Lenses and Waves

Archimedes
Volume 9

Archimedes
NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF
SCIENCE AND TECHNOLOGY
VOLUME 9
EDITOR
JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology,
Pasadena, CA, USA.
ADVISORY BOARD
HENK BOS, University of Utrecht
MORDECHAI FEINGOLD, Virginia Polytechnic Institute
ALLAN D. FRANKLIN, University of Colorado at Boulder
KOSTAS GAVROGLU, National Technical University of Athens
ANTHONY GRAFTON, Princeton University
FREDERIC L. HOLMES, Yale University
PAUL HOYNINGEN-HUENE, University of Hannover
EVELYN FOX KELLER, MIT
TREVOR LEVERE, University of Toronto
JESPER LÜTZEN, Copenhagen University
WILLIAM NEWMAN, Harvard University
JÜRGEN RENN, Max-Planck-Institut für Wissenschaftsgeschichte
ALEX ROLAND, Duke University
ALAN SHAPIRO, University of Minnesota
NANCY SIRAISI, Hunter College of the City University of New York
NOEL SWERDLOW, University of Chicago
Archimedes has three fundamental goals; to further the integration of the histories of
science and technology with one another: to investigate the technical, social and practical
histories of specific developments in science and technology; and finally, where
possible and desirable, to bring the histories of science and technology into closer contact
with the philosophy of science. To these ends, each volume will have its own
theme and title and will be planned by one or more members of the Advisory Board in
consultation with the editor. Although the volumes have specific themes, the series itself
will not be limited to one or even to a few particular areas. Its subjects include any
of the sciences, ranging from biology through physics, all aspects of technology, broadly
construed, as well as historically-engaged philosophy of science or technology.
Taken as a whole, Archimedes will be of interest to historians, philosophers, and scientists,
as well as to those in business and industry who seek to understand how science
and industry have come to be so strongly linked.

Lenses and Waves
Christiaan Huygens and the Mathematical Science
of Optics in the Seventeenth Century
by
FOKKO JAN DIJKSTERHUIS
University of Twente,
Enschede, The Netherlands
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contents
CHAPTER 1 INTRODUCTION – ‘THE PERFECT CARTESIAN’ 1
A history of Traité de la Lumière 2
Huygens’ optics 4
New light on Huygens 8
CHAPTER 2 1653 - 'TRACTATUS' 11
2.1 The Tractatus of 1653 12
2.1.1 Ovals to lenses 13
2.1.2 A theory of the telescope 16
The focal distance of a bi-convex lens 17
Images 20
Conclusion 24
2.2 Dioptrics and the telescope 24
2.2.1 Kepler and the mathematics of lenses 26
Image formation 28
Lenses 29
Perspectiva and the telescope 33
2.2.2 The use of the sine law 35
Descartes and the ideal telescope 36
After Descartes 37
Dioptrics as mathematics 40
2.2.3 The need for theory 41
The micrometer and telescopic sights 42
Understanding the telescope 46
Huygens’ position 50
CHAPTER 3 1655-1672 - 'DE ABERRATIONE' 53
3.1 The use of theory 55
3.1.1 Huygens and the art of telescope making 57
Huygens’ skills 58
Alternative configurations 59
Experiential knowledge 60
3.1.2 Inventions on telescopes by Huygens 63
3.2 Dealing with aberrations 67
3.2.1 Properties of spherical aberration 67
Specilla circularia 70
Theory and its applications 72

vi CONTENTS
3.2.2 Putting theory to practice 77
A new design 80
3.2.3 Newton’s other look and Huygens’ response 83
3.3 Dioptrica in the context of Huygens’ mathematical science 92
3.3.1 The mathematics of things 92
Huygens ‘géomètre’ 95
3.3.2 Huygens the scholar & Huygens the craftsman 100
The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument 104
CHAPTER 4 THE 'PROJET' OF 1672 107
‘Projet du Contenu de la Dioptrique’ 109
4.1 The nature of light and the laws of optics 112
4.1.1 Alhacen on the cause of refraction 114
4.1.2 Kepler on the measure and the cause of refraction 117
The measure of refraction 118
True measures 123
Paralipomena and the seventeenth-century reconfiguration of optics 124
4.1.3 The laws of optics in corpuscular thinking 126
Refraction in La Dioptrique 128
Epistemic aspects of Descartes’ account in historical context 130
Historian’s assessment of Descartes’ optics 132
Reception of Descartes’ account of refraction 134
Barrow’s causal account of refraction 136
4.2 The mathematics of strange refraction 140
4.2.1 Bartholinus and Huygens on Iceland Crystal 142
Bartholinus’ experimenta 143
Huygens’ alternatives 147
4.2.2 Rays versus waves: the mathematics of things revisited 152
The particular problem of strange refraction: waves versus masses 155
CHAPTER 5 1677-1679 - WAVES OF LIGHT 159
5.1 A new theory of waves 161
5.1.1 A first EUPHKA 162
The solution of the ‘difficulté’ of Iceland Crystal 168
5.1.2 Undulatory theory 172
Explaining strange refraction 176
5.1.3 Traité de la Lumière and the ‘Projet’ 181
5.2 Comprehensible explanations 185
5.2.1 Mechanisms of light 186

CONTENTS vii
Hobbes, Hooke and the pitfalls of mechanistic philosophy:
rigid waves 189
5.2.2 ‘Raisons de mechanique’ 195
Newton’s speculations on the nature of light 196
The status of ‘raisons de mechanique’ 200
5.3 A second EUPHKA 204
5.3.1 Danish objections 205
Forced innovation 207
5.3.2 Hypotheses and deductions 209
CHAPTER 6 1690 - TRAITÉ DE LA LUMIÈRE 213
6.1 Creating Traité de la Lumière 214
6.1.1 Completing ‘Dioptrique’ 216
Huygens’ dioptrics in the 1680s 216
6.1.2 From ‘Dioptrique’ to Traité de la Lumière 219
The publication of Traité de la Lumière 222
6.2 Traité de la Lumière and the advent of physical optics 225
Mathematization by extending mathematics 227
The matter of rays 229
The mathematics of light 232
6.3 Traité de la Lumière and Huygens’ oeuvre 236
6.3.1 Huygens’ Cartesianism 237
The subtle matter of 1669 238
Huygens versus Newton 242
Huygens’ self-image 247
6.3.2 The reception of Huygens 249
CHAPTER 7 CONCLUSION: LENSES & WAVES 255
A seventeenth-century Archimedes 255
From mathematics to mechanisms 259
Huygens and Descartes 261
The small Archimedes 262
LIST OF FIGURES 265
BIBLIOGRAPHY 267
INDEX 285

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Preface
“Le doute fait peine a l’esprit.
C’est pourquoy tout le monde
se range volontiers a l’opinion de ceux
qui pretendent avoir trouvè la certitude.” 1
This book evolved out of the dissertation that I defended on April 1, 1999 at
the University of Twente. At the successive stages of its development critical
readers have cast doubts on my argument. It has not troubled my mind; on
the contrary, they enabled me to improve my argument in ways I could not
have managed on my own. So, I want to thank Floris Cohen, Alan Shapiro,
Jed Buchwald, Joella Yoder, and many others. Most of all, however, I have to
thank Casper Hakfoort, who saw the final text of my dissertation but did not
live to witness my defence and the further development of this study of
optics in the seventeenth century.
This book would not have been possible without NWO (Netherlands
Organisation for Scientific Research) and NACEE (Netherlands American
Commission for Educational Exchange) who supplied me with a travel grant
and a Fulbright grant, respectively, to work with Alan Shapiro in
Minneapolis. The book would also not have been possible without the
willingness of Kluwer Publishers and Jed Buchwald to include it in the
Archimedes Series, and the unrelenting efforts of Charles Erkelens to see it
through.
During the years this text accompanied my professional and personal doings,
numerous people have helped me grow professionally and personally. I want
to thank Peter-Paul Verbeek, John Heymans, Petra Bruulsema, Kai Barth,
Albert van Helden, Rienk Vermij, Paul Lauxtermann, Lissa Roberts, and
many, many others.
Still, the idea to study Huygens and his optics would not have even
germinated – let alone that this book would have matured – without my life
companion, Anne, with whom I now share a much more valuable creation.
Thank you.
Fokko Jan Dijksterhuis
Calgary, June 2004
This book is dedicated to Casper Hakfoort
In memory of Lies Dijksterhuis
1 Undated note by Christiaan Huygens (probably 1686 or 1687), OC21, 342

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

2 CHAPTER 1
Iceland crystal, including refractions that could only be observed by cutting
the crystal along unnatural sections. The sketch records the experimental
verification of Huygens’ elliptical waves and, with it, the confirmation of his
theory of light and refractions.
This brief synopsis explains what ‘actually’ happened on that 6 th of
August in 1679. The various terms and concepts will be explicated later on in
this book. For this moment, it suffices to make clear the core of Huygens’
wave theory and its historical significance. For Huygens the successful
experiment meant the confirmation of his explanation of strange refraction
and his wave theory in general. In the context of the history of seventeenthcentury
optics, and of the mathematical sciences in general, the importance
of the event lies in the twofold particular nature of Huygens’ theory: a
mathematized model of the mechanistic nature of light considered as a
hypothesis validated by experimental confirmation. With the mathematical
form of his theory, Huygens can be said to have restored the problematic
legacy of Descartes’ natural philosophy, by defining mathematical principles
for the mechanistic explanation of the physical nature of light. The
hypothetical-deductive structure of his theory implied the abandonment of
the quest for certainty of that same Cartesian legacy and of seventeenthcentury
science in general. Huygens presented waves of light, the inextricable
core of his account of optical phenomena, explicitly as hypothetical entities
whose certainty is inherently relative. In so doing, he set off from Descartes
in a direction diametrically opposite to Newton, the principal other restorer
of mechanistic science.
About a decade after the EUREKA of 6 August 1679, Huygens published
his wave theory of light and his explanations of ordinary and strange
refraction in Traité de la Lumière (1690). This book established his fame as a
pioneer of mathematical physics as evidenced by the fact his principle of
wave propagation is still known and used in various fields of modern physics
under the name ‘Huygens’ principle’. Its historical importance is also
generally acknowledged. According to Shapiro, Huygens stood out for his
“…continual ability to rise above mechanism and to treat the continuum
theory of light mathematically.” 1 E.J. Dijksterhuis calls it the high point of
mechanistic science and its creator the first ‘perfect Cartesian’: “In Huygens
does Cartesian physics for the first time take the shape its creator had in
mind.” 2 How did this come about? How did Christiaan Huygens came to
realize this historical landmark? Or more specifically, how did he arrive at his
wave theory of light? That is the central question of this study.
A history of Traité de la Lumière
Unlike its eventual formulation in Traité de la Lumière, the development of
Huygens’ wave theory has hardly been subject to historical investigation. The
1 Shapiro, “Kinematic optics”, 244. (For referencing see page 267)
2 Dijksterhuis, Mechanization, IV: 212 & 283 (references to this book will be made by section numbers). It
should be noted that Dijksterhuis mainly focuses on the mathematical model of wave propagation.

‘THE PERFECT CARTESIAN’ 3
first step for such a study is to go into the papers documenting Huygens’
optics. The historian who does so on the basis of existing literature, awaits a
surprise. There is much more to Huygens’ optics than waves. He elaborated
a comprehensive theory of the dioptrical properties of lenses and their
configurations in telescopes, that goes by the title of Dioptrica. A second
surprise is in store when one takes a closer looks to these papers on
geometrical optics. The papers on dioptrics cover the Huygens’ complete
scientific career and form the exclusive content of the first two decades of
his optical studies. The wave theory and related subjects are fully absent; not
before 1672 do they turn up. In other words, the optics that brought
Huygens future fame dates from a considerably late stage in the development
of Huygens’ optics.
In this way new and more specific questions arise regarding the question
‘how did Huygens arrive at his wave theory’? What exactly was his optics?
How did he move from Dioptrica to Traité de la Lumière? And what does this
teach us about the historical significance of his wave theory, Huygens’
creation of a physical optics, and the character of his science? The point is
that Huygens’ dioptrics turns his seemingly self-explanatory wave theory into
a historical problem. It did not develop from some innate cartesianism, for
he was no born Cartesian, certainly not in optics at least. Fully absent from
Dioptrica is the central question of Traité de la Lumière: what is the nature of
light and how can it explain the laws of optics. Huygens first raised this
question in 1672 – five years before he found his definite answer (which he
confirmed another two years later in 1679). His previous twenty years of
extensive dioptrical studies give scarcely any occasion to expect that this man
was to give the mechanistic explanation of light and its properties a wholly
new direction. In view of Dioptrica, the question is not only how Huygens
came to treat the mechanistic nature of light in his particular way, but even
how he came to consider the mechanistic nature of light in the first place.
In the literature on Huygens’ optics, mechanistic philosophy has been
customarily considered a natural part of his thinking. Only Bos, in his entry
in the Dictionary of Scientific Biography, points at the relatively minor role
mechanistic philosophizing played in his science before his move to Paris in
the late 1660s. 3 The question of how the wave theory took shape in Huygens’
mind thus becomes all the more intriguing. What caused Huygens to tackle
this subject he had consistently ignored throughout his earlier work on
optics? How do Dioptrica and Traité de la Lumière relate and what light does
the former shed on the latter? Part of the answer is given by the fact that
only at the very last moment, short before its publication, Huygens decided
to change the title of his treatise on the wave theory from ‘dioptrics’ to
‘treatise on light’. In his mind the two were closely connected, questions now
3 Bos, “Huygens”, 609. Van Berkel further alludes to the influence of Parisian circles on the prominence
of mechanistic philosophy in Huygens’ oeuvre: Van Berkel, “Legacy”, 55-59.

4 CHAPTER 1
are: how exactly and what does this mean for our understanding of Huygens’
optics?
In addition to the question of where in Huygens’ oeuvre Traité de la
Lumière properly belongs, a more general question may be asked: where in
seventeenth-century science may this kind of science be taken to belong?
Were the questions Huygens addressed in Traité de la Lumière part of any
particular scientific discipline or coherent field of study? In the course of my
investigation, it has became increasingly clear to me that the term ‘optics’ is
rather problematic regarding the study of light in the seventeenth century,
just like the term ‘science’ in general. Our modern understanding of ‘optics’
implies an investigation of phenomena of light much like Traité de la Lumière:
a mathematically formulated theory of the physical nature of light explaining
the mathematical regularities of those phenomena. Yet, optics in this sense
was only just beginning to develop during the seventeenth century. The term
‘optics’ in the seventeenth century denoted the mathematical study of the
behavior of light rays that we are used to identify with geometrical optics.
This is what Huygens pursued in Dioptrica, prior to developing his wave
theory. A general question regarding the history of optics raised by Huygens’
Traité de la Lumière is how a new kind of optics, a physical optics, came into
being in the seventeenth century and how this related to the older science of
geometrical optics. This transformation of the mathematical science of optics
is manifest in the title Huygens eventually chose for his treatise.
Huygens’ optics
This book offers in the first place an account of the development of
Huygens’ optics, from the first steps of Dioptrica in 1652 to the eventual
Traité de la Lumière of 1690. The following chapters take a chronological
course through his engagements with the study of light, whereby the
historical connection of its various parts sets the perspective. Terms like
‘optics’ and ‘science’ are problematic historically. Nevertheless, for sake of
convenience, I will freely use them to denote the study of light in general and
natural inquiry, except when this would give rise to (historical)
misunderstandings. When discussing their historical character and
development specifically, I will use appropriately historicizing phrases.
Chapter two discusses Tractatus – the unfinished treatise of 1653 on
dioptrics that marks the beginning of Huygens’ engagement with optics.
Tractatus contained an comprehensive and rigorous theory of the telescope
and I will argue that this makes it unique in seventeenth-century
mathematical optics. Huygens was one of the few to raise theoretical
questions regarding the properties and working of the telescope, and almost
the only one to direct his mathematical proficiency towards the actual
instruments used in astronomy. Kepler had preceded him, but he had not
known the law of refraction and therefore could not derive but an
approximate theory of lenses and their configurations. Some four decades
afterwards, and two decades after the publication of the sine law, Huygens

‘THE PERFECT CARTESIAN’ 5
was the first to apply it to spherical lenses and remained so for almost two
decades more. Chapter three discusses his practical pursuits in dioptrics
leading into his subsequent treatise on dioptrics, De Aberratione of 1665.
Huygens made a unique effort to employ dioptrical theory to improve the
telescope. The contrast with Descartes is particularly conspicuous, for
Huygens did not fit the telescope into the ideal mold prescribed by theory
but directed his theory towards the instruments that were practically feasible.
The effort was unsuccessful, for with his new theory of light and colors
Newton made him realize the futility of his design. Taking into account
Huygens’ background in dioptrics sheds, I will argue, new light on the
famous dispute with Newton in 1672.
These chapters are confined to what we would call geometrical optics and
to its relationship to practical matters of telescopy. 4 I try to explain what this
science was about and what was particular about the way Huygens pursued
it. These chapters offer a fairly detailed account of Dioptrica within the
context of seventeenth-century geometrical optics, and as such open fresh
ground in the history of science. At the turn of the century, Kepler had laid a
new foundation for geometrical optics. Image formation now became a
matter of determining where and how a bundle of diverging rays from each
point of the object is brought into focus again (or not) instead of tracing
single rays from object point to image point. Of old, the ray was the bearer
of the physical properties of light, but in the course of the seventeenth
century this began to be qualified and questioned. In the wake of Kepler and
Descartes the mathematical science of optics gradually transformed into new
ways of doing optics. The traditional, geometrical way of doing optics did
not vanish, though. It was ray optics in which the question of the nature of
light need not penetrate further than determining the physical properties of
rays in their interaction with opaque and transparent materials. This is the
mathematical optics Huygens grew up with and that set the tone in his
earliest dealings with the physics of light propagation. Only on second
thought did he focus on the new question what is light and how can this
explain its properties. This transformation is the subject of the next chapters.
In chapters four and five my focus shifts, along with Huygens’, to the
mechanistic nature of light. In 1672 a particular problem drew his attention
to the question what light is and how its properties can be explained: the
strange refraction in Iceland Crystal which created a puzzle regarding the
physics of refraction that Huygens wanted to solve. These two chapters
discuss the three stages of his investigation, his first analysis of the
mathematics of strange refraction in 1672 and his eventual solution by means
of elliptical waves in 1677 and its confirmation in 1679. Huygens’ first attack
on the problem of strange refraction is historically significant because he
approached it along traditional lines of geometrical optics. Only in second
4
Preliminary results are published in: Dijksterhuis, “Huygens’ Dioptrica” and Dijksterhuis, “Huygens’s
efforts”.

6 CHAPTER 1
instance did he turn to the actual question underlying the problem: how
exactly do waves of light propagate. And only in third instance, and forced
by critical reactions, did Huygens seek for experimental validation of the
theory he initially had developed primarily rationally. These chapters offer a
new, detailed reconstruction of the origin of the wave theory on the basis of
manuscript material that has not been taken into account earlier. It is also a
reconstruction of how Huygens got from Dioptrica to Traité de la Lumière, in
which I compare his approach to questions pertaining to the physical nature
of light in the mathematical science of optics to his predecessors and
contemporaries. Central themes are the way the nature of light was
accounted for in the mathematical study of light in the seventeenth century
and to relationships between explanatory theories of light and the laws of
optics. I will argue that Huygens was the first to successfully mathematize a
mechanistic conception of light. He was rivaled only by Newton, but for
epistemological reasons he kept his hypotheses private.
Chapter six reviews the development of Traité de la Lumière, its
significance for the history of seventeenth-century optics, and for our
understanding of Huygens’ science. After discussing the publication history
of Traité de la Lumière, which reveals that Huygens disconnected it from
Dioptrica only at the very last moment, I sketch some lines for a new
perspective of the history of seventeenth-century optics in which traditional
geometrical optics is taken into account as an important root. The
mathematico-physical consideration of light of Traité de la Lumière was a
particular answer to a new kind of question. A kind of question also
addressed by such diverse scholars as Kepler, Descartes and Newton. In this
sense, my study of the development of Traité de la Lumière, in particular in
relationship with Dioptrica, is also a study of the origins of a new science of
optics, nowadays denoted by the term ‘physical optics’. Some instances of
physical optics developed in the seventeenth-century, most notably by
Huygens and Newton. But the primacy of the question ‘what is the physical
nature of light and how may this explain its properties? first had to be
discovered and this only gradually came about in the pursuit of the
mathematical science of optics. In the case of Huygens this emergence was
particularly quiet. While solving the intriguing puzzle of strange refraction,
he developed a new way of doing mathematical optics but he seems to have
been hardly aware of the new ground he was breaking. At the close of this
chapter, I discuss his alleged Cartesianism and I will argue that Huygens
stumbled into becoming a ‘perfect Cartesian’ rather than determinedly and
systematically create it.
This study is based on the optical papers in the Oeuvres Complètes and
additional manuscript material. A large part of these have as yet not been
studied. The Oeuvres Complètes split up Huygens’ optics in two parts – volume
13 for Dioptrica and volume 19 for Traité de la Lumière. This subdivision along
modern disciplinary lines resounds in the historical literature. E.J.
Dijksterhuis, for example, separates explicitly ‘geometrical optics’ – where he

‘THE PERFECT CARTESIAN’ 7
merely mentions Huygens – and physical theories of optics. 5 No doubt all
this has contributed its share to the fact that the relationship between Traité
de la Lumière and Dioptrica – historical, conceptual as well as epistemical – has
gone unexamined so far. 6 As for Traité de la Lumière, most historical
interpretations are based on and confined to the published text. The
additional manuscript material published in OC19 has hardly been taken into
account and no-one to my knowledge has used the original manuscript
material in the Codices Hugeniorum in the Leiden university library. 7 Huygens’
wave theory has been the subject of several historical studies. Each in their
own way has been valuable for this study. E.J. Dijksterhuis gives an
illuminating analysis of the merits of Traité de la Lumière as a pioneering
instance of mathematical physics considered in the light of Descartes’
mechanistic program. Sabra includes an account of Huygens’ wave theory in
his study of the historical development of the interplay of theory and
observation in seventeenth-century optics. Shapiro offers a searching analysis
of the historical development of the physical concepts underlying Huygens’
wave theory. I intend to add to our growing historical understanding of Traité
de la Lumière by reconstructing its origin and development in the context of
his optical studies as a whole and of that of seventeenth-century optics in
general.
Huygens’ lifelong engagement with dioptrics as such has hardly been
studied. 8 Even Harting, the microscopist who by mid-nineteenth century
gives Huygens’ telescopic work a central place in his biographical sketch,
mentions dioptrical theory only in passing. 9 The editorial remarks in the
‘Avertissement’ of OC13 form the main exception and are one of the few
sources of information on the history of seventeenth-century geometrical
optics in general. Some topics pertaining to seventeenth-century geometrical
optics have been studied in considerable detail, but for the most part in the
context of the seventeenth-century development of physical science. These
are Kepler’s theory of image formation, the discovery of the sine law and
Newton’s mathematical theory of colors and they are integrated in my
5
Dijksterhuis, Mechanisering, IV: 168-171, 284-287.
6
Hashimoto hardly goes beyond noting that “… two works were closely related in Huygens’s mind.”:
Hashimoto, “Huygens”, 87-88.
7
Dijksterhuis, Mechanization, IV: 284-287 and Sabra, Theories, 159-230 are confined to Traité de la Lumière.
Shapiro uses some of the manuscripts published in Oeuvres Complètes. Ziggelaar, “How”, draws mainly on
OC19. Yoder has pointed out that the wave theory is no exception to the rule that in general, studies of
Huygens’ work tend to focus on his published works.
8
Hashimoto has published a not too satisfactory article in which he discusses Huygens’ dioptrics in
general terms. Apart from some substantial flaws in his analyses and argument, Hashimoto fails to
substantiate some of his main claims regarding Huygens’ ‘Baconianism’. Hashimoto, “Huygens”, 75-76;
86-87; 89-90. For example, he reads back into Tractatus the utilitarian goal of De aberratione (60, compare
my section 3.3.2), he thinks Huygens determined the configuration of his eyepiece theoretically (75,
compare my section 3.1.2), maintains that Systema saturnium grew out of his study of dioptrics (89,
compare my section 3.1.2) and that Huygens ‘went into the speculation about the cause of colors’ after his
study of spherical aberration (89, compare my section 3.2.3)
9
Harting, Christiaan Huygens, 13-14. Harting based himself on manuscript material disclosed in
Uylenbroek’s oration on the dioptrical work by the brothers Huygens: Uylenbroek, Oratio.

8 CHAPTER 1
accounts of, respectively, seventeenth-century dioptrics in chapter 2, the
epistemic role and status of explanations in optics in chapter 4, and Huygens’
own dealings with colors in chapter 3 as well as his specific approach to
mechanistic reasoning in chapter 5. Little literature on the history of the field
of geometrical optics and its context exist. 10
A substantial part of my argument is based upon comparisons with the
pursuits of other seventeenth-century students of optics. In order to come to
a historically sound understanding of what Huygens was doing, I find it
necessary to find out how his optics relates to the pursuits of his
predecessors and contemporaries. What questions did they ask (and what
not) and how did they answer them? Why did they ask these questions and
what answers did they find satisfactory? For example, in chapter 2 the
earliest part of Dioptrica is compared with, among other works, Kepler’s
Dioptrice and Descartes’ La Dioptrique. All bear the same title, yet the
differences are considerable. Descartes discussed ideal lenses and did so in
general terms only, rather than explaining their focusing and magnifying
properties as Kepler had done. Huygens, in his habitual search for practical
application, expressly focused on analyzing the dioptrical properties of real,
spherical lenses and their configurations, thus developing a rigorous and
general mathematical theory of the telescope. By means of such comparisons
it is possible to determine in what way Huygens marked himself off as a
seventeenth-century student of optics, or did not. These comparisons are
focused on Huygens’ optics, so I confine my discussions of seventeenthcentury
of optics to the mathematical aspects of dioptrics and physical
optics. Other themes like practical dioptrics and natural philosophy in
general will be treated only in relation to Huygens.
This is an intellectual history of Huygens’ optics and of seventeenthcentury
optics in general. The nature of the available sources – as well that of
the man – are not suitable for some kind of social or cultural history. He
operated rather autonomously, mainly because he was in the position to do
so, and he was no gatherer of allies and did not try very hard to propagate his
ideas about science and gain a following.
New light on Huygens
This study offers, in the first place, a concise history of Huygens’ optics. Yet
it is not a mere discussion of Huygens’ contributions to various parts of
optics. I also intend to shed more light on the character of Huygens’
scientific personality. The issue of getting a clear picture of his scientific
activity and its defining features is an acknowledged problem. In summing
up a 1979 symposium on the life and work of Huygens, Rupert Hall
10 For example, the precise application of the sine law to dioptrical problems, for example, has hardly been
studied. Shapiro, “The Optical Lectures” is a valuable exception, discussing Barrow’s lectures and their
historical context. The relationship between the development of the telescope and of dioptrical theory –
essential to my account of Dioptrica – has never been investigated in any detail. Van Helden has pointed
out the weak connection between both in general terms: Van Helden, “The telescope in the 17 th century”,
45-49; Van Helden, “Birth”, 63-68.

‘THE PERFECT CARTESIAN’ 9
concluded that “… it isn’t at all easy to understand how all the multifarious
activities of this man’s life fit together.” 11 Huygens has been called the true
heir of Galileo, the perfect Cartesian, and also a man deftly steering a middle
course between Baconian empiricism and Cartesian rationalism. 12 Huygens
himself has not been much of a help in this. He always was particularly
reticent about his own motives. He was an intermediate figure between
Galileo and Descartes on the one hand and Newton and Leibniz on the
other but, lacking as he did a pronounced conception of the aims and
methods of his science, he is difficult to situate among the protagonists of
the scientific revolution.
In 1979, the most apt characterization of Huygens seemed to be that of
an eclectic, who took up loose issues and solved them with the means he
considered appropriate without some sort of central direction becoming
apparent. 13 The original idea behind this study was that in Traité de la Lumière
this eclecticism grew into a fruitful synthesis of mathematical, mechanistic
and experimental approaches. This idea originates from the work my advisor,
the Casper Hakfoort. In his study of eighteenth-century optics he formulated
the idea when he pointed out the significance of natural philosophy for the
development of optics, which in his view was hithertho neglected. 14 I
consider it an honor to have been able to pursue this idea and to have had it
bear unanticipated fruit. By trying to understand how the multifarious
aspects of his optics fit together, I hope to be able to shed light on the
character of his science in general and on his place in seventeenth-century
science. Dioptrica, while adding to the standing impression of the great
versatility of Huygens’ oeuvre, has not changed my expectation that an
understanding of the way the possible coherence of these aspects evolved
may contribute to a better characterization of Huygens’ science and of his
place in seventeenth-century science as a whole.
In the final chapter of this book, I review my account of the development
of Huygens’ optics to see what light this may shed on his scientific
personality. By way of conclusion it offers a sketch of his science, based on
the previous chapters that go into the details – often technical – of his optics
and its development. This chapter can be read independently as an essay on
Huygens.
Huygens was a puzzle solver indeed, an avid seeker of rigorous, exact
solutions to intricate mathematical puzzles. But these puzzles do have
coherence, they all concerned questions regarding concrete, almost tangible
subjects in the various fields of seventeenth-century mathematics. He was an
eclectic, but only in comparison with the chief protagonists of the scientific
11
Hall, “Summary”, 311.
12
Westfall, Construction, 132-154; Dijksterhuis, Mechanization, 212; Elzinga, Research program and Westman,
“Problem”, 100-101.
13
Hall, “Summary”, 305-306. As regards his studies of motion, Yoder has further specified this
characterization; Yoder, Unrollling time, 169-179.
14
Hakfoort, Optics in the age of Euler, 183-184.

10 CHAPTER 1
revolution, that set up schemes to lay new foundations for natural inquiry.
Huygens did not have such a program and, as a result, his science seems to
lack coherence, unless a coherence is looked for on a different level of
seventeenth-century science. The essay therefore first forgets about
Huygens’ alleged Cartesianism to sketch the mathematician and his
idiosyncratic focus on instruments. He was not a half-baked philosopher but
a typical mathematician. A new Archimedes, as Mersenne foretold in 1647.
The incomparable Huygens, as Leibniz said in 1695 upon the news of his
death. 15 Then I ask anew how his Cartesianism fits into the picture. We may
have trouble getting a balanced idea of what he was doing, but it appears that
he was hardly aware of the size of the new ground he had been breaking. He
had, in fact, developed a new mathematical science of optics.
15 OC1, 47 and OC10, 721.

Chapter 2
1653 - 'Tractatus'
The mathematical understanding of telescopes
“Now, however, I am completely into dioptrics”, Huygens wrote on 29
October 1652 in a letter to his former teacher in mathematics, Frans van
Schooten, Jr. 1 His enthusiasm had been induced by a discovery in dioptrical
theory he had recently made. It was an addition to Descartes’ account of the
refracting properties of curves in La Géométrie, that promised a useful
extension of the plan for telescopes with perfect focusing properties
Descartes had set out in La Dioptrique. During his study at Leiden University
in 1645-6, Huygens had studied Descartes’ mathematical works, La Géométrie
in particular, intensively with Van Schooten. Although Van Schooten was
professor of ‘Duytsche Mathematique’ at the Engineering school, appointed
to teach practical mathematics in the vernacular to surveyors and the like.
Huygens was not the only patrician son he introduced to the new
mathematics: the future Pensionary Johan de Witt and the future Amsterdam
mayor Johannes Hudde.
From 1647 Huygens and Van Schooten had to resort to corresponding
over mathematics, when Huygens had to go to Breda to the newly
established ‘Collegium Auriacum’, the college of the Oranges to which the
Huygens family was closely connected politically. In 1649 Huygens had
returned home to The Hague and now, in 1652, he was ‘private citizen’. He
did not feel like pursuing the career in diplomacy his father had planned for
him and, with the Oranges out of power since 1650, not many duties were
left to call on him. Huygens could, in other words, freely pursue his one
interest, the study of the mathematical sciences. An appointment at a
university was out of order for someone of his standing and, as Holland
lacked a centrally organized church and a grand court, interesting options for
patronage were not directly available. 2 So, with a room in his parental home
at the ‘Plein’ in The Hague and a modest allowance from his father, he could
live the honorable life of an ‘amateur des sciences’. He enjoyed to company
of his older brother Constantijn, who joined him in his work in practical
dioptrics (see next chapter), and dedicated himself to mathematics.
Geometry and mechanics were his main focus in these years, with his
theories of impact and of pendulum motion and his invention of the
1
OC 1, 215. “Nunc autem in dioptricis totus sum ...”
2
Berkel, “Illusies”, 83-84. In the 1660s Huygens would start to seek patronage abroad, first in Florence
and then, successfully in Paris.

12 CHAPTER 2
pendulum clock as the most renowned achievements. Yet, the discovery
made late 1652 had sparked his interest in dioptrics, which largely dominated
his scholarly activities the next two years. The letter to Van Schooten was the
onset to a lifelong engagement with dioptrics, which nevertheless has little
been studied historically. 3
In the months following the letter to Van Schooten, Huygens elaborated
a treatise that contained a mathematical theory of the dioptrical properties of
lenses and telescopes. I will refer to this treatise as Tractatus and it is the
subject of this chapter. 4 In Tractatus Huygens treated a specific set of
dioptrical questions, directed at understanding the working of the telescope.
In the first section of this chapter the content and character of the treatise
are discussed. In the second section Huygens’ approach to dioptrics is
compared with that of contemporaries, by examining how other
mathematicians dealt with the questions that stood central in Tractatus. In this
discussion of seventeenth-century dioptrics the relationship between
dioptrical theory and the development of the telescope is the central topic. I
will argue that Huygens in his mathematical theory stood out for his focus
on questions that were relevant to actual telescopes. In this he was the first
to follow Kepler’s lead; other theorists were absorbed by abstract questions
emerging from mathematical theory for which men of practice, in their turn,
did not care. In the next chapter Huygens’ own telescopic practices are
discussed. Now first the theoretical considerations of Tractatus.
2.1 The Tractatus of 1653
The background to Huygens’ letter to Van Schooten was a problem with the
lenses used in the telescopes of those days. Lenses were spherical, i.e. their
cross section is circular. As a result they do not focus parallel rays perfectly.
Rays from a distant point source that are
refracted by a spherical surface do not
intersect in a single point, rays close to
the axis are refracted to a more distant
point on the axis than rays farther from
the axis (Figure 2). This is called spherical
aberration and results in slightly blurred Figure 2 Spherical aberration
3 The most thorough-going account still are the ‘avertissements’ by the editors of the Oeuvres Complètes.
Southall, “Some of Huygens’ contributions” reported on Huygens’ dioptrics after the publication of
volume 13. Harting, Christiaan Huygens had earlier discussed it briefly. In relationship with his astronomical
work and his practical dioptrics, Albert van Helden, “Development” and Anne van Helden/Van Gent,
The Huygens collection and “Lens production” discuss some topics. In the context of the history of
seventeenth-century geometrical optics – which in its own right has little been studied – Shapiro,
“’Optical Lectures’” mention Huygens’ contributions. They are remarkably absent from the Malet, “Isaac
Barrow” and “Kepler and the telescope”. Hashimoto, “Huygens, dioptrics” is the only effort to discuss
Huygens’ dioptrics in the context of his broader oeuvre.
4 OC13, 1-271. The editors of the Oeuvres Complètes have labeled it Dioptrica, Pars I. Tractatus de refractione et
telescopiis. Its content stems from the 1650s. The original version of Tractatus does not exist anymore. A
copy was made in Paris by Niquet – probably in 1666 or 1667, at the beginning of Huygens’ stay in Paris
– on which the text of the Oeuvres Complètes is based. The editors assume Niquet’s copy of Tractatus is
largely identical with the original 1653 manuscript; “Avertissement”, XXX.

1653 - TRACTATUS 13
images. In La Dioptrique (1637), Descartes had explained that surfaces whose
section is an ellipse or a hyperbola do not suffer this impediment. They are
called aplanatic surfaces. Descartes could demonstrate this by means of the
sine law, the exact law of refraction he had discovered some 10 years earlier.
According to the sine law, the sines of incident and of refracted rays are in
constant proportion. This ratio of sines is nowadays called index of
refraction, it depends upon the refracting medium.
The discovery Huygens made in late 1652 sprang from Descartes’ La
Géométrie. Together with La Dioptrique and Les Météores, this essay was
appended to Discours de la methode (1637). In La Géométrie, Descartes had
introduced his new analytic geometry. In La Géométrie mathematical proof
was given of the claim of La Dioptrique that the ellipse and hyperbola are
aplanatic curves. In his letter to Van Schooten, Huygens wrote that he had
discovered that under certain conditions circles also are aplanatic. This
discovery implied that spherical lenses could focus perfectly in particular
cases. Consequently, Huygens considered it of considerable importance for
the improvement of telescopes.
Huygens’ expectation that his discovery would be useful in practice, was
fostered by the fact that Descartes’ claims had turned out not to be
practically feasable. Around 1650, no one had succeeded in actually grinding
the lenses prescribed in La Dioptrique. 5 Apart from that, the treatise did not
discuss the spherical lenses actually employed in telescopes. Descartes had
applied his exact law only to theoretical lenses. When he made his discovery,
Huygens must have realized that no-one had applied the sine law to spherical
lenses yet. In the aftermath of his discovery, Huygens set out to correct this
and develop a dioptrical theory of real lenses.
2.1.1 OVALS TO LENSES
In his letter to Van Schooten, Huygens did not explain the details of his
discovery. He did so much later, in an appendix to a letter of 29 October
1654 that contained comments upon Van Schooten’s first Latin edition of
La Géométrie: Geometria à Renato Des Cartes (1649). 6 In book two, Descartes
had introduced a range of special curves, ovals as he called them. This was
not a mere abstract exercise, he said, for these curves were useful in optics:
“For the rest, so that you know that the consideration of the curved lines here
proposed is not without use, and that they have diverse properties that do not yield at
all to those of conic sections, I here want to add further the explanation of certain
ovals, that you will see to be very useful for the theory of catoptrics and of dioptrics.” 7
By means of the sine law, Descartes derived four classes of ovals that are
aplanatic curves. If such a curve is the section of a refracting surface, rays
5
With the possible exception of Descartes himself. See below, section 3.1
6
OC1, 305-305.
7
Descartes, Geometrie, 352 (AT6, 424). “Au reste affin que vous sçachiées que la consideration des lignes
courbes icy proposée n’est pas sans usage, & qu’elles ont diverses proprietés, qui ne cedent en rien a celles
des sections coniques, ie veux encore adiouster icy l’explication de certaines Ovales, que vous verrés estres
tres utiles pour la Theorie de la Catoptrique, & de la Dioptrique.”

14 CHAPTER 2
coming from a single point are refracted towards another single point. In
certain cases, the ovals reduce to the ellipses and hyperbolas of La Dioptrique.
Huygens in his turn discovered that a particular class of these ovals may also
reduce to a circle.
In La Géométrie Descartes introduced the
said class of ovals as follows (Figure 3). The
dotted line is an oval of this class. If the
right part 2X2 of the oval is the right
boundary of a refracting medium, rays
intersecting in point F are refracted to point
G. 8 The oval is constructed as follows. Lines
FA and AS intersect in A at an arbitrary
angle, F is an arbitrary point on FA. Draw a
circle with center F and radius F5. Line 56 is
drawn, so that A5 is to A6 as the ratio of
sines of the refracting medium. G is an arbitrary point between A and 5, S is
on A6 with AS = AG. A circle with center G and radius S6 cuts the first circle
in the points 2, 2. These are the first two points of the oval. This procedure
is repeated with points 7 and 8, et cetera until the oval 22X22 is completed. 9
Huygens’ discovered that the oval reduces to a circle when the ratio of AF to
AG is equal to the ratio of A5 to A6, the ratio of sines. 10 This means that with
respect to rays tending to F, a spherical surface 2X2 will focus them exactly in
G. Van Schooten was a bit skeptical about Huygens’ claim. Could such a
simple fact have escaped Descartes? Nevertheless, he included it in the
second edition of Geometria à Renatio Des Cartes (1659). 11
Discovering that a spherical surface is aplanatic in certain cases is one
thing, applying it to lenses in practice is another. It remained to be seen what
shape the second surface should have and how it might be employed in
telescopes. For one thing, it does not seem useful for objective lenses, the
front and most important lens of a telescope that receives parallel rays. It
appears the usefulness of the discovery was limited, for Huygens never
returned to it in his dioptrical studies. 12
The historical importance of the discovery lies in the fact that it aroused
Huygens’ interest in dioptrics. He did not exaggerate when he said he was
engrossed in dioptrics. Not only its theory, practice too. Five days after his
letter to Van Schooten, on 4 November, he wrote to Gerard Gutschoven, an
acquainted mathematician in Antwerp. 13 After some introductory remarks,
8 Descartes, Geometrie, 358-359 (AT6, 430-431). The left part 2A2 is a mirror that reflects rays intersecting
in G so that they (virtually) intersect in F, provided that it diminishes the ‘tendency’ of the rays to a given
degree.
9 Descartes, Geometrie, 353-354 (AT6, 424-426). The curve satisfies the equation F2 – FA = n(G2 – GA).
10 OC1, 305. See note 9: the equation becomes AF = nAG.
11 Reproduced in OC14, 419.
12 In Tractatus, he merely mentioned that a spherical surface is aplanatic for certain points: OC13, 64-67.
13 OC1, 190-192.
Figure 3 Cartesian oval.

1653 - TRACTATUS 15
Huygens launched a series of questions on the art of making lenses. What
material are grinding moulds made of, how is the spherical figure of a lens
checked, what glue is used to attach the lenses to a grip, et cetera. Only after
these questions did he explain to Gutschoven that he wanted to know all
these things because he had discovered something that would greatly
improve telescopes.
Figure 4 Focal distance of a bi-convex lens
The letter reveals that Huygens had already begun to investigate the
dioptrical properties of spherical lenses. It contained a theorem on the focal
distance of parallel rays refracted by a bi-convex lens CD (Figure 4). 14 AC and
DB are the radii of the anterior and posterior side of the lens. L and E are
determined by DL : LB =CE : EA = n, the index of refraction. O is found by
EL : LB = ED : EO. Rays parallel to the axis EL come from the direction of L.
Without proof Huygens said that O is the focus of the refracted rays. He said
that he could prove this and that he had found many more theorems. A
month later, in a letter of 10 December to André Tacquet, a Jesuit
mathematician in Louvain, he added an important insight. 15 As a result of
spherical aberration point O is not the exact focus. Nevertheless, it may be
taken as the focus: “… since beyond point O no converging rays intersect
with the axis.” 16 In later letters to Tacquet and Gutschoven he called this
point the ‘punctum concursus’. 17 It is the where rays closest to the axis are
refracted to. This definition would be fundamental to the theory of Tractatus,
which apparently was well under way. Huygens told Tacquet that he had
already written two books of a treatise on dioptrics: one on focal distances,
another one on magnification. A third one on telescopes was in preparation.
Within a month or two after his letter to van Schooten, Huygens’
understanding of dioptrics was rapidly developing. It was also developing in
a particular direction. Huygens was studying the dioptrical properties of
spherical lenses. He must have found out that little had been published on
the subject. The only mathematical theory of spherical lenses was Kepler’s
Dioptrice (1611), but it lacked an exact law of refraction. Only Descartes had
applied the sine law to lenses, but he had ignored spherical lenses. Huygens
had begun to develop an exact theory of spherical lenses by himself. He
combined this theoretical interest with an interest in practical matters of
telescope making. He reported to have seen a telescope made by the famous
craftsman Johann Wiesel of Augsburg. He was impressed and regretted that
14
OC1, 192.
15
OC1, 201-205.
16
OC1, 204. “… adeo ut nullius radij concursus cum axe contingat ultra punctum O.”
17
OC1, 224-226.

16 CHAPTER 2
Holland did not have such excellent craftsmen. 18 On 10 February, 1653,
Gutschoven finally informed him on the art of lens making. 19 Huygens did
not put the information to practice right-away, he first elaborated his
dioptrical theory.
2.1.2 A THEORY OF THE TELESCOPE
Huygens had written to Tacquet that his treatise would consist of three parts:
a theory of focal distances of lenses, a theory of the magnification produced
by configurations of lenses, and an account of the dioptrical properties of
telescopes based on the theory of the two preceding parts. The third part
was not yet finished when Huygens wrote Tacquet, in fact he never
elaborated in the form originally conceived. The third part of Tractatus as it is
found in the Oeuvres Complètes is a collection of dispersed propositions
collected by the editors. Only the first two seem to be from the 1650s. 20 In
the arrangements of manuscripts Huygens made in the late 1680s, part one
of Tractatus appears for the large part as it is found in the Oeuvres Complètes. 21
Judging from the various page numberings, Huygens has not edited it very
much, except that he inserted – probably in the late 1660s – parts of his
study of spherical aberration after the twentieth proposition. Part two of
Tractatus has been reshuffled somewhat more, but the main line appears to
be sufficiently original. In the following discussion of Tractatus, I follow the
text of the Oeuvres Complètes in so far as it appears to reflect the original
treatise.
Huygens coupled his orientation on the telescope with the mathematical
rigor typical of him. Although he singled out dioptrical problems that were
relevant to the telescope, he treated these with a generality and completeness
that often exceeded the direct needs of explaining the working of the
telescope. Huygens’ rigorous approach is clear from the very start of
Tractatus. Basic for his treatment of focal distances was the realization that
spherical surfaces do not focus exactly. This had been noticed earlier and had
been the rationale behind La Dioptrique. Nobody, however, had gone beyond
the mere observation of spherical aberration. Huygens got a firmer
mathematical grip on the imperfect focusing of lenses by defining which
point on the axis may count as the focus. Although he only discussed focal
points, Huygens took spherical aberration into account by consistently
determining the focus as the ‘punctum concursus’.
18 OC1, 215.
19 OC1, 219-223.
20 The decisions the editors made for the remaining propositions are sometimes somewhat mysterious.
For example, the fourth proposition has been assembled of fragments from various folios. And from
folio Hug29, 177 they put a diagram in part three of Tractatus, but they transferred the main contents to
‘De telescopiis’ (see section 6.1.2).
21 On this arrangement see page 221. By the way, the two first propositions of part three are inserted after
part one.

1653 - TRACTATUS 17
In part one of Tractatus he defined ‘punctum
concursus’ as follows. In the third proposition, he
defined the focus as the limit point of the intersections of
refracted rays with the axis (Figure 5). 22 ABC is a planoconvex
lens and parallel rays are incident from the
direction of D. Consequently, they are only refracted by
the spherical surface. Huygens showed that the closer
rays are to the axis DE, the closer to E they reach it.
Beyond E no refracted rays crosses the axis. This limit
point E he defined as the ‘punctum concursus’ of the
spherical surface ABC. If the surface is concave, rays do
not intersect at all after refraction, they diverge. In this
case, the ‘punctum concursus’ is the virtual focus, the
limit point of the intersections with the axis of the
backwards extended refracted rays.
In the first part of Tractatus, Huygens derived focal
distances of all types of spherical lenses by determining
exactly the ‘punctum concursus’ in each case. Refraction of parallel rays by a
lens consists in most cases of the two successive refractions by each side of
the lens. Determining the focal distance thus consists of three problems.
First, the refraction of parallel rays from air to glass by a spherical surface.
Second, that of the refraction of converging or diverging rays from glass to
air. Finally, combining both. Huygens built up his theory accordingly. He
first derived theorems expressing focal distance of spherical surfaces for
parallel rays in terms of their radii. Secondly, he derived theorems expressing
the focal distance for non-parallel rays in terms of the radius and the focal
distance for parallel rays. Finally, he expressed the focal distance of the
various kinds of lenses in terms of the radii of their sides. In each case he
took the thickness of the lens into account. Only afterwards did he derive
simplified theorems for thin lenses, in which their thickness is ignored. I now
sketch the typical case of a bi-convex lens, the theorem that Huygens
included without proof in his letters to Gutschoven and Tacquet. The
determination of the focal distances of other lenses – plano-convex, biconcave,
etc. – went along similar lines.
The focal distance of a bi-convex lens
The eighth proposition of Tractatus dealt with parallel rays refracted at the
convex surface of a denser medium (Figure 6). AC is the radius of ABP; Q is a
point on the axis AC so that AQ : QC = n, where n is the index of refraction.
Huygens demonstrated that Q is the ‘punctum concursus’ of parallel rays OB,
NP. A refracted ray BL intersects axis AC in a point L between A and Q. With
the sine law BL : LC = AQ : QC = n. For any ray OB, BL is smaller than AL and
AL is smaller than AQ. Therefore no refracted rays intersect the axis beyond
22 OC13, 16-19
Figure 5 Punctum
concursus

18 CHAPTER 2
Q. In order to prove that Q is the ‘punctum concursus’ of ABP, consider ray
NP and its refraction PK. PK is found with the sine law and KQ is therefore a
given interval. On KQ choose L and draw T, close to A, so that LT : CL =
AQ : CQ = n. Now PL : LC < PK : KC = n. PL is smaller than TL, which in its
turn is smaller AL. A circle with center L and radius TL intersects the
refracting surface ABP between A and P in a point B. Draw BL and BC and it
follows that BL : LC = TL : LC = n. Therefore BO is refracted to L. So, the
closer a paraxial ray is to the axis, the closer to Q the refracted ray will
intersect with the axis. Q is the limit point of these intersections and
therefore the ‘punctum concursus’. When the index of refraction
AQ : QC = 3 : 2 – the approximate value for glass – AQ is exactly three times
the radius AC.
The refraction at the
posterior side of the lens is dealt
with in the twelfth proposition.
This case is more complex as
the incident rays are converging
due to the refraction at the
anterior side. Huygens dealt
with eight cases of non-parallel
rays. 23 For all cases, he
expressed the focal distance of
the non-parallel rays in terms of
the focal distance of the surface
for paraxial rays. The case at
hand is the fourth part of the
proposition (Figure 7). 24 Rays
converge towards a point S,
outside the dense medium
bounded by a spherical surface
AB with radius AC. Q is the ‘punctum concursus’ of paraxial rays coming
from R. With SQ : SA = SC : SD, the ‘punctum concursus’ D of the converging
rays LB is found. Huygens’ proof consisted of a reversal of the first case
treated in this proposition: rays diverging from D are refracted so that they
(virtually) intersect in S. 25 This proof is similar to the one above.
Finally, in the sixteenth proposition of Tractatus, Huygens determined the
focal distance of a convex lens by combining the preceding results. It was
equal to the theorem he put forward in his letters to Tacquet and
Gutschoven. CD is a bi-convex lens with radii of curvature AC and BD
(Figure 8). The foci for paraxial rays are respectively E and L. According to
the eighth proposition CE : EA = DL : LB = n. With the twelfth proposition,
23 OC13, 40-79.
24 OC13, 70-73.
25 OC13, 42-47.
Figure 6 Refraction at
the anterior side of a
bi-convex lens
Figure 7 Refraction
at the posterior side
of a bi-convex lens.

1653 - TRACTATUS 19
the ‘punctum concursus’ N for parallel rays from the direction of L is found
with EL : ED = EB : EN. After the refraction at the surface C, the rays
converge towards E; they are then refracted at the surface D towards N. In
modern notation:
nAC BD
BC CD
DN
n 1
n(
AC BD)
(
n 1)
CD
, where DN is the focal distance
measured from the anterior face of the lens. For rays coming from the other
direction O is the ‘punctum concursus’.
The case of a bi-convex lens was only one out of many cases Huygens
treated in the fourteenth to seventeenth proposition of Tractatus. Taking both
spherical aberration and the thickness of the lens into account, he derived
exact theorems for the focal distance of each type of lens. In each case, he
also showed how to simplify the theorem when the thickness of the lens is
not taken into account. In the case of a bi-convex lens, he started by
comparing the focal distances CO and DN when the radii of both sides of the
lens are not equal. Their difference vanishes when the thickness of the lens
CD is ignored and both refractions are assumed to take place simultaneously.
The focal distance N is then easily found by first determining point L with
AC BD
DL : LB = n and then AB : AD = DE : EA, or DN
AC BD
2
. 26 In the case of
a glass lens (n = 3 : 2) LB is twice BD and (AC + BD) : AC = 2BD : DN. It
follows directly that the focal distance is equal to the radius in the case of an
equi-convex lens. In the twentieth proposition of Tractatus, Huygens
extended the results for thin lenses to non-parallel rays. In this case rays
diverge from a point on the axis relatively close to the lens and are refracted
towards a point P found by DO · DP = DC2 (DO is the focal distance for
parallel rays coming from the opposite direction). Huygens had to treat all
cases of positive and negative lens sides separately, but the result comes
down to the modern formula 1 1 1 . 27
p p'
f
In the remainder of the first book of Tractutus, Huygens completed his
theory of focal distances by determining the image of an extended object,
rounded off in the twenty-fourth proposition (Figure 9). The diameter of the
image IG is to the diameter of the object KF as the distance HL of the image
26 OC13, 88-89. Equivalent to the modern formula 1
f
27 OC13, 98-109.
Figure 8 Focal distance DN of a bi-convex lens
= (n -1)( + )
1 1
R1
R 2
.

20 CHAPTER 2
to the lens is to the distance EL of the object to the lens. 28
The point L has a special property that Huygens had
established in the preceding proposition. An arbitrary ray
that passes through this point leaves the lens parallel to the
incident ray. 29 In the twenty-second proposition, Huygens
had demonstrated that the focal distance LG of rays from a
point K of the axis is more or less equal to that of a point E
on the axis. 30 The triangles KLF and GLI are therefore
similar, which proves the theorem. 31
Images
The theory of focal distances formed the basis of Huygens’ discussion of the
properties of images formed by lenses and lens-systems in the second book
of Tractatus. Huygens’ theory of images is once again both rigorous and
general. The central questions in book two were how to determine the
orientation of the image and the degree of magnification. For the time being,
Huygens ignored the question whether an image is in focus. In this way he
could derive general theorems on the relationship between the shape of
lenses and their magnifying properties. He then showed how these reduced
to simpler theorems in particular cases, for example for a distant object. In
the third book he showed what configurations produced focused images.
Figure 10 Magnification by a convex lens.
Figure 9 Extended
image.
In the second and third propositions of book two, Huygens discussed a
convex lens. His aim was to determine the magnification of the image for the
various positions of eye D, lens ACB and object MEN (Figure 10). In order to
distinguish between upright and reversed images, Huygens defined the
‘punctum correspondens’ (later called ‘punctum dirigens’). 32 This is the focus
of rays emanating from the point where the eye is situated and is thus found
by means of the theory of the first book. First, the eye D is between a convex
28
OC13, 122-125.
29
OC13, 118-123. In modern terms, L is the optical center.
30
OC13, 114-119.
31
Huygens added that when the thickness of the less is taken into account, point V in the lens instead of
L should be taken as the vertex of the triangle.
32
OC13, 176n1.

1653 - TRACTATUS 21
lens ACB and its focus O. In this case, the object MEN is seen upright and
magnified. The lens refracts a ray NBP to BD so that point N of the object is
seen in B, whereas it would be seen in C without the lens. AB is larger than
AC and on the same side of the axis. Huygens then showed that
AB : AC = (AO : OD)·(ED : EP), which in the case of a distant object reduces
to AB : AC = AO : OD. 33 If, on the other hand, the eye is placed in the focus
(so that AD = AO) and NB is taken parallel to the axis, AB : AC = EO : AO
which becomes infinitely large when the object is placed at large distance. In
the next proposition, Huygens considered the cases where the eye is placed
beyond the focus O. In this case the ‘punctum correspondens’ P is on the
other side of the lens and the image will be reversed when the object is
placed beyond it.
With the same degree of generality, in the fifth proposition Huygens
discussed the images produced by a configuration of two lenses. 34 He figured
(Figure 11, most left one) two lenses A and B with focal distances GA and HB,
the eye C and the object DEF, all arbitrarily positioned on a common axis. He
then constructed point K on the axis, the ‘punctum correspondens’ of the
eye with respect to lens B, the ocular lens. Next, he constructed point L on
the axis, the ‘punctum correspondens’ of point K with respect to lens A, the
objective lens. In this way, a ray LD will be refracted by the two lenses to the
eye via points M on lens A and N on lens B. Without lenses, the eye sees
point F of the object – where DE = DF – along line COF. The degree of
magnification is therefore determined by the proportion BN : BO. In this
general case, the magnification follows from
BN : BO = (HB : HC)·(AG : GK)·(EC : EL). Huygens derived this proportion for
the case of a concave ocular and a convex objective, but the same applied to
a system of two convex lenses.
In the adjoining drawings, Huygens sketched various positions of eye,
lenses and object (Figure 11 gives four cases). These showed whether the
image was upright or reversed. In addition, he showed how the general
theorem reduced to a simpler one in particular cases. For example, when the
‘punctum correspondens’ of the ocular K and the focus of the objective G
coincide, it reduces to (HB : HC)·(EC : AK). Likewise, the configurations used
in practice were only a special case that Huygens discussed as he went along.
If a concave ocular and a convex objective are positioned in such a way that
BG = BH, where the ocular is between the objective and its focus, the
magnification of a distant object is AG : BH. The same applies to two convex
lenses that are positioned with their foci coinciding in between. In other
words, the magnification is equal to the quotient of the focal distances of
both lenses. In this roundabout way, Huygens proved what had been, and
33 OC13, 174-179.
34 OC13, 186-197.

22 CHAPTER 2
continued to be, assumed for quite some time, as Molyneux was to remark in
1690. 35
Figure 11 Four of the cases discussed (additional lettering
added).
With the magnifying properties of lens-systems thus established in a most
general way in parts one and two of Tractatus, Huygens’ subsequent account
of actual telescopes came down to a rather straightforward application to a
few specific cases. The state in which he left the third part of Tractatus in
1653 is hard to determine. It probably consisted of only two or three
theorems. Huygens did not discuss optimal configurations of lenses in
telescopes systematically, but only explained under what conditions ocular
and objective produced sharp images. The solution was simple, as he stated
in the first proposition. In order to see a sharp image, the rays from the
35 “This is the great Proposition asserted by most Dioptrick Writers, but hitherto proved by none (for as much
as I know) …” Molyneux, Dioptrica nova, 161.

1653 - TRACTATUS 23
object should leave the ocular parallel to the axis. 36 The foci of the lenses
should therefore coincide. For myopic people and those using a telescope to
project images things are different. In these cases the rays should be brought
to focus after they have passed the ocular and the foci of the lenses should
not coincide. In the second proposition, Huygens discussed the
configuration of two lenses required to project images and determined their
magnification. 37
Huygens aimed at providing a general and exact theory of the properties
of lenses and their configurations. The generality of Huygens’ theory reached
its high-point in a theorem that is inserted in part two of Tractatus as the sixth
proposition. It may be of a later date, as the manuscript is on a different kind
of paper and written with a different pen than the rest of this part. 38
Nevertheless, the theorem states that the magnification of an arbitrary
system of lenses remains the same when eye and object switch place. 39 This
theorem, so Huygens concluded his demonstration, would be useful in
determining the magnification and distinctness of images.
Figure 12 Analysis of Keplerian telescope with erector lens. See also Figure 13.
Huygens applied the theorem in the third and fourth proposition included by
the editors of Oeuvres Complètes in book three. The third proposition is
certainly of a later date, as it analyses the eyepiece Huygens invented in
1662. 40 The fourth proposition discusses a configuration of three convex
lenses proposed by Kepler in 1611 (Figure 12). 41 A telescope of two convex
lenses ordinarily produces a reversed image, but a third lens inserted between
the ocular and the objective may re-erect the image.
Huygens explained that an upright and sharp image is attained as follows
(Figure 13). 42 AC is the focal distance of the objective lens YAB, and HF the
focal distance of the ocular QHR. The third lens DET is identical with the
ocular with a focal distance EL = HF. It is placed so that EC = 2EL and EH =
3EL. In this case, point C on the axis is the ‘punctum correspondens’ for rays
through focus F of the ocular. Therefore a ray from S at a large distance is
refracted by the lenses in such a way that it leaves the ocular parallel to the
axis towards the eye PN. In order to determine the magnification by the
36 OC13, 244-247.
37 OC13, 246-253.
38 Hug29, 151-167.
39 OC13, 198-199.
40 OC13, 252n1. See below, section 3.1.2.
41 Dating this theorem is difficult. It may have been written in 1653, as the configuration was well-known.
Yet, Huygens also discussed the enlarged field of such a configuration, which may imply that it is of a
later date. See note 20 on page 16 above.
42 OC13, 258-261.

24 CHAPTER 2
system, Huygens applied
proposition six of book
two. The eye is imagined
at S and the object at PN.
In this way the
magnification is
determined by the Figure 13 Diagram for the analysis in Figure 12.
proportion YB : PN. It easily follows that this proportion is equal to AC : EL,
the proportion of the focal distances of the objective and the ocular.
Conclusion
In Tractatus, Huygens addressed a specific question: how can the working of
the telescope be understood mathematically? Regarding thin glass lenses his
answers, as we shall see in the next section, were not that new. Yet, he had
arrived at these answers by way of a rigorous mathematical analysis of the
properties of lenses. With the sine law, Huygens derived general and exact
theorems regarding the focal distances of thick lenses for both parallel and
non-parallel rays, irrespective of the material lenses are made of. On the basis
of this exact theory, he showed that these theorems reduce to the familiar,
simpler ones when the thickness of the lens is ignored and a specific index of
refraction is chosen. In the same way, he first established a general theorem
regarding the magnification by a lens-system and then showed that, in the
cases of actual telescopes, it reduced to the simple and familiar one. If the
elaboration of the theory of Tractatus was markedly mathematical, its
rationale was the telescope. Its goal was a ‘theory of the telescope’: an
account of the working of the telescope on the basis of dioptrical theory. In
this sense, the theory of the first two books was almost too elaborate. All in
all, in his Tractatus, Huygens gave a rigorous answer to the question how the
working of the telescope can be understood mathematically.
Huygens was the first one to elaborate a theory of the teleoscope by
means of the exact law of refraction. He knew that his treatise would fill gaps
left by others, in particular Descartes, so we would expect him to publish it
soon. However, as contrasted to other mathematical treatises he published in
this period, he did not press ahead with Tractatus. He inquired with
publishers and Van Schooten even proposed to append Huygens’ treatise to
a Latin edition of Descartes’ Discours de la Methode, La Dioptrique and Les
Météores, but nothing came of it. 43 Despite repeated announcements between
1655 and 1665 that he was publishing Tractatus, Huygens never did. 44
2.2 Dioptrics and the telescope
The orientation on the telescope is essential to Tractatus. If Huygens was the
first to apply the sine law to questions regarding the telescope, what had
other students of dioptrics been doing? In this section, I sketch the
43
OC1, 280; 301-303; 321-322. Huygens did not pin much faith in Van Schooten’s proposal.
44
I will say a bit more about his publishing pattern on page 174.

1653 - TRACTATUS 25
development of seventeenth-century dioptrics, with a particular emphasis on
the way questions regarding the telescope were addressed.
The telescope was made public when in September 1608 a spectacle
maker from Middelburg, Hans Lipperhey, came to The Hague to request a
patent for a “… certain device by means of which all things at a very great
distance can be seen as if they were nearby, …” 45 It was a configuration of a
convex and a concave lens fitted appropriately in a tube and turned out to
magnify things seen through it. The patent was denied, as within a couple of
week two other claimants turned up. It is doubtful whether Lipperhey had
made the invention himself. He may have learned it from his neighbour
Sacharias Janssen, who in his turn seems to have learned the secret of the
device from an itinerant Italian. 46 The history of the invention of the
telescope is an intricate one, in which Jacob Metius of Alkmaar was the first
to be publicly named its true inventor by Descartes. The first doubts were
raised in the 1650s through the publication of Pierre Borel. Huygens himself
was one of the first to perform some archival research on the matter,
claiming that the credit should go to either Lipperhey or Janssen. 47 The news
of the device spread quickly through Europe and by the summer of 1609
simple telescopes were commonly for sale in the major cities of Europe. 48
The news also reached the ears of scholars, who realized the device could
be of use in astronomical observation. Most successful among them was
Galileo in Venice, whose interest in the telescope was aroused in the spring
of 1609. He figured out how to make one and how to improve it. Among the
earliest telescopists, Galileo was the only one who not only knew how the
telescope could be improved, but also had the means to do so. In August, he
had made a telescope that magnified nine times, as opposed to the ordinary
three-powered spyglasses. A couple of months later he had made telescopes
that were even more powerful. 49 In this way, Galileo turned the spyglass into
a powerful instrument of astronomical observation. 50 He observed the
heavens and saw spectacular things: mountains on the Moon, satellites
around Jupiter, and more. In March 1610, he published his observations in
Sidereus nuncius. Galileo also sent a copy to the Prague court with a specific
request for a comment by Kepler. 51
In May, Kepler published his comment in Dissertatio cum nuncio sidereo. He
primarily responded to Galileo’s observations, but he also said a few things
about the instrument. In Sidereus nuncius, Galileo had explained its
construction and use, but he had left out any mathematical account. 52 In
45
Van Helden, Invention, 35-36; Galileo, Sidereus nuncius, 3-4 (Van Helden’s introduction).
46
De Waard, Uitvinding, 105-225; Van Helden, Invention, 20-25.
47
OC13, 436-437.
48
Van Helden, Invention, 21, 36.
49
Van Helden, Invention, 26; Galileo, Sidereus nuncius, 6, 9 (Van Helden’s Introduction).
50
Van Helden, “Galileo and the telescope”, 153-157.
51
Galileo, Sidereus nuncius, 94 (Van Helden’s Conclusion).
52
Galileo, Sidereus nuncius, 37-39.

26 CHAPTER 2
reply, Kepler briefly explained how lenses refract rays of light so that they
can produced magnified images. 53 The explanation in Dissertatio was only a
sketch, but the message was clear. The telescope was a remarkable invention,
but its working needed mathematical clarification. A theory of the telescope
was called for. Within a few months, Kepler developed one. In September
1610, he finished the manuscript of Dioptrice, published the next year.
“Some have disputed over the priority of its invention, others rather applied themselves
to the perfection of the instrument, as there chance mainly counted, here reason
dominated. But Galileo scored the greatest triumph by exploring its use to disclose
secrets, because zeal procured him with the design and fortune has not withheld him
the success. I, driven by an honest emulation, have shown the mathematicians a new
field to expose their acuteness, in which the causes and principles are retraced to the
laws of geometry, the effects of which are so awaited with much impatience and are of
such pleasing diversity.” 54
The goal of Dioptrice was to provide a mathematical account of the working
of the telescope. In Kepler’s view, the working of any instrument used in
astronomy should be understood precisely. A decade earlier, he had
approached the puzzling properties of the pinhole images used in the
observation of solar eclipses. His answer had been a new theory of image
formation, which he had published in Paralipomena (1604). In Dioptrice, Kepler
applied this theory to lenses in order to determine the dioptrical properties
of the telescope. Dioptrice had one substantial shortcoming: Kepler knew that
he did not know the exact law of refraction. He used an approximate rule
instead.
2.2.1 KEPLER AND THE MATHEMATICS OF LENSES
Kepler’s concerns in Paralipomena were induced by a problem of astronomical
observation. In 1598, Tycho Brahe had reported an anomalous observation
of the apparent size of the moon during a solar eclipse. 55 Brahe used a
pinhole to project the image of the eclipsed sun. When he measured the
diameter of the projection he realized that “the moon during a solar eclipse
does not appear to be the same size as it appears at other times during full
moons when it is equally far away”. 56 He tried to produce consistent values
by applying some ad hoc corrections to his measurements. 57 Kepler took a
different approach, analyzing mathematically the way the image was
produced. He had known the anomaly of pinhole images for some time
53
Kepler, Conversation, [19-21].
54
Kepler, Dioptrice, dedication (KGW4, 331). “… circaque eam alij de palma primae inventionis certarent,
alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio
dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum
triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego
doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est
causarum lege geometrica demonstrandarum, quibus tam exoptati, tam jucunda varietate multiplices
effectus inniterentur.”
55
Straker, “Kepler’s theory of pinhole images”, 276-278.
56
Cited and translated in: Straker, “Kepler’s theory of pinhole images”, 278.
57
Straker, “Kepler’s theory of pinhole images”, 275-276; 280-282.

1653 - TRACTATUS 27
when in 1600 he set himself to see whether an ‘optical cause’ might account
for it. The solution to the apparent anomaly of pinhole projections of solar
eclipses would be the copestone of Paralipomena. I will only discuss Kepler’s
theory of image formation and its application to the eye.
The optical theory available around 1600 was the medieval tradition of
perspectiva and it did not provide Kepler with an answer for the anomaly of
pinhole observations. Perspectiva built on the great synthetic work from the
eleventh century of the Arab mathematician Alhacen, when it was adopted
by a line of thirteenth-century Christian mathematicians, Bacon, Witelo and
Pecham. They elaborated a mathematical theory of optics, in addition to the
natural philosophical and medical theories, in which vision was analyzed in
terms of the behavior of light rays. 58 The designation ‘perspectiva’ derives
from the common title for their works and it constituted the canon of
mathematical optics well into the seventeenth century. In the sixteenth
century perspectiva texts had been published, with the 1672 edition by
Friedrich Risner of Alhacen’s Optica and Witelo’s Perspectiva as the most
important. 59
The problem of pinhole images was well-known in perspectiva. It was
known since Antiquity that the image of the sun, projected by a square
aperture, can still be round. This seemed to contradict the basic principle of
optics: the rectilinearity of light rays. The solutions given by perspectivist
writers did not satisfy Kepler. Each had in the end sacrificed the principle of
rectilinearity – the foundation of geometrical
optics. 60 Kepler had to resolve the problem by
himself. His solution consisted of a new theory
of the way rays form images of objects. This
theory, in its turn, would be the foundation of
his dioptrics as well as of seventeenth-century
geometrical optics in general.
Kepler approached the problem anew and
did so by uncompromisingly applying the
principle of rectilinearity. In Paralipomena, he
describes how he replaced a ray of light by a
thread. He took a book, attached a thread to
one of its corner and guided it along the edges
of a many-cornered aperture, thus tracing out
the figure of the aperture. Repeating this for
the other corners of the book, and many more
points, he ended up with a multitude of
overlapping figures that formed an image of
Figure 14 Kepler’s solution to
the pinhole problem
58 Further discussed in section 4.1.1.
59 Dupré points out Risner’s programmatic discussion of the science of optics in the preface to the edition
which constitute an important, yet still little studied, agenda for seventeenth-century optics. Dupré,
Galileo, the Telescope, 54.
60 See Lindberg, “Laying the foundations”, 14-29.

28 CHAPTER 2
the book. In the same way, he argued, all the points of the sun project
overlapping images of the aperture (Figure 14). The resulting image has the
shape of the sun, albeit with a blurred edge. In the projection of an eclipse,
the image of the shadow of the moon is partially overlapped by the image of
the sun. Consequently, the diameter of the moon seems too small. In chapter
two of Paralipomena, Kepler had solved the apparent anomaly of pinhole
observations in principle, building on the previous chapter, he elaborated the
exact solution in the eleventh and final chapter.
Image formation
Kepler came to the conclusion that there were more problems in
perspectiva, in particular its core, the theory of vision. In chapter five of
Paralipomena, he elaborated a new theory of vision on the basis of his newly
gained understanding of image formation. In its fourth section, Kepler listed
the defects of existing theories of vision, the most important being a wrong
understanding of the anatomy of the eye and of the mathematics of image
formation. Perspectivist theories considered the lens the sensitive organ of
the eye, whereas recent anatomical investigations had demonstrated,
convincingly according to Kepler, that the retina receives images from
objects. He himself had shown the defects of the perspectivist understanding
of image formation, calling Witelo by name, and he now went on to
reconsider the optics of the eye.
In perspectivist theory, each point of an object emits rays of light in each
direction. This, however, raises the problem how a sharp image can be
perceived, that is: how a one-to-one relationship between a point of the
object and a point of the image in the eye is established. According to
Alhacen there can be only one point in the eye where a ray from a point of
the object can be perceived. He stated that this must be the one entering the
eye perpendicularly (and thus perpendicular to the lens). He explained that
the other rays are refracted by the eye, therefore weakened, and thus do not
partake in the formation of the image. 61 In medieval optics, images were
therefore taken to be produced by single rays from each point of the object.
Kepler saw no reason to differentiate between weak and strong rays. He did
not see, for that matter, why refraction would weaken a ray. In his view, all
rays emitted by a point should somehow partake in the formation of an
image. In the case of pinholes this resulted in a fuzzy image, but what about
the sharp images by which we generally see the world?
Kepler’s answer was that the cone of rays coming from one point is
somehow brought to focus on the retina. Following certain recent
anatomical observations he considered the retina as the sensitive organ of
the eye, in contrast to perspectivist theory that had assigned the power of
visual perception to the crystalline humor. According to Kepler, the various
humors of the eye can be regarded as one refracting sphere. In the fifth
61 Alhacen, Optics I, 68 (book 1, section 17) and 77 (book 1, section 46).

1653 - TRACTATUS 29
chapter of Paralipomena, Kepler explained how images are formed on the
retina. In order to account for spherical aberration, he argued that the pupil
as well a the slightly a-spherical shape of the posterior side of the humors
diminish the severest aberrations. Kepler’s analysis was based on his study of
refraction in the fourth chapter of Paralipomena. In this chapter, he had tried
unsuccessfully to find an exact law of refraction, but his understanding of
refraction at plane surface sufficed for discussing the focusing properties of
spheres at least qualitatively. 62
With this Kepler completed his theory of image formation. It had
originated in the solution of an anomalous astronomical observation and its
ultimate rationale was astronomical observation. With his definition of optics
and its indispensability to cosmology, Kepler fits in a Ramist trend in the
sixteenth century that Dupré refers to with ‘the art of seeing well’ and to
which Risner also belongs. 63 The full title of Paralipomena starts with Ad
Vitellionem paralipomena, quibus astronomiae pars optica traditur, …. In his preface,
Kepler proclaimed eclipses to be the most noble and ancient part of
astronomy: “… these darknesses are the astronomers’ eyes, the defects are a
cornucopia of theory, these blemishes illuminate the minds of mortals with the
most precious pictures.” 64 The eye being the fundamental instrument of
observation, to Kepler a reliable theory of visual perception was
indispensable for astronomers. His perspectivist forebears had not treated
the matter satisfactorily and thus he had provided the necessary additions to
Witelo. Revolutionary additions, to be sure. The eye perceives dots rather
than things and in the analysis of vision “… we should not look to entire
objects, but to individual points of objects, …” 65 Kepler had made it clear
that all rays from an object point partake in the formation of images, whose
sharpness is not evident beforehand. Image formation was no longer a
matter of tracing individual rays from object to image. The task of the
optician now became to determine exactly how a bundle of rays is brought to
focus again after it is emitted by a point of an object.
Lenses
Kepler approached the newly invented telescope in the same manner as the
pinhole and the eye. The working of the telescope should be properly
understood if it were to be used in astronomical observation. For Kepler,
this meant that a mathematical theory was required, a mathematical theory of
the telescope so to say. He had already treated lenses briefly in the final
proposition of chapter five of Paralipomena. At that moment spectacle glasses
were new topic in optical literature. Kepler expressed his amazement that no
62
Kepler’s efforts to find a law of refraction are discussed below, in section 4.1.2.
63
Dupré, Galileo, the telescope, 31.
64
Kepler, Paralipomena, 4 (KGW2, 16). “… hae t e n e b r a e sint Astronomorum o c u l i , hi d e f e c t u s
doctrinae sint a b u n d a n t i a , hi n a e v i mentes mortalium preciosissimis p i c t u r i s illustrent.”
Translation Donahue, Optics, 16.
65
Kepler, Paralipomena, 201 (KGW2, 181). “Itaque non oportet nos ad res totas respicere, sed ad rerum
singular puncta, …” Translation Donahue, Optics, 217.

30 CHAPTER 2
mathematical account of such an important and widespread device existed.
We can understand his surprise, for spectacles had already been invented
around 1300. 66 A brief account by Francesco Maurolyco, that dated back to
around 1521, was not to be published before 1611, in Diaphaneon seu
transparentium libellus. Kepler would not have found much in it to his liking,
for it was a qualitative theory based on a somewhat confusing variant of the
perspectivist theory of vision and refraction. 67 Kepler knew that Della Porta
had written a study of refraction, but he had not been able to lay hands on
De refractione (1593). He dismissed what Della Porta had written in Magia
naturalis, namely that spectacles correct vision because they magnify images.
Kepler elaborated his own account of lenses, dedicating it in Paralipomena to
his patron Ludwig von Dietrichstein, whom he said had kept him busy for
three years with the question of the secret of spectacles. 68 Kepler explained
the beneficial effects of spectacles as follows. Myopic and presbyotic vision
occurs when rays are not brought to focus on the retina but in front of it or
beyond. He gave a short, qualitative discussion of the effect of lenses on a
bundle of parallel rays coming from a distant point. Convex and concave
lenses – for myopics and presbyotics respectively – move the focus of rays to
the retina. Some magnification may occur, but this is not the reason why
spectacle lenses correct vision.
With the introduction of the telescope in astronomy, the qualitative
account of single lenses in Paralipomena did not suffice any more. In Dioptrice,
Kepler extended his theory of image formation to a quantitative analysis of
the properties of lenses and their configurations. 69 As a matter of fact, he was
the one to coin the term ‘dioptrics’. 70 Compared to Huygens’ Tractatus,
Kepler’s dioptrical theory was of more limited scope. His goal was to explain
the formation of images by a telescope. He therefore restricted his theory to
a few types of lenses and mainly confined himself to object points at infinite
distance when incident rays are parallel. The basic concept was the focus of a
lens, the point where parallel rays intersect after refraction. Kepler could not
determine the focal distance with the exactness we have seen with Huygens.
He could not, for example, determine the exact route of a ray through the
refractions at both surfaces of a lens. The main obstacle in the way of a more
extensive treatment was the fact that Kepler did not know the exact law of
refraction. In Dioptrice, he used an approximation that was valid only for
angles of incidence below 30º, and that, even so, applied solely to glass.
According to this rule the angle of deviation is one third of the angle of
66
Rosen, “The invention of eyeglasses”, 13-46.
67
Lindberg, “Optics in 16th century Italy”136-141. Maurolyco had preceded Kepler in his analysis of the
pinhole image: Lindberg, “Optics in 16th century Italy”, 132-135; Lindberg, “Laying the foundations”.
68
Kepler, Paralipomena, 200-202 (KGW2, 181-183).
69
Malet, “Kepler and the telescope” offers a detailed discussion of Dioptrice, without however presenting it
as a part of the ‘optical part of astronomy’.
70
Kepler, Dioptrice, dedication (KGW4, 331).

1653 - TRACTATUS 31
incidence; the angle between the incident ray, produced beyond the
refracting surface, and the refracted ray.
Kepler began with a discussion
of the focal distances of planoconvex
lenses (Figure 15). A ray
HG is incident on a convex surface
with radius AC, the angle of
incidence is GAC. As the angle of
deviation is one third of this, HG
will be refracted towards F, with AC : AF = 1 : 2. 71 The focal distance is
therefore approximately three times the radius of the convex face.
Analogously, he argued that the focal distance of a plano-convex lens, the
plane face turned towards the incident rays, is approximately twice the radius
of curvature. For other cases Kepler established only rough estimations. If
convergent rays are incident on the plane side of a plano-convex lens, the
refracted rays intersect the axis within the focal distance. Combining these
three theorems, Kepler showed that the focal distance of a bi-convex lens is
both smaller than three times the radius of the anterior side and twice the
radius of the posterior side. In the case of an equi-convex lens, this comes
down to a focal distance approximately equal to the radius of its sides. 72
Kepler did not determine the focal distance of a concave lens, he only
showed that rays diverge after refraction. 73
On this basis, the properties of images formed by lenses are easily found.
The image DBF of an extended object CAE through a bi-convex lens GH is
formed at focal distance (Figure 16). The picture is inversed as the rays from
C are refracted towards D, etcetera. As the focal distance is roughly the radius
of any side, the magnitudes of object and image will be in a proportion equal
to their respective distances to the lens. 74 In Dioptrice, Kepler briefly reiterated
his theory of vision. On the one hand, so he said in the dedication, he did so
for the sake of completeness, on the other hand because some readers had
trouble understanding his account in Paralipomena. 75 He explained that a
perfectly focusing surface was not spherical, but should be hyperbolic, like
the crystalline humor of the eye was. 76 On the basis of his theory of the
retinal image, he explained the effect of a lens placed before the eye once
more. Depending upon the position of the eye with respect to the focal
distance, the object will be perceived sharply. 77 When the eye is placed not
too far from the focus, a magnified image will be perceived.
71 Kepler, Dioptrice, 11 (KGW4, 363).
72 Kepler, Dioptrice, 12-15 (KGW4, 363-367).
73 Kepler, Dioptrice, 45-49 (KGW4, 388-393).
74 Kepler, Dioptrice, 16-18 (KGW4, 367-369).
75 Kepler, Dioptrice, dedication (KGW4, 335).
76 Kepler, Dioptrice, 21-24 (KGW4, 371-372).
77 Kepler, Dioptrice, 35-42 (KGW4, 381-387).
Figure 15 Focal distance of a plano-convex lens

32 CHAPTER 2
Kepler proceeded to discuss the combination of
two convex lenses. He explained how these should
be configured in order to perceive a sharp,
magnified image. 78 This is achieved when the foci of
both lenses coincide. It is remarkable that that
Kepler discussed the configuration of two convex
lenses, because in 1611 only the combination of a
convex objective and a concave ocular was known
to produce a telescopic effect. Kepler probably
arrived at this alternative configuration by
theoretical considerations. 79 He never manufactured
this kind of telescope himself. The configuration has
come to be known as a Keplerian or Astronomical
telescope, as opposed to the Dutch or Galilean
telescope with a concave ocular. Much later it
became clear that the Keplerian type has the
advantage of a larger field of view, but Kepler
himself did not know this. He did realize that this
configuration had a drawback, it produced inverted
images. This could be corrected, he said, by inserting
a third lens at an appropriate place between ocular
and objective lens.
Kepler then turned to an account of concave
lenses and finally to a discussion of so-called Dutch
telescopes. He explained the configuration of a
Figure 16 Image formation
by a lens
convex objective and a concave ocular only in broad lines. As he did not
speak of the focus of a concave lens, he could only roughly point out where
the ocular should be placed with respect to the focus of the objective. His
discussion of the configurations and the resulting properties of the images
remained mainly qualitative. Kepler offered a wealth of practical guidelines as
to the configurations of lenses and the way the best effects are achieved.
Dioptrice does not consist of rigorously demonstrated theorems. Without an
exact law of refraction, a quantitative and exact theory could hardly be
attained. This was not necessarily Kepler’s intention. Rather, he intended to
explain the working of the telescope mathematically. He did so by analyzing,
on the basis of his theory of image formation, how it forms magnified
images.
All this may lead us to conclude that Huygens’ Tractatus can be seen as an
up-to-date answer to the question Kepler had originally addressed in
Dioptrice; updated in the sense that the analysis of lenses was based on the
sine law. It established the dioptrical properties of spherical lenses and
78 Kepler, Dioptrice, 42-43 (KGW4, 387-388).
79 A possible source of inspiration may have come from the analogous configuration of the eye and a
convex spectacle glass, as the eye acts as a convex lens does. See also Malet, “Kepler and the telescope”,
119-120.

1653 - TRACTATUS 33
focused on problems pertaining to their configurations in actual telescopes.
Like Kepler, Huygens intended to found the dioptrical properties on a sound
mathematical basis. Whether a continuation of Dioptrice was his actual goal,
can only be surmised as he did not explicitly refer to it in such a
programmatic sense. Huygens did know Dioptrice, it had been on the reading
list of his mathematics tutor Stampioen and much later he commended it to
his brother Constantijn as the best introduction to dioptrics. 80 Huygens did
not have much to offer that was not already known. Tractatus covered more
types of lenses but the eventual results regarding the focal distances of lenses
and the magnifying properties did not differ much from Dioptrice. The crucial
difference is that Huygens founded his results on a general and exact theory
of focal distances. It rigorously proved Kepler’s results. He had the exact law
of refraction at his disposal and thus could be exact where Kepler necessarily
had to leave his readers with approximate answers.
Perspectiva and the telescope
At the same time when Kepler wrote Dioptrice, two other scholars devised an
account of the telescope. Della Porta’s ‘De telescopio’ remained
unpublished, De Domini’s De Radiis Visus et Lucis was published in 1611.
Both were based on perspectivist theory of image formation. Before I go on
to discuss the impact of the sine law on dioptrics, I briefly discuss these in
order to make it clear why that perspectivist theory was intrinsically
inadequate to account fully for the effect of lenses.
Shortly before his death, Della Porta extended his theory of lenses of De
Refractione to telescopes in a manuscript ‘De telescopio’. 81 It reveals the
problems lenses posed for perspectivist theory of image formation. In order
to determine the place where an object is seen, perspectiva used the cathetus
rule. The cathetus is the line through the object point, perpendicular to the
reflecting or refracting surface. The cathetus rule states that the image is the
intersection of the ray entering the eye and the cathetus. Modern Keplerian
theory shows that, although valid in many cases, this rule turns out to break
down for curved surfaces in particular. To account for images of lenses
another problem turns up. As a lens refracts a ray twice, this seems to imply
that the rule has to be applied twice also. Della Porta avoided this problem
by considering only one cathetus.
Della Porta considered a lens in terms of refracting spheres, as he had
done in De Refractione (Figure 17). The dotted lines indicate such spheres and
the lens dcgf is formed by their overlap. 82 The object ab is perceived as
follows: a ray from point a is refracted along cd to the eye. Della Porta drew
the cathetus ka of the lower surface of the lens, which also is its radius.
When produced, the ray entering the eye intersects the cathetus in point h,
80
OC1, 6 (Stampioen’s list of recommended readings spans pages 5-10) and OC6, 215.
81
Della Porta’s account of refraction by spheres and lenses in De refractione is discussed in Lindberg,
“Optics in 16th century Italy”, 143-146.
82
Della Porta, De Telescopio, 113-114.

34 CHAPTER 2
where point a is seen. In the same manner point i is constructed and hi is the
object as perceived through the lens. The question is why only the ray acd
emanating from point a is singled out. Della Porta seemed to assume that
this is the one that enters the eye perpendicularly. Yet, in the case of a distant
object, he no longer chooses rays parallel to the axis of the system, but the
crossing rays ad and bg (Figure 18). He probably did so to account for the
reversing of the image, but he lacked a theoretical justification. Della Porta’s
account of concave lenses was even more troublesome, as he ignored the
implication of his reasoning that the eye cannot perceive the whole object at
once. Moreover, he persistently has rays refracted from the perpendicular at
the first surface (for example de in Figure 19). 83 ‘De telescopio’ culminated in
an account of a Galilean telescope (Figure 19). Della Porta traced the path of
a ray emanation from point a of the object. He then chose the cathetus with
respect to the upper surface of the concave lens and argued that qr is the
image perceived.
Figure 17 Image of a near object
Figure 18 Image of
distant object
All in all, from Kepler’s perspective Della Porta’s theory of lenses was
fraught with difficulties and mathematically it was riddled with ambiguities.
Part of these arose from his sloppiness and lack of understanding of certain
problems. Part of the problem lies also with the perspectivist foundation of
his account. How, for example, should the cathetus rule be applied to two or
more refractions? More important, perspectivist theory offers no means of
differentiating between sharp and fuzzy images, quite a relevant issue with
respect to the telescope. 84 Della Porta made no attempt to deal with it.
Whether he chose ignore it or wass unaware of it is unclear. He was quite
content with what he had written. As the inventor of the telescope – so he
83 Della Porta, De telescopio, 141-142.
84 Compare Lindberg, “Optics in 16 th century Italy”, 146-147.
Figure 19 Image by a
telescope

1653 - TRACTATUS 35
fancied – he regarded himself as the only authority in these matters. 85 Shortly
after he wrote ‘De telescopio’ he died, and the text remained unknown until
1940.
De Radiis Visus et Lucis of De Dominis is well-known for its discussion of
the rainbow, but it also contains an account of lenses and the telescope. Like
Della Porta, De Dominis maintained perspectivist theory. His theory did not
go beyond a brief, qualitative theory of the refraction of visual rays by lenses.
In this way it avoided the problems revealed by Della Porta’s theory. It does
not seem to have counted as a serious alternative to Dioptrice. Unlike Kepler,
De Dominis was rarely referred to in matters dioptrical. In the widely read
Rosa Ursina (1630), Scheiner adopted Kepler’s theory of image formation. He
elaborately treated the construction and use of telescopes. Scheiner discussed
the properties of lenses and their configurations, but he did not incorporate
the quantitative part of Dioptrice – his account remained qualitative. Another
authoritative book on geometrical optics at the time, Opticorum Libri Sex
(1611) by the Antwerp mathematician Aguilón, did not discuss refraction or
lenses at all.
Dioptrice had been a reaction to Galileo’s neglect to explain the telescope
dioptrically in Sidereus Nuncius. Although quite an able mathematician, Galileo
never developed a theory of dioptrics. He applied himself to the
improvement of the instrument by making better lenses and optimizing the
quality of telescopic images. His friend Sagredo did take an interest in the
dioptrics of lenses, but was not encouraged to pursue his study. Galileo
wanted him to concentrate on matters of glass-making and lens-grinding. 86
On Dioptrice Galileo kept silent altogether. 87 Apparently, this self-styled
mathematical philosopher was not interested in the mathematical properties
of the instrument that had brought him fame. He did, however, have a clear
understanding of the working of lenses and telescopes. Dupré has recently
argued that Galileo relied on a tradition of practical knowledge, of mirrors in
particular, that had developed in the sixteenth century next to the
mathematical strand on which my account focuses. 88
2.2.2 THE USE OF THE SINE LAW
The exact law of refraction Kepler had to make do without, was soon found.
More than that, it had been within his reach. The English astronomer
Thomas Harriot had discovered it in 1601. After the publication of
Paralipomena, he and Kepler had corresponded on optical matters. However,
the correspondence broke off before Harriot had revealed his discovery. 89
Long before that, but unknown until the late twentieth century, the tenth-
85
Ronchi, “Refractione au Telescopio”, 56 and 34. “They know nothing of perspective.” and “... and it
pleases me that the idea of the telescope in a tube has been mine; ...”
86
Pedersen, “Sagredo’s optical researches”, 144-148.
87
KGW4, “Nachbericht”, 476.
88
Dupré, Galileo, the Telescope, chapters 4 to 6 in particular.
89
Harriot is discussed in section 4.1.2.

36 CHAPTER 2
century student of burning glasses Ibn Sahl had used a rule equivalent to the
sine law. 90 Around 1620, the Leiden professor of mathematics Willebrord
Snel was next and in the late 1620s Descartes closed the ranks of discoverers
of the law of refraction. 91 He published it in La Dioptrique (1637), shortly after
Pierre Hérigone had done so in the fifth volume of Cursus Mathematicus.
Hérigone did not use it in his dioptrical account, which summarized Dioptrice.
Harriot and Snel have left no trace of applying their find to lenses. Which
leaves La Dioptrique for further inspection.
Descartes and the ideal telescope
La Dioptrique was the fruit of Descartes’ involvement in the activities of
Parisian savants regarding (non-spherical) mirrors and lenses, which also
places him in the sixteenth-century tradition of mirror-making. 92 Descartes,
however, added his natural philosophical leanings and Kepler’s optical
teachings. In collaboration with the mathematician Mydorge and the artisan
Ferrier, he allegedly managed to produce a hyperbolic lens and in the course
of events he discovered the law of refraction. La Dioptrique had much
influence on seventeenth-century optics, especially through its second
discourse where Descartes derived the sine law. 93 In the following discourses,
Descartes first discussed the eye and vision – summarizing Kepler’s theory
of the retinal image – and then went on to a consideration “Of the means of
perfecting vision”. 94 This seventh discourse anticipated his discussion of
telescopes. He laid stress on the way spectacles enhance vision, instead of
correcting it. The telescope itself was introduced in a peculiar way. Descartes
explained how an elongated lens may further enhance vision. He then
replaced the solid middle part by air, thus arriving at a telescope consisting of
two lenses. 95 The argument was clear, but the discussion of focal and
magnifying properties of lenses was entirely qualitative and the sine law
played no role in it.
In the eighth discourse of La Dioptrique, Descartes applied the sine law to
lenses under the title: “Of the figures transparent bodies must have to divert
the rays by refraction in all manners that serve vision”. 96 Its sole purpose was
to show that lenses ought to have an elliptic or hyperbolic surface in order to
bring rays to a perfect focus. Avoiding the subtleties of geometry he
explained how these lines could be drawn by practical means and
demonstrated the relevant properties of the ellipse and hyperbola. As regards
the focal distances of lenses thus obtained with respect to configuration and
90
Rashed, “Pioneer”, 478-486.
91
For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat
later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250.
92
Gaukroger, Descartes, 138-146. Dupré, Galileo, the Telescope, 53-54.
93
Discussed in section 4.1.3
94
Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.”
95
Descartes, AT6, 155-160.
96
Descartes, AT6, 165. “Des figures que doivent avoir les corps transparens pour detourner les rayons par
refraction en toutes les façons qui servent a la veuë”

1653 - TRACTATUS 37
magnification, the account remained qualitative. La Dioptrique was written,
Descartes said in the opening discourse, for the benefit of craftsmen who
would have to grind and apply his elliptic and hyperbolic lenses. Therefore
the mathematical content was kept to a minimum. 97 Apparently this implied
that Descartes need not elaborate a theory of the dioptrical properties of
lenses.
Descartes adopted the term Kepler had coined for the mathematical
study of lenses. He had not, however, adopted the spirit of Kepler’s study.
Dioptrice and La Dioptrique approached the telescope from opposite
directions. Kepler had discussed actual telescopes and drudged on properties
of lenses that did not fit mathematics so neatly. Descartes prescribed what
the telescope should be according to mathematical theory. The telescope,
having been invented and thus far cultivated by experience and fortune,
could now reach a state of perfection by explaining its difficulties. 98 Huygens
was harsh in his judgment of La Dioptrique. In 1693, he wrote:
“Monsieur Descartes did not know what would be the effect of his hyperbolic
telescopes, and assumed incomparably more about it than he should have. He did not
understand sufficiently the theory of dioptrics, as his poor build-up demonstration of
the telescope reveals.” 99
We can say that Descartes, according to Huygens, had failed to develop a
theory of the telescope. He had ignored the questions that really mattered
according to Huygens: an exact theory of the dioptrical properties of lenses
and their configurations. La Dioptrique glanced over a telescope that existed
only in the ideal world of mathematics.
Unfortunately for Descartes, no one during the following decades
succeeded in actually grinding the a-spherical lenses of his design. Of mere
anecdotal interest is the irony with which Huygens’ tutor Stampioen had in
1640 pointed out to Descartes’ yet unfulfilled promise of a perfect telescope:
“… my servant Research will turn him a better spyglass without circles …
But nevertheless, what this Mathematicien has promised to do for six years
is still not satisfied.” 100 But Stampioen was in the middle of a terrible dispute
with Descartes at that moment.
After Descartes
Hobbes, Descartes’ most ardent rival in matters of mechanistic philosophy,
developed an alternative derivation of the sine law too. In the elaboration of
his dioptrical theory he also discussed spherical lenses. The unpublished “A
97
Descartes, AT6, 82-83. Ribe, “Cartesian optics” offers an enlightening account of the artisanal roots of
La Dioptrique.
98
Descartes, AT6, 82.
99
OC10, 402-403. “Mr. des Cartes n’a connu quel seroit l’effet de ses Lunettes hyperboliques, et en a
presumè incomparablement plus qu’il ne devoit. n’entendant pas assez cette Theorie de la dioptrique, ce
qui paroit par sa demonstration très mal bastie des Telescopes.”
100
Stampioen, Wis-konstigh ende reden-maetigh bewys, 58. “… mijn Knecht Ondersoeck sal hem eens een beter
Verre-kijcker sonder cirkeltjes daer toe weten te drayen : … Maer niettemin ’t geen dese Mathematicien al
over 6 Iaren belooft heeft te doen, blijft nog on-vol-daen.”

38 CHAPTER 2
minute or First Draught of the Optiques” of 1646 (the most complete
elaboration of his optics) included several chapters on lenses and
telescopes. 101 He did not, however, make the most of his knowledge of the
sine law. The account consisted of qualitative theorems – without proof and
often dubious – which applied mostly to single rays refracted by lenses.
Despite the presence of an exact law of refraction, Hobbes’ account (if
published) would have been no match for Dioptrice.
With the exact law of refraction established and published, the road
might seem open for a follow-up of Dioptrice in the form of an exact theory
of the dioptrical properties of spherical lenses. It was not to be, for various
reasons. First of all the sine law became generally known and accepted only
around 1660. 102 This delay may have been caused by a slow distribution of
Descartes’ works – and this maybe partly because La Dioptrique was written
in French – or the bad odor his ideas were in. As late as 1663, in Optica
promota, Gregory showed that the ellipse and hyperbola are aplanatic without
using the sine law. In 1647, Cavalieri extended the theory of Dioptrice to some
more types of lenses, using Kepler’s original rule. As the title Exercitationes
geometricae sex suggests, this was an exercise in mathematics not aimed at
furthering the understanding of the telescope. In this regard, Cavalieri was
not an exception.
Further, and more importantly, mathematicians addressed questions
raised in Kepler’s Paralipomena rather than in his Dioptrice, to wit abstract
optical imagery pertaining to Kepler’s theory of image formation, and the
‘anaclastic’ problem that had been put in a different light by that theory. The
anaclastic problem, or ‘Alhacen’s problem’, is closely related to the
determination of aplanatic surfaces: to find the point of reflection or
refraction of a ray passing from a given point to another. 103 When all rays are
considered, as is relevant in Kepler’s theory of image formation, to find these
points means determining the aplanatic surface. In this theory images are
formed by the focusing of bundles of rays, and in most cases of reflection
and refraction the image of a point source will not be a point. The properties
of these images became an important subject of study in seventeenth-century
geometrical optics. In Optica promota, Gregory extended the theory of
Paralipomena with his contributions to the theory of optical imagery and his
determination of aplanatic surfaces. From this viewpoint, La Dioptrique
embroidered on Paralipomena rather than Dioptrice.
The seventeenth-century study of these topics reached its highpoint in
the lectures Barrow and, later, Newton delivered at the university of
Cambridge. Barrow’s lectures were published in 1669, those of Newton
remained unpublished during his lifetime. With Huygens’ dioptrical work
101
Stroud, Minute, 20; Prins, “Hobbes on light and vision”, 129-132. On Hobbes’ derivation of the sine
law, see section 5.2.1.
102
Lohne, ”Geschichte des Brechungsgesetzes”, 166.
103
Huygens worked on it in 1671-2, see page 160.

1653 - TRACTATUS 39
remaining uncompleted and unpublished as well, Lectiones XVIII was the most
advanced treatise on geometrical optics published until the end of the
seventeenth century. The core of Lectiones XVIII consists of lectures IV
through XIII, in which Barrow determined the image of a point source in any
reflection or refraction in plane and spherical surfaces. Barrow developed a
mathematical theory of imagery by analyzing the intersections the refractions
of a bundle of rays.
For example, a point A is seen by
an eye off the axis AB, with COD being
the pupil of the eye. 104 (Figure 20). The
pupil is perpendicular to the refracted
ray NO, which passes through the
center of the pupil. The extension
KNO is called the principal ray. Now,
draw the refracted rays MC and RD
that pass through the edge of the
pupil. Produced backwards, MC and
RD will not intersect the principal ray
NKO in one point, but in points X and
V. Barrow demonstrated that point Z
on the principal ray is the limit of
these intersections. According to his
definition of the image, Z is the place
of the image. Consequently, the
cathetus rule does not apply here, as it
Figure 20 Barrow’s analysis of image
formation in refraction.
would have point K as the image. 105 Barrow applied this determination of the
image point to various problems in refraction. In the case of spherical
surfaces he derived expressions for the place of the image point for the eye
being both on and off the axis of the surface.
Barrow defined the image point in a similar way as Huygens defined the
‘punctum concursus’ and applied it with comparable rigor to study the
refracting properties of spherical surfaces. Many of their results were
equivalent. Yet, they had different goals. Huygens intended to explain the
dioptrical properties of the telescope and therefore confined himself to
paraxial rays, not discussing optical imagery. He ignored mathematically
sophisticated problems that had no relevance to the telescope, like the focus
of an oblique cone of rays. Barrow’s aim was to develop a general theory of
optical imagery. He had no intention of explaining the telescope and many of
the problems he treated had no relevance to it. 106 Still, in lecture XIV, he also
discussed spherical lenses. He gave, without proof, a series of equations for
the focal and image points of all kinds of lenses by way of an example of the
104 Barrow, Lectiones, [82-83].
105 Compare Shapiro, “The Optical Lectures”, 130 & 133-134.
106 Compare Shapiro, “The Optical Lectures”, 150-151; and Malet, “Isaac Barrow”, 286.

40 CHAPTER 2
application of his preceding discussion of single spherical surfaces. It was an
exact solution to the problem of Dioptrice, yet a complex and cumbersome
one. 107 Barrow had chosen the example “… with a view to common use, and
particularly aimed at reducing the labour of anyone who comes across
them.” 108 Barrow’s exact theory of focal distances was the first in print, but it
was no more than a theory of focal distances. He did not discuss
magnification and configurations of lenses.
Barrow’s footsteps were followed by Newton in his lectures. Their
central subject was his mathematical theory of colors. In a section “On the
Refractions of Curved Surfaces” he also treated some topics regarding
monochromatic rays. Newton extended Barrow’s theory of image formation
to three-dimensional bundles of rays. He demonstrated the existence of a
second image point at the intersection of the axis and the principal ray.
Newton had developed his own solution of the anaclastic problem –
although in the lectures he abandoned it in favor of Barrow’s – and found a
new way to derive Descartes’ ovals. The final goal of the lectures were,
however, colors. So, when Newton, in his 31 st proposition, determined the
spherical aberration of a ray, he did so to compare it to chromatic aberration.
The latter was larger and “Consequently, the heterogeneity of light and not
the unsuitability of a spherical shape is the reason why we have not yet
advanced telescopes to a greater degree of perfection.” 109 In this way,
Newton dismissed Descartes proposal as a dead-end. This was an important
result for the understanding of the working of the telescope. In order to
overcome the disturbing effects of aberration, Newton proposed to use
mirrors instead of lenses. Newton’s theory of colors and his reflector are
further discussed in section 3.2.3.
Dioptrics as mathematics
The discovery of an exact law of refraction had supplied geometrical optics
with a foundation for the mathematical study of the behavior of refracted
rays. This study consisted of deducing theorems from the postulates and
definitions of dioptrics in a rigorous way aimed at generality. With Kepler’s
new theory of image formation, a range of new problems were raised relating
to the perfect and imperfect focusing of rays. To the ones already mentioned
was added, at the end of the century, that of caustics; the locus of
intersections of rays refracted by a curved surface. 110 These problems were
markedly theoretical, mathematical puzzles tackled without practical
objectives. Halley, for example, in a paper of 1693 solved “the problem of
finding the foci of optick glasses universally” by means of a single algebraic
107
Shapiro, “The Optical Lectures”, 149-150.
108
Barrow, Lectiones, [168].
109
Newton, Optical Papers 1, 427.
110
In a series of papers of the 1680s and 1690s Tschirnhaus, Jakob and Johann Bernoulli and Hermann
attacked the problem. They were preceded by Huygens in 1677, but he did not publish his account until
1690, see section 5.1. Jakob Bernoulli published a general solution in Acta Eruditorum in 1693. In his
Analyse des infiniments petits (1696), L’Hopital gave a definitive solution on basis of the differential calculus.

1653 - TRACTATUS 41
equation. 111 Despite his involvement in practical matters of telescopes Halley,
like Barrow, did not further apply his finding to the effect of lenses. His
principal goal seems to have been to supplement Molyneux’ theory of focal
distances by means of giving “An instance of the excellence of modern
algebra, …” 112 All in all, the telescope rarely directed the dioptrical studies
undertaken by mathematicians.
Kepler is rightly regarded as the founder of seventeenth-century
geometrical optics, yet it was Paralipomena rather than Dioptrice that
constituted the starting-point for later studies. Similarly, Descartes’ La
Géométrie was the starting-point for later studies of aplanatic surfaces rather
than La Dioptrique. I find it remarkable that an instrument that had
revolutionized astronomy was ignored by students of geometrical optics in
the same way as spectacles had been previously. Kepler alone had, right
upon its invention, insisted that a mathematical understanding of the
telescope was needed for its use in observation, and Huygens was the only
one to take the instruction to heart. His approach was that of a
mathematician, yet he applied his mathematical abilities to a practical
question: understanding the working of the telescope. In Tractatus, he used
the sine law to derive an exact and general theory of the properties of
spherical lenses and their configurations. It remains to be seen, however,
whether such a mathematical theory of the telescope was really of any use.
Tractatus remained unpublished, those interested had to do with Dioptrice.
2.2.3 THE NEED FOR THEORY
Dioptrice had arisen from Kepler’s conviction that, in order to make reliable
observations, and astronomical instrument should be understood precisely.
The mathematicians I have discussed in the preceding section did not follow
his lead. Even Descartes and Newton, who proposed innovations in
telescope design, did not bother to elaborate theories of the way telescopes
produce sharp, magnified images. Maybe this was so because they, like the
others mathematicians that have been discussed, did were not much involved
in telescopic observation. Could the case be different for the mathematicians
who were, the observers? We have seen that Galileo, the most renowned
telescopist, was not really interested in mathematical questions of dioptrics.
He applied himself rather to practical matters of the manufacture and
improvement of the telescope. It does not seem that Galileo had to invoke
dioptrical arguments to defend the reality of telescopic observations, at least
not arguments from the mathematical tradition of perspective and Kepler. 113
To be sure, as a pioneer in astronomical telescopy Galileo was confronted
with suspicions about the reality of heavenly things seen through the tube,
but these soon wore off. Likewise, telescopists like Scheiner and Hevelius in
111
Halley, “Instance”, 960.
112
See: Albury, “Halley, Huygens, and Newton”, 455-457.
113
Galileo, Sidereus nuncius, 112-113 and 92-93 (Van Helden’s conclusion). See Dupré, Galileo and the
telescope, 175-178.

42 CHAPTER 2
their books on telescopic observation contented themselves with a cursory,
qualitative account of the telescope, drawing on Kepler’s lessons. Dioptrice
was the standard theory referred to well until the close of the century, but
mostly as regards the basic theorems on focal lengths and configurations.
Apparently this sufficed the needs of practical dioptrics leaving the
mathematical details superfluous. Had Kepler made things more difficult
than they really were?
This theme may be illustrated with the example of Isaac Beeckman, a
savant who combined an interest in practical affairs with a theoretical
outlook. He was interested in many things, including optics in all its
manifestations, and kept an elaborate diary of his ideas and observations. It
enables us to get an idea what a knowledgeable man would do with the
mathematics of dioptrics. The diary contains numerous notes on visual
observation that show that he read the literature – Aguilón, Kepler –
attentively. In addition, he was familiar with Descartes’ optical ideas and their
development, being in close contact with him on and off since 1618. 114 In the
1620s, Beeckman became interested in telescopes and he acquired some
lenses and instruments and later, in the 1630s, he put much effort in grinding
lenses and building telescopes. 115 Working on them, he encountered the
problem of spherical aberration (and later chromatic aberration) for which
he considered several remedies. The notes concerned are interesting for they
show a basic understanding of the working of lenses – as he would have
acquired from Paralipomena and Dioptrice – but the actual problem, that the
aberration is inherent to the spherical shape of a lens, seems to have eluded
him. Besides the common use of diaphragms to decrease the disturbance,
Beeckman thought up some sagacious ideas like combining lenses on a
spherical surface in order to emulate one large lens or a lens built up in thin
rings like a Fresnel lens. 116 The first idea he tested, just to discover soon that
it did not work and that he had overlooked a basic property of lenses. 117 He
was enough of an experimentalist not to trust ideas blindly. When Descartes
informed him in 1629 of his project of a hyperbolic lens, Beeckman reacted
skeptical. 118
The micrometer and telescopic sights
The principal reason why astronomers did not show much interest in
dioptrics lies, I think, in the fact that the telescope was a qualitative
instrument during the first decades after its introduction. It had revealed
new, spectacular phenomena in the sky, but it had not been deployed in the
114
Schuster, “Descartes opticien” and Van Berkel, “Descartes’ debt”.
115
Beeckman, Journal, II, 209-211; 294-296. For lens grinding see down, page 57.
116
For the second idea see Beeckman, Journal, II, 367-368. For a later consideration see for example: III,
296.
117
Beeckman, Journal, II, 296; 357.
118
Beeckman, Journal, III, 109-110.

1653 - TRACTATUS 43
exact description of the universe. 119 After all, the telescope was an artful
means to reveal new things in the heavens, whereas astronomical
measurement instruments aided the naked eye. 120 The step to combine these
two by aiding the artificial eyes with quadrants and the like, was not taken
immediately. Around the middle of the century, astronomical measurements
were still made by using the pre-telescopic methods and instruments
developed by Tycho Brahe. Hevelius used telescopes extensively to study the
surface of the Moon, but he turned to open-sight instruments when making
measurements. Efforts had been made to use the telescope for
measurements, but in vain. Until the 1670s, the accuracy of telescopic
observations was determined by the acuity of the human eye. But then
change set in. With the introduction of the micrometer, the telescope was
transformed into an instrument of precision. Significantly, the men closely
involved in that development were the ones to seek a more precise account
of the working of the telescope.
The configuration Kepler had thought up in 1611 had the drawback that
it reversed the image. Given the quality of lenses made at that time, it was
not advisable to add a third lens to re-erect the image. The two-lens
Keplerian telescope was therefore used only to project images, whereas the
Galilean type continued to be used for direct observation. In the course of
time the first advantage of Kepler’s configuration was discovered: its wider
field of view. When the length of a Galilean telescope is increased the field
of view quickly diminishes, which makes it very difficult to use. Towards the
1640s, the Keplerian telescope was gaining ground, in particular through the
good craftsmanship of telescope makers like Fontana in Naples and Wiesel
in Augsburg. 121 At some point in the early 1640s, the second advantage of
this type was discovered by the Lancashire astronomer William Gascoigne.
The Keplerian configuration has a positive focus inside the telescope; an
object inserted into it will cast a sharp shadow over the object seen through
the tube. Gascoigne relates that he discovered this by accident after a spider
had spun its web in his telescope. 122 Inserting some kind of ruler makes it
possible to make measurements of telescopic images. He died in 1644,
before he could publish his discovery and his measurements of the diameters
of planets. 123
Gascoigne’s accomplishments were made public in 1667 when Richard
Towneley, backed by Christopher Wren and Robert Hooke, claimed British
priority for the invention of the micrometer. This happened after a letter of
119 Van Helden, Measure, 118-119.
120 Compare Dear, Discipline and Experience, 210-216.
121 Van Helden, “Astronomical telescope”, 26-32. See also below section 3.1.1.
122 Rigaud, Correspondence, 46: “This is that admirable secret, which, as all other things, appeared when it
pleased the All Disposer, at whose direction a spider’s line drawn in an opened case could first give me by
its perfect apparition, when I was with two convexes trying experiments about the sun, the unexpected
knowledge.”
123 McKeon, “Les débuts I”, 258-266.

44 CHAPTER 2
Adrien Auzout was published in Philosophical Transactions, in which he
described a method of determining the diameters of planets. 124 He had
devised – possibly with the help of Pierre Petit – and used – together with
Jean Picard – a grate of thin wires and a moveable reference frame inserted
in the focal plane of a telescope. In two letters, also published in Philosophical
Transactions, Towneley argued that Gascoigne had made and used a
micrometer much earlier. He described a pair of moveable fillets that could
be inserted into the focal plane. 125 He himself had used and improved the
device – probably since late 1665 – to make accurate observations. 126
The principle of the micrometer, however, had already been published in
1659; by Huygens in Systema Saturnium, his astronomical work in which he
presented his discoveries regarding the ring and the satellites of Saturn. In its
final section he explained the principle and described how to use it to make
measurements. He inserted a ring in the focal plane and then measured the
angular magnitude of the opening thus produced by timing the passage of a
star. Next, he inserted a cuneiform strip through a hole in the tube until it
just covered a planet. The angular diameter of the planet was determined by
taking out the strip and comparing its width at the point found with the
inner diameter of the ring. 127 It was not a real micrometer, but Huygens’
rather cumbersome method did produce reliable, accurate data. 128 It was a
convenient method for measuring the size of planets, Huygens said, as one
did not have to wait for a conjunction of the planet with the Moon or a
star. 129 Huygens had been acquainted with Auzout and Petit since 1660 and
had come to Paris in 1666 to give leadership to the Académie. His
explanation of the principle of the micrometer certainly inspired their work
on the micrometer, but the precise nature of Huygens’ contribution is hard
to determine. 130
The principle of the micrometer had another important application:
telescopic sights. By inserting crosshairs in the focal plane, a telescope could
reliably be aligned on a measuring arc. 131 With the telescopic sight the
accuracy of Brahe’s measurements could finally be improved. Several
programs of astronomical measurement now set off. In Paris, Picard and
other members of the Académie – completed in 1669 by Cassini – put into
use a new, well-equipped observatory. 132 Picard’s work on cartographic
124
OldCorr3, 293: “… prendre les diametres du soleil, de la lune et des planetes par une methode que nous
avons, Monsieur Picard et moy, que ie croy la meilleure de toutes celles que l’on a pratiquer Jusques a
present, ...”
125
McKeon, “Les débuts I”, 266-269.
126
McKeon, “Les débuts I”, 286. In Micrographia (1665) Hooke had suggested that a scale may be inserted
into the focal plane of telescopes. Hooke, Micrographia, 237.
127
OC21, 348-351.
128
Van Helden, Measure, 120-121.
129
OC21, 352-353.
130
McKeon, “Les débuts I”, 286; Van Helden, Measure, 118.
131
McKeon, “Renouvellement”, 122.
132
McKeon, “Renouvellement”, 126.

1653 - TRACTATUS 45
measurements resulted in the determination of the arc of the meridian,
published in Mesure de la Terre (1671). In London, Hooke and Wren devoted
themselves to carrying out the idea of telescopic sights. In 1669, Hooke
announced that he had established the motion of the earth by means of a
mural quadrant thus equipped. His claim met with great skepticism. In 1675,
Flamsteed was appointed Astronomer Royal at the London counterpart to
the Paris observatory. At the Royal Observatory, he erected a wealth of
precision instruments and set up a program of astronomical measurements,
eventually resulting in Historia coelestis brittanica (1725).
The usefulness of telescopic sights was not, however, beyond all doubt.
Hevelius, the most renowned astronomer in those days, was suspicious. He
believed that telescopic sights were unreliable and therefore preferred naked
eye views. 133 In 1672, a letter by Flamsteed was published in Philosophical
Transactions in which he defended the use of telescopes for astronomical
measurements. 134 He praised Hevelius for having improved Brahe’s
astronomical data, but doubted whether any further progress could be
possible as long as the latter refrained from using ‘glasses’. Hevelius took
offense at Flamsteed’s allegations, and responded in Machina coelestis pars prior
(1673) and in a letter that was published in part in Philosophical Transactions of
April 1674:
“For it is not only a question of seeing the stars somewhat more distinctly (…) but
whether the instruments point correctly in every direction, whether the telescopic sights
of the instrument can be accurately directed many times to any observations, and can
be reliably maintained; but I very much doubt whether this can be done with equal
precision every time.” 135
The argument went a bit out of hand when, later in 1674, Hooke interfered
with a vehement attack on Hevelius in Animadversions on the first part of the
machina coelestis. Deeply hurt, Hevelius sent Flamsteed a letter in which he
once more explained his doubts about the reliability of telescopic sights. The
dispute was settled only five years later after a visit to Gdansk by Halley. He
reported that Hevelius’ naked eye observations were indeed incredibly
accurate.
Hevelius had fought a lost battle – so it can be said with hindsight – but
he was right in his suspicions about the reliability of telescopic sights. He
knew from experience how difficult it is to align instruments reliably. Already
in 1668 – right after the announcement of the micrometer – he had written
to Oldenburg: “For many things seem most certain in theory, which in
practice often fall far enough from truth.” 136 He was astonished that Hooke
would claim great accuracy for his measurements on the basis of just single
observations. Hevelius knew that accuracy was gained by hard and systematic
work. Picard, Cassini and Flamsteed undertook such an arduous task, but
133
Flamsteed, Gresham lectures, 34-39 (Forbes’s introduction).
134
OldCorr9, 326-327.
135
OldCorr10, 520.
136
OldCorr4, 448.

46 CHAPTER 2
were convinced that the new optical devices were useful. The telescopic sight
and the micrometer, together with the pendulum clock, brought about a
revolution in positional astronomy between 1665 and 1680. 137 In dioptrics it
raised the question of the exact properties of lenses anew.
Understanding the telescope
Apart from the practical problems of mounting and aligning, the theoretical
problem of the working of the telescope now became a matter of sustained
interest. As a result of his discussion with Hevelius over the reliability of
telescopic sights, Flamsteed realized that a theoretical justification of his
claims was also needed: “… to prove that optick glasses did not impose
upon or senses. then to shew that they might be applyed to instruments &
rectified as well as plaine sights.” 138 His chance to elaborate a dioptrical
account of the telescope came in the early 1680s, when, appointed Gresham
professor of Astronomy, he could deliver a series of lectures on astronomy.
In these lectures, he discussed instruments and their use at length and
included an account of dioptrics.
“Yet such has beene the fault of or time that hitherto very little materiall on this subject
has been published in or language. [Tho severall learned persons have done well
concerning opticks in ye latine Tongue. Yet how glasses may be applyed to instruments
& how the faults commonly committed in theire applycation might be amended or
rather shund & how all the difficultys suggested by ingenious persons who had not the
good to understand them aright might be avoyded the best authors of Dioptricks have
been hitherto silent. … I shall therefore make it my businesse in this & my following
lectures of this terme fully to explain the Nature of telescopes the reason of their
performances, how they may be applyed to Levells, Quadrants, & Sextants. & how the
instruments furnished with them may be so rectified & adjusted that they may be free
from all suspicion of errors]” 139
Flamsteed began with a discussion of the focal distances of convex lenses. It
has two notable features. First, he took the consequences of Newton’s
theory of colors into account by pointing out the chromatic aberration of
lenses. Second, the paucity of his demonstrations shows that he was not an
outstanding geometer. 140 By means of the sine law, he calculated the
refraction of single rays numerically and then compared the result with the
Keplerian rules for focal distances of a bundle of rays. By calculating
spherical aberration he discovered – as Huygens had done earlier – that the
aberration of a plano-convex lens varies considerably depending on which
side is turned towards the incident rays. He gave only a qualitative account of
chromatic abberation. On this basis he argued that only telescopes consisting
of two convex lenses are useful in astronomy, because these admit the
137
Van Helden, “Huygens and the astronomers”, 156-157; Van Helden, Measure, 127-129.
138
Flamsteed, Gresham lectures, 154.
139
Flamsteed, Gresham lectures, 119 & 132. Flamsteed later deleted the part between brackets.
140 Flamsteed, Gresham lectures, 120-127.

1653 - TRACTATUS 47
insertion of a micrometer or crosshairs. 141 He then went on to explain in
detail how to mount a telescope on quadrants and other things. 142
Flamsteed did not achieve the exactness and rigor of Huygens. His
analysis of the properties of lenses consisted of numerical calculations rather
than of general theorems. His account was larded with solutions to practical
problems, and here indeed resided the eventual goal of giving a dioptrical
account of the properties and effects of telescopes. Given the scarcity of
suitable publications on these matters, Flamsteed did not have much to start
from. He confessed that he had not taken the time to peruse Kepler’s
Paralipomena and he claimed never to have read Dioptrice. 143 He based himself
instead on some letters in which Gascoigne discussed the foci of planoconvex
and plano-concave lenses. 144 He considered his own account of other
lenses and of telescopes “… but a superstructure on yt foundation”. 145 It
sufficed to free the telescope of the imputation that “… all observations
made with glasses [are] more doubtfull & uncerteine …” 146 Flamsteed’s
lectures attracted only a small audience and did not go to print until this
century. 147
Some of Flamsteed’s ideas were passed on by Molyneux. During the
1680s, the men had corresponded extensively on dioptrics, among other
things. In 1692 Molyneux published Dioptrica Nova, an elaborate dioptrical
account of the telescope. In its preface, he acknowledged his debt to
Flamsteed:
“… the Geometrical Method of calculating a Rays Progress, which in many particulars is so
amply delivered hereafter, is wholly new, and never before publish’d. And for the first
Intimation thereof, I must acknowledg my self obliged to my worthy Friend Mr.
Flamsteed Astron. Reg. who had it from some unpublished Papers of Mr. Gascoignes.” 148
Dioptrica Nova was a compilation of dioptrical works published during the
seventeenth century. 149 Molyneux’ own contribution consists of his particular
presentation of the material, arranging theoretical knowledge in such a way
that it was useful for understanding the working of telescopes. He gave his
own demonstrations of many of its theorems, but he did not aim at
mathematical rigor or completeness:
“… [the Reader] is not to expect Geometrical Strictness in several Particulars of this
Doctrine. … ; as being more desirous of shewing in gross the Properties of Glasses and
141
Flamsteed, Gresham lectures, 136.
142
Flamsteed, Gresham lectures, 140-143.
143
Flamsteed, Gresham lectures, 40; 146n2 (Forbes’ introduction).
144
Flamsteed, Gresham lectures, 8-9; 40 (Forbes’ introduction).
145
Flamsteed, Gresham lectures, 39 (Forbes’ introduction).
146
Flamsteed, Gresham lectures, 149.
147
Flamsteed, Gresham lectures, 4-5 (Forbes’ introduction).
148
Molyneux, Dioptrica nova, (Admonition to the reader).
149
Molyneux mentioned Kepler, Cavalieri, Hérigone, Dechales, Fabri, Gregory and Barrow.

48 CHAPTER 2
their Effects in Telescopes, than of affecting a Nicety, which would be of little Use in
Practice.” 150
The limitations of Molyneux’ mathematics are easily noted. In proposition
III, for example, he discussed refraction by a bi-convex lens of a ray parallel
to the axis. 151 Taking into account both the distance of the ray from the axis
and the thickness of the lens, he derived by means of the sine law the point
where the refracted ray intersects the axis. In generalizing this to the focal
distance of the lens, Molyneux was less exact:
“If by this Method we calculate the Progress of a Ray through a Double Convex-Glass
of equal Convexities; and the thickness of the Glass be little or nothing in comparison
of the Radius of the Convexity; and the Distance of the Point of Incidence from the
Axis be but small, we shall find the Point of Concourse to be distant from the Glass
about the Radius of the Convexity nearly.” 152
He then gave Kepler’s theorem and reproduced the latter’s proof. For a lens
with unequal curvatures, he stated that the refracted ray could also be
constructed exactly. He confined himself, however, to a “… Shorter Rule
laid down by most Optick Writers”, which is identical with Huygens’ rule for
a thin lens cited on page 19. 153 This pattern of exact constructions for single
rays and questionable generalizations to bundles of rays recurs throughout
Dioptrica Nova. His problem was that he somehow had to link Flamsteed’s
discussion of single rays with the Keplerian rules of focal distances found in
most published treatises. He was not able enough a mathematician to derive
general theorems on focal distances by means of the sine law.
The problem with Molyneux’ generalizations is that he thought that the
intersection of single refracted ray with the axis was an approximation of the
focal distance. He did not fully understand that the focus of a refracting
body is the (limit) point of the refracted rays of a pencil of rays. His
definition of ‘focus’ in terms of the intersection of a single ray with the axis
makes this clear. 154 Taken literally, this would mean that a spherical surface
has many foci for one point object. In a scholium following proposition III
of Dioptrica Nova, he discussed spherical aberration. He began by reproducing
Flamsteed’s calculations for single rays as well as his conclusions concerning
the use of lenses. He then defined the distance between the ‘focus’ and the
intersection of the refracted ray with the axis as the ‘depth of the focus’. 155
Again, he mixed up the refraction of a single ray with the focusing of a pencil
of rays.
It is not difficult to point out flaws in Molyneux’ demonstrations, but we
should bear in mind the practical aim of Dioptrica Nova. In his discussion of
150
Molyneux, Dioptrica nova, (Admonition to the reader).
151
Molyneux, Dioptrica nova, 19-23.
152
Molyneux, Dioptrica nova, 20.
153
Molyneux, Dioptrica nova, 22.
154
Molyneux, Dioptrica nova, 9.
155
Molyneux, Dioptrica nova, 24. From the preceding it will be clear, that following Molyneux's line of
thought this distance should be zero, for both points are by definition the same.

1653 - TRACTATUS 49
images of extended objects, Molyneux displayed a better understanding of
the focusing of rays. In a section on “… the Representation of outward
Objects in a Dark Chamber; a Convex-Glass”, he described how the image is
formed by the focusing of pencils of rays originating in the points of an
object. 156 He then remarked that “… tho all the Rays from each point are not
united in an answerable Point in the Image, yet there are a sufficient quantity
of them to render the Representation very perfect.” 157 Rather than
mathematically precise, this was a practical definition of focus. It explained
why in practice images may appear sharp. Besides all the objections that can
be raised against Molyneux’ theory from a mathematical point of view we
should bear in mind that Dioptrica Nova was the first published dioptrical
account of telescopes, since Dioptrice. It was up-to-date with developments in
telescope making and was intended to be useful for practice.
Before coming to a conclusion of this chapter, we go back in time and
cross back over the Channel. Flamsteed’s ally in the debate over telescopic
sights, Picard, also saw the importance of theory. In a letter to Hevelius, he
had briefly explained the working of the telescopic sight in dioptrical terms. 158
Somewhat earlier – probably in 1668 – he had pointed out the need for such
an analysis: “[optical devices] can also be subject to certain refractions that
should be known well.” 159 In Mesure de la Terre, he had briefly discussed
matters of aligning and rectifying telescopic sights in these terms. Picard was
known for his interest and ability in matters dioptrical. At the Académie, he
frequently discoursed of dioptrical theory. 160 In this, the telescope stood
central:
“What we have just explained about the construction of telescopes, concerns only its
use in instruments made for observation, …” 161
Picard never published his dioptrics, but a collections of papers he had read
at the Académie was published posthumously in 1693 under the title
‘Fragmens de Dioptrique’. 162 Picard had a major advantage over Flamsteed.
He was acquainted with one of the most knowledgeable men in dioptrics:
Huygens. Besides his learning, in 1666 Huygens had brought a copy of the
manuscript of Tractatus to Paris. 163 At the Académie, Huygens had also
discoursed on dioptrics. “Fragmens de Dioptrique” make it clear that Picard
must have been among Huygens’ most attentive listeners. They are for the
156 Molyneux, Dioptrica nova, 36-38.
157 Molyneux, Dioptrica nova, 38.
158 Picolet, “Correspondence”, 38-39.
159 “… peuuent aussi estre sujets a certaines refractions qu’il faut bien connoistre.” Quoted in McKeon,
“Renouvellement”, 126-128. It is found in: A. Ac. Sc., Registres, t. 3, fol 156 r o - 164 v o spéc. 157 v o.
160 Blay, ”Travaux de Picard”, 329-332. Blay cites several references.
161 Blay, “Travaux de Picard” 343. “Ce que nous venons d’expliquer touchant la construction des lunettes
d’approche, n’est que par rapport à l’usage que l’on en fait dans les instruments qui servent à l’observer,
…”
162 Divers Ouvrages de Mathematique et de Physique, par Messieurs de l’Academie Royale des Sciences (1693), 375-412.
163 OC13, “Avertissement”, 7.

50 CHAPTER 2
most part derived from Huygens’ dioptrical theories, and I will not discuss
them in further detail. 164
Huygens’ position
Picard’s dioptrical fragments bring us back to Huygens. What had he been
doing in the meantime? In Systema saturnium he had alluded to an elaborate
theory of dioptrics, which we know he possessed indeed. Yet, despite
ongoing requests to publish it, he had kept it to himself. It may be clear by
now that Tractatus is a unique work in the development of seventeenthcentury
dioptrics. Huygens was the first and only man to follow the lead of
Dioptrice. Like Kepler, he combined the two things necessary to develop a
theory of the telescope: mathematical proficiency and an orientation on the
instrument. Unlike Kepler, he had the exact law of refraction and thus he
could rigorously develop an exact theory of the telescope.
But did Huygens really follow Kepler? Did he want to understand the
telescope in view of its use in astronomy? Tractatus came into being well
before Huygens commenced his practical activities of telescope making and
astronomical observation (discussed in the next chapter). Unlike Flamsteed
and Picard, he did not seek answers to questions that had arisen in practice.
Nevertheless, his orientation on the telescope is clear. He passed by all those
sophisticated problems not relevant to the understanding of the telescope
that preoccupied mathematicians like Barrow. However, nowhere does
Huygens mention Kepler as an example. It looks as if developing a theory of
the telescope on the basis of the sine law was to him an interesting
mathematical puzzle, maybe just to correct Descartes’ useless approach to
dioptrics. The problem had not yet been solved and Huygens only too gladly
seized the opportunity. Which makes his exclusive orientation on the
instrument all the more interesting.
The transformation of the telescope into an instrument of precision
brought back an interest in the dioptrical properties of the telescope. In this
regard, one might say that Kepler had prematurely raised the question after a
mathematical understanding of the telescope. In 1611, it was a qualitative
instrument and remained so for another half century. To understand its
working, a qualitative account of the effects of lenses therefore sufficed.
Similarly, we can ask whether an exact theory like Huygens’ was really
needed. It seems that Kepler’s or Keplerian theories satisfied the needs of
men like Flamsteed and Molyneux pretty well. They lacked sufficient
proficiency in mathematics to treat lenses in exact terms, but they may also
have been perfectly satisfied with their approximate results.
Huygens himself did not put much work in applying his theory to the
questions that occupied Picard and Flamsteed. The principle of the
micrometer may or may not have been the result of his theoretical
understanding, in Systema saturnium he explained it only briefly. Huygens did
164 Blay, “Travaux de Picard”, 340.

1653 - TRACTATUS 51
expect that theory could be useful. The discovery that a sphere is an
aplanatic surface in some cases had given the initial impulse to his interest in
dioptrics. In his letter to Van Schooten, he expressed the expectation that
this theoretical insight would contribute to the improvement of the
telescope. From the very start, Huygens saw a connection between the
theory of dioptrics and the practice of telescope making. In the next chapter
we shall see what he would make of it.

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Chapter 3
1655-1672 - 'De Aberratione'
Huygens' practical optics and the aspirations of dioptrical theory
In the decade following Tractatus, Huygens was at home were his
mathematical virtuosity grew to full stature. These are the years of his most
renowned achievements: the invention – in 1656 – improvement and
employment of the pendulum clock and the theory of pendulum that were
the basis of his master piece Horologium Oscillatorium (1673); the discovery in
1655 of a satellite of Saturn and the identification of its ring. Through his
correspondence and publications Huygens increasingly gained recognition
among Europe’s scholars. He traveled abroad, first to Paris in 1655 to meet
the leading French mathematicians, then to Paris and London in 1660-1, and
again in 1663-4, the last time being elected fellow of the Royal Society. There
were squabbles as well, in Italy in particular, over the priority of the
pendulum clock with Florentine sympathizers of the late Galileo and with
the Roman telescope maker Divini over the superiority of his telescopes.
Probably as a result of the clock dispute, he did not obtain a position at the
court of prince Leopold, but in 1666 Huygens realized his learned assets. At
the instigation of Colbert he came to Paris to help organize an ‘académie des
sciences’, thus confirming his status as Europe’s leading mathematician. Life
in Paris, with it competitive milieu, was no unqualified pleasure. Huygens
correspondence shows symptoms of homesickness, he particularly missed
his brother Constantijn, and in 1670 he was was smitten with ‘melancholie’
for the first time. In these years he also experienced the first major setback in
his science: a design for a perfect telescope proved useless. The design was
the outcome of Huygens’ practical activities in telescopy of the late 1650s
and his subsequent theoretical reflections thereupon of the 1660s. These are
the subject of this chapter.
When Huygens’ interest in dioptrics was sparked late 1652, it was both its
theoretical and practical aspects. He immediately began inquiring about the
art of lens making, but he engaged in practical dioptrics only after he put
aside the manuscript of Tractatus. Around 1655, he and his brother
Constantijn acquired the art of lens making and started building telescopes. 1
The practice bore fruit almost immediately. In 1656, Christiaan published a
pamphlet De saturni luna observatio nova on the discovery of a satellite around
Saturn. It was the first new celestial body in the solar system to be
1 Editor’s comment, OC15, 10. See also Anne van Helden, “Lens production”, 70.

54 CHAPTER 3
discovered since Galileo. 2 The tract ended with an anagram holding
Huygens’ second discovery: the true nature of the inexplicable appearance of
Saturn. Three years later, he elaborated his explanation in Systema saturnium.
The strange attachments to the planet that disappeared from time to time
were manifestations of a solid ring around the planet. 3
He owed much to his instruments, Huygens wrote in Systema saturnium.
He prided himself on his practical skills of telescope making and claimed that
his success proved the unmatched quality of his telescopes. Van Helden
explains that his discovery owed at least as much to his talents for
geometrical and physical reasoning. 4 Initially, Huygens had used a 12-foot
telescope of their own make. After a trip to Paris, where he probably
discussed his observations, the brothers built a new, 23-foot telescope which
he started using in February 1656. 5 He illustrated the difference between
both pieces in Systema saturnium (Figure 21). Everyone could see for himself
that Huygens could hold his own with the best of telescope makers. At least,
that is how he saw it himself. His boasting offended Eustachio Divini in
Rome, who saw his fame of being the best telescope maker in Europe
challenged. In 1660 he published Brevis annotatio in systema saturnium, disputing
the observational results as well as Huygens’ claims regarding his
instruments. 6 The tract was actually written by the Roman astronomer Fabri.
In the ensuing dispute Divini/Fabri were no match for Huygens, at least not
as regards the structure of the system of Saturn. 7
Figure 21 Observations of Saturn with the 12- and a 23-foot telescope.
The dispute itself is less interesting than the fact that Huygens did not feel
above at entering a dispute with a craftsman. It raises questions about the
relationship between his theoretical and his practical pursuits, how he valued
his mathematical expertise and his skilful craftsmanship. The last part of this
2
Huygens did not name it, he called it ‘saturni luna’ and sometimes ‘comes meus’. The name Titan was
given by Herschel in 1847.
3
OC15, 296-299.
4
Van Helden, “Huygens and the astronomers”, 150-154. Van Helden, “Divini vs Huygens”, 48-50.
5
OC15, 177; 230. Huygens employed Rhineland feet (0,3139 meters) and inches (0,026 meters).
6
It is reprinted in OC15, 403-437.
7
Van Helden, “Divini vs Huygens”, 36-40.

1655-1672 - DE ABERRATIONE 55
chapter examines these themes in a broader context of the scientific
revolution, and forms a conclusion of this account of Huygens’ dioptrics
prior to the metamorphosis of his optics discussed in the subsequent
chapters. So much can be said that Huygens’ passion was with the
instrument, not its employment. For Huygens telescopic astronomy was a
pastime rather than a full-time job. Although he had solved the puzzle of
Saturn’s bulges by systematic observation, this never became his vocation.
His fascination was with its design and manufacture of telescopes. 8 To this
we may also count his interest in dioptrical theory, being a means of
tinkering with the instrument and contemplating its workings.
In the ten or so years after 1653 when the brothers engaged in practical
pursuits, Huygens did not work on dioptrical theory (at least no traces ar
left). During the 1660s he returned to theory and set out for what should
have been the crowning glory of his dioptrical work: the design of a
telescope in which spherical aberration was nullified. Not by means of
imaginary lenses of the kind Descartes had thought up, but by means of
actual spherical lenses. In the design came together Huygens’ theoretical
understanding and practical experience with lenses and it brought him closer
to bridging the gap between theory and practice than any other in the
seventeenth century. Newton’s ‘New Theory’ of colors eventually
shipwrecked the project. Newton’s approach of mathematical optics
essentially differed from Huygens’. These differences shed light on the
character of the Huygens’ dioptrics and may explain why Huygens did not
manage to bridge the said gap completely.
3.1 The use of theory
Around 1600, spectacle makers had advanced their art far enough to enable
the discovery of the telescopic effect. 9 Astronomers in their turn discovered
the possibilities of this chance invention. Their pursuit demanded far greater
power than the first spyglasses offered. They needed skillful hands:
sometimes their own, but usually those of a craftsman. Galileo, not
particularly all fingers and thumbs himself, had the advantage of living close
to Venice, the center of European glass industry. After the success of Sidereus
nuncius he established a workshop for telescopes. Simon Marius, in Germany,
was less lucky: he had great trouble finding a good lens maker and could not
put the new invention to fruitful use. 10 During the first half of the
seventeenth century, the manufacture of telescopes for astronomy developed
into a small trade of specialized craftsmen. This section will not treat the
history of lens and telescope manufacture, it focuses on the relationship
between dioptrical theory and the art of telescope making. Central questions
are: to what extent was theoretical knowledge used in practical dioptrics, if it
8
Van Helden, “Huygens and the astronomers” 148, 157-158.
9
Van Helden, Invention, 16-20.
10
Van Helden, Invention, 26; 47-48.

56 CHAPTER 3
was useful at all; did the scholarly world contribute to the art of telescope
making besides revenue, status, and stimuli for progress?
René Descartes definitely believed art could learn from philosophy, and
that it should. In La Dioptrique he intended to show the benefits of
philosophy. The telescope, he said, was a product of practical wit and skills,
but the explanation of its difficulties could bring it to a higher level of
perfection. 11 Just as the telescope surpassed the natural limitations of vision,
so the scholar could teach the craftsman how to overcome his limitations.
The sine law dictated that lenses should have a conical rather than a spherical
shape, as we have seen in the previous chapter. Descartes had also
considered the way his design could be put to practice. In the tenth and final
discourse of La Dioptrique he described the way his lenses could be made. His
account included an ingenious lathe to grind hyperbolic lenses. It reflected
his efforts, during the late 1620s, to make a hyperbolic lens. Descartes was
never lavish to point out his debt to others – to put it mildly – so he did not
tell his readers that he owed much to his cooperation with the lens maker
Jean Ferrier and the mathematician Claude Mydorge. 12 Allegedly, the
threesome succeeded in grinding a convex hyperbolic lens. “And it turned
out perfectly well …”, Descartes wrote in 1635 to Huygens’ father
Constantijn. 13 It had been made by hand. The next step was to design a lathe.
Towards the end of 1628 Descartes left for Holland. In the fall of 1629, he
and Ferrier exchanged some letters in which an earlier design for a lathe was
mentioned. 14 They discussed a machine Descartes had contrived for making a
cutting blade with a hyperbolic edge. 15 Ferrier proposed several modifications
and improvements that turn up in La Dioptrique. 16
Throughout the seventeenth century, Descartes’ account gave rise to
efforts to make elliptic and hyperbolic lenses. Around 1635, Constantijn
Huygens arranged unsuccessful attempts to grind them. 17 In 1643, Rheita
claimed to have succeeded with tools he described in Novem stellae circum
Iovem. Wren described a device to make a hyperboloid surface in an article
published in Philosophical Transactions in 1669. During the 1670s, Huygens
corresponded with Smethwick over another design. 18 Much in these
suggestions never went beyond the paper stage. To execute them skills and
tools – as well as patience – were needed. To design a lens may have been a
scholarly challenge, actually to make it required craftsmanship. And then it
11
Descartes, Dioptrique, 2-3 (AT6, 82-83).
12
Shea, “Descartes and Ferrier”, 146-148.
13
AT1, 598-600. “Et il reussit parfaitement bien; …” It turned out that it was impossible to make a
concave lens in the same way.
14
AT1, lts 8, 11, 12,13,22,21,27. Shea, “Descartes and Ferrier”. The letters not only reveal Ferrier’s
mastery of the art but also his mathematical knowledge.
15
AT1, 33-35.
16
Descartes, Dioptrique, 141-150 (AT6, 215-224).
17
Ploeg, Constantijn Huygens, 34-38.
18
OC7, 111; 117; 487; 511-513. In 1654 Huygens described a mechanism to draw ellipses on the basis of a
circle, apparently aimed at making elliptic lenses out of spherical ones; OC17, 287-292.

1655-1672 - DE ABERRATIONE 57
remained to be seen whether it could be made to function properly. As for
Descartes’ ideal lenses, theory had not advanced practice yet.
3.1.1 HUYGENS AND THE ART OF TELESCOPE MAKING
In the middle the seventeenth century the art of lens making had progressed
enormously, allowing telescopes to be made with more than two lenses and
challenging the inventiveness of telescope makers. Skills, tools and materials
were the principal necessities for the manufacture of ordinary spherical
lenses. The manufacture was not fully under control: glass suffered from
various flaws, the produced faces of lenses were spherical at best, and so on.
Clever solutions were needed to obtain good telescopic images. The state of
the art of lensmaking can be more or less reconstructed from the quality of
surviving lenses, but this only provides indirect evidence of the art itself.
Lensmaking practice in the first decades after the invention of the telescope
is hardly documented. Some information can be distilled from Sirturus and
Scheiner, but their accounts are quite succinct and rather superficial.
A rare source of information is found in the diary of Isaac Beeckman,
which meticulously records his trials and errors with using and
manufacturing lenses. 19 His interest in telescopes was excited in the early
1620s, but not until 1632 did he embark on serious lensmaking himself. In
the meantime he recorded his dissatisfaction with the lenses he acquired with
established lensmakers. The number of notes on grinding, polishing and the
like, quickly grows in the early 1630s and in 1633 he acquired a grinding
mould and commenced manufacturing his own lenses. Beeckman visited
several artisans who taught him their art. The diary describes their techniques
in much detail, particularly noting the differences. 20 The attentive pupil was a
quick learner. In the autumn of 1635, Beeckman compared one of his lenses
with one from Johannes Sachariassen of Middelburg – one of his tutors and
son of one of the claimants of the invention of the telescope – and found
out it was better. 21 Beeckman’s journal was a hidden treasure. He showed it
only to Descartes, Mersenne and Hortensius and it remained unknown until
Cornelis de Waard discovered it in 1905.
An aspirant lens maker lacked published expositions to learn of the art.
Rheita in 1645 was somewhat more elaborate, but when Huygens took on to
lens making in the early 1650s, he had to consult, like Beeckman before him,
experts and acquire the art by trial and error. The questions he fired at
Gutschoven in 1652 display the diversity of the know-how involved in the
process of cutting, grinding, and polishing to make a good lens out of a lump
of glass. 22 How to make grinding moulds? How can a perfect spherical figure
19
Next to numerous short entries, the main body is collected under the heading “Notes sur le rodage et le
polisage des verres” in Beeckman, Journal, III, 371-431.
20
Beeckman, Journal, III, 69, 249, 308, 383.
21
Beeckman, Journal, III, 430.
22
OC1, 191. See also Anne van Helden, “Lens production”, 70-75.

58 CHAPTER 3
be created? What kind of sand is needed for grinding? Which glue is best to
attach the handle? Et cetera, et cetera.
Despite its orientation on the telescope, Huygens’ 1653 study of dioptrics
did not aim at its improvement. Apparently, he had not found much use in
the discovery of 1652. In the third part of Tractatus he had written down an
the idea to add a little mirror to a Keplerian telescope to re-erect the image
without the need to add an extra lens. 23 This was not a new idea and it did
not improve the defects of lenses directly. Having put aside the manuscript
on the mathematical properties of lenses, Huygens turned to their material
properties. In the practical work he pursued with his brother, he developed
an artisanal understanding of lenses. The question is what such an
understanding entailed and how it related to the theoretical understanding
Huygens had developed in Tractatus.
Huygens’ skills
Some notes survive, in which Huygens described the process of grinding
lenses. 24 In 1658 he recorded how he had made a “good” 4½-foot bi-convex
lens:
“Always kept it fairly wet to preserve the dust. But not too much water in the
beginning, or it will bump. Never forget to press evenly, and often lifted the hand and placed
it evenly again. It is best to be alone. … At first I ground the other side wrongly: the
reason for this was that I either took too much water in the beginning, or that I did not
polish on the right spot. I first corrected somewhat by polishing at the right spot again;
then with more polishing it got worse once again.” 25
Making lenses was first of all a matter of ‘Fingerspitzengefühl’ acquired
through much practice. Huygens and his brother did so and eventually
became pretty good at it. 26 One of the main problems of grinding and
polishing was to secure an optimal shape of a lens. Both surfaces should be
really spherical and the axes should coincide. As Huygens’ notes show this
entailed a good deal of accuracy and patience. Proper tools did not only ease
the laborious task but improve the control of the manufacture and thus the
quality of the lens. In his notes, Huygens described a device (Figure 22) that
relieved the hands and ensured a proper, even pressure on the glass. 27 A
similar device had been described by Beeckman, who added that it was a
23
OC1, 242. He distributed several telescopes of this design during the next decade. (OC1, 242; OC13,
264n3; OC4, 132-3; OC4, 224, 228-9)
24
OC17, 293-304.
25
OC17, 294. “altijdt redelijck nat gehouden om te beter de stof te bewaren. doch in ’t eerst niet al te veel
waters, want anders stoot het aen. altijdt dencken om gelijck te drucken, en dickwils de hand af gelicht en weer
gelijck aen geset. ’t is best alleen te sijn. … De andere sijde sleep ick eerst eens mis: daer de oorsaeck van
was, of dat ick in ’t eerst te veel water nam of dat ick niet op de goeije plaets en polijsten. ick
verbeterdense eerst wat met op de rechte plaets noch eens te polijsten; daer nae met noch meer polijsten
wierd het weer erger.”
26
The earliest lenses that remain – one in the Utrecht University Museum and two at Boerhaave Museum
in Leiden – are not very good. Their fame as lens makers stems from the 1680s. Anne van Helden, “Lens
production”, 75-78; Anne van Helden, Collection, IV; 22.
27
OC17, 299.

1655-1672 - DE ABERRATIONE 59
technique used by mirror-makers. 28 It is
not known where Huygens got the idea.
Such tools for improving the
grinding process had been thought up
earlier, in particular by the most
prestigious lens makers. In Galileo’s
workshop – reigning until the 1640s – a
lathe was introduced that permitted
greater precision than was attained by
ordinary spectacle makers. 29 During the
1660s, the Campani brothers in Rome
became the undisputed masters of the
art. They used a range of machines of
their own design, producing lenses
unsurpassed until the eighteenth
century. The Huygens brothers kept a
close eye on developments like these and around 1665 references to a type of
lathe designed by Campani turned up in their writings. The quality of lenses
seems to have depended to some degree on the lens maker’s inventiveness to
convert laborious and unsteady handiwork into reliable tools.
Huygens had learned how to make lenses. He knew the limitations of the
art and of its products. Even the best lenses might suffer from flaws like
bubbles, irregular density, faults in shape, etcetera. In the end, the proof of
the pudding was in the eating. The quality of telescopes was determined by
trial, sometimes literally. Campani beat Divini early 1664 by a series of
carefully organized ‘paragoni’: open contests in which printed sheets at a
distance were read by means of the instruments of both competitors. 30
Figure 22 Beam to facilitate lens grinding.
Alternative configurations
Besides the quality of lenses, telescopes could be improved by configuring
lenses alternatively. Kepler could not have known that the configuration of
two convex lenses he discussed in Dioptrice had several advantages over the
Galilean one. The positive focus that made possible the micrometer has
already been discussed in the previous chapter. Scheiner, who used a
Keplerian telescope to project images of the sun, discovered by coincidence
that it had a wider field of view. There are indications that Fontana was the
first to put Kepler’s idea into practice, although Scheiner was the first to
mention using it. 31 Around 1640 Fontana was the first to challenge Galileo’s
dominance in the trade and he did so with Keplerian telescopes. Around that
time, the Galilean configuration was beginning to reach the limits to which
its power could be increased without the field of view becoming too small.
28
Beeckman, Journal, III, 232.
29
Bedini, “Makers”, 108-110; Bedini, “Lens making”, 688-691.
30
Bonelli, “Divini and Campani”, 21-25.
31
Van Helden, “Astronomical telescope”, 20-25. Compare Malet, “Kepler and the telescope”, 120.

60 CHAPTER 3
The inversion of the picture a Keplerian telescope produced could be
overcome by adding a third lens. Given the quality of the earliest lenses it
was not advisable to ‘multiply’ glasses, but by the 1640s multi-lens telescopes
were beginning to become acceptable. With the increase of length and
magnification, however, the field of a Keplerian telescope also became
narrow. For example, the 23-foot telescope that Huygens used in his
observations of Saturn had a field of only 17'. It could not display the entire
Moon at once. The limited field of view could be overcome by adding a field
lens. The Augsburg telescope maker Johann Wiesel was probably the first to
make telescopes with such compound oculars. 32 In a letter of 17 December
1649, Wiesel described a four-lens telescope. It had an eyepiece consisting of
two plano-convex lenses fitted in a small tube. The eyepiece tube was
inserted in a composition of tubes which held an objective lens at the far side
and a plano-convex ocular which acted as a field lens. Wiesel added:
“Sir you may bee assured this is y. e first starrie tubus wch I have made of this manner &
so good yt it goes farre beyond all others wherof my selfe also doe not little rejoyce.” 33
The fame of Wiesel’s telescopes spread quickly and throughout Europe
telescope makers tried to equal them. On a visit to his relative Edelheer in
Antwerpen on New Year’s eve 1652, Huygens saw a Wiesel telescope and
was very impressed. It was a four-lens telescope, probably comparable to one
Wiesel described in 1649. Towards the end of 1654 Huygens acquired two
letters written by Wiesel - one to his cousin Vogelaer - describing the
construction and use of several optical instruments. 34 In the first letter a sixlens
telescope was described, which could be used for both terrestrial and
celestial purposes. Wiesel was an artisan, a very good one with an unmatched
understanding of lenses and their configurations. His was another kind of
understanding than the dioptrical theory Huygens developed in 1653. This
kind of experiential knowledge Huygens acquired the following years in his
practical dioptrics. Then, some ten years later, in an artisanal manner
Huygens made his own compound eyepiece with excellent dioptrical
properties.
Experiential knowledge
Telescope makers had a great deal of knowledge of dioptrics, as witnessed by
the fruits of their labor that are preserved. Like the process of production,
the thinking behind these products is more difficult to retrieve. It is barely
documented as craftsmen in general were reluctant to reveal the secrets of
their trade. There is reason to believe that their knowledge of dioptrics was
of a different kind than that of mathematicians. That much we can infer
from what little material that has been preserved. Lens makers knew very
32
Van Helden, “Compound”, 27-29; Keil, “Technology transfer”, 272-273. They are first mentioned in
Rheita’s Oculus Enoch et Eliae (1645), who referred his readers to Wiesel. For the relationship between
Rheita and Wiesel see Keill, Augustanus Opticus, 66-77.
33
Van Helden, “Compound eyepieces”, 34. The entire letter is reproduced on 34-35.
34
OC1, 308-311.

1655-1672 - DE ABERRATIONE 61
well how to grind lenses to suitable proportions and
configure appropriately. The nature of this knowledge
and to what extent they understood the dioptrical
properties of lenses will be discussed now. Note that
this is a discussion of very limited scope, determined
by the considerations and activities of Huygens, that
passes over the a wealth of historical knowledge that
can be, and has been, gathered regarding the telescope
making trade. 35
A booklet on spectacles written in 1623 gives an
indication of the understanding their manufacturers
had of glasses. Uso de los antojos was written by the
Andalusian licentiate Daza de Valdez. It explained
how to choose glasses of appropriate strength for a
patient. Daza described a procedure to determine the
‘grado’ of a given lens (Figure 23). 36 He drew two solid
circles X and Q of unequal diameter on a sheet of
paper as well as a specific scale at one of the circles. A
glass was then positioned on the scale in such a way
that both circles were seen equally large. The position
of the lens on the scale gave its ‘grado’. 37 Daza did not
explain the method, he only described how it was
employed. It was a practical procedure that did not require any
understanding of its effect on rays.
A manuscript written around 1670 in Rome by a certain Giovanni
Bolantio contains a similar kind of procedural knowledge. It discussed the
manufacture of telescopes and probably recorded the daily routine in some
workshop. 38 It contains two tables listing the ocular and objective lenses
needed to make a telescope of a specific strength, characterized by its length.
The lenses are characterized by the doubled radius of the pattern in which
they were ground. 39 Figure 23 Daza’s scale
With these tables at hand, the workman could choose the
patterns needed to make a telescope on order. Bolantio did not explain
whether he had constructed the tables himself nor how they were made.
Some dioptrical rules are implicit in them. For example, the length of a
35
See for example the recent, formidable study on Wiesel by Inge Keill which may serve as a guide to
themes and literature: Keil, Augustanus Opticus.
36
Daza, Uso de los antojos, 137-140. It appears that this classification in terms of ‘degrees’ was, at that time,
replacing an older one in terms of the common age of someone bearing spectacles of a particular
strength. The ‘grados’ Daza employs seem to be identical with the ‘punti’ Garzoni mentions in his
discussion of the craft in La piazza universale (1585). See also: Pflugk, “Beiträge”, 50-55.
37
Daza did not explain how the scale on the paper was established. Von Rohr has given an alluring
suggestion as to how such a scale might be construed. Spectacle makers knew that multiple lenses of a
given strength could be substituted by a stronger one to reach the same effect. Thus the first position on
the scale was determined by a weak lens and the other positions determined by the amount of equal lenses
which had to be put in those positions. Von Rohr, “Versuch”, 4.
38
Bedini & Bennet, “Treatise”.
39
Bedini & Bennet, “Treatise”, 120-121.

62 CHAPTER 3
Keplerian telescope is set equal to the doubled radius of its objective lens,
which - correctly - implies that the focal distance of the objective is twice the
radius of its convex side. Another table prescribed the size of the aperture
for a given objective, in each case 1
80
th of the tube’s length. Bolantio
explained that the objective should be partially covered by a ring so that no
light fell on the interior of the tube, which apparently implied the ratio used
in the table. 40 Whatever the dioptrical understanding implicit in Bolantio’s
manuscript, it was presented in a procedural, how-to style that did not
require further theoretical knowledge. 41
Some telescope makers published their own observations, to promote
their products. 42 They did not publish the secrets of their art, as their
revenues depended on them. Information on the manufacture of lenses and
telescopes could be found in books that were mostly written by scholars.
Examples are Telescopium (1618) by Girolamo Sirtori, Selenographia (1647) by
Hevelius and La Dioptrique oculaire (1672) by Cherubin d’Orleans. In 1685,
Huygens wrote a treatise on lens grinding in Dutch, Memorien aangaende het
slijpen van glasen tot verrekijckers, published posthumously in Latin in the
Opuscula posthuma (1703). 43 Memorien was the elaboration of notes like the one
cited above. Huygens described the process of lens making as a set of
directives, procedures, tips and tricks. No attempt is made to explain why
things work as they work: for example a geometrical account of the grinding
device is absent. Memorien supplied the kind of experiential knowledge also
found in Bolantio’s manuscript: a description of skills Huygens had acquired
through long-time practice.
To what extent a telescope maker like Campani understood the dioptrics
implicit in tables like those in Bolantio’s manuscript cannot be determined.
First rank, specialized telescope makers like Divini and Campani had
received some formal education, so they may have been able to read and
study a book like Dioptrice. It remains to be seen whether a question like this
is relevant at all. I doubt whether dioptrical knowledge would have been of
any use in the design and manufacture of telescopes. They knew very well
the effect of diverse types of lenses, but this probably was experiential
knowledge. Innovative craftsmen like Wiesel were able to find new
configurations with improved properties. These are likely to have been the
product of trial and error. It has been said that Kepler’s configuration was
the only contribution from the theory of dioptrics to the improvement of the
telescope. 44 Still, its advantages had to be discovered in practice. The
40
Bedini & Bennet, “Treatise”, 117.
41
Willach discusses dioptrical theory emerging from the correspondence of Rheita en Wiesel which
suggests similar lines. Willach, “Development of telescope optics”, 390-394.
42
For example: Fontana’s Novae coelestium (1646) and Campani, Lettere di Giuseppe Campani intorne all'ombre
delle Stelle Medicee (1665).
43
OC21, 252-290.
44
Van Helden, “The telescope in the 17th century”, 44-49.

1655-1672 - DE ABERRATIONE 63
improvement of the telescope was the result of the artisanal process of trialand-error.
Better configurations were designed by making them, not made by
designing them.
3.1.2 INVENTIONS ON TELESCOPES BY HUYGENS
After Tractatus followed a decade of practical dioptrics, that was crowned by
the publication of Systema Saturnium. Together with his brother, Huygens had
become a skilled telescope maker and could already pride himself on some
innovations of the instrument. In the previous chapter, one of these
innovations has been discussed: a device to make telescopic measurements.
It is not known how Huygens discovered the principle of the micrometer.
The discovery was probably related to an innovation of the telescope he had
developed somewhat earlier: the diaphragm.
The diaphragm improved the way images were enhanced by blocking part
of the light entering the telescope. Early in 1610 Galileo discovered that
telescopic images became more distinct when he covered the objective lens
with a paper ring. 45 He determined the optimal size and shape of the ring by
means of trial-and-effort and did not try – at least not on paper – to explain
the effect dioptrically. As contrasted to such an aperture stop, a diaphragm is
inserted into the focal plane. It has the advantage of diminishing the effect
we call chromatic aberration. In December 1659 Huygens first employed a
diaphragm in his 23-foot telescope. 46 As he related in 1684:
“N.B. In 1659 in my system of Saturn, I have taught the use of placing a diaphragm, as
it is called, in the focus of the ocular lens, without which those telescopes cannot be
freed from the defects of colors.” 47
Apparently, he recognized the combining a diaphragm with some measuring
device a bit later. 48 The fact that an object inserted in the focal plane casts a
sharp shadow over things seen through the telescope seems a logical
consequence of Huygens’ understanding of the dioptrics of a Keplerian
configuration. Still, it took him some time to recognize its usefulness and this
may well have been a chance discovery. The fact that a diaphragm reduces
‘the defects of colors’ did not follow from his dioptrical theory and had to be
discovered in practice.
Until the 1660s, Huygens’ approach to telescope making did not differ
substantially from that of an ordinary craftsman. We have seen his
unmatched understanding of dioptrical theory but it cannot be told what role
it played in his practical pursuits. In Systema saturnium, he described his
micrometer in a procedural way, without explaining it analytically in
dioptrical terms. The book contained only one dioptrical passage. He wrote
45
Bedini, “The tube of long vision”, 157-159.
46
OC15, 56.
47
OC13, 826. “N.B. me anno 1659 in Systemate Saturnio meo docuisse usum diaphragmatis quod vocant,
in foco ocularis lentis ponendi, absque quo colorum vitio haec telescopia carere non poterant.” In 1694 he
explicitly claimed that he was the first to use a diaphragm: OC13, 774.
48
McKeon, “Les débuts I”, 237.

64 CHAPTER 3
that the power of a telescope could better be determined by calculation than
using the ordinary ways of comparison. He referred to a theorem in
“Dioptricis nostris”: the magnification is equal to the proportion of the focal
distances of objective and ocular. 49
Figure 24 Huygens’ eyepiece. (see also the diagram in Figure 25)
The year 1662 marks a turn in Huygens’ dioptrics. He invented something
new and then turned to dioptrical theory again. The invention was a
particular configuration of three lenses in a compound ocular (Figure 24).
Nowadays called ‘Huygens’ eyepiece’, it had considerable advantages over
earlier solutions: it produced a large field of view and images that suffered
relatively little from aberrations. 50 Huygens had developed the eyepiece after
his trip to Paris and London in 1660-1, where he had talked much on
telescopes and related matters. In Paris he had seen the artisan Menard and
the ingeneer Pierre Petit, who had the best collection of telescopes in Paris.
In London he saw telescopes with compound eyepieces made by the
telescope makers Paul Neile and Richard Reeve. 51
In 1662, Huygens made his first telescopes with field lenses. Later that
year, he found out what configuration of lenses produced bright images and
a wide field. On 5 October he wrote to his brother Lodewijk in Paris:
“As for oculars, you will see that I have found something new that causes that
distinctness in daytime telescopes [i.e. terrestrial], and the same thing in the very long
ones, while giving them at the same time a wide opening.” 52
Huygens’ design quickly became known and was adopted widely. How
Huygens had found the precise configuration is unknown, yet everything
points at it being a matter of trial-and-error inspired by the examples he had
seen. 53
After the invention, however, Huygens did something others like Wiesel
and Reeve did not do. He set out to understand how it worked by analyzing
the dioptrical properties of his eyepiece. Huygens described its configuration
in a proposition inserted in the third part of Tractatus. 54
49
OC15, 230-233.
50
Van Helden, “Compound eyepieces”, 33; Van Helden, “Huygens and the astronomers”, 158.
51
OC22, 568-576.
52
OC4, 242-3: “car pour les oculaires vous voyez bien que j’y ay trouvè quelque chose de nouveau, qui
cause cette nettetè dans les lunettes du jour, et de mesme dans les plus longues, leur donnant en mesme
temps une grande ouverture.”
53
Van Helden, “Compound eyepieces”, 33.
54
OC13, 252-259. The text in Oeuvres Complètes is probably from 1666. The notes contain some previous
phrasing, probably from 1662. OC13, 252n1

1655-1672 - DE ABERRATIONE 65
“Although lenses should not be multiplied without necessity, because much light is lost
due to the thickness of the glass and the repeated reflections, experience has shown it is
nevertheless useful to do so here.” 55
When the single ocular lens is replaced by two lenses, so Huygens continued,
the field of the telescope can be enlarged. Moreover, the images produced
are less deformed and the irregularities of the lenses are less disturbing.
The precise configuration
of the eyepiece was as follows
(Figure 25). AB is the
objective lens, CD and EF
form the eyepiece; the focal
distances are LG, KT, and SH
respectively. Now, KT is
about four times SH, and the
distance KS between the ocular lenses is about twice the focal distance SH of
the outer one. Finally, the focus G of the objective lens AB should fall
between the outer ocular EF and its focus H in such a way that H is the
‘punctum correspondens’ of point G with respect to lens CD. Rays coming
from a distant point Q will therefore be parallel after refraction in the outer
ocular lens EF. Having determined the position of the eye M and the
magnification by the system, Huygens concluded by explaining that points P
and Q of the object are seen sharp but reversed.
Huygens had demonstrated that this configuration produced sharp,
magnified images. This was a rather straightforward application of the theory
he had developed in 1653. The text bears witness to the fact that Huygens
had gained much experience with actual lenses since the days of Tractatus. At
one point in the theorem, he indicated why images do not suffer much from
the irregularities in the lenses. Because the eye is so close to the outer ocular,
“the spots and tiny bubbles of air, that are always in the material of the glass, cannot be
perceived in lens EF. But one does not see them in lens CD either, because the eye
perceives objects placed there confusedly, but those that are located close to H
distinctly.” 56
Huygens did not, however, explicitly compare the field of his configuration
to that of a telescope with a single ocular, nor did he explain why images
were less deformed. 57 His analysis offered a dioptrical understanding of his
eyepiece but it did not improve it:
55 OC13, 252-253. “Quanquam lentes non frustra sint multiplicandae, quod et vitri crassitudine et iteratis
reflexionibus non parum lucis depereat; hic tamen utiliter id fieri experientia docuit.”
56 OC13, 256-257. “Atque ex hac oculi propinquitate sit primum ut naevi, seu bullulae minutissimae,
quibus vitri materia nunquam caret, in lente EF percipi non possint. Sed neque in lente CD; quoniam
oculus confuse cernit quae hic objiciuntur, distincte vero quae ad H.”
57 He developed a systematic theory of the field of view of a telescope much later, after 1685: OC13, 450-
461, 468-73.
Figure 25 Diagram for the eyepiece, accompanying
Figure 24.

66 CHAPTER 3
“We give here, if not the best combination of all lenses, the investigation of which
would take long and might be impossible, but one which experience has shown us to be
useful.” 58
The particular configuration of Huygens’ eyepiece was a product of trial-anderror,
and theory could not, or not yet, add to that. Huygens the scholar had
not yet been able to assist Huygens the craftsman.
As contrasted to other telescope makers, however, Huygens was able to
understand retrospectively and in mathematical terms, what he was doing
when configuring lenses. That is to say, he understood the dioptrical
properties of lenses and their configurations. He could explain whether and
how a configuration of lenses produced sharp, magnified images. But he
could not explain everything of the kind. In another proposition found in
part III of Tractatus and apparently following the one discussed above,
Huygens discussed a telescope with an erector-lens such as Kepler had
proposed. 59 He concluded with some remarks about the quality of images
produced by various configurations. With a telescope consisting of a convex
objective and a concave ocular – the Galilean configuration – images are
more distinct “and defiled by no colored rims that can hardly be prevented
in this composed of three lenses.” 60 A well-chosen combination of lenses
could counter these defects, but
“different people combine ocular lenses differently with regard to each other, looking
for the best combination with only the guide of experience. It would not be easy, to be
sure, to teach something about this that is grounded in certainty, since the consideration of
colors cannot be reduced to the laws of geometry, ….” 61 [italics added]
In his practical work Huygens had found out that lenses suffered from all
kinds of defects. Some of these eluded dioptrical analysis. But he had also
found out that nuisances caused by fogs, bubbles and colors could be
diminished. The diaphragm had already proven this. His eyepiece gave
another means to improve the quality of images. 62 He could not fully explain
its advantages, nor could he improve it by means of dioptrical analysis. Still,
the eyepiece had proven that a well-chosen configuration of lenses could be
advantageous. And it made him realize that even better configurations could
be found, even though he was as yet pessimistic about such an enterprise. If
Huygens the scholar could gain a thorough understanding of the defects of
58 OC13, 252-253. “Dabimus autem in his, etsi non omnium optimam lentium compositionem, quam
investigare longum esset ac forsan impossibile, at ejusmodi quam nobis experientia utilem esse ostendit.”
59 OC13, 258-265. Discussed above, section 2.1.2..
60 OC13, 262-263. “… res visas, atque etiam distinctiores efficere, nullisque colorum pigmentis infectas
quod in hic lentium trium compositione aegre vitari potest.”
61 OC13, 264-265. “Alij vero aliter lentes oculares in his inter se consociant, sola experientia duce quid
optimum sit quaerentes. nec sane facile foret certa ratione aliquid circa haec praecipere, quum colorum
consideratio ad geometriae leges revocari nequeat, …”
62 A way to reduce colors that was more commonly employed, was to make objective lenses with large
focal distances. These, however, had the drawback that telescopes became very long and tubes too heavy
to remain straight. In 1662, it occurred to Huygens that this could be circumvented by making a tubeless
telescope. He realized it much later and published a little tract on it, Astroscopia Compendiaria (1684). OC21,
201-231.

1655-1672 - DE ABERRATIONE 67
lenses, he might teach Huygens the craftsmen how to combine lenses in the
best possible way. The next decade he actually set out to do this.
3.2 Dealing with aberrations
According to Hugyens, not all defects of lenses could be explained
dioptrically. One particular defect, however, was subject to the laws of
geometry: spherical aberration. It could therefore be explained and, possibly,
prevented. Huygens was not the first to design a solution to prevent the
defects of lenses. Descartes had done so with his elliptic and hyperbolic
lenses. Newton would built a mirror telescope in order to avoid the defects
of (spherical) lenses. Huygens was the first to take spherical lenses as a
starting point for a theoretical design, instead of ruling them out beforehand.
In 1665, he began a study of spherical aberration with the intention to design
a telescope consisting of spherical lenses such as to neutralize each others’
aberrations.
The idea that the lenses of a telescope might cancel out their mutual
aberrations had occurred to no-one yet:
“Until this day it is believed that spherical surfaces are … less apt for this use [of
making telescopes]. Nobody has suspected that the defects of convex lenses can be
corrected by means of concave lenses.” 63
The project added a new dimension to Huygens’ dioptrical studies. No
longer did he just want to understand the telescope, but now he also wanted
to improve it by means of dioptrical theory. In so doing, he followed
Descartes’ ideal that the scholar could lead the craftsman, but it had taken on
a different form. Huygens started out with what was practically feasible instead of what
was theoretically desirable. Spherical lenses had been the focus of both his
theoretical investigations and his practical activities. It looks like Huygens
now wanted to combine these two sides of his involvement with telescopes.
3.2.1 PROPERTIES OF SPHERICAL ABERRATION
In order to be able to determine an optimal configuration of lenses, Huygens
first had to develop a theory of spherical aberration. The phenomenon had
been known for a long time. In perspectivist theory it was known that a
burning glass does not direct all sunrays to one point. No one, however, had
gone beyond the mere recognition of the phenomenon, and its exact
properties had not been studied. Kepler went farthest by pointing out the
connection between a ray’s distance from the axis and its deviation from the
focus, but this necessarily remained qualitative. 64 Mathematicians like
Descartes had focused on determining surfaces that did not suffer from such
aberration. With his concept of ‘punctum concursus’ of Tractatus, Huygens
had been the first to take spherical aberration into account in dioptrical
theory, defining the focus as the limit point of intersecting rays. In 1665 he
63 OC13, 318-319. “creditum est hactenus … sphaericae superficies minus aptae essent his usibus, nemine
suspicante vitium convexarum lentium lentibus cavis tolli posse.”
64 Kepler, Paralipomena, 185-186 (KGW2, 168-169). Kepler repeated his insights in Dioptrice.

68 CHAPTER 3
extended this by developing a theory of spherical aberration. He subjected,
so to say, the ‘punctum concursus’ to a closer examination to see how exactly
spherical aberration affected the imaging properties of lenses.
Huygens’ study of spherical aberration
had been preceded by a calculation of rays
refracted by a plano-convex lens he had
apparently carried out in 1653. In Tractatus,
he remarked that rays “reunite somewhat
better, i.e. that the points where they cut
the axis are closer to one point, …, when
the convex surface faces the incident rays,
than when the plane surface faces them.” 65
The 1653 calculation is lost, but was
probably identical to later ones. 66 The result
implied that the orientation of a lens
affected the degree of aberration. In 1665,
Huygens went to see whether the amount
of spherical aberration might deliberately
be decreased by a proper configuration of
lenses. He began a study of spherical
aberration under the heading “Adversaria
ad Dioptricen spectantia in quibus quæritur
aberratio a foco”. 67 After a decade of quiet,
the ‘Adversaria’ was the next chapter of
Huygens’ dioptrical studies.
In ‘Adversaria’ Huygens derived
expressions for the amount of spherical
aberration as it depends upon the
properties of a lens. The rigor familiar from
Tractatus returns immediately. On the basis
of the theorems of Tractatus, he took the
refractions at both faces of the lens as well
as its thickness into account. In the first
calculations Huygens returned to the claim
of 1653. He derived an expression for the aberration of the extreme ray
incident on a plano-convex lens GBC with focal distance GS (Figure 26). A
parallel ray is refracted at the extreme point C of the lens towards T on the
axis. The derivation of the aberration TS is straightforward and yields
65 OC13, 82-83. “…, accuratius aliquanto eos propiusque ad unum punctum convenire …, cum superficies
convexa venientibus opposita est radijs, quam si plana ad illos convertatur.” Huygens had also written this
to Gutschoven in his letter of 6 March 1653: OC1, 225. As we have seen above, Flamsteed carried out a
numerical calculation and came to the same conclusion, which returned in Molyneux’ Dioptrica nova.
Flamsteed, Gresham Lectures, 120-127. Molyneux, Dioptrica nova, 23-25.
66 OC13, LII (“Avertissement”), those later calculations are on pages 283-287.
67 OC13, 355-375.
Figure 26 Spherical aberration of a
plano-convex lens.

TS = 7 BG, where BG is the thickness
6
of the lens. 68 When the lens is reversed
and rays are incident on the plane
side, the aberration becomes
TS = 9
2 BG.69 The aberration is
therefore considerably smaller –
almost four times – when the convex
side faces the incident rays. This time
Huygens went further than the mere
observation that the orientation of a
lens affects the amount of aberration.
The faces of a lens are surfaces with
different radii – infinite in the case of
a plane face. The proportion between
these radii apparently determines how
large the aberration is. Consequently,
an ideal lens can be found by
determining the optimal proportion
of both radii.
To do so, Huygens derived an
expression for the aberration of a
parallel ray HC incident on the
extreme end of a lens IMCB (Figure
27). AB = a and NM = n are the radii
of the anterior and posterior side and
BG = b is the thickness of the anterior
half of the lens. The thickness of the
entire lens BM = q can be expressed as
q =
ba
b . The anterior face refracts
n
an extreme ray HC towards P, a little
off its focus R. The posterior face, in
its turn, refracts the extension CF of
ray CP towards D, a little off the focus
E of the lens. Huygens then expressed
the spherical aberration DE of the
extreme ray in terms of the radii of
the faces and the thickness of half the
2 2
7nq6anq27aq lens: DE = 2 “…
6(
an) 68 OC13, 357.
69 OC13, 359.
1655-1672 - DE ABERRATIONE 69
Figure 27 Aberration of a bi-convex lens

70 CHAPTER 3
the space on the axis within which all parallel rays are brought together,
which space DE is defined by this rule.” 70 Or, the aberration DE is found by
2 2
7n 6an27a multiplying the thickness of the lens q by the expression 2 ,
6(
an) which only depends on the radii of both faces. The shape of a lens that
produces minimal aberration can be found by determining the minimum of
this expression; this yields a : n = 1 : 6. 71 In this case the aberration of the
extreme ray DE = 15q. Huygens found the same for a bi-concave lens,
14
whereas a converging meniscus lens yielded a meaningless outcome. 72
Satisfied, he summarized the result:
“In the optimal lens the radius of the convex objective side is to the radius of the
convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.” 73
The ‘Adversaria’ provided general expressions for spherical aberration in
terms of the shape of a lens. It contained a set of derivations and calculations
without explanation. He did not, for example, point at certain simplifications
he had carried out. The results were not therefore fully exact, as will become
clear later on. Still, it was the most advanced account of spherical aberration
at the time. On the basis of his theory of spherical aberration he went on to
design a configuration of lenses that minimized the ‘aberrations from the
focus’.
A note of clarification needs to be made. Huygens did not yet call the
phenomenon he was investigating spherical aberration. Around 1665,
Huygens referred to it in a general way: “aberration from the focus” and
“Investigate which convex spherical lens brings parallel rays better
together.” 74 Only much later, when distinguishing the aberration caused by
colors, did he explicitly called it “the aberrations of rays that arise from the
spherical shape of the surfaces”. 75 We should bear this in mind when
interpreting Huygens’ study of aberrations and his designs for perfect
telescopes. That is, we do not know for certain what exactly he thought his
design would improve.
Specilla circularia
Before continuing with Huygens, mention has to be made of another study
of spherical aberration. Not because it mattered much for the mathematical
theory of spherical aberration – it did not – but because it approached the
70 OC13, 364. “DE spatium in axe intra quod radij omnes paralleli coguntur, quod spatium DE per regulam
hanc definitur.”
71 OC13, 366-367. Modern methods yield the same result.
72 OC13, 375 and 370. In the latter case the solution yields a negative value for the radius of the posterior
side.
73 OC13, 367. “Radius convexi objectivi ad radium convexi interioris in lente optima ut 1 ad 6. EUPHKA. 6
Aug. 1665.”
74 OC13, 280n2. “Quaenam lens sphaerica convexa melius radios parallelos coligat investigare.”
75 OC13, 280-281. “aberrationes radiorum quae ex figura superficierum sphaerica oriuntur”

1655-1672 - DE ABERRATIONE 71
problem central to La Dioptrique in an original way. Moreover, it preceded
Huygens’ study and he may have known it in some way. The study is found
in two manuscript copies of Specilla circularia, a tract presumed to have been
written in 1656 by Johannes Hudde, an acquaintance of Huygens. 76 The fact
that Hudde had written on dioptrics was known from his correspondence
with Spinoza. 77 Apparently Spinoza had a copy of Specilla circularia, as he
referred to a ‘small dioptrica’ by Hudde and some of his own figures and
calculations are clearly based on it. In addition, a tract called Specilla circularia
turns up in Huygens’ correspondence in 1656. On 30 May 1656, Van
Schooten wrote that he had recently bought an anonymously published
treatise called Specilla circularia. He supposed it was written by Huygens
“because of its accuracy”. 78 Huygens replied that it was not and that he had
never heard of it. 79 He asked for a copy, but it is not clear whether he ever
received one. Huygens corresponded with Hudde on mathematical topics,
but they did not discuss dioptrics. Huygens visited Spinoza several times
around 1665 and they discussed dioptrical matters extensively. Whether or
not he knew Specilla circularia, it would not have added to Huygens’
understanding of spherical aberration. Probably he would not have accepted
Hudde’s analysis and conclusions, either.
The main goal of Specilla
circularia was to show there
was no point in striving after
the manufacture of
Descartes’ aspherical lenses.
In practice one legitimately
makes do with spherical
lenses, because spherical
aberrations are sufficiently
small. 80 In order to
substantiate this claim,
Figure 28 Hudde’s calculation of spherical aberration
Hudde employed an original definition of the focus of a lens (Figure 28). AB
is a ray parallel to the axis DNI at distance BF. It is refracted to BI by a convex
surface with radius DN. Choosing DN = 1 and an index of refraction 20 : 13,
Hudde calculated the length of NI for various values of BF, concluding that
the smaller BF the larger NI (where I approaches K). 81 Considering
76
The original tract is lost, but has been identified by Vermij with two manuscript copies discovered in
London and Hannover. Both are dated 25 April 1656 and one gives the name of the author: “Huddenius
consul Amstelodamensis”, which suggests the copy itself was made in or after 1672. Vermij, “Bijdrage”,
27; Vermij and Atzema, “Specilla circularia”, 104-107.
77
Spinoza, “Briefwisseling”, 251. Spinoza’s letters contain calculations that are similar to those in Specilla
circularia. The letter can also be found in OC6, 36-39, where it is assumed to be addressed to Huygens.
78
OC1, 422. “propter accurationem”
79
OC1, 429.
80
Vermij and Atzema, “Specilla circularia”, 119.
81
Vermij and Atzema, “Specilla circularia”, 116: “Ex quibus patet, quanto x sive BF minor est, tanto etiam
punctum I longius distare ab N;”

72 CHAPTER 3
numerically all rays between B and F, he calculated the proportion of BF to
Im, through which all refracted rays pass. Seeing that IM is small compared to
BF, Hudde concluded that K could be regarded as the focus. 82 According to
Hudde, the focus was not an exact, geometrical point, but a ‘mechanical
point’, a point that cannot be divided mechanically or whose parts are not
truly discernable. 83 This practical outlook made him reject Descartes’
proposal as superfluous.
Hudde’s study lacked Huygens’ rigor. From a mathematical point of view,
he explained away spherical aberration. He attained ‘practical’ exactness,
rather than mathematical, much in the same way as a Flamsteed or
Molyneux. Hudde called in question whether spherical aberration was as
relevant a problem as Descartes had claimed it to be. In Specilla circularia, he
argued that in practice it was not. The spirit of Hudde’s study of dioptrics
was similar to that of Huygens’: to see what mathematics could teach about
the working of lenses in practice. Hudde’s conclusion was the opposite of
Huygens’. In Huygens’ view, an exact understanding of the phenomenon
might yield a telescope that actually smoothed aberrations away.
Theory and its applications
Sometime after writing the ‘Adversaria’, Huygens elaborated it into a
rounded essay on spherical aberration. It contained his first solution to the
problem his study was aimed at: a configuration of spherical lenses that
neutralized spherical aberration. The essay is found in the Oeuvres Complètes
under the title De Aberratione radiorum a foco. In De Aberratione Huygens
worked up and extended his earlier notes. He set up his argument with a
definition of the thickness of a lens and several auxiliary propositions. 84
Besides the expressions he had given in the ‘Adversaria’ for the aberration of
extreme rays, he established the relationship between the aberration of an
arbitrary ray and its distance from the axis. 85
In the fourth and fifth propositions of De Aberratione the results of the
‘Adversaria’ returned. Huygens now explicated the simplifications he had
carried out earlier. He first derived a more exact expression – which I will
not give – for the aberration of the extreme ray incident on the plane side of
a plano-convex lens. When the radius of the convex side is 72 inches and the
extreme ray is 1 inch from the axis, this expression yielded an aberration of
31253 inches. Huygens then stated – without proof – that the aberration
1000000
could be found more easily by multiplying the thickness of the lens by 9 , the 2
rule found in the ‘Adversaria’. 86 There is, he admitted, a slight difference
82
Vermij and Atzema, “Specilla circularia”, 117: “Unde constat, focum ipsum pro puncto mechanico
tantum habendum esse.”
83
Vermij and Atzema, “Specilla circularia”, 114: “Punctum autem mechanicum appello, quod in
mechanicis aut divisible non est, aut cujus partes hic non sunt considerata digna.”
84
OC13, 276-277.
85
OC13, 308-313.
86
OC13, 282-285. Each time he assumed an index of refraction 3 : 2.

1655-1672 - DE ABERRATIONE 73
1 ( 1000000 inches) but this was of no significance in actual telescopes. 87 When
the convex side of the same lens faced the incident rays, the exact calculation
yielded an aberration of 81021 inches. In this case, the easier rule of
10000000
‘Adversaria’ – multiplying the thickness of the lens by 7
6 – gave 81022
10000000 , a
1
difference of only inches. Again, the main goal of this exercise was to
1000000
show that the aberration of a plano-convex lens is least when its convex side
faces the incident rays. 88 Continuing with a bi-convex lens, Huygens sketched
out how the aberration might be calculated exactly, but immediately moved
on to an ‘abbreviated rule’ he had ‘found’. 89 This was the expression of the
‘Adversaria’, found by “ignoring very little quantities, but judiciously so as
needed.” 90 The rule applied to convex as well as to concave lenses and
yielded the optimal proportion of both radii of 1 : 6. The resulting bi-convex
15 91
times its thickness.
lens produces an aberration of only 14
Surprisingly, these laborious derivations were not of great value for
telescopes. After having explained the optimal proportions of bi-concave
lenses, Huygens wrote that they were not useful as ocular lenses. In
telescopes, he said, one should choose “… other, less perfect lenses, so that
the defects of the convex lens are compensated and corrected by their
defects.” 92 Those less perfect lenses were diverging concavo-convex lenses.
Huygens showed that these lenses always produce a larger aberration than biconvex
or bi-concave lenses.
As ocular lenses they could, however, be useful:
“With concave and convex spherical lenses, to make telescopes that are better than the
one made according to what we know now, and that emulate the perfection of those
that are made with elliptic or hyperbolic lenses.” 93
Here was what Huygens had been looking for: a configuration where lenses
mutually cancel out their aberration. He had designed a telescope in which
the ocular corrects for the aberration of the objective lens, thus equaling the
effect of a-spherical lenses.
The solution was as follows: given an objective lens and the required
magnification of a telescope, determine the shape of the ocular lens (Figure
29). On the axis BDFE of lens ABCD, divide the focal distance DE by point F
87
OC13, 284-285. “Exigua quidem differentiola, sed quae in illa lentium latitudine quae telescopiorum
usibus idonea est, nullius sit momenti.”
88
OC13, 284-287.
89
OC13, 290-291. “Et haec quidem methodus ad exactam supputationem adhibenda esset. Invenimus
autem et hic Regulam compendiosam …”
90
OC13, 290-291. “Quae regula … inventa est neglectis minimis, sed necessario cum delectu.”
91
OC13, 290-291& 302-303.
92
OC13, 302-303. “…, sed aliae minus perfectae, quarum nempe vitijs compensantur ac corrigentur vitia
lentis convexae, …”
93
OC13, 318-319. “Ex lentibus sphæricis cavis et convexis telesopia componere hactenus cognitis ejus
generis meliora, perfectionemque eorum quæ ellipticis hyperbolicisve lentibus constant æmulantia.”

74 CHAPTER 3
according to the chosen magnification. The ocular GFH is to be placed in F.
For example, DE : FE = 10 : 1 when the telescope should magnify ten times.
Because the foci of both lenses should coincide, the focal distance of the
ocular lens is given, namely FE. Due to its spherical aberration, the objective
lens does not refract parallel rays KK, CC to E but to N and O along KLN and
CHO. After refraction by the ocular lens rays LM, HI should all be parallel.
This can be accomplished when the aberrations NE, OE are the same for the
objective and ocular lenses with respect to the parallel rays KK, CC and LM,
HI respectively. With the expression for the aberrations of both lenses,
Huygens could determine the required radii of the ocular lens. The radius of
86 the convex side should be 100 times its focal distance FE, the radius of the
86 94 concave side 272 times FE. Next, he proved that this ocular indeed canceled
out the aberration of the objective. 95 Ergo, the proposed configuration would
produce almost perfect images.
Huygens supplied a table in which he listed telescopes with various
magnifications against the ocular lenses required according to his analysis.
These numerical examples were, so to say, the blueprint by means of which
his design could be realized by any skilled worker. By way of conclusion,
Huygens remarked that the advantages of his design could only be realized
by lenses that were truly spherical. The manufacture of spherical lenses
should therefore be resumed diligently.
Huygens had made clear at the outset of his exposition that the
usefulness of his design was limited. Only a concave ocular could correct for
the aberration of the objective lens. The design was therefore useful only for
Galilean telescopes. In astronomy, telescopes required a convex ocular:
“However, it is certain that this mutual correction is not found in the composition of
convex lenses. On the contrary, the defect of the exterior lens is always a bit augmented
by the ocular lens and it cannot be remedied in any way.” 96
Figure 29 Galilean configuration in which spherical aberration is neutralized.
Still, his theory was not entirely useless for the Keplerian telescopes required
for astronomical observation. In the final propositions of De Aberratione,
94 OC13, 320-323.
95 OC13, 324-327. For the rays KK and LM – that are not extreme rays – Huygens used the proposition on
the linear proportion between aberration of a ray and the square of its distance to the axis. OC13, 308-
313.
96 OC13, 318-319. “Sed certum est in convexis inter se compositis emendationem illam mutuam non
reperiri. Imo contra, vitium exterioris lentis a lente ocularis augetur semper nonnihil neque id ulla ratione
impediri potest.”

1655-1672 - DE ABERRATIONE 75
Huygens examined the means to enhance the quality of images produced by
telescopes with a convex ocular. That is, he took a theoretical look at the
matter. On the basis of this analysis, he could provide directions for
optimizing the quality of telescopes with convex oculars.
The magnifying power of a telescope depends upon the ratio of the focal
distances of objective and ocular lenses, and can therefore be increased by
reducing the focal distance of the ocular lens. This, however, simultaneously
decreases the clarity and distinctness of images. To maintain clarity at the
same time, the opening of the objective lens would have to be made larger. 97
He began by considering a naked eye in front of which a telescope is placed.
Assuming that an equal number of rays should enter the eye when a more
powerful telescope is taken, Huygens argued that the opening should be kept
proportional to the magnification. This implied that his 22-foot telescope
would need an opening of 125 times the area of the pupil. In reality, he
observed, a satisfactory telescope had a much smaller opening, only 15 times
the area of the pupil. Evidently, in astronomical observation one could do
with much smaller clarity. He therefore did not take the eye as starting-point,
but a telescope with satisfactory quality. If the ocular is replaced by an ocular
that magnifies twice as much, the clarity will be four times smaller. The
opening of the objective should therefore be increased accordingly.
Evidently, this cannot be done at will and one should “consider accurately
which magnification the opening of the exterior lens can support”. 98
Maintaining the clarity of images does not mean,
however, that their quality is maintained. Increasing the
opening of a lens renders images less distinct. Huygens
made it clear that only experience could tell which
configuration produced satisfactory images. Yet, when such
a telescope is known, theory can explain how the quality of
images is maintained when its strength increases. In his
account, Huygens applied a new conception of spherical
aberration that he had defined in an earlier proposition of
De Aberratione. He called it the ‘circle of aberration’. As
contrasted to the earlier conception, in which the aberration
GD is measured along the axis, the circle of aberration is
measured by the distance ED perpendicular to the axis
(Figure 30). In other words, the circle of aberration is the
spot produced by parallel rays coming from one point of a
distant object. Consequently, the images produced by two
lens systems are equally clear and equally distinct when the
respective circles of aberration are the same. 99
Figure 30 ‘Circle’
of aberration.
97
OC13, 332-335.
98
OC13, 336-337. “sed diligenter expendendum quale incrementum exterioris lentis apertura perferre
valeat”
99
OC13, 340-343.

76 CHAPTER 3
Huygens supposed that the circle of aberration XV of a lens system is
mainly produced by the objective lens AB (Figure 31). The ocular lens PO
barely increases the diameter of the circle and could therefore be considered
to have a perfect focus. He considered the opening BC of the objective lens
required to maintain a constant circle of aberration when the focal distance
CD of this lens is changed. He proved that the proportion CD 3 : CB 4 should
remain constant. 100 Finally, the quality of images will be maintained upon
changing the ocular lens, when the proportion OD : 4 CD between the focal
distances of both lenses is maintained. 101 Again, Huygens converted these
proportions into a table of numerical values, listing the optimal values of the
focal distances of both lenses and the opening of the objective, as well as the
resulting magnification of the system. 102 This table concluded De Aberratione.
Huygens’ theoretical accomplishments in De Aberratione are beyond dispute.
Like the theory of focal distances and magnification of Tractatus, his theory
of spherical aberration was rigorous and general. And again his theoretical
studies were aimed at understanding the telescope; in this case, at
understanding how a system of lenses produces spherical aberration.
Huygens could claim that he understood why an opening of such-and-such
dimensions maintained the quality of images.
His results were couched in two tables listing the required components to
make these optimal systems, in a way quite comparable to the ones found in
Bolantio’s manuscript. They prescribed how to assemble a telescope without
presupposing theoretical knowledge of dioptrics. The difference is that
Huygens’ tables were derived from his mathematical theory of lenses instead
of a record of experiential knowledge. The table prescribing the aperture of
telescopes was not gained by some implicit rule of thumb, but was based on
an explicit theorem derived from dioptrical properties of lenses. Huygens
could prove that the openings he prescribed were optimal. Whether this
worked in practice remains to be seen. At least he could claim that he could
calculate beforehand how to adjust the components of a telescope when its
length was changed, thus avoiding a renewed process of trial-and-error.
Huygens had realized the goal of De Aberratione. He had demonstrated
that the aberrations of spherical lenses could be made to cancel out.
100 OC13, 342-345.
101 OC13, 348-351.
102 OC13, 350-353.
Figure 31 Aberration produced by a Keplerian configuration.

1655-1672 - DE ABERRATIONE 77
Moreover, he had employed theory to improve the telescope. The design was
still a blueprint, and at this point his accomplishments were theoretical only.
He had developed a further understanding of the properties of spherical
lenses and found means to configure them optimally. He had not yet ‘tested’
his designs, nor had he verified his theory as a whole. For example, his
concept of circle of aberration suggests a way to study the observational
properties of spherical aberration, to see whether it correctly described the
defects of lenses. Nowhere, however, did he refer to something of the kind. I
will return to this point below. Huygens’ next step was an attempt to realize
his design of a telescope in which ocular and objective lenses cancelled out
their mutual aberrations. A test to the theory?
3.2.2 PUTTING THEORY TO PRACTICE
Not until 1668 did Huygens set about realizing his design. 103 By that time he
lived in Paris, where he had arrived in the summer of 1666. 104 In the
meantime, telescopes had not been out of his mind, though. They were
frequently discussed in his correspondence with Constantijn. He examined
the quality of glass and lenses made by Parisian craftsmen, not being
impressed. 105 He was particularly dissatisfied by a telescope he had bought for
his father – a campanine made by one Menard. 106 He equipped a campanine
of his own with lenses made by his brother and was pretty contented with
it. 107 In April 1668, he decided to have Constantijn make lenses for the design
of De Aberratione. 108
On 11 May 1668, Huygens gave his brother detailed instructions to grind
a set of lenses. For the objectives Constantijn made – plano-convex lenses of
2 feet and 8 inches – a concavo-convex ocular was required. The radii ought
to be 0,187 and 0,289 inches, respectively, and Huygens drew out the shapes
in his letter. 109 Combined, these lenses would perform like hyperbolic glasses,
he said,
“… because the concave lens corrects the defects arising from the spherical shape of
the objective lens. therefore I cannot determine the opening of the objective that
maybe might be 3 or 4 times larger than an ordinary one has, but if we can just double
it much would be gained and the clarity will be sufficiently large for the magnification
of 30.” 110
103
OC13, 303n4; 331n4.
104
OC5, 375; OC6, 23.
105
OC6, 151; 205; 207.
106
OC6, 86-87; 151; 205.
107
OC6, 207.
108
OC6, 209.
109
OC6, 214-215.
110
OC6, 214. “Ce composè, …, doibt faire autant que les verres hyperboliques, parce que le concave
corrige les defauts de l’objectif qui vienent de la figure spherique. c’est pourquoy je ne puis pas determiner
l’ouverture de l’objectif qui peut etre pourra estre 3 ou 4 fois plus grande qu’a l’ordinaire, mais si nous la
pouvons seulement faire double ce sera beaucoup gaignè et la clartè sera assez grande pour la
multiplication de 30.”

78 CHAPTER 3
Huygens did not explain the ‘secret’ of his new method to his brother. He
urged him not to tell anyone about the plans. Constantijn responded quickly.
On the first of June, Huygens answered two letters – now lost – his brother
had sent on May 18 and 24. 111 Constantijn had sent an ocular with only one
side ground according to his instructions, the other being plane. Apparently,
Constantijn had made some objections to his brother’s design. Huygens did
not agree and urged his brother to make a lens exactly to his directives.
Huygens did not await new lenses,
but immediately tried the one
Constantijn had sent him. A week
later he reported on the disappointing
results. When the objective lens was
covered in ordinary fashion, the
Figure 32 Rendering of Huygens’ sketch.
system performed reasonably well. Yet, the system fell short of his
expectations. According to his design, the quality of the image should be
maintained when the whole objective lens was exposed to light. (Figure 32)
“but uncovering the entire glass I see a bit of coloring which leads me to believe that
there is an inconvenience therein, which results from the angle made by the two
surfaces of the objective at the edges. This necessarily causes colors, in such a way that
by making hyperbolic glasses one encounters the same things when making them very
large.” 112
Huygens here tentatively drew an important conclusion. That is, we
recognize that he was on the right track by suspecting that those colors were
inherent to the refraction of rays and could not be prevented by hyperbolic
lenses. Moreover, his suggestion that the production of colors could be
linked to the angle of the lens’ surfaces was promising in light of Newton’s
later theory of colors. The remark may have been inspired by a measurement
Huygens had performed in November 1665. 113 Having read Hooke’s account,
in Micrographia, of colors produced in thin films of transparent material, he
set out to determine the thickness of the film, which Hooke had not been
able to do. He pressed two lenses together to produce colored rings. The
colors appear where the two lenses nearly meet, a situation comparable to
the thin rim of a glass lens. Whether this measurement and the remark of
1668 are connected is, however, mere speculation. In Micrographia, he also
would have found discussions of prism experiments, and the effect of a
prism may also explain the emphasis on the angle between the faces of the
lens at the edge. Whatever be the case, Huygens did not pursue this line of
thinking. He suspected that the proportions of Constantijn’s lens were not
the gist of the problem, “but before assuring that, I would be pleased to
111
OC6, 218-220.
112
OC6, 220-221. “mais en decouvrant tout le verre je vois un peu de couleurs ce qui me fait croire qu’il y
a un inconvenient de costè la, qui provient de l’angle que font les 2 surfaces de l’objectif vers les bords.
qui cause necessairement des couleurs, de sorte qu’en faisant des verres hyperboliques l’on trouueroit la
mesme chose en les faisant fort grands.”
113
OC17, 341. Huygens’ measurements, as well as the experiments Newton performed at the same time,
are amply discussed in Westfall, “Rings”.

1655-1672 - DE ABERRATIONE 79
carry out the plan with an entire glass, like I have asked you to make for
me.” 114
During the following months, Huygens kept reminding his brother that
he was waiting for the proper lens. 115 He even considered taking up his own
grinding work and started looking around in Paris for able craftsmen. 116 On
November 30, he sent his brother additional directives for oculars. 117 On 1
February 1669, Huygens brought his invention to Constantijn’s attention for
the last time: “You don’t talk anymore about the oculars you have promised
me.” 118 This was the final, somewhat aggrieved sentence of a letter in which
he informed his brother of another letter – one he had received from a
certain baron de Nulandt, an acquaintance of Constantijn living in The
Hague at that time. 119 The baron was engaged in making telescopes and also
had some ideas regarding dioptrical theory. On 20 December 1668, Nulandt
had written to Huygens. In the letter of 1 February to his brother, Huygens
wrote:
“The worthy Baron de Nulandt begins to talk like a great savant, and lets me coolly
know that he has found the same proportions of glasses to imitate the hyperbola of
which I have talked to him in my letter, although I am sure that this is infinitely beyond
his capacities. The calculations he sends me are far from the truth, and I will not refrain
from showing him this.” 120
Huygens had told Nulandt about his idea of nullifying spherical aberration
by means of spherical lenses in a letter now lost. On 18 January, Nulandt had
replied that he had also found that a concave meniscus lens could correct the
aberration of the objective lens, but had not given any details. 121 In that letter,
Nulandt calculated the amount of aberration for two lenses and had drawn
conclusions that were contrary to Huygens’ own. Huygens’ letters in reply
are lost, but it is clear that he easily convinced Nulandt of his mistakes. In his
next letter, Nulandt admitted that his configuration for nullifying spherical
aberration was faulty, because he had calculated the aberration of lenses in a
wrong way. 122
114
OC6, 221. “mais devant que de l’assurer je serois bien aise de faire l’essay avec un verre entier, que je
vous ay priè de me vouloir faire.”
115
OC6, 236; 266. He did not show consideration for the fact that Constantijn was getting ready for his
marriage on 28 August 1668.
116
OC6, 266; 300.
117
OC6, 299-300.
118
OC6, 353. “Vous ne parlez plus des oculaires que vous m’avez promis.”
119
Little is known about him. He published an anti-Cartesian treatise Elementa physica in 1669 in which he
included an extract of a letter written by Christiaan (OC6, 420-421). He first appears in a letter to Huygens
of 20 December 1668, which suggests that they had met, probably in Paris. OC6, 304-305.
120
OC6, 353, “Le Seigneur Baron de Nulandt commence a parler en grand docteur, et me mande
froidement, d’avoir trouvè les mesmes proportions de verres, pour imiter l’Hyperbole, dont je lui avois
parlè dans ma lettre, quoique je sasche bien que cela passe infiniment sa capacitè. Les calcus qu’il
m’envoye sont trop eloignez de la veritè, et je ne manqueray pas de le lui remontrer.”
121
OC6, 348-351; particularly 350.
122 OC6, 363-367; particularly 364.

80 CHAPTER 3
A new design
We could have passed over this episode with Nulandt, if its conclusion had
not coincided with the next phase in Huygens’ study of spherical aberration.
On that same 1st of February, he gloriously
wrote down “A composite lens emulating a
hyperbolic lens. EUPHKA” 123 He had found a
new solution to the problem of neutralizing
spherical aberration that made his earlier one
superfluous. It consisted of a combination of
two lenses that would replace one objective
lens. This composite lens could therefore be
used in telescopes for astronomical
observation, whereas the earlier solution was
useful for terrestrial telescopes only. On
February 22, he asked his brother Lodewijk to
tell Constantijn
“… that I abandon the little ocular I had asked
from him, because I have found something better
and more substantial in these matters, that I would
like to try out myself.” 124
Huygens’ idea was as follows (Figure 33).
The bi-concave lens VBC and the plano-convex
lens KSTG have the same focus E, with respect
to diverging rays MV coming from M, and
parallel rays QK, respectively. In addition, the
spherical aberration EN produced by each lens
is the same for these rays. An arbitrary ray QK,
parallel to the axis ASM, is refracted by lens KST
towards point N, a little off its focus E. Lens
VBC, in its turn, refracts a ray MV towards the
same point N, at the same distance from its
focus E. Ray CN is therefore refracted towards
M. As a result, the composite lens brings all
parallel rays QK to a perfect focus M and “…
will emulate a hyperbolical or elliptical lens
perfectly.” 125 The system acts as a converging
lens and can therefore replace the objective of
any telescope. In his proof, Huygens worked
Figure 33 The invention of 1669
123
OC13, 408. “Lens composita hyperbolicae aemula. EUPHKA 1 Febr. 1669.”
124
OC6, 377. “Vous pourrez luy dire que je le quite pour ce qui est du petit oculaire que je luy avois
demandè, ayant trouvè quelque chose meilleur et de plus considerable en cette matiere, dont j’ay envie de
faire moy mesme l’essay.”
125
OC13, 413. “…[lens] compositae ex duabus VBC, KST, quae Hyperbolicae aut Ellipticae perfectionem
aemulabitur.”

1655-1672 - DE ABERRATIONE 81
the other way around. 126 The bi-concave lens VBC is given. M is the center of
surface BV, so that rays from M are not refracted by it. Surface CB of this lens
refracts a ray MC to KCN, intersecting the axis in N, where EN is the spherical
aberration. The problem is to find a convex lens KST with the same focus E,
which refracts a parallel ray QK, at distance KS to the axis, to the same point
N. Huygens chose BE – nearly equal to GE – as the focal distance of this lens
KST. Its spherical aberration EN is – by the rule from the ‘Adversaria’ – 7
6
times its thickness GS. This length EN is also the spherical aberration of
surface BC of the bi-concave lens VBC. It can be expressed in terms of its
radius AB, the length BG (proportional to the distance CG of the ray to the
axis), and the length BM. Equating both expressions for EN, he found a
proportionality between the radius of KST and BC. It is 100 to 254, or nearly
2 to 5. In addition MB, the radius of the other surface BV of the bi-concave
lens, has to be twice that of BC or ten times that of KST. At the end of his
calculations Huygens summarized the solution:
“A lens composed of two emulates a hyperbolic lens, the one plano-convex the other
concave on both sides. The radii of the surfaces are nearly two, five, ten.” 127
Five days later, on 6 February, he sent a letter to Oldenburg to which he
appended an anagram containing his ‘important invention’: 128
a bc
d e h i l m n op
r s t u y
52
2 14
1 23
3 1 3 2 23
24
1
This second invention can be regarded as the final piece of Huygens’ project
of canceling out spherical aberration by means of spherical lenses. He had
shown that spherical lenses were indeed apt for telescopes by designing a
configuration that produced an almost perfect focus. As contrasted to the
earlier invention of 1665, this one could improve telescopes used for
astronomical observation. 129 What remained to be done, was to test the
design.
We should remember that it was not an ordinary project Huygens had
embarked on. His theoretical investigations of spherical aberration served
the practical goal of improving actual telescopes. With this he marked
himself off from both theoreticians and practitioners. Unlike other telescope
makers – as he manifested himself earlier – he had aimed at improving the
telescope by means of theoretical study. The configuration in which
aberration was to be neutralized was not the result of trial-and-error like his
eyepiece, but of mathematical analysis of lenses and calculating the optimal
126
OC13, 411-413.
127
OC13, 417n2. “Lens e duabus composita hyperbolicam aemulatur, altera planoconvexa altera cava
utrimque. Semidiametri superficierum sunt proximè duo, quinque, decem.”
128
OC4, 354-355 and OC13, 417. The solution of the anagram is: “Lens e duabus composita hyperbolicam
aemulatur”.
129
Huygens may have tested the idea to combine two lenses into an objective earlier, at the time of the
invention of 1665. Hug29, 76v and 77r contain sketches reminiscent of the earlier invention as well as
ones reminiscent of the 1669 invetion. The folios can date from any time between the two inventions, but
appear to reflect some intermediate stage in his thinking.

82 CHAPTER 3
combination. He had made a blueprint, a design by which his perfect
telescope should be made, instead of designing one by first making it. Unlike
earlier theorists like Descartes, however, Huygens had not started from the
ideal situation but from the actual materials available to a telescope maker.
He worked halfway between the scholar and the craftsman in an
unprecedented effort to combine their respective theoretical and practical
goals.
Apparently the earlier, unsuccessful test of his first invention had not
shaken his confidence that spherical lenses could cancel out their mutual
aberrations. He had not changed his theory of spherical aberration, including
the values he used to approximate the amount of aberration produced by a
lens. Did Huygens expect that the composite lens would not produce those
disturbing colors? He may have thought that his new design was of a
different kind. As contrasted to the earlier one, it was not the ocular lens that
canceled out the aberration of the objective lens, but the aberration was
neutralized within the composite objective. This raises the question how
Huygens had hit upon the idea not to consider the configuration of a
complete telescope, but of a single lens system. It may have dawned upon
him when he was pointing out the flaws in Nulandt’s statements. It would
indeed be ironical that Huygens would have drawn inspiration for this
remarkable invention from a man he held so low.
Despite the triumphant EUPHKA, little is heard of the invention after
February 1669. On 26 June 1669 he wrote Oldenburg that he had been
working on lenses for a couple of weeks. He pointed out difficulties of
attaining truly spherical figures and of the glass available to him. 130 It is not
clear whether he was trying to execute his design or that he was working on
the 60-foot lenses mentioned in several letters of this period.
In the meantime appeals to publish his dioptrical studies were numerous.
On 18 March, Oldenburg warned him not to wait too long: “Sir, allow me to
urge you to be willing to finish your Dioptrique for fear that you will not be
preceded in this by someone else.” 131 At the end of October it was too late.
Barrow published his Lectiones XVIII and Oldenburg sent Huygens a copy
on November 21. 132 With the publication of Lectiones XVIII, Huygens lost
priority on a basic accomplishment of Tractatus: the application of the sine
law to spherical lenses. Barrow’s lessons were, as we have seen, of a different
nature than Tractatus. Barrow himself was aware of the differences. In a letter
to Collins, written on Easter Eve 1669, he wrote:
“… had I known M. Huygens had been printing his Optics, I should hardly have sent
my book. He is one that hath had considerations a long time upon that subject, and is
used to be very exact in what he does, and hath joined much experience with his
130
OC6, 460. In November the Royal Society decided to send Huygens a piece of the excellent glass made
in England. OC6, 533 and note 5.
131
OC6, 389. “Monsieur permettez moy de vous presser de vouloir acheuer vostre Dioptrique de peur que
vous n’y soyez prevenu de quelque autre.” He warned him again on April 8. OC6, 416.
132
OC6, 534.

1655-1672 - DE ABERRATIONE 83
speculations. What I have done is only what, in a small time, my thoughts did suggest,
and I never had opportunity of any experience.” 133
Barrow was too humble about his mathematical abilities but he was right in
observing that Huygens had more ‘experience’ in dioptrical matters. Huygens
praised Barrow in a letter to Oldenburg of 22 January 1670, but added “…
someday you will see that what I have written about it is completely
different.” 134 Yet, he did not hurry. The publication of Lectiones XVIII may
have pushed his plans to the background.
In February 1670 Huygens fell ill and he went to The Hague in
September, with an explicit ban by his physician to engage in intellectual
labor. In June 1671 he returned to Paris. Huygens’ dioptrics are not
mentioned among the manuscripts he entrusted to Vernon in February 1670,
when he feared the worst. 135 In Holland, he was with Constantijn again and
we may speculate that they also discussed dioptrical matters. In general,
Huygens wrote little about dioptrics in these years. He exchanged letters with
de Sluse on Alhacen’s problem, a mathematical problem regarding spherical
mirrors. 136 Much of his correspondence was taken up by a discussion about
the laws of collision he had sent to Oldenburg. No trace is found that
Huygens worked on executing the design of February 1669. Not long after
his return to Paris in June 1671, Huygens received a letter that would
eventually mean the end of his plans.
3.2.3 NEWTON’S OTHER LOOK AND HUYGENS’ RESPONSE
The invention of February 1669 is found on two places in Huygens’
manuscripts. One is his notebook of that period, the other is in the folder
also containing ‘Adversaria’ and seems to be the original calculation. 137 Both
contain the sketch of his invention and the ‘EUPHKA 1 feb. 1669’. In the last
one, however, the EUPHKA is crossed out and a ‘P.S.’ is added: “This
invention is useless as a result of the Newtonian aberration that produces
colors.” 138 Along with his invention, Huygens discarded all parts of De
Aberratione dealing with the improvement of telescopic images, namely his
earlier invention and his rules for the opening of keplerian telescopes. He
tore them from his manuscript and put them in a cover which said: “Rejecta
ex dioptricis nostris”. 139 The P.S. is dated October 25, without a year, but it is
likely to be 1672. 140 Evidently, this drastic decision was occasioned by the
preceding correspondence with Newton on colors.
133
Rigaud, Correspondence II, 70.
134
OC7, 2-3. “… vous verrez quelque jour que ce que j’en ey escrit est encore tout different.”
135
OC7, 7-13; especially 10-11.
136
Discussed in: Bruins, “Problema Alhaseni”.
137
Hug2, 72r and Hug29, 87r respectively.
138
OC13, 409n2. “Hoc inutile est inventum propter Abberationem Niutoniana quae colores inducit.”
139
OC13, 314n1.
140
The editors of the Oeuvres Complètes date it 1673, but in a conversation Alan Shapiro and I came to the
conclusion that it must have been 1672. I will return to this on page 92.

84 CHAPTER 3
In a letter of 11 January 1672, Oldenburg first made mention of Newton to
Huygens. 141 This ‘mathematics professor in Cambridge’ had invented a small
telescope in which the objective lens was replaced by a mirror. According to
Oldenburg it represented an object “without any color and very distinct in all
its parts.” 142 In his next letter of 25 January, Oldenburg sent him a drawing
and a detailed description, and asked Huygens’ opinion. 143 At the bottom of a
relatively wide tube a concave mirror reflected rays to a plano-convex ocular
lens via a small plane mirror. Huygens promptly sent Oldenburg his opinion
on the device. In the 81st issue of Philosophical Transactions (15 March, O.S.),
Oldenburg published Newton’s description of his reflector along with some
of Huygens’ comments. 144 In the meantime, Huygens had also sent a letter on
Newton’s telescope to Gallois, the editor of the Journal des Sçavans, who
published an extract of it in the issue of February 29. 145
Huygens spoke in the highest terms of Newton’s telescope. He
enumerated no less than four advantages over ordinary telescopes: a mirror
suffers less from spherical aberration, it does not ‘impede rays at the edge of
the glass due to the inclination of both surfaces’, there is no loss of light due
to internal reflections, and inhomogeneities in the material which affect
lenses play no part in mirrors. 146 Figure 34 The crossed out EUREKA.
In short, the reflector was a promising
device. The main obstacle for its success, already pointed out by Oldenburg,
was to find a durable material for making reflecting surfaces which lent itself
141
N.S. All dates are New Style unless indicated otherwise.
142
OC7, 124-125. “… qui envoye l’object à l’oeil, et l’y represente sans aucune couleur et fort
distinctement en toutes ses parties.”
143
OC7, 129-131.
144
OC7, 131 Huygens’ note a; 140-143.
145
OC7, 134-136.
146
OC7, 134-136 (to Gallois); 140-141 (to Oldenburg). In a note added to the description of Newton’s
reflector, Huygens calculated the difference of spherical aberration produced by a spherical lens and a
spherical mirror. The aberrations produced by a lens and a mirror with the same focal distance and
aperture are 28 to 3. Therefore, he concluded, the aperture of a mirror can be three times as large. OC7,
132.

1655-1672 - DE ABERRATIONE 85
to good polishing. 147 The second of the advantages Huygens listed is
interesting. Although he did not mention colors, it is clear that he referred to
the observation he had made in 1668. In his letter to Oldenburg he almost
literally repeated it:
“Besides, by [the mirror] he avoids an inconvenience, which is inseparable from convex
Object-Glasses, which is the Obliquity of both their surfaces, which vitiateth the
refraction of the rays that pass towards the sides of the glass, and does more hurt than
men are aware of.” 148
In his letter to Gallois, Huygens added that this defect could not be
prevented by a-spherical lenses. Evidently, he still was aware that his earlier
observation was of consequence to the use of lenses. Still, we have no idea
how he thought it would affect his invention of 1669.
Those disturbing colors would eventually induce Huygens to discard his
invention, but not until Newton had convinced him about his own ideas on
their cause. In his letter of 21 March, Oldenburg notified Huygens of a paper
by Newton in the 80th issue of Philosophical Transactions (19 February, O.S.):
“In this print you will find a new theory of Mr. Newton, (…) regarding light and colors:
in which he maintains that light is not a similar thing, but a mixture of differently
refrangible rays …” 149
It was, of course, the famous paper in which Newton set forth his ‘New
theory about Light and Colors’. According to Newton, rays of different
colors have a different degree of refrangibility: to each color belongs one,
immutable index of refraction. Moreover, he argued that white light is not
homogeneous but a mixture of all colors. Colors therefore are produced
when this mixture is separated, for example by refraction, into its
components. In ‘New theory’, Newton described his experiments with
prisms to substantiate his claim that color is an original and immutable
property of light rays which depends solely upon a specific index of
refraction.
Newton also explained why he had developed his reflecting telescope.
After he introduced his idea of different refrangibility, he wrote that it had
made him realize that colors could not be prevented in any lens and that
mirrors should be used instead. On the basis of the measurement of the
spectrum produced by one of his prisms, he calculated that the difference
between the refractions of the red and blue rays is about a 25th part of the
mean refraction. Consequently chromatic aberration is about a 50th part of
the opening of the lens and therefore considerable larger than the spherical
aberration produced by the same lens. 150 After this ‘digression’, Newton went
147 OC7, 134 (to Gallois); 141 (to Oldenburg). Oldenburg had pointed this out to Huygens in the letter
accompanying the description of Newton’s reflector: OC7, 128.
148 Oldenburg’s translation of OC7, 140 in: OldCor8, 520.
149 OC7, 156. “Dans cet imprimé vous trouverez une theorie nouvelle de Monsieur Newton, (…) touchant
la lumiere et les couleurs: ou il maintient, que la lumiere n’est pas une chose similaire, mais un meslange de
rayons refrangibles differemment …” The paper was therefore published in the issue preceding the one
containing the description of his reflector.
150 Newton, Correspondence I, 95.

86 CHAPTER 3
on to lay down his doctrine of the origin of colors in the form of 13
propositions substantiated by the experiments he had described. 151
On 9 April, Huygens gave a first reaction to Newton’s theory:
“… I see that he has noticed like me the defect of the refraction of convex objective
glasses caused by the inclination of their surfaces. As regards his new Theory of colors,
I consider it quite ingenious, but it will have to be seen whether it is compatible with all
experiences.” 152
Two things stand out in this comment. In the first place, Huygens was
mainly interested in the significance of Newton’s findings for dioptrics. In
the second place, he seemed to miss the point of the theory of different
refrangibility. 153 In his view, Newton had merely confirmed what he had
observed earlier. Seemingly, he did not realize that Newton’s point was that
chromatic aberration is a consequence of the constitution of light, rather
than the shapes of lenses. In his next letter to Oldenburg, of July 1, Huygens
went more deeply into the matter, though still along the same lines. After
discussing Newton’s telescope a bit further, he wrote:
“As regards his new hypothesis of colors of which you ask my opinion, I admit that it
seems very plausible to me, and the experimentum crucis (if I understand it correctly, as
it is described somewhat obscurely) confirms it very much. But I don’t agree with what
he says about the aberration of rays through convex glasses. For while reading what he
writes, I find that following his principles this aberration must be twice as large as he
1
takes it, to wit 25 the opening of the glass, which experience however seems to
contradict. so that this aberration may not always be proportional to the angle of
inclination of rays.” 154
We see what kind of ‘experiences’ Huygens had in mind when he cast doubt
on the validity of Newton’s theory: the colors he had seen in lenses. He did
not believe that chromatic aberration was as large as Newton claimed.
Consequently, Newton’s explanation of the aberration was questionable. But
it does not appear that Huygens had considered Newton’s theory of colors in
much detail. It seems that he had mainly read the part on lenses. He did not
use the term or notion of different refrangibility and only talked in terms of
aberrations.
151
Newton, Correspondence I, 96-100.
152
OC7, 165. “… je vois qu’il a remarquè comme moy le defaut de la refraction des verres convexes
objectifs a cause de l’inclination de leurs surfaces. Pour ce qui est de sa nouvelle Theorie des couleurs, elle
me paroit fort ingenieuse, mais il faudra veoir si elle est compatible avec toutes les experiences.”
153
See also: Sabra, Theories of Light, 268-267.
154
OC7, 186. “Pour ce qui est de sa nouvelle hypothese des couleurs dont vous souhaittez scavoir mon
sentiment, j’avoue que jusqu’icy elle me paroist tres vraysemblable, et l’experimentum crucis (si j’entens
bien, car il est ecrit un peu obscurement) la confirme beaucoup. Mais sur ce qu’il dit de l’abberration des
rayons a travers des verres convexes je ne suis pas de son avis. Car je trouvay en lisant son ecrit que cette
1
aberration suivant son principe devroit estre double de ce qu’il la fait, scavoir
25
de l’ouverture du verre, a
quoy pourtant l’experience semble repugner. de sorte que peut estre cette aberration n’est pas tousjours
proportionelle aux angles d’inclinaison des rayons.”

1655-1672 - DE ABERRATIONE 87
Newton realized that Huygens did not grasp the full import of his theory.
Reacting to Huygens’ first comment on his theory, he had written Oldenburg
on 13 April (O.S.):
“Monsieur Hugenius has very well observed the confusion of refractions neare the
edges of a Lens where its two superficies are inclined much like the planes of a Prism
whose refractions are in like manner confused. But it is not from ye inclination of those
superficies so much as from ye heterogeneity of light that that confusion is caused.” 155
This remark was not,
however, communicated
to Huygens. On July 8
(O.S.), Newton replied to
Huygens’ second
comment in a letter
Oldenburg forwarded to
Huygens on 28 July. 156 He acknowledged that the presentation of his theory
might have been obscure for reasons of brevity. Newton also realized that
Huygens had misread his discussion of chromatic aberration. “But I see
not,” he wrote, “why the Aberration of a Telescope should be more than
about 1/50 of ye Glasses aperture”. He included a drawing of the way he
had calculated the proportion (Figure 35):
“Now, since by my principles y e difference of Refraction of y e most difforme rayes is
about y e 24 th or 25 th part of their whole refraction, y e Angle GDH will be about a 25 th
part of y e angle MDH, and consequently the subtense GH (which is y e diameter of y e least
space, in to which y e refracted rays converge) will be about a 25 th of y e subtense MH,
and therefore a 49 th part of the whole line MN, y e diameter of y e Lens; or, in round
numbers, about a fiftieth part, as I asserted.” 157
The same letter was accompanied by a copy of the 84th issue of Philosophical
Transactions (17 June, O.S.). It contained a letter in which Pardies criticized
Newton’s theory and a reply by the latter. Two weeks later, Oldenburg sent
Huygens the next issue of Philosophical Transactions (15 July, O.S.) containing
further correspondence of Pardies and Newton on the matter. 158 Pardies, a
Jesuit priest and a Parisian acquaintance of Huygens, also criticized Newton’s
claims, but in a more searching manner and with a different line of approach.
He questioned the core of Newton’s theory – different refrangibility – and
raised several objections to his experiments and his interpretations thereof.
For example, he initially doubted whether the oblong spectrum could not be
explained by the accepted rules of refraction. 159 He also questioned the very
idea of different refrangibility, which in his view depended upon a
corpuscular conception of light. In his view, colors could also be caused by a
‘diffusion’ of light, for example by a slight spreading of the waves he
155
Newton, Correspondence I, 137.
156
Newton, Correspondence I, 212-213; OC7, 207-208.
157
OC7, 207-208.
158
OC7, 215.
159
Newton, Correspondence 1, 131-132.
Figure 35 Newton’s determination of chromatic aberration.

88 CHAPTER 3
supposed light to consist of. 160 In his two successive replies, Newton clarified
his experiments and his claims. He fully convinced Pardies of his claims and
the father ended the discussion by saying that he was ‘very satisfied’. 161
One expects that by now, late July, it must have dawned upon Huygens
what Newton’s new theory was about. Still, the letter he sent Oldenburg on
27 September does not give the impression that he really grasped the essence
of different refrangibility. 162 He regarded Newton’s replies as a further
confirmation of the theory, but added that things could still be otherwise. It
had, however, dawned upon him that Newton had also something to say
about the nature of light, to wit the compound nature of white light. To this
he raised objections of a different kind:
“Besides, if it were true that the rays of light were, from their origin, some red, some
blue etc., there would still remain the great difficulty of explaining by the physics,
mechanics wherein this diversity of colors consists.” 163
The remark was clearly inspired by the objections Pardies had made. It does
not give the impression that Huygens had given the matter any further
thought. Let it be noted that this was the first moment Huygens raised
objections of a mechanistic nature against Newton, after their discussion had
progressed in several letters, and that the objections are not brought out
strongly. He still did not refer to the notion of different refrangibility. As a
conclusion he admitted his misreading of Newton’s discussion of chromatic
aberration.
Huygens’ next letter to Oldenburg, four months later on 14 January 1673,
displayed a drastic change in his attitude towards Newton’s theory. Not only
did he show to have considered the claims about the nature of white light
and colors, he also subjected them to a serious critique. 164 In addition, the
tone of his comments became sharper. In his view Newton unnecessarily
complicated matters:
“I also do not see why Monsieur Newton does not content himself with the two colors
yellow and blue, because it will be much easier to find some hypothesis by motion that
explains these two differences, than for so many diversities as there are of other colors.
And until he has found this hypothesis he will not have taught us wherein the nature
and diversity of colors consists but only this accident (which certainly is very
considerable) of their different refrangibility.” 165
160
Newton, Correspondence 1, 157.
161
Newton, Correspondence 1, 205. “Je suis tres satisfait de la derniere réponse que M. Newton a bien voulu
faire à mes instances.”
162
Sabra, Theories of Light, 270.
163
OC7, 228-229. “De plus quand il seroyt vray que les rayons de lumiere, des leur origine, fussent les uns
rouges, les autres bleus &c. il resteroit encor la grande difficultè d’expliquer par la physique, mechanique
en quoy consiste cette diversitè de couleurs.”
164
OC7, 242-244.
165
OC7, 243. “Je ne vois pas aussi pourquoy Monsieur Newton ne se contente pas des 2 couleurs jaune et
bleu, car il sera bien plus aisè de trouver quelque hypothese par le mouvement qui explique ces deux
differences que non pas pour tant de diversitez qu’il y a d’autres couleurs. Et jusqu’a ce qu’il ait trouvè
cette hypothese il ne nous aura pas appris en quoy consiste la nature et difference des couleurs mais
seulement cet accident (qui assurement est fort considerable) de leur differente refrangibilitè.”

1655-1672 - DE ABERRATIONE 89
In his view, white light may also be produced by mixing yellow and blue
alone. By maintaining that there are only two primary colors, Huygens drew
upon a letter published in the 88th issue of Philosophical Transactions (18
November 1672, O.S.) in which Newton responded to comments by Hooke.
Among other things, Hooke had claimed that two primary colors sufficed to
explain the diversity of colors. Hooke’s comments had not been published
and his name was not mentioned in Newton’s reply. Huygens referred to
Hooke’s prism experiments in Micrographia (1665). He suggested an
experiment to verify whether all colors are necessary to produce white light.
Evidently, Huygens had problems with Newton’s claims about the nature of
light. What these problems were exactly, why he would prefer just two
colors, and what he meant by ‘explaining by physics, mechanics’ and ‘some
hypothesis of motion’ is not explained in the letter. In chapter 6 we will be
able to reconstruct, in retrospect, the background to Huygens’ remarks. He
had a reasonably clear idea what mechanistic explanation ought to be, but it
appears that by 1672 he had not yet elaborated in much detail his conception
of the mechanistic nature of light.
Besides raising objections to Newton’s ideas on the nature of light and
colors, Huygens summed up his own treatment of chromatic aberration:
“Apart from that, as regards the effect of the different refractions of rays in telescope
glasses, it is certain that experience does not correspond with what Monsieur Newton
finds, because by considering only the distinct picture that an objective of 12 feet makes
in a dark room, one sees that it is too distinct and too sharp to be able to be produced
by rays that disperse from the 50th part of the aperture so that, as I believe to have
brought to your attention before, the difference of the refrangibility may not always
have the same proportion in the large and small inclinations of the rays on the surfaces
of the glass.” 166
There is no reason to assume that Huygens had not actually performed this
test. 167 We may only wonder why he had not done so in 1665. Then again, the
alleged observations remained qualitative. We may wonder what had
prompted Huygens to consider the issue of chromatic aberration anew.
Assuming that the full import of Newton’s theory had occurred to him as a
result of Pardies’ comments, he may have realized at some time by late 1672
that chromatic aberration was a problem of refraction and thus inherent to
lenses. As Newton emphasized in his reply to Hooke: “And for Dioptrique
Telescopes I told you that the difficulty consisted not in the figure of the
166 OC7, 243-244. “Au reste pour ce qui est de l’effect des differentes refractions des rayons dans les verres
de lunettes, il est certaine que l’experience ne s’accorde pas avec ce que trouve Monsieur Newton, car a
considerer seulement la peinture distincte que fait un objectif de 12 pieds dans une chambre obscure, l’on
voit qu’elle est trop distincte et trop bien terminée pour pouvoir estre produite par des rayons qui
s’escarteroient de la 50 me partie de l’ouverture de sorte que, comme je vous crois avoir mandè desia cy
devant la difference de la refrangibilité ne suit pas peut estre tousjours de la mesme proportion dans les
grandes et petites inclinations des rayons sur les surfaces du verre.”
167 Because he was a ‘devoted water-color painter’, Shapiro is puzzled about Huygens’ assertion that
yellow and blue may produce white, “… because this is contrary to all beliefs about color mixing held in
the seventeenth century.” Shapiro, “Evolving structure”, 223-224. We should bear in mind that Huygens
was also an experienced employer of magic lanterns.

90 CHAPTER 3
glasse but in ye difformity of refractions.” 168 Despite his doubts about the
true properties of different refrangibility, Huygens now recognized that the
disturbing colors in lenses are inherent to refraction. There is no word about
spherical aberration in his letter, and he may indeed already have realized at
this point that his project to design configurations to neutralize it had
become useless. Would this help account for the sharpening in his tone?
Newton partially granted Huygens both objections. He dropped the claim
that all colors are necessary to compound white light restricting it now to
sunlight. 169 In his reply of April 3 (O.S.), he strongly objected to Huygens’
claim that two primary colors are more easily explained, but he explicitly
refrained from proposing a ‘Mechanicall Hypothesis”. 170 As regards the actual
effect of chromatic aberration, he watered down his claim a bit. The rays that
are dispersed mostly
“… are but few in comparison to those, which are refracted more Justly; for, the rays
which fall on the middle parts of the Glass, are refracted with sufficient exactness, as
also are those that fall near the perimeter and have a mean degree of Refrangibility; So that
there remain only the rays, wich fall near the perimeter and are most or least refrangible
to cause any sensible confusion in the Picture. And these are yet so much further
weaken’d by the greater space, through which they are scatter’d, that the Light which
falls on the due point, is infinitely more dense than that which falls on any point about
it. …” 171
As a conclusion, Newton suggested a way to measure the chromatic
aberration of the extreme rays to verify his claims. Huygens accepted
Newton’s argument, but added that
“… he must also acknowledge that this abstraction [dispersion] of rays does not
therefore harm lenses as much as he seems to have wished to be believed, when he
proposed concave mirrors as the only hope for perfecting telescopes.” 172
Huygens was not, however, satisfied with Newton’s rebuttal concerning the
nature of white light and colors: “… but seeing that he maintains his opinion
with so much ardor, this deprives me of the appetite for disputing.” 173 Two
weeks later, he wrote to Oldenburg not to send Newton his last letter at all
and to tell him only that he did not want to dispute anymore. 174 Newton did
receive the letter anyhow and replied on 23 June (O.S.) by a more precise
reformulation of his theory, which was published in the 96th issue of
168
Newton, Correspondence I, 173.
169
Shapiro, “Evolving structure”, 224-225.
170
OC7, 265-266 and Newton, Correspondence I, 264-265.
171
OC7, 267 and Newton, Correspondence I, 266. In Opticks, he elaborated this argument a bit further and
mathematically, and reduced chromatic aberration to 1/250 of the aperture as contrasted to the original
1/50. Newton, Optical lectures, 429n15.
172
OC7, 302-303. “… mais aussi doit il avouer que cette abstraction des rayons ne nuit donc pas tant aux
verres qu’il semble avoir voulu faire accroire, lors qu’il a proposè les mirroirs concaves comme la seule
esperance de perfectionner les telescopes.”
173
OC7, 302. “…, mais voyant qu’il soustient son opinion avec tant de chaleur cela m’oste l’envie de
disputer.”
174
OC7, 315.

1655-1672 - DE ABERRATIONE 91
Philosophical Transactions (21 July O.S.). 175 Newton invited him once again to
compare by computation aberrations both of lenses and mirrors, but
Huygens did not respond anymore.
Thus came an end to a dispute that had run an odd course. In January
1672 Huygens had welcomed the newcomer on the scene of European
scholarship as a kindred spirit in matters dioptrical; in June 1673 he refrained
from discussing any further with someone who so obstinately clung to his
claims. But most striking about the state of affairs I find the relative late
moment at which Huygens recognized the purport of Newton’s paper. Until
the letter of September 1672, the fact that Newton’s theory concerned the
physical nature of light escaped him. And then again, he made only one –
apparently non-committal – objection. Only in the letter of January 1673 did
he engage in a dispute on Newton’s theory of colors, to break it off in the
next letter. Until the letters of Pardies were published, Huygens only paid
attention to what Newton had said about the aberrations of lenses. And even
at this point, he failed to grasp Newton’s message. He only talked of
chromatic aberration in the same terms as he had treated spherical
aberration. One gets a strong impression that in 1672 Huygens lacked a
certain sensibility for the kind of question Newton addressed, namely
concerning the physical nature of light. This is all the more surprising since
Huygens has become famous for a theory explaining the nature of light of
his own.
The preceding reconstruction sheds new light on this famous dispute.
Huygens was not a Cartesian that a priori rejected Newton’s theory for
reasons of its mechanistic inadequacy and untenability, like Hooke did and
Pardies too initially, and like he is usually presented in historical literature. 176
We should reconsider the his dispute from the perspective of Dioptrica.
Huygens entered the dispute from his background in dioptrics. He was
interested (and informed) in lenses and telescopes and he had something to
loose. At first he did not look beyond issues directly pertaining to lenses and
it took some time before he realized what Newton’s theory was about. He
began to raise mechanistic doubts only during the final stages of the dispute,
and probably when he realized the consequence for his project of nullifying
spherical aberration. For in anything may explain Huygens relative reluctance
in accepting Newton’s theory, it would be De Aberratione.
Somewhere along the line, Huygens must have realized that Newton’s
findings wrecked his project of perfecting the telescope. He crossed out the
‘Eureka’ of February 1669 and discarded a large part of his theory of
spherical aberration. ‘Newtonian’ aberration had rendered his designs
useless. Spherical aberration might be cancelled out by successive lenses,
chromatic aberration could never be prevented. Despite the objections he
raised in it, the letter of January 1673 reveals that Huygens had recognized
175
OC7, 328-333 and Newton, Correspondence I, 291-295. See also Shapiro, “Evolving structure”, 225-228.
176
For example Sabra, Theories of Light, 268-272.

92 CHAPTER 3
different refrangibility. I find it reasonable to presume that this had
happened at some time during the preceding months. He demonstrably had
contacts with Pardies, who had accepted the crux of different refrangibility
and might have pointed it out to Huygens. Moreover, at that time Mariotte
carried out experiments that corroborated Newton’s results. 177 I find it
therefore likely that Huygens took the drastic decision to discard his project
on 25 October 1672, rather than 1673 as the editors of the Oeuvres Complètes
have it. 178 I find it unlikely that Huygens would have taken the step when the
whole event had long passed. The only fact supporting this interpretation is
the publication of Newton’s and Huygens’ letters of 3 April and 10 June
respectively in the issue of Philosophical Transactions of 6 October 1673 (O.S.).
Although Huygens had long known their content, further reflection upon
the dispute might conceivably have triggered the decision to discard the main
part of De Aberratione.
3.3 Dioptrica in the context of Huygens’ mathematical science
The drastic decision of October 1672 brought Huygens’ study of lenses to a
temporary end. He was not to resume it, adjusting his theory of aberration
by taking ‘Newtonian’ aberration into account, until his return to Holland in
the 1680s. 179 This would not change the character of his dioptrical studies as
we have come to know it in these two chapters. Although he had lost the
ambition to design a perfect telescope, the orientation on the telescope
guided his dioptrical studies. He conducted his dioptrical studies in order to
understand the instrument. With a Huygens one tends, however, to overlook
the obvious. For him, understanding something meant mathematically
understanding something. Together with his orientation on the instrument,
his mathematical approach is the clue to Dioptrica.
In addition to this scholarly contemplation, Huygens had applied himself
to the craft of telescope making. De Aberratione can be seen as an effort to
combine his dual capacities as a scholar and a craftsman. In this sense, it
should have been the climax of his involvement with the telescope. It turned
into an anti-climax. In this concluding section on Dioptrica, I first go through
the nature of Huygens’ mathematical approach and its consequences for De
Aberratione. Then I consider his orientation on the telescope in the broader
context of our understanding of Huygens’ science.
3.3.1 THE MATHEMATICS OF THINGS
Ignoring for a moment the particular aims of Huygens’ dioptrical studies, we
may notice that De Aberratione has all the features of a geometrical treatise. It
is structured as a set of propositions and definitions regarding spherical
aberration. De Aberratione is not a geometrical treatise by appearance only. In
177 Shapiro, “Gradual acceptance”, 78-80.
178 Additional evidence for this dating I find in the fact that Pardies and Huygens discussed Iceland Crystal
in the summer of 1672. See further footnote 120 on page 140 below.
179 See section 6.1.

1655-1672 - DE ABERRATIONE 93
its elaboration it consisted of a geometrical derivation of the properties of
spherical aberration. Like Tractatus, it rested on little more than the sine law
and a generous dose of Euclidean geometry.
In the elaboration of the theory of spherical aberration, geometry had the
upper hand. This stands out clearest in the simplifications Huygens
employed. He used a simplified expression in order to determine the amount
of aberration produced by a particular lens. He justified this by comparing
the calculated differences between both expressions. What effects such
differences would have in actual lenses, he did not tell. Nowhere in De
Aberratione does Huygens give an indication that he had considered the
question how the calculated properties of spherical aberration related to its
observed properties. A modern reader would expect otherwise, but Huygens
went about by geometrical deduction exclusively. This geometrical analysis
resulted in a sophisticated theory of spherical aberration in which complex
problems were solved of neutralizing it by configuring spherical lenses
properly.
But Huygens’ goal was not mere theory, he aimed at its practical
application to real lenses and telescopes. This marked him off from his
fellow dioptricians. Had he not applied his theory to design better telescopes
and tested his design, he would not have been confronted with those
disturbing colors. The fact that Huygens was taken by surprise by those
disturbing colors need not surprise us. In his dioptrical study of lenses,
Huygens confined himself to their mathematical properties and excluded the
consideration of colors. Likewise, in his study of halos and parhelia, written
around 1663, he confined himself to tracing the paths of rays of light
through transparent particles in the atmosphere and left out any
consideration of the colors of these phenomena. 180 Colors eluded the laws of
geometry, so he wrote there with even greater conviction than in Tractatus:
“However, to investigate the cause of these colors further; to know why they are
generated in a prism, I want to undertake by no means, I admit on the contrary not to
know the cause at all, and I think that no one will comprehend their nature easily for as
long as some major light will not have enlightened the science of natural things.” 181
That major light had come, it was named Newton, and it had eclipsed
Huygens’ grand project of perfecting telescopes.
Huygens was well acquainted with the disturbing colors produced by
lenses. Dealing with them was, in his view, a matter of trial-and-error
configuring of lenses instead of purposive calculation. When colors came to
disturb the predicted optimal working of his design, he did not do anything
with them. Despite the importance of colors for his project, Huygens did not
elaborate upon his observation that colors might be related to the angle of
180 With connected reproduced in OC17, 364-516. On the dating see OC17, 359.
181 OC17, 373. “Doch de reden van dese couleuren verder te ondersoecken, te weten waerom die in een
prisma gegenereert worden, wil ick geensins ondernemen, emo fateor rationem eorum me prorsus
ignorare, neque facile quemquam ipsas perspecturum arbitror quandiu naturalium rerum scientiae major
aliqua lux non affulserit.”

94 CHAPTER 3
incidence of a ray of light. He did not adjust his theory of spherical
aberration, nor the way he intended to counter its effects in telescopes.
Apparently, he saw no possibility to extend dioptrics to the properties of
colors. Colors kept eluding his mathematical understanding. In other words,
he did not take the step to leave the established domain of mathematical
optics. This should not be strongly counted against him, for no-one in the
seventeenth century did so. Except for Newton, who had an extraordinary
scholarly disposition that combined a mathematical outlook with an interest
in material things fostered by experimental philosophy and the new natural
philosophies in general.
Newton did see geometry in colors, but he looked at them from an
entirely different perspective. His studies of prismatic colors had begun
around 1665 with an experiment described by Boyle – with a thread that was
half blue and half red and appeared broken when seen through a prism. 182
Unlike Boyle, he interpreted this in terms of the refraction of rays of light.
He realized that the rays coming from both parts of the thread are refracted
at different angles. In other words, Newton interpreted the phenomenon in
the geometrical terms of rays and angles. On this basis he began his
prismatic experiments, deliberately studying the differences of the angles
with which rays of various colors are refracted. Unlike Descartes, Boyle and
Hooke before him, he tried to make the spectrum as large as possible, by
projecting it as far as possible. 183 He passed the beam of light at minimum
deviation, so that the effect of the width of the beam was minimized. By
turning the prism into a precision instrument, Newton discovered that it was
the principles of geometrical optics that were violated by the spectrum. The
solution of the anomaly consisted of linking ‘color’ with ‘refractive index’
and thus with the sine law of refraction. Different refrangibility reduced
colors to the laws of geometry.
It was not only the mere recognition of geometry that led to different
refrangibility. In order to establish the laws to which colors were subject,
Newton employed experiment in a new way. Combining mathematical
thinking with a heuristic use of experiment, he developed the new
methodological means of quantitative experiment. By measuring the
phenomena produced in his prisms he was able to discover geometrical
properties where previously there had been none.
“But since I observe that geometers have hitherto erred with respect to a certain
property of light concerning its refractions, while they implicitly assume in their
demonstrations a certain not well established physical hypothesis, I judge it will not be
unappreciated if I subject the principles of this science to a rather strict examination,
adding what I have conceived concerning them and confirmed by numerous
experiments to what my reverend predecessor last delivered in this place.” 184
182
Newton, Certain philosophical questions, 467.
183
Westfall, Never at rest, 163-164.
184
Newton, Optical papers 1, 47 & 281.

1655-1672 - DE ABERRATIONE 95
Newton considered his discovery of different refrangibility an addition to
geometrical optics. A necessary addition because it explained “… how much
the perfection of dioptrics is impeded by this property and how that
obstacle, insofar as its nature allows, may be avoided.” 185 These lines could
have been addressed to Huygens personally, had Newton known of De
Aberratione. Newton was aware that he was breaking new ground. In the
letter he sent to Oldenburg he wrote:
“A naturalist would scearce expect to see ye science of [colours] become mathematicall,
& yet I dare affirm that there is as much certainty in it as in any other part of
Opticks.” 186
These lines were, however, omitted when his ‘New theory’ appeared in
Philosophical Transactions. With his theory Newton went beyond the recognized
boundaries of geometrical optics by extending it to the study of colored rays.
Huygens, on the other hand, stayed within the established domain of
optical phenomena to be studied mathematically. He elaborated his dioptrical
theories in the manner customary in geometrical optics. As a mathematical
theory, the content of Dioptrica did not deviate in any principal way from the
doctrines found in Aguilón or Barrow. In Paralipomena physical
considerations – albeit within the traditional domain of mathematical optics
– were much more integrated in mathematics, but Huygens did not follow
this line of Kepler at this moment. 187 As a topic of mixed mathematics,
geometrical optics was principally a matter of geometrical deduction. The
difference with ‘pure’ geometry was that lines and circles represented
physical objects like rays, reflecting and refracting surfaces. Geometrical
inference was preconditioned by a specific set of postulates: the laws of
optics describing the behavior of unimpeded, reflected and refracted rays.
Or, as Huygens would state it in Traité de la Lumière, optics is a science
“where geometry is applied to matter.” 188
Huygens ‘géomètre’
Thus Huygens treated spherical aberration as a geometrical problem which
ought to be solved by mathematical analysis. Despite the vital importance of
colors for his project, he did not go beyond the traditional boundaries of
mixed mathematics in order to tackle the problem. He confined his
investigation to effects known to be reducible to the laws of geometry.
Geometrical optics did not provide the means to deal with colors, so he left
them to the craftsman. In this sense Huygens’ Dioptrica fits in with his
mathematical science in general. In his studies of circular motion and
consonance he also focused on exploring their mathematical properties on
the basis of established (mathematical) principles.
185
Newton, Optical papers 1, 49 & 283.
186
Newton, Correspondence 1, 96.
187
See section 4.1.2.
188
Traité, 1. “… toutes les sciences où la Geometrie est appliquée à la matiere, …”

96 CHAPTER 3
During the final weeks of 1659, Huygens took up and solved a problem
that Mersenne had discussed 12 years earlier in Reflexiones physico-mathematicae
(1647). The problem was to determine the distance traversed by a body in its
first second of free fall, which amounts to determining half the value of the
constant of gravitational acceleration. After having tried Mersenne’s
experimental approach, Huygens abandoned it in favor of a theoretical
consideration of gravitational acceleration. He began a study of circular
motion which in his view was closely connected to gravity: “The weight of a
body is the same as the conatus of matter, equal to it and moved very swiftly,
to recede from a center.” 189 Circular motion had been discussed by both
Descartes and Galileo, but only in qualitative and fairly rough terms. 190
Huygens set out to analyze circular motion mathematically. He derived an
expression for the tension on a chord exerted by a body moving in a circle,
by equating it with the tension exerted by the weight of the body. 191 He then
considered the situation in which a body revolves on a chord in such a way
that a stable situation is created and centrifugal and gravitational tension are
counterbalanced. With the conical pendulum thus procured and reversing his
calculations, Huygens found an improved value for gravitational acceleration
and dismissed Mersenne’s original experiment. 192 Analyzing the experiment
mathematically and comparing the time of vertical fall to the time of fall
along an arc, he derived a theory of pendulum motion eventually resulting in
the discovery of the isochronity of the cycloid. 193
The aim of Huygens’ studies of curvilinear fall and circular motion was to
render these motions with the same exactness Galileo had achieved with free
fall. 194 In the case of curvilinear fall this meant to solve the tricky
mathematical problem of relating the times with which curved and straight
paths are traversed. In the case of circular motion, he quantitatively
compared centrifugal and gravitational acceleration. Huygens’ success came
from his proficiency in using infinitesimal analysis and his control of
geometrical reasoning. 195 He conceptualized the forces he was studying in a
way that could be geometrically represented, which in his view meant to treat
free fall and centrifugal force in terms of velocities. 196 He considered, for
example, gravity as mere weight, and acceleration as continuous alteration of
inertial motion. 197 In other words, rather than mathematizing these
189
Yoder, Unrolling time, 16-17.
190
Yoder, Unrolling time, 33-34.
191
Yoder, Unrolling time, 19-23. This expression for centrifugal tendency amounts to the modern formula:
F = mv2/r. 192
Yoder, Unrolling time, 27-32.
193
Yoder, Unrolling time, 48-59.
194
The first draft of De vi centrifuga opened with a quotation of Horace: “Freely I stepped into the void, the
first”, above his discovery of the isochronicity of the cycloid he wrote: “Great matters not investigated by
the men of genius among our forefathers; Yoder, Unrolling time, 42 and 61.
195
Yoder, Unrolling time, 62-64.
196
The same goes for his earlier study of impact, to be discussed in section 4.2.2.
197 Westfall, Force, 160-165.

1655-1672 - DE ABERRATIONE 97
phenomena, he reduced them to concepts already mathematized. To be
more specific: Huygens reduced these dynamical phenomena to the
kinematic groundwork laid by Galileo.
Yoder has pointed out Huygens’ talent for transferring physics to
geometry. 198 His proficiency in idealizing phenomena enabled him to
mathematize not only the abstract objects of mechanics but also concrete
bobs and cords. Once transformed into a geometrical picture, Huygens
could apply his geometrical skills. Just as in his study of spherical aberration,
the kind of experimentation by which Newton had mathematized colors was
absent from Huygens’ studies of circular motion. He was surely a careful
observer and capable of designing clever experiments as an independent
means to test theoretical conclusions. 199 Yet, the precision he achieved in
measuring the constant of gravitational acceleration was made possible by his
mathematical understanding of the matter. Exploring mathematical
properties of a phenomenon empirically was not the way he approached his
objects of study. On the contrary, he readily dismissed Mersenne’s
experiment as indecisive, aware of the imprecision and bias of observation. 200
He approached his subject first of all theoretically, interpreting concepts
geometrically and analyzing phenomena by means of his mathematical
mastery.
In his dioptrical studies, Huygens had likewise relied on his geometrical
proficiency. His theory of spherical aberration was the outcome of rigorous,
sometimes clever deduction. At the point he could have broken really new
ground – when colors emerged – Huygens halted. The process of
geometrizing new phenomena that had proven to be so fruitful in his study
of motion did not get going in dioptrics. Seemingly, he did not see
possibilities to transform those disturbing colors into a geometrical picture,
despite some promising observations he had made of them. However, we
should bear in mind that motion, as contrasted to colors, had already been
mathematized. In his geometrization of circular motion, Huygens could
build on the groundwork laid by Galileo.
Compared to his study of circular motion, De Aberratione was rather
straightforward geometrical reasoning. In this regard, it comes closer to his
study of consonance that occupied him, on and off, from 1661 onwards. 201
The first problem Huygens attacked was the order of consonance, an issue
that had arisen (anew) with the new theories of music of the sixteenth and
seventeenth centuries. In the theory of consonance Huygens adopted, the
coincidence theory of Mersenne and Galileo, the order of consonants was
not evident. He derived a clever rule that only left one problem. His rule
7 seemed to imply that 4 should be placed between the major third and the
198 Yoder, Unrolling time, 171-173.
199 Yoder, Unrolling time, 31-32.
200 Yoder, Unrolling time, 170-171.
201 Cohen, Quantifying music, 209-230 and Cohen, “Huygens and consonance”, 271-301.

98 CHAPTER 3
fourth which implied that “… the number 7, …, is not incapable of
producing consonance …”, a conclusion that ran in the face of all previous
musical theory. 202 At that time – around 1661 – Huygens decided not to
accept the consonance of intervals with 7 because they had no regular place
in the scale.
Next, Huygens addressed a problem in tuning. When keyboards are tuned
according to then customary mean tone temperament, the question was how
the fifths employed ought to be adjusted with respect to pure fifths. 203 In
order to determine a mathematical solution, Huygens started by deriving the
ratios of all twelve tones in terms of the string lengths of the octave and the
fifth. In the course of his investigation, Huygens found a new property of
mean tone temperament. It concerned the quantitative difference between
the diatonic and the chromatic semitones. 204 Calculating the ratio of both
kinds of semitones, he concluded that C-D can be divided into 5 equal parts
and, consequently, the octave into 31 equal parts. Thus Huygens arrived at
the 31-tone division of the octave he had found discussed by Mersenne and
Salinas. In a letter published 30 years later in Histoire des Ouvrages des Sçavans
(October 1691), Huygens elaborately explained how he calculated the various
string lengths and pointed out advantages of his 31-tone division. 205 The
paper did not contain a further consequence Huygens had drawn in his
private notes: the consonance of intervals based on the number 7. Thus,
Huygens’ ‘most original contribution to the science of music’ remained
unknown to the world until this century. 206
Huygens’ studies of consonance show, once more, his dexterity in
exploring and elaborating the mathematics of a topic. He added rigor and
precision to Mersenne’s science of music, using Galileo’s approach and
202 OC20, 37. Translation: Cohen, Quantifying music, 214.
203 The tones of the octave are found using the consonances; this is called the division of the octave. The
3
seven tones of the diatonic scale are found by means of the fifth (
2
) and its complement, the fourth
4
(
7
). Likewise the chromatic tones are found by addition of fifths. A problem arises, however, because a
complete octave cannot be reached again by continuous addition of fifths. A small difference, called the
3 12
2 7
Pythagorean comma, exists between 12 fifths ( 2)
and 7 octaves ( 1)
. As a result, the tones of the
octave ought to be tempered in musical practice, which means that the purity of some consonances is
sacrificed. In mean tone temperament most major thirds are pure and the fifths are made a bit too large;
in equal temperament all consonances save the octave are a bit impure. Huygens preferred the former, the
latter has become standard tuning in Western music since the early nineteenth century.
204 The diatonic semitone is the difference between E and F, B and c, etc.; the chromatic semitone is the
difference between, for example, C and C . The chromatically sharpened C and flattened D – C # and D b –
differ, whereby C-D b and C #-D have the size of a diatonic semitone. The difference between C-C # and C-
D b is the difference between both kinds of semitones.
205 Most of Huygens’ musical studies is reproduced in OC20, 1-173. The French and Latin versions of the
letter have been reprinted with Dutch and English translations by Rasch in: Huygens, Le cycle harmonique.
206 Cohen, Quantifying music, 225-226.

1655-1672 - DE ABERRATIONE 99
extending it to problems the latter had ignored. 207 As with his studies of
dioptrics and circular motion, Huygens’ study of consonance did not develop
in an empirical vacuum. He rejected Stevin’s theory, as purely mathematical
and ignoring the demands of sense perception. But he also rejected systems
that lacked theoretical foundation. 208 His aim was to develop a sound
mathematical theory that explained and founded his musical preferences.
Mean tone temperament therefore was his natural starting point, and the 31tone
division seems a natural outcome of his analysis of its mathematical
properties as it conformed to both his theoretical and practical preferences.
Like his studies of circular motion and consonance, Huygens’ study of
spherical aberration, and this is almost a truism, was predominantly
mathematical. Huygens fruitfully explored and rigorously examined
mathematical theory. More revealing in the context of the present study is
the relationship between mathematics and observation. Huygens was not
blind for the empirical facts. On the contrary, they constituted the main
directive of his investigation in such diverse ways as the measure of gravity,
pleasing temperament and workable lens-shapes. Huygens knew how to
check his theoretical conclusions empirically and he was not easily satisfied.
Exploratory observation of phenomena was not the way Huygens
approached a subject. In modern terms: he did not employ experiment
heuristically. In the case of gravity, he had soon found out that mere
observation did not yield reliable knowledge. The result proved him right:
the analysis of the mathematical properties of circular motion gave him a
better theory as well as a better means of measurement. Huygens successfully
extended the Galilean, mathematical approach to gravity and circular motion.
Newton likewise was a mathematician with a Galilean spirit, but in his
study of colors he linked it with the experimental approach of Baconianism.
Although he was favorably disposed to Bacon’s program for the organization
of science (see below), Huygens did not regard the experimental collecting of
data as a source for new theories, let alone a trustworthy basis for
mathematical derivation. He explored the underlying mathematical structure
of a phenomenon the results of which could be verified to see whether the
supposed structure was real. In the case of consonance, the empirical
foundation of the theory had already been established. In the case of
spherical aberration, however, such preliminary work had not yet been done,
unfortunately. It turned out that not all effects of lenses depended upon the
known mathematical properties of lenses.
Suppose he had pursued his idea that colors were related to the
inclination of the sides of a lens. He might have taken some objective lenses,
covered their center (instead of their circumference as was customary) and
207 Cohen, Quantifying music, 209. It should be noted that, unlike his predecessors, Huygens possessed
logarithms and was therefore readily able to calculate, for example, a 4 1
5
.
208
Cohen, “Huygens and consonance”, 293-294.

100 CHAPTER 3
see how the different inclinations affected the generation of colors. He might
even have taken a prism to study the effect of twofold refraction on colors.
He might even have measured the angles of the inclination and – even more
speculative – made some measurements on the colors themselves. He did
not, and left colors aside in De Aberratione. In short, he recognized the
importance of the colors displayed by his lenses, but did not know what to
do about them. Which amounts to saying that he did not know what to do
about them mathematically.
3.3.2 HUYGENS THE SCHOLAR &HUYGENS THE CRAFTSMAN
Which brings us back to what Huygens’ study of spherical aberration was all
about: the improvement of telescopes. From the viewpoint of dioptrics,
nothing was wrong with his theory of spherical aberration. It described the
properties, derived from the principles of dioptrics, of light rays when
refracted by spherical surfaces. From the viewpoint of Huygens’ project
there was, however, a serious problem. He did not develop his theory in
order merely to extend his dioptrical knowledge, but to find an improved
configuration of lenses. Huygens’ theory of spherical aberration could not
take colors into account – let alone explain how to minimize their disturbing
effects. From the viewpoint of dioptrical theory, colors were a further effect
yet to be understood; from the viewpoint of De Aberratione they were a fatal
blow. Without the practical goal of De Aberratione, Huygens probably would
never have run across the disturbing colors that spherical lenses also
produced.
I have amply argued that the orientation of Dioptrica on the telescope
marked off Huygens’ dioptrical studies from those of most other
seventeenth-century scholars. He was one of the very few who tried to
acquire a theoretical understanding of the telescope and, in addition, he
wanted to improve the instrument on this basis. That is not necessarily to say
that this practical orientation is characteristic of Huygens’ science in general.
Although applications of theory to instruments were never far from his
mind, his studies of consonance and circular motion were not guided by an
orientation on instruments as his studies of dioptrics were.
The problem of tuning keyboard instruments was important for Huygens’
musical studies but their main goal was the mathematical theory of
consonance. Having elaborated his 31-tone division, he readily saw the
practical application in the guise of a suitable organ, on which one could
switch easily between keys in mean tone temperament. Likewise, his study of
circular motion was aimed at a physical problem (measuring gravity) and
took the form of a thorough, mathematical analysis of circular motion in
many of its manifestations. It was not a analysis of the clock he had invented
earlier, nor did the question which pendulum would be isochronous guide
it. 209 Still, practical thinking of a kind was inherent in Huygens’ study of
209 Yoder, Unrolling time, 71-73.

1655-1672 - DE ABERRATIONE 101
circular motion. He often couched his thoughts on circular motion in some
mechanical form. And he designed several clocks that embodied his
theoretical findings. As regards his original pendulum clock he reaped the
rewards of his study by equipping it with cheeks that gave its bob an
isochronous path.
If instruments did not guide Huygens’ other studies the way they did in
dioptrics, his approach to them was nevertheless similar. Horologium
Oscillatorium of 1673 does not just describe the pendulum clock and the ideal
cycloidal path, but also gives the mathematical theory of motion embodied in
it. Going beyond the mere necessities of explaining its mechanical working –
as in Dioptrica – he elaborated his theories of circular motion, evolutes and
physical pendulums. Of the achievements of 1659, Horologium Oscillatorium
included the study of curvilinear fall and cycloidal motion, transformed into
a direct and refined derivation, but it listed only the resulting propositions of
his study of circular motion and the conical clock. In addition, it contained a
discussion of physical pendulums. Huygens imaginatively applied the insight
that a system of bodies can be considered as a single body concentrated in
the center of gravity, to a physical pendulum considered to be resolved into
its constituent parts independently. With this he could express the motion of
the pendulum by means of the accelerated motion of its parts. Next he
compared the physical pendulum to an isochronous simple pendulum,
deriving an expression for the length of the latter in terms of the length and
the weights of the parts of the former. 210
His organ likewise rested on an sound and even elegant theory of
consonance. In this way he showed the solid theoretical basis on which his
inventions rested, showing at the same time that he was not a mere
empiricist but a learned inventor. 211 De Aberratione stands out among
Huygens’ studies in that he developed theory with the explicit aim of
improving an instrument. Earlier, he had proven the working of his eyepiece
on a mathematical basis, but he had not been able to demonstrate that it was
the best configuration possible. In De Aberratione Huygens set out to design a
configuration of lenses that he could prove mathematically was the best one
possible.
Huygens was not unique for trying to solve a practical problem by means
of theory. Descartes’ a-spherical lenses were meant to serve as a solution to
the same problem Huygens attacked. Descartes had tried to realize his design
by thinking up a device fit for making those lenses. Examples from other
fields can be found without much effort; the problem of finding longitude at
sea is only the first to come to mind. The seventeenth century is pervaded by
scholars who believed theory could or should be of practical use. The special
thing about De Aberratione is the way Huygens set out to solve the problem
of spherical aberration. His starting point consisted of the mathematical
210 Westfall, Force, 165-167.
211 Cohen, Quantifying music, 224.

102 CHAPTER 3
theory of spherical lenses he had developed earlier. As contrasted to
Descartes and others, his design for a better – or even perfect – telescope
did not start out with the ideal lenses of geometry, but with the ‘poor’ lenses
of actual telescopes. He did not avoid or explain away the defects of
spherical lenses, like Descartes or Hudde. He analyzed these defects in order
to take them into account and eventually correct them. Huygens’ design of a
perfect telescope was not based on the theoretically desirable, but on the
practically feasible.
Although craftsmanship preconditioned De Aberratione, Huygens did not
go the craftsman’s way as in his earlier inventions. He wanted to derive a
blueprint for an improved configuration on the basis of his theoretical
understanding of lenses. Instead of tinkering with lenses, he would be
tinkering with mathematics. He replaced the trial-and-error configuring of
lenses by mathematical design. Whether consciously or not, Huygens was
trying to bridge the gap between craftsmanship and scholarship. It was an
effort to make science useful for the solution of practical problems. An
advanced one, as the limitations and possibilities of the actual art of
telescope making were at the very heart of Huygens’ project. De Aberratione
can be seen as an early effort to do science-based technology.
How did Huygens set about it? He tried to understand mathematically the
technical problem of imperfect focusing and to solve it by means of his
theory. The configuration of lenses was the only part of the artisanal process
of telescope making he replaced by theoretical investigation. Colors he left
for crafty hands. Despite this close tie to practice, the subsequent elaboration
of the project was a matter of plain mathematics. He reduced the problem of
the imperfect focusing of spherical lenses to the mathematical problem of
spherical aberration. He then designed a configuration of lenses that
overcame the latter problem, assuming that it also solved the original
practical problem. It did not, for the test of his design brought to light an
additional technical problem that escaped his mathematical theory of lenses.
In a way, it was not just a test of his design but of his theory of spherical
aberration as well. The trial of 1668 can be seen as an empirical test of his
theory of spherical aberration – the first one, as far as the sources reveal.
Whether Huygens also saw it in this way may be doubted. His second design
of 1669 was founded upon the same theory. We do not know whether he
expected it to be free of colors, had it been realized. With hindsight, we can
say that the failure of Huygens’ project is an example of the fact that
technology goes beyond the mere application of science.
Huygens had remarked earlier that colors were a technical problem.
Minimizing their effects was a matter of craftsmanship and eluded
mathematical understanding. Unlike Barrow, Huygens was not inexperienced
with the craft of telescope making at all. With his diaphragm and his eyepiece
he had shown that he was quite capable of handling such technical problems.
The remark in his letter to Constantijn shows that he must have known, in a
practical way, much more of the properties of those disturbing colors than

1655-1672 - DE ABERRATIONE 103
his dioptrical writings reveal. Still, he did or could not integrate this
knowledge into his theory of spherical aberration. Neither by adjusting it in
some appropriate way, nor by extending it by a mathematical theory of
colors. At the crucial point where colors thwarted his plans to design a
perfect telescope, he did not know how to fit his experiential knowledge of
lenses into his theoretical knowledge of them. Huygens did indeed appear as
a scholar as well as a craftsman, but he did not weld both roles.
Would it be reasonable to presume that Huygens’ project fell short of the
kind of method Newton had successfully used to mathematize colors? If
quantitative experimentation is the obvious way to get a mathematical grip
on colors, one may say that it was in the wrong hands as far as the sciencebased
improvement of the telescope is concerned. Newton’s methodological
innovation stemmed from an entirely different context from Huygens’
dioptrics. Newton was after the physical nature of light and colors, a nature
that in his view ought to be mathematically structured. His calculations of
spherical aberration gave the same theoretical results as those of Huygens,
but they were aimed at substantiating his claim that chromatic aberration was
much larger. 212 Newton saw the practical implications of his findings. He did
not stick to his negative conclusion and set out to show how lenses could be
replaced by mirrors. 213 Yet, telescopes had not been the goal of Newton’s
studies of light and colors. His original interest concerned their physical
properties and the nature of matter.
It may be questioned whether the kind of problem Huygens ran into – a
technical problem that escaped his theory – would have given rise to a
Newtonian quantification of those disturbing colors. The example of
Hudde’s Specilla circularia makes it clear that a practical approach may also
give cause for reasoning a problem away. Hudde explicitly distinguished
mathematical and ‘mechanical’ exactness. In practice mechanical exactness
would do, and Hudde accordingly simplified his mathematical analysis on the
strength of explicitly practical considerations. It may have been precisely
Huygens’ practical outlook that made him ignore colors in his theory of
aberrations. He was studying the geometrical properties of lenses and those
colors fell outside this domain. He knew that these could be dealt with by
other means: the crafty tinkering with lenses he was also competent in. We
can only speculate what form Huygens’ study would have taken, had he
212 In his lectures Newton derived a formula for spherical aberration. Newton, Optical papers 1, 405-411.
213 His discovery of dispersion led him to conclude that no lens could ever prevent the disturbing effects
of aberration and made him design his reflector. Shortly after he published his theory, he did consider the
possibility that chromatic aberration could be prevented in lenses. In a letter to Hooke (Newton,
Correspondence I, 172), he alluded to the possibility of constructing a compound lens that canceled out
chromatic aberration. Pursuing an idea of Hooke’s, he considered the possibility of using a lens
compounded of different refracting media in which chromatic aberration was cancelled in the course of
consecutive refractions. (Newton, Mathematical Papers I, 575-576). In Opticks he ruled out this possibility,
probably because it was at odds with the dispersion law he put forward in it. Shapiro, “Dispersion law”,
102-113; Bechler, “Disagreeable”, 107-119.

104 CHAPTER 3
pursued his thinking on the disturbing colors his configuration turned out to
produce.
The reality is that colors thwarted Huygens’ plan to design via theory a
configuration of spherical lenses that minimized the effect of spherical
aberration. It was not his own observation of those colors that made him
drop the project. He needed a Newton to point out that those colors were
inherent to lenses. And he only got the point when Newton made clear that
it was an aberration; a mathematical property inherent to refraction. Huygens
realized that his project was futile when he saw the ‘Abberationem
Niutonianam’. Disappointed, he stroke out the larger part of what was one
of the most advanced efforts in seventeenth-century science to do sciencebased
technology.
The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument
Huygens’ orientation on the telescope may explain the form and content of
Dioptrica, it does not explain it as such. Why did Huygens want to develop a
theory of the telescope? Why did he want to prove mathematically that his
eyepiece performed the way he knew by experience it did? Kepler’s motive
for creating Dioptrice had been his conviction that an exact understanding of
the telescope was needed for reliable observations. In the practice of midseventeenth-century
telescopy this need did not turn out to be as pressing as
Kepler had thought. Even when the telescope became an instrument of
precision, astronomers could go about it with a rather superficial
understanding of the dioptrics of the telescope.
Kepler’s point of view does not seem to have been Huygens’ main
motive to embark upon a study of the dioptrics of the telescope. In a preface
he wrote in the 1680s for Dioptrica, he expressed his surprise that no one had
explained the telescope theoretically. One would have expected this
marvelous, revolutionary invention to have aroused the interest of scholars.
“But it was far from that: the construction of this ingenious instrument was found by
chance and the best learned men have not yet been able to give a satisfactory theory.” 214
In this preface, Huygens did not explain what further use such a theory
would have. He wanted to explain the telescope and did not wonder whether
others also found this important. To Huygens, I believe, the dioptrics of the
telescope was a meaningful topic in its own right.
Huygens’ practical activities strengthen the impression that he was
fascinated by the instrument for its own sake. As we have seen, his interest in
telescopes went far beyond mere dioptrical theory. He made telescopes and
prided himself with the innovations he had made to the instrument as well as
to the craft. Yet, making telescopes seems to have been a goal in itself. 215
Despite his impressive discoveries around Saturn, Huygens never became a
telescopist. He did not – and could not and need not – make some sort of a
214
OC13, 435. “Sed hoc tam longe abest, ut fortuito reperti artificij rationem non adhuc satis explicare
potuerint viri doctissimi.”
215
Van Helden, “Huygens and the astronomers”, 148 & 158-159.

1655-1672 - DE ABERRATIONE 105
living out of the manufacture of telescopes. He prided himself with making
good instruments – the best, he claimed in Systema saturnium – but it seems
making them was his ultimate goal and pleasure. To this desire to make his
telescopes work, he added in Dioptrica the desire to figure out how they
worked. Both intellectually and practically, Huygens was fascinated by the
working of telescope in itself. During the 1660s he added an extra dimension
to his zygomorphic interest in the telescope. In De Aberratione he aimed to
put his theory to practice in order to design a perfect telescope. Deliberately
or not, he tried to join his practical and theoretical activities regarding the
telescope.
In the wider context of seventeenth-century science, De Aberratione can be
seen as an instance of the omnipresent utilitarian ideal. An ideal that took on
various forms, ranging from invoking science to the general benefit of
mankind to using it to understand and solve particular problems of river
hydraulics or gunnery. Or, the other way around, Bacon’s call for an alliance
between the sciences and the crafts would have scientists learn from and
turn to the experiential knowledge acquired by craftsmen. At the academies
in London and Paris programs were developed to take stock of the arts.
Little came of those plans and the rare times scholars set out to offer their
learning to practicians were even less successful. 216 De Aberratione is an
example of a specific application of science to a practical problem. Such
instances were the exception in the seventeenth century, as utilitarianism
often did not go beyond grand utopian schemes. 217 Not accidentally, Huygens
brought it about, as he combined the scholar and the craftsman in one
person. He did so, however, without wasting his breath in Dioptrica on
Baconian or otherwise inspired ideals.
The only place where utilitarian ideas are explicit is the plan he wrote
around 1663 for the Académie. 218 Huygens’ own interests – dioptrics,
harmonics, motion – were prominent in the plan, and he explicitly linked
them to practical issues of astronomy, navigation and geodesy. In these
plans, as in his own studies, it was a utilitarianism of sorts: centered around
scientific instruments and thus focused on the advancement of science. He
was no exception in this regard. Westfall has shown that almost all
interrelations that were established during the seventeenth century between
scholarship and craftsmanship concerned scientific instruments. 219 It is a kind
of utilitarianism that stays very close to science itself. In the plan for the
Académie those instruments served as mediators for a selected set of issues
for the common good. The general benefit of solving the problem of finding
longitude may be clear, but for the rest the development of instruments was
aimed at the advancement of science. This in its turn was apparently
216
Boas, “Oldenburg, the ‘Philosophical transactions’, and technology”, 27-35; Ochs, “Royal society”
217
Another example is Castelli’s attempt to engineer river hydraulics, discussed in Maffioli, Out of Galileo.
218
OC4, 325-329.
219
Westfall, “Science and technology”, 72.

106 CHAPTER 3
considered a useful task in itself. In the preface cited above, Huygens sings
the praise of the invention of the telescope. It had served the contemplation
of the heavenly bodies tremendously and had revealed the constitution of the
universe and our place in it. “What man, unless plain stupid, does not
acknowledge the grandeur and importance of these discoveries?” 220
Huygens’ interest in scientific instruments was not exceptional. The form
it took was exceptional. Huygens gave a particular twist to the idea that
theory could be used to improve the telescope. Instead of deriving an ideal
solution to the problem of spherical aberration, he applied his mathematical
understanding of real, spherical lenses. Gaining a theoretical understanding
of the telescope was not that hard for a Huygens; applying it to solve
practical problems proved a more tricky business. With his clocks he was
more successful. His theoretical knowledge of circular motion enabled him
to design an isochronous pendulum. Still, the usefulness of the cheeks was
rather limited. He had to rack his brains considerably to find means to make
his clocks seaworthy – with or without cheeks.
Instruments may not have guided Huygens’ other pursuits as they did in
dioptrics, they certainly were important to him. He published part of his
studies of circular motion in the guise of a treatise describing and explaining
his isochronous pendulum clock. One might say that Huygens used
instruments to present himself and his scientific knowledge. This would ally
with the way he emphasized, in Systema saturnium, the quality of his
telescopes. It would also offer a (partial) explanation of the fact that he did
not publish Dioptrica despite repeated requests. The book would lack a vital
element: an impressing innovation of the telescope. The invention he had
placed his hopes on – a configuration of spherical lenses neutralizing
spherical aberration – had turned out to be worthless.
220 OC13, 439. “Quae magna et praeclara esse quis nisi plane stupidus non agnoscit?”

Chapter 4
The 'Projet' of 1672
The puzzle of strange refraction and causes in geometrical optics
Huygens was in Paris in the autumn of 1672. He was still a leading scholar,
but some clouds had begun to appear in the sky. The discussions at the
‘Académie’ sometimes distressed him, in particular the interventions of
Roberval. 1 His status was challenged by aspiring newcomers. The previous
chapter described how Newton with his new theory thwarted his plans to
design a perfect telescope. And the successful entrée on the Parisian scene of
Cassini put serious pressure on his position as 'primes' under Louis' savants.
Cassini had arrived from Rome in 1669 and almost immediately had started
to adorn his patron with a series of astronomical observations, where
Huygens could set little against. 2
With all that, sickness had begun to plague him. In February 1670 he had
fallen ill and he went home to the ‘air natale’ of The Hague in September.
June 1671 Huygens returned to Paris to resume his activities, but in
December 1675 he would relapse into his ‘maladie’. Whether these illnesses
were caused by his ‘professional’ troubles is hard to tell. Huygens biographer
Cees Andriesse holds this view, developing a Freudian reading of Huygens’
personality, in which Christiaan identifies with his intellectual achievements
to make up for the early loss of his mother. 3 Still, going through his Paris
letters to his brother Constantijn gives the impression that Christiaan was
bothered by a good share of homesickness. And maybe the Paris
environment just was not that good for Huygens’ constitution. Whether or
not his failing health was related, it is certain that the move from the
confines of his parental home to the competitive milieu of Paris had put his
science under pressure in the early 1670s. Huygens did not stand by idly,
however. In 1672, he was in the middle of preparing the description and
explanation of his pendulum clock for publication. Horologium Oscillatorium,
his masterpiece, appeared in 1673 dedicated to his patron Louis. And
whoever might think that Huygens had given up on his dioptrics because of
Newton's interference, was dead wrong.
Huygens had discarded the results of his analysis of spherical aberration
in October 1672. Around the same time, he drew up a plan for a publication
1
Gabbey, “Huygens and mechanics”, 174-175; Andriesse, Titan, 235-243.
2
Van Helden, “Constrasting careers”, 97-101.
3
Andriesse, Titan, 244-247 and “The melancholic genius”, 8-11. I have discussed Andriesse’s account in
Dutch in Dijksterhuis, “Titan en Christiaan”.

108 CHAPTER 4
on dioptrics. Under the heading ‘Projet du Contenu de la Dioptrique’, he
first listed the topics he would discuss and then made an outline of the
chapters. 4 The treatise would contain a large part of the dioptrical theory he
had developed since 1653. It would be a comprehensive account of the
refraction of light rays in lenses and their configurations. With this, he finally
prepared to give in to the persistent demands of his correspondents to
publish his dioptrics. He would not be able to present an impressive
innovation, like the cycloïdal pendulum of Horologium Oscillatorium. But also
without the design of a flawless telescope, Huygens had something to offer.
The theory of Tractatus was still worth publishing, despite the fact that
Barrow had gotten ahead of him by publishing the derivation of the focal
distances of spherical lenses from the sine law. A theory elaborating the
dioptrical properties of telescopes was still not available. Huygens had
enough material left to fill up a treatise on dioptrics.
The ‘Projet’ – as I will refer to it – is a key text in the development of
Huygens’ optics. On the one hand, it straightened out the remains of his
previous studies of dioptrics. On the other hand, it pointed a new direction
for his optics that would eventually lead to the Traité de la Lumière. This
direction was sign-posted by two new topics the ‘Projet’ introduced to
Huygens’ optics. First, the treatise would contain a chapter on the nature of
light. Huygens planned to give an explanation of the sine law of refraction in
terms of waves of light. Secondly, he would discuss an optical phenomenon
recently discovered: the strange refraction of Iceland crystal. The topic bears
no relevance whatsoever to the questions about telescopes that had occupied
him in his previous dioptrical studies. The reason for treating strange
refraction was that it posed a problem for the kind of explanation of the sine
law he had in mind. All this is remarkable, for in his dioptrics Huygens had
never before bothered about the nature of light or the cause of refraction.
What is more, in his recent dispute with Newton, he appeared to have a
blind spot for these very subjects.
In this chapter, we follow Huygens’ switch from the mathematical
analysis of the behavior of refracted rays to the consideration of its causes
and the explanation of optical laws. The issue of causes became relevant for
Huygens through the phenomenon of strange refraction. The first attack of
the problem was inconclusive and, moreover, left the issue of the cause of
refraction untouched. Together, this attack and the ‘Projet’ are illuminating,
not only for the development of Huygens’ optics and his conception of
mathematical science, but also for seventeenth-century optics in general.
Optics was in the middle of a transition from medieval ‘perspectiva’ to new
way of dealing mathematically with phenomena of light. This chapter focuses
on the issue of causes and explanations in optics. Over the shoulder of
Huygens we look back to the way Alhacen, Kepler, Descartes dealt with the
4
OC13, 738-745. I date the sketch in 1672, instead of 1673 as the editors of Oeuvres Complètes have it. See
page 92 above and page 140 below.

THE 'PROJET' OF 1672 109
physical foundations of optical laws. Huygens’ opinions and conduct in 1672
turn out to be rather illustrative of the transition optics was going through.
The problem of strange refraction would be solved five years later, but not
without Huygens developing a different and innovative approach to the
nature of light. That will be discussed in the next chapter.
‘Projet du Contenu de la Dioptrique’
The ‘Projet’ sketchily fills up the two sides of a manuscript page, with all
kinds of additions and remarks inserted in and around a main line of
contents. 5 It begins with a short list of topics and continues with an outline
of the chapters and their content. The planned treatise on dioptrics would, of
course, be about the telescope: “my principal design is to show the reasons
and measures of the effects of telescopes and microscopes.” 6 The treatise
would open with a historical chapter on the invention and advancement of
the telescope and of telescopic discoveries, and was to include an account of
the development of the mathematical understanding of lenses and related
phenomena. 7 In chapters four to seven Huygens would expound his own
theory of dioptrics, the theory of the telescope of Tractatus. He would solely
discuss spherical lenses – “the only ones useful until now” – and leave out
the hyperbolic and elliptic lenses invented by Descartes. Huygens was clear
about the principal defect of Descartes’ treatment of dioptrics:
“What I have said about the necessity of the theory of spherical ones is so true that
Descartes, for not having examined it, has not known to determine the most important
thing in the effect of telescopes, which is the proportion of their magnification, for
what he says about it means nothing; …” 8
The final chapter of ‘Dioptrique’ would treat the structure and the working
of the eye. The main part of these four chapters was ready and only needed
some rewriting and restructuring. The second and the third are the chapters
of the ‘Projet’ that interest us here. They introduced two topics new to
Huygens’ dioptrics: the cause of refraction and the strange refraction of
Iceland crystal.
In chapter two, Huygens planned to treat the sine law and its causes. He
would start with a historical account of the discovery of the sine law – in his
view undeservedly attributed to Descartes – and discuss some features of
refraction. Next, he would give an explanation of refraction and discuss the
nature of light. Although sketchy, the gist of his plans is clear. He rejected
the explanation of the sine law Descartes had given in La Dioptrique:
5
Hug2, 188r-188v.
6
OC13, 740. “mon principal dessein est de faire voir les raisons et les mesures des effects des lunettes
d’approche et des microscopes.”
7
OC13, 740. Huygens mentions Archimedes (things seen under water), Alhacen, Kepler and Galileo by
name. He elaborated his historical account later during the 1680s.
8
OC13, 743. “Ce que j’ay dit de la necessitè de la theorie des spheriques est si vrai, que Descartes pour ne
l’avoir point examinée n’a sceu determiner la chose la plus importante dans l’effect des lunnetes qui est la
proportion de leur grossissement, car ce qu’il en dit ne signifie rien; …”

110 CHAPTER 4
“difficulties against Descartes. where would the acceleration come from. he makes light
a tendency to move [conatus movendi], which makes it difficult to understand
refraction as he explains it, at least in my view. … light extends circularly and not in an
instant,…” 9
The concluding words reveal Huygens’ own conception: the nature of light is
to spread out circularly over time. In other words, light consists of waves.
The notes also clarify where Huygens had got the idea to think of light as
waves. “Refraction as explained by Pardies.” 10 Ignace-Gaston Pardies was a
Jesuit father with a keen interest in the mathematical sciences, who actively
participated in the Parisian scientific life, and with whom Huygens
maintained good relations. Pardies had proposed the idea that light consists
of waves and had explained the sine law with it. Huygens listed some
essentials of a wave theory: “transparency without penetration. bodies
capable of this successive movement. Propagation perpendicular to circles.” 11
In the margin he added “vid. micrograph. Hookij”, a reminder to check
Hooke’s alternative wave theory of Micrographia. 12 The original formulation of
Pardies’ theory has been lost, so we cannot know what precisely Huygens
knew of it. He had known of “… the hypothesis of father Pardies …” at
least since August 1669, when he mentioned it in a discussion at the
Académie. 13 On 6 July 1672 Pardies sent him a treatise on refraction that
probably revealed some more details. After Pardies died in 1673, his confrere
Pierre Ango published his explanation of refraction – at least its main lines –
in L’Optique divisée en trois livres (1682). Ango had taken ‘the best parts’ of
Pardies’ theory and blended them with own ideas, but Huygens did not have
a high opinion of Ango’s work. 14 We do not know to what exact extent
Huygens knew Pardies’ theory and derived his own understanding of the
nature of light and refraction from it. We do know that they stood in close
contact over these matters, that Huygens openly acknowledged the
contributions of Pardies, and that the essentials of Pardies’ theory where
central to Huygens’ subsequent attack of strange refraction. He explicitly
recorded the main assumption of Pardies’ derivation of the sine law:
“Propagation perpendicular to circles.” In other words, rays are always
normal to waves. 15
9
OC13, 742. “difficultez contre des Cartes. d’où viendrait l’acceleration. il fait la lumiere un conatus
movendi, selon quoy il est malaisè d’entendre la refraction comme il l’explique, a mon avis au moins. …
lumiere s’estend circulairement et non dans l’instant, …”
10
OC13, 742. “Refraction comment expliquee par Pardies.”
11
OC13, 742. “transparance sans penetration. corps capable de ce mouvement successif. Propagation
perpendiculaire aux cercles.”
12
OC13, 742 note 1.
13
OC16, 184. “… l’hypothese du P. Pardies …” Pardies’ second letter to Newton in their dispute about
colors suggests that Pardies’ wave conception of light was rooted in Grimaldi’s ideas. Shapiro, “Newton’s
definition”, 197.
14
Shapiro, “Kinematic optics”, 209-210. OC10, 203-204.
15
This is discussed below, in section 4.2.2.

THE 'PROJET' OF 1672 111
In the ‘Projet’ the third chapter was only indicated by a title and a single
remark: “Iceland crystal” and “difficulty of the crystal or talc of Iceland. its
description. shape. properties.” 16 Iceland crystal was a rarity from the barren
nordic lands displaying remarkable properties. This had been known for
ages, but a sample had recently been brought to Copenhagen to increase the
collection of curiosities of the Danish king. Danmark’s leading
mathematician, Erasmus Bartholinus, then made a study of the crystal and its
phenomena and reported on its strange refraction properties in 1669 in a
treatise called Experimenta crystalli islandici disdiaclastici (1669). The strange
refraction of Iceland crystal contradicted the sine law. It refracts a
perpendicularly incident ray, which is impossible according to the sine law.
Still, Iceland crystal had no relevance whatsoever to telescopes. So why
would Huygens include it in his ‘Dioptrique’? The reason is that strange
refraction constituted a problem for Pardies’ explanation of refraction. The
‘difficulté’ of Iceland crystal was that the refraction of the perpendicularly
incident ray could not be reconciled with the assumption that rays are
normal to waves. Huygens did not say this explicitly, but the place where he
indicated the ‘difficulté’ makes it clear that Iceland crystal was a problem for
Pardies’ explanation of refraction. Moreover, in his first notes on the
phenomenon of around the same time, Huygens explicitly phrased the
problem this way. 17
We now see why Huygens would want to include strange refraction in a
treatise on the dioptrics of the telescope. If his explanation of refraction
were to be acceptable, it should not be contradicted by this particular kind of
refraction. But why would he care for the tenability of the explanation so
much? Huygens had not bothered to explain refraction before. Part of the
answer lies in the fact that it was customary to do so. Books on geometrical
optics usually contained a preliminary account of the nature of light and the
causes of the laws of optics. The explanation of the sine law was to complete
Huygens’ dioptrics so that it could be published as a proper treatise in
geometrical optics. It would also complete his critique of Descartes’ La
Dioptrique. As his theory of spherical lenses corrected the latter’s failure to
explain the telescope properly, the projected explanation of the sine law
would correct the difficulties in Descartes’ explanation.
Waves would do the job, assuming that the problem of strange refraction
could be settled. But what job exactly would they do? Just before the sketch
of his explanation of refraction Huygens added an epistemological remark.
An utterance of this kind is rare with Huygens, and this one is particularly
illuminating:
“Although it suffices to pose these laws as principles of this doctrine, as they are very
certain by experience, it will not be unbecoming to examine more profoundly the cause
of the refraction in order to try to give also that satisfaction to the curiosity of the mind
16
OC13, 743: “Cristal d’Islande” and 739: “difficultè du cristal ou talc de Islande. sa description. figure.
proprietez.”
17
See below at the beginning of section 4.2.

112 CHAPTER 4
that loves to know the reason of every thing. And to have at least the possible and
probable causes instead of remaining in an entire ignorance.” 18
Huygens here presents his causal account as only supplementary rather than
foundational. This raises the question what status waves of light had.
Apparently they were merely probable and did not convey some indisputable
truth. Still, they ought to explain refraction and do so in a better way than
Descartes’ ‘conatus’ had done. Moreover, the explanation of ordinary
refraction must not be contradicted by another instance of refraction, exotic
as it might be. Just like any mathematical theory, an explanatory theory ought
to be consistent. Despite the limited importance of waves, Huygens took the
problem that strange refraction posed for waves seriously. He went on to get
it out of the way before publishing his ‘Dioptrique’.
4.1 The nature of light and the laws of optics
The problem that Huygens recognized is historically significant. Optics was
in the middle of a transformation initiated by Kepler and Descartes, in which
the rising corpuscular view on essences was shifting the understanding of the
nature of light as well as the relationship between causal explanations and
mathematical descriptions of its properties. Whether he fully realized it or
not, with the ‘difficulté’ of strange refraction Huygens found himself at the
heart of this remapping of the scholarly treatment of light. Before turning to
Huygens’ first efforts to reconcile strange refraction with light waves, I will
sketch the historical context of the epistemological issues raised by the
‘Projet’, in particular the relationship between physics and mathematics of
light. To this end, I sketch the way Huygens’ most significant precursors in
optics treated the issue of causality with respect to reflection and refraction:
Alhacen, Kepler, Descartes and Barrow. This will bring into perspective
Huygens’ specific, and rather non-committal approach to the explanation of
the law of refraction.
In this regard, it should be noted that the phrase ‘law of refraction’ was
rarely used in seventeenth-century optics. 19 In the project Huygens spoke of
‘loix de refraction’, which included for example reciprocity as well, as he also
would do in Traité de la Lumière. In Dioptrica he called the sine law the
proportion or ratio of sines. 20 The concept of a law of nature aroses in
18 OC13, 741. “Quoy qu’il suffise de poser ces loix pour principes de cette doctrine, comme estant tres
certains par l’experience, il ne sera pas hors de propos de rechercher plus profondement la cause de la
refraction pour tascher de donner encore cette satisfaction a la curiositè de l’esprit qui aime a scavoir
raison de toute chose. Et d’avoir au moins les causes possibles et vraysemblables que de demeurer dans
une entiere ignorance.”
19 Kepler used ‘mensura’ (Kepler, KGW2, 78; see below). Descartes spoke of the laws of motion but of
‘mesurer les refractions’ (Descartes, AT6, 102). In his optical lectures of 1670 Newton used ‘regula’ and
‘mensura’ (Newton, Optical papers 1, I, 168-171 & 310-311). In Opticks Newton, like Huygens in Dioptrica,
used ‘proportion’ or ‘ratio’ of sines (Newton, Opticks, 5-6 & 79-82).
20 Huygens, OC13, 143-145. In Traité de la Lumière he simply called the sine law the ‘principale proprieté’ of
refraction (others are its reciprocity and total reflection); Huygens, Traité de la Lumière, 32-33. In his notes
he sometimes spoke of ‘laws’ or ‘principles’ (OC13, 741) as he did in the draft of Dioptrica prepared
around 1666 (OC13, 2-9).

THE 'PROJET' OF 1672 113
seventeenth-century natural philosophy and entered the mathematical
sciences only gradually.
In the opening lines of the first chapter of Paralipomena, which treats the
nature of light, Kepler points out a disciplinary division between physical and
mathematical aspects of light.
“Albeit that since, for the time being, we here verge away from Geometry to a physical
consideration, our discussion will accordingly be somewhat freer, and not everywhere
assisted by diagrams and letters or bound by the chains of proofs, but, looser in its
conjectures, will pursue a certain freedom in philosophizing - despite this, I shall exert
myself, if it can be done, to see that even this part be divided into propositions.” 21
In the subsequent chapters Kepler naturally returned to the firm grounds of
geometry, but not before he had pointed out an unfortunate side effect of
this division. In the appendix to the chapter he complains that the insights
mathematicians have acquired regarding light are neglected and undeservedly
underrated by natural philosophers. Therefor, in this appendix, Kepler
explains the common misunderstandings of them - notably the followers of
Aristotle - although they could have corrected themselves had they taken
notice of the writings of opticians. 22
The gap between physical and mathematical accounts of the cosmos in
pre-Keplerian astronomy is well-documented. It is tempting to take stock of
Scholastic views on the nature and function of mathematical inquiry and
generalize the status of mathematical astronomy towards that of the
mathematical sciences as a whole. Smith argues that a historical link between
classical astronomy and classical optics existed, consisting of shared
conceptions, commitment and methodologies. 23 Still, when considering the
relationship between mathematical descriptions of light and its nature,
caution should be taken.
Compared to the other fields of mathematics, the development of
geometrical optics followed a rather idiosyncratic course up to the early
seventeenth century. Since Greek antiquity it was realized that the central
object of study – the light ray – combines almost naturally physical and
mathematical conceptualization. In this way geometrical optics had
incorporated a realistic mode of geometrical reasoning since the very
founding of the science by Euclid and Ptolemy. Through the influential work
of Alhacen the onset of a physico-mathematical conception of optics was
established at a much earlier time than would be the case in the other
mathematical sciences. In its transmission through medieval perspectiva,
Alhacen’s optics was the starting point for Kepler and Descartes and
profoundly affected their innovations of the science. As a consequence,
21
Kepler, Paralipomena, 5 (KGW2, 18. Translation Donahue: Kepler, Optics, 17). “Caeterum cum hic à
Geometria interdum in physicam contemplationem deflectamus: sermo quoque erit paulò liberior, non
ubique literis et figuris accommodatus, aut demonstrationum vinculis astrictus, sed coniecturis dissolutior,
libertatem aliquam philosophandi sectabitur: Dabo tamen operam, si fieri potest, ut in Propositiones et
ipse dividâtur.”
22
Kepler, Paralipomena, 29 (KGW2, 38)
23
Smith, “Saving the appearances”, 73-91.

114 CHAPTER 4
methodological, epistemological and conceptual features of perspectivist
optics are perceptible throughout seventeenth-century optics. 24
4.1.1 ALHACEN ON THE CAUSE OF REFRACTION
In the eleventh century, the Islamic scholar Ibn al-Haytham composed a
voluminous work on optics, Kitb al-Manzir. The Optics of Alhacen, as they
are commonly referred to in the West, was intended to bring together
mathematicians’ and physicists’ accounts of light and vision by giving a
systematic treatment of optics that met the demands of both. 25 This required
the combination of the Aristotelian doctrine of forms received by the eye
and Ptolemy’s ray-wise analysis of the perception of shape and position.
According to Alhacen both these notions were partly true but incomplete. 26
The synthesis he had in mind - ‘tarkb’ - consisted of a theory in which the
forms of light and color issue from every point of the object and extend
rectilinearly in all directions. 27 It met the demands of the Aristotelian doctrine
by considering light rays as the direction in which light extended and those
of Ptolemy by appointing rays as the ultimate tool of analysis. As contrasted
to his mathematical precursors, Alhacen regarded a ray as a purely
mathematical entity: “Thus the radial lines are imaginary lines that determine
the direction in which the eye is affected by the form.” 28 In a later work on
optics, the Discourse on light, Alhacen expounded his conception of light and
reflected on the character of the science of optics by discussing the
distinction between mathematical and physical aspects of light. In his view,
each provided answers to different kinds of questions: in physics one
investigates the essence of light; in mathematics the radiation or spatial
behavior of light. Physical theory classified the various kinds of bodies:
luminous, shining, transparent, opaque. Mathematical theory described the
perception of things by means of rays, rectilinear and inflected. 29 In the Optics
Alhacen adopted the Aristotelian concept of forms without further
philosophical inquiry. His exposition on the nature of light in book 1 served
as a physical foundation for the mathematical and experimental investigation
of light and vision that constituted the heart of the Optics.
Alhacen provided the basis for the flourishing of the study of light and
vision in thirteenth-century Europe given shape to by Robert Grosseteste,
Roger Bacon, John Pecham and Witelo. 30 Alhacen’s work reached the west in
24
This theme is amplified by, among others, Schuster, Descartes, 332-334: Smith, Descartes’s Theory of Light
and Refraction, 4-12.
25
Alhacen, Optics I, 3-6 (book 1). The content and scope of Alhacen’s optics is discussed in Sabra’s
introduction to his translation of its first three books: Alhacen, Optics II, xix-lxiii. See further Lindberg,
Theories, 85-86.
26
Alhacen, Optics I, 81 (book 1, section 61).
27
Alhacen’s account for the subsequent one-to-one correspondence between the points of the object and
the image in the eye is discussed in section 2.2.1 above.
28
Alhacen, Optics I, 82 (book 1, section 62)
29
Alhacen, Optics I, li (Sabra’s introduction). See also Sabra, “Physical and mathematical”, 7-9.
30
Lindberg, Theories, 120-121 and 109-116.

THE 'PROJET' OF 1672 115
truncated form, for the Latin translation, in both manuscript and printed
form, lacks his first three books. It was translated in the thirteenth (possibly
twelfth) century and became known as Perspectiva (or De aspectibus). 31 The
Perspectiva communis (ca. 1279) of Pecham and the Perspectiva (ca. 1275) of
Witelo are to be understood primarily as compendia of Alhacen’s optics.
These works became textbooks of perspectiva - a common denominator of
medieval optics. Friedrich Risner published (the remaining books of)
Alhacen’s Optics together with Witelo’s Perspectiva in 1572, an edition that
remained authoritative well into the seventeenth century. To provide for the
now lacking physical foundation of Alhacen’s optics, perspectivist writers
drew on the ideas of Grosseteste and Bacon. Through Bacon, Grosseteste’s
theory of the multiplication of species was incorporated into perspectivist
theories. Although it provided a mathematically structured account of the
nature of light and its propagation, it served no function in perspectivist
accounts of the behavior of light rays interacting with various media. The
perspectivist writers reiterated Alhacen’s analysis of reflection and refraction,
adding some clarifications on its basic assumptions. 32
In his accounts of reflection and refraction Alhacen also discussed the
causes of these phenomena. These appealed only to light qua radiation,
however, not its physical essence. The core of Alhacen’s account consists of
a mathematical analysis of rays in their components perpendicular and
parallel to the reflecting or refracting surface. In reflection the parallel
component remains unaltered whereas the perpendicular component is
inverted, which readily yields the law of reflection. 33 By differentiating the
parallel and perpendicular components of a light ray he extended Ptolemy’s
analysis, who had only considered the angles before and after reflection. 34 In
his account Alhacen appealed to an analogy between reflected rays and a
rebounding ball: “We can see the same thing in natural and accidental
motion, …”. 35 He pictured a sphere attached to an arrow projected
perpendicularly or obliquely to a mirror. This mechanical analogy applied to
the mathematical analysis of the motion of light rays and did not appeal to
the form-like nature of light. Light is reflected because its motion is fully or
partially ‘terminated’ by an obstacle.
Alhacen’s causal account of refraction proceeded along similar lines. Rays
are refracted because their motion changes when they enter a medium of
different density. In the case of refraction towards the normal – into a denser
medium – he assumed that part of the parallel component was altered. He
did so implicitly, in an comparison with a ball striking a thin slate. “For
31
Alhacen, Optics I, lxxiii-lxxix (Sabra’s introduction).
32
Lindberg, “Cause”, 30-31.
33
Risner, Optica thesaurus, 112-113. Witelo relied the argument of the shortest path: Risner, Optica thesaurus,
198.
34
Lindberg, “Cause”, 25-29. Sabra, “Explanation”, 551-552.
35
Risner, Optica thesaurus, 112-113. “Huius aút rei simile in naturalibus motibus videre possumus, & etiá in
accidentalibus.” Translation: Lindberg, in Grant, Source book, 418.

116 CHAPTER 4
things moved naturally in a straight line through some substance that will
receive them, passage along the perpendicular to the surface of the body in
which passage takes place is the easiest.” 36 A couple of lines further, Alhacen
continued: “Therefore, the motion [of the light] will be deviated toward a
direction in which it is more easily moved than in its original direction. But
the easier motion is along the perpendicular, and that motion which is closer
to the perpendicular is easier than the more remote.” 37 In the case of
refraction away from the normal Alhacen abandoned the appeal to the
easiest path. He considered the components of the ‘motion’ again and stated
without argument that the parallel component is increased. Besides being
inconsistent, Alhacen’s account of refraction remained qualitative, as he did
not attempt to determine to what degree a refraction ray was bent towards
the normal, nor to what proportion the parallel component was altered.
Alhacen’s account of refraction primarily consists of an experimental
analysis. In Risner’s edition it covers the first eleven or twelve propositions
of the seventh book, which return in the second chapter of Witelo’s part. In
the tenth chapter the latter added to the quantitative account of refraction by
providing a table – supposedly observational – of angles of refraction for a
set of incident rays.
In Alhacen’s accounts of reflection and refraction two levels of inference
can be distinguished. In the first place, the analysis of rays in their
perpendicular and parallel components revealed some deeper lying
mathematical structure of both phenomena. It unified his accounts to some
extent, although he did not assume the parallel component unaltered in all
cases like Descartes would later do. The second level involves mechanical
analogies that illuminate rather than prove the mathematical analyses of
reflected and refracted rays. The causal account provided additional support
for the properties of reflection and refraction, but the ultimate justification
was empirical. 38 In this regard the analogies can be considered to serve
didactical purposes.
Alhacen’s analogies do not - and were not intended to - explain refraction
and reflection by deriving their properties from an account of the nature of
light. That is the way Huygens and his fellow seventeenth-century students
of optics understood ‘explaining the properties of light’ and which his waves
of light would have to bring about. Whereas the rectilinearity of rays
followed rather naturally from Alhacen’s understanding of forms, reflection
and refraction are discussed in terms of light rays instead of interactions
between forms with reflecting and refracting substances. The ideals of
36 Risner, Optica thesaurus, 241. “Omnium autem moterum naturaliter, que recte moventur per aliquod
corpus passibile, transitus super perpendicularem, que est in superficie corperis in quo est transitus, erit
facilior.” Translation: Lindberg, “Cause”, 26.
37 Risner, Optica thesaurus, 241. “...: accidit ergo, ut declinetur ad partem motus, in quam facilius movebitur,
quàm in partem, in quam movebatur : sed facilior motuum est super perpendicularem: & quod vicinius est
perpendiculari, est facilius remotiore.” Translation (amended): Lindberg, “Cause”, 27.
38 Alhacen, Optics I, lxi (Sabra’s introduction); Risner, Optica thesaurus, XVII-XIX (Lindberg’s introduction).

THE 'PROJET' OF 1672 117
mechanical philosophy notwithstanding, this ‘physical’ understanding of rays
and their behavior would crucially affect the investigations of Huygens and
other seventeenth-century opticians. Kepler and Descartes set off where
Alhacen and Witelo had left off. A law of reflection was known, as well as
diverse mathematical properties of radiated light, but refraction remained to
be understood only qualitatively. The thirteenth-century synthesis left
perspectiva as a comprehensive body of knowledge – Alhacen’s theory of
vision, solutions to various problems of reflection and so on – riddled with
some persistent, well-known problems like the pinhole image. 39 The sixteenth
century witnessed major developments in optics, but mainly in its practical
parts that bore on Galileo’s telescopic achievements rather than Kepler’s and
Descartes’ theoretical pursuits. 40
4.1.2 KEPLER ON THE MEASURE AND THE CAUSE OF REFRACTION
The heritage of medieval perspectiva Kepler received, consisted of a welldefined
set of aims and criteria for geometrical optics: mathematical analysis
of the behavior of light rays. 41 In Paralipomena he took up this heritage and
transformed it radically. In chapter two we have seen how he created a new
theory of image formation by rigorously applying the principle of rectilinear
propagation of light rays. We now turn to his account of the causes
underlying the behavior of light rays. Here the same approach is
recognizable. In Kepler’s view, the mathematically established properties of
things are real and should be directive in physical considerations. Kepler’s
conception of the nature of light can be seen as a realist reading of
perspectivist’s mathematical ideas which he then rigorously employed in the
investigation of the behavior of light. 42
At the start of this section I cited the opening lines of Paralipomena, where
Kepler pointed out the relative freedom of reasoning he would employ in
these matters pertaining to physics. In the first chapter, ‘De Natura Lucis’, he
expounded the general concepts and principles pertaining to his account of
optics. On the whole, his theory of light was the perspectivists’ theory of
multiplication of species enriched with neoplatonist metaphysics. 43 According
to Kepler, light is an incorporeal substance which has two aspects, essence
and quantity, by which it has two operations (‘energias’), illumination and
local motion, respectively. 44 Radiation is the form of propagation: light
spreads in all directions and does so spherically. Light rays are the radii of
this sphere and thus rectilinear. Light itself can be regarded as the twodimensional
surface of an expanding sphere. The mathematical structure of
39
Lindberg, Theories, 122-132.
40
Dupré, Galileo, 17-19.
41
Lindberg, “Roger Bacon”, 249-250.
42
The following discussion owes much to Buchdahl’s illuminating discussion of Kepler’s method:
Buchdahl “Methodological aspects”. References are to the original text, corresponding pages in the
Gesammelte Werke in parentheses. Except where noted, translations are by Donahue from Kepler, Optics.
43
Lindberg, “Incorporeality”, 240-243.
44
Kepler, Paralipomena, 13 (KGW2, 24)

118 CHAPTER 4
this theory was clearly perspectivist in origin, but to Kepler it represented the
physical nature of light, not only its mathematical behavior.
On the basis of this theory of light, Kepler discussed the behavior of
propagated light. The rectilinearity of light rays is a direct outcome of
Kepler’s conviction that light ‘strives to attain the configuration of the
spherical’. 45 Where light is deflected from its straight path this must be the
effect of the interaction of light and matter. As light is a two-dimensional
surface this interaction can only occur with the surface of reflecting and
refracting media. Kepler attributed a form of density to surfaces and argued
that light is hindered in its passage through the surface of a body
proportionally to its density. 46 In the case of reflection the density of the
surface is so high that light falling upon it “... is made to rebound in the
direction opposite to that whence it approached.” 47 Kepler specified that this
applied to the perpendicular component of a ray - i.e. the part of the motion
towards the surface. The law of reflection now followed naturally, thus
clearing the way for an exact analysis of the properties of reflection in
chapter 3 of Paralipomena. In this chapter Kepler took up the classical topic
of perspectiva to determine mathematically the location where a reflected
image is perceived. 48
The measure of refraction
For refraction things were more complicated. The causal account in chapter
1 did not yield an exact law, so in the fourth chapter of Paralipomena Kepler
could not readily embark on an analysis of refractional phenomena. Instead,
he first had to find such a ‘measure’ of refraction. The course of the chapter
reveals Kepler’s conception of the distinction between causes and measures
in optics. Initially he stuck to the epistemic organization of his treatise by
analyzing refraction in term of rays, the components of their motion, and
regularities in the various angles at which they are refracted. However, when
all this yielded no satisfactory results he took the nature of light into
consideration to see where these ‘proper’ causes could lead him to find the
measure of refraction.
In proposition XX of chapter 1 Kepler derived from his suppositions
about the interactions of light (surfaces) with (the surface of) a dense
medium that a ray is refracted towards the perpendicular. 49 His argument
comes down to the idea that the surface impedes the spreading of the sphere
of light. Kepler explained that this understanding is based on the fact that
motion belongs to light and that said interaction is general for moving
45
Kepler, Paralipomena, 8 (KGW2, 20). “Nam diximus affectari à luce figurationem Sphaerici.”
46
Propositions XII-XIV: Kepler, Paralipomena, 10-11 (KGW2, 22-23)
47
Kepler, Paralipomena, 13 (KGW2, 25). “Lux in superficiem illapsa repercutitur in plagam oppositam,
unde advenit.”
48
An important part of this was Kepler’s negation of the generality of the cathetus rule and the
introduction of his new theory of image formation, which have been discussed above in section 2.2.1.
49
Proposition XX: Kepler, Paralipomena, 15-21 (KGW2, 26-31).

THE 'PROJET' OF 1672 119
matter. In this way he followed up on the mechanical analogies employed in
perspectivist causal accounts, but he did not do so without appropriating that
line of reasoning to his own means. “For it may be permissible here for me
to use the words of the optical writers in a sense contrary to their own
opinion, and carry them over into a better one.” 50 Kepler went on to develop
the analysis of a ball spun into water by distinguishing between the dynamics
of the parallel and perpendicular components of its motion, whereby light is
rarified in the former direction and merely transported in the latter direction.
He then proceeded with a short discussion of the underlying physics, to wit
the statics of a balance. In this way Kepler transformed the mechanical
analogies employed by his perspectivist forebears to illuminate the
mathematics of refraction into a physical foundation of the analysis of
refraction. The account in chapter 1 only yielded a qualitative understanding
of refraction, and only partial for that matter, for Kepler did not discuss the
passage of light into a rarer medium.
At the opening of chapter 4, ‘De Refractionum Mensura’, Kepler still
lacked an exact law of refraction. He needed this ‘measure’ in the first place
for his account of the dioptrics of the eye in the next chapter (see above
section 2.1.1.), but in the end principally for his account of atmospheric
refraction later in Paralipomena. After all, it was a treatise in the optical part of
astronomy for which the laws of optics were instrumental. Nevertheless my
discussion will be confined to the optics per se: Kepler’s tour the force to
tackle the mathematics of refraction.
Kepler began with a review of the received opinions regarding the
measure of refraction. In this section, he tied in with the traditional approach
of considering the physical properties of light rays and their components.
After negating several opinions, Kepler laid down the - in his view - generally
established understanding: first, that the density of the refracting medium is
the cause of refraction and, second, the angle of incidence contributes to its
cause. The question therefor was how these two aspects are connected.
Kepler ran through several options as they had been set forth, rejecting each
as insufficient. Next, he contemplated how the two said aspects could
correctly be combined. 51 Kepler proceeded to represent these conditions
geometrically (Figure 36). BC is the refracting surface of a medium BCED and
AB, AG, AF are incident rays. Kepler now extended the medium to DEKL,
thus representing the greater density of its surface. He then constructed a
refracted ray FQ by drawing HN perpendicular to the lower surface and
joining N at the imaginary bottom with F. 52 Comparing the results of this
method with Witelo’s table, Kepler simply concluded that it was refuted by
50
Kepler, Paralipomena, 16 (KGW2, 27). “Liceat enim hîc mihi verba Opticorum contra mentem ipsorum
usurpare, et in meliorem sensum traducere.”
51
Kepler, Paralipomena, 85-87 (KGW2, 85-86)
52
This is equivalent with sini : tanr = constant. Lohne, “Kepler und Harriot”, 197. Compare Buchdahl,
“Methodological aspects”, 283.

120 CHAPTER 4
experience. 53 He tried some more ideas flowing
from this geometry, including some ways of
evaluating the ‘refractaria’ - the locus of images
where the points of a line are percieved, D for
point L, I for point M, etcetera. 54 All ideas were
refuted by experience and Kepler abandoned his
attempt of finding a measure of refraction on the
basis of an analysis of the physics of light rays.
In the next three sections, Kepler temporarily
ignored the causes of refraction and focused on
finding mathematical regularities in the given
angles of incidence and refraction. Building on the
known properties of reflection, he tried certain
analogies between reflection and refraction. Kepler
argued that in the case of refraction in a medium
with infinite density, all rays must be refracted into
the perpendicular. He then correlated this case to
reflection by a parabolic mirror with rays coming
from its focus. This led him to consider the
relationship of conic sections with refraction. He
constructed a diagram of angles of incidence and
Figure 36 The first stage of
Kepler’s attack of
refraction.
refraction and considered the intersection of the accompanying rays. The
resulting curve is similar to a hyperbola, but points from where the rays
come are not the matching foci, so Kepler dismissed this attempt as well.
This and other trials with conic sections – including the effort to construct
an anaclastic curve – still did not give Kepler a correct ‘measure of
refractions’ and he abandoned this line of thought as well.
Finally, Kepler returned to his causal analysis of refraction of chapter 1 to
query whether - “may God look kindly upon us” - this would yield the
measure of refraction. 55 As contrasted to the ray analysis of the first stage, he
now considered the interaction of the surface of light with the surface of the
refracting medium. 56 Kepler warned beforehand he would perhaps stray
somewhat from his goal of finding the measure of refractions in its causes,
and halfway through his exercise he would acknowledge “In demonstrating
the true cause of this directly and a priori, I am stuck.” 57 He did not formally
deduce a ‘measure’ from the causes of refraction, but rather had employed
(in Buchdahl’s words) “physical considerations to guide the intuitive search
for responsible factors relevant to the result.” 58
53
Kepler, Paralipomena, 86 (KGW2, 86). “Hic modus refutatur experientiâ: ...”.
54
Kepler, Paralipomena, 88-89 (KGW2, 87-88).
55
Kepler, Paralipomena, 110 (KGW2, 104). “quod Deus benè vertat”
56
Kepler, Paralipomena, 110-114 (KGW2, 104-108).
57
Kepler, Paralipomena, 110 and 113 (KGW2, 104 and 107). “Etsi enim à scopo forsan etiamnum nonnihil
aberrabimus: ...” and “In genuina huius rei causa directè et à priori demonstranda haereo.”
58
Buchdahl, “Methodological aspects”, 291.

Kepler began with the understanding
of the nature of light as the surface of an
expanding sphere laid down in the
opening chapter of Paralipomena (Figure
37). ABMK is the section of a physical ray
obliquely incident on the surface BC and
refracted towards QBMR. According to
Kepler the angle of deviation must be
proportional to the angle of incidence.
THE 'PROJET' OF 1672 121
Figure 37 The final stage of Kepler’s
analysis of refraction
This condition is met when only the (surface)density of the refracting
medium is assumed to be effective. With increasing obliquity, BM increases
and therefore the resistance met by the light increases. Now “… there is
more density in BM than in LM ...” so that the proportion LM to BM must be
added as a factor of refraction onto the proportionality of angles of
incidence and deviation. 59 However, the proportion LM to BM – or sec i –
implied a paradox. Horizontal rays would be refracted at an infinitely large
angle. Kepler therefore changed his perspective and now considered BR of
the refracted ray. He concluded that the secans of the angle at the upper
surface of the denser medium ‘plays a part’ in refraction. 60 Refraction was
thus a composite of two factors: the proportionality of i-r to i and the
proportionality of i-r to sec r – in other words: i-r = c·i·sec r, where c is some
constant.
In proposition 8, Kepler gave instructions how to apply this analysis to
calculate angles of refraction. It is in the form of a ‘problem’, a procedural
statement of the sort the later Dioptrice was composed of, as we saw above in
section 2.2.1. After all, Kepler’s struggle had not yielded a general ‘measure
of refraction’ independent of specific media and transcending measurements.
First, both factors are determined for the medium by means of one known
pair of incident and refracted rays. Then the angle of refraction for any other
angle of incidence is computed. By means of an example, Kepler calculated a
table for refraction from air into water. The values differed somewhat from
Witelo’s data which Kepler had plied so rigorously in the previous sections.
This time he was more tolerant: “This tiny discrepancy should not move you;
believe me: below such a degree of precision, experience does not go in this
not very well-fitted business.” 61 Moreover, he (correctly) suspected that
Witelo had modified his table on the basis of Ptolemy’s false supposition
that the secondary differences of the angles are constant. “Therefore, the
fault lies in Witelo’s refractions”, and Kepler proceeded to use his own result
to consider atmospheric refraction. 62 Although the empirical correctness was
59 Kepler, Paralipomena, 111 (KGW2, 105). “Plùs igitur densitatis est in BM, quàm in LM.”
60 Kepler, Paralipomena, 113 (KGW2, 107). “..., sciendum igitur, eorum angulorum incidentiae secantes concurrere ad
mensuram refractionum, qui constituuntur ad superficiem in medio densiori.”
61 Kepler, Paralipomena, 116 (KGW2, 109). “Neque te moveat tantilla discrepantia, credas mihi, infra tantam
subtilitatem, experientiam in hac minus apt materia non descendere.”
62 Kepler, Paralipomena, 116 (KGW2, 109). “Ergò in Vitellionis refractionibus culpa haeret.”

122 CHAPTER 4
not beyond doubt, Kepler preferred his own data over Witelo’s because it
was based on “regularity and order”. In the final propositions of this section
and the remaining sections of the chapter, Kepler was now able to dealt with
proper subject of the chapter: the quantitative treatment of atmospheric
refraction.
Kepler’s search for a ‘measure’ refraction clearly reveals the idiosyncrasies
of his thinking. He laboriously reported on his persistent efforts to find a
satisfactory law, and although – so we can see with hindsight – he came
tantalizingly close he did not succeed. The successive stages of his attack
display his ever inventive mathematical reasoning, mixed with those typical
Renaissance conceptions of his that make it hard for a modern reader to
distinguish mathematics and physical ideas. In the light of ensuing
developments in seventeenth-century optics, the final stage of Kepler’s
struggle with refraction is the most interesting. Here he took his conception
of the nature of light into account in order to find a law of refraction. In a
kind of microphysical, though far from corpuscular, analysis he considered
the interaction of a surface of light and the refracting medium. At this stage
he move farthest away from traditional approaches. Although the resulting
‘rule’ was phrased in terms of rays, he had taken the true nature of light into
account while analyzing the interaction of rays and (refracting) media. As I
see it, this was possible because of his realist view of mathematical
description. With Kepler, the mathematics of light propagation necessarily
reflected the nature of light.
One may argue that mathematics took the lead in his thinking. Kepler
more or less reduced light to a mathematical entity, a two-dimensional
surface. The geometry of refraction was rather autonomous in his final
attempt to derive a law. 63 Yet, pure formalisms would have been meaningless
for him. Kepler maintained geometrical optics as a mathematical theory
explaining the behavior of light rays. He adopted many concepts of
perspectivist theories of light and refraction, but he applied them in a radical
and sometimes radically different way. On the level of methodology, all
relevant components – physics, mathematics, observation – had been
present in perspectivist optics, but Kepler sought a closer connection
between them and often used these means in a much stricter way. He
repeatedly allowed Witelo’s data to refute the outcome of his trials. Kepler’s
wanted to establish a closer tie between the nature of light and the laws of
optics and derive ‘measure’ from ‘cause’. He openly acknowledged that he
could not realize this ideal. He resorted to a freer mode of reasoning
because, as I see it, he was far too creative a thinker to stick too rigidly to his
ideals.
63 See for example: Buchdahl, “Methodological aspects”, 291.

THE 'PROJET' OF 1672 123
True measures
Even without a true measure of refraction, an inventive mathematician like
Kepler could solve problems in the behavior of refracted rays. In Dioptrice, he
determined properties of spherical lenses in a less rigorous way, pragmatically
applying a rule that had only limited validity. Likewise, his predecessors had
used their limited knowledge to discuss isolated problems regarding
refraction. In order to turn ‘dioptrics’ into a genuine part of the
mathematical science of optics, a true measure of refraction was still needed.
However impressive Kepler’s persistence to find a true measure of
refraction, his efforts will always have a tragic side. Around the same time he
was struggling with the phenomenon, across the Channel the exact law had
already been found by the very man Kepler had been corresponding with:
Thomas Harriot.
Harriot had done so by traditional means that were accessible to Kepler
too: analysis of the observed propagation of light rays. The difference was
that Harriot made new observations and had a lucky hand in this. Harriot’s
success shows that, in the case of refraction, traditional methods could yield
the required result. Around 1597, Harriot had begun looking for a law of
refraction. Initially, he also tried to find a law on the basis of Witelo’s tables.
As these efforts were unsuccessful, he decided that Witelo was unreliable and
started to measure angles of refractions anew. 64 After some fruitless attempts,
he chose a way of measurement that proved very lucky.
In 1601, he measured refraction
by means of an astrolabe suspended
in water (Figure 38). Viewing along
the center R of the astrolabe, he
determined the positions O where a
point was seen when moved along
the lower edge of the astrolabe. Then
he determined the image points B.
The cathetus rule (see page 33) said
that the image point is the
intersection of the normal to the
refracting surface and the incident
ray. All image points were on a circle.
This meant that RO and RB were in constant proportion, and likewise were
the sines of i and r. In a table Harriot compared angles of deviation as he had
measured them with calculated ones, but he did not reveal how he had used
the figure with two concentric circles – which he called ‘Regium’ – for his
calculations. The calculated values give reason to believe that it was the sine
relation he used. 65
64
Lohne, “Geschichte des Brechungsgesetzes”, 159-160.
65
Lohne, “Kepler und Harriot”, 202-203.
Figure 38 Harriot’s measurements (Lohne).

124 CHAPTER 4
Harriot had reconsidered and reapplied traditional methods anew and
found – what might be called – an empirical law of refraction. As contrasted
to Kepler, he had turned to the measurement of refraction, instead of the
theoretical trench-plowing of his hapless correspondent. Harriot does not
seem to have considered the ‘proper cause’ of refraction with which his law
may have been understood. His accomplishments were known only to a
small circle of acquaintances. It is possible that they spread through
correspondence, but he became known as a discoverer of the law of
refraction only in the twentieth century. 66
Around 1620, Willebrord Snel was the next to discover the exact measure
of refraction - again by means readily available to Kepler. He did not publish
his discovery, but it became generally known in the 1660s. How he
discovered the law will remain a matter of conjecture. Snel’s papers on optics
are lost, except for the notes he made in Risner’s Opticae libri quatuor (1606)
and an outline of a treatise on optics discovered in the 1930s. 67 Hentschel has
been the first to make a thorough attempt at reconstruction. In his view, Snel
was inspired by an ‘experimentum elegans’ described by Alhacen and copied
by Witelo that involved a segmented disc lowered into water. This led him to
study the refractaria and, facilitated by his geodetic expertise, to the law of
refraction in secans form. 68 I do not fully agree with Hentschel’s analysis, for
I think that the idea of a contraction of the unrefracted perpendicular ray
may have opened to Snel a more direct route to his discovery. Whichever
interpretation is preferable, the main point is that Snel employed means
readily available to Kepler. What is more, his approach of rational analysis of
mathematical regularities in a set of refracted rays was precisely how Kepler
set about initially. He even analyzed the refractaria from various perspectives,
which makes it all the more surprising that Snel was seemingly unfamiliar
with Paralipomena. 69 It remains to be seen why Snel was successful - or: why
he was satisfied with what he found, as contrasted to Kepler’s fruitless
struggle. Maybe he was less strict in empirical matters or he was - like Harriot
- just lucky with looking at the issue from the right perspective.
Paralipomena and the seventeenth-century reconfiguration of optics
The central concept of perspectiva was the visual ray, which established the
visual relation between objects and observer. 70 In seventeenth-century optics
the concept of ray underwent two substantial changes, both anticipated by
Kepler: the subordination of vision to light and the physicalization of the ray.
66
Lohne, “Geschichte des Brechungsgesetzes”, 160-161. Harriot corresponded with Kepler after the
publication of Paralipomena. The correspondence broke off, however, before Harriot could reveal his
findings. KGW2, 425.
67
The notes are in Vollgraff, Risneri Opticam. The outline was discovered by Cornelis de Waard, who
transcribed and translated it in Waard, “Le manuscript perdu de Snellius”. A German translation is given
in Hentschel, “Das Brechungsgesetz”, 313-319.
68
Hentschel, “Das Brechungsgesetz”, 302-308.
69
Hentschel, “Das Brechungsgesetz”, 334 note 22.
70
Smith, “Saving the appearances”, 86-89.

THE 'PROJET' OF 1672 125
The most important change in the mathematical study of light was the
abandonment of questions of cognition. Perspectivist theory not only
consisted of a theory of perception but also seized epistemological and
psychological problems of visual cognition. 71 The eye was crucial in that the
behavior of rays was understood on the basis of an understanding of visual
cognition. In the seventeenth-century optics the eye became a subordinate
topic in the mathematical study of optics and questions of cognition were
abandoned altogether. Kepler’s theory of image formation was a theory of
rays painting pictures on a dead, passive surface. His theory of the retinal
image was a theory only of ray tracing and he passed over physiological and
psychological issues. Only in the fifth chapter of Paralipomena did he explain
how the eye paints pictures on the retina, after he had explained image
formation, reflection and refraction. The mathematical analysis of the
behavior of light rays was turned into the study of the paths of light rays
without an eye necessarily being present. Instead of the foundation of geometrical
optics, vision became an application of it. The ray became a light ray instead of a
visual ray.
Kepler’s theory was readily assimilated in the first decades of the
seventeenth century. The subordination of vision to the theory of image
formation is clear in most seventeenth-century works on geometrical optics.
This includes Huygens, who deferred his discussion of the eye to the last
chapter of his projected ‘Dioptrique’. Shapiro has pointed out that Barrow’s
thinking in terms of images as the eye perceives them was crucial to his
extension of Kepler’s theory of image formation, as had been the case with
Gregory. 72 Yet, they too confined themselves to retinal imagery and adopted
the Keplerian understanding of the eye as an optical instrument that painted
images on the retina.
Closely connected with the changing role of the eye and vision in the
mathematical study of light is the changing meaning of the optician’s
elementary tool: the ray. Whereas the mathematical line used in optical
analysis in perspectiva represented a real line in space, it came to represent an
imaginary line in time in the course of the seventeenth century. 73 Instead of
constituting light itself, a ray of light became – in various ways – the path
traced out by some substance that constituted light. The corpuscular
conceptions in the new philosophies of the seventeenth century transformed
the light ray into an effect of some material action. Kepler’s conception of
light and his analysis of reflection and refraction anticipated this, but with
him light remained expressly incorporeal. One may say that the combined
subordination of questions of vision and ‘physicalization’ of light constitutes
the transition from medieval perspectiva to seventeenth-century geometrical
optics.
71 Smith, “Big picture”, 587-589.
72 Shapiro, “The Optical lectures”, 137.
73 Smith, “Ptolemy’s search”, 239-240.

126 CHAPTER 4
It is beyond dispute that Kepler was crucial to the development of
seventeenth-century optics. With his seminal work, he gave the study of
optics a new start at the beginning of the seventeenth century. What his
influence was exactly is harder to determine. As a result of the advent of
corpuscular conceptions of nature, his explanation of the nature of light was
outdated almost immediately. On the level of theories and mathematical
concepts his influence is clear: his theory of image formation and of vision
were the starting-point of all subsequent studies. However, his contribution
was largely obscured by the uncredited adoption of his ideas by Descartes
most notably. On the level of methodology the matter is less clear. Descartes
called Kepler his “first teacher in Optics”, but what he had been taught he
did not say. 74 He did not, for one thing, adopt Kepler’s candor as regards the
way he discovered things. Seventeenth-century savants found Kepler’s
Renaissance conceptions hard to take and the odor of mysticism that
surrounded him seems to have been responsible for the fact that few
referred to Kepler directly. As regards the way mathematical reasoning could
be applied to understand natural phenomena, he was quickly overshadowed
by Descartes and Galileo. Huygens, in particular, was silent on Kepler as
regards his approach to optics.
4.1.3 THE LAWS OF OPTICS IN CORPUSCULAR THINKING
The new philosophies of the seventeenth century came to see light as an
effect of some material action. As a consequence, the mechanical analogies
used in perspectivist accounts of reflection and refraction were put in a
different light. Discussions of motions and impact regarding the causes of
reflection and refraction were now connected directly with the essence of
light. Yet, accounting for the nature of light was not integrated with
mathematical analysis of the behavior of light rays at one go. This is evident
in Descartes’ account of refraction in La Dioptrique, a peculiar amalgam of
perspectivist and mechanistic reasoning. In La Dioptrique Descartes made
public the sine law, which he had discovered in Paris in the late 1620s during
his collaborative efforts to realize non-spherical lenses (see section 3.1). How
exactly he arrived at the sine law remains a subject for debate, but it is certain
Descartes did not discover it along the lines of his account in La Dioptrique.
Descartes’ account of refraction is difficult to comprehend in twentiethcentury
parlance. A quick detour via the correspondence of Claude Mydorge,
one of his Parisian collaborators, will be enlightening for modern readers. In
a letter to Mersenne from around 1627, Mydorge used a rule to calculate
angles of refraction, given the angles of one pair of incident and refracted
rays (Figure 39). If FE-GE is the given pair, the refraction EN of HE is found
in the following way. Draw a semicircle around E that cuts EF in F. Draw IF
parallel to AB, and from I drop IG parallel to CE, cutting EG in G. Draw a
second semicircle around E through G. Now draw HM, cutting the first
74 AT 2, 86 (to Mersenne, 31 March 1638).

semicircle in M, and drop MN,
cutting the second semicircle in N.
EN is the required refracted ray.
The rule comes down to a
cosecant ‘law’: cosec i : cosec r =
FE : EG. Later in the letter,
Mydorge applied this rule to lenses
and transformed it into sine form. 75
Mydorge’s rule embodies the
two assumptions that formed the
core of Descartes’ derivation of
the sine law in La Dioptrique. First,
a constant ratio between the
incident and refracted rays,
represented by the constant ratio
of the radii of the two semi-circles.
THE 'PROJET' OF 1672 127
Second, the constant length of the parallel components FO and OI before
and after refraction. In the diagram of Mydorge these assumptions are
represented directly by the lengths of the respective lines. However, instead
of a distance diagram, in La Dioptrique Descartes used a time diagram where
the lengths of lines represent duration (Figure 40). Instead of the two semicircles
representing the constant ratio of the effect of the media, it shows a
single circle. As a consequence the constancy of the parallel component was
represented by lines of differing length (AH and HF).
I have begun with Mydorge’s
rule because it somewhat bridges
the gap between Cartesian
conceptualization of refraction and
our understanding of the sine law.
It gives the modern reader a clear
idea of the assumptions
fundamental to Descartes’
derivation as well as the way he
adapted it to his own line of
thinking. As I will argue below, the
diagrams Descartes used fitted
Figure 39 Mydorge's rule
Figure 40 Descartes’ analysis of refraction
perspectivist analysis of refraction rather than his own account, and he chose
them deliberately. The account of La Dioptrique, with all its complicating
facets, was how seventeenth-century readers got to know the sine law and
the mechanistic interpretation of refraction. It formed the starting-point of
all subsequent accounts of the causes of refraction, although few adopted
Descartes’ conceptual and methodological notions in full.
75 Mersenne, Correspondence I, 404-415. This letter and its import for Descartes’ optics is discussed
thoroughly in Schuster, “Descartes opticien”, 272-277 and Schuster, Descartes and the Scientific Revolution, 304-
308.

128 CHAPTER 4
Refraction in La Dioptrique
Descartes began La Dioptrique with an explication of the way rays of light
enter the eye and are deflected on their way to it. He did not intend to
explain the true nature of light, he said, as the essay ought to be intelligible to
the common reader. He took the liberty, he said, to employ a threesome of
comparisons between the behavior of light and everyday phenomena:
“…; imitating in this the Astronomers, who, although their assumptions are almost all
false or uncertain, nevertheless, because these assumptions refer to different
observations they have made, do not fail to draw many true and well-assured
conclusions from them.” 76
First, light acts like the white stick that enables a blind man to sense objects;
it is an action instantaneously propagated through a medium without matter
being transported. Second, this action is like the tendency of a portion of
wine in a barrel of half-pressed grapes to move to a hole in the bottom. It
works along straight lines that can cross each other without hindrance. In
other words, light is not a motion but a tendency to motion:
“And in the same way, considering that it is not so much the movement as the action
of luminous bodies that must be taken for their light, you must judge that the rays of
this light are nothing else but the lines along which this action tends.” 77
Although essentially light is a tendency to movement rather than actual
motion, with respect to the deflections from its straight path rays of light
follow the laws of motion, Descartes maintained. So, in the third
comparison, the way light interacts with mediums of different nature is
compared to the deflections of a moving ball encountering hard or liquid
bodies. Thus the three comparisons of the first discourse of La Dioptrique
established a qualitative basis for the mathematical account of refraction in
the next.
The second discourse ‘Of refraction’ opens with an account of reflection
providing the conceptual basis for Descartes’ explanation of the ‘way in
which refractions ought to be measured’. 78 It introduces a crucial distinction
with regard to the powers governing the motion of an object: one that works
to continue the ball’s motion and one that determines the particular direction
in which the ball moves. 79 Instead of the more accurate ‘absolute quantity of
force of motion’ and ‘directional quantity of force of motion’, for sake of
convenience I will speak of ‘quantity’ and ‘direction’ both of which may
76 Descartes, AT6, 83. “imitant en cecy les Astronomes, qui, bien que leurs suppositions soyent presque
toutes fausses ou incertaines, toutefois, a cause qu’elles se rapportent a diverses observations qu’ils ont
faites, ne laissent pas d’en tirer plusieurs consequences tres vrayes & tres assurées.” (Translation based on
Olscamp)
77 Descartes, AT6, 88. “& ainsy, pensant que ce n’est pas tant le mouvement, comme l’action des cors
lumineus qu’il faut prendre pour leur lumiere, vous devés iuger que les rayons de cete lumiere ne sont
autre chose, que les lignes suivant lesquelles tend cete action.” (Translation based on Olscamp)
78 “… en quelle sorte se doivent mesurer les refractions”, AT6, 101-102.
79 “Seulement faut il remarquer, que la puissance, telle qu’elle soit, qui fait continuer le mouvement de cete
balle, est differente de celle que la determine a se mouvoir plustost vers un costé que vers un autre, …”
AT6, 94.

THE 'PROJET' OF 1672 129
apply to Cartesian motion proper as well as to tendency to movement. When
a ball rebounds from the surface of an impenetrable body the following
happens. The quantity of its motion is unaffected because it remains moving
through the same medium - the air surrounding the body - and only the
direction changes. Regarding the parallel and perpendicular components of
the direction, Descartes noted that the body offers resistance only in the
direction perpendicular to its surface. Thus the parallel component is
unaltered.
To determine the path of the
ball after the impact, Descartes
switched to a derivation in which
he graphically mathematized the
assumptions just established
(Figure 41). In circle AFD radius
AB represents the path along
which the ball approaches the
surface where it rebounds from B
in some direction. As the quantity
of motion is constant, the ball
must traverse the same distance
Figure 41 Descartes’ analysis of reflection
after reflection. It thus reaches the circumference of the circle somewhere.
Since the parallel component of its direction is also constant, it follows that
the horizontal distance traversed after reflection must be equal too.
Therefore, BE is equal to BC. Under these conditions the ball can either
arrive at point D or point F on the circle. It cannot penetrate the body below
GE and so F is the only option left. “And thus you will easily see how
reflection occurs, namely according to an angle always equal to the one that
is called angle of incidence”, Descartes concluded without much further
ado. 80
Like reflection, refraction is understood as the combined effect on the
quantity and the direction of motion. The only difference is that in refraction
the ball penetrates the medium. In other words, it enters a medium of
different density. Therefore the quantity of motion changes. It does so at the
passing of the surface separating both mediums. This can be compared to
smashing a ball through a thin cloth. It loses part of its speed, say half. Again
only the perpendicular component of the direction of the motion is affected
and the parallel component remains unaltered. As in the case of reflection,
Descartes switched to a mathematical derivation in the form of a diagram to
determine the exact path of the ball after impact (Figure 40). As a result of
the loss of speed, it takes the ball twice as long to reach the circumference of
the circle after impact at B. However, as its determination to advance parallel
to the surface is unchanged, it moves twice as far to the right in this time.
80 “Et ainsy vous voyés facilement comment se fait la reflexion, a sçavoir selon un angle tousiours esgal a
celuy qu’on nomme l’angle d’incidence.” AT6, 96.

130 CHAPTER 4
Therefore the distance between lines FE and HB must be twice a large as that
between AC and HB. As a result, the ball reaches point I on the circle. The
same is the case when instead of a cloth the ball hits the surface of a body of
water. For the water does not alter the motion of the ball any further after it
has passed the surface, according to Descartes. When the ball passes a
boundary where in some way or another its quantity of motion is augmented,
it reaches the circumference of the circle earlier and is deflected towards the
normal of the surface. Note that Descartes did not specify the change of the
perpendicular component, a point that is often overlooked. He did not know
that amount and he did not need to, for the two assumptions he used suffice
for the derivation of the sine law. 81
As Descartes took the motions of the ball to reflect the deflections of
light, he could now draw his main conclusion. Rays of light are deflected in
exact proportion to the ease with which a transparent medium receives them
compared to the medium from which they come. The only remaining
difference between the motion of a ball and the action of light is that a
denser medium like water allows rays of light to pass more easily. The
deflection caused by the passage from one medium into another ought to be
measured, not by the angles made with the refracting surface, but by the lines
CB and BE. Unlike the proportion between the angles of incidence and
refraction, the proportion between these sines remains the same for any
refraction caused by a pair of mediums, irrespective of the angle of
incidence. Et voilà, the law of sines.
Epistemic aspects of Descartes’ account in historical context
Both historically and intrinsically, Descartes’ account of refraction is a key
text in the transition from medieval perspectiva to seventeenth-century
optics. Yet, the line of inference is subtle and, at many points, implicitly
pursued. I will have to enlarge in some detail on its epistemic aspects in their
historical context.
At least three levels of inference can be distinguished in Descartes’
account. In the first place the level of mathematics. This holds the derivation
of the sine law from the two assumptions conveyed in the diagrams
accompanying his discourse. First, the passage to another medium alters in a
fixed ratio the quantity of motion. This ratio is represented by the radius of
the circle. Second, the parallel component of the direction of motion is
unaffected. This is represented by drawing horizontal lines in proportion to
the successive times to travel to and from the center of the circle. The
mathematical inference of Descartes’ account constituted a successful
culmination of perspectivist optics, in that Descartes was the first to derive a
law of refraction on the analytical groundwork laid by Kepler and his
forebears. He brought consistency to the analysis of reflection and refraction
by having the parallel component constant in all cases. More important, in
81 When both aspects of the motion are interpreted as speeds the assumptions can be written as: vr = nvi
and vi sini = vr sinr, which directly yield sini = n sinr. See Sabra, Theories of Light, 111.

THE 'PROJET' OF 1672 131
the first assumption, he stated an exact relationship between the medium and
the length of a ray. Combined with the second assumption – which was not
new – the sine law could be derived. Mathematically speaking, the proof – as
Newton later phrased it – was not inelegant. It was fairly undisputed in the
seventeenth century and the starting point for much optical investigations. 82
Descartes’ first assumption was more than a purely mathematical
assumption, which brings us to the second level of inference that holds the
physical properties of rays. The physics of rays had been central in
perspectivist optics, but the content of Descartes’ assumptions was
innovative. According to Sabra and Schuster, stating a positive dependence
of the motion of light on the density of the medium, irrespective of the
direction of propagation, made up the decisive break with tradition. 83
Descartes may have drawn inspiration for this from his reading of
Paralipomena (which he did not acknowledge at all in La Dioptrique). In
proposition XX of chapter 1 and the sequel section of chapter 4, Kepler also
associated the propagation of a ray with the medium. Descartes may have
read Kepler’s diagrams physically, so that the length of the rays represent the
action of light as affected by the media. 84 Descartes’ diagram represented the
actions involved when a ray enters a refracting medium and served to justify
his assumptions. He did so by drawing an analogy between a refracted ray
and a tennis ball struck through a frail canvas by the man in the diagram
(Figure 40).
As we have seen, these mechanical analogies had a long history in optics
with a direct line from Alhacen to Kepler and, now, Descartes. The
mechanical analogies had a different meaning for Descartes than for his
perspectivist forebears. To an Alhacen the motions of bodies compared to
light only with respect to its propagation, not its essence. According to
Descartes light was essentially corpuscular. He made clear that they went
further than a mere analogy:
“… when [rays] meet certain other bodies they are liable to be deflected by them, or
weakened, in the same way as the movement of a ball or a rock thrown in the air is
deflected by those bodies it encounters. For it is quite easy to believe that the action or
the inclination to move which I have said must be taken for light, must follow in this
the same laws as does movement.” 85
However, Descartes took care not to transgress the conceptual and
methodological boundaries of perspectiva openly. He presented his account
82
Huygens’ case is discussed below in section 4.2.1., Newton in section 5.2.2. of the next chapter. This
theme is leading in Dijksterhuis, “Once Snel breaks down”. Newton’s view is cited below on page 133,
footnote 98.
83
Sabra, Theories, 97-107; Schuster, Descartes, 333-334.
84
Schuster, “Descartes opticien”, 279-285; Schuster, Descartes, 334-336.
85
Descartes, AT6, 88-89. “mais, lors qu’ils rencontrent quelques autres cors, ils sont sujets a estre
détournés par eux, ou amortis, en mesme façon que l’est le mouvement d’une balle, ou d’une pierre iettée
dans l’air, par ceux qu’elle rencontre. Car il est bien aysé a croire que l’action ou inclination a se mouvoir,
que j’ay dit devoir estre prise pour la lumiere, doit suivre en cecy les mesmes loys que le mouvement.”
(Translation based on Olscamp)

132 CHAPTER 4
in terms of analogies and explicitly said these did not reflect the true nature
of light. Restricting in this way his account to the behavior of rays, he
methodologically tied in with tradition. Still, mechanistic thinking was at the
heart of La Dioptrique. Assuming a proportionality between density and
motion is almost unthinkable outside a corpuscular framework. Indeed, at
the close of the second discourse Descartes showed his hand. The
comparisons had a much higher content of realism than suggested by his
circumspect introduction of them.
“For finally I dare to say that the three comparisons which I have just used are so
correct, that all the particularities that that can be noted in them correspond to certain
others which are found to be very similar in light; …” 86
If the mechanisms Descartes employed in the analogies and to which he
ascribed a fair degree of realism do little to persuade our post-Galilean
minds, one ought to remember that they were modeled on an understanding
of motion that was rooted in a hydrostatics of pressures rather than a
kinematics of velocities. Probably this was also one of the reasons the
analogies did not convince his seventeenth-century readers either. 87
Descartes’ intricate employment of mechanical analogies brings us to the
third level of inference in his account of refraction, where the physical nature
is involved in the analysis. Although he did not elaborate his theory of light
and circumspectly presented the mechanics of deflected motion as analogy,
Descartes’ line of reasoning strongly suggests that the laws of optics to be
derived from his mechanistic understanding of light. In Sabra’s words: “As
repeatedly asserted by Descartes, the ‘suppositions’ at the beginning of the
Dioptric belong to this [domain of a priori truth]”. 88 This is substantiated by
the fact that Descartes deviated from perspectivist tradition in a second
important respect as well. In La Dioptrique he did not explicitly call for an
empirical foundation of the sine law. In this way, Descartes’ derivation of the
sine law was intended as a derivation from the true nature of light.
Historian’s assessment of Descartes’ optics
The question whether or not Descartes actually succeeded in deriving the
sine law from his mechanistic theory of light has been a matter of incessant
debate among historians of science. Although few seventeenth-century
students of optics were convinced by Descartes argument, I think it
appropriate to digress somewhat to contemporary evaluations because these
are illuminating as regard the exact purport of his account.
Many have argued that Descartes’ claim, that a tendency to move is
subject to the same laws as motion itself, was mere rhetoric. Schuster, on the
other hand, argues that Descartes’ theory of light did provide the basis of the
86 “Car enfin j’ose dire que les trois comparaisons, dont je viens de me servir, sont si propres, que toutes
les particularités que s’y peuvent remarquer, se raportent a quelques autres qui se trouvent toutes
semblables en la lumiere; …” AT6, 104.
87 Except Clerselier who expressly defended Descartes’ mechanistic models; Sabra, Theories, 116-135.
88 Sabra, Theories, 44.

THE 'PROJET' OF 1672 133
analogies, despite the fact that it hardly appears in La Dioptrique. 89 Drawing
on the work of Mahoney, he says that the analogies provided a ‘heuristic
model’ that legitimately compared the action of light with the motion of a
ball. By leaving specific material factors in the motion of the ball aside,
Descartes could single out the ball’s tendency to move rather than its
motion. He then was ready to consider this tendency and distinguish
between “… the power, …, which causes the movement of this ball to
continue …” and “… that which determines it to move in one direction
rather than in another, …” 90 According to Schuster this does not refer to a
distinction between force of motion and direction of motion, but to a
distinction between quantity of force of motion and directional magnitude of
force of motion. The two assumptions of Descartes’ derivation are based on
this distinction: the quantity depended on the medium and the parallel
component of directional magnitude was constant. In La Dioptrique
Descartes labeled the directional magnitude with the term ‘determination’ in
order to analyze the components of the action without implicating the
notion of velocity. 91
With this interpretation of the analogies, Descartes’ analysis is not directly
at odds with the system he expounded in Principia Philosophiae and Le Monde.
There he had made the same distinction between quantity and directional
magnitude. The first law of nature states that the quantity of force of motion
is constant when a body is in uniform rectilinear motion; the third law states
that a force of motion is conserved in a unique direction (tangent to the path
of motion). 92 According to Schuster, the tension between the analogies and
the tendency theory can be resolved when Descartes’ heuristic use of the
analogies is interpreted in the terms of his theory of motion. 93 In the light of
the Galilean conception of motion Huygens and Newton employed (as do
we), Descartes’ claim that he derived the laws of optics from his mechanistic
principles was untenable. Sabra has sufficiently pointed this out. 94 Yet, this
was not so much because his system was incoherent or inconsistent as,
rather, because the interpretation of the underlying principles had changed.
Descartes usually considered motion at an instant of impact and discussed it
in terms of the body’s force to move. In the light of this science of motion,
the mathematical derivation of the sine law can indeed be physically
interpreted in a plausible manner. Yet, through his crude presentation in La
Dioptrique Descartes made little effort to prevent misunderstandings and
misinterpretations.
89
Schuster, “Descartes opticien”, 261-272 Schuster, Descartes, 273; Mahoney, Fermat, 387-393; Sabra,
Theories, 78-89.
90
Descartes, AT6, 94-95.
91
Schuster, “Descartes opticien”, 258-261; Schuster, Descartes, 293
92
Schuster, Descartes, 288.
93
Schuster, “Descartes opticien” 261-265.
94
Sabra, Theories, 112-116.

134 CHAPTER 4
Schuster has proposed a possible route along which Descartes’ discovery
of the sine law may have taken place. 95 In it, his collaboration with Mydorge
plays an important role - the cosecant rule being a crucial step towards the
law of sines. According to Schuster, the actual discovery was independent of
Descartes’ mechanistic ‘predilections’; rather the other way around: the latter
were triggered by the former. 96 Shea has argued for a different route to the
discovery, via measurements of angles of refraction by means of a prism. 97 In
this variant too, the discovery was the result of an analysis of the behavior of
rays. Descartes developed his mechanistic interpretation of his analysis of
refraction after the discovery. With the presentation in La Dioptrique, he then
obscured his analysis and explanation considerably. He adopted the use of
analogies and adapted his derivation of the sine law to the perspectivist
analysis of refraction. It appears as if Descartes tried to make his theory look
as traditional as possible.
Yet, he deviated from tradition by reversing the way in which he justified
the law. Descartes suggested that the laws of optics ought to be based on
prior principles regarding the nature of light. And despite the circumspection
of his presentation, it was clear that he regarded the mechanistic causes of
refraction an important, if not crucial, matter. In this way, he shifted the
focus of the mathematical study of light towards the nature of light and the
causes of the laws of optics. La Dioptrique is hard to characterize in terms of
seventeenth-century geometrical optics. On the one hand, it was obviously a
treatise on geometrical optics. It discussed the behavior of light in terms of
the mathematical laws of the propagation of rays, in particular as they are
refracted by lenses. Still, it did not offer quite as thorough an account of
lenses as one would expect from a mathematical treatise. As we have seen in
chapter two, La Dioptrique did not elaborate the mathematical theory of
refraction in a way modeled after Kepler’s Dioptrice. Its main goal was to
establish the law of refraction and explain its main consequences for the
working of the telescope.
Reception of Descartes’ account of refraction
In Descartes’ system of natural philosophy, natural phenomena were
explained from mechanistic principles. His optics was the most elaborate
example of this project. Even if this elaboration was not fully unproblematic,
it made clear what a new, mechanistic science of optics should be about. It
did not halt at the mathematical description of natural phenomena, nor at
depicting micro-mechanisms to explain them, but sought to explain the
mathematical laws of nature by its mechanistic nature. Notwithstanding
recent pleas by historians of science for Descartes’ integrity, few
contemporaries accepted his explanation of refraction. “The author would
have demonstrated not inelegantly the truth of this, if only he had not left
95
Schuster, Descartes, 321-326. See also: Costabel, “Refraction et La Dioptrique”.
96
Schuster, Descartes, 343-346.
97
Shea, Magic, 156-157.

THE 'PROJET' OF 1672 135
room for doubt concerning the physical causes he assumed”, Newton wrote
30 years later. 98 In view of Galileo’s science of motion it is doubtful whether
the motion of a ball struck through a frail canvas is subject to the
assumptions Descartes made. His explanation of refraction into a denser
medium – towards the normal – was regarded most problematic. In order to
account for the necessary increase of motion, he introduced the rather ad
hoc assumption that the ball was struck again at the refracting surface.
Besides rejecting Descartes’ theory of light on the conviction that the speed
of light is finite, in the ‘Projet’ Huygens explicitly mentioned this extra
assumption as one of the difficulties in Descartes’ derivation. 99
Newton and Huygens wrote at a time when the law of sines as such had
been generally accepted. This had taken some twenty years, during which it
only slowly became widely known. Cavalieri in 1647 did not employ the law
of sines and Gregory seems to have been ignorant of it as late as 1663. As we
have seen in the previous chapters, Huygens was one of the very few to
pursue the study of dioptrics in this period. Compared to the preceding
decades, the 1660s witnessed a true upsurge of the study of light. The
investigations of Grimaldi, Boyle, Hooke, Newton, Bartholinus, brought to
light a collection of new properties shaking the foundations of optics.
Remarkably, the final acceptance of the law of sines coincided with
accusations of plagiarism directed at Descartes. In De natura lucis et proprietate
(1662) Isaac Vossius said that Descartes had seen Snel’s papers and
concocted his own proof. We now know this charge to be undeserved but it
has been adopted by many since. Descartes may have heard of Snel’s
achievement through his contacts with the circle that included Golius (Snel’s
successor) and Constantijn Huygens sr. around 1632, but he had found the
law much earlier. Christiaan Huygens started to display doubts regarding
Descartes’ originality since the early 1660s. Probably spurred by Vossius’
claims, he traced and examined Snel’s papers. As late as 1693 he voiced his
opinion as follows: “It is true that from all appearances these laws of
refraction aren’t the invention of Mr. des Cartes, because it is certain that he
has seen the manuscript book of Snel, which I also have seen.” 100 Most
remarkable about this is that Huygens could have known, through his father,
much earlier about Snel’s achievement. Constantijn sr. had heard of it
through a letter from Golius of 1 November 1632. Apparently the topic had
never entered their conversation.
The slow adoption of the sine law may have been brought about by the
bad odor of Descartes’ philosophy, or simply the slow diffusion of his
works. Fermat was convinced of the sine law’s validity only after he found
98 Newton, Optical lectures, 170-171 & 310-313.
99 A similar conclusion can be drawn with respect to Descartes’ theory of hydrostatics on which his
concept of ‘conatus’ was based. Shapiro, “Light, pressure”, 260-266.
100 “Il est vray que ces loix de la refraction ne sont pas l’invention de Mr. des Cartes selon toutes les
apparences, car il est certain qu’il a vu le livre manuscrit de Snellius, que j’ay vu aussi; ...” OC10, 405-6. See
also OC13, 9 note 1.

136 CHAPTER 4
his own demonstration. Right after the publication of La Dioptrique he had
severely criticized Descartes’ derivation, and maintained his objections when
supporters of Descartes reopened the debate in 1657. 101 Employing the
principle of natural economy, previously used by Hero and Witelo for
reflection, Fermat deduced the law of sines in 1662, thus strengthening his
conviction that Descartes’ mechanistic line of reasoning had been false. To
know that the law was independent of Descartes’ mechanistic reasoning may
have facilitated its acceptance, although it may well be that the ostensible
non-acceptance was simply a matter of inactivity on the front of optics
during the 1640s and 1650s.
The reception of La Dioptrique makes clear that the treatise is hard to
situate in the development of seventeenth-century optics. It formed the
starting-point of most subsequent investigations in optics, and has therefore
been the focus of many historical studies. 102 La Dioptrique showed how the
properties of light could be discussed in corpuscular terms and its readers
got this message. Although few agreed with the details of Descartes’
derivation of the sine law, nor with his system of mechanistic philosophy in
full, he set the idiom for the all-prevailing thinking on light in corpuscular
terms. As a consequence, the traditional analogies between light and motion
implied a potential claim about the true nature of light and could not be used
as informally as before. Descartes had intended to found the laws of optics
in the mechanistic nature of light, but his derivation was not free from
ambiguities and obscurities. A mathematician like Barrow adopted the
corpuscular understanding of nature but not Descartes’ approach to
explanation. We now turn to him, to see how he dealt with questions
regarding the status of the corpuscular nature of light and how it ought to
explain the laws of optics.
Barrow’s causal account of refraction
Barrow was a mathematician with a clear awareness of the epistemological
intricacies of mathematics and its applications to nature. The lectures on
mathematics which he delivered at Cambridge between 1664 and 1666 dealt
at great length with the status of mathematical concepts and methods and
their relevance for the study of nature. 103 His subsequent lectures on optics
are likewise riddled with epistemic statements. Lectiones XVIII of 1669 is
illuminating with respect to Huygens’ ‘Projet’ as it assigns a similar role to
explanations of the causes of the laws of optics. The subject of the lectures
was ‘Optics’, one of the fields that are “… bright with the flowers of Physics
and sown with the harvest of Mechanics,…” 104 The core of this science
101
The debate is listed in Smith, Descartes’s theory of light and refraction, 81-82 and discussed in detail in Sabra,
Theories of Light, 116-135.
102
Sabra, Theories, 12.
103
Published in 1666 as Lectiones mathematicae XXIII. They were translated by John Kirkby and published in
1734 under the title The usefulness of mathematical learning etc. It is cited in Shapiro, Fits providing improved
translations.
104
Barrow, Lectiones, [10].

THE 'PROJET' OF 1672 137
consisted of six generally accepted principles required to elaborate
mathematical theory. In a way reminiscent of the ‘Projet’, Barrow said that
these hypotheses, as he called them, were empirically founded but also
needed some sort of explanation:
“The hypotheses agree with observation, but we must also fortify them with some
support of reason, by treading on the foundations and suppositions laid down.” 105
In his first lecture Barrow discussed these foundations and suppositions,
although he mainly defined terms like ‘light’ (in relation to illumination,
images, ‘phasmata’ and the like), ‘refraction’ and ‘opaque’. He then proposed
a theory of light that is a hybrid of viewing light as a pulse and as a pressure
propagated simultaneously in the first and second matter of the Cartesian
scheme. 106 Whatever he meant precisely, Barrow did not lend much weight to
this theory.
“Still, since it is desirable for me to lay some preliminary foundations about the nature of
light, to agree with my explanation of hypotheses which I shall later offer, I conceive
the facts to be these, or something like them: …” 107
These preliminary foundations merely needed to be consistent with the
ensuing explanations of the laws of optics and Barrow expressly did not
claim any authority in these matters. 108 In what followed, Barrow’s theory of
light came down to considering a ray to be the path traced out by a pulse-like
entity, “… two-dimensional and like a sort of rectangular parallelogram lying
in a plane at right angles to the surface of the inflecting medium, …” 109 This
conception of a physical ray traced out by a line of light emitted by a shining
object went back to Hobbes’ theory of light. Barrow’s derivation of the law
of sines can likewise be traced back to Hobbes. 110 With this definition of a
ray, Barrow now could make ‘some attempt to explain’ the laws of optics,
stressing once more that they were empirically founded:
“… I need practically nothing else to explain the hypotheses which all opticians in
common with each other assume and which must necessarily be laid down as a
foundation for building up this science. I shall make no effort to prove what I have
said, since … it seems clearer than light itself that such proofs cannot be given,
although a number of experiments show that they are given in actuality.” 111
Besides accounting for the rectilinearity of light rays, he discussed some basic
assumptions of geometrical optics, like the fact that ‘inflections’ take place in
a plane perpendicular to the surface of the ‘inflecting’ medium. Then, in the
second lecture, he moved on to these ‘inflections’ proper, reflection and
105
Barrow, Lectiones, [26].
106
Barrow, Lectiones, [15-16].
107
Barrow, Lectiones, [15]; (emphasis in original).
108
Barrow, Lectiones, [8, 15].
109
Barrow, Lectiones, [26].
110
Shapiro, “Kinematic optics”, 177-181. Hobbes’ optics is discussed in the next chapter, section 5.2.1.
111 Barrow, Lectiones, [17]

138 CHAPTER 4
refraction. 112 For reflection, he
considered BD – a line of light in the
most realist sense of the word –
colliding obliquely with a reflecting
surface EF (Figure 42). He argued
that, after B hits the surface, this end
of the line of light rebounds while end
D continues its way, resulting in the
rotation of BD around its center Z.
This rotation lasts until D hits the
surface and the line of light is in position . The line of light then continues
towards . From the symmetry of the situation the equivalence of the
angles of incidence and reflection follows directly.
To substantiate his claim, Barrow invoked a general ‘law’ of motion:
“… that it is constantly found in nature, when a straight movement degenerates into a
circular one, that it is the extreme parts of the moving objects that direct and control all
motion.” 113
He applied the same law to derive
the sine law (Figure 43). On
entering the more resisting
medium below EF, point B of the
line of light BD will be slowed
down while D continues with the
original speed. As a consequence
DB will be rotated around a point
Z until D also reaches the ‘denser’
medium. Then the line of light
will continue along a straight path.
Now, the proportion between ZD
and ZB is constant for any angle
of incidence and depends upon
the particular difference of the
densities. From this it easily
follows that for i =GBM and
r =N, sin i : sin r = ZD : ZB. 114 After thus explaining refraction into a rarer
medium and total reflection, Barrow was ready to elaborate the ‘Optic
Science’ of his lectures in the common manner:
“…considering rays as one-dimensional (seeing that the other dimensions, in which
physicists delight, have no importance for the calculations here undertaken).” 115
Figure 42 Barrow’s explanation of reflection.
Figure 43 Explanation of refraction.
112 Like Maignan in his Perspectiva Horaria (1648), Barrow added a derivation of the law of reflection which
Hobbes had not provided. See Shapiro, “Kinematic optics”, 175-178.
113 Barrow, Lectiones, [28].
114 Barrow, Lectiones, [29-31]
115 Barrow, Lectiones, [39-41]

THE 'PROJET' OF 1672 139
Lectiones XVIII treated optics as the mathematical science aimed at the
analysis of the behavior of light rays. Priority was with the laws of optics,
being laws of rays that were justified empirically and generally accepted. In
this sense the lectures stood with both feet in traditional geometrical optics.
Yet, Barrow was too conscious of epistemic issues regarding mathematics
and of the new developments in natural philosophy to treat optics in the
outright traditional manner of other contemporary works. A good example is
the Opera mathematica, a mathematics textbook from 1669 by the Flemish
Jesuit Andreas Tacquet, a correspondent of Huygens. In its catoptrical
chapters, Tacquet makes room for a noncommittal survey of explanations of
reflection: some give natural economy as the ‘ratio’ of reflection, others
maintain that the perpendicular component of a ray’s motion is inverted, and
so on. 116 Even Descartes is reviewed, stripped of all corpuscular trimmings to
be sure. Tacquet did not show preference for any of the alternatives, he only
explained the various ways in which the law of reflection could be deduced.
The business of a mathematical student of light was to establish those
properties of rays interacting with varying mediums so that the laws
describing its behavior could be derived logically.
For Barrow mixed mathematics - where natural things are considered in
their quantitative aspects - was a genuine part of mathematics. In his lectures
on mathematics, Barrow effectively discarded the distinction between
sensible and intelligible matter, so that a science like optics could approach
the certainty of geometry. The certainty of inferences only depended on the
certainty of the presuppositions - axioms, postulates, principles. 117 Barrow
presented his explanations as a non-committal elucidation of empirically
founded laws, similar to the mechanical analogies of perspectivist theory.
The new mode of thought regarding the nature of things had changed the
understanding of the nature of light and the causes of reflection and
refraction. Yet, compared to these, corpuscular accounts of the causes of
reflection and refraction obtained a different meaning, as it implied a
potential claim about the true nature of light. This, combined with his
epistemic awareness, may explain Barrow’s reluctance to make strong claims
about his explanations.
In his comments, Barrow considerably qualified the status of his theory
of light and his causal accounts. His focus was on the laws and he did not
elaborate his account of the mechanistic nature of light in any detail or
explore its consequences. He was rather vague about the necessity and role
of such an account. The laws of optics should ‘not be repugnant to reason’
and be given ‘some support of reason’. He invoked a law of motion, but did
not intend to prove the laws like Descartes, by deriving them from his theory
of light. He offered a physical rationale for the laws, without making clear
the exact purpose of his explanations. As a consequence, he parried the
116
Tacquet, Opera mathematica (Antwerp, 1669), Catoptricae libri tres, 217-218
117
Shapiro, Fits, 31-36.

140 CHAPTER 4
question raised by the new philosophies of what status the corpuscular
nature of light should have and how it ought to explain the laws of optics.
4.2 The mathematics of strange refraction
Kepler and Descartes had drawn attention to the problem of the relationship
between a theory expounding the true nature of light and the mathematical
behavior of light rays. It remains to be seen how Huygens considered this
issue. What exactly did he mean by explaining refraction with waves? What
were those waves and how would he proceed from there to the sine law?
The statements in the ‘Projet’ suggest that his opinion about causal accounts
was similar to Barrow’s. Explaining refraction was a rather non-committal
affair to satisfy the minds of the particularly curious. Still, he wanted to solve
the problem strange refraction posed for Pardies’ explanation of ordinary
refraction. Apparently, the nature of light was serious enough a matter for
Huygens first to wish to get this inconsistency out of the way. 118 Given the
definition of the problem, the line of his first attack of strange refraction is
rather surprizing.
Huygens’ first attempt at understanding strange refraction is found on
some ten pages in his notebook. 119 In my view, it must have taken place
around the same time he noted down the ‘Projet’, somewhere during the
second half of 1672. 120 On the first pages Huygens jotted down some
sketches characterizing the phenomenon. The first shows five pairs of
incident and refracted rays (Figure 44). One of each pair, indicated by the
letter r is refracted regularly (‘regelmatig’) according to the sine law, the other
one indicated by the letter o is refracted irregularly (‘onregelmatig’). 121 Below,
Huygens wrote what is irregular about it:
“The perpendicularly incident [ray] is refracted It does not make a double reflection.” 122
118 Ziggelaar correctly points out that the problem of strange refraction was a reason Huygens did not
directly elaborate ‘Projet’ (which he sees as a new plan for a treatise on dioptrics), but he does not discuss
his first attempt to solve it beyond a single, and incorrect, characterization. Ziggelaar, “How”, 181-182.
See also page 162.
119 Hug2, 173v-178v. It consists of seven pages numbered by Huygens (175r-178r), preceded by two and a
half pages with some notes and followed by a page containing a further note plus the record of an
experiment performed in 1679 (discussed in section 5.3.1) Parts of their content are reproduced in OC19,
407-415.
120 I disagree with the editors of the Oeuvres Complètes regarding the dating of the papers. I think this first
study took place around the time of Pardies’ letter, much earlier than they presume. On 4 September
1672, hardly a month later, Huygens wrote to his brother Constantijn, saying he was not yet going to
publish “what I have observed of the crystal or talc of Iceland” (OC7, 219. “…ce que j’ay observè du
Chrystal ou Talc d’Islande; …”). I think this remark refers to his discovery of another peculiar
phenomenon displayed by Iceland crystal – polarization – recorded on the final pages of his investigation.
The discovery is in OC19, 412-414. The editors date these between December 1672 and June 1673, but it
is possible that they – or similar notes now lost – were written at the same, earlier date.
121 Hug2, 173v. One half of Hug2, 174r is torn away; the page contains a remark that seems of a later date.
122 Hug2, 173v; OC19, 407. “Perpendiculariter incidens refringitur Non facit duplicem reflexionem.” The
editors combine this with a remark written on Hug2, 175v.

THE 'PROJET' OF 1672 141
Figure 44 Sketch of refracted rays in Iceland crystal: r (‘regelmatig’) for
ordinary refraction; o (‘onregelmatig’) for strange refraction.
On 8 July 1672, Pardies wrote Huygens about strange refraction. He had
visited Picard and taken a look at a piece of Iceland crystal brought from
Denmark. Pardies did not believe the phenomenon contradicted the sine
law, as he thought it could be explained from the composition of the crystal.
“… it seems to me that it is not as troublesome as I had imagined to explain this effect.
… I am very much mistaken if one cannot demonstrate that, if one were to cut various
pieces of glass in rhomboid shape and simply put one on the other to make a total
rhomboid out of them, two refractions would present themselves.” 123
Some sketches Huygens made in his notebook
around the same time are reminiscent of Pardies’
view (Figure 45). They seem to explore how the
composition of the crystal may explain strange
refraction. The surface is drawn indented, so that
part of the perpendicularly incident light actually
falls on an oblique surface. A perpendicular ray
falls upon the indented surface so that part of the
wave is divided into many small wavelets, that
proceed obliquely to the surface. 124
Apparently, Huygens did not accept Pardies’ idea, for he did not
elaborate it beyond these sketches. Moreover, he ended his first study with
the conclusion that the refracted perpendicular contradicted the wave
explanation of refraction. 125 A sketch on the next page of his notebook makes
123 OC7, 193. “… il me semble qu’il n’est pas si malaisé que je m’estois imaginé, d’expliquer cét effet. Je
suis fort trompé si l’on ne peut démonstrer que si l’on taillait plusiers pieces de verre en rhomboide et
qu’on les mit simplement l’une sur l’autre pour en faire un rhomboide total, il s’y feroit deux refractions.”
124 Hug2, 178v; OC19, 415.
125 See below page 151 footnote 148.
Figure 45 A refracted
perpendicular caused by the
composition of the crystal.

142 CHAPTER 4
it clear what kind of problem strange refraction constituted
for the wave theory (Figure 46). It shows the strange
refraction of a perpendicular ray along with, what seems to
be, the propagation of waves. 126 After having passed the
refracting surface, the waves proceed obliquely to their
direction of propagation, which contradicts the assumptions
of Pardies' theory. Thus this tiny sketch illustrates what
Huygens called the ‘difficulté’ of strange refraction.
Strange refraction posed a problem for the explanation of