

Flamsteed, Gresham lectures, 140-143.<br />

143<br />

Flamsteed, Gresham lectures, 40; 146n2 (Forbes’ introduction).<br />

144<br />

Flamsteed, Gresham lectures, 8-9; 40 (Forbes’ introduction).<br />

145<br />

Flamsteed, Gresham lectures, 39 (Forbes’ introduction).<br />

146<br />

Flamsteed, Gresham lectures, 149.<br />

147<br />

Flamsteed, Gresham lectures, 4-5 (Forbes’ introduction).<br />

148<br />

Molyneux, Dioptrica nova, (Admonition to the reader).<br />

149<br />

Molyneux mentioned Kepler, Cavalieri, Hérigone, Dechales, Fabri, Gregory <strong>and</strong> Barrow.

48 CHAPTER 2<br />

their Effects in Telescopes, than of affecting a Nicety, which would be of little Use in<br />

Practice.” 150<br />

The limitations of Molyneux’ mathematics are easily noted. In proposition<br />

III, for example, he discussed refraction by a bi-convex lens of a ray parallel<br />

to the axis. 151 Taking into account both the distance of the ray from the axis<br />

<strong>and</strong> the thickness of the lens, he derived by means of the sine law the point<br />

where the refracted ray intersects the axis. In generalizing this to the focal<br />

distance of the lens, Molyneux was less exact:<br />

“If by this Method we calculate the Progress of a Ray through a Double Convex-Glass<br />

of equal Convexities; <strong>and</strong> the thickness of the Glass be little or nothing in comparison<br />

of the Radius of the Convexity; <strong>and</strong> the Distance of the Point of Incidence from the<br />

Axis be but small, we shall find the Point of Concourse to be distant from the Glass<br />

about the Radius of the Convexity nearly.” 152<br />

He then gave Kepler’s theorem <strong>and</strong> reproduced the latter’s proof. For a lens<br />

with unequal curvatures, he stated that the refracted ray could also be<br />

constructed exactly. He confined himself, however, to a “… Shorter Rule<br />

laid down by most Optick Writers”, which is identical with Huygens’ rule for<br />

a thin lens cited on page 19. 153 This pattern of exact constructions for single<br />

rays <strong>and</strong> questionable generalizations to bundles of rays recurs throughout<br />

Dioptrica Nova. His problem was that he somehow had to link Flamsteed’s<br />

discussion of single rays with the Keplerian rules of focal distances found in<br />

most published treatises. He was not able enough a mathematician to derive<br />

general theorems on focal distances by means of the sine law.<br />

The problem with Molyneux’ generalizations is that he thought that the<br />

intersection of single refracted ray with the axis was an approximation of the<br />

focal distance. He did not fully underst<strong>and</strong> that the focus of a refracting<br />

body is the (limit) point of the refracted rays of a pencil of rays. His<br />

definition of ‘focus’ in terms of the intersection of a single ray with the axis<br />

makes this clear. 154 Taken literally, this would mean that a spherical surface<br />

has many foci for one point object. In a scholium following proposition III<br />

of Dioptrica Nova, he discussed spherical aberration. He began by reproducing<br />

Flamsteed’s calculations for single rays as well as his conclusions concerning<br />

the use of lenses. He then defined the distance between the ‘focus’ <strong>and</strong> the<br />

intersection of the refracted ray with the axis as the ‘depth of the focus’. 155<br />

Again, he mixed up the refraction of a single ray with the focusing of a pencil<br />

of rays.<br />

It is not difficult to point out flaws in Molyneux’ demonstrations, but we<br />

should bear in mind the practical aim of Dioptrica Nova. In his discussion of<br />

150<br />

Molyneux, Dioptrica nova, (Admonition to the reader).<br />

151<br />

Molyneux, Dioptrica nova, 19-23.<br />

152<br />

Molyneux, Dioptrica nova, 20.<br />

153<br />

Molyneux, Dioptrica nova, 22.<br />

154<br />

Molyneux, Dioptrica nova, 9.<br />

155<br />

Molyneux, Dioptrica nova, 24. From the preceding it will be clear, that following Molyneux's line of<br />

thought this distance should be zero, for both points are by definition the same.

1653 - TRACTATUS 49<br />

images of extended objects, Molyneux displayed a better underst<strong>and</strong>ing of<br />

the focusing of rays. In a section on “… the Representation of outward<br />

Objects in a Dark Chamber; a Convex-Glass”, he described how the image is<br />

formed by the focusing of pencils of rays originating in the points of an<br />

object. 156 He then remarked that “… tho all the Rays from each point are not<br />

united in an answerable Point in the Image, yet there are a sufficient quantity<br />

of them to render the Representation very perfect.” 157 Rather than<br />

mathematically precise, this was a practical definition of focus. It explained<br />

why in practice images may appear sharp. Besides all the objections that can<br />

be raised against Molyneux’ theory from a mathematical point of view we<br />

should bear in mind that Dioptrica Nova was the first published dioptrical<br />

account of telescopes, since Dioptrice. It was up-to-date with developments in<br />

telescope making <strong>and</strong> was intended to be useful for practice.<br />

Before coming to a conclusion of this chapter, we go back in time <strong>and</strong><br />

cross back over the Channel. Flamsteed’s ally in the debate over telescopic<br />

sights, Picard, also saw the importance of theory. In a letter to Hevelius, he<br />

had briefly explained the working of the telescopic sight in dioptrical terms. 158<br />

Somewhat earlier – probably in 1668 – he had pointed out the need for such<br />

an analysis: “[optical devices] can also be subject to certain refractions that<br />

should be known well.” 159 In Mesure de la Terre, he had briefly discussed<br />

matters of aligning <strong>and</strong> rectifying telescopic sights in these terms. Picard was<br />

known for his interest <strong>and</strong> ability in matters dioptrical. At the Académie, he<br />

frequently discoursed of dioptrical theory. 160 In this, the telescope stood<br />

central:<br />

“What we have just explained about the construction of telescopes, concerns only its<br />

use in instruments made for observation, …” 161<br />

Picard never published his dioptrics, but a collections of papers he had read<br />

at the Académie was published posthumously in 1693 under the title<br />

‘Fragmens de Dioptrique’. 162 Picard had a major advantage over Flamsteed.<br />

He was acquainted with one of the most knowledgeable men in dioptrics:<br />

Huygens. Besides his learning, in 1666 Huygens had brought a copy of the<br />

manuscript of Tractatus to Paris. 163 At the Académie, Huygens had also<br />

discoursed on dioptrics. “Fragmens de Dioptrique” make it clear that Picard<br />

must have been among Huygens’ most attentive listeners. They are for the<br />

156 Molyneux, Dioptrica nova, 36-38.<br />

157 Molyneux, Dioptrica nova, 38.<br />

158 Picolet, “Correspondence”, 38-39.<br />

159 “… peuuent aussi estre sujets a certaines refractions qu’il faut bien connoistre.” Quoted in McKeon,<br />

“Renouvellement”, 126-128. It is found in: A. Ac. Sc., Registres, t. 3, fol 156 r o - 164 v o spéc. 157 v o.<br />

160 Blay, ”Travaux de Picard”, 329-332. Blay cites several references.<br />

161 Blay, “Travaux de Picard” 343. “Ce que nous venons d’expliquer touchant la construction des lunettes<br />

d’approche, n’est que par rapport à l’usage que l’on en fait dans les instruments qui servent à l’observer,<br />

…”<br />

162 Divers Ouvrages de Mathematique et de Physique, par Messieurs de l’Academie Royale des Sciences (1693), 375-412.<br />

163 OC13, “Avertissement”, 7.

50 CHAPTER 2<br />

most part derived from Huygens’ dioptrical theories, <strong>and</strong> I will not discuss<br />

them in further detail. 164<br />

Huygens’ position<br />

Picard’s dioptrical fragments bring us back to Huygens. What had he been<br />

doing in the meantime? In Systema saturnium he had alluded to an elaborate<br />

theory of dioptrics, which we know he possessed indeed. Yet, despite<br />

ongoing requests to publish it, he had kept it to himself. It may be clear by<br />

now that Tractatus is a unique work in the development of seventeenthcentury<br />

dioptrics. Huygens was the first <strong>and</strong> only man to follow the lead of<br />

Dioptrice. Like Kepler, he combined the two things necessary to develop a<br />

theory of the telescope: mathematical proficiency <strong>and</strong> an orientation on the<br />

instrument. Unlike Kepler, he had the exact law of refraction <strong>and</strong> thus he<br />

could rigorously develop an exact theory of the telescope.<br />

But did Huygens really follow Kepler? Did he want to underst<strong>and</strong> the<br />

telescope in view of its use in astronomy? Tractatus came into being well<br />

before Huygens commenced his practical activities of telescope making <strong>and</strong><br />

astronomical observation (discussed in the next chapter). Unlike Flamsteed<br />

<strong>and</strong> Picard, he did not seek answers to questions that had arisen in practice.<br />

Nevertheless, his orientation on the telescope is clear. He passed by all those<br />

sophisticated problems not relevant to the underst<strong>and</strong>ing of the telescope<br />

that preoccupied mathematicians like Barrow. However, nowhere does<br />

Huygens mention Kepler as an example. It looks as if developing a theory of<br />

the telescope on the basis of the sine law was to him an interesting<br />

mathematical puzzle, maybe just to correct Descartes’ useless approach to<br />

dioptrics. The problem had not yet been solved <strong>and</strong> Huygens only too gladly<br />

seized the opportunity. Which makes his exclusive orientation on the<br />

instrument all the more interesting.<br />

The transformation of the telescope into an instrument of precision<br />

brought back an interest in the dioptrical properties of the telescope. In this<br />

regard, one might say that Kepler had prematurely raised the question after a<br />

mathematical underst<strong>and</strong>ing of the telescope. In 1611, it was a qualitative<br />

instrument <strong>and</strong> remained so for another half century. To underst<strong>and</strong> its<br />

working, a qualitative account of the effects of lenses therefore sufficed.<br />

Similarly, we can ask whether an exact theory like Huygens’ was really<br />

needed. It seems that Kepler’s or Keplerian theories satisfied the needs of<br />

men like Flamsteed <strong>and</strong> Molyneux pretty well. They lacked sufficient<br />

proficiency in mathematics to treat lenses in exact terms, but they may also<br />

have been perfectly satisfied with their approximate results.<br />

Huygens himself did not put much work in applying his theory to the<br />

questions that occupied Picard <strong>and</strong> Flamsteed. The principle of the<br />

micrometer may or may not have been the result of his theoretical<br />

underst<strong>and</strong>ing, in Systema saturnium he explained it only briefly. Huygens did<br />

164 Blay, “Travaux de Picard”, 340.

1653 - TRACTATUS 51<br />

expect that theory could be useful. The discovery that a sphere is an<br />

aplanatic surface in some cases had given the initial impulse to his interest in<br />

dioptrics. In his letter to Van Schooten, he expressed the expectation that<br />

this theoretical insight would contribute to the improvement of the<br />

telescope. From the very start, Huygens saw a connection between the<br />

theory of dioptrics <strong>and</strong> the practice of telescope making. In the next chapter<br />

we shall see what he would make of it.

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Chapter 3<br />

1655-1672 - 'De Aberratione'<br />

Huygens' practical optics <strong>and</strong> the aspirations of dioptrical theory<br />

In the decade following Tractatus, Huygens was at home were his<br />

mathematical virtuosity grew to full stature. These are the years of his most<br />

renowned achievements: the invention – in 1656 – improvement <strong>and</strong><br />

employment of the pendulum clock <strong>and</strong> the theory of pendulum that were<br />

the basis of his master piece Horologium Oscillatorium (1673); the discovery in<br />

1655 of a satellite of Saturn <strong>and</strong> the identification of its ring. Through his<br />

correspondence <strong>and</strong> publications Huygens increasingly gained recognition<br />

among Europe’s scholars. He traveled abroad, first to Paris in 1655 to meet<br />

the leading French mathematicians, then to Paris <strong>and</strong> London in 1660-1, <strong>and</strong><br />

again in 1663-4, the last time being elected fellow of the Royal Society. There<br />

were squabbles as well, in Italy in particular, over the priority of the<br />

pendulum clock with Florentine sympathizers of the late Galileo <strong>and</strong> with<br />

the Roman telescope maker Divini over the superiority of his telescopes.<br />

Probably as a result of the clock dispute, he did not obtain a position at the<br />

court of prince Leopold, but in 1666 Huygens realized his learned assets. At<br />

the instigation of Colbert he came to Paris to help organize an ‘académie des<br />

sciences’, thus confirming his status as Europe’s leading mathematician. Life<br />

in Paris, with it competitive milieu, was no unqualified pleasure. Huygens<br />

correspondence shows symptoms of homesickness, he particularly missed<br />

his brother Constantijn, <strong>and</strong> in 1670 he was was smitten with ‘melancholie’<br />

for the first time. In these years he also experienced the first major setback in<br />

his science: a design for a perfect telescope proved useless. The design was<br />

the outcome of Huygens’ practical activities in telescopy of the late 1650s<br />

<strong>and</strong> his subsequent theoretical reflections thereupon of the 1660s. These are<br />

the subject of this chapter.<br />

When Huygens’ interest in dioptrics was sparked late 1652, it was both its<br />

theoretical <strong>and</strong> practical aspects. He immediately began inquiring about the<br />

art of lens making, but he engaged in practical dioptrics only after he put<br />

aside the manuscript of Tractatus. Around 1655, he <strong>and</strong> his brother<br />

Constantijn acquired the art of lens making <strong>and</strong> started building telescopes. 1<br />

The practice bore fruit almost immediately. In 1656, Christiaan published a<br />

pamphlet De saturni luna observatio nova on the discovery of a satellite around<br />

Saturn. It was the first new celestial body in the solar system to be<br />

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

54 CHAPTER 3<br />

discovered since Galileo. 2 The tract ended with an anagram holding<br />

Huygens’ second discovery: the true nature of the inexplicable appearance of<br />

Saturn. Three years later, he elaborated his explanation in Systema saturnium.<br />

The strange attachments to the planet that disappeared from time to time<br />

were manifestations of a solid ring around the planet. 3<br />

He owed much to his instruments, Huygens wrote in Systema saturnium.<br />

He prided himself on his practical skills of telescope making <strong>and</strong> claimed that<br />

his success proved the unmatched quality of his telescopes. Van Helden<br />

explains that his discovery owed at least as much to his talents for<br />

geometrical <strong>and</strong> physical reasoning. 4 Initially, Huygens had used a 12-foot<br />

telescope of their own make. After a trip to Paris, where he probably<br />

discussed his observations, the brothers built a new, 23-foot telescope which<br />

he started using in February 1656. 5 He illustrated the difference between<br />

both pieces in Systema saturnium (Figure 21). Everyone could see for himself<br />

that Huygens could hold his own with the best of telescope makers. At least,<br />

that is how he saw it himself. His boasting offended Eustachio Divini in<br />

Rome, who saw his fame of being the best telescope maker in Europe<br />

challenged. In 1660 he published Brevis annotatio in systema saturnium, disputing<br />

the observational results as well as Huygens’ claims regarding his<br />

instruments. 6 The tract was actually written by the Roman astronomer Fabri.<br />

In the ensuing dispute Divini/Fabri were no match for Huygens, at least not<br />

as regards the structure of the system of Saturn. 7<br />

Figure 21 Observations of Saturn with the 12- <strong>and</strong> a 23-foot telescope.<br />

The dispute itself is less interesting than the fact that Huygens did not feel<br />

above at entering a dispute with a craftsman. It raises questions about the<br />

relationship between his theoretical <strong>and</strong> his practical pursuits, how he valued<br />

his mathematical expertise <strong>and</strong> his skilful craftsmanship. The last part of this<br />

2<br />

Huygens did not name it, he called it ‘saturni luna’ <strong>and</strong> sometimes ‘comes meus’. The name Titan was<br />

given by Herschel in 1847.<br />

3<br />

OC15, 296-299.<br />

4<br />

Van Helden, “Huygens <strong>and</strong> the astronomers”, 150-154. Van Helden, “Divini vs Huygens”, 48-50.<br />

5<br />

OC15, 177; 230. Huygens employed Rhinel<strong>and</strong> feet (0,3139 meters) <strong>and</strong> inches (0,026 meters).<br />

6<br />

It is reprinted in OC15, 403-437.<br />

7<br />

Van Helden, “Divini vs Huygens”, 36-40.

1655-1672 - DE ABERRATIONE 55<br />

chapter examines these themes in a broader context of the scientific<br />

revolution, <strong>and</strong> forms a conclusion of this account of Huygens’ dioptrics<br />

prior to the metamorphosis of his optics discussed in the subsequent<br />

chapters. So much can be said that Huygens’ passion was with the<br />

instrument, not its employment. For Huygens telescopic astronomy was a<br />

pastime rather than a full-time job. Although he had solved the puzzle of<br />

Saturn’s bulges by systematic observation, this never became his vocation.<br />

His fascination was with its design <strong>and</strong> manufacture of telescopes. 8 To this<br />

we may also count his interest in dioptrical theory, being a means of<br />

tinkering with the instrument <strong>and</strong> contemplating its workings.<br />

In the ten or so years after 1653 when the brothers engaged in practical<br />

pursuits, Huygens did not work on dioptrical theory (at least no traces ar<br />

left). During the 1660s he returned to theory <strong>and</strong> set out for what should<br />

have been the crowning glory of his dioptrical work: the design of a<br />

telescope in which spherical aberration was nullified. Not by means of<br />

imaginary lenses of the kind Descartes had thought up, but by means of<br />

actual spherical lenses. In the design came together Huygens’ theoretical<br />

underst<strong>and</strong>ing <strong>and</strong> practical experience with lenses <strong>and</strong> it brought him closer<br />

to bridging the gap between theory <strong>and</strong> practice than any other in the<br />

seventeenth century. Newton’s ‘New Theory’ of colors eventually<br />

shipwrecked the project. Newton’s approach of mathematical optics<br />

essentially differed from Huygens’. These differences shed light on the<br />

character of the Huygens’ dioptrics <strong>and</strong> may explain why Huygens did not<br />

manage to bridge the said gap completely.<br />

3.1 The use of theory<br />

Around 1600, spectacle makers had advanced their art far enough to enable<br />

the discovery of the telescopic effect. 9 Astronomers in their turn discovered<br />

the possibilities of this chance invention. Their pursuit dem<strong>and</strong>ed far greater<br />

power than the first spyglasses offered. They needed skillful h<strong>and</strong>s:<br />

sometimes their own, but usually those of a craftsman. Galileo, not<br />

particularly all fingers <strong>and</strong> thumbs himself, had the advantage of living close<br />

to Venice, the center of European glass industry. After the success of Sidereus<br />

nuncius he established a workshop for telescopes. Simon Marius, in Germany,<br />

was less lucky: he had great trouble finding a good lens maker <strong>and</strong> could not<br />

put the new invention to fruitful use. 10 During the first half of the<br />

seventeenth century, the manufacture of telescopes for astronomy developed<br />

into a small trade of specialized craftsmen. This section will not treat the<br />

history of lens <strong>and</strong> telescope manufacture, it focuses on the relationship<br />

between dioptrical theory <strong>and</strong> the art of telescope making. Central questions<br />

are: to what extent was theoretical knowledge used in practical dioptrics, if it<br />

8<br />

Van Helden, “Huygens <strong>and</strong> the astronomers” 148, 157-158.<br />

9<br />

Van Helden, Invention, 16-20.<br />

10<br />

Van Helden, Invention, 26; 47-48.

56 CHAPTER 3<br />

was useful at all; did the scholarly world contribute to the art of telescope<br />

making besides revenue, status, <strong>and</strong> stimuli for progress?<br />

René Descartes definitely believed art could learn from philosophy, <strong>and</strong><br />

that it should. In La Dioptrique he intended to show the benefits of<br />

philosophy. The telescope, he said, was a product of practical wit <strong>and</strong> skills,<br />

but the explanation of its difficulties could bring it to a higher level of<br />

perfection. 11 Just as the telescope surpassed the natural limitations of vision,<br />

so the scholar could teach the craftsman how to overcome his limitations.<br />

The sine law dictated that lenses should have a conical rather than a spherical<br />

shape, as we have seen in the previous chapter. Descartes had also<br />

considered the way his design could be put to practice. In the tenth <strong>and</strong> final<br />

discourse of La Dioptrique he described the way his lenses could be made. His<br />

account included an ingenious lathe to grind hyperbolic lenses. It reflected<br />

his efforts, during the late 1620s, to make a hyperbolic lens. Descartes was<br />

never lavish to point out his debt to others – to put it mildly – so he did not<br />

tell his readers that he owed much to his cooperation with the lens maker<br />

Jean Ferrier <strong>and</strong> the mathematician Claude Mydorge. 12 Allegedly, the<br />

threesome succeeded in grinding a convex hyperbolic lens. “And it turned<br />

out perfectly well …”, Descartes wrote in 1635 to Huygens’ father<br />

Constantijn. 13 It had been made by h<strong>and</strong>. The next step was to design a lathe.<br />

Towards the end of 1628 Descartes left for Holl<strong>and</strong>. In the fall of 1629, he<br />

<strong>and</strong> Ferrier exchanged some letters in which an earlier design for a lathe was<br />

mentioned. 14 They discussed a machine Descartes had contrived for making a<br />

cutting blade with a hyperbolic edge. 15 Ferrier proposed several modifications<br />

<strong>and</strong> improvements that turn up in La Dioptrique. 16<br />

Throughout the seventeenth century, Descartes’ account gave rise to<br />

efforts to make elliptic <strong>and</strong> hyperbolic lenses. Around 1635, Constantijn<br />

Huygens arranged unsuccessful attempts to grind them. 17 In 1643, Rheita<br />

claimed to have succeeded with tools he described in Novem stellae circum<br />

Iovem. Wren described a device to make a hyperboloid surface in an article<br />

published in Philosophical Transactions in 1669. During the 1670s, Huygens<br />

corresponded with Smethwick over another design. 18 Much in these<br />

suggestions never went beyond the paper stage. To execute them skills <strong>and</strong><br />

tools – as well as patience – were needed. To design a lens may have been a<br />

scholarly challenge, actually to make it required craftsmanship. And then it<br />

11<br />

Descartes, Dioptrique, 2-3 (AT6, 82-83).<br />

12<br />

Shea, “Descartes <strong>and</strong> Ferrier”, 146-148.<br />

13<br />

AT1, 598-600. “Et il reussit parfaitement bien; …” It turned out that it was impossible to make a<br />

concave lens in the same way.<br />

14<br />

AT1, lts 8, 11, 12,13,22,21,27. Shea, “Descartes <strong>and</strong> Ferrier”. The letters not only reveal Ferrier’s<br />

mastery of the art but also his mathematical knowledge.<br />

15<br />

AT1, 33-35.<br />

16<br />

Descartes, Dioptrique, 141-150 (AT6, 215-224).<br />

17<br />

Ploeg, Constantijn Huygens, 34-38.<br />

18<br />

OC7, 111; 117; 487; 511-513. In 1654 Huygens described a mechanism to draw ellipses on the basis of a<br />

circle, apparently aimed at making elliptic lenses out of spherical ones; OC17, 287-292.

1655-1672 - DE ABERRATIONE 57<br />

remained to be seen whether it could be made to function properly. As for<br />

Descartes’ ideal lenses, theory had not advanced practice yet.<br />

3.1.1 HUYGENS AND THE ART OF TELESCOPE MAKING<br />

In the middle the seventeenth century the art of lens making had progressed<br />

enormously, allowing telescopes to be made with more than two lenses <strong>and</strong><br />

challenging the inventiveness of telescope makers. Skills, tools <strong>and</strong> materials<br />

were the principal necessities for the manufacture of ordinary spherical<br />

lenses. The manufacture was not fully under control: glass suffered from<br />

various flaws, the produced faces of lenses were spherical at best, <strong>and</strong> so on.<br />

Clever solutions were needed to obtain good telescopic images. The state of<br />

the art of lensmaking can be more or less reconstructed from the quality of<br />

surviving lenses, but this only provides indirect evidence of the art itself.<br />

Lensmaking practice in the first decades after the invention of the telescope<br />

is hardly documented. Some information can be distilled from Sirturus <strong>and</strong><br />

Scheiner, but their accounts are quite succinct <strong>and</strong> rather superficial.<br />

A rare source of information is found in the diary of Isaac Beeckman,<br />

which meticulously records his trials <strong>and</strong> errors with using <strong>and</strong><br />

manufacturing lenses. 19 His interest in telescopes was excited in the early<br />

1620s, but not until 1632 did he embark on serious lensmaking himself. In<br />

the meantime he recorded his dissatisfaction with the lenses he acquired with<br />

established lensmakers. The number of notes on grinding, polishing <strong>and</strong> the<br />

like, quickly grows in the early 1630s <strong>and</strong> in 1633 he acquired a grinding<br />

mould <strong>and</strong> commenced manufacturing his own lenses. Beeckman visited<br />

several artisans who taught him their art. The diary describes their techniques<br />

in much detail, particularly noting the differences. 20 The attentive pupil was a<br />

quick learner. In the autumn of 1635, Beeckman compared one of his lenses<br />

with one from Johannes Sachariassen of Middelburg – one of his tutors <strong>and</strong><br />

son of one of the claimants of the invention of the telescope – <strong>and</strong> found<br />

out it was better. 21 Beeckman’s journal was a hidden treasure. He showed it<br />

only to Descartes, Mersenne <strong>and</strong> Hortensius <strong>and</strong> it remained unknown until<br />

Cornelis de Waard discovered it in 1905.<br />

An aspirant lens maker lacked published expositions to learn of the art.<br />

Rheita in 1645 was somewhat more elaborate, but when Huygens took on to<br />

lens making in the early 1650s, he had to consult, like Beeckman before him,<br />

experts <strong>and</strong> acquire the art by trial <strong>and</strong> error. The questions he fired at<br />

Gutschoven in 1652 display the diversity of the know-how involved in the<br />

process of cutting, grinding, <strong>and</strong> polishing to make a good lens out of a lump<br />

of glass. 22 How to make grinding moulds? How can a perfect spherical figure<br />

19<br />

Next to numerous short entries, the main body is collected under the heading “Notes sur le rodage et le<br />

polisage des verres” in Beeckman, Journal, III, 371-431.<br />

20<br />

Beeckman, Journal, III, 69, 249, 308, 383.<br />

21<br />

Beeckman, Journal, III, 430.<br />

22<br />

OC1, 191. See also Anne van Helden, “Lens production”, 70-75.

58 CHAPTER 3<br />

be created? What kind of s<strong>and</strong> is needed for grinding? Which glue is best to<br />

attach the h<strong>and</strong>le? Et cetera, et cetera.<br />

Despite its orientation on the telescope, Huygens’ 1653 study of dioptrics<br />

did not aim at its improvement. Apparently, he had not found much use in<br />

the discovery of 1652. In the third part of Tractatus he had written down an<br />

the idea to add a little mirror to a Keplerian telescope to re-erect the image<br />

without the need to add an extra lens. 23 This was not a new idea <strong>and</strong> it did<br />

not improve the defects of lenses directly. Having put aside the manuscript<br />

on the mathematical properties of lenses, Huygens turned to their material<br />

properties. In the practical work he pursued with his brother, he developed<br />

an artisanal underst<strong>and</strong>ing of lenses. The question is what such an<br />

underst<strong>and</strong>ing entailed <strong>and</strong> how it related to the theoretical underst<strong>and</strong>ing<br />

Huygens had developed in Tractatus.<br />

Huygens’ skills<br />

Some notes survive, in which Huygens described the process of grinding<br />

lenses. 24 In 1658 he recorded how he had made a “good” 4½-foot bi-convex<br />

lens:<br />

“Always kept it fairly wet to preserve the dust. But not too much water in the<br />

beginning, or it will bump. Never forget to press evenly, <strong>and</strong> often lifted the h<strong>and</strong> <strong>and</strong> placed<br />

it evenly again. It is best to be alone. … At first I ground the other side wrongly: the<br />

reason for this was that I either took too much water in the beginning, or that I did not<br />

polish on the right spot. I first corrected somewhat by polishing at the right spot again;<br />

then with more polishing it got worse once again.” 25<br />

Making lenses was first of all a matter of ‘Fingerspitzengefühl’ acquired<br />

through much practice. Huygens <strong>and</strong> his brother did so <strong>and</strong> eventually<br />

became pretty good at it. 26 One of the main problems of grinding <strong>and</strong><br />

polishing was to secure an optimal shape of a lens. Both surfaces should be<br />

really spherical <strong>and</strong> the axes should coincide. As Huygens’ notes show this<br />

entailed a good deal of accuracy <strong>and</strong> patience. Proper tools did not only ease<br />

the laborious task but improve the control of the manufacture <strong>and</strong> thus the<br />

quality of the lens. In his notes, Huygens described a device (Figure 22) that<br />

relieved the h<strong>and</strong>s <strong>and</strong> ensured a proper, even pressure on the glass. 27 A<br />

similar device had been described by Beeckman, who added that it was a<br />

23<br />

OC1, 242. He distributed several telescopes of this design during the next decade. (OC1, 242; OC13,<br />

264n3; OC4, 132-3; OC4, 224, 228-9)<br />

24<br />

OC17, 293-304.<br />

25<br />

OC17, 294. “altijdt redelijck nat gehouden om te beter de stof te bewaren. doch in ’t eerst niet al te veel<br />

waters, want <strong>and</strong>ers stoot het aen. altijdt dencken om gelijck te drucken, en dickwils de h<strong>and</strong> af gelicht en weer<br />

gelijck aen geset. ’t is best alleen te sijn. … De <strong>and</strong>ere sijde sleep ick eerst eens mis: daer de oorsaeck van<br />

was, of dat ick in ’t eerst te veel water nam of dat ick niet op de goeije plaets en polijsten. ick<br />

verbeterdense eerst wat met op de rechte plaets noch eens te polijsten; daer nae met noch meer polijsten<br />

wierd het weer erger.”<br />

26<br />

The earliest lenses that remain – one in the Utrecht University Museum <strong>and</strong> two at Boerhaave Museum<br />

in Leiden – are not very good. Their fame as lens makers stems from the 1680s. Anne van Helden, “Lens<br />

production”, 75-78; Anne van Helden, Collection, IV; 22.<br />

27<br />

OC17, 299.

1655-1672 - DE ABERRATIONE 59<br />

technique used by mirror-makers. 28 It is<br />

not known where Huygens got the idea.<br />

Such tools for improving the<br />

grinding process had been thought up<br />

earlier, in particular by the most<br />

prestigious lens makers. In Galileo’s<br />

workshop – reigning until the 1640s – a<br />

lathe was introduced that permitted<br />

greater precision than was attained by<br />

ordinary spectacle makers. 29 During the<br />

1660s, the Campani brothers in Rome<br />

became the undisputed masters of the<br />

art. They used a range of machines of<br />

their own design, producing lenses<br />

unsurpassed until the eighteenth<br />

century. The Huygens brothers kept a<br />

close eye on developments like these <strong>and</strong> around 1665 references to a type of<br />

lathe designed by Campani turned up in their writings. The quality of lenses<br />

seems to have depended to some degree on the lens maker’s inventiveness to<br />

convert laborious <strong>and</strong> unsteady h<strong>and</strong>iwork into reliable tools.<br />

Huygens had learned how to make lenses. He knew the limitations of the<br />

art <strong>and</strong> of its products. Even the best lenses might suffer from flaws like<br />

bubbles, irregular density, faults in shape, etcetera. In the end, the proof of<br />

the pudding was in the eating. The quality of telescopes was determined by<br />

trial, sometimes literally. Campani beat Divini early 1664 by a series of<br />

carefully organized ‘paragoni’: open contests in which printed sheets at a<br />

distance were read by means of the instruments of both competitors. 30<br />

Figure 22 Beam to facilitate lens grinding.<br />

Alternative configurations<br />

Besides the quality of lenses, telescopes could be improved by configuring<br />

lenses alternatively. Kepler could not have known that the configuration of<br />

two convex lenses he discussed in Dioptrice had several advantages over the<br />

Galilean one. The positive focus that made possible the micrometer has<br />

already been discussed in the previous chapter. Scheiner, who used a<br />

Keplerian telescope to project images of the sun, discovered by coincidence<br />

that it had a wider field of view. There are indications that Fontana was the<br />

first to put Kepler’s idea into practice, although Scheiner was the first to<br />

mention using it. 31 Around 1640 Fontana was the first to challenge Galileo’s<br />

dominance in the trade <strong>and</strong> he did so with Keplerian telescopes. Around that<br />

time, the Galilean configuration was beginning to reach the limits to which<br />

its power could be increased without the field of view becoming too small.<br />

28<br />

Beeckman, Journal, III, 232.<br />

29<br />

Bedini, “Makers”, 108-110; Bedini, “Lens making”, 688-691.<br />

30<br />

Bonelli, “Divini <strong>and</strong> Campani”, 21-25.<br />

31<br />

Van Helden, “Astronomical telescope”, 20-25. Compare Malet, “Kepler <strong>and</strong> the telescope”, 120.

60 CHAPTER 3<br />

The inversion of the picture a Keplerian telescope produced could be<br />

overcome by adding a third lens. Given the quality of the earliest lenses it<br />

was not advisable to ‘multiply’ glasses, but by the 1640s multi-lens telescopes<br />

were beginning to become acceptable. With the increase of length <strong>and</strong><br />

magnification, however, the field of a Keplerian telescope also became<br />

narrow. For example, the 23-foot telescope that Huygens used in his<br />

observations of Saturn had a field of only 17'. It could not display the entire<br />

Moon at once. The limited field of view could be overcome by adding a field<br />

lens. The Augsburg telescope maker Johann Wiesel was probably the first to<br />

make telescopes with such compound oculars. 32 In a letter of 17 December<br />

1649, Wiesel described a four-lens telescope. It had an eyepiece consisting of<br />

two plano-convex lenses fitted in a small tube. The eyepiece tube was<br />

inserted in a composition of tubes which held an objective lens at the far side<br />

<strong>and</strong> a plano-convex ocular which acted as a field lens. Wiesel added:<br />

“Sir you may bee assured this is y. e first starrie tubus wch I have made of this manner &<br />

so good yt it goes farre beyond all others wherof my selfe also doe not little rejoyce.” 33<br />

The fame of Wiesel’s telescopes spread quickly <strong>and</strong> throughout Europe<br />

telescope makers tried to equal them. On a visit to his relative Edelheer in<br />

Antwerpen on New Year’s eve 1652, Huygens saw a Wiesel telescope <strong>and</strong><br />

was very impressed. It was a four-lens telescope, probably comparable to one<br />

Wiesel described in 1649. Towards the end of 1654 Huygens acquired two<br />

letters written by Wiesel - one to his cousin Vogelaer - describing the<br />

construction <strong>and</strong> use of several optical instruments. 34 In the first letter a sixlens<br />

telescope was described, which could be used for both terrestrial <strong>and</strong><br />

celestial purposes. Wiesel was an artisan, a very good one with an unmatched<br />

underst<strong>and</strong>ing of lenses <strong>and</strong> their configurations. His was another kind of<br />

underst<strong>and</strong>ing than the dioptrical theory Huygens developed in 1653. This<br />

kind of experiential knowledge Huygens acquired the following years in his<br />

practical dioptrics. Then, some ten years later, in an artisanal manner<br />

Huygens made his own compound eyepiece with excellent dioptrical<br />

properties.<br />

Experiential knowledge<br />

Telescope makers had a great deal of knowledge of dioptrics, as witnessed by<br />

the fruits of their labor that are preserved. Like the process of production,<br />

the thinking behind these products is more difficult to retrieve. It is barely<br />

documented as craftsmen in general were reluctant to reveal the secrets of<br />

their trade. There is reason to believe that their knowledge of dioptrics was<br />

of a different kind than that of mathematicians. That much we can infer<br />

from what little material that has been preserved. Lens makers knew very<br />

32<br />

Van Helden, “Compound”, 27-29; Keil, “Technology transfer”, 272-273. They are first mentioned in<br />

Rheita’s Oculus Enoch et Eliae (1645), who referred his readers to Wiesel. For the relationship between<br />

Rheita <strong>and</strong> Wiesel see Keill, Augustanus Opticus, 66-77.<br />

33<br />

Van Helden, “Compound eyepieces”, 34. The entire letter is reproduced on 34-35.<br />

34<br />

OC1, 308-311.

1655-1672 - DE ABERRATIONE 61<br />

well how to grind lenses to suitable proportions <strong>and</strong><br />

configure appropriately. The nature of this knowledge<br />

<strong>and</strong> to what extent they understood the dioptrical<br />

properties of lenses will be discussed now. Note that<br />

this is a discussion of very limited scope, determined<br />

by the considerations <strong>and</strong> activities of Huygens, that<br />

passes over the a wealth of historical knowledge that<br />

can be, <strong>and</strong> has been, gathered regarding the telescope<br />

making trade. 35<br />

A booklet on spectacles written in 1623 gives an<br />

indication of the underst<strong>and</strong>ing their manufacturers<br />

had of glasses. Uso de los antojos was written by the<br />

Andalusian licentiate Daza de Valdez. It explained<br />

how to choose glasses of appropriate strength for a<br />

patient. Daza described a procedure to determine the<br />

‘grado’ of a given lens (Figure 23). 36 He drew two solid<br />

circles X <strong>and</strong> Q of unequal diameter on a sheet of<br />

paper as well as a specific scale at one of the circles. A<br />

glass was then positioned on the scale in such a way<br />

that both circles were seen equally large. The position<br />

of the lens on the scale gave its ‘grado’. 37 Daza did not<br />

explain the method, he only described how it was<br />

employed. It was a practical procedure that did not require any<br />

underst<strong>and</strong>ing of its effect on rays.<br />

A manuscript written around 1670 in Rome by a certain Giovanni<br />

Bolantio contains a similar kind of procedural knowledge. It discussed the<br />

manufacture of telescopes <strong>and</strong> probably recorded the daily routine in some<br />

workshop. 38 It contains two tables listing the ocular <strong>and</strong> objective lenses<br />

needed to make a telescope of a specific strength, characterized by its length.<br />

The lenses are characterized by the doubled radius of the pattern in which<br />

they were ground. 39 Figure 23 Daza’s scale<br />

With these tables at h<strong>and</strong>, the workman could choose the<br />

patterns needed to make a telescope on order. Bolantio did not explain<br />

whether he had constructed the tables himself nor how they were made.<br />

Some dioptrical rules are implicit in them. For example, the length of a<br />

35<br />

See for example the recent, formidable study on Wiesel by Inge Keill which may serve as a guide to<br />

themes <strong>and</strong> literature: Keil, Augustanus Opticus.<br />

36<br />

Daza, Uso de los antojos, 137-140. It appears that this classification in terms of ‘degrees’ was, at that time,<br />

replacing an older one in terms of the common age of someone bearing spectacles of a particular<br />

strength. The ‘grados’ Daza employs seem to be identical with the ‘punti’ Garzoni mentions in his<br />

discussion of the craft in La piazza universale (1585). See also: Pflugk, “Beiträge”, 50-55.<br />

37<br />

Daza did not explain how the scale on the paper was established. Von Rohr has given an alluring<br />

suggestion as to how such a scale might be construed. Spectacle makers knew that multiple lenses of a<br />

given strength could be substituted by a stronger one to reach the same effect. Thus the first position on<br />

the scale was determined by a weak lens <strong>and</strong> the other positions determined by the amount of equal lenses<br />

which had to be put in those positions. Von Rohr, “Versuch”, 4.<br />

38<br />

Bedini & Bennet, “Treatise”.<br />

39<br />

Bedini & Bennet, “Treatise”, 120-121.

62 CHAPTER 3<br />

Keplerian telescope is set equal to the doubled radius of its objective lens,<br />

which - correctly - implies that the focal distance of the objective is twice the<br />

radius of its convex side. Another table prescribed the size of the aperture<br />

for a given objective, in each case 1<br />

80<br />

th of the tube’s length. Bolantio<br />

explained that the objective should be partially covered by a ring so that no<br />

light fell on the interior of the tube, which apparently implied the ratio used<br />

in the table. 40 Whatever the dioptrical underst<strong>and</strong>ing implicit in Bolantio’s<br />

manuscript, it was presented in a procedural, how-to style that did not<br />

require further theoretical knowledge. 41<br />

Some telescope makers published their own observations, to promote<br />

their products. 42 They did not publish the secrets of their art, as their<br />

revenues depended on them. Information on the manufacture of lenses <strong>and</strong><br />

telescopes could be found in books that were mostly written by scholars.<br />

Examples are Telescopium (1618) by Girolamo Sirtori, Selenographia (1647) by<br />

Hevelius <strong>and</strong> La Dioptrique oculaire (1672) by Cherubin d’Orleans. In 1685,<br />

Huygens wrote a treatise on lens grinding in Dutch, Memorien aangaende het<br />

slijpen van glasen tot verrekijckers, published posthumously in Latin in the<br />

Opuscula posthuma (1703). 43 Memorien was the elaboration of notes like the one<br />

cited above. Huygens described the process of lens making as a set of<br />

directives, procedures, tips <strong>and</strong> tricks. No attempt is made to explain why<br />

things work as they work: for example a geometrical account of the grinding<br />

device is absent. Memorien supplied the kind of experiential knowledge also<br />

found in Bolantio’s manuscript: a description of skills Huygens had acquired<br />

through long-time practice.<br />

To what extent a telescope maker like Campani understood the dioptrics<br />

implicit in tables like those in Bolantio’s manuscript cannot be determined.<br />

First rank, specialized telescope makers like Divini <strong>and</strong> Campani had<br />

received some formal education, so they may have been able to read <strong>and</strong><br />

study a book like Dioptrice. It remains to be seen whether a question like this<br />

is relevant at all. I doubt whether dioptrical knowledge would have been of<br />

any use in the design <strong>and</strong> manufacture of telescopes. They knew very well<br />

the effect of diverse types of lenses, but this probably was experiential<br />

knowledge. Innovative craftsmen like Wiesel were able to find new<br />

configurations with improved properties. These are likely to have been the<br />

product of trial <strong>and</strong> error. It has been said that Kepler’s configuration was<br />

the only contribution from the theory of dioptrics to the improvement of the<br />

telescope. 44 Still, its advantages had to be discovered in practice. The<br />

40<br />

Bedini & Bennet, “Treatise”, 117.<br />

41<br />

Willach discusses dioptrical theory emerging from the correspondence of Rheita en Wiesel which<br />

suggests similar lines. Willach, “Development of telescope optics”, 390-394.<br />

42<br />

For example: Fontana’s Novae coelestium (1646) <strong>and</strong> Campani, Lettere di Giuseppe Campani intorne all'ombre<br />

delle Stelle Medicee (1665).<br />

43<br />

OC21, 252-290.<br />

44<br />

Van Helden, “The telescope in the 17th century”, 44-49.

1655-1672 - DE ABERRATIONE 63<br />

improvement of the telescope was the result of the artisanal process of trial<strong>and</strong>-error.<br />

Better configurations were designed by making them, not made by<br />

designing them.<br />

3.1.2 INVENTIONS ON TELESCOPES BY HUYGENS<br />

After Tractatus followed a decade of practical dioptrics, that was crowned by<br />

the publication of Systema Saturnium. Together with his brother, Huygens had<br />

become a skilled telescope maker <strong>and</strong> could already pride himself on some<br />

innovations of the instrument. In the previous chapter, one of these<br />

innovations has been discussed: a device to make telescopic measurements.<br />

It is not known how Huygens discovered the principle of the micrometer.<br />

The discovery was probably related to an innovation of the telescope he had<br />

developed somewhat earlier: the diaphragm.<br />

The diaphragm improved the way images were enhanced by blocking part<br />

of the light entering the telescope. Early in 1610 Galileo discovered that<br />

telescopic images became more distinct when he covered the objective lens<br />

with a paper ring. 45 He determined the optimal size <strong>and</strong> shape of the ring by<br />

means of trial-<strong>and</strong>-effort <strong>and</strong> did not try – at least not on paper – to explain<br />

the effect dioptrically. As contrasted to such an aperture stop, a diaphragm is<br />

inserted into the focal plane. It has the advantage of diminishing the effect<br />

we call chromatic aberration. In December 1659 Huygens first employed a<br />

diaphragm in his 23-foot telescope. 46 As he related in 1684:<br />

“N.B. In 1659 in my system of Saturn, I have taught the use of placing a diaphragm, as<br />

it is called, in the focus of the ocular lens, without which those telescopes cannot be<br />

freed from the defects of colors.” 47<br />

Apparently, he recognized the combining a diaphragm with some measuring<br />

device a bit later. 48 The fact that an object inserted in the focal plane casts a<br />

sharp shadow over things seen through the telescope seems a logical<br />

consequence of Huygens’ underst<strong>and</strong>ing of the dioptrics of a Keplerian<br />

configuration. Still, it took him some time to recognize its usefulness <strong>and</strong> this<br />

may well have been a chance discovery. The fact that a diaphragm reduces<br />

‘the defects of colors’ did not follow from his dioptrical theory <strong>and</strong> had to be<br />

discovered in practice.<br />

Until the 1660s, Huygens’ approach to telescope making did not differ<br />

substantially from that of an ordinary craftsman. We have seen his<br />

unmatched underst<strong>and</strong>ing of dioptrical theory but it cannot be told what role<br />

it played in his practical pursuits. In Systema saturnium, he described his<br />

micrometer in a procedural way, without explaining it analytically in<br />

dioptrical terms. The book contained only one dioptrical passage. He wrote<br />

45<br />

Bedini, “The tube of long vision”, 157-159.<br />

46<br />

OC15, 56.<br />

47<br />

OC13, 826. “N.B. me anno 1659 in Systemate Saturnio meo docuisse usum diaphragmatis quod vocant,<br />

in foco ocularis lentis ponendi, absque quo colorum vitio haec telescopia carere non poterant.” In 1694 he<br />

explicitly claimed that he was the first to use a diaphragm: OC13, 774.<br />

48<br />

McKeon, “Les débuts I”, 237.

64 CHAPTER 3<br />

that the power of a telescope could better be determined by calculation than<br />

using the ordinary ways of comparison. He referred to a theorem in<br />

“Dioptricis nostris”: the magnification is equal to the proportion of the focal<br />

distances of objective <strong>and</strong> ocular. 49<br />

Figure 24 Huygens’ eyepiece. (see also the diagram in Figure 25)<br />

The year 1662 marks a turn in Huygens’ dioptrics. He invented something<br />

new <strong>and</strong> then turned to dioptrical theory again. The invention was a<br />

particular configuration of three lenses in a compound ocular (Figure 24).<br />

Nowadays called ‘Huygens’ eyepiece’, it had considerable advantages over<br />

earlier solutions: it produced a large field of view <strong>and</strong> images that suffered<br />

relatively little from aberrations. 50 Huygens had developed the eyepiece after<br />

his trip to Paris <strong>and</strong> London in 1660-1, where he had talked much on<br />

telescopes <strong>and</strong> related matters. In Paris he had seen the artisan Menard <strong>and</strong><br />

the ingeneer Pierre Petit, who had the best collection of telescopes in Paris.<br />

In London he saw telescopes with compound eyepieces made by the<br />

telescope makers Paul Neile <strong>and</strong> Richard Reeve. 51<br />

In 1662, Huygens made his first telescopes with field lenses. Later that<br />

year, he found out what configuration of lenses produced bright images <strong>and</strong><br />

a wide field. On 5 October he wrote to his brother Lodewijk in Paris:<br />

“As for oculars, you will see that I have found something new that causes that<br />

distinctness in daytime telescopes [i.e. terrestrial], <strong>and</strong> the same thing in the very long<br />

ones, while giving them at the same time a wide opening.” 52<br />

Huygens’ design quickly became known <strong>and</strong> was adopted widely. How<br />

Huygens had found the precise configuration is unknown, yet everything<br />

points at it being a matter of trial-<strong>and</strong>-error inspired by the examples he had<br />

seen. 53<br />

After the invention, however, Huygens did something others like Wiesel<br />

<strong>and</strong> Reeve did not do. He set out to underst<strong>and</strong> how it worked by analyzing<br />

the dioptrical properties of his eyepiece. Huygens described its configuration<br />

in a proposition inserted in the third part of Tractatus. 54<br />

49<br />

OC15, 230-233.<br />

50<br />

Van Helden, “Compound eyepieces”, 33; Van Helden, “Huygens <strong>and</strong> the astronomers”, 158.<br />

51<br />

OC22, 568-576.<br />

52<br />

OC4, 242-3: “car pour les oculaires vous voyez bien que j’y ay trouvè quelque chose de nouveau, qui<br />

cause cette nettetè dans les lunettes du jour, et de mesme dans les plus longues, leur donnant en mesme<br />

temps une gr<strong>and</strong>e ouverture.”<br />

53<br />

Van Helden, “Compound eyepieces”, 33.<br />

54<br />

OC13, 252-259. The text in Oeuvres Complètes is probably from 1666. The notes contain some previous<br />

phrasing, probably from 1662. OC13, 252n1

1655-1672 - DE ABERRATIONE 65<br />

“Although lenses should not be multiplied without necessity, because much light is lost<br />

due to the thickness of the glass <strong>and</strong> the repeated reflections, experience has shown it is<br />

nevertheless useful to do so here.” 55<br />

When the single ocular lens is replaced by two lenses, so Huygens continued,<br />

the field of the telescope can be enlarged. Moreover, the images produced<br />

are less deformed <strong>and</strong> the irregularities of the lenses are less disturbing.<br />

The precise configuration<br />

of the eyepiece was as follows<br />

(Figure 25). AB is the<br />

objective lens, CD <strong>and</strong> EF<br />

form the eyepiece; the focal<br />

distances are LG, KT, <strong>and</strong> SH<br />

respectively. Now, KT is<br />

about four times SH, <strong>and</strong> the<br />

distance KS between the ocular lenses is about twice the focal distance SH of<br />

the outer one. Finally, the focus G of the objective lens AB should fall<br />

between the outer ocular EF <strong>and</strong> its focus H in such a way that H is the<br />

‘punctum correspondens’ of point G with respect to lens CD. Rays coming<br />

from a distant point Q will therefore be parallel after refraction in the outer<br />

ocular lens EF. Having determined the position of the eye M <strong>and</strong> the<br />

magnification by the system, Huygens concluded by explaining that points P<br />

<strong>and</strong> Q of the object are seen sharp but reversed.<br />

Huygens had demonstrated that this configuration produced sharp,<br />

magnified images. This was a rather straightforward application of the theory<br />

he had developed in 1653. The text bears witness to the fact that Huygens<br />

had gained much experience with actual lenses since the days of Tractatus. At<br />

one point in the theorem, he indicated why images do not suffer much from<br />

the irregularities in the lenses. Because the eye is so close to the outer ocular,<br />

“the spots <strong>and</strong> tiny bubbles of air, that are always in the material of the glass, cannot be<br />

perceived in lens EF. But one does not see them in lens CD either, because the eye<br />

perceives objects placed there confusedly, but those that are located close to H<br />

distinctly.” 56<br />

Huygens did not, however, explicitly compare the field of his configuration<br />

to that of a telescope with a single ocular, nor did he explain why images<br />

were less deformed. 57 His analysis offered a dioptrical underst<strong>and</strong>ing of his<br />

eyepiece but it did not improve it:<br />

55 OC13, 252-253. “Quanquam lentes non frustra sint multiplic<strong>and</strong>ae, quod et vitri crassitudine et iteratis<br />

reflexionibus non parum lucis depereat; hic tamen utiliter id fieri experientia docuit.”<br />

56 OC13, 256-257. “Atque ex hac oculi propinquitate sit primum ut naevi, seu bullulae minutissimae,<br />

quibus vitri materia nunquam caret, in lente EF percipi non possint. Sed neque in lente CD; quoniam<br />

oculus confuse cernit quae hic objiciuntur, distincte vero quae ad H.”<br />

57 He developed a systematic theory of the field of view of a telescope much later, after 1685: OC13, 450-<br />

461, 468-73.<br />

Figure 25 Diagram for the eyepiece, accompanying<br />

Figure 24.

66 CHAPTER 3<br />

“We give here, if not the best combination of all lenses, the investigation of which<br />

would take long <strong>and</strong> might be impossible, but one which experience has shown us to be<br />

useful.” 58<br />

The particular configuration of Huygens’ eyepiece was a product of trial-<strong>and</strong>error,<br />

<strong>and</strong> theory could not, or not yet, add to that. Huygens the scholar had<br />

not yet been able to assist Huygens the craftsman.<br />

As contrasted to other telescope makers, however, Huygens was able to<br />

underst<strong>and</strong> retrospectively <strong>and</strong> in mathematical terms, what he was doing<br />

when configuring lenses. That is to say, he understood the dioptrical<br />

properties of lenses <strong>and</strong> their configurations. He could explain whether <strong>and</strong><br />

how a configuration of lenses produced sharp, magnified images. But he<br />

could not explain everything of the kind. In another proposition found in<br />

part III of Tractatus <strong>and</strong> apparently following the one discussed above,<br />

Huygens discussed a telescope with an erector-lens such as Kepler had<br />

proposed. 59 He concluded with some remarks about the quality of images<br />

produced by various configurations. With a telescope consisting of a convex<br />

objective <strong>and</strong> a concave ocular – the Galilean configuration – images are<br />

more distinct “<strong>and</strong> defiled by no colored rims that can hardly be prevented<br />

in this composed of three lenses.” 60 A well-chosen combination of lenses<br />

could counter these defects, but<br />

“different people combine ocular lenses differently with regard to each other, looking<br />

for the best combination with only the guide of experience. It would not be easy, to be<br />

sure, to teach something about this that is grounded in certainty, since the consideration of<br />

colors cannot be reduced to the laws of geometry, ….” 61 [italics added]<br />

In his practical work Huygens had found out that lenses suffered from all<br />

kinds of defects. Some of these eluded dioptrical analysis. But he had also<br />

found out that nuisances caused by fogs, bubbles <strong>and</strong> colors could be<br />

diminished. The diaphragm had already proven this. His eyepiece gave<br />

another means to improve the quality of images. 62 He could not fully explain<br />

its advantages, nor could he improve it by means of dioptrical analysis. Still,<br />

the eyepiece had proven that a well-chosen configuration of lenses could be<br />

advantageous. And it made him realize that even better configurations could<br />

be found, even though he was as yet pessimistic about such an enterprise. If<br />

Huygens the scholar could gain a thorough underst<strong>and</strong>ing of the defects of<br />

58 OC13, 252-253. “Dabimus autem in his, etsi non omnium optimam lentium compositionem, quam<br />

investigare longum esset ac forsan impossibile, at ejusmodi quam nobis experientia utilem esse ostendit.”<br />

59 OC13, 258-265. Discussed above, section 2.1.2..<br />

60 OC13, 262-263. “… res visas, atque etiam distinctiores efficere, nullisque colorum pigmentis infectas<br />

quod in hic lentium trium compositione aegre vitari potest.”<br />

61 OC13, 264-265. “Alij vero aliter lentes oculares in his inter se consociant, sola experientia duce quid<br />

optimum sit quaerentes. nec sane facile foret certa ratione aliquid circa haec praecipere, quum colorum<br />

consideratio ad geometriae leges revocari nequeat, …”<br />

62 A way to reduce colors that was more commonly employed, was to make objective lenses with large<br />

focal distances. These, however, had the drawback that telescopes became very long <strong>and</strong> tubes too heavy<br />

to remain straight. In 1662, it occurred to Huygens that this could be circumvented by making a tubeless<br />

telescope. He realized it much later <strong>and</strong> published a little tract on it, Astroscopia Compendiaria (1684). OC21,<br />

201-231.

1655-1672 - DE ABERRATIONE 67<br />

lenses, he might teach Huygens the craftsmen how to combine lenses in the<br />

best possible way. The next decade he actually set out to do this.<br />

3.2 Dealing with aberrations<br />

According to Hugyens, not all defects of lenses could be explained<br />

dioptrically. One particular defect, however, was subject to the laws of<br />

geometry: spherical aberration. It could therefore be explained <strong>and</strong>, possibly,<br />

prevented. Huygens was not the first to design a solution to prevent the<br />

defects of lenses. Descartes had done so with his elliptic <strong>and</strong> hyperbolic<br />

lenses. Newton would built a mirror telescope in order to avoid the defects<br />

of (spherical) lenses. Huygens was the first to take spherical lenses as a<br />

starting point for a theoretical design, instead of ruling them out beforeh<strong>and</strong>.<br />

In 1665, he began a study of spherical aberration with the intention to design<br />

a telescope consisting of spherical lenses such as to neutralize each others’<br />

aberrations.<br />

The idea that the lenses of a telescope might cancel out their mutual<br />

aberrations had occurred to no-one yet:<br />

“Until this day it is believed that spherical surfaces are … less apt for this use [of<br />

making telescopes]. Nobody has suspected that the defects of convex lenses can be<br />

corrected by means of concave lenses.” 63<br />

The project added a new dimension to Huygens’ dioptrical studies. No<br />

longer did he just want to underst<strong>and</strong> the telescope, but now he also wanted<br />

to improve it by means of dioptrical theory. In so doing, he followed<br />

Descartes’ ideal that the scholar could lead the craftsman, but it had taken on<br />

a different form. Huygens started out with what was practically feasible instead of what<br />

was theoretically desirable. Spherical lenses had been the focus of both his<br />

theoretical investigations <strong>and</strong> his practical activities. It looks like Huygens<br />

now wanted to combine these two sides of his involvement with telescopes.<br />

3.2.1 PROPERTIES OF SPHERICAL ABERRATION<br />

In order to be able to determine an optimal configuration of lenses, Huygens<br />

first had to develop a theory of spherical aberration. The phenomenon had<br />

been known for a long time. In perspectivist theory it was known that a<br />

burning glass does not direct all sunrays to one point. No one, however, had<br />

gone beyond the mere recognition of the phenomenon, <strong>and</strong> its exact<br />

properties had not been studied. Kepler went farthest by pointing out the<br />

connection between a ray’s distance from the axis <strong>and</strong> its deviation from the<br />

focus, but this necessarily remained qualitative. 64 Mathematicians like<br />

Descartes had focused on determining surfaces that did not suffer from such<br />

aberration. With his concept of ‘punctum concursus’ of Tractatus, Huygens<br />

had been the first to take spherical aberration into account in dioptrical<br />

theory, defining the focus as the limit point of intersecting rays. In 1665 he<br />

63 OC13, 318-319. “creditum est hactenus … sphaericae superficies minus aptae essent his usibus, nemine<br />

suspicante vitium convexarum lentium lentibus cavis tolli posse.”<br />

64 Kepler, Paralipomena, 185-186 (KGW2, 168-169). Kepler repeated his insights in Dioptrice.

68 CHAPTER 3<br />

extended this by developing a theory of spherical aberration. He subjected,<br />

so to say, the ‘punctum concursus’ to a closer examination to see how exactly<br />

spherical aberration affected the imaging properties of lenses.<br />

Huygens’ study of spherical aberration<br />

had been preceded by a calculation of rays<br />

refracted by a plano-convex lens he had<br />

apparently carried out in 1653. In Tractatus,<br />

he remarked that rays “reunite somewhat<br />

better, i.e. that the points where they cut<br />

the axis are closer to one point, …, when<br />

the convex surface faces the incident rays,<br />

than when the plane surface faces them.” 65<br />

The 1653 calculation is lost, but was<br />

probably identical to later ones. 66 The result<br />

implied that the orientation of a lens<br />

affected the degree of aberration. In 1665,<br />

Huygens went to see whether the amount<br />

of spherical aberration might deliberately<br />

be decreased by a proper configuration of<br />

lenses. He began a study of spherical<br />

aberration under the heading “Adversaria<br />

ad Dioptricen spectantia in quibus quæritur<br />

aberratio a foco”. 67 After a decade of quiet,<br />

the ‘Adversaria’ was the next chapter of<br />

Huygens’ dioptrical studies.<br />

In ‘Adversaria’ Huygens derived<br />

expressions for the amount of spherical<br />

aberration as it depends upon the<br />

properties of a lens. The rigor familiar from<br />

Tractatus returns immediately. On the basis<br />

of the theorems of Tractatus, he took the<br />

refractions at both faces of the lens as well<br />

as its thickness into account. In the first<br />

calculations Huygens returned to the claim<br />

of 1653. He derived an expression for the aberration of the extreme ray<br />

incident on a plano-convex lens GBC with focal distance GS (Figure 26). A<br />

parallel ray is refracted at the extreme point C of the lens towards T on the<br />

axis. The derivation of the aberration TS is straightforward <strong>and</strong> yields<br />

65 OC13, 82-83. “…, accuratius aliquanto eos propiusque ad unum punctum convenire …, cum superficies<br />

convexa venientibus opposita est radijs, quam si plana ad illos convertatur.” Huygens had also written this<br />

to Gutschoven in his letter of 6 March 1653: OC1, 225. As we have seen above, Flamsteed carried out a<br />

numerical calculation <strong>and</strong> came to the same conclusion, which returned in Molyneux’ Dioptrica nova.<br />

Flamsteed, Gresham Lectures, 120-127. Molyneux, Dioptrica nova, 23-25.<br />

66 OC13, LII (“Avertissement”), those later calculations are on pages 283-287.<br />

67 OC13, 355-375.<br />

Figure 26 Spherical aberration of a<br />

plano-convex lens.

TS = 7 BG, where BG is the thickness<br />

6<br />

of the lens. 68 When the lens is reversed<br />

<strong>and</strong> rays are incident on the plane<br />

side, the aberration becomes<br />

TS = 9<br />

2 BG.69 The aberration is<br />

therefore considerably smaller –<br />

almost four times – when the convex<br />

side faces the incident rays. This time<br />

Huygens went further than the mere<br />

observation that the orientation of a<br />

lens affects the amount of aberration.<br />

The faces of a lens are surfaces with<br />

different radii – infinite in the case of<br />

a plane face. The proportion between<br />

these radii apparently determines how<br />

large the aberration is. Consequently,<br />

an ideal lens can be found by<br />

determining the optimal proportion<br />

of both radii.<br />

To do so, Huygens derived an<br />

expression for the aberration of a<br />

parallel ray HC incident on the<br />

extreme end of a lens IMCB (Figure<br />

27). AB = a <strong>and</strong> NM = n are the radii<br />

of the anterior <strong>and</strong> posterior side <strong>and</strong><br />

BG = b is the thickness of the anterior<br />

half of the lens. The thickness of the<br />

entire lens BM = q can be expressed as<br />

q =<br />

ba<br />

b . The anterior face refracts<br />

n<br />

an extreme ray HC towards P, a little<br />

off its focus R. The posterior face, in<br />

its turn, refracts the extension CF of<br />

ray CP towards D, a little off the focus<br />

E of the lens. Huygens then expressed<br />

the spherical aberration DE of the<br />

extreme ray in terms of the radii of<br />

the faces <strong>and</strong> the thickness of half the<br />

2 2<br />

7nq6anq27aq lens: DE = 2 “…<br />

6(<br />

an) 68 OC13, 357.<br />

69 OC13, 359.<br />

1655-1672 - DE ABERRATIONE 69<br />

Figure 27 Aberration of a bi-convex lens

70 CHAPTER 3<br />

the space on the axis within which all parallel rays are brought together,<br />

which space DE is defined by this rule.” 70 Or, the aberration DE is found by<br />

2 2<br />

7n 6an27a multiplying the thickness of the lens q by the expression 2 ,<br />

6(<br />

an) which only depends on the radii of both faces. The shape of a lens that<br />

produces minimal aberration can be found by determining the minimum of<br />

this expression; this yields a : n = 1 : 6. 71 In this case the aberration of the<br />

extreme ray DE = 15q. Huygens found the same for a bi-concave lens,<br />

14<br />

whereas a converging meniscus lens yielded a meaningless outcome. 72<br />

Satisfied, he summarized the result:<br />

“In the optimal lens the radius of the convex objective side is to the radius of the<br />

convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.” 73<br />

The ‘Adversaria’ provided general expressions for spherical aberration in<br />

terms of the shape of a lens. It contained a set of derivations <strong>and</strong> calculations<br />

without explanation. He did not, for example, point at certain simplifications<br />

he had carried out. The results were not therefore fully exact, as will become<br />

clear later on. Still, it was the most advanced account of spherical aberration<br />

at the time. On the basis of his theory of spherical aberration he went on to<br />

design a configuration of lenses that minimized the ‘aberrations from the<br />

focus’.<br />

A note of clarification needs to be made. Huygens did not yet call the<br />

phenomenon he was investigating spherical aberration. Around 1665,<br />

Huygens referred to it in a general way: “aberration from the focus” <strong>and</strong><br />

“Investigate which convex spherical lens brings parallel rays better<br />

together.” 74 Only much later, when distinguishing the aberration caused by<br />

colors, did he explicitly called it “the aberrations of rays that arise from the<br />

spherical shape of the surfaces”. 75 We should bear this in mind when<br />

interpreting Huygens’ study of aberrations <strong>and</strong> his designs for perfect<br />

telescopes. That is, we do not know for certain what exactly he thought his<br />

design would improve.<br />

Specilla circularia<br />

Before continuing with Huygens, mention has to be made of another study<br />

of spherical aberration. Not because it mattered much for the mathematical<br />

theory of spherical aberration – it did not – but because it approached the<br />

70 OC13, 364. “DE spatium in axe intra quod radij omnes paralleli coguntur, quod spatium DE per regulam<br />

hanc definitur.”<br />

71 OC13, 366-367. Modern methods yield the same result.<br />

72 OC13, 375 <strong>and</strong> 370. In the latter case the solution yields a negative value for the radius of the posterior<br />

side.<br />

73 OC13, 367. “Radius convexi objectivi ad radium convexi interioris in lente optima ut 1 ad 6. EUPHKA. 6<br />

Aug. 1665.”<br />

74 OC13, 280n2. “Quaenam lens sphaerica convexa melius radios parallelos coligat investigare.”<br />

75 OC13, 280-281. “aberrationes radiorum quae ex figura superficierum sphaerica oriuntur”

1655-1672 - DE ABERRATIONE 71<br />

problem central to La Dioptrique in an original way. Moreover, it preceded<br />

Huygens’ study <strong>and</strong> he may have known it in some way. The study is found<br />

in two manuscript copies of Specilla circularia, a tract presumed to have been<br />

written in 1656 by Johannes Hudde, an acquaintance of Huygens. 76 The fact<br />

that Hudde had written on dioptrics was known from his correspondence<br />

with Spinoza. 77 Apparently Spinoza had a copy of Specilla circularia, as he<br />

referred to a ‘small dioptrica’ by Hudde <strong>and</strong> some of his own figures <strong>and</strong><br />

calculations are clearly based on it. In addition, a tract called Specilla circularia<br />

turns up in Huygens’ correspondence in 1656. On 30 May 1656, Van<br />

Schooten wrote that he had recently bought an anonymously published<br />

treatise called Specilla circularia. He supposed it was written by Huygens<br />

“because of its accuracy”. 78 Huygens replied that it was not <strong>and</strong> that he had<br />

never heard of it. 79 He asked for a copy, but it is not clear whether he ever<br />

received one. Huygens corresponded with Hudde on mathematical topics,<br />

but they did not discuss dioptrics. Huygens visited Spinoza several times<br />

around 1665 <strong>and</strong> they discussed dioptrical matters extensively. Whether or<br />

not he knew Specilla circularia, it would not have added to Huygens’<br />

underst<strong>and</strong>ing of spherical aberration. Probably he would not have accepted<br />

Hudde’s analysis <strong>and</strong> conclusions, either.<br />

The main goal of Specilla<br />

circularia was to show there<br />

was no point in striving after<br />

the manufacture of<br />

Descartes’ aspherical lenses.<br />

In practice one legitimately<br />

makes do with spherical<br />

lenses, because spherical<br />

aberrations are sufficiently<br />

small. 80 In order to<br />

substantiate this claim,<br />

Figure 28 Hudde’s calculation of spherical aberration<br />

Hudde employed an original definition of the focus of a lens (Figure 28). AB<br />

is a ray parallel to the axis DNI at distance BF. It is refracted to BI by a convex<br />

surface with radius DN. Choosing DN = 1 <strong>and</strong> an index of refraction 20 : 13,<br />

Hudde calculated the length of NI for various values of BF, concluding that<br />

the smaller BF the larger NI (where I approaches K). 81 Considering<br />

76<br />

The original tract is lost, but has been identified by Vermij with two manuscript copies discovered in<br />

London <strong>and</strong> Hannover. Both are dated 25 April 1656 <strong>and</strong> one gives the name of the author: “Huddenius<br />

consul Amstelodamensis”, which suggests the copy itself was made in or after 1672. Vermij, “Bijdrage”,<br />

27; Vermij <strong>and</strong> Atzema, “Specilla circularia”, 104-107.<br />

77<br />

Spinoza, “Briefwisseling”, 251. Spinoza’s letters contain calculations that are similar to those in Specilla<br />

circularia. The letter can also be found in OC6, 36-39, where it is assumed to be addressed to Huygens.<br />

78<br />

OC1, 422. “propter accurationem”<br />

79<br />

OC1, 429.<br />

80<br />

Vermij <strong>and</strong> Atzema, “Specilla circularia”, 119.<br />

81<br />

Vermij <strong>and</strong> Atzema, “Specilla circularia”, 116: “Ex quibus patet, quanto x sive BF minor est, tanto etiam<br />

punctum I longius distare ab N;”

72 CHAPTER 3<br />

numerically all rays between B <strong>and</strong> F, he calculated the proportion of BF to<br />

Im, through which all refracted rays pass. Seeing that IM is small compared to<br />

BF, Hudde concluded that K could be regarded as the focus. 82 According to<br />

Hudde, the focus was not an exact, geometrical point, but a ‘mechanical<br />

point’, a point that cannot be divided mechanically or whose parts are not<br />

truly discernable. 83 This practical outlook made him reject Descartes’<br />

proposal as superfluous.<br />

Hudde’s study lacked Huygens’ rigor. From a mathematical point of view,<br />

he explained away spherical aberration. He attained ‘practical’ exactness,<br />

rather than mathematical, much in the same way as a Flamsteed or<br />

Molyneux. Hudde called in question whether spherical aberration was as<br />

relevant a problem as Descartes had claimed it to be. In Specilla circularia, he<br />

argued that in practice it was not. The spirit of Hudde’s study of dioptrics<br />

was similar to that of Huygens’: to see what mathematics could teach about<br />

the working of lenses in practice. Hudde’s conclusion was the opposite of<br />

Huygens’. In Huygens’ view, an exact underst<strong>and</strong>ing of the phenomenon<br />

might yield a telescope that actually smoothed aberrations away.<br />

Theory <strong>and</strong> its applications<br />

Sometime after writing the ‘Adversaria’, Huygens elaborated it into a<br />

rounded essay on spherical aberration. It contained his first solution to the<br />

problem his study was aimed at: a configuration of spherical lenses that<br />

neutralized spherical aberration. The essay is found in the Oeuvres Complètes<br />

under the title De Aberratione radiorum a foco. In De Aberratione Huygens<br />

worked up <strong>and</strong> extended his earlier notes. He set up his argument with a<br />

definition of the thickness of a lens <strong>and</strong> several auxiliary propositions. 84<br />

Besides the expressions he had given in the ‘Adversaria’ for the aberration of<br />

extreme rays, he established the relationship between the aberration of an<br />

arbitrary ray <strong>and</strong> its distance from the axis. 85<br />

In the fourth <strong>and</strong> fifth propositions of De Aberratione the results of the<br />

‘Adversaria’ returned. Huygens now explicated the simplifications he had<br />

carried out earlier. He first derived a more exact expression – which I will<br />

not give – for the aberration of the extreme ray incident on the plane side of<br />

a plano-convex lens. When the radius of the convex side is 72 inches <strong>and</strong> the<br />

extreme ray is 1 inch from the axis, this expression yielded an aberration of<br />

31253 inches. Huygens then stated – without proof – that the aberration<br />

1000000<br />

could be found more easily by multiplying the thickness of the lens by 9 , the 2<br />

rule found in the ‘Adversaria’. 86 There is, he admitted, a slight difference<br />

82<br />

Vermij <strong>and</strong> Atzema, “Specilla circularia”, 117: “Unde constat, focum ipsum pro puncto mechanico<br />

tantum habendum esse.”<br />

83<br />

Vermij <strong>and</strong> Atzema, “Specilla circularia”, 114: “Punctum autem mechanicum appello, quod in<br />

mechanicis aut divisible non est, aut cujus partes hic non sunt considerata digna.”<br />

84<br />

OC13, 276-277.<br />

85<br />

OC13, 308-313.<br />

86<br />

OC13, 282-285. Each time he assumed an index of refraction 3 : 2.

1655-1672 - DE ABERRATIONE 73<br />

1 ( 1000000 inches) but this was of no significance in actual telescopes. 87 When<br />

the convex side of the same lens faced the incident rays, the exact calculation<br />

yielded an aberration of 81021 inches. In this case, the easier rule of<br />

10000000<br />

‘Adversaria’ – multiplying the thickness of the lens by 7<br />

6 – gave 81022<br />

10000000 , a<br />

1<br />

difference of only inches. Again, the main goal of this exercise was to<br />

1000000<br />

show that the aberration of a plano-convex lens is least when its convex side<br />

faces the incident rays. 88 Continuing with a bi-convex lens, Huygens sketched<br />

out how the aberration might be calculated exactly, but immediately moved<br />

on to an ‘abbreviated rule’ he had ‘found’. 89 This was the expression of the<br />

‘Adversaria’, found by “ignoring very little quantities, but judiciously so as<br />

needed.” 90 The rule applied to convex as well as to concave lenses <strong>and</strong><br />

yielded the optimal proportion of both radii of 1 : 6. The resulting bi-convex<br />

15 91<br />

times its thickness.<br />

lens produces an aberration of only 14<br />

Surprisingly, these laborious derivations were not of great value for<br />

telescopes. After having explained the optimal proportions of bi-concave<br />

lenses, Huygens wrote that they were not useful as ocular lenses. In<br />

telescopes, he said, one should choose “… other, less perfect lenses, so that<br />

the defects of the convex lens are compensated <strong>and</strong> corrected by their<br />

defects.” 92 Those less perfect lenses were diverging concavo-convex lenses.<br />

Huygens showed that these lenses always produce a larger aberration than biconvex<br />

or bi-concave lenses.<br />

As ocular lenses they could, however, be useful:<br />

“With concave <strong>and</strong> convex spherical lenses, to make telescopes that are better than the<br />

one made according to what we know now, <strong>and</strong> that emulate the perfection of those<br />

that are made with elliptic or hyperbolic lenses.” 93<br />

Here was what Huygens had been looking for: a configuration where lenses<br />

mutually cancel out their aberration. He had designed a telescope in which<br />

the ocular corrects for the aberration of the objective lens, thus equaling the<br />

effect of a-spherical lenses.<br />

The solution was as follows: given an objective lens <strong>and</strong> the required<br />

magnification of a telescope, determine the shape of the ocular lens (Figure<br />

29). On the axis BDFE of lens ABCD, divide the focal distance DE by point F<br />

87<br />

OC13, 284-285. “Exigua quidem differentiola, sed quae in illa lentium latitudine quae telescopiorum<br />

usibus idonea est, nullius sit momenti.”<br />

88<br />

OC13, 284-287.<br />

89<br />

OC13, 290-291. “Et haec quidem methodus ad exactam supputationem adhibenda esset. Invenimus<br />

autem et hic Regulam compendiosam …”<br />

90<br />

OC13, 290-291. “Quae regula … inventa est neglectis minimis, sed necessario cum delectu.”<br />

91<br />

OC13, 290-291& 302-303.<br />

92<br />

OC13, 302-303. “…, sed aliae minus perfectae, quarum nempe vitijs compensantur ac corrigentur vitia<br />

lentis convexae, …”<br />

93<br />

OC13, 318-319. “Ex lentibus sphæricis cavis et convexis telesopia componere hactenus cognitis ejus<br />

generis meliora, perfectionemque eorum quæ ellipticis hyperbolicisve lentibus constant æmulantia.”

74 CHAPTER 3<br />

according to the chosen magnification. The ocular GFH is to be placed in F.<br />

For example, DE : FE = 10 : 1 when the telescope should magnify ten times.<br />

Because the foci of both lenses should coincide, the focal distance of the<br />

ocular lens is given, namely FE. Due to its spherical aberration, the objective<br />

lens does not refract parallel rays KK, CC to E but to N <strong>and</strong> O along KLN <strong>and</strong><br />

CHO. After refraction by the ocular lens rays LM, HI should all be parallel.<br />

This can be accomplished when the aberrations NE, OE are the same for the<br />

objective <strong>and</strong> ocular lenses with respect to the parallel rays KK, CC <strong>and</strong> LM,<br />

HI respectively. With the expression for the aberrations of both lenses,<br />

Huygens could determine the required radii of the ocular lens. The radius of<br />

86 the convex side should be 100 times its focal distance FE, the radius of the<br />

86 94 concave side 272 times FE. Next, he proved that this ocular indeed canceled<br />

out the aberration of the objective. 95 Ergo, the proposed configuration would<br />

produce almost perfect images.<br />

Huygens supplied a table in which he listed telescopes with various<br />

magnifications against the ocular lenses required according to his analysis.<br />

These numerical examples were, so to say, the blueprint by means of which<br />

his design could be realized by any skilled worker. By way of conclusion,<br />

Huygens remarked that the advantages of his design could only be realized<br />

by lenses that were truly spherical. The manufacture of spherical lenses<br />

should therefore be resumed diligently.<br />

Huygens had made clear at the outset of his exposition that the<br />

usefulness of his design was limited. Only a concave ocular could correct for<br />

the aberration of the objective lens. The design was therefore useful only for<br />

Galilean telescopes. In astronomy, telescopes required a convex ocular:<br />

“However, it is certain that this mutual correction is not found in the composition of<br />

convex lenses. On the contrary, the defect of the exterior lens is always a bit augmented<br />

by the ocular lens <strong>and</strong> it cannot be remedied in any way.” 96<br />

Figure 29 Galilean configuration in which spherical aberration is neutralized.<br />

Still, his theory was not entirely useless for the Keplerian telescopes required<br />

for astronomical observation. In the final propositions of De Aberratione,<br />

94 OC13, 320-323.<br />

95 OC13, 324-327. For the rays KK <strong>and</strong> LM – that are not extreme rays – Huygens used the proposition on<br />

the linear proportion between aberration of a ray <strong>and</strong> the square of its distance to the axis. OC13, 308-<br />

313.<br />

96 OC13, 318-319. “Sed certum est in convexis inter se compositis emendationem illam mutuam non<br />

reperiri. Imo contra, vitium exterioris lentis a lente ocularis augetur semper nonnihil neque id ulla ratione<br />

impediri potest.”

1655-1672 - DE ABERRATIONE 75<br />

Huygens examined the means to enhance the quality of images produced by<br />

telescopes with a convex ocular. That is, he took a theoretical look at the<br />

matter. On the basis of this analysis, he could provide directions for<br />

optimizing the quality of telescopes with convex oculars.<br />

The magnifying power of a telescope depends upon the ratio of the focal<br />

distances of objective <strong>and</strong> ocular lenses, <strong>and</strong> can therefore be increased by<br />

reducing the focal distance of the ocular lens. This, however, simultaneously<br />

decreases the clarity <strong>and</strong> distinctness of images. To maintain clarity at the<br />

same time, the opening of the objective lens would have to be made larger. 97<br />

He began by considering a naked eye in front of which a telescope is placed.<br />

Assuming that an equal number of rays should enter the eye when a more<br />

powerful telescope is taken, Huygens argued that the opening should be kept<br />

proportional to the magnification. This implied that his 22-foot telescope<br />

would need an opening of 125 times the area of the pupil. In reality, he<br />

observed, a satisfactory telescope had a much smaller opening, only 15 times<br />

the area of the pupil. Evidently, in astronomical observation one could do<br />

with much smaller clarity. He therefore did not take the eye as starting-point,<br />

but a telescope with satisfactory quality. If the ocular is replaced by an ocular<br />

that magnifies twice as much, the clarity will be four times smaller. The<br />

opening of the objective should therefore be increased accordingly.<br />

Evidently, this cannot be done at will <strong>and</strong> one should “consider accurately<br />

which magnification the opening of the exterior lens can support”. 98<br />

Maintaining the clarity of images does not mean,<br />

however, that their quality is maintained. Increasing the<br />

opening of a lens renders images less distinct. Huygens<br />

made it clear that only experience could tell which<br />

configuration produced satisfactory images. Yet, when such<br />

a telescope is known, theory can explain how the quality of<br />

images is maintained when its strength increases. In his<br />

account, Huygens applied a new conception of spherical<br />

aberration that he had defined in an earlier proposition of<br />

De Aberratione. He called it the ‘circle of aberration’. As<br />

contrasted to the earlier conception, in which the aberration<br />

GD is measured along the axis, the circle of aberration is<br />

measured by the distance ED perpendicular to the axis<br />

(Figure 30). In other words, the circle of aberration is the<br />

spot produced by parallel rays coming from one point of a<br />

distant object. Consequently, the images produced by two<br />

lens systems are equally clear <strong>and</strong> equally distinct when the<br />

respective circles of aberration are the same. 99<br />

Figure 30 ‘Circle’<br />

of aberration.<br />

97<br />

OC13, 332-335.<br />

98<br />

OC13, 336-337. “sed diligenter expendendum quale incrementum exterioris lentis apertura perferre<br />

valeat”<br />

99<br />

OC13, 340-343.

76 CHAPTER 3<br />

Huygens supposed that the circle of aberration XV of a lens system is<br />

mainly produced by the objective lens AB (Figure 31). The ocular lens PO<br />

barely increases the diameter of the circle <strong>and</strong> could therefore be considered<br />

to have a perfect focus. He considered the opening BC of the objective lens<br />

required to maintain a constant circle of aberration when the focal distance<br />

CD of this lens is changed. He proved that the proportion CD 3 : CB 4 should<br />

remain constant. 100 Finally, the quality of images will be maintained upon<br />

changing the ocular lens, when the proportion OD : 4 CD between the focal<br />

distances of both lenses is maintained. 101 Again, Huygens converted these<br />

proportions into a table of numerical values, listing the optimal values of the<br />

focal distances of both lenses <strong>and</strong> the opening of the objective, as well as the<br />

resulting magnification of the system. 102 This table concluded De Aberratione.<br />

Huygens’ theoretical accomplishments in De Aberratione are beyond dispute.<br />

Like the theory of focal distances <strong>and</strong> magnification of Tractatus, his theory<br />

of spherical aberration was rigorous <strong>and</strong> general. And again his theoretical<br />

studies were aimed at underst<strong>and</strong>ing the telescope; in this case, at<br />

underst<strong>and</strong>ing how a system of lenses produces spherical aberration.<br />

Huygens could claim that he understood why an opening of such-<strong>and</strong>-such<br />

dimensions maintained the quality of images.<br />

His results were couched in two tables listing the required components to<br />

make these optimal systems, in a way quite comparable to the ones found in<br />

Bolantio’s manuscript. They prescribed how to assemble a telescope without<br />

presupposing theoretical knowledge of dioptrics. The difference is that<br />

Huygens’ tables were derived from his mathematical theory of lenses instead<br />

of a record of experiential knowledge. The table prescribing the aperture of<br />

telescopes was not gained by some implicit rule of thumb, but was based on<br />

an explicit theorem derived from dioptrical properties of lenses. Huygens<br />

could prove that the openings he prescribed were optimal. Whether this<br />

worked in practice remains to be seen. At least he could claim that he could<br />

calculate beforeh<strong>and</strong> how to adjust the components of a telescope when its<br />

length was changed, thus avoiding a renewed process of trial-<strong>and</strong>-error.<br />

Huygens had realized the goal of De Aberratione. He had demonstrated<br />

that the aberrations of spherical lenses could be made to cancel out.<br />

100 OC13, 342-345.<br />

101 OC13, 348-351.<br />

102 OC13, 350-353.<br />

Figure 31 Aberration produced by a Keplerian configuration.

1655-1672 - DE ABERRATIONE 77<br />

Moreover, he had employed theory to improve the telescope. The design was<br />

still a blueprint, <strong>and</strong> at this point his accomplishments were theoretical only.<br />

He had developed a further underst<strong>and</strong>ing of the properties of spherical<br />

lenses <strong>and</strong> found means to configure them optimally. He had not yet ‘tested’<br />

his designs, nor had he verified his theory as a whole. For example, his<br />

concept of circle of aberration suggests a way to study the observational<br />

properties of spherical aberration, to see whether it correctly described the<br />

defects of lenses. Nowhere, however, did he refer to something of the kind. I<br />

will return to this point below. Huygens’ next step was an attempt to realize<br />

his design of a telescope in which ocular <strong>and</strong> objective lenses cancelled out<br />

their mutual aberrations. A test to the theory?<br />

3.2.2 PUTTING THEORY TO PRACTICE<br />

Not until 1668 did Huygens set about realizing his design. 103 By that time he<br />

lived in Paris, where he had arrived in the summer of 1666. 104 In the<br />

meantime, telescopes had not been out of his mind, though. They were<br />

frequently discussed in his correspondence with Constantijn. He examined<br />

the quality of glass <strong>and</strong> lenses made by Parisian craftsmen, not being<br />

impressed. 105 He was particularly dissatisfied by a telescope he had bought for<br />

his father – a campanine made by one Menard. 106 He equipped a campanine<br />

of his own with lenses made by his brother <strong>and</strong> was pretty contented with<br />

it. 107 In April 1668, he decided to have Constantijn make lenses for the design<br />

of De Aberratione. 108<br />

On 11 May 1668, Huygens gave his brother detailed instructions to grind<br />

a set of lenses. For the objectives Constantijn made – plano-convex lenses of<br />

2 feet <strong>and</strong> 8 inches – a concavo-convex ocular was required. The radii ought<br />

to be 0,187 <strong>and</strong> 0,289 inches, respectively, <strong>and</strong> Huygens drew out the shapes<br />

in his letter. 109 Combined, these lenses would perform like hyperbolic glasses,<br />

he said,<br />

“… because the concave lens corrects the defects arising from the spherical shape of<br />

the objective lens. therefore I cannot determine the opening of the objective that<br />

maybe might be 3 or 4 times larger than an ordinary one has, but if we can just double<br />

it much would be gained <strong>and</strong> the clarity will be sufficiently large for the magnification<br />

of 30.” 110<br />

103<br />

OC13, 303n4; 331n4.<br />

104<br />

OC5, 375; OC6, 23.<br />

105<br />

OC6, 151; 205; 207.<br />

106<br />

OC6, 86-87; 151; 205.<br />

107<br />

OC6, 207.<br />

108<br />

OC6, 209.<br />

109<br />

OC6, 214-215.<br />

110<br />

OC6, 214. “Ce composè, …, doibt faire autant que les verres hyperboliques, parce que le concave<br />

corrige les defauts de l’objectif qui vienent de la figure spherique. c’est pourquoy je ne puis pas determiner<br />

l’ouverture de l’objectif qui peut etre pourra estre 3 ou 4 fois plus gr<strong>and</strong>e qu’a l’ordinaire, mais si nous la<br />

pouvons seulement faire double ce sera beaucoup gaignè et la clartè sera assez gr<strong>and</strong>e pour la<br />

multiplication de 30.”

78 CHAPTER 3<br />

Huygens did not explain the ‘secret’ of his new method to his brother. He<br />

urged him not to tell anyone about the plans. Constantijn responded quickly.<br />

On the first of June, Huygens answered two letters – now lost – his brother<br />

had sent on May 18 <strong>and</strong> 24. 111 Constantijn had sent an ocular with only one<br />

side ground according to his instructions, the other being plane. Apparently,<br />

Constantijn had made some objections to his brother’s design. Huygens did<br />

not agree <strong>and</strong> urged his brother to make a lens exactly to his directives.<br />

Huygens did not await new lenses,<br />

but immediately tried the one<br />

Constantijn had sent him. A week<br />

later he reported on the disappointing<br />

results. When the objective lens was<br />

covered in ordinary fashion, the<br />

Figure 32 Rendering of Huygens’ sketch.<br />

system performed reasonably well. Yet, the system fell short of his<br />

expectations. According to his design, the quality of the image should be<br />

maintained when the whole objective lens was exposed to light. (Figure 32)<br />

“but uncovering the entire glass I see a bit of coloring which leads me to believe that<br />

there is an inconvenience therein, which results from the angle made by the two<br />

surfaces of the objective at the edges. This necessarily causes colors, in such a way that<br />

by making hyperbolic glasses one encounters the same things when making them very<br />

large.” 112<br />

Huygens here tentatively drew an important conclusion. That is, we<br />

recognize that he was on the right track by suspecting that those colors were<br />

inherent to the refraction of rays <strong>and</strong> could not be prevented by hyperbolic<br />

lenses. Moreover, his suggestion that the production of colors could be<br />

linked to the angle of the lens’ surfaces was promising in light of Newton’s<br />

later theory of colors. The remark may have been inspired by a measurement<br />

Huygens had performed in November 1665. 113 Having read Hooke’s account,<br />

in Micrographia, of colors produced in thin films of transparent material, he<br />

set out to determine the thickness of the film, which Hooke had not been<br />

able to do. He pressed two lenses together to produce colored rings. The<br />

colors appear where the two lenses nearly meet, a situation comparable to<br />

the thin rim of a glass lens. Whether this measurement <strong>and</strong> the remark of<br />

1668 are connected is, however, mere speculation. In Micrographia, he also<br />

would have found discussions of prism experiments, <strong>and</strong> the effect of a<br />

prism may also explain the emphasis on the angle between the faces of the<br />

lens at the edge. Whatever be the case, Huygens did not pursue this line of<br />

thinking. He suspected that the proportions of Constantijn’s lens were not<br />

the gist of the problem, “but before assuring that, I would be pleased to<br />

111<br />

OC6, 218-220.<br />

112<br />

OC6, 220-221. “mais en decouvrant tout le verre je vois un peu de couleurs ce qui me fait croire qu’il y<br />

a un inconvenient de costè la, qui provient de l’angle que font les 2 surfaces de l’objectif vers les bords.<br />

qui cause necessairement des couleurs, de sorte qu’en faisant des verres hyperboliques l’on trouueroit la<br />

mesme chose en les faisant fort gr<strong>and</strong>s.”<br />

113<br />

OC17, 341. Huygens’ measurements, as well as the experiments Newton performed at the same time,<br />

are amply discussed in Westfall, “Rings”.

1655-1672 - DE ABERRATIONE 79<br />

carry out the plan with an entire glass, like I have asked you to make for<br />

me.” 114<br />

During the following months, Huygens kept reminding his brother that<br />

he was waiting for the proper lens. 115 He even considered taking up his own<br />

grinding work <strong>and</strong> started looking around in Paris for able craftsmen. 116 On<br />

November 30, he sent his brother additional directives for oculars. 117 On 1<br />

February 1669, Huygens brought his invention to Constantijn’s attention for<br />

the last time: “You don’t talk anymore about the oculars you have promised<br />

me.” 118 This was the final, somewhat aggrieved sentence of a letter in which<br />

he informed his brother of another letter – one he had received from a<br />

certain baron de Nul<strong>and</strong>t, an acquaintance of Constantijn living in The<br />

Hague at that time. 119 The baron was engaged in making telescopes <strong>and</strong> also<br />

had some ideas regarding dioptrical theory. On 20 December 1668, Nul<strong>and</strong>t<br />

had written to Huygens. In the letter of 1 February to his brother, Huygens<br />

wrote:<br />

“The worthy Baron de Nul<strong>and</strong>t begins to talk like a great savant, <strong>and</strong> lets me coolly<br />

know that he has found the same proportions of glasses to imitate the hyperbola of<br />

which I have talked to him in my letter, although I am sure that this is infinitely beyond<br />

his capacities. The calculations he sends me are far from the truth, <strong>and</strong> I will not refrain<br />

from showing him this.” 120<br />

Huygens had told Nul<strong>and</strong>t about his idea of nullifying spherical aberration<br />

by means of spherical lenses in a letter now lost. On 18 January, Nul<strong>and</strong>t had<br />

replied that he had also found that a concave meniscus lens could correct the<br />

aberration of the objective lens, but had not given any details. 121 In that letter,<br />

Nul<strong>and</strong>t calculated the amount of aberration for two lenses <strong>and</strong> had drawn<br />

conclusions that were contrary to Huygens’ own. Huygens’ letters in reply<br />

are lost, but it is clear that he easily convinced Nul<strong>and</strong>t of his mistakes. In his<br />

next letter, Nul<strong>and</strong>t admitted that his configuration for nullifying spherical<br />

aberration was faulty, because he had calculated the aberration of lenses in a<br />

wrong way. 122<br />

114<br />

OC6, 221. “mais devant que de l’assurer je serois bien aise de faire l’essay avec un verre entier, que je<br />

vous ay priè de me vouloir faire.”<br />

115<br />

OC6, 236; 266. He did not show consideration for the fact that Constantijn was getting ready for his<br />

marriage on 28 August 1668.<br />

116<br />

OC6, 266; 300.<br />

117<br />

OC6, 299-300.<br />

118<br />

OC6, 353. “Vous ne parlez plus des oculaires que vous m’avez promis.”<br />

119<br />

Little is known about him. He published an anti-Cartesian treatise Elementa physica in 1669 in which he<br />

included an extract of a letter written by Christiaan (OC6, 420-421). He first appears in a letter to Huygens<br />

of 20 December 1668, which suggests that they had met, probably in Paris. OC6, 304-305.<br />

120<br />

OC6, 353, “Le Seigneur Baron de Nul<strong>and</strong>t commence a parler en gr<strong>and</strong> docteur, et me m<strong>and</strong>e<br />

froidement, d’avoir trouvè les mesmes proportions de verres, pour imiter l’Hyperbole, dont je lui avois<br />

parlè dans ma lettre, quoique je sasche bien que cela passe infiniment sa capacitè. Les calcus qu’il<br />

m’envoye sont trop eloignez de la veritè, et je ne manqueray pas de le lui remontrer.”<br />

121<br />

OC6, 348-351; particularly 350.<br />

122 OC6, 363-367; particularly 364.

80 CHAPTER 3<br />

A new design<br />

We could have passed over this episode with Nul<strong>and</strong>t, if its conclusion had<br />

not coincided with the next phase in Huygens’ study of spherical aberration.<br />

On that same 1st of February, he gloriously<br />

wrote down “A composite lens emulating a<br />

hyperbolic lens. EUPHKA” 123 He had found a<br />

new solution to the problem of neutralizing<br />

spherical aberration that made his earlier one<br />

superfluous. It consisted of a combination of<br />

two lenses that would replace one objective<br />

lens. This composite lens could therefore be<br />

used in telescopes for astronomical<br />

observation, whereas the earlier solution was<br />

useful for terrestrial telescopes only. On<br />

February 22, he asked his brother Lodewijk to<br />

tell Constantijn<br />

“… that I ab<strong>and</strong>on the little ocular I had asked<br />

from him, because I have found something better<br />

<strong>and</strong> more substantial in these matters, that I would<br />

like to try out myself.” 124<br />

Huygens’ idea was as follows (Figure 33).<br />

The bi-concave lens VBC <strong>and</strong> the plano-convex<br />

lens KSTG have the same focus E, with respect<br />

to diverging rays MV coming from M, <strong>and</strong><br />

parallel rays QK, respectively. In addition, the<br />

spherical aberration EN produced by each lens<br />

is the same for these rays. An arbitrary ray QK,<br />

parallel to the axis ASM, is refracted by lens KST<br />

towards point N, a little off its focus E. Lens<br />

VBC, in its turn, refracts a ray MV towards the<br />

same point N, at the same distance from its<br />

focus E. Ray CN is therefore refracted towards<br />

M. As a result, the composite lens brings all<br />

parallel rays QK to a perfect focus M <strong>and</strong> “…<br />

will emulate a hyperbolical or elliptical lens<br />

perfectly.” 125 The system acts as a converging<br />

lens <strong>and</strong> can therefore replace the objective of<br />

any telescope. In his proof, Huygens worked<br />

Figure 33 The invention of 1669<br />

123<br />

OC13, 408. “Lens composita hyperbolicae aemula. EUPHKA 1 Febr. 1669.”<br />

124<br />

OC6, 377. “Vous pourrez luy dire que je le quite pour ce qui est du petit oculaire que je luy avois<br />

dem<strong>and</strong>è, ayant trouvè quelque chose meilleur et de plus considerable en cette matiere, dont j’ay envie de<br />

faire moy mesme l’essay.”<br />

125<br />

OC13, 413. “…[lens] compositae ex duabus VBC, KST, quae Hyperbolicae aut Ellipticae perfectionem<br />

aemulabitur.”

1655-1672 - DE ABERRATIONE 81<br />

the other way around. 126 The bi-concave lens VBC is given. M is the center of<br />

surface BV, so that rays from M are not refracted by it. Surface CB of this lens<br />

refracts a ray MC to KCN, intersecting the axis in N, where EN is the spherical<br />

aberration. The problem is to find a convex lens KST with the same focus E,<br />

which refracts a parallel ray QK, at distance KS to the axis, to the same point<br />

N. Huygens chose BE – nearly equal to GE – as the focal distance of this lens<br />

KST. Its spherical aberration EN is – by the rule from the ‘Adversaria’ – 7<br />

6<br />

times its thickness GS. This length EN is also the spherical aberration of<br />

surface BC of the bi-concave lens VBC. It can be expressed in terms of its<br />

radius AB, the length BG (proportional to the distance CG of the ray to the<br />

axis), <strong>and</strong> the length BM. Equating both expressions for EN, he found a<br />

proportionality between the radius of KST <strong>and</strong> BC. It is 100 to 254, or nearly<br />

2 to 5. In addition MB, the radius of the other surface BV of the bi-concave<br />

lens, has to be twice that of BC or ten times that of KST. At the end of his<br />

calculations Huygens summarized the solution:<br />

“A lens composed of two emulates a hyperbolic lens, the one plano-convex the other<br />

concave on both sides. The radii of the surfaces are nearly two, five, ten.” 127<br />

Five days later, on 6 February, he sent a letter to Oldenburg to which he<br />

appended an anagram containing his ‘important invention’: 128<br />

a bc<br />

d e h i l m n op<br />

r s t u y<br />

52<br />

2 14<br />

1 23<br />

3 1 3 2 23<br />

24<br />

1<br />

This second invention can be regarded as the final piece of Huygens’ project<br />

of canceling out spherical aberration by means of spherical lenses. He had<br />

shown that spherical lenses were indeed apt for telescopes by designing a<br />

configuration that produced an almost perfect focus. As contrasted to the<br />

earlier invention of 1665, this one could improve telescopes used for<br />

astronomical observation. 129 What remained to be done, was to test the<br />

design.<br />

We should remember that it was not an ordinary project Huygens had<br />

embarked on. His theoretical investigations of spherical aberration served<br />

the practical goal of improving actual telescopes. With this he marked<br />

himself off from both theoreticians <strong>and</strong> practitioners. Unlike other telescope<br />

makers – as he manifested himself earlier – he had aimed at improving the<br />

telescope by means of theoretical study. The configuration in which<br />

aberration was to be neutralized was not the result of trial-<strong>and</strong>-error like his<br />

eyepiece, but of mathematical analysis of lenses <strong>and</strong> calculating the optimal<br />

126<br />

OC13, 411-413.<br />

127<br />

OC13, 417n2. “Lens e duabus composita hyperbolicam aemulatur, altera planoconvexa altera cava<br />

utrimque. Semidiametri superficierum sunt proximè duo, quinque, decem.”<br />

128<br />

OC4, 354-355 <strong>and</strong> OC13, 417. The solution of the anagram is: “Lens e duabus composita hyperbolicam<br />

aemulatur”.<br />

129<br />

Huygens may have tested the idea to combine two lenses into an objective earlier, at the time of the<br />

invention of 1665. Hug29, 76v <strong>and</strong> 77r contain sketches reminiscent of the earlier invention as well as<br />

ones reminiscent of the 1669 invetion. The folios can date from any time between the two inventions, but<br />

appear to reflect some intermediate stage in his thinking.

82 CHAPTER 3<br />

combination. He had made a blueprint, a design by which his perfect<br />

telescope should be made, instead of designing one by first making it. Unlike<br />

earlier theorists like Descartes, however, Huygens had not started from the<br />

ideal situation but from the actual materials available to a telescope maker.<br />

He worked halfway between the scholar <strong>and</strong> the craftsman in an<br />

unprecedented effort to combine their respective theoretical <strong>and</strong> practical<br />

goals.<br />

Apparently the earlier, unsuccessful test of his first invention had not<br />

shaken his confidence that spherical lenses could cancel out their mutual<br />

aberrations. He had not changed his theory of spherical aberration, including<br />

the values he used to approximate the amount of aberration produced by a<br />

lens. Did Huygens expect that the composite lens would not produce those<br />

disturbing colors? He may have thought that his new design was of a<br />

different kind. As contrasted to the earlier one, it was not the ocular lens that<br />

canceled out the aberration of the objective lens, but the aberration was<br />

neutralized within the composite objective. This raises the question how<br />

Huygens had hit upon the idea not to consider the configuration of a<br />

complete telescope, but of a single lens system. It may have dawned upon<br />

him when he was pointing out the flaws in Nul<strong>and</strong>t’s statements. It would<br />

indeed be ironical that Huygens would have drawn inspiration for this<br />

remarkable invention from a man he held so low.<br />

Despite the triumphant EUPHKA, little is heard of the invention after<br />

February 1669. On 26 June 1669 he wrote Oldenburg that he had been<br />

working on lenses for a couple of weeks. He pointed out difficulties of<br />

attaining truly spherical figures <strong>and</strong> of the glass available to him. 130 It is not<br />

clear whether he was trying to execute his design or that he was working on<br />

the 60-foot lenses mentioned in several letters of this period.<br />

In the meantime appeals to publish his dioptrical studies were numerous.<br />

On 18 March, Oldenburg warned him not to wait too long: “Sir, allow me to<br />

urge you to be willing to finish your Dioptrique for fear that you will not be<br />

preceded in this by someone else.” 131 At the end of October it was too late.<br />

Barrow published his Lectiones XVIII <strong>and</strong> Oldenburg sent Huygens a copy<br />

on November 21. 132 With the publication of Lectiones XVIII, Huygens lost<br />

priority on a basic accomplishment of Tractatus: the application of the sine<br />

law to spherical lenses. Barrow’s lessons were, as we have seen, of a different<br />

nature than Tractatus. Barrow himself was aware of the differences. In a letter<br />

to Collins, written on Easter Eve 1669, he wrote:<br />

“… had I known M. Huygens had been printing his Optics, I should hardly have sent<br />

my book. He is one that hath had considerations a long time upon that subject, <strong>and</strong> is<br />

used to be very exact in what he does, <strong>and</strong> hath joined much experience with his<br />

130<br />

OC6, 460. In November the Royal Society decided to send Huygens a piece of the excellent glass made<br />

in Engl<strong>and</strong>. OC6, 533 <strong>and</strong> note 5.<br />

131<br />

OC6, 389. “Monsieur permettez moy de vous presser de vouloir acheuer vostre Dioptrique de peur que<br />

vous n’y soyez prevenu de quelque autre.” He warned him again on April 8. OC6, 416.<br />

132<br />

OC6, 534.

1655-1672 - DE ABERRATIONE 83<br />

speculations. What I have done is only what, in a small time, my thoughts did suggest,<br />

<strong>and</strong> I never had opportunity of any experience.” 133<br />

Barrow was too humble about his mathematical abilities but he was right in<br />

observing that Huygens had more ‘experience’ in dioptrical matters. Huygens<br />

praised Barrow in a letter to Oldenburg of 22 January 1670, but added “…<br />

someday you will see that what I have written about it is completely<br />

different.” 134 Yet, he did not hurry. The publication of Lectiones XVIII may<br />

have pushed his plans to the background.<br />

In February 1670 Huygens fell ill <strong>and</strong> he went to The Hague in<br />

September, with an explicit ban by his physician to engage in intellectual<br />

labor. In June 1671 he returned to Paris. Huygens’ dioptrics are not<br />

mentioned among the manuscripts he entrusted to Vernon in February 1670,<br />

when he feared the worst. 135 In Holl<strong>and</strong>, he was with Constantijn again <strong>and</strong><br />

we may speculate that they also discussed dioptrical matters. In general,<br />

Huygens wrote little about dioptrics in these years. He exchanged letters with<br />

de Sluse on Alhacen’s problem, a mathematical problem regarding spherical<br />

mirrors. 136 Much of his correspondence was taken up by a discussion about<br />

the laws of collision he had sent to Oldenburg. No trace is found that<br />

Huygens worked on executing the design of February 1669. Not long after<br />

his return to Paris in June 1671, Huygens received a letter that would<br />

eventually mean the end of his plans.<br />

3.2.3 NEWTON’S OTHER LOOK AND HUYGENS’ RESPONSE<br />

The invention of February 1669 is found on two places in Huygens’<br />

manuscripts. One is his notebook of that period, the other is in the folder<br />

also containing ‘Adversaria’ <strong>and</strong> seems to be the original calculation. 137 Both<br />

contain the sketch of his invention <strong>and</strong> the ‘EUPHKA 1 feb. 1669’. In the last<br />

one, however, the EUPHKA is crossed out <strong>and</strong> a ‘P.S.’ is added: “This<br />

invention is useless as a result of the Newtonian aberration that produces<br />

colors.” 138 Along with his invention, Huygens discarded all parts of De<br />

Aberratione dealing with the improvement of telescopic images, namely his<br />

earlier invention <strong>and</strong> his rules for the opening of keplerian telescopes. He<br />

tore them from his manuscript <strong>and</strong> put them in a cover which said: “Rejecta<br />

ex dioptricis nostris”. 139 The P.S. is dated October 25, without a year, but it is<br />

likely to be 1672. 140 Evidently, this drastic decision was occasioned by the<br />

preceding correspondence with Newton on colors.<br />

133<br />

Rigaud, Correspondence II, 70.<br />

134<br />

OC7, 2-3. “… vous verrez quelque jour que ce que j’en ey escrit est encore tout different.”<br />

135<br />

OC7, 7-13; especially 10-11.<br />

136<br />

Discussed in: Bruins, “Problema Alhaseni”.<br />

137<br />

Hug2, 72r <strong>and</strong> Hug29, 87r respectively.<br />

138<br />

OC13, 409n2. “Hoc inutile est inventum propter Abberationem Niutoniana quae colores inducit.”<br />

139<br />

OC13, 314n1.<br />

140<br />

The editors of the Oeuvres Complètes date it 1673, but in a conversation Alan Shapiro <strong>and</strong> I came to the<br />

conclusion that it must have been 1672. I will return to this on page 92.

84 CHAPTER 3<br />

In a letter of 11 January 1672, Oldenburg first made mention of Newton to<br />

Huygens. 141 This ‘mathematics professor in Cambridge’ had invented a small<br />

telescope in which the objective lens was replaced by a mirror. According to<br />

Oldenburg it represented an object “without any color <strong>and</strong> very distinct in all<br />

its parts.” 142 In his next letter of 25 January, Oldenburg sent him a drawing<br />

<strong>and</strong> a detailed description, <strong>and</strong> asked Huygens’ opinion. 143 At the bottom of a<br />

relatively wide tube a concave mirror reflected rays to a plano-convex ocular<br />

lens via a small plane mirror. Huygens promptly sent Oldenburg his opinion<br />

on the device. In the 81st issue of Philosophical Transactions (15 March, O.S.),<br />

Oldenburg published Newton’s description of his reflector along with some<br />

of Huygens’ comments. 144 In the meantime, Huygens had also sent a letter on<br />

Newton’s telescope to Gallois, the editor of the Journal des Sçavans, who<br />

published an extract of it in the issue of February 29. 145<br />

Huygens spoke in the highest terms of Newton’s telescope. He<br />

enumerated no less than four advantages over ordinary telescopes: a mirror<br />

suffers less from spherical aberration, it does not ‘impede rays at the edge of<br />

the glass due to the inclination of both surfaces’, there is no loss of light due<br />

to internal reflections, <strong>and</strong> inhomogeneities in the material which affect<br />

lenses play no part in mirrors. 146 Figure 34 The crossed out EUREKA.<br />

In short, the reflector was a promising<br />

device. The main obstacle for its success, already pointed out by Oldenburg,<br />

was to find a durable material for making reflecting surfaces which lent itself<br />

141<br />

N.S. All dates are New Style unless indicated otherwise.<br />

142<br />

OC7, 124-125. “… qui envoye l’object à l’oeil, et l’y represente sans aucune couleur et fort<br />

distinctement en toutes ses parties.”<br />

143<br />

OC7, 129-131.<br />

144<br />

OC7, 131 Huygens’ note a; 140-143.<br />

145<br />

OC7, 134-136.<br />

146<br />

OC7, 134-136 (to Gallois); 140-141 (to Oldenburg). In a note added to the description of Newton’s<br />

reflector, Huygens calculated the difference of spherical aberration produced by a spherical lens <strong>and</strong> a<br />

spherical mirror. The aberrations produced by a lens <strong>and</strong> a mirror with the same focal distance <strong>and</strong><br />

aperture are 28 to 3. Therefore, he concluded, the aperture of a mirror can be three times as large. OC7,<br />

132.

1655-1672 - DE ABERRATIONE 85<br />

to good polishing. 147 The second of the advantages Huygens listed is<br />

interesting. Although he did not mention colors, it is clear that he referred to<br />

the observation he had made in 1668. In his letter to Oldenburg he almost<br />

literally repeated it:<br />

“Besides, by [the mirror] he avoids an inconvenience, which is inseparable from convex<br />

Object-Glasses, which is the Obliquity of both their surfaces, which vitiateth the<br />

refraction of the rays that pass towards the sides of the glass, <strong>and</strong> does more hurt than<br />

men are aware of.” 148<br />

In his letter to Gallois, Huygens added that this defect could not be<br />

prevented by a-spherical lenses. Evidently, he still was aware that his earlier<br />

observation was of consequence to the use of lenses. Still, we have no idea<br />

how he thought it would affect his invention of 1669.<br />

Those disturbing colors would eventually induce Huygens to discard his<br />

invention, but not until Newton had convinced him about his own ideas on<br />

their cause. In his letter of 21 March, Oldenburg notified Huygens of a paper<br />

by Newton in the 80th issue of Philosophical Transactions (19 February, O.S.):<br />

“In this print you will find a new theory of Mr. Newton, (…) regarding light <strong>and</strong> colors:<br />

in which he maintains that light is not a similar thing, but a mixture of differently<br />

refrangible rays …” 149<br />

It was, of course, the famous paper in which Newton set forth his ‘New<br />

theory about Light <strong>and</strong> Colors’. According to Newton, rays of different<br />

colors have a different degree of refrangibility: to each color belongs one,<br />

immutable index of refraction. Moreover, he argued that white light is not<br />

homogeneous but a mixture of all colors. Colors therefore are produced<br />

when this mixture is separated, for example by refraction, into its<br />

components. In ‘New theory’, Newton described his experiments with<br />

prisms to substantiate his claim that color is an original <strong>and</strong> immutable<br />

property of light rays which depends solely upon a specific index of<br />

refraction.<br />

Newton also explained why he had developed his reflecting telescope.<br />

After he introduced his idea of different refrangibility, he wrote that it had<br />

made him realize that colors could not be prevented in any lens <strong>and</strong> that<br />

mirrors should be used instead. On the basis of the measurement of the<br />

spectrum produced by one of his prisms, he calculated that the difference<br />

between the refractions of the red <strong>and</strong> blue rays is about a 25th part of the<br />

mean refraction. Consequently chromatic aberration is about a 50th part of<br />

the opening of the lens <strong>and</strong> therefore considerable larger than the spherical<br />

aberration produced by the same lens. 150 After this ‘digression’, Newton went<br />

147 OC7, 134 (to Gallois); 141 (to Oldenburg). Oldenburg had pointed this out to Huygens in the letter<br />

accompanying the description of Newton’s reflector: OC7, 128.<br />

148 Oldenburg’s translation of OC7, 140 in: OldCor8, 520.<br />

149 OC7, 156. “Dans cet imprimé vous trouverez une theorie nouvelle de Monsieur Newton, (…) touchant<br />

la lumiere et les couleurs: ou il maintient, que la lumiere n’est pas une chose similaire, mais un meslange de<br />

rayons refrangibles differemment …” The paper was therefore published in the issue preceding the one<br />

containing the description of his reflector.<br />

150 Newton, Correspondence I, 95.

86 CHAPTER 3<br />

on to lay down his doctrine of the origin of colors in the form of 13<br />

propositions substantiated by the experiments he had described. 151<br />

On 9 April, Huygens gave a first reaction to Newton’s theory:<br />

“… I see that he has noticed like me the defect of the refraction of convex objective<br />

glasses caused by the inclination of their surfaces. As regards his new Theory of colors,<br />

I consider it quite ingenious, but it will have to be seen whether it is compatible with all<br />

experiences.” 152<br />

Two things st<strong>and</strong> out in this comment. In the first place, Huygens was<br />

mainly interested in the significance of Newton’s findings for dioptrics. In<br />

the second place, he seemed to miss the point of the theory of different<br />

refrangibility. 153 In his view, Newton had merely confirmed what he had<br />

observed earlier. Seemingly, he did not realize that Newton’s point was that<br />

chromatic aberration is a consequence of the constitution of light, rather<br />

than the shapes of lenses. In his next letter to Oldenburg, of July 1, Huygens<br />

went more deeply into the matter, though still along the same lines. After<br />

discussing Newton’s telescope a bit further, he wrote:<br />

“As regards his new hypothesis of colors of which you ask my opinion, I admit that it<br />

seems very plausible to me, <strong>and</strong> the experimentum crucis (if I underst<strong>and</strong> it correctly, as<br />

it is described somewhat obscurely) confirms it very much. But I don’t agree with what<br />

he says about the aberration of rays through convex glasses. For while reading what he<br />

writes, I find that following his principles this aberration must be twice as large as he<br />

1<br />

takes it, to wit 25 the opening of the glass, which experience however seems to<br />

contradict. so that this aberration may not always be proportional to the angle of<br />

inclination of rays.” 154<br />

We see what kind of ‘experiences’ Huygens had in mind when he cast doubt<br />

on the validity of Newton’s theory: the colors he had seen in lenses. He did<br />

not believe that chromatic aberration was as large as Newton claimed.<br />

Consequently, Newton’s explanation of the aberration was questionable. But<br />

it does not appear that Huygens had considered Newton’s theory of colors in<br />

much detail. It seems that he had mainly read the part on lenses. He did not<br />

use the term or notion of different refrangibility <strong>and</strong> only talked in terms of<br />

aberrations.<br />

151<br />

Newton, Correspondence I, 96-100.<br />

152<br />

OC7, 165. “… je vois qu’il a remarquè comme moy le defaut de la refraction des verres convexes<br />

objectifs a cause de l’inclination de leurs surfaces. Pour ce qui est de sa nouvelle Theorie des couleurs, elle<br />

me paroit fort ingenieuse, mais il faudra veoir si elle est compatible avec toutes les experiences.”<br />

153<br />

See also: Sabra, Theories of Light, 268-267.<br />

154<br />

OC7, 186. “Pour ce qui est de sa nouvelle hypothese des couleurs dont vous souhaittez scavoir mon<br />

sentiment, j’avoue que jusqu’icy elle me paroist tres vraysemblable, et l’experimentum crucis (si j’entens<br />

bien, car il est ecrit un peu obscurement) la confirme beaucoup. Mais sur ce qu’il dit de l’abberration des<br />

rayons a travers des verres convexes je ne suis pas de son avis. Car je trouvay en lisant son ecrit que cette<br />

1<br />

aberration suivant son principe devroit estre double de ce qu’il la fait, scavoir<br />

25<br />

de l’ouverture du verre, a<br />

quoy pourtant l’experience semble repugner. de sorte que peut estre cette aberration n’est pas tousjours<br />

proportionelle aux angles d’inclinaison des rayons.”

1655-1672 - DE ABERRATIONE 87<br />

Newton realized that Huygens did not grasp the full import of his theory.<br />

Reacting to Huygens’ first comment on his theory, he had written Oldenburg<br />

on 13 April (O.S.):<br />

“Monsieur Hugenius has very well observed the confusion of refractions neare the<br />

edges of a Lens where its two superficies are inclined much like the planes of a Prism<br />

whose refractions are in like manner confused. But it is not from ye inclination of those<br />

superficies so much as from ye heterogeneity of light that that confusion is caused.” 155<br />

This remark was not,<br />

however, communicated<br />

to Huygens. On July 8<br />

(O.S.), Newton replied to<br />

Huygens’ second<br />

comment in a letter<br />

Oldenburg forwarded to<br />

Huygens on 28 July. 156 He acknowledged that the presentation of his theory<br />

might have been obscure for reasons of brevity. Newton also realized that<br />

Huygens had misread his discussion of chromatic aberration. “But I see<br />

not,” he wrote, “why the Aberration of a Telescope should be more than<br />

about 1/50 of ye Glasses aperture”. He included a drawing of the way he<br />

had calculated the proportion (Figure 35):<br />

“Now, since by my principles y e difference of Refraction of y e most difforme rayes is<br />

about y e 24 th or 25 th part of their whole refraction, y e Angle GDH will be about a 25 th<br />

part of y e angle MDH, <strong>and</strong> consequently the subtense GH (which is y e diameter of y e least<br />

space, in to which y e refracted rays converge) will be about a 25 th of y e subtense MH,<br />

<strong>and</strong> therefore a 49 th part of the whole line MN, y e diameter of y e Lens; or, in round<br />

numbers, about a fiftieth part, as I asserted.” 157<br />

The same letter was accompanied by a copy of the 84th issue of Philosophical<br />

Transactions (17 June, O.S.). It contained a letter in which Pardies criticized<br />

Newton’s theory <strong>and</strong> a reply by the latter. Two weeks later, Oldenburg sent<br />

Huygens the next issue of Philosophical Transactions (15 July, O.S.) containing<br />

further correspondence of Pardies <strong>and</strong> Newton on the matter. 158 Pardies, a<br />

Jesuit priest <strong>and</strong> a Parisian acquaintance of Huygens, also criticized Newton’s<br />

claims, but in a more searching manner <strong>and</strong> with a different line of approach.<br />

He questioned the core of Newton’s theory – different refrangibility – <strong>and</strong><br />

raised several objections to his experiments <strong>and</strong> his interpretations thereof.<br />

For example, he initially doubted whether the oblong spectrum could not be<br />

explained by the accepted rules of refraction. 159 He also questioned the very<br />

idea of different refrangibility, which in his view depended upon a<br />

corpuscular conception of light. In his view, colors could also be caused by a<br />

‘diffusion’ of light, for example by a slight spreading of the waves he<br />

155<br />

Newton, Correspondence I, 137.<br />

156<br />

Newton, Correspondence I, 212-213; OC7, 207-208.<br />

157<br />

OC7, 207-208.<br />

158<br />

OC7, 215.<br />

159<br />

Newton, Correspondence 1, 131-132.<br />

Figure 35 Newton’s determination of chromatic aberration.

88 CHAPTER 3<br />

supposed light to consist of. 160 In his two successive replies, Newton clarified<br />

his experiments <strong>and</strong> his claims. He fully convinced Pardies of his claims <strong>and</strong><br />

the father ended the discussion by saying that he was ‘very satisfied’. 161<br />

One expects that by now, late July, it must have dawned upon Huygens<br />

what Newton’s new theory was about. Still, the letter he sent Oldenburg on<br />

27 September does not give the impression that he really grasped the essence<br />

of different refrangibility. 162 He regarded Newton’s replies as a further<br />

confirmation of the theory, but added that things could still be otherwise. It<br />

had, however, dawned upon him that Newton had also something to say<br />

about the nature of light, to wit the compound nature of white light. To this<br />

he raised objections of a different kind:<br />

“Besides, if it were true that the rays of light were, from their origin, some red, some<br />

blue etc., there would still remain the great difficulty of explaining by the physics,<br />

mechanics wherein this diversity of colors consists.” 163<br />

The remark was clearly inspired by the objections Pardies had made. It does<br />

not give the impression that Huygens had given the matter any further<br />

thought. Let it be noted that this was the first moment Huygens raised<br />

objections of a mechanistic nature against Newton, after their discussion had<br />

progressed in several letters, <strong>and</strong> that the objections are not brought out<br />

strongly. He still did not refer to the notion of different refrangibility. As a<br />

conclusion he admitted his misreading of Newton’s discussion of chromatic<br />

aberration.<br />

Huygens’ next letter to Oldenburg, four months later on 14 January 1673,<br />

displayed a drastic change in his attitude towards Newton’s theory. Not only<br />

did he show to have considered the claims about the nature of white light<br />

<strong>and</strong> colors, he also subjected them to a serious critique. 164 In addition, the<br />

tone of his comments became sharper. In his view Newton unnecessarily<br />

complicated matters:<br />

“I also do not see why Monsieur Newton does not content himself with the two colors<br />

yellow <strong>and</strong> blue, because it will be much easier to find some hypothesis by motion that<br />

explains these two differences, than for so many diversities as there are of other colors.<br />

And until he has found this hypothesis he will not have taught us wherein the nature<br />

<strong>and</strong> diversity of colors consists but only this accident (which certainly is very<br />

considerable) of their different refrangibility.” 165<br />

160<br />

Newton, Correspondence 1, 157.<br />

161<br />

Newton, Correspondence 1, 205. “Je suis tres satisfait de la derniere réponse que M. Newton a bien voulu<br />

faire à mes instances.”<br />

162<br />

Sabra, Theories of Light, 270.<br />

163<br />

OC7, 228-229. “De plus qu<strong>and</strong> il seroyt vray que les rayons de lumiere, des leur origine, fussent les uns<br />

rouges, les autres bleus &c. il resteroit encor la gr<strong>and</strong>e difficultè d’expliquer par la physique, mechanique<br />

en quoy consiste cette diversitè de couleurs.”<br />

164<br />

OC7, 242-244.<br />

165<br />

OC7, 243. “Je ne vois pas aussi pourquoy Monsieur Newton ne se contente pas des 2 couleurs jaune et<br />

bleu, car il sera bien plus aisè de trouver quelque hypothese par le mouvement qui explique ces deux<br />

differences que non pas pour tant de diversitez qu’il y a d’autres couleurs. Et jusqu’a ce qu’il ait trouvè<br />

cette hypothese il ne nous aura pas appris en quoy consiste la nature et difference des couleurs mais<br />

seulement cet accident (qui assurement est fort considerable) de leur differente refrangibilitè.”

1655-1672 - DE ABERRATIONE 89<br />

In his view, white light may also be produced by mixing yellow <strong>and</strong> blue<br />

alone. By maintaining that there are only two primary colors, Huygens drew<br />

upon a letter published in the 88th issue of Philosophical Transactions (18<br />

November 1672, O.S.) in which Newton responded to comments by Hooke.<br />

Among other things, Hooke had claimed that two primary colors sufficed to<br />

explain the diversity of colors. Hooke’s comments had not been published<br />

<strong>and</strong> his name was not mentioned in Newton’s reply. Huygens referred to<br />

Hooke’s prism experiments in Micrographia (1665). He suggested an<br />

experiment to verify whether all colors are necessary to produce white light.<br />

Evidently, Huygens had problems with Newton’s claims about the nature of<br />

light. What these problems were exactly, why he would prefer just two<br />

colors, <strong>and</strong> what he meant by ‘explaining by physics, mechanics’ <strong>and</strong> ‘some<br />

hypothesis of motion’ is not explained in the letter. In chapter 6 we will be<br />

able to reconstruct, in retrospect, the background to Huygens’ remarks. He<br />

had a reasonably clear idea what mechanistic explanation ought to be, but it<br />

appears that by 1672 he had not yet elaborated in much detail his conception<br />

of the mechanistic nature of light.<br />

Besides raising objections to Newton’s ideas on the nature of light <strong>and</strong><br />

colors, Huygens summed up his own treatment of chromatic aberration:<br />

“Apart from that, as regards the effect of the different refractions of rays in telescope<br />

glasses, it is certain that experience does not correspond with what Monsieur Newton<br />

finds, because by considering only the distinct picture that an objective of 12 feet makes<br />

in a dark room, one sees that it is too distinct <strong>and</strong> too sharp to be able to be produced<br />

by rays that disperse from the 50th part of the aperture so that, as I believe to have<br />

brought to your attention before, the difference of the refrangibility may not always<br />

have the same proportion in the large <strong>and</strong> small inclinations of the rays on the surfaces<br />

of the glass.” 166<br />

There is no reason to assume that Huygens had not actually performed this<br />

test. 167 We may only wonder why he had not done so in 1665. Then again, the<br />

alleged observations remained qualitative. We may wonder what had<br />

prompted Huygens to consider the issue of chromatic aberration anew.<br />

Assuming that the full import of Newton’s theory had occurred to him as a<br />

result of Pardies’ comments, he may have realized at some time by late 1672<br />

that chromatic aberration was a problem of refraction <strong>and</strong> thus inherent to<br />

lenses. As Newton emphasized in his reply to Hooke: “And for Dioptrique<br />

Telescopes I told you that the difficulty consisted not in the figure of the<br />

166 OC7, 243-244. “Au reste pour ce qui est de l’effect des differentes refractions des rayons dans les verres<br />

de lunettes, il est certaine que l’experience ne s’accorde pas avec ce que trouve Monsieur Newton, car a<br />

considerer seulement la peinture distincte que fait un objectif de 12 pieds dans une chambre obscure, l’on<br />

voit qu’elle est trop distincte et trop bien terminée pour pouvoir estre produite par des rayons qui<br />

s’escarteroient de la 50 me partie de l’ouverture de sorte que, comme je vous crois avoir m<strong>and</strong>è desia cy<br />

devant la difference de la refrangibilité ne suit pas peut estre tousjours de la mesme proportion dans les<br />

gr<strong>and</strong>es et petites inclinations des rayons sur les surfaces du verre.”<br />

167 Because he was a ‘devoted water-color painter’, Shapiro is puzzled about Huygens’ assertion that<br />

yellow <strong>and</strong> blue may produce white, “… because this is contrary to all beliefs about color mixing held in<br />

the seventeenth century.” Shapiro, “Evolving structure”, 223-224. We should bear in mind that Huygens<br />

was also an experienced employer of magic lanterns.

90 CHAPTER 3<br />

glasse but in ye difformity of refractions.” 168 Despite his doubts about the<br />

true properties of different refrangibility, Huygens now recognized that the<br />

disturbing colors in lenses are inherent to refraction. There is no word about<br />

spherical aberration in his letter, <strong>and</strong> he may indeed already have realized at<br />

this point that his project to design configurations to neutralize it had<br />

become useless. Would this help account for the sharpening in his tone?<br />

Newton partially granted Huygens both objections. He dropped the claim<br />

that all colors are necessary to compound white light restricting it now to<br />

sunlight. 169 In his reply of April 3 (O.S.), he strongly objected to Huygens’<br />

claim that two primary colors are more easily explained, but he explicitly<br />

refrained from proposing a ‘Mechanicall Hypothesis”. 170 As regards the actual<br />

effect of chromatic aberration, he watered down his claim a bit. The rays that<br />

are dispersed mostly<br />

“… are but few in comparison to those, which are refracted more Justly; for, the rays<br />

which fall on the middle parts of the Glass, are refracted with sufficient exactness, as<br />

also are those that fall near the perimeter <strong>and</strong> have a mean degree of Refrangibility; So that<br />

there remain only the rays, wich fall near the perimeter <strong>and</strong> are most or least refrangible<br />

to cause any sensible confusion in the Picture. And these are yet so much further<br />

weaken’d by the greater space, through which they are scatter’d, that the Light which<br />

falls on the due point, is infinitely more dense than that which falls on any point about<br />

it. …” 171<br />

As a conclusion, Newton suggested a way to measure the chromatic<br />

aberration of the extreme rays to verify his claims. Huygens accepted<br />

Newton’s argument, but added that<br />

“… he must also acknowledge that this abstraction [dispersion] of rays does not<br />

therefore harm lenses as much as he seems to have wished to be believed, when he<br />

proposed concave mirrors as the only hope for perfecting telescopes.” 172<br />

Huygens was not, however, satisfied with Newton’s rebuttal concerning the<br />

nature of white light <strong>and</strong> colors: “… but seeing that he maintains his opinion<br />

with so much ardor, this deprives me of the appetite for disputing.” 173 Two<br />

weeks later, he wrote to Oldenburg not to send Newton his last letter at all<br />

<strong>and</strong> to tell him only that he did not want to dispute anymore. 174 Newton did<br />

receive the letter anyhow <strong>and</strong> replied on 23 June (O.S.) by a more precise<br />

reformulation of his theory, which was published in the 96th issue of<br />

168<br />

Newton, Correspondence I, 173.<br />

169<br />

Shapiro, “Evolving structure”, 224-225.<br />

170<br />

OC7, 265-266 <strong>and</strong> Newton, Correspondence I, 264-265.<br />

171<br />

OC7, 267 <strong>and</strong> Newton, Correspondence I, 266. In Opticks, he elaborated this argument a bit further <strong>and</strong><br />

mathematically, <strong>and</strong> reduced chromatic aberration to 1/250 of the aperture as contrasted to the original<br />

1/50. Newton, Optical lectures, 429n15.<br />

172<br />

OC7, 302-303. “… mais aussi doit il avouer que cette abstraction des rayons ne nuit donc pas tant aux<br />

verres qu’il semble avoir voulu faire accroire, lors qu’il a proposè les mirroirs concaves comme la seule<br />

esperance de perfectionner les telescopes.”<br />

173<br />

OC7, 302. “…, mais voyant qu’il soustient son opinion avec tant de chaleur cela m’oste l’envie de<br />

disputer.”<br />

174<br />

OC7, 315.

1655-1672 - DE ABERRATIONE 91<br />

Philosophical Transactions (21 July O.S.). 175 Newton invited him once again to<br />

compare by computation aberrations both of lenses <strong>and</strong> mirrors, but<br />

Huygens did not respond anymore.<br />

Thus came an end to a dispute that had run an odd course. In January<br />

1672 Huygens had welcomed the newcomer on the scene of European<br />

scholarship as a kindred spirit in matters dioptrical; in June 1673 he refrained<br />

from discussing any further with someone who so obstinately clung to his<br />

claims. But most striking about the state of affairs I find the relative late<br />

moment at which Huygens recognized the purport of Newton’s paper. Until<br />

the letter of September 1672, the fact that Newton’s theory concerned the<br />

physical nature of light escaped him. And then again, he made only one –<br />

apparently non-committal – objection. Only in the letter of January 1673 did<br />

he engage in a dispute on Newton’s theory of colors, to break it off in the<br />

next letter. Until the letters of Pardies were published, Huygens only paid<br />

attention to what Newton had said about the aberrations of lenses. And even<br />

at this point, he failed to grasp Newton’s message. He only talked of<br />

chromatic aberration in the same terms as he had treated spherical<br />

aberration. One gets a strong impression that in 1672 Huygens lacked a<br />

certain sensibility for the kind of question Newton addressed, namely<br />

concerning the physical nature of light. This is all the more surprising since<br />

Huygens has become famous for a theory explaining the nature of light of<br />

his own.<br />

The preceding reconstruction sheds new light on this famous dispute.<br />

Huygens was not a Cartesian that a priori rejected Newton’s theory for<br />

reasons of its mechanistic inadequacy <strong>and</strong> untenability, like Hooke did <strong>and</strong><br />

Pardies too initially, <strong>and</strong> like he is usually presented in historical literature. 176<br />

We should reconsider the his dispute from the perspective of Dioptrica.<br />

Huygens entered the dispute from his background in dioptrics. He was<br />

interested (<strong>and</strong> informed) in lenses <strong>and</strong> telescopes <strong>and</strong> he had something to<br />

loose. At first he did not look beyond issues directly pertaining to lenses <strong>and</strong><br />

it took some time before he realized what Newton’s theory was about. He<br />

began to raise mechanistic doubts only during the final stages of the dispute,<br />

<strong>and</strong> probably when he realized the consequence for his project of nullifying<br />

spherical aberration. For in anything may explain Huygens relative reluctance<br />

in accepting Newton’s theory, it would be De Aberratione.<br />

Somewhere along the line, Huygens must have realized that Newton’s<br />

findings wrecked his project of perfecting the telescope. He crossed out the<br />

‘Eureka’ of February 1669 <strong>and</strong> discarded a large part of his theory of<br />

spherical aberration. ‘Newtonian’ aberration had rendered his designs<br />

useless. Spherical aberration might be cancelled out by successive lenses,<br />

chromatic aberration could never be prevented. Despite the objections he<br />

raised in it, the letter of January 1673 reveals that Huygens had recognized<br />

175<br />

OC7, 328-333 <strong>and</strong> Newton, Correspondence I, 291-295. See also Shapiro, “Evolving structure”, 225-228.<br />

176<br />

For example Sabra, Theories of Light, 268-272.

92 CHAPTER 3<br />

different refrangibility. I find it reasonable to presume that this had<br />

happened at some time during the preceding months. He demonstrably had<br />

contacts with Pardies, who had accepted the crux of different refrangibility<br />

<strong>and</strong> might have pointed it out to Huygens. Moreover, at that time Mariotte<br />

carried out experiments that corroborated Newton’s results. 177 I find it<br />

therefore likely that Huygens took the drastic decision to discard his project<br />

on 25 October 1672, rather than 1673 as the editors of the Oeuvres Complètes<br />

have it. 178 I find it unlikely that Huygens would have taken the step when the<br />

whole event had long passed. The only fact supporting this interpretation is<br />

the publication of Newton’s <strong>and</strong> Huygens’ letters of 3 April <strong>and</strong> 10 June<br />

respectively in the issue of Philosophical Transactions of 6 October 1673 (O.S.).<br />

Although Huygens had long known their content, further reflection upon<br />

the dispute might conceivably have triggered the decision to discard the main<br />

part of De Aberratione.<br />

3.3 Dioptrica in the context of Huygens’ mathematical science<br />

The drastic decision of October 1672 brought Huygens’ study of lenses to a<br />

temporary end. He was not to resume it, adjusting his theory of aberration<br />

by taking ‘Newtonian’ aberration into account, until his return to Holl<strong>and</strong> in<br />

the 1680s. 179 This would not change the character of his dioptrical studies as<br />

we have come to know it in these two chapters. Although he had lost the<br />

ambition to design a perfect telescope, the orientation on the telescope<br />

guided his dioptrical studies. He conducted his dioptrical studies in order to<br />

underst<strong>and</strong> the instrument. With a Huygens one tends, however, to overlook<br />

the obvious. For him, underst<strong>and</strong>ing something meant mathematically<br />

underst<strong>and</strong>ing something. Together with his orientation on the instrument,<br />

his mathematical approach is the clue to Dioptrica.<br />

In addition to this scholarly contemplation, Huygens had applied himself<br />

to the craft of telescope making. De Aberratione can be seen as an effort to<br />

combine his dual capacities as a scholar <strong>and</strong> a craftsman. In this sense, it<br />

should have been the climax of his involvement with the telescope. It turned<br />

into an anti-climax. In this concluding section on Dioptrica, I first go through<br />

the nature of Huygens’ mathematical approach <strong>and</strong> its consequences for De<br />

Aberratione. Then I consider his orientation on the telescope in the broader<br />

context of our underst<strong>and</strong>ing of Huygens’ science.<br />

3.3.1 THE MATHEMATICS OF THINGS<br />

Ignoring for a moment the particular aims of Huygens’ dioptrical studies, we<br />

may notice that De Aberratione has all the features of a geometrical treatise. It<br />

is structured as a set of propositions <strong>and</strong> definitions regarding spherical<br />

aberration. De Aberratione is not a geometrical treatise by appearance only. In<br />

177 Shapiro, “Gradual acceptance”, 78-80.<br />

178 Additional evidence for this dating I find in the fact that Pardies <strong>and</strong> Huygens discussed Icel<strong>and</strong> Crystal<br />

in the summer of 1672. See further footnote 120 on page 140 below.<br />

179 See section 6.1.

1655-1672 - DE ABERRATIONE 93<br />

its elaboration it consisted of a geometrical derivation of the properties of<br />

spherical aberration. Like Tractatus, it rested on little more than the sine law<br />

<strong>and</strong> a generous dose of Euclidean geometry.<br />

In the elaboration of the theory of spherical aberration, geometry had the<br />

upper h<strong>and</strong>. This st<strong>and</strong>s out clearest in the simplifications Huygens<br />

employed. He used a simplified expression in order to determine the amount<br />

of aberration produced by a particular lens. He justified this by comparing<br />

the calculated differences between both expressions. What effects such<br />

differences would have in actual lenses, he did not tell. Nowhere in De<br />

Aberratione does Huygens give an indication that he had considered the<br />

question how the calculated properties of spherical aberration related to its<br />

observed properties. A modern reader would expect otherwise, but Huygens<br />

went about by geometrical deduction exclusively. This geometrical analysis<br />

resulted in a sophisticated theory of spherical aberration in which complex<br />

problems were solved of neutralizing it by configuring spherical lenses<br />

properly.<br />

But Huygens’ goal was not mere theory, he aimed at its practical<br />

application to real lenses <strong>and</strong> telescopes. This marked him off from his<br />

fellow dioptricians. Had he not applied his theory to design better telescopes<br />

<strong>and</strong> tested his design, he would not have been confronted with those<br />

disturbing colors. The fact that Huygens was taken by surprise by those<br />

disturbing colors need not surprise us. In his dioptrical study of lenses,<br />

Huygens confined himself to their mathematical properties <strong>and</strong> excluded the<br />

consideration of colors. Likewise, in his study of halos <strong>and</strong> parhelia, written<br />

around 1663, he confined himself to tracing the paths of rays of light<br />

through transparent particles in the atmosphere <strong>and</strong> left out any<br />

consideration of the colors of these phenomena. 180 Colors eluded the laws of<br />

geometry, so he wrote there with even greater conviction than in Tractatus:<br />

“However, to investigate the cause of these colors further; to know why they are<br />

generated in a prism, I want to undertake by no means, I admit on the contrary not to<br />

know the cause at all, <strong>and</strong> I think that no one will comprehend their nature easily for as<br />

long as some major light will not have enlightened the science of natural things.” 181<br />

That major light had come, it was named Newton, <strong>and</strong> it had eclipsed<br />

Huygens’ gr<strong>and</strong> project of perfecting telescopes.<br />

Huygens was well acquainted with the disturbing colors produced by<br />

lenses. Dealing with them was, in his view, a matter of trial-<strong>and</strong>-error<br />

configuring of lenses instead of purposive calculation. When colors came to<br />

disturb the predicted optimal working of his design, he did not do anything<br />

with them. Despite the importance of colors for his project, Huygens did not<br />

elaborate upon his observation that colors might be related to the angle of<br />

180 With connected reproduced in OC17, 364-516. On the dating see OC17, 359.<br />

181 OC17, 373. “Doch de reden van dese couleuren verder te ondersoecken, te weten waerom die in een<br />

prisma gegenereert worden, wil ick geensins ondernemen, emo fateor rationem eorum me prorsus<br />

ignorare, neque facile quemquam ipsas perspecturum arbitror qu<strong>and</strong>iu naturalium rerum scientiae major<br />

aliqua lux non affulserit.”

94 CHAPTER 3<br />

incidence of a ray of light. He did not adjust his theory of spherical<br />

aberration, nor the way he intended to counter its effects in telescopes.<br />

Apparently, he saw no possibility to extend dioptrics to the properties of<br />

colors. Colors kept eluding his mathematical underst<strong>and</strong>ing. In other words,<br />

he did not take the step to leave the established domain of mathematical<br />

optics. This should not be strongly counted against him, for no-one in the<br />

seventeenth century did so. Except for Newton, who had an extraordinary<br />

scholarly disposition that combined a mathematical outlook with an interest<br />

in material things fostered by experimental philosophy <strong>and</strong> the new natural<br />

philosophies in general.<br />

Newton did see geometry in colors, but he looked at them from an<br />

entirely different perspective. His studies of prismatic colors had begun<br />

around 1665 with an experiment described by Boyle – with a thread that was<br />

half blue <strong>and</strong> half red <strong>and</strong> appeared broken when seen through a prism. 182<br />

Unlike Boyle, he interpreted this in terms of the refraction of rays of light.<br />

He realized that the rays coming from both parts of the thread are refracted<br />

at different angles. In other words, Newton interpreted the phenomenon in<br />

the geometrical terms of rays <strong>and</strong> angles. On this basis he began his<br />

prismatic experiments, deliberately studying the differences of the angles<br />

with which rays of various colors are refracted. Unlike Descartes, Boyle <strong>and</strong><br />

Hooke before him, he tried to make the spectrum as large as possible, by<br />

projecting it as far as possible. 183 He passed the beam of light at minimum<br />

deviation, so that the effect of the width of the beam was minimized. By<br />

turning the prism into a precision instrument, Newton discovered that it was<br />

the principles of geometrical optics that were violated by the spectrum. The<br />

solution of the anomaly consisted of linking ‘color’ with ‘refractive index’<br />

<strong>and</strong> thus with the sine law of refraction. Different refrangibility reduced<br />

colors to the laws of geometry.<br />

It was not only the mere recognition of geometry that led to different<br />

refrangibility. In order to establish the laws to which colors were subject,<br />

Newton employed experiment in a new way. Combining mathematical<br />

thinking with a heuristic use of experiment, he developed the new<br />

methodological means of quantitative experiment. By measuring the<br />

phenomena produced in his prisms he was able to discover geometrical<br />

properties where previously there had been none.<br />

“But since I observe that geometers have hitherto erred with respect to a certain<br />

property of light concerning its refractions, while they implicitly assume in their<br />

demonstrations a certain not well established physical hypothesis, I judge it will not be<br />

unappreciated if I subject the principles of this science to a rather strict examination,<br />

adding what I have conceived concerning them <strong>and</strong> confirmed by numerous<br />

experiments to what my reverend predecessor last delivered in this place.” 184<br />

182<br />

Newton, Certain philosophical questions, 467.<br />

183<br />

Westfall, Never at rest, 163-164.<br />

184<br />

Newton, Optical papers 1, 47 & 281.

1655-1672 - DE ABERRATIONE 95<br />

Newton considered his discovery of different refrangibility an addition to<br />

geometrical optics. A necessary addition because it explained “… how much<br />

the perfection of dioptrics is impeded by this property <strong>and</strong> how that<br />

obstacle, insofar as its nature allows, may be avoided.” 185 These lines could<br />

have been addressed to Huygens personally, had Newton known of De<br />

Aberratione. Newton was aware that he was breaking new ground. In the<br />

letter he sent to Oldenburg he wrote:<br />

“A naturalist would scearce expect to see ye science of [colours] become mathematicall,<br />

& yet I dare affirm that there is as much certainty in it as in any other part of<br />

Opticks.” 186<br />

These lines were, however, omitted when his ‘New theory’ appeared in<br />

Philosophical Transactions. With his theory Newton went beyond the recognized<br />

boundaries of geometrical optics by extending it to the study of colored rays.<br />

Huygens, on the other h<strong>and</strong>, stayed within the established domain of<br />

optical phenomena to be studied mathematically. He elaborated his dioptrical<br />

theories in the manner customary in geometrical optics. As a mathematical<br />

theory, the content of Dioptrica did not deviate in any principal way from the<br />

doctrines found in Aguilón or Barrow. In Paralipomena physical<br />

considerations – albeit within the traditional domain of mathematical optics<br />

– were much more integrated in mathematics, but Huygens did not follow<br />

this line of Kepler at this moment. 187 As a topic of mixed mathematics,<br />

geometrical optics was principally a matter of geometrical deduction. The<br />

difference with ‘pure’ geometry was that lines <strong>and</strong> circles represented<br />

physical objects like rays, reflecting <strong>and</strong> refracting surfaces. Geometrical<br />

inference was preconditioned by a specific set of postulates: the laws of<br />

optics describing the behavior of unimpeded, reflected <strong>and</strong> refracted rays.<br />

Or, as Huygens would state it in Traité de la Lumière, optics is a science<br />

“where geometry is applied to matter.” 188<br />

Huygens ‘géomètre’<br />

Thus Huygens treated spherical aberration as a geometrical problem which<br />

ought to be solved by mathematical analysis. Despite the vital importance of<br />

colors for his project, he did not go beyond the traditional boundaries of<br />

mixed mathematics in order to tackle the problem. He confined his<br />

investigation to effects known to be reducible to the laws of geometry.<br />

Geometrical optics did not provide the means to deal with colors, so he left<br />

them to the craftsman. In this sense Huygens’ Dioptrica fits in with his<br />

mathematical science in general. In his studies of circular motion <strong>and</strong><br />

consonance he also focused on exploring their mathematical properties on<br />

the basis of established (mathematical) principles.<br />

185<br />

Newton, Optical papers 1, 49 & 283.<br />

186<br />

Newton, Correspondence 1, 96.<br />

187<br />

See section 4.1.2.<br />

188<br />

Traité, 1. “… toutes les sciences où la Geometrie est appliquée à la matiere, …”

96 CHAPTER 3<br />

During the final weeks of 1659, Huygens took up <strong>and</strong> solved a problem<br />

that Mersenne had discussed 12 years earlier in Reflexiones physico-mathematicae<br />

(1647). The problem was to determine the distance traversed by a body in its<br />

first second of free fall, which amounts to determining half the value of the<br />

constant of gravitational acceleration. After having tried Mersenne’s<br />

experimental approach, Huygens ab<strong>and</strong>oned it in favor of a theoretical<br />

consideration of gravitational acceleration. He began a study of circular<br />

motion which in his view was closely connected to gravity: “The weight of a<br />

body is the same as the conatus of matter, equal to it <strong>and</strong> moved very swiftly,<br />

to recede from a center.” 189 Circular motion had been discussed by both<br />

Descartes <strong>and</strong> Galileo, but only in qualitative <strong>and</strong> fairly rough terms. 190<br />

Huygens set out to analyze circular motion mathematically. He derived an<br />

expression for the tension on a chord exerted by a body moving in a circle,<br />

by equating it with the tension exerted by the weight of the body. 191 He then<br />

considered the situation in which a body revolves on a chord in such a way<br />

that a stable situation is created <strong>and</strong> centrifugal <strong>and</strong> gravitational tension are<br />

counterbalanced. With the conical pendulum thus procured <strong>and</strong> reversing his<br />

calculations, Huygens found an improved value for gravitational acceleration<br />

<strong>and</strong> dismissed Mersenne’s original experiment. 192 Analyzing the experiment<br />

mathematically <strong>and</strong> comparing the time of vertical fall to the time of fall<br />

along an arc, he derived a theory of pendulum motion eventually resulting in<br />

the discovery of the isochronity of the cycloid. 193<br />

The aim of Huygens’ studies of curvilinear fall <strong>and</strong> circular motion was to<br />

render these motions with the same exactness Galileo had achieved with free<br />

fall. 194 In the case of curvilinear fall this meant to solve the tricky<br />

mathematical problem of relating the times with which curved <strong>and</strong> straight<br />

paths are traversed. In the case of circular motion, he quantitatively<br />

compared centrifugal <strong>and</strong> gravitational acceleration. Huygens’ success came<br />

from his proficiency in using infinitesimal analysis <strong>and</strong> his control of<br />

geometrical reasoning. 195 He conceptualized the forces he was studying in a<br />

way that could be geometrically represented, which in his view meant to treat<br />

free fall <strong>and</strong> centrifugal force in terms of velocities. 196 He considered, for<br />

example, gravity as mere weight, <strong>and</strong> acceleration as continuous alteration of<br />

inertial motion. 197 In other words, rather than mathematizing these<br />

189<br />

Yoder, Unrolling time, 16-17.<br />

190<br />

Yoder, Unrolling time, 33-34.<br />

191<br />

Yoder, Unrolling time, 19-23. This expression for centrifugal tendency amounts to the modern formula:<br />

F = mv2/r. 192<br />

Yoder, Unrolling time, 27-32.<br />

193<br />

Yoder, Unrolling time, 48-59.<br />

194<br />

The first draft of De vi centrifuga opened with a quotation of Horace: “Freely I stepped into the void, the<br />

first”, above his discovery of the isochronicity of the cycloid he wrote: “Great matters not investigated by<br />

the men of genius among our forefathers; Yoder, Unrolling time, 42 <strong>and</strong> 61.<br />

195<br />

Yoder, Unrolling time, 62-64.<br />

196<br />

The same goes for his earlier study of impact, to be discussed in section 4.2.2.<br />

197 Westfall, Force, 160-165.

1655-1672 - DE ABERRATIONE 97<br />

phenomena, he reduced them to concepts already mathematized. To be<br />

more specific: Huygens reduced these dynamical phenomena to the<br />

kinematic groundwork laid by Galileo.<br />

Yoder has pointed out Huygens’ talent for transferring physics to<br />

geometry. 198 His proficiency in idealizing phenomena enabled him to<br />

mathematize not only the abstract objects of mechanics but also concrete<br />

bobs <strong>and</strong> cords. Once transformed into a geometrical picture, Huygens<br />

could apply his geometrical skills. Just as in his study of spherical aberration,<br />

the kind of experimentation by which Newton had mathematized colors was<br />

absent from Huygens’ studies of circular motion. He was surely a careful<br />

observer <strong>and</strong> capable of designing clever experiments as an independent<br />

means to test theoretical conclusions. 199 Yet, the precision he achieved in<br />

measuring the constant of gravitational acceleration was made possible by his<br />

mathematical underst<strong>and</strong>ing of the matter. Exploring mathematical<br />

properties of a phenomenon empirically was not the way he approached his<br />

objects of study. On the contrary, he readily dismissed Mersenne’s<br />

experiment as indecisive, aware of the imprecision <strong>and</strong> bias of observation. 200<br />

He approached his subject first of all theoretically, interpreting concepts<br />

geometrically <strong>and</strong> analyzing phenomena by means of his mathematical<br />

mastery.<br />

In his dioptrical studies, Huygens had likewise relied on his geometrical<br />

proficiency. His theory of spherical aberration was the outcome of rigorous,<br />

sometimes clever deduction. At the point he could have broken really new<br />

ground – when colors emerged – Huygens halted. The process of<br />

geometrizing new phenomena that had proven to be so fruitful in his study<br />

of motion did not get going in dioptrics. Seemingly, he did not see<br />

possibilities to transform those disturbing colors into a geometrical picture,<br />

despite some promising observations he had made of them. However, we<br />

should bear in mind that motion, as contrasted to colors, had already been<br />

mathematized. In his geometrization of circular motion, Huygens could<br />

build on the groundwork laid by Galileo.<br />

Compared to his study of circular motion, De Aberratione was rather<br />

straightforward geometrical reasoning. In this regard, it comes closer to his<br />

study of consonance that occupied him, on <strong>and</strong> off, from 1661 onwards. 201<br />

The first problem Huygens attacked was the order of consonance, an issue<br />

that had arisen (anew) with the new theories of music of the sixteenth <strong>and</strong><br />

seventeenth centuries. In the theory of consonance Huygens adopted, the<br />

coincidence theory of Mersenne <strong>and</strong> Galileo, the order of consonants was<br />

not evident. He derived a clever rule that only left one problem. His rule<br />

7 seemed to imply that 4 should be placed between the major third <strong>and</strong> the<br />

198 Yoder, Unrolling time, 171-173.<br />

199 Yoder, Unrolling time, 31-32.<br />

200 Yoder, Unrolling time, 170-171.<br />

201 Cohen, Quantifying music, 209-230 <strong>and</strong> Cohen, “Huygens <strong>and</strong> consonance”, 271-301.

98 CHAPTER 3<br />

fourth which implied that “… the number 7, …, is not incapable of<br />

producing consonance …”, a conclusion that ran in the face of all previous<br />

musical theory. 202 At that time – around 1661 – Huygens decided not to<br />

accept the consonance of intervals with 7 because they had no regular place<br />

in the scale.<br />

Next, Huygens addressed a problem in tuning. When keyboards are tuned<br />

according to then customary mean tone temperament, the question was how<br />

the fifths employed ought to be adjusted with respect to pure fifths. 203 In<br />

order to determine a mathematical solution, Huygens started by deriving the<br />

ratios of all twelve tones in terms of the string lengths of the octave <strong>and</strong> the<br />

fifth. In the course of his investigation, Huygens found a new property of<br />

mean tone temperament. It concerned the quantitative difference between<br />

the diatonic <strong>and</strong> the chromatic semitones. 204 Calculating the ratio of both<br />

kinds of semitones, he concluded that C-D can be divided into 5 equal parts<br />

<strong>and</strong>, consequently, the octave into 31 equal parts. Thus Huygens arrived at<br />

the 31-tone division of the octave he had found discussed by Mersenne <strong>and</strong><br />

Salinas. In a letter published 30 years later in Histoire des Ouvrages des Sçavans<br />

(October 1691), Huygens elaborately explained how he calculated the various<br />

string lengths <strong>and</strong> pointed out advantages of his 31-tone division. 205 The<br />

paper did not contain a further consequence Huygens had drawn in his<br />

private notes: the consonance of intervals based on the number 7. Thus,<br />

Huygens’ ‘most original contribution to the science of music’ remained<br />

unknown to the world until this century. 206<br />

Huygens’ studies of consonance show, once more, his dexterity in<br />

exploring <strong>and</strong> elaborating the mathematics of a topic. He added rigor <strong>and</strong><br />

precision to Mersenne’s science of music, using Galileo’s approach <strong>and</strong><br />

202 OC20, 37. Translation: Cohen, Quantifying music, 214.<br />

203 The tones of the octave are found using the consonances; this is called the division of the octave. The<br />

3<br />

seven tones of the diatonic scale are found by means of the fifth (<br />

2<br />

) <strong>and</strong> its complement, the fourth<br />

4<br />

(<br />

7<br />

). Likewise the chromatic tones are found by addition of fifths. A problem arises, however, because a<br />

complete octave cannot be reached again by continuous addition of fifths. A small difference, called the<br />

3 12<br />

2 7<br />

Pythagorean comma, exists between 12 fifths ( 2)<br />

<strong>and</strong> 7 octaves ( 1)<br />

. As a result, the tones of the<br />

octave ought to be tempered in musical practice, which means that the purity of some consonances is<br />

sacrificed. In mean tone temperament most major thirds are pure <strong>and</strong> the fifths are made a bit too large;<br />

in equal temperament all consonances save the octave are a bit impure. Huygens preferred the former, the<br />

latter has become st<strong>and</strong>ard tuning in Western music since the early nineteenth century.<br />

204 The diatonic semitone is the difference between E <strong>and</strong> F, B <strong>and</strong> c, etc.; the chromatic semitone is the<br />

difference between, for example, C <strong>and</strong> C . The chromatically sharpened C <strong>and</strong> flattened D – C # <strong>and</strong> D b –<br />

differ, whereby C-D b <strong>and</strong> C #-D have the size of a diatonic semitone. The difference between C-C # <strong>and</strong> C-<br />

D b is the difference between both kinds of semitones.<br />

205 Most of Huygens’ musical studies is reproduced in OC20, 1-173. The French <strong>and</strong> Latin versions of the<br />

letter have been reprinted with Dutch <strong>and</strong> English translations by Rasch in: Huygens, Le cycle harmonique.<br />

206 Cohen, Quantifying music, 225-226.

1655-1672 - DE ABERRATIONE 99<br />

extending it to problems the latter had ignored. 207 As with his studies of<br />

dioptrics <strong>and</strong> circular motion, Huygens’ study of consonance did not develop<br />

in an empirical vacuum. He rejected Stevin’s theory, as purely mathematical<br />

<strong>and</strong> ignoring the dem<strong>and</strong>s of sense perception. But he also rejected systems<br />

that lacked theoretical foundation. 208 His aim was to develop a sound<br />

mathematical theory that explained <strong>and</strong> founded his musical preferences.<br />

Mean tone temperament therefore was his natural starting point, <strong>and</strong> the 31tone<br />

division seems a natural outcome of his analysis of its mathematical<br />

properties as it conformed to both his theoretical <strong>and</strong> practical preferences.<br />

Like his studies of circular motion <strong>and</strong> consonance, Huygens’ study of<br />

spherical aberration, <strong>and</strong> this is almost a truism, was predominantly<br />

mathematical. Huygens fruitfully explored <strong>and</strong> rigorously examined<br />

mathematical theory. More revealing in the context of the present study is<br />

the relationship between mathematics <strong>and</strong> observation. Huygens was not<br />

blind for the empirical facts. On the contrary, they constituted the main<br />

directive of his investigation in such diverse ways as the measure of gravity,<br />

pleasing temperament <strong>and</strong> workable lens-shapes. Huygens knew how to<br />

check his theoretical conclusions empirically <strong>and</strong> he was not easily satisfied.<br />

Exploratory observation of phenomena was not the way Huygens<br />

approached a subject. In modern terms: he did not employ experiment<br />

heuristically. In the case of gravity, he had soon found out that mere<br />

observation did not yield reliable knowledge. The result proved him right:<br />

the analysis of the mathematical properties of circular motion gave him a<br />

better theory as well as a better means of measurement. Huygens successfully<br />

extended the Galilean, mathematical approach to gravity <strong>and</strong> circular motion.<br />

Newton likewise was a mathematician with a Galilean spirit, but in his<br />

study of colors he linked it with the experimental approach of Baconianism.<br />

Although he was favorably disposed to Bacon’s program for the organization<br />

of science (see below), Huygens did not regard the experimental collecting of<br />

data as a source for new theories, let alone a trustworthy basis for<br />

mathematical derivation. He explored the underlying mathematical structure<br />

of a phenomenon the results of which could be verified to see whether the<br />

supposed structure was real. In the case of consonance, the empirical<br />

foundation of the theory had already been established. In the case of<br />

spherical aberration, however, such preliminary work had not yet been done,<br />

unfortunately. It turned out that not all effects of lenses depended upon the<br />

known mathematical properties of lenses.<br />

Suppose he had pursued his idea that colors were related to the<br />

inclination of the sides of a lens. He might have taken some objective lenses,<br />

covered their center (instead of their circumference as was customary) <strong>and</strong><br />

207 Cohen, Quantifying music, 209. It should be noted that, unlike his predecessors, Huygens possessed<br />

logarithms <strong>and</strong> was therefore readily able to calculate, for example, a 4 1<br />

5<br />

.<br />

208<br />

Cohen, “Huygens <strong>and</strong> consonance”, 293-294.

100 CHAPTER 3<br />

see how the different inclinations affected the generation of colors. He might<br />

even have taken a prism to study the effect of twofold refraction on colors.<br />

He might even have measured the angles of the inclination <strong>and</strong> – even more<br />

speculative – made some measurements on the colors themselves. He did<br />

not, <strong>and</strong> left colors aside in De Aberratione. In short, he recognized the<br />

importance of the colors displayed by his lenses, but did not know what to<br />

do about them. Which amounts to saying that he did not know what to do<br />

about them mathematically.<br />

3.3.2 HUYGENS THE SCHOLAR &HUYGENS THE CRAFTSMAN<br />

Which brings us back to what Huygens’ study of spherical aberration was all<br />

about: the improvement of telescopes. From the viewpoint of dioptrics,<br />

nothing was wrong with his theory of spherical aberration. It described the<br />

properties, derived from the principles of dioptrics, of light rays when<br />

refracted by spherical surfaces. From the viewpoint of Huygens’ project<br />

there was, however, a serious problem. He did not develop his theory in<br />

order merely to extend his dioptrical knowledge, but to find an improved<br />

configuration of lenses. Huygens’ theory of spherical aberration could not<br />

take colors into account – let alone explain how to minimize their disturbing<br />

effects. From the viewpoint of dioptrical theory, colors were a further effect<br />

yet to be understood; from the viewpoint of De Aberratione they were a fatal<br />

blow. Without the practical goal of De Aberratione, Huygens probably would<br />

never have run across the disturbing colors that spherical lenses also<br />

produced.<br />

I have amply argued that the orientation of Dioptrica on the telescope<br />

marked off Huygens’ dioptrical studies from those of most other<br />

seventeenth-century scholars. He was one of the very few who tried to<br />

acquire a theoretical underst<strong>and</strong>ing of the telescope <strong>and</strong>, in addition, he<br />

wanted to improve the instrument on this basis. That is not necessarily to say<br />

that this practical orientation is characteristic of Huygens’ science in general.<br />

Although applications of theory to instruments were never far from his<br />

mind, his studies of consonance <strong>and</strong> circular motion were not guided by an<br />

orientation on instruments as his studies of dioptrics were.<br />

The problem of tuning keyboard instruments was important for Huygens’<br />

musical studies but their main goal was the mathematical theory of<br />

consonance. Having elaborated his 31-tone division, he readily saw the<br />

practical application in the guise of a suitable organ, on which one could<br />

switch easily between keys in mean tone temperament. Likewise, his study of<br />

circular motion was aimed at a physical problem (measuring gravity) <strong>and</strong><br />

took the form of a thorough, mathematical analysis of circular motion in<br />

many of its manifestations. It was not a analysis of the clock he had invented<br />

earlier, nor did the question which pendulum would be isochronous guide<br />

it. 209 Still, practical thinking of a kind was inherent in Huygens’ study of<br />

209 Yoder, Unrolling time, 71-73.

1655-1672 - DE ABERRATIONE 101<br />

circular motion. He often couched his thoughts on circular motion in some<br />

mechanical form. And he designed several clocks that embodied his<br />

theoretical findings. As regards his original pendulum clock he reaped the<br />

rewards of his study by equipping it with cheeks that gave its bob an<br />

isochronous path.<br />

If instruments did not guide Huygens’ other studies the way they did in<br />

dioptrics, his approach to them was nevertheless similar. Horologium<br />

Oscillatorium of 1673 does not just describe the pendulum clock <strong>and</strong> the ideal<br />

cycloidal path, but also gives the mathematical theory of motion embodied in<br />

it. Going beyond the mere necessities of explaining its mechanical working –<br />

as in Dioptrica – he elaborated his theories of circular motion, evolutes <strong>and</strong><br />

physical pendulums. Of the achievements of 1659, Horologium Oscillatorium<br />

included the study of curvilinear fall <strong>and</strong> cycloidal motion, transformed into<br />

a direct <strong>and</strong> refined derivation, but it listed only the resulting propositions of<br />

his study of circular motion <strong>and</strong> the conical clock. In addition, it contained a<br />

discussion of physical pendulums. Huygens imaginatively applied the insight<br />

that a system of bodies can be considered as a single body concentrated in<br />

the center of gravity, to a physical pendulum considered to be resolved into<br />

its constituent parts independently. With this he could express the motion of<br />

the pendulum by means of the accelerated motion of its parts. Next he<br />

compared the physical pendulum to an isochronous simple pendulum,<br />

deriving an expression for the length of the latter in terms of the length <strong>and</strong><br />

the weights of the parts of the former. 210<br />

His organ likewise rested on an sound <strong>and</strong> even elegant theory of<br />

consonance. In this way he showed the solid theoretical basis on which his<br />

inventions rested, showing at the same time that he was not a mere<br />

empiricist but a learned inventor. 211 De Aberratione st<strong>and</strong>s out among<br />

Huygens’ studies in that he developed theory with the explicit aim of<br />

improving an instrument. Earlier, he had proven the working of his eyepiece<br />

on a mathematical basis, but he had not been able to demonstrate that it was<br />

the best configuration possible. In De Aberratione Huygens set out to design a<br />

configuration of lenses that he could prove mathematically was the best one<br />

possible.<br />

Huygens was not unique for trying to solve a practical problem by means<br />

of theory. Descartes’ a-spherical lenses were meant to serve as a solution to<br />

the same problem Huygens attacked. Descartes had tried to realize his design<br />

by thinking up a device fit for making those lenses. Examples from other<br />

fields can be found without much effort; the problem of finding longitude at<br />

sea is only the first to come to mind. The seventeenth century is pervaded by<br />

scholars who believed theory could or should be of practical use. The special<br />

thing about De Aberratione is the way Huygens set out to solve the problem<br />

of spherical aberration. His starting point consisted of the mathematical<br />

210 Westfall, Force, 165-167.<br />

211 Cohen, Quantifying music, 224.

102 CHAPTER 3<br />

theory of spherical lenses he had developed earlier. As contrasted to<br />

Descartes <strong>and</strong> others, his design for a better – or even perfect – telescope<br />

did not start out with the ideal lenses of geometry, but with the ‘poor’ lenses<br />

of actual telescopes. He did not avoid or explain away the defects of<br />

spherical lenses, like Descartes or Hudde. He analyzed these defects in order<br />

to take them into account <strong>and</strong> eventually correct them. Huygens’ design of a<br />

perfect telescope was not based on the theoretically desirable, but on the<br />

practically feasible.<br />

Although craftsmanship preconditioned De Aberratione, Huygens did not<br />

go the craftsman’s way as in his earlier inventions. He wanted to derive a<br />

blueprint for an improved configuration on the basis of his theoretical<br />

underst<strong>and</strong>ing of lenses. Instead of tinkering with lenses, he would be<br />

tinkering with mathematics. He replaced the trial-<strong>and</strong>-error configuring of<br />

lenses by mathematical design. Whether consciously or not, Huygens was<br />

trying to bridge the gap between craftsmanship <strong>and</strong> scholarship. It was an<br />

effort to make science useful for the solution of practical problems. An<br />

advanced one, as the limitations <strong>and</strong> possibilities of the actual art of<br />

telescope making were at the very heart of Huygens’ project. De Aberratione<br />

can be seen as an early effort to do science-based technology.<br />

How did Huygens set about it? He tried to underst<strong>and</strong> mathematically the<br />

technical problem of imperfect focusing <strong>and</strong> to solve it by means of his<br />

theory. The configuration of lenses was the only part of the artisanal process<br />

of telescope making he replaced by theoretical investigation. Colors he left<br />

for crafty h<strong>and</strong>s. Despite this close tie to practice, the subsequent elaboration<br />

of the project was a matter of plain mathematics. He reduced the problem of<br />

the imperfect focusing of spherical lenses to the mathematical problem of<br />

spherical aberration. He then designed a configuration of lenses that<br />

overcame the latter problem, assuming that it also solved the original<br />

practical problem. It did not, for the test of his design brought to light an<br />

additional technical problem that escaped his mathematical theory of lenses.<br />

In a way, it was not just a test of his design but of his theory of spherical<br />

aberration as well. The trial of 1668 can be seen as an empirical test of his<br />

theory of spherical aberration – the first one, as far as the sources reveal.<br />

Whether Huygens also saw it in this way may be doubted. His second design<br />

of 1669 was founded upon the same theory. We do not know whether he<br />

expected it to be free of colors, had it been realized. With hindsight, we can<br />

say that the failure of Huygens’ project is an example of the fact that<br />

technology goes beyond the mere application of science.<br />

Huygens had remarked earlier that colors were a technical problem.<br />

Minimizing their effects was a matter of craftsmanship <strong>and</strong> eluded<br />

mathematical underst<strong>and</strong>ing. Unlike Barrow, Huygens was not inexperienced<br />

with the craft of telescope making at all. With his diaphragm <strong>and</strong> his eyepiece<br />

he had shown that he was quite capable of h<strong>and</strong>ling such technical problems.<br />

The remark in his letter to Constantijn shows that he must have known, in a<br />

practical way, much more of the properties of those disturbing colors than

1655-1672 - DE ABERRATIONE 103<br />

his dioptrical writings reveal. Still, he did or could not integrate this<br />

knowledge into his theory of spherical aberration. Neither by adjusting it in<br />

some appropriate way, nor by extending it by a mathematical theory of<br />

colors. At the crucial point where colors thwarted his plans to design a<br />

perfect telescope, he did not know how to fit his experiential knowledge of<br />

lenses into his theoretical knowledge of them. Huygens did indeed appear as<br />

a scholar as well as a craftsman, but he did not weld both roles.<br />

Would it be reasonable to presume that Huygens’ project fell short of the<br />

kind of method Newton had successfully used to mathematize colors? If<br />

quantitative experimentation is the obvious way to get a mathematical grip<br />

on colors, one may say that it was in the wrong h<strong>and</strong>s as far as the sciencebased<br />

improvement of the telescope is concerned. Newton’s methodological<br />

innovation stemmed from an entirely different context from Huygens’<br />

dioptrics. Newton was after the physical nature of light <strong>and</strong> colors, a nature<br />

that in his view ought to be mathematically structured. His calculations of<br />

spherical aberration gave the same theoretical results as those of Huygens,<br />

but they were aimed at substantiating his claim that chromatic aberration was<br />

much larger. 212 Newton saw the practical implications of his findings. He did<br />

not stick to his negative conclusion <strong>and</strong> set out to show how lenses could be<br />

replaced by mirrors. 213 Yet, telescopes had not been the goal of Newton’s<br />

studies of light <strong>and</strong> colors. His original interest concerned their physical<br />

properties <strong>and</strong> the nature of matter.<br />

It may be questioned whether the kind of problem Huygens ran into – a<br />

technical problem that escaped his theory – would have given rise to a<br />

Newtonian quantification of those disturbing colors. The example of<br />

Hudde’s Specilla circularia makes it clear that a practical approach may also<br />

give cause for reasoning a problem away. Hudde explicitly distinguished<br />

mathematical <strong>and</strong> ‘mechanical’ exactness. In practice mechanical exactness<br />

would do, <strong>and</strong> Hudde accordingly simplified his mathematical analysis on the<br />

strength of explicitly practical considerations. It may have been precisely<br />

Huygens’ practical outlook that made him ignore colors in his theory of<br />

aberrations. He was studying the geometrical properties of lenses <strong>and</strong> those<br />

colors fell outside this domain. He knew that these could be dealt with by<br />

other means: the crafty tinkering with lenses he was also competent in. We<br />

can only speculate what form Huygens’ study would have taken, had he<br />

212 In his lectures Newton derived a formula for spherical aberration. Newton, Optical papers 1, 405-411.<br />

213 His discovery of dispersion led him to conclude that no lens could ever prevent the disturbing effects<br />

of aberration <strong>and</strong> made him design his reflector. Shortly after he published his theory, he did consider the<br />

possibility that chromatic aberration could be prevented in lenses. In a letter to Hooke (Newton,<br />

Correspondence I, 172), he alluded to the possibility of constructing a compound lens that canceled out<br />

chromatic aberration. Pursuing an idea of Hooke’s, he considered the possibility of using a lens<br />

compounded of different refracting media in which chromatic aberration was cancelled in the course of<br />

consecutive refractions. (Newton, Mathematical Papers I, 575-576). In Opticks he ruled out this possibility,<br />

probably because it was at odds with the dispersion law he put forward in it. Shapiro, “Dispersion law”,<br />

102-113; Bechler, “Disagreeable”, 107-119.

104 CHAPTER 3<br />

pursued his thinking on the disturbing colors his configuration turned out to<br />

produce.<br />

The reality is that colors thwarted Huygens’ plan to design via theory a<br />

configuration of spherical lenses that minimized the effect of spherical<br />

aberration. It was not his own observation of those colors that made him<br />

drop the project. He needed a Newton to point out that those colors were<br />

inherent to lenses. And he only got the point when Newton made clear that<br />

it was an aberration; a mathematical property inherent to refraction. Huygens<br />

realized that his project was futile when he saw the ‘Abberationem<br />

Niutonianam’. Disappointed, he stroke out the larger part of what was one<br />

of the most advanced efforts in seventeenth-century science to do sciencebased<br />

technology.<br />

The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument<br />

Huygens’ orientation on the telescope may explain the form <strong>and</strong> content of<br />

Dioptrica, it does not explain it as such. Why did Huygens want to develop a<br />

theory of the telescope? Why did he want to prove mathematically that his<br />

eyepiece performed the way he knew by experience it did? Kepler’s motive<br />

for creating Dioptrice had been his conviction that an exact underst<strong>and</strong>ing of<br />

the telescope was needed for reliable observations. In the practice of midseventeenth-century<br />

telescopy this need did not turn out to be as pressing as<br />

Kepler had thought. Even when the telescope became an instrument of<br />

precision, astronomers could go about it with a rather superficial<br />

underst<strong>and</strong>ing of the dioptrics of the telescope.<br />

Kepler’s point of view does not seem to have been Huygens’ main<br />

motive to embark upon a study of the dioptrics of the telescope. In a preface<br />

he wrote in the 1680s for Dioptrica, he expressed his surprise that no one had<br />

explained the telescope theoretically. One would have expected this<br />

marvelous, revolutionary invention to have aroused the interest of scholars.<br />

“But it was far from that: the construction of this ingenious instrument was found by<br />

chance <strong>and</strong> the best learned men have not yet been able to give a satisfactory theory.” 214<br />

In this preface, Huygens did not explain what further use such a theory<br />

would have. He wanted to explain the telescope <strong>and</strong> did not wonder whether<br />

others also found this important. To Huygens, I believe, the dioptrics of the<br />

telescope was a meaningful topic in its own right.<br />

Huygens’ practical activities strengthen the impression that he was<br />

fascinated by the instrument for its own sake. As we have seen, his interest in<br />

telescopes went far beyond mere dioptrical theory. He made telescopes <strong>and</strong><br />

prided himself with the innovations he had made to the instrument as well as<br />

to the craft. Yet, making telescopes seems to have been a goal in itself. 215<br />

Despite his impressive discoveries around Saturn, Huygens never became a<br />

telescopist. He did not – <strong>and</strong> could not <strong>and</strong> need not – make some sort of a<br />

214<br />

OC13, 435. “Sed hoc tam longe abest, ut fortuito reperti artificij rationem non adhuc satis explicare<br />

potuerint viri doctissimi.”<br />

215<br />

Van Helden, “Huygens <strong>and</strong> the astronomers”, 148 & 158-159.

1655-1672 - DE ABERRATIONE 105<br />

living out of the manufacture of telescopes. He prided himself with making<br />

good instruments – the best, he claimed in Systema saturnium – but it seems<br />

making them was his ultimate goal <strong>and</strong> pleasure. To this desire to make his<br />

telescopes work, he added in Dioptrica the desire to figure out how they<br />

worked. Both intellectually <strong>and</strong> practically, Huygens was fascinated by the<br />

working of telescope in itself. During the 1660s he added an extra dimension<br />

to his zygomorphic interest in the telescope. In De Aberratione he aimed to<br />

put his theory to practice in order to design a perfect telescope. Deliberately<br />

or not, he tried to join his practical <strong>and</strong> theoretical activities regarding the<br />

telescope.<br />

In the wider context of seventeenth-century science, De Aberratione can be<br />

seen as an instance of the omnipresent utilitarian ideal. An ideal that took on<br />

various forms, ranging from invoking science to the general benefit of<br />

mankind to using it to underst<strong>and</strong> <strong>and</strong> solve particular problems of river<br />

hydraulics or gunnery. Or, the other way around, Bacon’s call for an alliance<br />

between the sciences <strong>and</strong> the crafts would have scientists learn from <strong>and</strong><br />

turn to the experiential knowledge acquired by craftsmen. At the academies<br />

in London <strong>and</strong> Paris programs were developed to take stock of the arts.<br />

Little came of those plans <strong>and</strong> the rare times scholars set out to offer their<br />

learning to practicians were even less successful. 216 De Aberratione is an<br />

example of a specific application of science to a practical problem. Such<br />

instances were the exception in the seventeenth century, as utilitarianism<br />

often did not go beyond gr<strong>and</strong> utopian schemes. 217 Not accidentally, Huygens<br />

brought it about, as he combined the scholar <strong>and</strong> the craftsman in one<br />

person. He did so, however, without wasting his breath in Dioptrica on<br />

Baconian or otherwise inspired ideals.<br />

The only place where utilitarian ideas are explicit is the plan he wrote<br />

around 1663 for the Académie. 218 Huygens’ own interests – dioptrics,<br />

harmonics, motion – were prominent in the plan, <strong>and</strong> he explicitly linked<br />

them to practical issues of astronomy, navigation <strong>and</strong> geodesy. In these<br />

plans, as in his own studies, it was a utilitarianism of sorts: centered around<br />

scientific instruments <strong>and</strong> thus focused on the advancement of science. He<br />

was no exception in this regard. Westfall has shown that almost all<br />

interrelations that were established during the seventeenth century between<br />

scholarship <strong>and</strong> craftsmanship concerned scientific instruments. 219 It is a kind<br />

of utilitarianism that stays very close to science itself. In the plan for the<br />

Académie those instruments served as mediators for a selected set of issues<br />

for the common good. The general benefit of solving the problem of finding<br />

longitude may be clear, but for the rest the development of instruments was<br />

aimed at the advancement of science. This in its turn was apparently<br />

216<br />

Boas, “Oldenburg, the ‘Philosophical transactions’, <strong>and</strong> technology”, 27-35; Ochs, “Royal society”<br />

217<br />

Another example is Castelli’s attempt to engineer river hydraulics, discussed in Maffioli, Out of Galileo.<br />

218<br />

OC4, 325-329.<br />

219<br />

Westfall, “Science <strong>and</strong> technology”, 72.

106 CHAPTER 3<br />

considered a useful task in itself. In the preface cited above, Huygens sings<br />

the praise of the invention of the telescope. It had served the contemplation<br />

of the heavenly bodies tremendously <strong>and</strong> had revealed the constitution of the<br />

universe <strong>and</strong> our place in it. “What man, unless plain stupid, does not<br />

acknowledge the gr<strong>and</strong>eur <strong>and</strong> importance of these discoveries?” 220<br />

Huygens’ interest in scientific instruments was not exceptional. The form<br />

it took was exceptional. Huygens gave a particular twist to the idea that<br />

theory could be used to improve the telescope. Instead of deriving an ideal<br />

solution to the problem of spherical aberration, he applied his mathematical<br />

underst<strong>and</strong>ing of real, spherical lenses. Gaining a theoretical underst<strong>and</strong>ing<br />

of the telescope was not that hard for a Huygens; applying it to solve<br />

practical problems proved a more tricky business. With his clocks he was<br />

more successful. His theoretical knowledge of circular motion enabled him<br />

to design an isochronous pendulum. Still, the usefulness of the cheeks was<br />

rather limited. He had to rack his brains considerably to find means to make<br />

his clocks seaworthy – with or without cheeks.<br />

Instruments may not have guided Huygens’ other pursuits as they did in<br />

dioptrics, they certainly were important to him. He published part of his<br />

studies of circular motion in the guise of a treatise describing <strong>and</strong> explaining<br />

his isochronous pendulum clock. One might say that Huygens used<br />

instruments to present himself <strong>and</strong> his scientific knowledge. This would ally<br />

with the way he emphasized, in Systema saturnium, the quality of his<br />

telescopes. It would also offer a (partial) explanation of the fact that he did<br />

not publish Dioptrica despite repeated requests. The book would lack a vital<br />

element: an impressing innovation of the telescope. The invention he had<br />

placed his hopes on – a configuration of spherical lenses neutralizing<br />

spherical aberration – had turned out to be worthless.<br />

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

Chapter 4<br />

The 'Projet' of 1672<br />

The puzzle of strange refraction <strong>and</strong> causes in geometrical optics<br />

Huygens was in Paris in the autumn of 1672. He was still a leading scholar,<br />

but some clouds had begun to appear in the sky. The discussions at the<br />

‘Académie’ sometimes distressed him, in particular the interventions of<br />

Roberval. 1 His status was challenged by aspiring newcomers. The previous<br />

chapter described how Newton with his new theory thwarted his plans to<br />

design a perfect telescope. And the successful entrée on the Parisian scene of<br />

Cassini put serious pressure on his position as 'primes' under Louis' savants.<br />

Cassini had arrived from Rome in 1669 <strong>and</strong> almost immediately had started<br />

to adorn his patron with a series of astronomical observations, where<br />

Huygens could set little against. 2<br />

With all that, sickness had begun to plague him. In February 1670 he had<br />

fallen ill <strong>and</strong> he went home to the ‘air natale’ of The Hague in September.<br />

June 1671 Huygens returned to Paris to resume his activities, but in<br />

December 1675 he would relapse into his ‘maladie’. Whether these illnesses<br />

were caused by his ‘professional’ troubles is hard to tell. Huygens biographer<br />

Cees Andriesse holds this view, developing a Freudian reading of Huygens’<br />

personality, in which Christiaan identifies with his intellectual achievements<br />

to make up for the early loss of his mother. 3 Still, going through his Paris<br />

letters to his brother Constantijn gives the impression that Christiaan was<br />

bothered by a good share of homesickness. And maybe the Paris<br />

environment just was not that good for Huygens’ constitution. Whether or<br />

not his failing health was related, it is certain that the move from the<br />

confines of his parental home to the competitive milieu of Paris had put his<br />

science under pressure in the early 1670s. Huygens did not st<strong>and</strong> by idly,<br />

however. In 1672, he was in the middle of preparing the description <strong>and</strong><br />

explanation of his pendulum clock for publication. Horologium Oscillatorium,<br />

his masterpiece, appeared in 1673 dedicated to his patron Louis. And<br />

whoever might think that Huygens had given up on his dioptrics because of<br />

Newton's interference, was dead wrong.<br />

Huygens had discarded the results of his analysis of spherical aberration<br />

in October 1672. Around the same time, he drew up a plan for a publication<br />

1<br />

Gabbey, “Huygens <strong>and</strong> mechanics”, 174-175; Andriesse, Titan, 235-243.<br />

2<br />

Van Helden, “Constrasting careers”, 97-101.<br />

3<br />

Andriesse, Titan, 244-247 <strong>and</strong> “The melancholic genius”, 8-11. I have discussed Andriesse’s account in<br />

Dutch in Dijksterhuis, “Titan en Christiaan”.

108 CHAPTER 4<br />

on dioptrics. Under the heading ‘Projet du Contenu de la Dioptrique’, he<br />

first listed the topics he would discuss <strong>and</strong> then made an outline of the<br />

chapters. 4 The treatise would contain a large part of the dioptrical theory he<br />

had developed since 1653. It would be a comprehensive account of the<br />

refraction of light rays in lenses <strong>and</strong> their configurations. With this, he finally<br />

prepared to give in to the persistent dem<strong>and</strong>s of his correspondents to<br />

publish his dioptrics. He would not be able to present an impressive<br />

innovation, like the cycloïdal pendulum of Horologium Oscillatorium. But also<br />

without the design of a flawless telescope, Huygens had something to offer.<br />

The theory of Tractatus was still worth publishing, despite the fact that<br />

Barrow had gotten ahead of him by publishing the derivation of the focal<br />

distances of spherical lenses from the sine law. A theory elaborating the<br />

dioptrical properties of telescopes was still not available. Huygens had<br />

enough material left to fill up a treatise on dioptrics.<br />

The ‘Projet’ – as I will refer to it – is a key text in the development of<br />

Huygens’ optics. On the one h<strong>and</strong>, it straightened out the remains of his<br />

previous studies of dioptrics. On the other h<strong>and</strong>, it pointed a new direction<br />

for his optics that would eventually lead to the Traité de la Lumière. This<br />

direction was sign-posted by two new topics the ‘Projet’ introduced to<br />

Huygens’ optics. First, the treatise would contain a chapter on the nature of<br />

light. Huygens planned to give an explanation of the sine law of refraction in<br />

terms of waves of light. Secondly, he would discuss an optical phenomenon<br />

recently discovered: the strange refraction of Icel<strong>and</strong> crystal. The topic bears<br />

no relevance whatsoever to the questions about telescopes that had occupied<br />

him in his previous dioptrical studies. The reason for treating strange<br />

refraction was that it posed a problem for the kind of explanation of the sine<br />

law he had in mind. All this is remarkable, for in his dioptrics Huygens had<br />

never before bothered about the nature of light or the cause of refraction.<br />

What is more, in his recent dispute with Newton, he appeared to have a<br />

blind spot for these very subjects.<br />

In this chapter, we follow Huygens’ switch from the mathematical<br />

analysis of the behavior of refracted rays to the consideration of its causes<br />

<strong>and</strong> the explanation of optical laws. The issue of causes became relevant for<br />

Huygens through the phenomenon of strange refraction. The first attack of<br />

the problem was inconclusive <strong>and</strong>, moreover, left the issue of the cause of<br />

refraction untouched. Together, this attack <strong>and</strong> the ‘Projet’ are illuminating,<br />

not only for the development of Huygens’ optics <strong>and</strong> his conception of<br />

mathematical science, but also for seventeenth-century optics in general.<br />

Optics was in the middle of a transition from medieval ‘perspectiva’ to new<br />

way of dealing mathematically with phenomena of light. This chapter focuses<br />

on the issue of causes <strong>and</strong> explanations in optics. Over the shoulder of<br />

Huygens we look back to the way Alhacen, Kepler, Descartes dealt with the<br />

4<br />

OC13, 738-745. I date the sketch in 1672, instead of 1673 as the editors of Oeuvres Complètes have it. See<br />

page 92 above <strong>and</strong> page 140 below.

THE 'PROJET' OF 1672 109<br />

physical foundations of optical laws. Huygens’ opinions <strong>and</strong> conduct in 1672<br />

turn out to be rather illustrative of the transition optics was going through.<br />

The problem of strange refraction would be solved five years later, but not<br />

without Huygens developing a different <strong>and</strong> innovative approach to the<br />

nature of light. That will be discussed in the next chapter.<br />

‘Projet du Contenu de la Dioptrique’<br />

The ‘Projet’ sketchily fills up the two sides of a manuscript page, with all<br />

kinds of additions <strong>and</strong> remarks inserted in <strong>and</strong> around a main line of<br />

contents. 5 It begins with a short list of topics <strong>and</strong> continues with an outline<br />

of the chapters <strong>and</strong> their content. The planned treatise on dioptrics would, of<br />

course, be about the telescope: “my principal design is to show the reasons<br />

<strong>and</strong> measures of the effects of telescopes <strong>and</strong> microscopes.” 6 The treatise<br />

would open with a historical chapter on the invention <strong>and</strong> advancement of<br />

the telescope <strong>and</strong> of telescopic discoveries, <strong>and</strong> was to include an account of<br />

the development of the mathematical underst<strong>and</strong>ing of lenses <strong>and</strong> related<br />

phenomena. 7 In chapters four to seven Huygens would expound his own<br />

theory of dioptrics, the theory of the telescope of Tractatus. He would solely<br />

discuss spherical lenses – “the only ones useful until now” – <strong>and</strong> leave out<br />

the hyperbolic <strong>and</strong> elliptic lenses invented by Descartes. Huygens was clear<br />

about the principal defect of Descartes’ treatment of dioptrics:<br />

“What I have said about the necessity of the theory of spherical ones is so true that<br />

Descartes, for not having examined it, has not known to determine the most important<br />

thing in the effect of telescopes, which is the proportion of their magnification, for<br />

what he says about it means nothing; …” 8<br />

The final chapter of ‘Dioptrique’ would treat the structure <strong>and</strong> the working<br />

of the eye. The main part of these four chapters was ready <strong>and</strong> only needed<br />

some rewriting <strong>and</strong> restructuring. The second <strong>and</strong> the third are the chapters<br />

of the ‘Projet’ that interest us here. They introduced two topics new to<br />

Huygens’ dioptrics: the cause of refraction <strong>and</strong> the strange refraction of<br />

Icel<strong>and</strong> crystal.<br />

In chapter two, Huygens planned to treat the sine law <strong>and</strong> its causes. He<br />

would start with a historical account of the discovery of the sine law – in his<br />

view undeservedly attributed to Descartes – <strong>and</strong> discuss some features of<br />

refraction. Next, he would give an explanation of refraction <strong>and</strong> discuss the<br />

nature of light. Although sketchy, the gist of his plans is clear. He rejected<br />

the explanation of the sine law Descartes had given in La Dioptrique:<br />

5<br />

Hug2, 188r-188v.<br />

6<br />

OC13, 740. “mon principal dessein est de faire voir les raisons et les mesures des effects des lunettes<br />

d’approche et des microscopes.”<br />

7<br />

OC13, 740. Huygens mentions Archimedes (things seen under water), Alhacen, Kepler <strong>and</strong> Galileo by<br />

name. He elaborated his historical account later during the 1680s.<br />

8<br />

OC13, 743. “Ce que j’ay dit de la necessitè de la theorie des spheriques est si vrai, que Descartes pour ne<br />

l’avoir point examinée n’a sceu determiner la chose la plus importante dans l’effect des lunnetes qui est la<br />

proportion de leur grossissement, car ce qu’il en dit ne signifie rien; …”

110 CHAPTER 4<br />

“difficulties against Descartes. where would the acceleration come from. he makes light<br />

a tendency to move [conatus movendi], which makes it difficult to underst<strong>and</strong><br />

refraction as he explains it, at least in my view. … light extends circularly <strong>and</strong> not in an<br />

instant,…” 9<br />

The concluding words reveal Huygens’ own conception: the nature of light is<br />

to spread out circularly over time. In other words, light consists of waves.<br />

The notes also clarify where Huygens had got the idea to think of light as<br />

waves. “Refraction as explained by Pardies.” 10 Ignace-Gaston Pardies was a<br />

Jesuit father with a keen interest in the mathematical sciences, who actively<br />

participated in the Parisian scientific life, <strong>and</strong> with whom Huygens<br />

maintained good relations. Pardies had proposed the idea that light consists<br />

of waves <strong>and</strong> had explained the sine law with it. Huygens listed some<br />

essentials of a wave theory: “transparency without penetration. bodies<br />

capable of this successive movement. Propagation perpendicular to circles.” 11<br />

In the margin he added “vid. micrograph. Hookij”, a reminder to check<br />

Hooke’s alternative wave theory of Micrographia. 12 The original formulation of<br />

Pardies’ theory has been lost, so we cannot know what precisely Huygens<br />

knew of it. He had known of “… the hypothesis of father Pardies …” at<br />

least since August 1669, when he mentioned it in a discussion at the<br />

Académie. 13 On 6 July 1672 Pardies sent him a treatise on refraction that<br />

probably revealed some more details. After Pardies died in 1673, his confrere<br />

Pierre Ango published his explanation of refraction – at least its main lines –<br />

in L’Optique divisée en trois livres (1682). Ango had taken ‘the best parts’ of<br />

Pardies’ theory <strong>and</strong> blended them with own ideas, but Huygens did not have<br />

a high opinion of Ango’s work. 14 We do not know to what exact extent<br />

Huygens knew Pardies’ theory <strong>and</strong> derived his own underst<strong>and</strong>ing of the<br />

nature of light <strong>and</strong> refraction from it. We do know that they stood in close<br />

contact over these matters, that Huygens openly acknowledged the<br />

contributions of Pardies, <strong>and</strong> that the essentials of Pardies’ theory where<br />

central to Huygens’ subsequent attack of strange refraction. He explicitly<br />

recorded the main assumption of Pardies’ derivation of the sine law:<br />

“Propagation perpendicular to circles.” In other words, rays are always<br />

normal to waves. 15<br />

9<br />

OC13, 742. “difficultez contre des Cartes. d’où viendrait l’acceleration. il fait la lumiere un conatus<br />

movendi, selon quoy il est malaisè d’entendre la refraction comme il l’explique, a mon avis au moins. …<br />

lumiere s’estend circulairement et non dans l’instant, …”<br />

10<br />

OC13, 742. “Refraction comment expliquee par Pardies.”<br />

11<br />

OC13, 742. “transparance sans penetration. corps capable de ce mouvement successif. Propagation<br />

perpendiculaire aux cercles.”<br />

12<br />

OC13, 742 note 1.<br />

13<br />

OC16, 184. “… l’hypothese du P. Pardies …” Pardies’ second letter to Newton in their dispute about<br />

colors suggests that Pardies’ wave conception of light was rooted in Grimaldi’s ideas. Shapiro, “Newton’s<br />

definition”, 197.<br />

14<br />

Shapiro, “Kinematic optics”, 209-210. OC10, 203-204.<br />

15<br />

This is discussed below, in section 4.2.2.

THE 'PROJET' OF 1672 111<br />

In the ‘Projet’ the third chapter was only indicated by a title <strong>and</strong> a single<br />

remark: “Icel<strong>and</strong> crystal” <strong>and</strong> “difficulty of the crystal or talc of Icel<strong>and</strong>. its<br />

description. shape. properties.” 16 Icel<strong>and</strong> crystal was a rarity from the barren<br />

nordic l<strong>and</strong>s displaying remarkable properties. This had been known for<br />

ages, but a sample had recently been brought to Copenhagen to increase the<br />

collection of curiosities of the Danish king. Danmark’s leading<br />

mathematician, Erasmus Bartholinus, then made a study of the crystal <strong>and</strong> its<br />

phenomena <strong>and</strong> reported on its strange refraction properties in 1669 in a<br />

treatise called Experimenta crystalli isl<strong>and</strong>ici disdiaclastici (1669). The strange<br />

refraction of Icel<strong>and</strong> crystal contradicted the sine law. It refracts a<br />

perpendicularly incident ray, which is impossible according to the sine law.<br />

Still, Icel<strong>and</strong> crystal had no relevance whatsoever to telescopes. So why<br />

would Huygens include it in his ‘Dioptrique’? The reason is that strange<br />

refraction constituted a problem for Pardies’ explanation of refraction. The<br />

‘difficulté’ of Icel<strong>and</strong> crystal was that the refraction of the perpendicularly<br />

incident ray could not be reconciled with the assumption that rays are<br />

normal to waves. Huygens did not say this explicitly, but the place where he<br />

indicated the ‘difficulté’ makes it clear that Icel<strong>and</strong> crystal was a problem for<br />

Pardies’ explanation of refraction. Moreover, in his first notes on the<br />

phenomenon of around the same time, Huygens explicitly phrased the<br />

problem this way. 17<br />

We now see why Huygens would want to include strange refraction in a<br />

treatise on the dioptrics of the telescope. If his explanation of refraction<br />

were to be acceptable, it should not be contradicted by this particular kind of<br />

refraction. But why would he care for the tenability of the explanation so<br />

much? Huygens had not bothered to explain refraction before. Part of the<br />

answer lies in the fact that it was customary to do so. Books on geometrical<br />

optics usually contained a preliminary account of the nature of light <strong>and</strong> the<br />

causes of the laws of optics. The explanation of the sine law was to complete<br />

Huygens’ dioptrics so that it could be published as a proper treatise in<br />

geometrical optics. It would also complete his critique of Descartes’ La<br />

Dioptrique. As his theory of spherical lenses corrected the latter’s failure to<br />

explain the telescope properly, the projected explanation of the sine law<br />

would correct the difficulties in Descartes’ explanation.<br />

<strong>Waves</strong> would do the job, assuming that the problem of strange refraction<br />

could be settled. But what job exactly would they do? Just before the sketch<br />

of his explanation of refraction Huygens added an epistemological remark.<br />

An utterance of this kind is rare with Huygens, <strong>and</strong> this one is particularly<br />

illuminating:<br />

“Although it suffices to pose these laws as principles of this doctrine, as they are very<br />

certain by experience, it will not be unbecoming to examine more profoundly the cause<br />

of the refraction in order to try to give also that satisfaction to the curiosity of the mind<br />

16<br />

OC13, 743: “Cristal d’Isl<strong>and</strong>e” <strong>and</strong> 739: “difficultè du cristal ou talc de Isl<strong>and</strong>e. sa description. figure.<br />

proprietez.”<br />

17<br />

See below at the beginning of section 4.2.

112 CHAPTER 4<br />

that loves to know the reason of every thing. And to have at least the possible <strong>and</strong><br />

probable causes instead of remaining in an entire ignorance.” 18<br />

Huygens here presents his causal account as only supplementary rather than<br />

foundational. This raises the question what status waves of light had.<br />

Apparently they were merely probable <strong>and</strong> did not convey some indisputable<br />

truth. Still, they ought to explain refraction <strong>and</strong> do so in a better way than<br />

Descartes’ ‘conatus’ had done. Moreover, the explanation of ordinary<br />

refraction must not be contradicted by another instance of refraction, exotic<br />

as it might be. Just like any mathematical theory, an explanatory theory ought<br />

to be consistent. Despite the limited importance of waves, Huygens took the<br />

problem that strange refraction posed for waves seriously. He went on to get<br />

it out of the way before publishing his ‘Dioptrique’.<br />

4.1 The nature of light <strong>and</strong> the laws of optics<br />

The problem that Huygens recognized is historically significant. Optics was<br />

in the middle of a transformation initiated by Kepler <strong>and</strong> Descartes, in which<br />

the rising corpuscular view on essences was shifting the underst<strong>and</strong>ing of the<br />

nature of light as well as the relationship between causal explanations <strong>and</strong><br />

mathematical descriptions of its properties. Whether he fully realized it or<br />

not, with the ‘difficulté’ of strange refraction Huygens found himself at the<br />

heart of this remapping of the scholarly treatment of light. Before turning to<br />

Huygens’ first efforts to reconcile strange refraction with light waves, I will<br />

sketch the historical context of the epistemological issues raised by the<br />

‘Projet’, in particular the relationship between physics <strong>and</strong> mathematics of<br />

light. To this end, I sketch the way Huygens’ most significant precursors in<br />

optics treated the issue of causality with respect to reflection <strong>and</strong> refraction:<br />

Alhacen, Kepler, Descartes <strong>and</strong> Barrow. This will bring into perspective<br />

Huygens’ specific, <strong>and</strong> rather non-committal approach to the explanation of<br />

the law of refraction.<br />

In this regard, it should be noted that the phrase ‘law of refraction’ was<br />

rarely used in seventeenth-century optics. 19 In the project Huygens spoke of<br />

‘loix de refraction’, which included for example reciprocity as well, as he also<br />

would do in Traité de la Lumière. In Dioptrica he called the sine law the<br />

proportion or ratio of sines. 20 The concept of a law of nature aroses in<br />

18 OC13, 741. “Quoy qu’il suffise de poser ces loix pour principes de cette doctrine, comme estant tres<br />

certains par l’experience, il ne sera pas hors de propos de rechercher plus profondement la cause de la<br />

refraction pour tascher de donner encore cette satisfaction a la curiositè de l’esprit qui aime a scavoir<br />

raison de toute chose. Et d’avoir au moins les causes possibles et vraysemblables que de demeurer dans<br />

une entiere ignorance.”<br />

19 Kepler used ‘mensura’ (Kepler, KGW2, 78; see below). Descartes spoke of the laws of motion but of<br />

‘mesurer les refractions’ (Descartes, AT6, 102). In his optical lectures of 1670 Newton used ‘regula’ <strong>and</strong><br />

‘mensura’ (Newton, Optical papers 1, I, 168-171 & 310-311). In Opticks Newton, like Huygens in Dioptrica,<br />

used ‘proportion’ or ‘ratio’ of sines (Newton, Opticks, 5-6 & 79-82).<br />

20 Huygens, OC13, 143-145. In Traité de la Lumière he simply called the sine law the ‘principale proprieté’ of<br />

refraction (others are its reciprocity <strong>and</strong> total reflection); Huygens, Traité de la Lumière, 32-33. In his notes<br />

he sometimes spoke of ‘laws’ or ‘principles’ (OC13, 741) as he did in the draft of Dioptrica prepared<br />

around 1666 (OC13, 2-9).

THE 'PROJET' OF 1672 113<br />

seventeenth-century natural philosophy <strong>and</strong> entered the mathematical<br />

sciences only gradually.<br />

In the opening lines of the first chapter of Paralipomena, which treats the<br />

nature of light, Kepler points out a disciplinary division between physical <strong>and</strong><br />

mathematical aspects of light.<br />

“Albeit that since, for the time being, we here verge away from Geometry to a physical<br />

consideration, our discussion will accordingly be somewhat freer, <strong>and</strong> not everywhere<br />

assisted by diagrams <strong>and</strong> letters or bound by the chains of proofs, but, looser in its<br />

conjectures, will pursue a certain freedom in philosophizing - despite this, I shall exert<br />

myself, if it can be done, to see that even this part be divided into propositions.” 21<br />

In the subsequent chapters Kepler naturally returned to the firm grounds of<br />

geometry, but not before he had pointed out an unfortunate side effect of<br />

this division. In the appendix to the chapter he complains that the insights<br />

mathematicians have acquired regarding light are neglected <strong>and</strong> undeservedly<br />

underrated by natural philosophers. Therefor, in this appendix, Kepler<br />

explains the common misunderst<strong>and</strong>ings of them - notably the followers of<br />

Aristotle - although they could have corrected themselves had they taken<br />

notice of the writings of opticians. 22<br />

The gap between physical <strong>and</strong> mathematical accounts of the cosmos in<br />

pre-Keplerian astronomy is well-documented. It is tempting to take stock of<br />

Scholastic views on the nature <strong>and</strong> function of mathematical inquiry <strong>and</strong><br />

generalize the status of mathematical astronomy towards that of the<br />

mathematical sciences as a whole. Smith argues that a historical link between<br />

classical astronomy <strong>and</strong> classical optics existed, consisting of shared<br />

conceptions, commitment <strong>and</strong> methodologies. 23 Still, when considering the<br />

relationship between mathematical descriptions of light <strong>and</strong> its nature,<br />

caution should be taken.<br />

Compared to the other fields of mathematics, the development of<br />

geometrical optics followed a rather idiosyncratic course up to the early<br />

seventeenth century. Since Greek antiquity it was realized that the central<br />

object of study – the light ray – combines almost naturally physical <strong>and</strong><br />

mathematical conceptualization. In this way geometrical optics had<br />

incorporated a realistic mode of geometrical reasoning since the very<br />

founding of the science by Euclid <strong>and</strong> Ptolemy. Through the influential work<br />

of Alhacen the onset of a physico-mathematical conception of optics was<br />

established at a much earlier time than would be the case in the other<br />

mathematical sciences. In its transmission through medieval perspectiva,<br />

Alhacen’s optics was the starting point for Kepler <strong>and</strong> Descartes <strong>and</strong><br />

profoundly affected their innovations of the science. As a consequence,<br />

21<br />

Kepler, Paralipomena, 5 (KGW2, 18. Translation Donahue: Kepler, Optics, 17). “Caeterum cum hic à<br />

Geometria interdum in physicam contemplationem deflectamus: sermo quoque erit paulò liberior, non<br />

ubique literis et figuris accommodatus, aut demonstrationum vinculis astrictus, sed coniecturis dissolutior,<br />

libertatem aliquam philosoph<strong>and</strong>i sectabitur: Dabo tamen operam, si fieri potest, ut in Propositiones et<br />

ipse dividâtur.”<br />

22<br />

Kepler, Paralipomena, 29 (KGW2, 38)<br />

23<br />

Smith, “Saving the appearances”, 73-91.

114 CHAPTER 4<br />

methodological, epistemological <strong>and</strong> conceptual features of perspectivist<br />

optics are perceptible throughout seventeenth-century optics. 24<br />

4.1.1 ALHACEN ON THE CAUSE OF REFRACTION<br />

In the eleventh century, the Islamic scholar Ibn al-Haytham composed a<br />

voluminous work on optics, Kitb al-Manzir. The Optics of Alhacen, as they<br />

are commonly referred to in the West, was intended to bring together<br />

mathematicians’ <strong>and</strong> physicists’ accounts of light <strong>and</strong> vision by giving a<br />

systematic treatment of optics that met the dem<strong>and</strong>s of both. 25 This required<br />

the combination of the Aristotelian doctrine of forms received by the eye<br />

<strong>and</strong> Ptolemy’s ray-wise analysis of the perception of shape <strong>and</strong> position.<br />

According to Alhacen both these notions were partly true but incomplete. 26<br />

The synthesis he had in mind - ‘tarkb’ - consisted of a theory in which the<br />

forms of light <strong>and</strong> color issue from every point of the object <strong>and</strong> extend<br />

rectilinearly in all directions. 27 It met the dem<strong>and</strong>s of the Aristotelian doctrine<br />

by considering light rays as the direction in which light extended <strong>and</strong> those<br />

of Ptolemy by appointing rays as the ultimate tool of analysis. As contrasted<br />

to his mathematical precursors, Alhacen regarded a ray as a purely<br />

mathematical entity: “Thus the radial lines are imaginary lines that determine<br />

the direction in which the eye is affected by the form.” 28 In a later work on<br />

optics, the Discourse on light, Alhacen expounded his conception of light <strong>and</strong><br />

reflected on the character of the science of optics by discussing the<br />

distinction between mathematical <strong>and</strong> physical aspects of light. In his view,<br />

each provided answers to different kinds of questions: in physics one<br />

investigates the essence of light; in mathematics the radiation or spatial<br />

behavior of light. Physical theory classified the various kinds of bodies:<br />

luminous, shining, transparent, opaque. Mathematical theory described the<br />

perception of things by means of rays, rectilinear <strong>and</strong> inflected. 29 In the Optics<br />

Alhacen adopted the Aristotelian concept of forms without further<br />

philosophical inquiry. His exposition on the nature of light in book 1 served<br />

as a physical foundation for the mathematical <strong>and</strong> experimental investigation<br />

of light <strong>and</strong> vision that constituted the heart of the Optics.<br />

Alhacen provided the basis for the flourishing of the study of light <strong>and</strong><br />

vision in thirteenth-century Europe given shape to by Robert Grosseteste,<br />

Roger Bacon, John Pecham <strong>and</strong> Witelo. 30 Alhacen’s work reached the west in<br />

24<br />

This theme is amplified by, among others, Schuster, Descartes, 332-334: Smith, Descartes’s Theory of Light<br />

<strong>and</strong> Refraction, 4-12.<br />

25<br />

Alhacen, Optics I, 3-6 (book 1). The content <strong>and</strong> scope of Alhacen’s optics is discussed in Sabra’s<br />

introduction to his translation of its first three books: Alhacen, Optics II, xix-lxiii. See further Lindberg,<br />

Theories, 85-86.<br />

26<br />

Alhacen, Optics I, 81 (book 1, section 61).<br />

27<br />

Alhacen’s account for the subsequent one-to-one correspondence between the points of the object <strong>and</strong><br />

the image in the eye is discussed in section 2.2.1 above.<br />

28<br />

Alhacen, Optics I, 82 (book 1, section 62)<br />

29<br />

Alhacen, Optics I, li (Sabra’s introduction). See also Sabra, “Physical <strong>and</strong> mathematical”, 7-9.<br />

30<br />

Lindberg, Theories, 120-121 <strong>and</strong> 109-116.

THE 'PROJET' OF 1672 115<br />

truncated form, for the Latin translation, in both manuscript <strong>and</strong> printed<br />

form, lacks his first three books. It was translated in the thirteenth (possibly<br />

twelfth) century <strong>and</strong> became known as Perspectiva (or De aspectibus). 31 The<br />

Perspectiva communis (ca. 1279) of Pecham <strong>and</strong> the Perspectiva (ca. 1275) of<br />

Witelo are to be understood primarily as compendia of Alhacen’s optics.<br />

These works became textbooks of perspectiva - a common denominator of<br />

medieval optics. Friedrich Risner published (the remaining books of)<br />

Alhacen’s Optics together with Witelo’s Perspectiva in 1572, an edition that<br />

remained authoritative well into the seventeenth century. To provide for the<br />

now lacking physical foundation of Alhacen’s optics, perspectivist writers<br />

drew on the ideas of Grosseteste <strong>and</strong> Bacon. Through Bacon, Grosseteste’s<br />

theory of the multiplication of species was incorporated into perspectivist<br />

theories. Although it provided a mathematically structured account of the<br />

nature of light <strong>and</strong> its propagation, it served no function in perspectivist<br />

accounts of the behavior of light rays interacting with various media. The<br />

perspectivist writers reiterated Alhacen’s analysis of reflection <strong>and</strong> refraction,<br />

adding some clarifications on its basic assumptions. 32<br />

In his accounts of reflection <strong>and</strong> refraction Alhacen also discussed the<br />

causes of these phenomena. These appealed only to light qua radiation,<br />

however, not its physical essence. The core of Alhacen’s account consists of<br />

a mathematical analysis of rays in their components perpendicular <strong>and</strong><br />

parallel to the reflecting or refracting surface. In reflection the parallel<br />

component remains unaltered whereas the perpendicular component is<br />

inverted, which readily yields the law of reflection. 33 By differentiating the<br />

parallel <strong>and</strong> perpendicular components of a light ray he extended Ptolemy’s<br />

analysis, who had only considered the angles before <strong>and</strong> after reflection. 34 In<br />

his account Alhacen appealed to an analogy between reflected rays <strong>and</strong> a<br />

rebounding ball: “We can see the same thing in natural <strong>and</strong> accidental<br />

motion, …”. 35 He pictured a sphere attached to an arrow projected<br />

perpendicularly or obliquely to a mirror. This mechanical analogy applied to<br />

the mathematical analysis of the motion of light rays <strong>and</strong> did not appeal to<br />

the form-like nature of light. Light is reflected because its motion is fully or<br />

partially ‘terminated’ by an obstacle.<br />

Alhacen’s causal account of refraction proceeded along similar lines. Rays<br />

are refracted because their motion changes when they enter a medium of<br />

different density. In the case of refraction towards the normal – into a denser<br />

medium – he assumed that part of the parallel component was altered. He<br />

did so implicitly, in an comparison with a ball striking a thin slate. “For<br />

31<br />

Alhacen, Optics I, lxxiii-lxxix (Sabra’s introduction).<br />

32<br />

Lindberg, “Cause”, 30-31.<br />

33<br />

Risner, Optica thesaurus, 112-113. Witelo relied the argument of the shortest path: Risner, Optica thesaurus,<br />

198.<br />

34<br />

Lindberg, “Cause”, 25-29. Sabra, “Explanation”, 551-552.<br />

35<br />

Risner, Optica thesaurus, 112-113. “Huius aút rei simile in naturalibus motibus videre possumus, & etiá in<br />

accidentalibus.” Translation: Lindberg, in Grant, Source book, 418.

116 CHAPTER 4<br />

things moved naturally in a straight line through some substance that will<br />

receive them, passage along the perpendicular to the surface of the body in<br />

which passage takes place is the easiest.” 36 A couple of lines further, Alhacen<br />

continued: “Therefore, the motion [of the light] will be deviated toward a<br />

direction in which it is more easily moved than in its original direction. But<br />

the easier motion is along the perpendicular, <strong>and</strong> that motion which is closer<br />

to the perpendicular is easier than the more remote.” 37 In the case of<br />

refraction away from the normal Alhacen ab<strong>and</strong>oned the appeal to the<br />

easiest path. He considered the components of the ‘motion’ again <strong>and</strong> stated<br />

without argument that the parallel component is increased. Besides being<br />

inconsistent, Alhacen’s account of refraction remained qualitative, as he did<br />

not attempt to determine to what degree a refraction ray was bent towards<br />

the normal, nor to what proportion the parallel component was altered.<br />

Alhacen’s account of refraction primarily consists of an experimental<br />

analysis. In Risner’s edition it covers the first eleven or twelve propositions<br />

of the seventh book, which return in the second chapter of Witelo’s part. In<br />

the tenth chapter the latter added to the quantitative account of refraction by<br />

providing a table – supposedly observational – of angles of refraction for a<br />

set of incident rays.<br />

In Alhacen’s accounts of reflection <strong>and</strong> refraction two levels of inference<br />

can be distinguished. In the first place, the analysis of rays in their<br />

perpendicular <strong>and</strong> parallel components revealed some deeper lying<br />

mathematical structure of both phenomena. It unified his accounts to some<br />

extent, although he did not assume the parallel component unaltered in all<br />

cases like Descartes would later do. The second level involves mechanical<br />

analogies that illuminate rather than prove the mathematical analyses of<br />

reflected <strong>and</strong> refracted rays. The causal account provided additional support<br />

for the properties of reflection <strong>and</strong> refraction, but the ultimate justification<br />

was empirical. 38 In this regard the analogies can be considered to serve<br />

didactical purposes.<br />

Alhacen’s analogies do not - <strong>and</strong> were not intended to - explain refraction<br />

<strong>and</strong> reflection by deriving their properties from an account of the nature of<br />

light. That is the way Huygens <strong>and</strong> his fellow seventeenth-century students<br />

of optics understood ‘explaining the properties of light’ <strong>and</strong> which his waves<br />

of light would have to bring about. Whereas the rectilinearity of rays<br />

followed rather naturally from Alhacen’s underst<strong>and</strong>ing of forms, reflection<br />

<strong>and</strong> refraction are discussed in terms of light rays instead of interactions<br />

between forms with reflecting <strong>and</strong> refracting substances. The ideals of<br />

36 Risner, Optica thesaurus, 241. “Omnium autem moterum naturaliter, que recte moventur per aliquod<br />

corpus passibile, transitus super perpendicularem, que est in superficie corperis in quo est transitus, erit<br />

facilior.” Translation: Lindberg, “Cause”, 26.<br />

37 Risner, Optica thesaurus, 241. “...: accidit ergo, ut declinetur ad partem motus, in quam facilius movebitur,<br />

quàm in partem, in quam movebatur : sed facilior motuum est super perpendicularem: & quod vicinius est<br />

perpendiculari, est facilius remotiore.” Translation (amended): Lindberg, “Cause”, 27.<br />

38 Alhacen, Optics I, lxi (Sabra’s introduction); Risner, Optica thesaurus, XVII-XIX (Lindberg’s introduction).

THE 'PROJET' OF 1672 117<br />

mechanical philosophy notwithst<strong>and</strong>ing, this ‘physical’ underst<strong>and</strong>ing of rays<br />

<strong>and</strong> their behavior would crucially affect the investigations of Huygens <strong>and</strong><br />

other seventeenth-century opticians. Kepler <strong>and</strong> Descartes set off where<br />

Alhacen <strong>and</strong> Witelo had left off. A law of reflection was known, as well as<br />

diverse mathematical properties of radiated light, but refraction remained to<br />

be understood only qualitatively. The thirteenth-century synthesis left<br />

perspectiva as a comprehensive body of knowledge – Alhacen’s theory of<br />

vision, solutions to various problems of reflection <strong>and</strong> so on – riddled with<br />

some persistent, well-known problems like the pinhole image. 39 The sixteenth<br />

century witnessed major developments in optics, but mainly in its practical<br />

parts that bore on Galileo’s telescopic achievements rather than Kepler’s <strong>and</strong><br />

Descartes’ theoretical pursuits. 40<br />

4.1.2 KEPLER ON THE MEASURE AND THE CAUSE OF REFRACTION<br />

The heritage of medieval perspectiva Kepler received, consisted of a welldefined<br />

set of aims <strong>and</strong> criteria for geometrical optics: mathematical analysis<br />

of the behavior of light rays. 41 In Paralipomena he took up this heritage <strong>and</strong><br />

transformed it radically. In chapter two we have seen how he created a new<br />

theory of image formation by rigorously applying the principle of rectilinear<br />

propagation of light rays. We now turn to his account of the causes<br />

underlying the behavior of light rays. Here the same approach is<br />

recognizable. In Kepler’s view, the mathematically established properties of<br />

things are real <strong>and</strong> should be directive in physical considerations. Kepler’s<br />

conception of the nature of light can be seen as a realist reading of<br />

perspectivist’s mathematical ideas which he then rigorously employed in the<br />

investigation of the behavior of light. 42<br />

At the start of this section I cited the opening lines of Paralipomena, where<br />

Kepler pointed out the relative freedom of reasoning he would employ in<br />

these matters pertaining to physics. In the first chapter, ‘De Natura Lucis’, he<br />

expounded the general concepts <strong>and</strong> principles pertaining to his account of<br />

optics. On the whole, his theory of light was the perspectivists’ theory of<br />

multiplication of species enriched with neoplatonist metaphysics. 43 According<br />

to Kepler, light is an incorporeal substance which has two aspects, essence<br />

<strong>and</strong> quantity, by which it has two operations (‘energias’), illumination <strong>and</strong><br />

local motion, respectively. 44 Radiation is the form of propagation: light<br />

spreads in all directions <strong>and</strong> does so spherically. Light rays are the radii of<br />

this sphere <strong>and</strong> thus rectilinear. Light itself can be regarded as the twodimensional<br />

surface of an exp<strong>and</strong>ing sphere. The mathematical structure of<br />

39<br />

Lindberg, Theories, 122-132.<br />

40<br />

Dupré, Galileo, 17-19.<br />

41<br />

Lindberg, “Roger Bacon”, 249-250.<br />

42<br />

The following discussion owes much to Buchdahl’s illuminating discussion of Kepler’s method:<br />

Buchdahl “Methodological aspects”. References are to the original text, corresponding pages in the<br />

Gesammelte Werke in parentheses. Except where noted, translations are by Donahue from Kepler, Optics.<br />

43<br />

Lindberg, “Incorporeality”, 240-243.<br />

44<br />

Kepler, Paralipomena, 13 (KGW2, 24)

118 CHAPTER 4<br />

this theory was clearly perspectivist in origin, but to Kepler it represented the<br />

physical nature of light, not only its mathematical behavior.<br />

On the basis of this theory of light, Kepler discussed the behavior of<br />

propagated light. The rectilinearity of light rays is a direct outcome of<br />

Kepler’s conviction that light ‘strives to attain the configuration of the<br />

spherical’. 45 Where light is deflected from its straight path this must be the<br />

effect of the interaction of light <strong>and</strong> matter. As light is a two-dimensional<br />

surface this interaction can only occur with the surface of reflecting <strong>and</strong><br />

refracting media. Kepler attributed a form of density to surfaces <strong>and</strong> argued<br />

that light is hindered in its passage through the surface of a body<br />

proportionally to its density. 46 In the case of reflection the density of the<br />

surface is so high that light falling upon it “... is made to rebound in the<br />

direction opposite to that whence it approached.” 47 Kepler specified that this<br />

applied to the perpendicular component of a ray - i.e. the part of the motion<br />

towards the surface. The law of reflection now followed naturally, thus<br />

clearing the way for an exact analysis of the properties of reflection in<br />

chapter 3 of Paralipomena. In this chapter Kepler took up the classical topic<br />

of perspectiva to determine mathematically the location where a reflected<br />

image is perceived. 48<br />

The measure of refraction<br />

For refraction things were more complicated. The causal account in chapter<br />

1 did not yield an exact law, so in the fourth chapter of Paralipomena Kepler<br />

could not readily embark on an analysis of refractional phenomena. Instead,<br />

he first had to find such a ‘measure’ of refraction. The course of the chapter<br />

reveals Kepler’s conception of the distinction between causes <strong>and</strong> measures<br />

in optics. Initially he stuck to the epistemic organization of his treatise by<br />

analyzing refraction in term of rays, the components of their motion, <strong>and</strong><br />

regularities in the various angles at which they are refracted. However, when<br />

all this yielded no satisfactory results he took the nature of light into<br />

consideration to see where these ‘proper’ causes could lead him to find the<br />

measure of refraction.<br />

In proposition XX of chapter 1 Kepler derived from his suppositions<br />

about the interactions of light (surfaces) with (the surface of) a dense<br />

medium that a ray is refracted towards the perpendicular. 49 His argument<br />

comes down to the idea that the surface impedes the spreading of the sphere<br />

of light. Kepler explained that this underst<strong>and</strong>ing is based on the fact that<br />

motion belongs to light <strong>and</strong> that said interaction is general for moving<br />

45<br />

Kepler, Paralipomena, 8 (KGW2, 20). “Nam diximus affectari à luce figurationem Sphaerici.”<br />

46<br />

Propositions XII-XIV: Kepler, Paralipomena, 10-11 (KGW2, 22-23)<br />

47<br />

Kepler, Paralipomena, 13 (KGW2, 25). “Lux in superficiem illapsa repercutitur in plagam oppositam,<br />

unde advenit.”<br />

48<br />

An important part of this was Kepler’s negation of the generality of the cathetus rule <strong>and</strong> the<br />

introduction of his new theory of image formation, which have been discussed above in section 2.2.1.<br />

49<br />

Proposition XX: Kepler, Paralipomena, 15-21 (KGW2, 26-31).

THE 'PROJET' OF 1672 119<br />

matter. In this way he followed up on the mechanical analogies employed in<br />

perspectivist causal accounts, but he did not do so without appropriating that<br />

line of reasoning to his own means. “For it may be permissible here for me<br />

to use the words of the optical writers in a sense contrary to their own<br />

opinion, <strong>and</strong> carry them over into a better one.” 50 Kepler went on to develop<br />

the analysis of a ball spun into water by distinguishing between the dynamics<br />

of the parallel <strong>and</strong> perpendicular components of its motion, whereby light is<br />

rarified in the former direction <strong>and</strong> merely transported in the latter direction.<br />

He then proceeded with a short discussion of the underlying physics, to wit<br />

the statics of a balance. In this way Kepler transformed the mechanical<br />

analogies employed by his perspectivist forebears to illuminate the<br />

mathematics of refraction into a physical foundation of the analysis of<br />

refraction. The account in chapter 1 only yielded a qualitative underst<strong>and</strong>ing<br />

of refraction, <strong>and</strong> only partial for that matter, for Kepler did not discuss the<br />

passage of light into a rarer medium.<br />

At the opening of chapter 4, ‘De Refractionum Mensura’, Kepler still<br />

lacked an exact law of refraction. He needed this ‘measure’ in the first place<br />

for his account of the dioptrics of the eye in the next chapter (see above<br />

section 2.1.1.), but in the end principally for his account of atmospheric<br />

refraction later in Paralipomena. After all, it was a treatise in the optical part of<br />

astronomy for which the laws of optics were instrumental. Nevertheless my<br />

discussion will be confined to the optics per se: Kepler’s tour the force to<br />

tackle the mathematics of refraction.<br />

Kepler began with a review of the received opinions regarding the<br />

measure of refraction. In this section, he tied in with the traditional approach<br />

of considering the physical properties of light rays <strong>and</strong> their components.<br />

After negating several opinions, Kepler laid down the - in his view - generally<br />

established underst<strong>and</strong>ing: first, that the density of the refracting medium is<br />

the cause of refraction <strong>and</strong>, second, the angle of incidence contributes to its<br />

cause. The question therefor was how these two aspects are connected.<br />

Kepler ran through several options as they had been set forth, rejecting each<br />

as insufficient. Next, he contemplated how the two said aspects could<br />

correctly be combined. 51 Kepler proceeded to represent these conditions<br />

geometrically (Figure 36). BC is the refracting surface of a medium BCED <strong>and</strong><br />

AB, AG, AF are incident rays. Kepler now extended the medium to DEKL,<br />

thus representing the greater density of its surface. He then constructed a<br />

refracted ray FQ by drawing HN perpendicular to the lower surface <strong>and</strong><br />

joining N at the imaginary bottom with F. 52 Comparing the results of this<br />

method with Witelo’s table, Kepler simply concluded that it was refuted by<br />

50<br />

Kepler, Paralipomena, 16 (KGW2, 27). “Liceat enim hîc mihi verba Opticorum contra mentem ipsorum<br />

usurpare, et in meliorem sensum traducere.”<br />

51<br />

Kepler, Paralipomena, 85-87 (KGW2, 85-86)<br />

52<br />

This is equivalent with sini : tanr = constant. Lohne, “Kepler und Harriot”, 197. Compare Buchdahl,<br />

“Methodological aspects”, 283.

120 CHAPTER 4<br />

experience. 53 He tried some more ideas flowing<br />

from this geometry, including some ways of<br />

evaluating the ‘refractaria’ - the locus of images<br />

where the points of a line are percieved, D for<br />

point L, I for point M, etcetera. 54 All ideas were<br />

refuted by experience <strong>and</strong> Kepler ab<strong>and</strong>oned his<br />

attempt of finding a measure of refraction on the<br />

basis of an analysis of the physics of light rays.<br />

In the next three sections, Kepler temporarily<br />

ignored the causes of refraction <strong>and</strong> focused on<br />

finding mathematical regularities in the given<br />

angles of incidence <strong>and</strong> refraction. Building on the<br />

known properties of reflection, he tried certain<br />

analogies between reflection <strong>and</strong> refraction. Kepler<br />

argued that in the case of refraction in a medium<br />

with infinite density, all rays must be refracted into<br />

the perpendicular. He then correlated this case to<br />

reflection by a parabolic mirror with rays coming<br />

from its focus. This led him to consider the<br />

relationship of conic sections with refraction. He<br />

constructed a diagram of angles of incidence <strong>and</strong><br />

Figure 36 The first stage of<br />

Kepler’s attack of<br />

refraction.<br />

refraction <strong>and</strong> considered the intersection of the accompanying rays. The<br />

resulting curve is similar to a hyperbola, but points from where the rays<br />

come are not the matching foci, so Kepler dismissed this attempt as well.<br />

This <strong>and</strong> other trials with conic sections – including the effort to construct<br />

an anaclastic curve – still did not give Kepler a correct ‘measure of<br />

refractions’ <strong>and</strong> he ab<strong>and</strong>oned this line of thought as well.<br />

Finally, Kepler returned to his causal analysis of refraction of chapter 1 to<br />

query whether - “may God look kindly upon us” - this would yield the<br />

measure of refraction. 55 As contrasted to the ray analysis of the first stage, he<br />

now considered the interaction of the surface of light with the surface of the<br />

refracting medium. 56 Kepler warned beforeh<strong>and</strong> he would perhaps stray<br />

somewhat from his goal of finding the measure of refractions in its causes,<br />

<strong>and</strong> halfway through his exercise he would acknowledge “In demonstrating<br />

the true cause of this directly <strong>and</strong> a priori, I am stuck.” 57 He did not formally<br />

deduce a ‘measure’ from the causes of refraction, but rather had employed<br />

(in Buchdahl’s words) “physical considerations to guide the intuitive search<br />

for responsible factors relevant to the result.” 58<br />

53<br />

Kepler, Paralipomena, 86 (KGW2, 86). “Hic modus refutatur experientiâ: ...”.<br />

54<br />

Kepler, Paralipomena, 88-89 (KGW2, 87-88).<br />

55<br />

Kepler, Paralipomena, 110 (KGW2, 104). “quod Deus benè vertat”<br />

56<br />

Kepler, Paralipomena, 110-114 (KGW2, 104-108).<br />

57<br />

Kepler, Paralipomena, 110 <strong>and</strong> 113 (KGW2, 104 <strong>and</strong> 107). “Etsi enim à scopo forsan etiamnum nonnihil<br />

aberrabimus: ...” <strong>and</strong> “In genuina huius rei causa directè et à priori demonstr<strong>and</strong>a haereo.”<br />

58<br />

Buchdahl, “Methodological aspects”, 291.

Kepler began with the underst<strong>and</strong>ing<br />

of the nature of light as the surface of an<br />

exp<strong>and</strong>ing sphere laid down in the<br />

opening chapter of Paralipomena (Figure<br />

37). ABMK is the section of a physical ray<br />

obliquely incident on the surface BC <strong>and</strong><br />

refracted towards QBMR. According to<br />

Kepler the angle of deviation must be<br />

proportional to the angle of incidence.<br />

THE 'PROJET' OF 1672 121<br />

Figure 37 The final stage of Kepler’s<br />

analysis of refraction<br />

This condition is met when only the (surface)density of the refracting<br />

medium is assumed to be effective. With increasing obliquity, BM increases<br />

<strong>and</strong> therefore the resistance met by the light increases. Now “… there is<br />

more density in BM than in LM ...” so that the proportion LM to BM must be<br />

added as a factor of refraction onto the proportionality of angles of<br />

incidence <strong>and</strong> deviation. 59 However, the proportion LM to BM – or sec i –<br />

implied a paradox. Horizontal rays would be refracted at an infinitely large<br />

angle. Kepler therefore changed his perspective <strong>and</strong> now considered BR of<br />

the refracted ray. He concluded that the secans of the angle at the upper<br />

surface of the denser medium ‘plays a part’ in refraction. 60 Refraction was<br />

thus a composite of two factors: the proportionality of i-r to i <strong>and</strong> the<br />

proportionality of i-r to sec r – in other words: i-r = c·i·sec r, where c is some<br />

constant.<br />

In proposition 8, Kepler gave instructions how to apply this analysis to<br />

calculate angles of refraction. It is in the form of a ‘problem’, a procedural<br />

statement of the sort the later Dioptrice was composed of, as we saw above in<br />

section 2.2.1. After all, Kepler’s struggle had not yielded a general ‘measure<br />

of refraction’ independent of specific media <strong>and</strong> transcending measurements.<br />

First, both factors are determined for the medium by means of one known<br />

pair of incident <strong>and</strong> refracted rays. Then the angle of refraction for any other<br />

angle of incidence is computed. By means of an example, Kepler calculated a<br />

table for refraction from air into water. The values differed somewhat from<br />

Witelo’s data which Kepler had plied so rigorously in the previous sections.<br />

This time he was more tolerant: “This tiny discrepancy should not move you;<br />

believe me: below such a degree of precision, experience does not go in this<br />

not very well-fitted business.” 61 Moreover, he (correctly) suspected that<br />

Witelo had modified his table on the basis of Ptolemy’s false supposition<br />

that the secondary differences of the angles are constant. “Therefore, the<br />

fault lies in Witelo’s refractions”, <strong>and</strong> Kepler proceeded to use his own result<br />

to consider atmospheric refraction. 62 Although the empirical correctness was<br />

59 Kepler, Paralipomena, 111 (KGW2, 105). “Plùs igitur densitatis est in BM, quàm in LM.”<br />

60 Kepler, Paralipomena, 113 (KGW2, 107). “..., sciendum igitur, eorum angulorum incidentiae secantes concurrere ad<br />

mensuram refractionum, qui constituuntur ad superficiem in medio densiori.”<br />

61 Kepler, Paralipomena, 116 (KGW2, 109). “Neque te moveat tantilla discrepantia, credas mihi, infra tantam<br />

subtilitatem, experientiam in hac minus apt materia non descendere.”<br />

62 Kepler, Paralipomena, 116 (KGW2, 109). “Ergò in Vitellionis refractionibus culpa haeret.”

122 CHAPTER 4<br />

not beyond doubt, Kepler preferred his own data over Witelo’s because it<br />

was based on “regularity <strong>and</strong> order”. In the final propositions of this section<br />

<strong>and</strong> the remaining sections of the chapter, Kepler was now able to dealt with<br />

proper subject of the chapter: the quantitative treatment of atmospheric<br />

refraction.<br />

Kepler’s search for a ‘measure’ refraction clearly reveals the idiosyncrasies<br />

of his thinking. He laboriously reported on his persistent efforts to find a<br />

satisfactory law, <strong>and</strong> although – so we can see with hindsight – he came<br />

tantalizingly close he did not succeed. The successive stages of his attack<br />

display his ever inventive mathematical reasoning, mixed with those typical<br />

Renaissance conceptions of his that make it hard for a modern reader to<br />

distinguish mathematics <strong>and</strong> physical ideas. In the light of ensuing<br />

developments in seventeenth-century optics, the final stage of Kepler’s<br />

struggle with refraction is the most interesting. Here he took his conception<br />

of the nature of light into account in order to find a law of refraction. In a<br />

kind of microphysical, though far from corpuscular, analysis he considered<br />

the interaction of a surface of light <strong>and</strong> the refracting medium. At this stage<br />

he move farthest away from traditional approaches. Although the resulting<br />

‘rule’ was phrased in terms of rays, he had taken the true nature of light into<br />

account while analyzing the interaction of rays <strong>and</strong> (refracting) media. As I<br />

see it, this was possible because of his realist view of mathematical<br />

description. With Kepler, the mathematics of light propagation necessarily<br />

reflected the nature of light.<br />

One may argue that mathematics took the lead in his thinking. Kepler<br />

more or less reduced light to a mathematical entity, a two-dimensional<br />

surface. The geometry of refraction was rather autonomous in his final<br />

attempt to derive a law. 63 Yet, pure formalisms would have been meaningless<br />

for him. Kepler maintained geometrical optics as a mathematical theory<br />

explaining the behavior of light rays. He adopted many concepts of<br />

perspectivist theories of light <strong>and</strong> refraction, but he applied them in a radical<br />

<strong>and</strong> sometimes radically different way. On the level of methodology, all<br />

relevant components – physics, mathematics, observation – had been<br />

present in perspectivist optics, but Kepler sought a closer connection<br />

between them <strong>and</strong> often used these means in a much stricter way. He<br />

repeatedly allowed Witelo’s data to refute the outcome of his trials. Kepler’s<br />

wanted to establish a closer tie between the nature of light <strong>and</strong> the laws of<br />

optics <strong>and</strong> derive ‘measure’ from ‘cause’. He openly acknowledged that he<br />

could not realize this ideal. He resorted to a freer mode of reasoning<br />

because, as I see it, he was far too creative a thinker to stick too rigidly to his<br />

ideals.<br />

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

THE 'PROJET' OF 1672 123<br />

True measures<br />

Even without a true measure of refraction, an inventive mathematician like<br />

Kepler could solve problems in the behavior of refracted rays. In Dioptrice, he<br />

determined properties of spherical lenses in a less rigorous way, pragmatically<br />

applying a rule that had only limited validity. Likewise, his predecessors had<br />

used their limited knowledge to discuss isolated problems regarding<br />

refraction. In order to turn ‘dioptrics’ into a genuine part of the<br />

mathematical science of optics, a true measure of refraction was still needed.<br />

However impressive Kepler’s persistence to find a true measure of<br />

refraction, his efforts will always have a tragic side. Around the same time he<br />

was struggling with the phenomenon, across the Channel the exact law had<br />

already been found by the very man Kepler had been corresponding with:<br />

Thomas Harriot.<br />

Harriot had done so by traditional means that were accessible to Kepler<br />

too: analysis of the observed propagation of light rays. The difference was<br />

that Harriot made new observations <strong>and</strong> had a lucky h<strong>and</strong> in this. Harriot’s<br />

success shows that, in the case of refraction, traditional methods could yield<br />

the required result. Around 1597, Harriot had begun looking for a law of<br />

refraction. Initially, he also tried to find a law on the basis of Witelo’s tables.<br />

As these efforts were unsuccessful, he decided that Witelo was unreliable <strong>and</strong><br />

started to measure angles of refractions anew. 64 After some fruitless attempts,<br />

he chose a way of measurement that proved very lucky.<br />

In 1601, he measured refraction<br />

by means of an astrolabe suspended<br />

in water (Figure 38). Viewing along<br />

the center R of the astrolabe, he<br />

determined the positions O where a<br />

point was seen when moved along<br />

the lower edge of the astrolabe. Then<br />

he determined the image points B.<br />

The cathetus rule (see page 33) said<br />

that the image point is the<br />

intersection of the normal to the<br />

refracting surface <strong>and</strong> the incident<br />

ray. All image points were on a circle.<br />

This meant that RO <strong>and</strong> RB were in constant proportion, <strong>and</strong> likewise were<br />

the sines of i <strong>and</strong> r. In a table Harriot compared angles of deviation as he had<br />

measured them with calculated ones, but he did not reveal how he had used<br />

the figure with two concentric circles – which he called ‘Regium’ – for his<br />

calculations. The calculated values give reason to believe that it was the sine<br />

relation he used. 65<br />

64<br />

Lohne, “Geschichte des Brechungsgesetzes”, 159-160.<br />

65<br />

Lohne, “Kepler und Harriot”, 202-203.<br />

Figure 38 Harriot’s measurements (Lohne).

124 CHAPTER 4<br />

Harriot had reconsidered <strong>and</strong> reapplied traditional methods anew <strong>and</strong><br />

found – what might be called – an empirical law of refraction. As contrasted<br />

to Kepler, he had turned to the measurement of refraction, instead of the<br />

theoretical trench-plowing of his hapless correspondent. Harriot does not<br />

seem to have considered the ‘proper cause’ of refraction with which his law<br />

may have been understood. His accomplishments were known only to a<br />

small circle of acquaintances. It is possible that they spread through<br />

correspondence, but he became known as a discoverer of the law of<br />

refraction only in the twentieth century. 66<br />

Around 1620, Willebrord Snel was the next to discover the exact measure<br />

of refraction - again by means readily available to Kepler. He did not publish<br />

his discovery, but it became generally known in the 1660s. How he<br />

discovered the law will remain a matter of conjecture. Snel’s papers on optics<br />

are lost, except for the notes he made in Risner’s Opticae libri quatuor (1606)<br />

<strong>and</strong> an outline of a treatise on optics discovered in the 1930s. 67 Hentschel has<br />

been the first to make a thorough attempt at reconstruction. In his view, Snel<br />

was inspired by an ‘experimentum elegans’ described by Alhacen <strong>and</strong> copied<br />

by Witelo that involved a segmented disc lowered into water. This led him to<br />

study the refractaria <strong>and</strong>, facilitated by his geodetic expertise, to the law of<br />

refraction in secans form. 68 I do not fully agree with Hentschel’s analysis, for<br />

I think that the idea of a contraction of the unrefracted perpendicular ray<br />

may have opened to Snel a more direct route to his discovery. Whichever<br />

interpretation is preferable, the main point is that Snel employed means<br />

readily available to Kepler. What is more, his approach of rational analysis of<br />

mathematical regularities in a set of refracted rays was precisely how Kepler<br />

set about initially. He even analyzed the refractaria from various perspectives,<br />

which makes it all the more surprising that Snel was seemingly unfamiliar<br />

with Paralipomena. 69 It remains to be seen why Snel was successful - or: why<br />

he was satisfied with what he found, as contrasted to Kepler’s fruitless<br />

struggle. Maybe he was less strict in empirical matters or he was - like Harriot<br />

- just lucky with looking at the issue from the right perspective.<br />

Paralipomena <strong>and</strong> the seventeenth-century reconfiguration of optics<br />

The central concept of perspectiva was the visual ray, which established the<br />

visual relation between objects <strong>and</strong> observer. 70 In seventeenth-century optics<br />

the concept of ray underwent two substantial changes, both anticipated by<br />

Kepler: the subordination of vision to light <strong>and</strong> the physicalization of the ray.<br />

66<br />

Lohne, “Geschichte des Brechungsgesetzes”, 160-161. Harriot corresponded with Kepler after the<br />

publication of Paralipomena. The correspondence broke off, however, before Harriot could reveal his<br />

findings. KGW2, 425.<br />

67<br />

The notes are in Vollgraff, Risneri Opticam. The outline was discovered by Cornelis de Waard, who<br />

transcribed <strong>and</strong> translated it in Waard, “Le manuscript perdu de Snellius”. A German translation is given<br />

in Hentschel, “Das Brechungsgesetz”, 313-319.<br />

68<br />

Hentschel, “Das Brechungsgesetz”, 302-308.<br />

69<br />

Hentschel, “Das Brechungsgesetz”, 334 note 22.<br />

70<br />

Smith, “Saving the appearances”, 86-89.

THE 'PROJET' OF 1672 125<br />

The most important change in the mathematical study of light was the<br />

ab<strong>and</strong>onment of questions of cognition. Perspectivist theory not only<br />

consisted of a theory of perception but also seized epistemological <strong>and</strong><br />

psychological problems of visual cognition. 71 The eye was crucial in that the<br />

behavior of rays was understood on the basis of an underst<strong>and</strong>ing of visual<br />

cognition. In the seventeenth-century optics the eye became a subordinate<br />

topic in the mathematical study of optics <strong>and</strong> questions of cognition were<br />

ab<strong>and</strong>oned altogether. Kepler’s theory of image formation was a theory of<br />

rays painting pictures on a dead, passive surface. His theory of the retinal<br />

image was a theory only of ray tracing <strong>and</strong> he passed over physiological <strong>and</strong><br />

psychological issues. Only in the fifth chapter of Paralipomena did he explain<br />

how the eye paints pictures on the retina, after he had explained image<br />

formation, reflection <strong>and</strong> refraction. The mathematical analysis of the<br />

behavior of light rays was turned into the study of the paths of light rays<br />

without an eye necessarily being present. Instead of the foundation of geometrical<br />

optics, vision became an application of it. The ray became a light ray instead of a<br />

visual ray.<br />

Kepler’s theory was readily assimilated in the first decades of the<br />

seventeenth century. The subordination of vision to the theory of image<br />

formation is clear in most seventeenth-century works on geometrical optics.<br />

This includes Huygens, who deferred his discussion of the eye to the last<br />

chapter of his projected ‘Dioptrique’. Shapiro has pointed out that Barrow’s<br />

thinking in terms of images as the eye perceives them was crucial to his<br />

extension of Kepler’s theory of image formation, as had been the case with<br />

Gregory. 72 Yet, they too confined themselves to retinal imagery <strong>and</strong> adopted<br />

the Keplerian underst<strong>and</strong>ing of the eye as an optical instrument that painted<br />

images on the retina.<br />

Closely connected with the changing role of the eye <strong>and</strong> vision in the<br />

mathematical study of light is the changing meaning of the optician’s<br />

elementary tool: the ray. Whereas the mathematical line used in optical<br />

analysis in perspectiva represented a real line in space, it came to represent an<br />

imaginary line in time in the course of the seventeenth century. 73 Instead of<br />

constituting light itself, a ray of light became – in various ways – the path<br />

traced out by some substance that constituted light. The corpuscular<br />

conceptions in the new philosophies of the seventeenth century transformed<br />

the light ray into an effect of some material action. Kepler’s conception of<br />

light <strong>and</strong> his analysis of reflection <strong>and</strong> refraction anticipated this, but with<br />

him light remained expressly incorporeal. One may say that the combined<br />

subordination of questions of vision <strong>and</strong> ‘physicalization’ of light constitutes<br />

the transition from medieval perspectiva to seventeenth-century geometrical<br />

optics.<br />

71 Smith, “Big picture”, 587-589.<br />

72 Shapiro, “The Optical lectures”, 137.<br />

73 Smith, “Ptolemy’s search”, 239-240.

126 CHAPTER 4<br />

It is beyond dispute that Kepler was crucial to the development of<br />

seventeenth-century optics. With his seminal work, he gave the study of<br />

optics a new start at the beginning of the seventeenth century. What his<br />

influence was exactly is harder to determine. As a result of the advent of<br />

corpuscular conceptions of nature, his explanation of the nature of light was<br />

outdated almost immediately. On the level of theories <strong>and</strong> mathematical<br />

concepts his influence is clear: his theory of image formation <strong>and</strong> of vision<br />

were the starting-point of all subsequent studies. However, his contribution<br />

was largely obscured by the uncredited adoption of his ideas by Descartes<br />

most notably. On the level of methodology the matter is less clear. Descartes<br />

called Kepler his “first teacher in Optics”, but what he had been taught he<br />

did not say. 74 He did not, for one thing, adopt Kepler’s c<strong>and</strong>or as regards the<br />

way he discovered things. Seventeenth-century savants found Kepler’s<br />

Renaissance conceptions hard to take <strong>and</strong> the odor of mysticism that<br />

surrounded him seems to have been responsible for the fact that few<br />

referred to Kepler directly. As regards the way mathematical reasoning could<br />

be applied to underst<strong>and</strong> natural phenomena, he was quickly overshadowed<br />

by Descartes <strong>and</strong> Galileo. Huygens, in particular, was silent on Kepler as<br />

regards his approach to optics.<br />

4.1.3 THE LAWS OF OPTICS IN CORPUSCULAR THINKING<br />

The new philosophies of the seventeenth century came to see light as an<br />

effect of some material action. As a consequence, the mechanical analogies<br />

used in perspectivist accounts of reflection <strong>and</strong> refraction were put in a<br />

different light. Discussions of motions <strong>and</strong> impact regarding the causes of<br />

reflection <strong>and</strong> refraction were now connected directly with the essence of<br />

light. Yet, accounting for the nature of light was not integrated with<br />

mathematical analysis of the behavior of light rays at one go. This is evident<br />

in Descartes’ account of refraction in La Dioptrique, a peculiar amalgam of<br />

perspectivist <strong>and</strong> mechanistic reasoning. In La Dioptrique Descartes made<br />

public the sine law, which he had discovered in Paris in the late 1620s during<br />

his collaborative efforts to realize non-spherical lenses (see section 3.1). How<br />

exactly he arrived at the sine law remains a subject for debate, but it is certain<br />

Descartes did not discover it along the lines of his account in La Dioptrique.<br />

Descartes’ account of refraction is difficult to comprehend in twentiethcentury<br />

parlance. A quick detour via the correspondence of Claude Mydorge,<br />

one of his Parisian collaborators, will be enlightening for modern readers. In<br />

a letter to Mersenne from around 1627, Mydorge used a rule to calculate<br />

angles of refraction, given the angles of one pair of incident <strong>and</strong> refracted<br />

rays (Figure 39). If FE-GE is the given pair, the refraction EN of HE is found<br />

in the following way. Draw a semicircle around E that cuts EF in F. Draw IF<br />

parallel to AB, <strong>and</strong> from I drop IG parallel to CE, cutting EG in G. Draw a<br />

second semicircle around E through G. Now draw HM, cutting the first<br />

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

semicircle in M, <strong>and</strong> drop MN,<br />

cutting the second semicircle in N.<br />

EN is the required refracted ray.<br />

The rule comes down to a<br />

cosecant ‘law’: cosec i : cosec r =<br />

FE : EG. Later in the letter,<br />

Mydorge applied this rule to lenses<br />

<strong>and</strong> transformed it into sine form. 75<br />

Mydorge’s rule embodies the<br />

two assumptions that formed the<br />

core of Descartes’ derivation of<br />

the sine law in La Dioptrique. First,<br />

a constant ratio between the<br />

incident <strong>and</strong> refracted rays,<br />

represented by the constant ratio<br />

of the radii of the two semi-circles.<br />

THE 'PROJET' OF 1672 127<br />

Second, the constant length of the parallel components FO <strong>and</strong> OI before<br />

<strong>and</strong> after refraction. In the diagram of Mydorge these assumptions are<br />

represented directly by the lengths of the respective lines. However, instead<br />

of a distance diagram, in La Dioptrique Descartes used a time diagram where<br />

the lengths of lines represent duration (Figure 40). Instead of the two semicircles<br />

representing the constant ratio of the effect of the media, it shows a<br />

single circle. As a consequence the constancy of the parallel component was<br />

represented by lines of differing length (AH <strong>and</strong> HF).<br />

I have begun with Mydorge’s<br />

rule because it somewhat bridges<br />

the gap between Cartesian<br />

conceptualization of refraction <strong>and</strong><br />

our underst<strong>and</strong>ing of the sine law.<br />

It gives the modern reader a clear<br />

idea of the assumptions<br />

fundamental to Descartes’<br />

derivation as well as the way he<br />

adapted it to his own line of<br />

thinking. As I will argue below, the<br />

diagrams Descartes used fitted<br />

Figure 39 Mydorge's rule<br />

Figure 40 Descartes’ analysis of refraction<br />

perspectivist analysis of refraction rather than his own account, <strong>and</strong> he chose<br />

them deliberately. The account of La Dioptrique, with all its complicating<br />

facets, was how seventeenth-century readers got to know the sine law <strong>and</strong><br />

the mechanistic interpretation of refraction. It formed the starting-point of<br />

all subsequent accounts of the causes of refraction, although few adopted<br />

Descartes’ conceptual <strong>and</strong> methodological notions in full.<br />

75 Mersenne, Correspondence I, 404-415. This letter <strong>and</strong> its import for Descartes’ optics is discussed<br />

thoroughly in Schuster, “Descartes opticien”, 272-277 <strong>and</strong> Schuster, Descartes <strong>and</strong> the Scientific Revolution, 304-<br />

308.

128 CHAPTER 4<br />

Refraction in La Dioptrique<br />

Descartes began La Dioptrique with an explication of the way rays of light<br />

enter the eye <strong>and</strong> are deflected on their way to it. He did not intend to<br />

explain the true nature of light, he said, as the essay ought to be intelligible to<br />

the common reader. He took the liberty, he said, to employ a threesome of<br />

comparisons between the behavior of light <strong>and</strong> everyday phenomena:<br />

“…; imitating in this the Astronomers, who, although their assumptions are almost all<br />

false or uncertain, nevertheless, because these assumptions refer to different<br />

observations they have made, do not fail to draw many true <strong>and</strong> well-assured<br />

conclusions from them.” 76<br />

First, light acts like the white stick that enables a blind man to sense objects;<br />

it is an action instantaneously propagated through a medium without matter<br />

being transported. Second, this action is like the tendency of a portion of<br />

wine in a barrel of half-pressed grapes to move to a hole in the bottom. It<br />

works along straight lines that can cross each other without hindrance. In<br />

other words, light is not a motion but a tendency to motion:<br />

“And in the same way, considering that it is not so much the movement as the action<br />

of luminous bodies that must be taken for their light, you must judge that the rays of<br />

this light are nothing else but the lines along which this action tends.” 77<br />

Although essentially light is a tendency to movement rather than actual<br />

motion, with respect to the deflections from its straight path rays of light<br />

follow the laws of motion, Descartes maintained. So, in the third<br />

comparison, the way light interacts with mediums of different nature is<br />

compared to the deflections of a moving ball encountering hard or liquid<br />

bodies. Thus the three comparisons of the first discourse of La Dioptrique<br />

established a qualitative basis for the mathematical account of refraction in<br />

the next.<br />

The second discourse ‘Of refraction’ opens with an account of reflection<br />

providing the conceptual basis for Descartes’ explanation of the ‘way in<br />

which refractions ought to be measured’. 78 It introduces a crucial distinction<br />

with regard to the powers governing the motion of an object: one that works<br />

to continue the ball’s motion <strong>and</strong> one that determines the particular direction<br />

in which the ball moves. 79 Instead of the more accurate ‘absolute quantity of<br />

force of motion’ <strong>and</strong> ‘directional quantity of force of motion’, for sake of<br />

convenience I will speak of ‘quantity’ <strong>and</strong> ‘direction’ both of which may<br />

76 Descartes, AT6, 83. “imitant en cecy les Astronomes, qui, bien que leurs suppositions soyent presque<br />

toutes fausses ou incertaines, toutefois, a cause qu’elles se rapportent a diverses observations qu’ils ont<br />

faites, ne laissent pas d’en tirer plusieurs consequences tres vrayes & tres assurées.” (Translation based on<br />

Olscamp)<br />

77 Descartes, AT6, 88. “& ainsy, pensant que ce n’est pas tant le mouvement, comme l’action des cors<br />

lumineus qu’il faut prendre pour leur lumiere, vous devés iuger que les rayons de cete lumiere ne sont<br />

autre chose, que les lignes suivant lesquelles tend cete action.” (Translation based on Olscamp)<br />

78 “… en quelle sorte se doivent mesurer les refractions”, AT6, 101-102.<br />

79 “Seulement faut il remarquer, que la puissance, telle qu’elle soit, qui fait continuer le mouvement de cete<br />

balle, est differente de celle que la determine a se mouvoir plustost vers un costé que vers un autre, …”<br />

AT6, 94.

THE 'PROJET' OF 1672 129<br />

apply to Cartesian motion proper as well as to tendency to movement. When<br />

a ball rebounds from the surface of an impenetrable body the following<br />

happens. The quantity of its motion is unaffected because it remains moving<br />

through the same medium - the air surrounding the body - <strong>and</strong> only the<br />

direction changes. Regarding the parallel <strong>and</strong> perpendicular components of<br />

the direction, Descartes noted that the body offers resistance only in the<br />

direction perpendicular to its surface. Thus the parallel component is<br />

unaltered.<br />

To determine the path of the<br />

ball after the impact, Descartes<br />

switched to a derivation in which<br />

he graphically mathematized the<br />

assumptions just established<br />

(Figure 41). In circle AFD radius<br />

AB represents the path along<br />

which the ball approaches the<br />

surface where it rebounds from B<br />

in some direction. As the quantity<br />

of motion is constant, the ball<br />

must traverse the same distance<br />

Figure 41 Descartes’ analysis of reflection<br />

after reflection. It thus reaches the circumference of the circle somewhere.<br />

Since the parallel component of its direction is also constant, it follows that<br />

the horizontal distance traversed after reflection must be equal too.<br />

Therefore, BE is equal to BC. Under these conditions the ball can either<br />

arrive at point D or point F on the circle. It cannot penetrate the body below<br />

GE <strong>and</strong> so F is the only option left. “And thus you will easily see how<br />

reflection occurs, namely according to an angle always equal to the one that<br />

is called angle of incidence”, Descartes concluded without much further<br />

ado. 80<br />

Like reflection, refraction is understood as the combined effect on the<br />

quantity <strong>and</strong> the direction of motion. The only difference is that in refraction<br />

the ball penetrates the medium. In other words, it enters a medium of<br />

different density. Therefore the quantity of motion changes. It does so at the<br />

passing of the surface separating both mediums. This can be compared to<br />

smashing a ball through a thin cloth. It loses part of its speed, say half. Again<br />

only the perpendicular component of the direction of the motion is affected<br />

<strong>and</strong> the parallel component remains unaltered. As in the case of reflection,<br />

Descartes switched to a mathematical derivation in the form of a diagram to<br />

determine the exact path of the ball after impact (Figure 40). As a result of<br />

the loss of speed, it takes the ball twice as long to reach the circumference of<br />

the circle after impact at B. However, as its determination to advance parallel<br />

to the surface is unchanged, it moves twice as far to the right in this time.<br />

80 “Et ainsy vous voyés facilement comment se fait la reflexion, a sçavoir selon un angle tousiours esgal a<br />

celuy qu’on nomme l’angle d’incidence.” AT6, 96.

130 CHAPTER 4<br />

Therefore the distance between lines FE <strong>and</strong> HB must be twice a large as that<br />

between AC <strong>and</strong> HB. As a result, the ball reaches point I on the circle. The<br />

same is the case when instead of a cloth the ball hits the surface of a body of<br />

water. For the water does not alter the motion of the ball any further after it<br />

has passed the surface, according to Descartes. When the ball passes a<br />

boundary where in some way or another its quantity of motion is augmented,<br />

it reaches the circumference of the circle earlier <strong>and</strong> is deflected towards the<br />

normal of the surface. Note that Descartes did not specify the change of the<br />

perpendicular component, a point that is often overlooked. He did not know<br />

that amount <strong>and</strong> he did not need to, for the two assumptions he used suffice<br />

for the derivation of the sine law. 81<br />

As Descartes took the motions of the ball to reflect the deflections of<br />

light, he could now draw his main conclusion. Rays of light are deflected in<br />

exact proportion to the ease with which a transparent medium receives them<br />

compared to the medium from which they come. The only remaining<br />

difference between the motion of a ball <strong>and</strong> the action of light is that a<br />

denser medium like water allows rays of light to pass more easily. The<br />

deflection caused by the passage from one medium into another ought to be<br />

measured, not by the angles made with the refracting surface, but by the lines<br />

CB <strong>and</strong> BE. Unlike the proportion between the angles of incidence <strong>and</strong><br />

refraction, the proportion between these sines remains the same for any<br />

refraction caused by a pair of mediums, irrespective of the angle of<br />

incidence. Et voilà, the law of sines.<br />

Epistemic aspects of Descartes’ account in historical context<br />

Both historically <strong>and</strong> intrinsically, Descartes’ account of refraction is a key<br />

text in the transition from medieval perspectiva to seventeenth-century<br />

optics. Yet, the line of inference is subtle <strong>and</strong>, at many points, implicitly<br />

pursued. I will have to enlarge in some detail on its epistemic aspects in their<br />

historical context.<br />

At least three levels of inference can be distinguished in Descartes’<br />

account. In the first place the level of mathematics. This holds the derivation<br />

of the sine law from the two assumptions conveyed in the diagrams<br />

accompanying his discourse. First, the passage to another medium alters in a<br />

fixed ratio the quantity of motion. This ratio is represented by the radius of<br />

the circle. Second, the parallel component of the direction of motion is<br />

unaffected. This is represented by drawing horizontal lines in proportion to<br />

the successive times to travel to <strong>and</strong> from the center of the circle. The<br />

mathematical inference of Descartes’ account constituted a successful<br />

culmination of perspectivist optics, in that Descartes was the first to derive a<br />

law of refraction on the analytical groundwork laid by Kepler <strong>and</strong> his<br />

forebears. He brought consistency to the analysis of reflection <strong>and</strong> refraction<br />

by having the parallel component constant in all cases. More important, in<br />

81 When both aspects of the motion are interpreted as speeds the assumptions can be written as: vr = nvi<br />

<strong>and</strong> vi sini = vr sinr, which directly yield sini = n sinr. See Sabra, Theories of Light, 111.

THE 'PROJET' OF 1672 131<br />

the first assumption, he stated an exact relationship between the medium <strong>and</strong><br />

the length of a ray. Combined with the second assumption – which was not<br />

new – the sine law could be derived. Mathematically speaking, the proof – as<br />

Newton later phrased it – was not inelegant. It was fairly undisputed in the<br />

seventeenth century <strong>and</strong> the starting point for much optical investigations. 82<br />

Descartes’ first assumption was more than a purely mathematical<br />

assumption, which brings us to the second level of inference that holds the<br />

physical properties of rays. The physics of rays had been central in<br />

perspectivist optics, but the content of Descartes’ assumptions was<br />

innovative. According to Sabra <strong>and</strong> Schuster, stating a positive dependence<br />

of the motion of light on the density of the medium, irrespective of the<br />

direction of propagation, made up the decisive break with tradition. 83<br />

Descartes may have drawn inspiration for this from his reading of<br />

Paralipomena (which he did not acknowledge at all in La Dioptrique). In<br />

proposition XX of chapter 1 <strong>and</strong> the sequel section of chapter 4, Kepler also<br />

associated the propagation of a ray with the medium. Descartes may have<br />

read Kepler’s diagrams physically, so that the length of the rays represent the<br />

action of light as affected by the media. 84 Descartes’ diagram represented the<br />

actions involved when a ray enters a refracting medium <strong>and</strong> served to justify<br />

his assumptions. He did so by drawing an analogy between a refracted ray<br />

<strong>and</strong> a tennis ball struck through a frail canvas by the man in the diagram<br />

(Figure 40).<br />

As we have seen, these mechanical analogies had a long history in optics<br />

with a direct line from Alhacen to Kepler <strong>and</strong>, now, Descartes. The<br />

mechanical analogies had a different meaning for Descartes than for his<br />

perspectivist forebears. To an Alhacen the motions of bodies compared to<br />

light only with respect to its propagation, not its essence. According to<br />

Descartes light was essentially corpuscular. He made clear that they went<br />

further than a mere analogy:<br />

“… when [rays] meet certain other bodies they are liable to be deflected by them, or<br />

weakened, in the same way as the movement of a ball or a rock thrown in the air is<br />

deflected by those bodies it encounters. For it is quite easy to believe that the action or<br />

the inclination to move which I have said must be taken for light, must follow in this<br />

the same laws as does movement.” 85<br />

However, Descartes took care not to transgress the conceptual <strong>and</strong><br />

methodological boundaries of perspectiva openly. He presented his account<br />

82<br />

Huygens’ case is discussed below in section 4.2.1., Newton in section 5.2.2. of the next chapter. This<br />

theme is leading in Dijksterhuis, “Once Snel breaks down”. Newton’s view is cited below on page 133,<br />

footnote 98.<br />

83<br />

Sabra, Theories, 97-107; Schuster, Descartes, 333-334.<br />

84<br />

Schuster, “Descartes opticien”, 279-285; Schuster, Descartes, 334-336.<br />

85<br />

Descartes, AT6, 88-89. “mais, lors qu’ils rencontrent quelques autres cors, ils sont sujets a estre<br />

détournés par eux, ou amortis, en mesme façon que l’est le mouvement d’une balle, ou d’une pierre iettée<br />

dans l’air, par ceux qu’elle rencontre. Car il est bien aysé a croire que l’action ou inclination a se mouvoir,<br />

que j’ay dit devoir estre prise pour la lumiere, doit suivre en cecy les mesmes loys que le mouvement.”<br />

(Translation based on Olscamp)

132 CHAPTER 4<br />

in terms of analogies <strong>and</strong> explicitly said these did not reflect the true nature<br />

of light. Restricting in this way his account to the behavior of rays, he<br />

methodologically tied in with tradition. Still, mechanistic thinking was at the<br />

heart of La Dioptrique. Assuming a proportionality between density <strong>and</strong><br />

motion is almost unthinkable outside a corpuscular framework. Indeed, at<br />

the close of the second discourse Descartes showed his h<strong>and</strong>. The<br />

comparisons had a much higher content of realism than suggested by his<br />

circumspect introduction of them.<br />

“For finally I dare to say that the three comparisons which I have just used are so<br />

correct, that all the particularities that that can be noted in them correspond to certain<br />

others which are found to be very similar in light; …” 86<br />

If the mechanisms Descartes employed in the analogies <strong>and</strong> to which he<br />

ascribed a fair degree of realism do little to persuade our post-Galilean<br />

minds, one ought to remember that they were modeled on an underst<strong>and</strong>ing<br />

of motion that was rooted in a hydrostatics of pressures rather than a<br />

kinematics of velocities. Probably this was also one of the reasons the<br />

analogies did not convince his seventeenth-century readers either. 87<br />

Descartes’ intricate employment of mechanical analogies brings us to the<br />

third level of inference in his account of refraction, where the physical nature<br />

is involved in the analysis. Although he did not elaborate his theory of light<br />

<strong>and</strong> circumspectly presented the mechanics of deflected motion as analogy,<br />

Descartes’ line of reasoning strongly suggests that the laws of optics to be<br />

derived from his mechanistic underst<strong>and</strong>ing of light. In Sabra’s words: “As<br />

repeatedly asserted by Descartes, the ‘suppositions’ at the beginning of the<br />

Dioptric belong to this [domain of a priori truth]”. 88 This is substantiated by<br />

the fact that Descartes deviated from perspectivist tradition in a second<br />

important respect as well. In La Dioptrique he did not explicitly call for an<br />

empirical foundation of the sine law. In this way, Descartes’ derivation of the<br />

sine law was intended as a derivation from the true nature of light.<br />

Historian’s assessment of Descartes’ optics<br />

The question whether or not Descartes actually succeeded in deriving the<br />

sine law from his mechanistic theory of light has been a matter of incessant<br />

debate among historians of science. Although few seventeenth-century<br />

students of optics were convinced by Descartes argument, I think it<br />

appropriate to digress somewhat to contemporary evaluations because these<br />

are illuminating as regard the exact purport of his account.<br />

Many have argued that Descartes’ claim, that a tendency to move is<br />

subject to the same laws as motion itself, was mere rhetoric. Schuster, on the<br />

other h<strong>and</strong>, argues that Descartes’ theory of light did provide the basis of the<br />

86 “Car enfin j’ose dire que les trois comparaisons, dont je viens de me servir, sont si propres, que toutes<br />

les particularités que s’y peuvent remarquer, se raportent a quelques autres qui se trouvent toutes<br />

semblables en la lumiere; …” AT6, 104.<br />

87 Except Clerselier who expressly defended Descartes’ mechanistic models; Sabra, Theories, 116-135.<br />

88 Sabra, Theories, 44.

THE 'PROJET' OF 1672 133<br />

analogies, despite the fact that it hardly appears in La Dioptrique. 89 Drawing<br />

on the work of Mahoney, he says that the analogies provided a ‘heuristic<br />

model’ that legitimately compared the action of light with the motion of a<br />

ball. By leaving specific material factors in the motion of the ball aside,<br />

Descartes could single out the ball’s tendency to move rather than its<br />

motion. He then was ready to consider this tendency <strong>and</strong> distinguish<br />

between “… the power, …, which causes the movement of this ball to<br />

continue …” <strong>and</strong> “… that which determines it to move in one direction<br />

rather than in another, …” 90 According to Schuster this does not refer to a<br />

distinction between force of motion <strong>and</strong> direction of motion, but to a<br />

distinction between quantity of force of motion <strong>and</strong> directional magnitude of<br />

force of motion. The two assumptions of Descartes’ derivation are based on<br />

this distinction: the quantity depended on the medium <strong>and</strong> the parallel<br />

component of directional magnitude was constant. In La Dioptrique<br />

Descartes labeled the directional magnitude with the term ‘determination’ in<br />

order to analyze the components of the action without implicating the<br />

notion of velocity. 91<br />

With this interpretation of the analogies, Descartes’ analysis is not directly<br />

at odds with the system he expounded in Principia Philosophiae <strong>and</strong> Le Monde.<br />

There he had made the same distinction between quantity <strong>and</strong> directional<br />

magnitude. The first law of nature states that the quantity of force of motion<br />

is constant when a body is in uniform rectilinear motion; the third law states<br />

that a force of motion is conserved in a unique direction (tangent to the path<br />

of motion). 92 According to Schuster, the tension between the analogies <strong>and</strong><br />

the tendency theory can be resolved when Descartes’ heuristic use of the<br />

analogies is interpreted in the terms of his theory of motion. 93 In the light of<br />

the Galilean conception of motion Huygens <strong>and</strong> Newton employed (as do<br />

we), Descartes’ claim that he derived the laws of optics from his mechanistic<br />

principles was untenable. Sabra has sufficiently pointed this out. 94 Yet, this<br />

was not so much because his system was incoherent or inconsistent as,<br />

rather, because the interpretation of the underlying principles had changed.<br />

Descartes usually considered motion at an instant of impact <strong>and</strong> discussed it<br />

in terms of the body’s force to move. In the light of this science of motion,<br />

the mathematical derivation of the sine law can indeed be physically<br />

interpreted in a plausible manner. Yet, through his crude presentation in La<br />

Dioptrique Descartes made little effort to prevent misunderst<strong>and</strong>ings <strong>and</strong><br />

misinterpretations.<br />

89<br />

Schuster, “Descartes opticien”, 261-272 Schuster, Descartes, 273; Mahoney, Fermat, 387-393; Sabra,<br />

Theories, 78-89.<br />

90<br />

Descartes, AT6, 94-95.<br />

91<br />

Schuster, “Descartes opticien”, 258-261; Schuster, Descartes, 293<br />

92<br />

Schuster, Descartes, 288.<br />

93<br />

Schuster, “Descartes opticien” 261-265.<br />

94<br />

Sabra, Theories, 112-116.

134 CHAPTER 4<br />

Schuster has proposed a possible route along which Descartes’ discovery<br />

of the sine law may have taken place. 95 In it, his collaboration with Mydorge<br />

plays an important role - the cosecant rule being a crucial step towards the<br />

law of sines. According to Schuster, the actual discovery was independent of<br />

Descartes’ mechanistic ‘predilections’; rather the other way around: the latter<br />

were triggered by the former. 96 Shea has argued for a different route to the<br />

discovery, via measurements of angles of refraction by means of a prism. 97 In<br />

this variant too, the discovery was the result of an analysis of the behavior of<br />

rays. Descartes developed his mechanistic interpretation of his analysis of<br />

refraction after the discovery. With the presentation in La Dioptrique, he then<br />

obscured his analysis <strong>and</strong> explanation considerably. He adopted the use of<br />

analogies <strong>and</strong> adapted his derivation of the sine law to the perspectivist<br />

analysis of refraction. It appears as if Descartes tried to make his theory look<br />

as traditional as possible.<br />

Yet, he deviated from tradition by reversing the way in which he justified<br />

the law. Descartes suggested that the laws of optics ought to be based on<br />

prior principles regarding the nature of light. And despite the circumspection<br />

of his presentation, it was clear that he regarded the mechanistic causes of<br />

refraction an important, if not crucial, matter. In this way, he shifted the<br />

focus of the mathematical study of light towards the nature of light <strong>and</strong> the<br />

causes of the laws of optics. La Dioptrique is hard to characterize in terms of<br />

seventeenth-century geometrical optics. On the one h<strong>and</strong>, it was obviously a<br />

treatise on geometrical optics. It discussed the behavior of light in terms of<br />

the mathematical laws of the propagation of rays, in particular as they are<br />

refracted by lenses. Still, it did not offer quite as thorough an account of<br />

lenses as one would expect from a mathematical treatise. As we have seen in<br />

chapter two, La Dioptrique did not elaborate the mathematical theory of<br />

refraction in a way modeled after Kepler’s Dioptrice. Its main goal was to<br />

establish the law of refraction <strong>and</strong> explain its main consequences for the<br />

working of the telescope.<br />

Reception of Descartes’ account of refraction<br />

In Descartes’ system of natural philosophy, natural phenomena were<br />

explained from mechanistic principles. His optics was the most elaborate<br />

example of this project. Even if this elaboration was not fully unproblematic,<br />

it made clear what a new, mechanistic science of optics should be about. It<br />

did not halt at the mathematical description of natural phenomena, nor at<br />

depicting micro-mechanisms to explain them, but sought to explain the<br />

mathematical laws of nature by its mechanistic nature. Notwithst<strong>and</strong>ing<br />

recent pleas by historians of science for Descartes’ integrity, few<br />

contemporaries accepted his explanation of refraction. “The author would<br />

have demonstrated not inelegantly the truth of this, if only he had not left<br />

95<br />

Schuster, Descartes, 321-326. See also: Costabel, “Refraction et La Dioptrique”.<br />

96<br />

Schuster, Descartes, 343-346.<br />

97<br />

Shea, Magic, 156-157.

THE 'PROJET' OF 1672 135<br />

room for doubt concerning the physical causes he assumed”, Newton wrote<br />

30 years later. 98 In view of Galileo’s science of motion it is doubtful whether<br />

the motion of a ball struck through a frail canvas is subject to the<br />

assumptions Descartes made. His explanation of refraction into a denser<br />

medium – towards the normal – was regarded most problematic. In order to<br />

account for the necessary increase of motion, he introduced the rather ad<br />

hoc assumption that the ball was struck again at the refracting surface.<br />

Besides rejecting Descartes’ theory of light on the conviction that the speed<br />

of light is finite, in the ‘Projet’ Huygens explicitly mentioned this extra<br />

assumption as one of the difficulties in Descartes’ derivation. 99<br />

Newton <strong>and</strong> Huygens wrote at a time when the law of sines as such had<br />

been generally accepted. This had taken some twenty years, during which it<br />

only slowly became widely known. Cavalieri in 1647 did not employ the law<br />

of sines <strong>and</strong> Gregory seems to have been ignorant of it as late as 1663. As we<br />

have seen in the previous chapters, Huygens was one of the very few to<br />

pursue the study of dioptrics in this period. Compared to the preceding<br />

decades, the 1660s witnessed a true upsurge of the study of light. The<br />

investigations of Grimaldi, Boyle, Hooke, Newton, Bartholinus, brought to<br />

light a collection of new properties shaking the foundations of optics.<br />

Remarkably, the final acceptance of the law of sines coincided with<br />

accusations of plagiarism directed at Descartes. In De natura lucis et proprietate<br />

(1662) Isaac Vossius said that Descartes had seen Snel’s papers <strong>and</strong><br />

concocted his own proof. We now know this charge to be undeserved but it<br />

has been adopted by many since. Descartes may have heard of Snel’s<br />

achievement through his contacts with the circle that included Golius (Snel’s<br />

successor) <strong>and</strong> Constantijn Huygens sr. around 1632, but he had found the<br />

law much earlier. Christiaan Huygens started to display doubts regarding<br />

Descartes’ originality since the early 1660s. Probably spurred by Vossius’<br />

claims, he traced <strong>and</strong> examined Snel’s papers. As late as 1693 he voiced his<br />

opinion as follows: “It is true that from all appearances these laws of<br />

refraction aren’t the invention of Mr. des Cartes, because it is certain that he<br />

has seen the manuscript book of Snel, which I also have seen.” 100 Most<br />

remarkable about this is that Huygens could have known, through his father,<br />

much earlier about Snel’s achievement. Constantijn sr. had heard of it<br />

through a letter from Golius of 1 November 1632. Apparently the topic had<br />

never entered their conversation.<br />

The slow adoption of the sine law may have been brought about by the<br />

bad odor of Descartes’ philosophy, or simply the slow diffusion of his<br />

works. Fermat was convinced of the sine law’s validity only after he found<br />

98 Newton, Optical lectures, 170-171 & 310-313.<br />

99 A similar conclusion can be drawn with respect to Descartes’ theory of hydrostatics on which his<br />

concept of ‘conatus’ was based. Shapiro, “Light, pressure”, 260-266.<br />

100 “Il est vray que ces loix de la refraction ne sont pas l’invention de Mr. des Cartes selon toutes les<br />

apparences, car il est certain qu’il a vu le livre manuscrit de Snellius, que j’ay vu aussi; ...” OC10, 405-6. See<br />

also OC13, 9 note 1.

136 CHAPTER 4<br />

his own demonstration. Right after the publication of La Dioptrique he had<br />

severely criticized Descartes’ derivation, <strong>and</strong> maintained his objections when<br />

supporters of Descartes reopened the debate in 1657. 101 Employing the<br />

principle of natural economy, previously used by Hero <strong>and</strong> Witelo for<br />

reflection, Fermat deduced the law of sines in 1662, thus strengthening his<br />

conviction that Descartes’ mechanistic line of reasoning had been false. To<br />

know that the law was independent of Descartes’ mechanistic reasoning may<br />

have facilitated its acceptance, although it may well be that the ostensible<br />

non-acceptance was simply a matter of inactivity on the front of optics<br />

during the 1640s <strong>and</strong> 1650s.<br />

The reception of La Dioptrique makes clear that the treatise is hard to<br />

situate in the development of seventeenth-century optics. It formed the<br />

starting-point of most subsequent investigations in optics, <strong>and</strong> has therefore<br />

been the focus of many historical studies. 102 La Dioptrique showed how the<br />

properties of light could be discussed in corpuscular terms <strong>and</strong> its readers<br />

got this message. Although few agreed with the details of Descartes’<br />

derivation of the sine law, nor with his system of mechanistic philosophy in<br />

full, he set the idiom for the all-prevailing thinking on light in corpuscular<br />

terms. As a consequence, the traditional analogies between light <strong>and</strong> motion<br />

implied a potential claim about the true nature of light <strong>and</strong> could not be used<br />

as informally as before. Descartes had intended to found the laws of optics<br />

in the mechanistic nature of light, but his derivation was not free from<br />

ambiguities <strong>and</strong> obscurities. A mathematician like Barrow adopted the<br />

corpuscular underst<strong>and</strong>ing of nature but not Descartes’ approach to<br />

explanation. We now turn to him, to see how he dealt with questions<br />

regarding the status of the corpuscular nature of light <strong>and</strong> how it ought to<br />

explain the laws of optics.<br />

Barrow’s causal account of refraction<br />

Barrow was a mathematician with a clear awareness of the epistemological<br />

intricacies of mathematics <strong>and</strong> its applications to nature. The lectures on<br />

mathematics which he delivered at Cambridge between 1664 <strong>and</strong> 1666 dealt<br />

at great length with the status of mathematical concepts <strong>and</strong> methods <strong>and</strong><br />

their relevance for the study of nature. 103 His subsequent lectures on optics<br />

are likewise riddled with epistemic statements. Lectiones XVIII of 1669 is<br />

illuminating with respect to Huygens’ ‘Projet’ as it assigns a similar role to<br />

explanations of the causes of the laws of optics. The subject of the lectures<br />

was ‘Optics’, one of the fields that are “… bright with the flowers of Physics<br />

<strong>and</strong> sown with the harvest of Mechanics,…” 104 The core of this science<br />

101<br />

The debate is listed in Smith, Descartes’s theory of light <strong>and</strong> refraction, 81-82 <strong>and</strong> discussed in detail in Sabra,<br />

Theories of Light, 116-135.<br />

102<br />

Sabra, Theories, 12.<br />

103<br />

Published in 1666 as Lectiones mathematicae XXIII. They were translated by John Kirkby <strong>and</strong> published in<br />

1734 under the title The usefulness of mathematical learning etc. It is cited in Shapiro, Fits providing improved<br />

translations.<br />

104<br />

Barrow, Lectiones, [10].

THE 'PROJET' OF 1672 137<br />

consisted of six generally accepted principles required to elaborate<br />

mathematical theory. In a way reminiscent of the ‘Projet’, Barrow said that<br />

these hypotheses, as he called them, were empirically founded but also<br />

needed some sort of explanation:<br />

“The hypotheses agree with observation, but we must also fortify them with some<br />

support of reason, by treading on the foundations <strong>and</strong> suppositions laid down.” 105<br />

In his first lecture Barrow discussed these foundations <strong>and</strong> suppositions,<br />

although he mainly defined terms like ‘light’ (in relation to illumination,<br />

images, ‘phasmata’ <strong>and</strong> the like), ‘refraction’ <strong>and</strong> ‘opaque’. He then proposed<br />

a theory of light that is a hybrid of viewing light as a pulse <strong>and</strong> as a pressure<br />

propagated simultaneously in the first <strong>and</strong> second matter of the Cartesian<br />

scheme. 106 Whatever he meant precisely, Barrow did not lend much weight to<br />

this theory.<br />

“Still, since it is desirable for me to lay some preliminary foundations about the nature of<br />

light, to agree with my explanation of hypotheses which I shall later offer, I conceive<br />

the facts to be these, or something like them: …” 107<br />

These preliminary foundations merely needed to be consistent with the<br />

ensuing explanations of the laws of optics <strong>and</strong> Barrow expressly did not<br />

claim any authority in these matters. 108 In what followed, Barrow’s theory of<br />

light came down to considering a ray to be the path traced out by a pulse-like<br />

entity, “… two-dimensional <strong>and</strong> like a sort of rectangular parallelogram lying<br />

in a plane at right angles to the surface of the inflecting medium, …” 109 This<br />

conception of a physical ray traced out by a line of light emitted by a shining<br />

object went back to Hobbes’ theory of light. Barrow’s derivation of the law<br />

of sines can likewise be traced back to Hobbes. 110 With this definition of a<br />

ray, Barrow now could make ‘some attempt to explain’ the laws of optics,<br />

stressing once more that they were empirically founded:<br />

“… I need practically nothing else to explain the hypotheses which all opticians in<br />

common with each other assume <strong>and</strong> which must necessarily be laid down as a<br />

foundation for building up this science. I shall make no effort to prove what I have<br />

said, since … it seems clearer than light itself that such proofs cannot be given,<br />

although a number of experiments show that they are given in actuality.” 111<br />

Besides accounting for the rectilinearity of light rays, he discussed some basic<br />

assumptions of geometrical optics, like the fact that ‘inflections’ take place in<br />

a plane perpendicular to the surface of the ‘inflecting’ medium. Then, in the<br />

second lecture, he moved on to these ‘inflections’ proper, reflection <strong>and</strong><br />

105<br />

Barrow, Lectiones, [26].<br />

106<br />

Barrow, Lectiones, [15-16].<br />

107<br />

Barrow, Lectiones, [15]; (emphasis in original).<br />

108<br />

Barrow, Lectiones, [8, 15].<br />

109<br />

Barrow, Lectiones, [26].<br />

110<br />

Shapiro, “Kinematic optics”, 177-181. Hobbes’ optics is discussed in the next chapter, section 5.2.1.<br />

111 Barrow, Lectiones, [17]

138 CHAPTER 4<br />

refraction. 112 For reflection, he<br />

considered BD – a line of light in the<br />

most realist sense of the word –<br />

colliding obliquely with a reflecting<br />

surface EF (Figure 42). He argued<br />

that, after B hits the surface, this end<br />

of the line of light rebounds while end<br />

D continues its way, resulting in the<br />

rotation of BD around its center Z.<br />

This rotation lasts until D hits the<br />

surface <strong>and</strong> the line of light is in position . The line of light then continues<br />

towards . From the symmetry of the situation the equivalence of the<br />

angles of incidence <strong>and</strong> reflection follows directly.<br />

To substantiate his claim, Barrow invoked a general ‘law’ of motion:<br />

“… that it is constantly found in nature, when a straight movement degenerates into a<br />

circular one, that it is the extreme parts of the moving objects that direct <strong>and</strong> control all<br />

motion.” 113<br />

He applied the same law to derive<br />

the sine law (Figure 43). On<br />

entering the more resisting<br />

medium below EF, point B of the<br />

line of light BD will be slowed<br />

down while D continues with the<br />

original speed. As a consequence<br />

DB will be rotated around a point<br />

Z until D also reaches the ‘denser’<br />

medium. Then the line of light <br />

will continue along a straight path.<br />

Now, the proportion between ZD<br />

<strong>and</strong> ZB is constant for any angle<br />

of incidence <strong>and</strong> depends upon<br />

the particular difference of the<br />

densities. From this it easily<br />

follows that for i =GBM <strong>and</strong><br />

r =N, sin i : sin r = ZD : ZB. 114 After thus explaining refraction into a rarer<br />

medium <strong>and</strong> total reflection, Barrow was ready to elaborate the ‘Optic<br />

Science’ of his lectures in the common manner:<br />

“…considering rays as one-dimensional (seeing that the other dimensions, in which<br />

physicists delight, have no importance for the calculations here undertaken).” 115<br />

Figure 42 Barrow’s explanation of reflection.<br />

Figure 43 Explanation of refraction.<br />

112 Like Maignan in his Perspectiva Horaria (1648), Barrow added a derivation of the law of reflection which<br />

Hobbes had not provided. See Shapiro, “Kinematic optics”, 175-178.<br />

113 Barrow, Lectiones, [28].<br />

114 Barrow, Lectiones, [29-31]<br />

115 Barrow, Lectiones, [39-41]

THE 'PROJET' OF 1672 139<br />

Lectiones XVIII treated optics as the mathematical science aimed at the<br />

analysis of the behavior of light rays. Priority was with the laws of optics,<br />

being laws of rays that were justified empirically <strong>and</strong> generally accepted. In<br />

this sense the lectures stood with both feet in traditional geometrical optics.<br />

Yet, Barrow was too conscious of epistemic issues regarding mathematics<br />

<strong>and</strong> of the new developments in natural philosophy to treat optics in the<br />

outright traditional manner of other contemporary works. A good example is<br />

the Opera mathematica, a mathematics textbook from 1669 by the Flemish<br />

Jesuit Andreas Tacquet, a correspondent of Huygens. In its catoptrical<br />

chapters, Tacquet makes room for a noncommittal survey of explanations of<br />

reflection: some give natural economy as the ‘ratio’ of reflection, others<br />

maintain that the perpendicular component of a ray’s motion is inverted, <strong>and</strong><br />

so on. 116 Even Descartes is reviewed, stripped of all corpuscular trimmings to<br />

be sure. Tacquet did not show preference for any of the alternatives, he only<br />

explained the various ways in which the law of reflection could be deduced.<br />

The business of a mathematical student of light was to establish those<br />

properties of rays interacting with varying mediums so that the laws<br />

describing its behavior could be derived logically.<br />

For Barrow mixed mathematics - where natural things are considered in<br />

their quantitative aspects - was a genuine part of mathematics. In his lectures<br />

on mathematics, Barrow effectively discarded the distinction between<br />

sensible <strong>and</strong> intelligible matter, so that a science like optics could approach<br />

the certainty of geometry. The certainty of inferences only depended on the<br />

certainty of the presuppositions - axioms, postulates, principles. 117 Barrow<br />

presented his explanations as a non-committal elucidation of empirically<br />

founded laws, similar to the mechanical analogies of perspectivist theory.<br />

The new mode of thought regarding the nature of things had changed the<br />

underst<strong>and</strong>ing of the nature of light <strong>and</strong> the causes of reflection <strong>and</strong><br />

refraction. Yet, compared to these, corpuscular accounts of the causes of<br />

reflection <strong>and</strong> refraction obtained a different meaning, as it implied a<br />

potential claim about the true nature of light. This, combined with his<br />

epistemic awareness, may explain Barrow’s reluctance to make strong claims<br />

about his explanations.<br />

In his comments, Barrow considerably qualified the status of his theory<br />

of light <strong>and</strong> his causal accounts. His focus was on the laws <strong>and</strong> he did not<br />

elaborate his account of the mechanistic nature of light in any detail or<br />

explore its consequences. He was rather vague about the necessity <strong>and</strong> role<br />

of such an account. The laws of optics should ‘not be repugnant to reason’<br />

<strong>and</strong> be given ‘some support of reason’. He invoked a law of motion, but did<br />

not intend to prove the laws like Descartes, by deriving them from his theory<br />

of light. He offered a physical rationale for the laws, without making clear<br />

the exact purpose of his explanations. As a consequence, he parried the<br />

116<br />

Tacquet, Opera mathematica (Antwerp, 1669), Catoptricae libri tres, 217-218<br />

117<br />

Shapiro, Fits, 31-36.

140 CHAPTER 4<br />

question raised by the new philosophies of what status the corpuscular<br />

nature of light should have <strong>and</strong> how it ought to explain the laws of optics.<br />

4.2 The mathematics of strange refraction<br />

Kepler <strong>and</strong> Descartes had drawn attention to the problem of the relationship<br />

between a theory expounding the true nature of light <strong>and</strong> the mathematical<br />

behavior of light rays. It remains to be seen how Huygens considered this<br />

issue. What exactly did he mean by explaining refraction with waves? What<br />

were those waves <strong>and</strong> how would he proceed from there to the sine law?<br />

The statements in the ‘Projet’ suggest that his opinion about causal accounts<br />

was similar to Barrow’s. Explaining refraction was a rather non-committal<br />

affair to satisfy the minds of the particularly curious. Still, he wanted to solve<br />

the problem strange refraction posed for Pardies’ explanation of ordinary<br />

refraction. Apparently, the nature of light was serious enough a matter for<br />

Huygens first to wish to get this inconsistency out of the way. 118 Given the<br />

definition of the problem, the line of his first attack of strange refraction is<br />

rather surprizing.<br />

Huygens’ first attempt at underst<strong>and</strong>ing strange refraction is found on<br />

some ten pages in his notebook. 119 In my view, it must have taken place<br />

around the same time he noted down the ‘Projet’, somewhere during the<br />

second half of 1672. 120 On the first pages Huygens jotted down some<br />

sketches characterizing the phenomenon. The first shows five pairs of<br />

incident <strong>and</strong> refracted rays (Figure 44). One of each pair, indicated by the<br />

letter r is refracted regularly (‘regelmatig’) according to the sine law, the other<br />

one indicated by the letter o is refracted irregularly (‘onregelmatig’). 121 Below,<br />

Huygens wrote what is irregular about it:<br />

“The perpendicularly incident [ray] is refracted It does not make a double reflection.” 122<br />

118 Ziggelaar correctly points out that the problem of strange refraction was a reason Huygens did not<br />

directly elaborate ‘Projet’ (which he sees as a new plan for a treatise on dioptrics), but he does not discuss<br />

his first attempt to solve it beyond a single, <strong>and</strong> incorrect, characterization. Ziggelaar, “How”, 181-182.<br />

See also page 162.<br />

119 Hug2, 173v-178v. It consists of seven pages numbered by Huygens (175r-178r), preceded by two <strong>and</strong> a<br />

half pages with some notes <strong>and</strong> followed by a page containing a further note plus the record of an<br />

experiment performed in 1679 (discussed in section 5.3.1) Parts of their content are reproduced in OC19,<br />

407-415.<br />

120 I disagree with the editors of the Oeuvres Complètes regarding the dating of the papers. I think this first<br />

study took place around the time of Pardies’ letter, much earlier than they presume. On 4 September<br />

1672, hardly a month later, Huygens wrote to his brother Constantijn, saying he was not yet going to<br />

publish “what I have observed of the crystal or talc of Icel<strong>and</strong>” (OC7, 219. “…ce que j’ay observè du<br />

Chrystal ou Talc d’Isl<strong>and</strong>e; …”). I think this remark refers to his discovery of another peculiar<br />

phenomenon displayed by Icel<strong>and</strong> crystal – polarization – recorded on the final pages of his investigation.<br />

The discovery is in OC19, 412-414. The editors date these between December 1672 <strong>and</strong> June 1673, but it<br />

is possible that they – or similar notes now lost – were written at the same, earlier date.<br />

121 Hug2, 173v. One half of Hug2, 174r is torn away; the page contains a remark that seems of a later date.<br />

122 Hug2, 173v; OC19, 407. “Perpendiculariter incidens refringitur Non facit duplicem reflexionem.” The<br />

editors combine this with a remark written on Hug2, 175v.

THE 'PROJET' OF 1672 141<br />

Figure 44 Sketch of refracted rays in Icel<strong>and</strong> crystal: r (‘regelmatig’) for<br />

ordinary refraction; o (‘onregelmatig’) for strange refraction.<br />

On 8 July 1672, Pardies wrote Huygens about strange refraction. He had<br />

visited Picard <strong>and</strong> taken a look at a piece of Icel<strong>and</strong> crystal brought from<br />

Denmark. Pardies did not believe the phenomenon contradicted the sine<br />

law, as he thought it could be explained from the composition of the crystal.<br />

“… it seems to me that it is not as troublesome as I had imagined to explain this effect.<br />

… I am very much mistaken if one cannot demonstrate that, if one were to cut various<br />

pieces of glass in rhomboid shape <strong>and</strong> simply put one on the other to make a total<br />

rhomboid out of them, two refractions would present themselves.” 123<br />

Some sketches Huygens made in his notebook<br />

around the same time are reminiscent of Pardies’<br />

view (Figure 45). They seem to explore how the<br />

composition of the crystal may explain strange<br />

refraction. The surface is drawn indented, so that<br />

part of the perpendicularly incident light actually<br />

falls on an oblique surface. A perpendicular ray<br />

falls upon the indented surface so that part of the<br />

wave is divided into many small wavelets, that<br />

proceed obliquely to the surface. 124<br />

Apparently, Huygens did not accept Pardies’ idea, for he did not<br />

elaborate it beyond these sketches. Moreover, he ended his first study with<br />

the conclusion that the refracted perpendicular contradicted the wave<br />

explanation of refraction. 125 A sketch on the next page of his notebook makes<br />

123 OC7, 193. “… il me semble qu’il n’est pas si malaisé que je m’estois imaginé, d’expliquer cét effet. Je<br />

suis fort trompé si l’on ne peut démonstrer que si l’on taillait plusiers pieces de verre en rhomboide et<br />

qu’on les mit simplement l’une sur l’autre pour en faire un rhomboide total, il s’y feroit deux refractions.”<br />

124 Hug2, 178v; OC19, 415.<br />

125 See below page 151 footnote 148.<br />

Figure 45 A refracted<br />

perpendicular caused by the<br />

composition of the crystal.

142 CHAPTER 4<br />

it clear what kind of problem strange refraction constituted<br />

for the wave theory (Figure 46). It shows the strange<br />

refraction of a perpendicular ray along with, what seems to<br />

be, the propagation of waves. 126 After having passed the<br />

refracting surface, the waves proceed obliquely to their<br />

direction of propagation, which contradicts the assumptions<br />

of Pardies' theory. Thus this tiny sketch illustrates what<br />

Huygens called the ‘difficulté’ of strange refraction.<br />

Strange refraction posed a problem for the explanation of<br />