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Volume 9
JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology,
Pasadena, CA, USA.
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
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
University of Twente,
Enschede, The Netherlands
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A history of Traité de la Lumière
Huygens’ optics
New light on Huygens
The Tractatus of 1653
Ovals to lenses
A theory of the telescope
The focal distance of a bi-convex lens
Dioptrics and the telescope
Kepler and the mathematics of lenses
Image formation
Perspectiva and the telescope
The use of the sine law
Descartes and the ideal telescope
After Descartes
Dioptrics as mathematics
The need for theory
The micrometer and telescopic sights
Understanding the telescope
Huygens’ position
The use of theory
Huygens and the art of telescope making
Huygens’ skills
Alternative configurations
Experiential knowledge
Inventions on telescopes by Huygens
Dealing with aberrations
Properties of spherical aberration
Specilla circularia
Theory and its applications
Putting theory to practice
A new design
Newton’s other look and Huygens’ response
Dioptrica in the context of Huygens’ mathematical science
The mathematics of things
Huygens ‘géomètre’
The ‘raison d’être’ of Dioptrica: l’instrument pour l’instrument
Huygens the scholar & Huygens the craftsman
‘Projet du Contenu de la Dioptrique’
The nature of light and the laws of optics
Alhacen on the cause of refraction
Kepler on the measure and the cause of refraction
The measure of refraction
True measures
Paralipomena and the seventeenth-century reconfiguration of optics 124
The laws of optics in corpuscular thinking
Refraction in La Dioptrique
Epistemic aspects of Descartes’ account in historical context
Historian’s assessment of Descartes’ optics
Reception of Descartes’ account of refraction
Barrow’s causal account of refraction
The mathematics of strange refraction
Bartholinus and Huygens on Iceland Crystal
Bartholinus’ experimenta
Huygens’ alternatives
Rays versus waves: the mathematics of things revisited
The particular problem of strange refraction: waves versus masses
A new theory of waves
A first EUPHKA
The solution of the ‘difficulté’ of Iceland Crystal
Undulatory theory
Explaining strange refraction
Traité de la Lumière and the ‘Projet’
Comprehensible explanations
Mechanisms of light
Hobbes, Hooke and the pitfalls of mechanistic philosophy:
rigid waves
‘Raisons de mechanique’
Newton’s speculations on the nature of light
The status of ‘raisons de mechanique’
A second EUPHKA
Danish objections
Forced innovation
Hypotheses and deductions
Creating Traité de la Lumière
Completing ‘Dioptrique’
Huygens’ dioptrics in the 1680s
From ‘Dioptrique’ to Traité de la Lumière
The publication of Traité de la Lumière
Traité de la Lumière and the advent of physical optics
Mathematization by extending mathematics
The matter of rays
The mathematics of light
Traité de la Lumière and Huygens’ oeuvre
Huygens’ Cartesianism
The subtle matter of 1669
Huygens versus Newton
Huygens’ self-image
The reception of Huygens
A seventeenth-century Archimedes
From mathematics to mechanisms
Huygens and Descartes
The small Archimedes
This page intentionally left blank
“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
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
Undated note by Christiaan Huygens (probably 1686 or 1687), OC21, 342
This page intentionally left blank
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
Iceland crystal, including refractions that could only be observed by cutting
the crystal along unnatural sections. The sketch records the experimental
verification of Huygens’ elliptical waves and, with it, the confirmation of his
theory of light and refractions.
This brief synopsis explains what ‘actually’ happened on that 6th of
August in 1679. The various terms and concepts will be explicated later on in
this book. For this moment, it suffices to make clear the core of Huygens’
wave theory and its historical significance. For Huygens the successful
experiment meant the confirmation of his explanation of strange refraction
and his wave theory in general. In the context of the history of seventeenthcentury optics, and of the mathematical sciences in general, the importance
of the event lies in the twofold particular nature of Huygens’ theory: a
mathematized model of the mechanistic nature of light considered as a
hypothesis validated by experimental confirmation. With the mathematical
form of his theory, Huygens can be said to have restored the problematic
legacy of Descartes’ natural philosophy, by defining mathematical principles
for the mechanistic explanation of the physical nature of light. The
hypothetical-deductive structure of his theory implied the abandonment of
the quest for certainty of that same Cartesian legacy and of seventeenthcentury science in general. Huygens presented waves of light, the inextricable
core of his account of optical phenomena, explicitly as hypothetical entities
whose certainty is inherently relative. In so doing, he set off from Descartes
in a direction diametrically opposite to Newton, the principal other restorer
of mechanistic science.
About a decade after the EUREKA of 6 August 1679, Huygens published
his wave theory of light and his explanations of ordinary and strange
refraction in Traité de la Lumière (1690). This book established his fame as a
pioneer of mathematical physics as evidenced by the fact his principle of
wave propagation is still known and used in various fields of modern physics
under the name ‘Huygens’ principle’. Its historical importance is also
generally acknowledged. According to Shapiro, Huygens stood out for his
“…continual ability to rise above mechanism and to treat the continuum
theory of light mathematically.”1 E.J. Dijksterhuis calls it the high point of
mechanistic science and its creator the first ‘perfect Cartesian’: “In Huygens
does Cartesian physics for the first time take the shape its creator had in
mind.”2 How did this come about? How did Christiaan Huygens came to
realize this historical landmark? Or more specifically, how did he arrive at his
wave theory of light? That is the central question of this study.
A history of Traité de la Lumière
Unlike its eventual formulation in Traité de la Lumière, the development of
Huygens’ wave theory has hardly been subject to historical investigation. The
Shapiro, “Kinematic optics”, 244. (For referencing see page 267)
Dijksterhuis, Mechanization, IV: 212 & 283 (references to this book will be made by section numbers). It
should be noted that Dijksterhuis mainly focuses on the mathematical model of wave propagation.
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
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.
are: how exactly and what does this mean for our understanding of Huygens’
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
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
Preliminary results are published in: Dijksterhuis, “Huygens’ Dioptrica” and Dijksterhuis, “Huygens’s
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
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
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
Dijksterhuis, Mechanisering, IV: 168-171, 284-287.
Hashimoto hardly goes beyond noting that “… two works were closely related in Huygens’s mind.”:
Hashimoto, “Huygens”, 87-88.
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.
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)
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.
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
For example, the precise application of the sine law to dioptrical problems, for example, has hardly been
studied. Shapiro, “The Optical Lectures” is a valuable exception, discussing Barrow’s lectures and their
historical context. The relationship between the development of the telescope and of dioptrical theory –
essential to my account of Dioptrica – has never been investigated in any detail. Van Helden has pointed
out the weak connection between both in general terms: Van Helden, “The telescope in the 17th century”,
45-49; Van Helden, “Birth”, 63-68.
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 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
Hall, “Summary”, 311.
Westfall, Construction, 132-154; Dijksterhuis, Mechanization, 212; Elzinga, Research program and Westman,
“Problem”, 100-101.
Hall, “Summary”, 305-306. As regards his studies of motion, Yoder has further specified this
characterization; Yoder, Unrollling time, 169-179.
Hakfoort, Optics in the age of Euler, 183-184.
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.
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
OC 1, 215. “Nunc autem in dioptricis totus sum ...”
Berkel, “Illusies”, 83-84. In the 1660s Huygens would start to seek patronage abroad, first in Florence
and then, successfully in Paris.
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
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.
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.
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.
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
With the possible exception of Descartes himself. See below, section 3.1
OC1, 305-305.
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.”
coming from a single point are refracted towards another single point. In
certain cases, the ovals reduce to the ellipses and hyperbolas of La Dioptrique.
Huygens in his turn discovered that a particular class of these ovals may also
reduce to a circle.
In La Géométrie Descartes introduced the
said class of ovals as follows (Figure 3). The
dotted line is an oval of this class. If the
right part 2X2 of the oval is the right
boundary of a refracting medium, rays
intersecting in point F are refracted to point
G.8 The oval is constructed as follows. Lines
FA and AS intersect in A at an arbitrary
angle, F is an arbitrary point on FA. Draw a
circle with center F and radius F5. Line 56 is
Figure 3 Cartesian oval.
drawn, so that A5 is to A6 as the ratio of
sines of the refracting medium. G is an arbitrary point between A and 5, S is
on A6 with AS = AG. A circle with center G and radius S6 cuts the first circle
in the points 2, 2. These are the first two points of the oval. This procedure
is repeated with points 7 and 8, et cetera until the oval 22X22 is completed.9
Huygens’ discovered that the oval reduces to a circle when the ratio of AF to
AG is equal to the ratio of A5 to A6, the ratio of sines.10 This means that with
respect to rays tending to F, a spherical surface 2X2 will focus them exactly in
G. Van Schooten was a bit skeptical about Huygens’ claim. Could such a
simple fact have escaped Descartes? Nevertheless, he included it in the
second edition of Geometria à Renatio Des Cartes (1659).11
Discovering that a spherical surface is aplanatic in certain cases is one
thing, applying it to lenses in practice is another. It remained to be seen what
shape the second surface should have and how it might be employed in
telescopes. For one thing, it does not seem useful for objective lenses, the
front and most important lens of a telescope that receives parallel rays. It
appears the usefulness of the discovery was limited, for Huygens never
returned to it in his dioptrical studies.12
The historical importance of the discovery lies in the fact that it aroused
Huygens’ interest in dioptrics. He did not exaggerate when he said he was
engrossed in dioptrics. Not only its theory, practice too. Five days after his
letter to Van Schooten, on 4 November, he wrote to Gerard Gutschoven, an
acquainted mathematician in Antwerp.13 After some introductory remarks,
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
Descartes, Geometrie, 353-354 (AT6, 424-426). The curve satisfies the equation F2 – FA = n(G2 – GA).
OC1, 305. See note 9: the equation becomes AF = nAG.
Reproduced in OC14, 419.
In Tractatus, he merely mentioned that a spherical surface is aplanatic for certain points: OC13, 64-67.
OC1, 190-192.
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
OC1, 192.
OC1, 201-205.
OC1, 204. “… adeo ut nullius radij concursus cum axe contingat ultra punctum O.”
OC1, 224-226.
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.
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
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’.
OC1, 215.
OC1, 219-223.
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).
On this arrangement see page 221. By the way, the two first propositions of part three are inserted after
part one.
In part one of Tractatus he defined ‘punctum
concursus’ as follows. In the third proposition, he
defined the focus as the limit point of the intersections of
refracted rays with the axis (Figure 5).22 ABC is a planoconvex lens and parallel rays are incident from the
direction of D. Consequently, they are only refracted by
the spherical surface. Huygens showed that the closer
rays are to the axis DE, the closer to E they reach it.
Beyond E no refracted rays crosses the axis. This limit
point E he defined as the ‘punctum concursus’ of the
spherical surface ABC. If the surface is concave, rays do
not intersect at all after refraction, they diverge. In this
case, the ‘punctum concursus’ is the virtual focus, the
limit point of the intersections with the axis of the
backwards extended refracted rays.
Figure 5 Punctum
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
OC13, 16-19
Q. In order to prove that Q is the ‘punctum concursus’ of ABP, consider ray
NP and its refraction PK. PK is found with the sine law and KQ is therefore a
given interval. On KQ choose L and draw T, close to A, so that LT : CL =
AQ : CQ = n. Now PL : LC < PK : KC = n. PL is smaller than TL, which in its
turn is smaller AL. A circle with center L and radius TL intersects the
refracting surface ABP between A and P in a point B. Draw BL and BC and it
follows that BL : LC = TL : LC = n. Therefore BO is refracted to L. So, the
closer a paraxial ray is to the axis, the closer to Q the refracted ray will
intersect with the axis. Q is the limit point of these intersections and
therefore the ‘punctum concursus’. When the index of refraction
= 3 : 2 – the approximate value for glass – AQ is exactly three times
the radius AC.
The refraction at the
posterior side of the lens is dealt
with in the twelfth proposition.
This case is more complex as
the incident rays are converging
due to the refraction at the
anterior side. Huygens dealt
with eight cases of non-parallel
rays. 23 For all cases, he
expressed the focal distance of
the non-parallel rays in terms of
the focal distance of the surface
for paraxial rays. The case at
hand is the fourth part of the
proposition (Figure 7).24 Rays Figure 6 Refraction at
Figure 7 Refraction
converge towards a point S, the anterior side of a
at the posterior side
bi-convex lens
of a bi-convex lens.
outside the dense medium
bounded by a spherical surface
AB with radius AC. Q is the ‘punctum concursus’ of paraxial rays coming
from R. With SQ : SA = SC : SD, the ‘punctum concursus’ D of the converging
rays LB is found. Huygens’ proof consisted of a reversal of the first case
treated in this proposition: rays diverging from D are refracted so that they
(virtually) intersect in S.25 This proof is similar to the one above.
Finally, in the sixteenth proposition of Tractatus, Huygens determined the
focal distance of a convex lens by combining the preceding results. It was
equal to the theorem he put forward in his letters to Tacquet and
Gutschoven. CD is a bi-convex lens with radii of curvature AC and BD
(Figure 8). The foci for paraxial rays are respectively E and L. According to
the eighth proposition CE : EA = DL : LB = n. With the twelfth proposition,
OC13, 40-79.
OC13, 70-73.
OC13, 42-47.
the ‘punctum concursus’ N for parallel rays from the direction of L is found
with EL : ED = EB : EN. After the refraction at the surface C, the rays
converge towards E; they are then refracted at the surface D towards N. In
modern notation: DN
nAC ˜ BD
n 1
n( AC BD ) ( n 1)CD
, where DN is the focal distance
measured from the anterior face of the lens. For rays coming from the other
direction O is the ‘punctum concursus’.
Figure 8 Focal distance DN of a bi-convex lens
The case of a bi-convex lens was only one out of many cases Huygens
treated in the fourteenth to seventeenth proposition of Tractatus. Taking both
spherical aberration and the thickness of the lens into account, he derived
exact theorems for the focal distance of each type of lens. In each case, he
also showed how to simplify the theorem when the thickness of the lens is
not taken into account. In the case of a bi-convex lens, he started by
comparing the focal distances CO and DN when the radii of both sides of the
lens are not equal. Their difference vanishes when the thickness of the lens
CD is ignored and both refractions are assumed to take place simultaneously.
The focal distance N is then easily found by first determining point L with
= n and then AB : AD = DE : EA, or DN
2 AC ˜ BD 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
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
OC13, 88-89. Equivalent to the modern formula 1f
OC13, 98-109.
= (n -1)(
+ R12
to the lens is to the distance EL of the object to the lens.28
The point L has a special property that Huygens had
established in the preceding proposition. An arbitrary ray
that passes through this point leaves the lens parallel to the
incident ray.29 In the twenty-second proposition, Huygens
had demonstrated that the focal distance LG of rays from a
point K of the axis is more or less equal to that of a point E
on the axis.30 The triangles KLF and GLI are therefore
similar, which proves the theorem.31
Figure 9 Extended
The theory of focal distances formed the basis of Huygens’ discussion of the
properties of images formed by lenses and lens-systems in the second book
of Tractatus. Huygens’ theory of images is once again both rigorous and
general. The central questions in book two were how to determine the
orientation of the image and the degree of magnification. For the time being,
Huygens ignored the question whether an image is in focus. In this way he
could derive general theorems on the relationship between the shape of
lenses and their magnifying properties. He then showed how these reduced
to simpler theorems in particular cases, for example for a distant object. In
the third book he showed what configurations produced focused images.
Figure 10 Magnification by a convex lens.
In the second and third propositions of book two, Huygens discussed a
convex lens. His aim was to determine the magnification of the image for the
various positions of eye D, lens ACB and object MEN (Figure 10). In order to
distinguish between upright and reversed images, Huygens defined the
‘punctum correspondens’ (later called ‘punctum dirigens’).32 This is the focus
of rays emanating from the point where the eye is situated and is thus found
by means of the theory of the first book. First, the eye D is between a convex
OC13, 122-125.
OC13, 118-123. In modern terms, L is the optical center.
OC13, 114-119.
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.
OC13, 176n1.
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
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
OC13, 174-179.
OC13, 186-197.
Figure 11 Four of the cases discussed (additional lettering
continued to be, assumed for quite some time, as Molyneux was to remark in
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
“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.
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
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
OC13, 244-247.
OC13, 246-253.
Hug29, 151-167.
OC13, 198-199.
OC13, 252n1. See below, section 3.1.2.
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.
OC13, 258-261.
system, Huygens applied
proposition six of book
two. The eye is imagined
at S and the object at PN.
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.
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
OC1, 280; 301-303; 321-322. Huygens did not pin much faith in Van Schooten’s proposal.
I will say a bit more about his publishing pattern on page 174.
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
Van Helden, Invention, 35-36; Galileo, Sidereus nuncius, 3-4 (Van Helden’s introduction).
De Waard, Uitvinding, 105-225; Van Helden, Invention, 20-25.
OC13, 436-437.
Van Helden, Invention, 21, 36.
Van Helden, Invention, 26; Galileo, Sidereus nuncius, 6, 9 (Van Helden’s Introduction).
Van Helden, “Galileo and the telescope”, 153-157.
Galileo, Sidereus nuncius, 94 (Van Helden’s Conclusion).
Galileo, Sidereus nuncius, 37-39.
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
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
Kepler, Conversation, [19-21].
Kepler, Dioptrice, dedication (KGW4, 331). “… circaque eam alij de palma primae inventionis certarent,
alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio
dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum
triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego
doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est
causarum lege geometrica demonstrandarum, quibus tam exoptati, tam jucunda varietate multiplices
effectus inniterentur.”
Straker, “Kepler’s theory of pinhole images”, 276-278.
Cited and translated in: Straker, “Kepler’s theory of pinhole images”, 278.
Straker, “Kepler’s theory of pinhole images”, 275-276; 280-282.
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
The problem of pinhole images was well-known in perspectiva. It was
known since Antiquity that the image of the sun, projected by a square
aperture, can still be round. This seemed to contradict the basic principle of
optics: the rectilinearity of light rays. The solutions given by perspectivist
writers did not satisfy Kepler. Each had in the end sacrificed the principle of
rectilinearity – the foundation of geometrical
optics.60 Kepler had to resolve the problem by
himself. His solution consisted of a new theory
of the way rays form images of objects. This
theory, in its turn, would be the foundation of
his dioptrics as well as of seventeenth-century
geometrical optics in general.
Kepler approached the problem anew and
did so by uncompromisingly applying the
principle of rectilinearity. In Paralipomena, he
describes how he replaced a ray of light by a
thread. He took a book, attached a thread to
one of its corner and guided it along the edges
of a many-cornered aperture, thus tracing out
the figure of the aperture. Repeating this for
the other corners of the book, and many more
points, he ended up with a multitude of Figure 14 Kepler’s solution to
overlapping figures that formed an image of
the pinhole problem
Further discussed in section 4.1.1.
Dupré points out Risner’s programmatic discussion of the science of optics in the preface to the edition
which constitute an important, yet still little studied, agenda for seventeenth-century optics. Dupré,
Galileo, the Telescope, 54.
See Lindberg, “Laying the foundations”, 14-29.
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
Alhacen, Optics I, 68 (book 1, section 17) and 77 (book 1, section 46).
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.
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
Kepler’s efforts to find a law of refraction are discussed below, in section 4.1.2.
Dupré, Galileo, the telescope, 31.
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.
Kepler, Paralipomena, 201 (KGW2, 181). “Itaque non oportet nos ad res totas respicere, sed ad rerum
singular puncta, …” Translation Donahue, Optics, 217.
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
Rosen, “The invention of eyeglasses”, 13-46.
Lindberg, “Optics in 16th century Italy”136-141. Maurolyco had preceded Kepler in his analysis of the
pinhole image: Lindberg, “Optics in 16th century Italy”, 132-135; Lindberg, “Laying the foundations”.
Kepler, Paralipomena, 200-202 (KGW2, 181-183).
Malet, “Kepler and the telescope” offers a detailed discussion of Dioptrice, without however presenting it
as a part of the ‘optical part of astronomy’.
Kepler, Dioptrice, dedication (KGW4, 331).
incidence; the angle between the incident ray, produced beyond the
refracting surface, and the refracted ray.
Kepler began with a discussion
of the focal distances of planoconvex lenses (Figure 15). A ray
HG is incident on a convex surface
with radius AC, the angle of
incidence is GAC. As the angle of Figure 15 Focal distance of a plano-convex lens
deviation is one third of this, HG
will be refracted towards F, with AC : AF = 1 : 2.71 The focal distance is
therefore approximately three times the radius of the convex face.
Analogously, he argued that the focal distance of a plano-convex lens, the
plane face turned towards the incident rays, is approximately twice the radius
of curvature. For other cases Kepler established only rough estimations. If
convergent rays are incident on the plane side of a plano-convex lens, the
refracted rays intersect the axis within the focal distance. Combining these
three theorems, Kepler showed that the focal distance of a bi-convex lens is
both smaller than three times the radius of the anterior side and twice the
radius of the posterior side. In the case of an equi-convex lens, this comes
down to a focal distance approximately equal to the radius of its sides.72
Kepler did not determine the focal distance of a concave lens, he only
showed that rays diverge after refraction.73
On this basis, the properties of images formed by lenses are easily found.
The image DBF of an extended object CAE through a bi-convex lens GH is
formed at focal distance (Figure 16). The picture is inversed as the rays from
C are refracted towards D, etcetera. As the focal distance is roughly the radius
of any side, the magnitudes of object and image will be in a proportion equal
to their respective distances to the lens.74 In Dioptrice, Kepler briefly reiterated
his theory of vision. On the one hand, so he said in the dedication, he did so
for the sake of completeness, on the other hand because some readers had
trouble understanding his account in Paralipomena.75 He explained that a
perfectly focusing surface was not spherical, but should be hyperbolic, like
the crystalline humor of the eye was.76 On the basis of his theory of the
retinal image, he explained the effect of a lens placed before the eye once
more. Depending upon the position of the eye with respect to the focal
distance, the object will be perceived sharply.77 When the eye is placed not
too far from the focus, a magnified image will be perceived.
Kepler, Dioptrice, 11 (KGW4, 363).
Kepler, Dioptrice, 12-15 (KGW4, 363-367).
Kepler, Dioptrice, 45-49 (KGW4, 388-393).
Kepler, Dioptrice, 16-18 (KGW4, 367-369).
Kepler, Dioptrice, dedication (KGW4, 335).
Kepler, Dioptrice, 21-24 (KGW4, 371-372).
Kepler, Dioptrice, 35-42 (KGW4, 381-387).
Kepler proceeded to discuss the combination of
two convex lenses. He explained how these should
be configured in order to perceive a sharp,
magnified image.78 This is achieved when the foci of
both lenses coincide. It is remarkable that that
Kepler discussed the configuration of two convex
lenses, because in 1611 only the combination of a
convex objective and a concave ocular was known
to produce a telescopic effect. Kepler probably
arrived at this alternative configuration by
theoretical considerations.79 He never manufactured
this kind of telescope himself. The configuration has
come to be known as a Keplerian or Astronomical
telescope, as opposed to the Dutch or Galilean
telescope with a concave ocular. Much later it
became clear that the Keplerian type has the
advantage of a larger field of view, but Kepler
himself did not know this. He did realize that this
configuration had a drawback, it produced inverted
images. This could be corrected, he said, by inserting
a third lens at an appropriate place between ocular
and objective lens.
Kepler then turned to an account of concave
lenses and finally to a discussion of so-called Dutch Figure 16 Image formation
by a lens
telescopes. He explained the configuration of a
convex objective and a concave ocular only in broad lines. As he did not
speak of the focus of a concave lens, he could only roughly point out where
the ocular should be placed with respect to the focus of the objective. His
discussion of the configurations and the resulting properties of the images
remained mainly qualitative. Kepler offered a wealth of practical guidelines as
to the configurations of lenses and the way the best effects are achieved.
Dioptrice does not consist of rigorously demonstrated theorems. Without an
exact law of refraction, a quantitative and exact theory could hardly be
attained. This was not necessarily Kepler’s intention. Rather, he intended to
explain the working of the telescope mathematically. He did so by analyzing,
on the basis of his theory of image formation, how it forms magnified
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
Kepler, Dioptrice, 42-43 (KGW4, 387-388).
A possible source of inspiration may have come from the analogous configuration of the eye and a
convex spectacle glass, as the eye acts as a convex lens does. See also Malet, “Kepler and the telescope”,
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,
OC1, 6 (Stampioen’s list of recommended readings spans pages 5-10) and OC6, 215.
Della Porta’s account of refraction by spheres and lenses in De refractione is discussed in Lindberg,
“Optics in 16th century Italy”, 143-146.
Della Porta, De Telescopio, 113-114.
where point a is seen. In the same manner point i is constructed and hi is the
object as perceived through the lens. The question is why only the ray acd
emanating from point a is singled out. Della Porta seemed to assume that
this is the one that enters the eye perpendicularly. Yet, in the case of a distant
object, he no longer chooses rays parallel to the axis of the system, but the
crossing rays ad and bg (Figure 18). He probably did so to account for the
reversing of the image, but he lacked a theoretical justification. Della Porta’s
account of concave lenses was even more troublesome, as he ignored the
implication of his reasoning that the eye cannot perceive the whole object at
once. Moreover, he persistently has rays refracted from the perpendicular at
the first surface (for example de in Figure 19).83 ‘De telescopio’ culminated in
an account of a Galilean telescope (Figure 19). Della Porta traced the path of
a ray emanation from point a of the object. He then chose the cathetus with
respect to the upper surface of the concave lens and argued that qr is the
image perceived.
Figure 17 Image of a near object
Figure 18 Image of
distant object
Figure 19 Image by a
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
Della Porta, De telescopio, 141-142.
Compare Lindberg, “Optics in 16th century Italy”, 146-147.
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
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
The exact law of refraction Kepler had to make do without, was soon found.
More than that, it had been within his reach. The English astronomer
Thomas Harriot had discovered it in 1601. After the publication of
Paralipomena, he and Kepler had corresponded on optical matters. However,
the correspondence broke off before Harriot had revealed his discovery.89
Long before that, but unknown until the late twentieth century, the tenth85
Ronchi, “Refractione au Telescopio”, 56 and 34. “They know nothing of perspective.” and “... and it
pleases me that the idea of the telescope in a tube has been mine; ...”
Pedersen, “Sagredo’s optical researches”, 144-148.
KGW4, “Nachbericht”, 476.
Dupré, Galileo, the Telescope, chapters 4 to 6 in particular.
Harriot is discussed in section 4.1.2.
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
Rashed, “Pioneer”, 478-486.
For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat
later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250.
Gaukroger, Descartes, 138-146. Dupré, Galileo, the Telescope, 53-54.
Discussed in section 4.1.3
Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.”
Descartes, AT6, 155-160.
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”
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
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
Descartes, AT6, 82-83. Ribe, “Cartesian optics” offers an enlightening account of the artisanal roots of
La Dioptrique.
Descartes, AT6, 82.
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.”
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.”
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
Stroud, Minute, 20; Prins, “Hobbes on light and vision”, 129-132. On Hobbes’ derivation of the sine
law, see section 5.2.1.
Lohne, ”Geschichte des Brechungsgesetzes”, 166.
Huygens worked on it in 1671-2, see page 160.
remaining uncompleted and unpublished as well, Lectiones XVIII was the most
advanced treatise on geometrical optics published until the end of the
seventeenth century. The core of Lectiones XVIII consists of lectures IV
through XIII, in which Barrow determined the image of a point source in any
reflection or refraction in plane and spherical surfaces. Barrow developed a
mathematical theory of imagery by analyzing the intersections the refractions
of a bundle of rays.
For example, a point A is seen by
an eye off the axis AB, with COD being
the pupil of the eye.104 (Figure 20). The
pupil is perpendicular to the refracted
ray NO, which passes through the
center of the pupil. The extension
KNO is called the principal ray. Now,
draw the refracted rays MC and RD
that pass through the edge of the
pupil. Produced backwards, MC and
RD will not intersect the principal ray
NKO in one point, but in points X and
V. Barrow demonstrated that point Z
on the principal ray is the limit of
these intersections. According to his
definition of the image, Z is the place
Figure 20 Barrow’s analysis of image
of the image. Consequently, the
formation in refraction.
cathetus rule does not apply here, as it
would have point K as the image. Barrow applied this determination of the
image point to various problems in refraction. In the case of spherical
surfaces he derived expressions for the place of the image point for the eye
being both on and off the axis of the surface.
Barrow defined the image point in a similar way as Huygens defined the
‘punctum concursus’ and applied it with comparable rigor to study the
refracting properties of spherical surfaces. Many of their results were
equivalent. Yet, they had different goals. Huygens intended to explain the
dioptrical properties of the telescope and therefore confined himself to
paraxial rays, not discussing optical imagery. He ignored mathematically
sophisticated problems that had no relevance to the telescope, like the focus
of an oblique cone of rays. Barrow’s aim was to develop a general theory of
optical imagery. He had no intention of explaining the telescope and many of
the problems he treated had no relevance to it.106 Still, in lecture XIV, he also
discussed spherical lenses. He gave, without proof, a series of equations for
the focal and image points of all kinds of lenses by way of an example of the
Barrow, Lectiones, [82-83].
Compare Shapiro, “The Optical Lectures”, 130 & 133-134.
Compare Shapiro, “The Optical Lectures”, 150-151; and Malet, “Isaac Barrow”, 286.
application of his preceding discussion of single spherical surfaces. It was an
exact solution to the problem of Dioptrice, yet a complex and cumbersome
one.107 Barrow had chosen the example “… with a view to common use, and
particularly aimed at reducing the labour of anyone who comes across
them.”108 Barrow’s exact theory of focal distances was the first in print, but it
was no more than a theory of focal distances. He did not discuss
magnification and configurations of lenses.
Barrow’s footsteps were followed by Newton in his lectures. Their
central subject was his mathematical theory of colors. In a section “On the
Refractions of Curved Surfaces” he also treated some topics regarding
monochromatic rays. Newton extended Barrow’s theory of image formation
to three-dimensional bundles of rays. He demonstrated the existence of a
second image point at the intersection of the axis and the principal ray.
Newton had developed his own solution of the anaclastic problem –
although in the lectures he abandoned it in favor of Barrow’s – and found a
new way to derive Descartes’ ovals. The final goal of the lectures were,
however, colors. So, when Newton, in his 31st proposition, determined the
spherical aberration of a ray, he did so to compare it to chromatic aberration.
The latter was larger and “Consequently, the heterogeneity of light and not
the unsuitability of a spherical shape is the reason why we have not yet
advanced telescopes to a greater degree of perfection.”109 In this way,
Newton dismissed Descartes proposal as a dead-end. This was an important
result for the understanding of the working of the telescope. In order to
overcome the disturbing effects of aberration, Newton proposed to use
mirrors instead of lenses. Newton’s theory of colors and his reflector are
further discussed in section 3.2.3.
Dioptrics as mathematics
The discovery of an exact law of refraction had supplied geometrical optics
with a foundation for the mathematical study of the behavior of refracted
rays. This study consisted of deducing theorems from the postulates and
definitions of dioptrics in a rigorous way aimed at generality. With Kepler’s
new theory of image formation, a range of new problems were raised relating
to the perfect and imperfect focusing of rays. To the ones already mentioned
was added, at the end of the century, that of caustics; the locus of
intersections of rays refracted by a curved surface.110 These problems were
markedly theoretical, mathematical puzzles tackled without practical
objectives. Halley, for example, in a paper of 1693 solved “the problem of
finding the foci of optick glasses universally” by means of a single algebraic
Shapiro, “The Optical Lectures”, 149-150.
Barrow, Lectiones, [168].
Newton, Optical Papers 1, 427.
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.
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.
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
Halley, “Instance”, 960.
See: Albury, “Halley, Huygens, and Newton”, 455-457.
Galileo, Sidereus nuncius, 112-113 and 92-93 (Van Helden’s conclusion). See Dupré, Galileo and the
telescope, 175-178.
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
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
Schuster, “Descartes opticien” and Van Berkel, “Descartes’ debt”.
Beeckman, Journal, II, 209-211; 294-296. For lens grinding see down, page 57.
For the second idea see Beeckman, Journal, II, 367-368. For a later consideration see for example: III,
Beeckman, Journal, II, 296; 357.
Beeckman, Journal, III, 109-110.
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
Van Helden, Measure, 118-119.
Compare Dear, Discipline and Experience, 210-216.
Van Helden, “Astronomical telescope”, 26-32. See also below section 3.1.1.
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
McKeon, “Les débuts I”, 258-266.
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
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, ...”
McKeon, “Les débuts I”, 266-269.
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.
OC21, 348-351.
Van Helden, Measure, 120-121.
OC21, 352-353.
McKeon, “Les débuts I”, 286; Van Helden, Measure, 118.
McKeon, “Renouvellement”, 122.
McKeon, “Renouvellement”, 126.
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
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
Flamsteed, Gresham lectures, 34-39 (Forbes’s introduction).
OldCorr9, 326-327.
OldCorr10, 520.
OldCorr4, 448.
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
Van Helden, “Huygens and the astronomers”, 156-157; Van Helden, Measure, 127-129.
Flamsteed, Gresham lectures, 154.
Flamsteed, Gresham lectures, 119 & 132. Flamsteed later deleted the part between brackets.
Flamsteed, Gresham lectures, 120-127.
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
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
“… 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
Flamsteed, Gresham lectures, 136.
Flamsteed, Gresham lectures, 140-143.
Flamsteed, Gresham lectures, 40; 146n2 (Forbes’ introduction).
Flamsteed, Gresham lectures, 8-9; 40 (Forbes’ introduction).
Flamsteed, Gresham lectures, 39 (Forbes’ introduction).
Flamsteed, Gresham lectures, 149.
Flamsteed, Gresham lectures, 4-5 (Forbes’ introduction).
Molyneux, Dioptrica nova, (Admonition to the reader).
Molyneux mentioned Kepler, Cavalieri, Hérigone, Dechales, Fabri, Gregory and Barrow.
their Effects in Telescopes, than of affecting a Nicety, which would be of little Use in
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
Molyneux, Dioptrica nova, (Admonition to the reader).
Molyneux, Dioptrica nova, 19-23.
Molyneux, Dioptrica nova, 20.
Molyneux, Dioptrica nova, 22.
Molyneux, Dioptrica nova, 9.
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.
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
“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
Molyneux, Dioptrica nova, 36-38.
Molyneux, Dioptrica nova, 38.
Picolet, “Correspondence”, 38-39.
“… peuuent aussi estre sujets a certaines refractions qu’il faut bien connoistre.” Quoted in McKeon,
“Renouvellement”, 126-128. It is found in: A. Ac. Sc., Registres, t. 3, fol 156 ro - 164 vo spéc. 157 vo.
Blay, ”Travaux de Picard”, 329-332. Blay cites several references.
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,
Divers Ouvrages de Mathematique et de Physique, par Messieurs de l’Academie Royale des Sciences (1693), 375-412.
OC13, “Avertissement”, 7.
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
Blay, “Travaux de Picard”, 340.
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
Editor’s comment, OC15, 10. See also Anne van Helden, “Lens production”, 70.
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
Huygens did not name it, he called it ‘saturni luna’ and sometimes ‘comes meus’. The name Titan was
given by Herschel in 1847.
OC15, 296-299.
Van Helden, “Huygens and the astronomers”, 150-154. Van Helden, “Divini vs Huygens”, 48-50.
OC15, 177; 230. Huygens employed Rhineland feet (0,3139 meters) and inches (0,026 meters).
It is reprinted in OC15, 403-437.
Van Helden, “Divini vs Huygens”, 36-40.
1655-1672 - DE ABERRATIONE
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
Van Helden, “Huygens and the astronomers” 148, 157-158.
Van Helden, Invention, 16-20.
Van Helden, Invention, 26; 47-48.
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
Descartes, Dioptrique, 2-3 (AT6, 82-83).
Shea, “Descartes and Ferrier”, 146-148.
AT1, 598-600. “Et il reussit parfaitement bien; …” It turned out that it was impossible to make a
concave lens in the same way.
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.
AT1, 33-35.
Descartes, Dioptrique, 141-150 (AT6, 215-224).
Ploeg, Constantijn Huygens, 34-38.
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
remained to be seen whether it could be made to function properly. As for
Descartes’ ideal lenses, theory had not advanced practice yet.
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
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.
Beeckman, Journal, III, 69, 249, 308, 383.
Beeckman, Journal, III, 430.
OC1, 191. See also Anne van Helden, “Lens production”, 70-75.
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
“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
OC1, 242. He distributed several telescopes of this design during the next decade. (OC1, 242; OC13,
264n3; OC4, 132-3; OC4, 224, 228-9)
OC17, 293-304.
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.”
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.
OC17, 299.
1655-1672 - DE ABERRATIONE
technique used by mirror-makers.28 It is
not known where Huygens got the idea.
Such tools for improving the
grinding process had been thought up
earlier, in particular by the most
prestigious lens makers. In Galileo’s
workshop – reigning until the 1640s – a
lathe was introduced that permitted
greater precision than was attained by
ordinary spectacle makers.29 During the
1660s, the Campani brothers in Rome
became the undisputed masters of the
art. They used a range of machines of
their own design, producing lenses
unsurpassed until the eighteenth
century. The Huygens brothers kept a Figure 22 Beam to facilitate lens grinding.
close eye on developments like these and around 1665 references to a type of
lathe designed by Campani turned up in their writings. The quality of lenses
seems to have depended to some degree on the lens maker’s inventiveness to
convert laborious and unsteady handiwork into reliable tools.
Huygens had learned how to make lenses. He knew the limitations of the
art and of its products. Even the best lenses might suffer from flaws like
bubbles, irregular density, faults in shape, etcetera. In the end, the proof of
the pudding was in the eating. The quality of telescopes was determined by
trial, sometimes literally. Campani beat Divini early 1664 by a series of
carefully organized ‘paragoni’: open contests in which printed sheets at a
distance were read by means of the instruments of both competitors.30
Alternative configurations
Besides the quality of lenses, telescopes could be improved by configuring
lenses alternatively. Kepler could not have known that the configuration of
two convex lenses he discussed in Dioptrice had several advantages over the
Galilean one. The positive focus that made possible the micrometer has
already been discussed in the previous chapter. Scheiner, who used a
Keplerian telescope to project images of the sun, discovered by coincidence
that it had a wider field of view. There are indications that Fontana was the
first to put Kepler’s idea into practice, although Scheiner was the first to
mention using it.31 Around 1640 Fontana was the first to challenge Galileo’s
dominance in the trade and he did so with Keplerian telescopes. Around that
time, the Galilean configuration was beginning to reach the limits to which
its power could be increased without the field of view becoming too small.
Beeckman, Journal, III, 232.
Bedini, “Makers”, 108-110; Bedini, “Lens making”, 688-691.
Bonelli, “Divini and Campani”, 21-25.
Van Helden, “Astronomical telescope”, 20-25. Compare Malet, “Kepler and the telescope”, 120.
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
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
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.
Van Helden, “Compound eyepieces”, 34. The entire letter is reproduced on 34-35.
OC1, 308-311.
1655-1672 - DE ABERRATIONE
well how to grind lenses to suitable proportions and
configure appropriately. The nature of this knowledge
and to what extent they understood the dioptrical
properties of lenses will be discussed now. Note that
this is a discussion of very limited scope, determined
by the considerations and activities of Huygens, that
passes over the a wealth of historical knowledge that
can be, and has been, gathered regarding the telescope
making trade.35
A booklet on spectacles written in 1623 gives an
indication of the understanding their manufacturers
had of glasses. Uso de los antojos was written by the
Andalusian licentiate Daza de Valdez. It explained
how to choose glasses of appropriate strength for a
patient. Daza described a procedure to determine the
‘grado’ of a given lens (Figure 23).36 He drew two solid
circles X and Q of unequal diameter on a sheet of
paper as well as a specific scale at one of the circles. A
glass was then positioned on the scale in such a way
that both circles were seen equally large. The position
of the lens on the scale gave its ‘grado’.37 Daza did not
explain the method, he only described how it was Figure 23 Daza’s scale
employed. It was a practical procedure that did not require any
understanding of its effect on rays.
A manuscript written around 1670 in Rome by a certain Giovanni
Bolantio contains a similar kind of procedural knowledge. It discussed the
manufacture of telescopes and probably recorded the daily routine in some
workshop.38 It contains two tables listing the ocular and objective lenses
needed to make a telescope of a specific strength, characterized by its length.
The lenses are characterized by the doubled radius of the pattern in which
they were ground.39 With these tables at hand, the workman could choose the
patterns needed to make a telescope on order. Bolantio did not explain
whether he had constructed the tables himself nor how they were made.
Some dioptrical rules are implicit in them. For example, the length of a
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.
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.
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.
Bedini & Bennet, “Treatise”.
Bedini & Bennet, “Treatise”, 120-121.
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 801 of the tube’s length. Bolantio
explained that the objective should be partially covered by a ring so that no
light fell on the interior of the tube, which apparently implied the ratio used
in the table.40 Whatever the dioptrical understanding implicit in Bolantio’s
manuscript, it was presented in a procedural, how-to style that did not
require further theoretical knowledge.41
Some telescope makers published their own observations, to promote
their products.42 They did not publish the secrets of their art, as their
revenues depended on them. Information on the manufacture of lenses and
telescopes could be found in books that were mostly written by scholars.
Examples are Telescopium (1618) by Girolamo Sirtori, Selenographia (1647) by
Hevelius and La Dioptrique oculaire (1672) by Cherubin d’Orleans. In 1685,
Huygens wrote a treatise on lens grinding in Dutch, Memorien aangaende het
slijpen van glasen tot verrekijckers, published posthumously in Latin in the
Opuscula posthuma (1703).43 Memorien was the elaboration of notes like the one
cited above. Huygens described the process of lens making as a set of
directives, procedures, tips and tricks. No attempt is made to explain why
things work as they work: for example a geometrical account of the grinding
device is absent. Memorien supplied the kind of experiential knowledge also
found in Bolantio’s manuscript: a description of skills Huygens had acquired
through long-time practice.
To what extent a telescope maker like Campani understood the dioptrics
implicit in tables like those in Bolantio’s manuscript cannot be determined.
First rank, specialized telescope makers like Divini and Campani had
received some formal education, so they may have been able to read and
study a book like Dioptrice. It remains to be seen whether a question like this
is relevant at all. I doubt whether dioptrical knowledge would have been of
any use in the design and manufacture of telescopes. They knew very well
the effect of diverse types of lenses, but this probably was experiential
knowledge. Innovative craftsmen like Wiesel were able to find new
configurations with improved properties. These are likely to have been the
product of trial and error. It has been said that Kepler’s configuration was
the only contribution from the theory of dioptrics to the improvement of the
telescope.44 Still, its advantages had to be discovered in practice. The
Bedini & Bennet, “Treatise”, 117.
Willach discusses dioptrical theory emerging from the correspondence of Rheita en Wiesel which
suggests similar lines. Willach, “Development of telescope optics”, 390-394.
For example: Fontana’s Novae coelestium (1646) and Campani, Lettere di Giuseppe Campani intorne all'ombre
delle Stelle Medicee (1665).
OC21, 252-290.
Van Helden, “The telescope in the 17th century”, 44-49.
1655-1672 - DE ABERRATIONE
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.
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
Bedini, “The tube of long vision”, 157-159.
OC15, 56.
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.
McKeon, “Les débuts I”, 237.
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
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
OC15, 230-233.
Van Helden, “Compound eyepieces”, 33; Van Helden, “Huygens and the astronomers”, 158.
OC22, 568-576.
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.”
Van Helden, “Compound eyepieces”, 33.
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
“Although lenses should not be multiplied without necessity, because much light is lost
due to the thickness of the glass and the repeated reflections, experience has shown it is
nevertheless useful to do so here.”55
When the single ocular lens is replaced by two lenses, so Huygens continued,
the field of the telescope can be enlarged. Moreover, the images produced
are less deformed and the irregularities of the lenses are less disturbing.
The precise configuration
of the eyepiece was as follows
(Figure 25). AB is the
objective lens, CD and EF
form the eyepiece; the focal
distances are LG, KT, and SH
Figure 25 Diagram for the eyepiece, accompanying
respectively. Now, KT is Figure 24.
about four times SH, and the
distance KS between the ocular lenses is about twice the focal distance SH of
the outer one. Finally, the focus G of the objective lens AB should fall
between the outer ocular EF and its focus H in such a way that H is the
‘punctum correspondens’ of point G with respect to lens CD. Rays coming
from a distant point Q will therefore be parallel after refraction in the outer
ocular lens EF. Having determined the position of the eye M and the
magnification by the system, Huygens concluded by explaining that points P
and Q of the object are seen sharp but reversed.
Huygens had demonstrated that this configuration produced sharp,
magnified images. This was a rather straightforward application of the theory
he had developed in 1653. The text bears witness to the fact that Huygens
had gained much experience with actual lenses since the days of Tractatus. At
one point in the theorem, he indicated why images do not suffer much from
the irregularities in the lenses. Because the eye is so close to the outer ocular,
“the spots and tiny bubbles of air, that are always in the material of the glass, cannot be
perceived in lens EF. But one does not see them in lens CD either, because the eye
perceives objects placed there confusedly, but those that are located close to H
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:
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.”
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.”
He developed a systematic theory of the field of view of a telescope much later, after 1685: OC13, 450461, 468-73.
“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
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
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.”
OC13, 258-265. Discussed above, section 2.1.2..
OC13, 262-263. “… res visas, atque etiam distinctiores efficere, nullisque colorum pigmentis infectas
quod in hic lentium trium compositione aegre vitari potest.”
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, …”
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,
1655-1672 - DE ABERRATIONE
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’
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.
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
OC13, 318-319. “creditum est hactenus … sphaericae superficies minus aptae essent his usibus, nemine
suspicante vitium convexarum lentium lentibus cavis tolli posse.”
Kepler, Paralipomena, 185-186 (KGW2, 168-169). Kepler repeated his insights in Dioptrice.
extended this by developing a theory of spherical aberration. He subjected,
so to say, the ‘punctum concursus’ to a closer examination to see how exactly
spherical aberration affected the imaging properties of lenses.
Huygens’ study of spherical aberration
had been preceded by a calculation of rays
refracted by a plano-convex lens he had
apparently carried out in 1653. In Tractatus,
he remarked that rays “reunite somewhat
better, i.e. that the points where they cut
the axis are closer to one point, …, when
the convex surface faces the incident rays,
than when the plane surface faces them.”65
The 1653 calculation is lost, but was
probably identical to later ones.66 The result
implied that the orientation of a lens
affected the degree of aberration. In 1665,
Huygens went to see whether the amount
of spherical aberration might deliberately
be decreased by a proper configuration of
lenses. He began a study of spherical
aberration under the heading “Adversaria
ad Dioptricen spectantia in quibus quæritur
aberratio a foco”.67 After a decade of quiet,
the ‘Adversaria’ was the next chapter of
Huygens’ dioptrical studies.
In ‘Adversaria’ Huygens derived
expressions for the amount of spherical
aberration as it depends upon the
properties of a lens. The rigor familiar from
Tractatus returns immediately. On the basis
of the theorems of Tractatus, he took the
refractions at both faces of the lens as well
Figure 26 Spherical aberration of a
as its thickness into account. In the first
plano-convex lens.
calculations Huygens returned to the claim
of 1653. He derived an expression for the aberration of the extreme ray
incident on a plano-convex lens GBC with focal distance GS (Figure 26). A
parallel ray is refracted at the extreme point C of the lens towards T on the
axis. The derivation of the aberration TS is straightforward and yields
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.
OC13, LII (“Avertissement”), those later calculations are on pages 283-287.
OC13, 355-375.
1655-1672 - DE ABERRATIONE
= 76 BG, where BG is the thickness
of the lens.68 When the lens is reversed
and rays are incident on the plane
TS = 2 BG.
The aberration is
therefore considerably smaller –
almost four times – when the convex
side faces the incident rays. This time
Huygens went further than the mere
observation that the orientation of a
lens affects the amount of aberration.
The faces of a lens are surfaces with
different radii – infinite in the case of
a plane face. The proportion between
these radii apparently determines how
large the aberration is. Consequently,
an ideal lens can be found by
determining the optimal proportion
of both radii.
To do so, Huygens derived an
expression for the aberration of a
parallel ray HC incident on the
extreme end of a lens IMCB (Figure
27). AB = a and NM = n are the radii
of the anterior and posterior side and
BG = b is the thickness of the anterior
half of the lens. The thickness of the
entire lens BM = q can be expressed as
q = b ba . The anterior face refracts
an extreme ray HC towards P, a little
off its focus R. The posterior face, in
its turn, refracts the extension CF of
ray CP towards D, a little off the focus
E of the lens. Huygens then expressed
the spherical aberration DE of the
extreme ray in terms of the radii of
the faces and the thickness of half the
lens: DE =
OC13, 357.
OC13, 359.
7n 2 q 6anq 27a 2 q
6(a n )2
Figure 27 Aberration of a bi-convex lens
the space on the axis within which all parallel rays are brought together,
which space DE is defined by this rule.”70 Or, the aberration DE is found by
multiplying the thickness of the lens q by the expression
7n 2 6an 27a 2
6( a n )2
which only depends on the radii of both faces. The shape of a lens that
produces minimal aberration can be found by determining the minimum of
this expression; this yields a : n = 1 : 6.71 In this case the aberration of the
extreme ray DE = 15 q. Huygens found the same for a bi-concave lens,
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
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
OC13, 364. “DE spatium in axe intra quod radij omnes paralleli coguntur, quod spatium DE per regulam
hanc definitur.”
OC13, 366-367. Modern methods yield the same result.
OC13, 375 and 370. In the latter case the solution yields a negative value for the radius of the posterior
OC13, 367. “Radius convexi objectivi ad radium convexi interioris in lente optima ut 1 ad 6. EUPHKA. 6
Aug. 1665.”
OC13, 280n2. “Quaenam lens sphaerica convexa melius radios parallelos coligat investigare.”
OC13, 280-281. “aberrationes radiorum quae ex figura superficierum sphaerica oriuntur”
1655-1672 - DE ABERRATIONE
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
Descartes’ aspherical lenses.
In practice one legitimately
makes do with spherical
lenses, because spherical
aberrations are sufficiently
to 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
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.
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.
OC1, 422. “propter accurationem”
OC1, 429.
Vermij and Atzema, “Specilla circularia”, 119.
Vermij and Atzema, “Specilla circularia”, 116: “Ex quibus patet, quanto x sive BF minor est, tanto etiam
punctum I longius distare ab N;”
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
inches. Huygens then stated – without proof – that the aberration
could be found more easily by multiplying the thickness of the lens by 29 , the
rule found in the ‘Adversaria’.86 There is, he admitted, a slight difference
Vermij and Atzema, “Specilla circularia”, 117: “Unde constat, focum ipsum pro puncto mechanico
tantum habendum esse.”
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.”
OC13, 276-277.
OC13, 308-313.
OC13, 282-285. Each time he assumed an index of refraction 3 : 2.
1655-1672 - DE ABERRATIONE
inches) but this was of no significance in actual telescopes.87 When
( 1000000
the convex side of the same lens faced the incident rays, the exact calculation
yielded an aberration of 10000000
inches. In this case, the easier rule of
‘Adversaria’ – multiplying the thickness of the lens by
– gave
difference of only 1000000
inches. Again, the main goal of this exercise was to
show that the aberration of a plano-convex lens is least when its convex side
faces the incident rays. 88 Continuing with a bi-convex lens, Huygens sketched
out how the aberration might be calculated exactly, but immediately moved
on to an ‘abbreviated rule’ he had ‘found’.89 This was the expression of the
‘Adversaria’, found by “ignoring very little quantities, but judiciously so as
needed.”90 The rule applied to convex as well as to concave lenses and
yielded the optimal proportion of both radii of 1 : 6. The resulting bi-convex
lens produces an aberration of only 15 times its thickness.91
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
OC13, 284-285. “Exigua quidem differentiola, sed quae in illa lentium latitudine quae telescopiorum
usibus idonea est, nullius sit momenti.”
OC13, 284-287.
OC13, 290-291. “Et haec quidem methodus ad exactam supputationem adhibenda esset. Invenimus
autem et hic Regulam compendiosam …”
OC13, 290-291. “Quae regula … inventa est neglectis minimis, sed necessario cum delectu.”
OC13, 290-291& 302-303.
OC13, 302-303. “…, sed aliae minus perfectae, quarum nempe vitijs compensantur ac corrigentur vitia
lentis convexae, …”
OC13, 318-319. “Ex lentibus sphæricis cavis et convexis telesopia componere hactenus cognitis ejus
generis meliora, perfectionemque eorum quæ ellipticis hyperbolicisve lentibus constant æmulantia.”
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
times its focal distance FE, the radius of the
the convex side should be 100
concave side 272
times FE.94 Next, he proved that this ocular indeed canceled
out the aberration of the objective.95 Ergo, the proposed configuration would
produce almost perfect images.
Figure 29 Galilean configuration in which spherical aberration is neutralized.
Huygens supplied a table in which he listed telescopes with various
magnifications against the ocular lenses required according to his analysis.
These numerical examples were, so to say, the blueprint by means of which
his design could be realized by any skilled worker. By way of conclusion,
Huygens remarked that the advantages of his design could only be realized
by lenses that were truly spherical. The manufacture of spherical lenses
should therefore be resumed diligently.
Huygens had made clear at the outset of his exposition that the
usefulness of his design was limited. Only a concave ocular could correct for
the aberration of the objective lens. The design was therefore useful only for
Galilean telescopes. In astronomy, telescopes required a convex ocular:
“However, it is certain that this mutual correction is not found in the composition of
convex lenses. On the contrary, the defect of the exterior lens is always a bit augmented
by the ocular lens and it cannot be remedied in any way.”96
Still, his theory was not entirely useless for the Keplerian telescopes required
for astronomical observation. In the final propositions of De Aberratione,
OC13, 320-323.
OC13, 324-327. For the rays KK and LM – that are not extreme rays – Huygens used the proposition on
the linear proportion between aberration of a ray and the square of its distance to the axis. OC13, 308313.
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
Huygens examined the means to enhance the quality of images produced by
telescopes with a convex ocular. That is, he took a theoretical look at the
matter. On the basis of this analysis, he could provide directions for
optimizing the quality of telescopes with convex oculars.
The magnifying power of a telescope depends upon the ratio of the focal
distances of objective and ocular lenses, and can therefore be increased by
reducing the focal distance of the ocular lens. This, however, simultaneously
decreases the clarity and distinctness of images. To maintain clarity at the
same time, the opening of the objective lens would have to be made larger.97
He began by considering a naked eye in front of which a telescope is placed.
Assuming that an equal number of rays should enter the eye when a more
powerful telescope is taken, Huygens argued that the opening should be kept
proportional to the magnification. This implied that his 22-foot telescope
would need an opening of 125 times the area of the pupil. In reality, he
observed, a satisfactory telescope had a much smaller opening, only 15 times
the area of the pupil. Evidently, in astronomical observation one could do
with much smaller clarity. He therefore did not take the eye as starting-point,
but a telescope with satisfactory quality. If the ocular is replaced by an ocular
that magnifies twice as much, the clarity will be four times smaller. The
opening of the objective should therefore be increased accordingly.
Evidently, this cannot be done at will and one should “consider accurately
which magnification the opening of the exterior lens can support”.98
Maintaining the clarity of images does not mean,
however, that their quality is maintained. Increasing the
opening of a lens renders images less distinct. Huygens
made it clear that only experience could tell which
configuration produced satisfactory images. Yet, when such
a telescope is known, theory can explain how the quality of
images is maintained when its strength increases. In his
account, Huygens applied a new conception of spherical
aberration that he had defined in an earlier proposition of
De Aberratione. He called it the ‘circle of aberration’. As
contrasted to the earlier conception, in which the aberration
GD is measured along the axis, the circle of aberration is
measured by the distance ED perpendicular to the axis
(Figure 30). In other words, the circle of aberration is the
spot produced by parallel rays coming from one point of a
distant object. Consequently, the images produced by two Figure 30 ‘Circle’
lens systems are equally clear and equally distinct when the
of aberration.
respective circles of aberration are the same.99
OC13, 332-335.
OC13, 336-337. “sed diligenter expendendum quale incrementum exterioris lentis apertura perferre
OC13, 340-343.
Huygens supposed that the circle of aberration XV of a lens system is
mainly produced by the objective lens AB (Figure 31). The ocular lens PO
barely increases the diameter of the circle and could therefore be considered
to have a perfect focus. He considered the opening BC of the objective lens
required to maintain a constant circle of aberration when the focal distance
CD of this lens is changed. He proved that the proportion CD3 : CB4 should
remain constant.100 Finally, the quality of images will be maintained upon
changing the ocular lens, when the proportion OD : 4 CD between the focal
distances of both lenses is maintained.101 Again, Huygens converted these
proportions into a table of numerical values, listing the optimal values of the
focal distances of both lenses and the opening of the objective, as well as the
resulting magnification of the system.102 This table concluded De Aberratione.
Figure 31 Aberration produced by a Keplerian configuration.
Huygens’ theoretical accomplishments in De Aberratione are beyond dispute.
Like the theory of focal distances and magnification of Tractatus, his theory
of spherical aberration was rigorous and general. And again his theoretical
studies were aimed at understanding the telescope; in this case, at
understanding how a system of lenses produces spherical aberration.
Huygens could claim that he understood why an opening of such-and-such
dimensions maintained the quality of images.
His results were couched in two tables listing the required components to
make these optimal systems, in a way quite comparable to the ones found in
Bolantio’s manuscript. They prescribed how to assemble a telescope without
presupposing theoretical knowledge of dioptrics. The difference is that
Huygens’ tables were derived from his mathematical theory of lenses instead
of a record of experiential knowledge. The table prescribing the aperture of
telescopes was not gained by some implicit rule of thumb, but was based on
an explicit theorem derived from dioptrical properties of lenses. Huygens
could prove that the openings he prescribed were optimal. Whether this
worked in practice remains to be seen. At least he could claim that he could
calculate beforehand how to adjust the components of a telescope when its
length was changed, thus avoiding a renewed process of trial-and-error.
Huygens had realized the goal of De Aberratione. He had demonstrated
that the aberrations of spherical lenses could be made to cancel out.
OC13, 342-345.
OC13, 348-351.
OC13, 350-353.
1655-1672 - DE ABERRATIONE
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?
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
OC13, 303n4; 331n4.
OC5, 375; OC6, 23.
OC6, 151; 205; 207.
OC6, 86-87; 151; 205.
OC6, 207.
OC6, 209.
OC6, 214-215.
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.”
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
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
OC6, 218-220.
OC6, 220-221. “mais en decouvrant tout le verre je vois un peu de couleurs ce qui me fait croire qu’il y
a un inconvenient de costè la, qui provient de l’angle que font les 2 surfaces de l’objectif vers les bords.
qui cause necessairement des couleurs, de sorte qu’en faisant des verres hyperboliques l’on trouueroit la
mesme chose en les faisant fort grands.”
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
carry out the plan with an entire glass, like I have asked you to make for
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
“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
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.”
OC6, 236; 266. He did not show consideration for the fact that Constantijn was getting ready for his
marriage on 28 August 1668.
OC6, 266; 300.
OC6, 299-300.
OC6, 353. “Vous ne parlez plus des oculaires que vous m’avez promis.”
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.
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.”
OC6, 348-351; particularly 350.
OC6, 363-367; particularly 364.
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
OC13, 408. “Lens composita hyperbolicae aemula. EUPHKA 1 Febr. 1669.”
OC6, 377. “Vous pourrez luy dire que je le quite pour ce qui est du petit oculaire que je luy avois
demandè, ayant trouvè quelque chose meilleur et de plus considerable en cette matiere, dont j’ay envie de
faire moy mesme l’essay.”
OC13, 413. “…[lens] compositae ex duabus VBC, KST, quae Hyperbolicae aut Ellipticae perfectionem
1655-1672 - DE ABERRATIONE
the other way around.126 The bi-concave lens VBC is given. M is the center of
surface BV, so that rays from M are not refracted by it. Surface CB of this lens
refracts a ray MC to KCN, intersecting the axis in N, where EN is the spherical
aberration. The problem is to find a convex lens KST with the same focus E,
which refracts a parallel ray QK, at distance KS to the axis, to the same point
N. Huygens chose BE – nearly equal to GE – as the focal distance of this lens
KST. Its spherical aberration EN is – by the rule from the ‘Adversaria’ – 76
times its thickness GS. This length EN is also the spherical aberration of
surface BC of the bi-concave lens VBC. It can be expressed in terms of its
radius AB, the length BG (proportional to the distance CG of the ray to the
axis), and the length BM. Equating both expressions for EN, he found a
proportionality between the radius of KST and BC. It is 100 to 254, or nearly
2 to 5. In addition MB, the radius of the other surface BV of the bi-concave
lens, has to be twice that of BC or ten times that of KST. At the end of his
calculations Huygens summarized the solution:
“A lens composed of two emulates a hyperbolic lens, the one plano-convex the other
concave on both sides. The radii of the surfaces are nearly two, five, ten.”127
Five days later, on 6 February, he sent a letter to Oldenburg to which he
appended an anagram containing his ‘important invention’:128
a bc d e h i l m nop r s t u y
5 2 2 1 4 1 23 3 1 3 2 232 4 1
This second invention can be regarded as the final piece of Huygens’ project
of canceling out spherical aberration by means of spherical lenses. He had
shown that spherical lenses were indeed apt for telescopes by designing a
configuration that produced an almost perfect focus. As contrasted to the
earlier invention of 1665, this one could improve telescopes used for
astronomical observation.129 What remained to be done, was to test the
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
OC13, 411-413.
OC13, 417n2. “Lens e duabus composita hyperbolicam aemulatur, altera planoconvexa altera cava
utrimque. Semidiametri superficierum sunt proximè duo, quinque, decem.”
OC4, 354-355 and OC13, 417. The solution of the anagram is: “Lens e duabus composita hyperbolicam
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.
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
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
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.
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.
OC6, 534.
1655-1672 - DE ABERRATIONE
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.
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.
Rigaud, Correspondence II, 70.
OC7, 2-3. “… vous verrez quelque jour que ce que j’en ey escrit est encore tout different.”
OC7, 7-13; especially 10-11.
Discussed in: Bruins, “Problema Alhaseni”.
Hug2, 72r and Hug29, 87r respectively.
OC13, 409n2. “Hoc inutile est inventum propter Abberationem Niutoniana quae colores inducit.”
OC13, 314n1.
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.
Figure 34 The crossed out EUREKA.
In a letter of 11 January 1672, Oldenburg first made mention of Newton to
Huygens.141 This ‘mathematics professor in Cambridge’ had invented a small
telescope in which the objective lens was replaced by a mirror. According to
Oldenburg it represented an object “without any color and very distinct in all
its parts.”142 In his next letter of 25 January, Oldenburg sent him a drawing
and a detailed description, and asked Huygens’ opinion.143 At the bottom of a
relatively wide tube a concave mirror reflected rays to a plano-convex ocular
lens via a small plane mirror. Huygens promptly sent Oldenburg his opinion
on the device. In the 81st issue of Philosophical Transactions (15 March, O.S.),
Oldenburg published Newton’s description of his reflector along with some
of Huygens’ comments.144 In the meantime, Huygens had also sent a letter on
Newton’s telescope to Gallois, the editor of the Journal des Sçavans, who
published an extract of it in the issue of February 29.145
Huygens spoke in the highest terms of Newton’s telescope. He
enumerated no less than four advantages over ordinary telescopes: a mirror
suffers less from spherical aberration, it does not ‘impede rays at the edge of
the glass due to the inclination of both surfaces’, there is no loss of light due
to internal reflections, and inhomogeneities in the material which affect
lenses play no part in mirrors.146 In short, the reflector was a promising
device. The main obstacle for its success, already pointed out by Oldenburg,
was to find a durable material for making reflecting surfaces which lent itself
N.S. All dates are New Style unless indicated otherwise.
OC7, 124-125. “… qui envoye l’object à l’oeil, et l’y represente sans aucune couleur et fort
distinctement en toutes ses parties.”
OC7, 129-131.
OC7, 131 Huygens’ note a; 140-143.
OC7, 134-136.
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,
1655-1672 - DE ABERRATIONE
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
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
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.
Oldenburg’s translation of OC7, 140 in: OldCor8, 520.
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.
Newton, Correspondence I, 95.
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
Two things stand out in this comment. In the first place, Huygens was
mainly interested in the significance of Newton’s findings for dioptrics. In
the second place, he seemed to miss the point of the theory of different
refrangibility.153 In his view, Newton had merely confirmed what he had
observed earlier. Seemingly, he did not realize that Newton’s point was that
chromatic aberration is a consequence of the constitution of light, rather
than the shapes of lenses. In his next letter to Oldenburg, of July 1, Huygens
went more deeply into the matter, though still along the same lines. After
discussing Newton’s telescope a bit further, he wrote:
“As regards his new hypothesis of colors of which you ask my opinion, I admit that it
seems very plausible to me, and the experimentum crucis (if I understand it correctly, as
it is described somewhat obscurely) confirms it very much. But I don’t agree with what
he says about the aberration of rays through convex glasses. For while reading what he
writes, I find that following his principles this aberration must be twice as large as he
takes it, to wit
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
Newton, Correspondence I, 96-100.
OC7, 165. “… je vois qu’il a remarquè comme moy le defaut de la refraction des verres convexes
objectifs a cause de l’inclination de leurs surfaces. Pour ce qui est de sa nouvelle Theorie des couleurs, elle
me paroit fort ingenieuse, mais il faudra veoir si elle est compatible avec toutes les experiences.”
See also: Sabra, Theories of Light, 268-267.
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
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
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
comment in a letter
Oldenburg forwarded to Figure 35 Newton’s determination of chromatic aberration.
Huygens on 28 July.156 He acknowledged that the presentation of his theory
might have been obscure for reasons of brevity. Newton also realized that
Huygens had misread his discussion of chromatic aberration. “But I see
not,” he wrote, “why the Aberration of a Telescope should be more than
about 1/50 of ye Glasses aperture”. He included a drawing of the way he
had calculated the proportion (Figure 35):
“Now, since by my principles ye difference of Refraction of ye most difforme rayes is
about ye 24th or 25th part of their whole refraction, ye Angle GDH will be about a 25th
part of ye angle MDH, and consequently the subtense GH (which is ye diameter of ye least
space, in to which ye refracted rays converge) will be about a 25th of ye subtense MH,
and therefore a 49th part of the whole line MN, ye diameter of ye Lens; or, in round
numbers, about a fiftieth part, as I asserted.” 157
The same letter was accompanied by a copy of the 84th issue of Philosophical
Transactions (17 June, O.S.). It contained a letter in which Pardies criticized
Newton’s theory and a reply by the latter. Two weeks later, Oldenburg sent
Huygens the next issue of Philosophical Transactions (15 July, O.S.) containing
further correspondence of Pardies and Newton on the matter.158 Pardies, a
Jesuit priest and a Parisian acquaintance of Huygens, also criticized Newton’s
claims, but in a more searching manner and with a different line of approach.
He questioned the core of Newton’s theory – different refrangibility – and
raised several objections to his experiments and his interpretations thereof.
For example, he initially doubted whether the oblong spectrum could not be
explained by the accepted rules of refraction.159 He also questioned the very
idea of different refrangibility, which in his view depended upon a
corpuscular conception of light. In his view, colors could also be caused by a
‘diffusion’ of light, for example by a slight spreading of the waves he
Newton, Correspondence I, 137.
Newton, Correspondence I, 212-213; OC7, 207-208.
OC7, 207-208.
OC7, 215.
Newton, Correspondence 1, 131-132.
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
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
Newton, Correspondence 1, 157.
Newton, Correspondence 1, 205. “Je suis tres satisfait de la derniere réponse que M. Newton a bien voulu
faire à mes instances.”
Sabra, Theories of Light, 270.
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.”
OC7, 242-244.
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
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
OC7, 243-244. “Au reste pour ce qui est de l’effect des differentes refractions des rayons dans les verres
de lunettes, il est certaine que l’experience ne s’accorde pas avec ce que trouve Monsieur Newton, car a
considerer seulement la peinture distincte que fait un objectif de 12 pieds dans une chambre obscure, l’on
voit qu’elle est trop distincte et trop bien terminée pour pouvoir estre produite par des rayons qui
s’escarteroient de la 50me partie de l’ouverture de sorte que, comme je vous crois avoir mandè desia cy
devant la difference de la refrangibilité ne suit pas peut estre tousjours de la mesme proportion dans les
grandes et petites inclinations des rayons sur les surfaces du verre.”
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.
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
Newton, Correspondence I, 173.
Shapiro, “Evolving structure”, 224-225.
OC7, 265-266 and Newton, Correspondence I, 264-265.
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.
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.”
OC7, 302. “…, mais voyant qu’il soustient son opinion avec tant de chaleur cela m’oste l’envie de
OC7, 315.
1655-1672 - DE ABERRATIONE
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
OC7, 328-333 and Newton, Correspondence I, 291-295. See also Shapiro, “Evolving structure”, 225-228.
For example Sabra, Theories of Light, 268-272.
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.
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
Shapiro, “Gradual acceptance”, 78-80.
Additional evidence for this dating I find in the fact that Pardies and Huygens discussed Iceland Crystal
in the summer of 1672. See further footnote 120 on page 140 below.
See section 6.1.
1655-1672 - DE ABERRATIONE
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
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
With connected reproduced in OC17, 364-516. On the dating see OC17, 359.
OC17, 373. “Doch de reden van dese couleuren verder te ondersoecken, te weten waerom die in een
prisma gegenereert worden, wil ick geensins ondernemen, emo fateor rationem eorum me prorsus
ignorare, neque facile quemquam ipsas perspecturum arbitror quandiu naturalium rerum scientiae major
aliqua lux non affulserit.”
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
Newton, Certain philosophical questions, 467.
Westfall, Never at rest, 163-164.
Newton, Optical papers 1, 47 & 281.
1655-1672 - DE ABERRATIONE
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
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.
Newton, Optical papers 1, 49 & 283.
Newton, Correspondence 1, 96.
See section 4.1.2.
Traité, 1. “… toutes les sciences où la Geometrie est appliquée à la matiere, …”
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
Yoder, Unrolling time, 16-17.
Yoder, Unrolling time, 33-34.
Yoder, Unrolling time, 19-23. This expression for centrifugal tendency amounts to the modern formula:
F = mv2/r.
Yoder, Unrolling time, 27-32.
Yoder, Unrolling time, 48-59.
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.
Yoder, Unrolling time, 62-64.
The same goes for his earlier study of impact, to be discussed in section 4.2.2.
Westfall, Force, 160-165.
1655-1672 - DE ABERRATIONE
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
In his dioptrical studies, Huygens had likewise relied on his geometrical
proficiency. His theory of spherical aberration was the outcome of rigorous,
sometimes clever deduction. At the point he could have broken really new
ground – when colors emerged – Huygens halted. The process of
geometrizing new phenomena that had proven to be so fruitful in his study
of motion did not get going in dioptrics. Seemingly, he did not see
possibilities to transform those disturbing colors into a geometrical picture,
despite some promising observations he had made of them. However, we
should bear in mind that motion, as contrasted to colors, had already been
mathematized. In his geometrization of circular motion, Huygens could
build on the groundwork laid by Galileo.
Compared to his study of circular motion, De Aberratione was rather
straightforward geometrical reasoning. In this regard, it comes closer to his
study of consonance that occupied him, on and off, from 1661 onwards.201
The first problem Huygens attacked was the order of consonance, an issue
that had arisen (anew) with the new theories of music of the sixteenth and
seventeenth centuries. In the theory of consonance Huygens adopted, the
coincidence theory of Mersenne and Galileo, the order of consonants was
not evident. He derived a clever rule that only left one problem. His rule
seemed to imply that 74 should be placed between the major third and the
Yoder, Unrolling time, 171-173.
Yoder, Unrolling time, 31-32.
Yoder, Unrolling time, 170-171.
Cohen, Quantifying music, 209-230 and Cohen, “Huygens and consonance”, 271-301.
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
OC20, 37. Translation: Cohen, Quantifying music, 214.
The tones of the octave are found using the consonances; this is called the division of the octave. The
seven tones of the diatonic scale are found by means of the fifth ( 2 ) and its complement, the fourth
( 74 ). Likewise the chromatic tones are found by addition of fifths. A problem arises, however, because a
complete octave cannot be reached again by continuous addition of fifths. A small difference, called the
Pythagorean comma, exists between 12 fifths
( 32 )12
and 7 octaves
( 12 ) 7 . As a result, the tones of the
octave ought to be tempered in musical practice, which means that the purity of some consonances is
sacrificed. In mean tone temperament most major thirds are pure and the fifths are made a bit too large;
in equal temperament all consonances save the octave are a bit impure. Huygens preferred the former, the
latter has become standard tuning in Western music since the early nineteenth century.
The diatonic semitone is the difference between E and F, B and c, etc.; the chromatic semitone is the
difference between, for example, C and C. The chromatically sharpened C and flattened D – C# and Db –
differ, whereby C-Db and C#-D have the size of a diatonic semitone. The difference between C-C# and CDb is the difference between both kinds of semitones.
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.
Cohen, Quantifying music, 225-226.
1655-1672 - DE ABERRATIONE
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
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 5 .
Cohen, “Huygens and consonance”, 293-294.
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.
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
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
Yoder, Unrolling time, 71-73.
1655-1672 - DE ABERRATIONE
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
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
Westfall, Force, 165-167.
Cohen, Quantifying music, 224.
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
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
In his lectures Newton derived a formula for spherical aberration. Newton, Optical papers 1, 405-411.
His discovery of dispersion led him to conclude that no lens could ever prevent the disturbing effects
of aberration and made him design his reflector. Shortly after he published his theory, he did consider the
possibility that chromatic aberration could be prevented in lenses. In a letter to Hooke (Newton,
Correspondence I, 172), he alluded to the possibility of constructing a compound lens that canceled out
chromatic aberration. Pursuing an idea of Hooke’s, he considered the possibility of using a lens
compounded of different refracting media in which chromatic aberration was cancelled in the course of
consecutive refractions. (Newton, Mathematical Papers I, 575-576). In Opticks he ruled out this possibility,
probably because it was at odds with the dispersion law he put forward in it. Shapiro, “Dispersion law”,
102-113; Bechler, “Disagreeable”, 107-119.
pursued his thinking on the disturbing colors his configuration turned out to
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
OC13, 435. “Sed hoc tam longe abest, ut fortuito reperti artificij rationem non adhuc satis explicare
potuerint viri doctissimi.”
Van Helden, “Huygens and the astronomers”, 148 & 158-159.
1655-1672 - DE ABERRATIONE
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
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
Boas, “Oldenburg, the ‘Philosophical transactions’, and technology”, 27-35; Ochs, “Royal society”
Another example is Castelli’s attempt to engineer river hydraulics, discussed in Maffioli, Out of Galileo.
OC4, 325-329.
Westfall, “Science and technology”, 72.
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.
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
Gabbey, “Huygens and mechanics”, 174-175; Andriesse, Titan, 235-243.
Van Helden, “Constrasting careers”, 97-101.
Andriesse, Titan, 244-247 and “The melancholic genius”, 8-11. I have discussed Andriesse’s account in
Dutch in Dijksterhuis, “Titan en Christiaan”.
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
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.
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:
Hug2, 188r-188v.
OC13, 740. “mon principal dessein est de faire voir les raisons et les mesures des effects des lunettes
d’approche et des microscopes.”
OC13, 740. Huygens mentions Archimedes (things seen under water), Alhacen, Kepler and Galileo by
name. He elaborated his historical account later during the 1680s.
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; …”
“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
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
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, …”
OC13, 742. “Refraction comment expliquee par Pardies.”
OC13, 742. “transparance sans penetration. corps capable de ce mouvement successif. Propagation
perpendiculaire aux cercles.”
OC13, 742 note 1.
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.
Shapiro, “Kinematic optics”, 209-210. OC10, 203-204.
This is discussed below, in section 4.2.2.
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
“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
OC13, 743: “Cristal d’Islande” and 739: “difficultè du cristal ou talc de Islande. sa description. figure.
See below at the beginning of section 4.2.
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
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.”
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).
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).
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,
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.”
Kepler, Paralipomena, 29 (KGW2, 38)
Smith, “Saving the appearances”, 73-91.
methodological, epistemological and conceptual features of perspectivist
optics are perceptible throughout seventeenth-century optics.24
In the eleventh century, the Islamic scholar Ibn al-Haytham composed a
voluminous work on optics, KitĆb al-ManĆzir. The Optics of Alhacen, as they
are commonly referred to in the West, was intended to bring together
mathematicians’ and physicists’ accounts of light and vision by giving a
systematic treatment of optics that met the demands of both.25 This required
the combination of the Aristotelian doctrine of forms received by the eye
and Ptolemy’s ray-wise analysis of the perception of shape and position.
According to Alhacen both these notions were partly true but incomplete. 26
The synthesis he had in mind - ‘tarkĩb’ - consisted of a theory in which the
forms of light and color issue from every point of the object and extend
rectilinearly in all directions.27 It met the demands of the Aristotelian doctrine
by considering light rays as the direction in which light extended and those
of Ptolemy by appointing rays as the ultimate tool of analysis. As contrasted
to his mathematical precursors, Alhacen regarded a ray as a purely
mathematical entity: “Thus the radial lines are imaginary lines that determine
the direction in which the eye is affected by the form.”28 In a later work on
optics, the Discourse on light, Alhacen expounded his conception of light and
reflected on the character of the science of optics by discussing the
distinction between mathematical and physical aspects of light. In his view,
each provided answers to different kinds of questions: in physics one
investigates the essence of light; in mathematics the radiation or spatial
behavior of light. Physical theory classified the various kinds of bodies:
luminous, shining, transparent, opaque. Mathematical theory described the
perception of things by means of rays, rectilinear and inflected.29 In the Optics
Alhacen adopted the Aristotelian concept of forms without further
philosophical inquiry. His exposition on the nature of light in book 1 served
as a physical foundation for the mathematical and experimental investigation
of light and vision that constituted the heart of the Optics.
Alhacen provided the basis for the flourishing of the study of light and
vision in thirteenth-century Europe given shape to by Robert Grosseteste,
Roger Bacon, John Pecham and Witelo.30 Alhacen’s work reached the west in
This theme is amplified by, among others, Schuster, Descartes, 332-334: Smith, Descartes’s Theory of Light
and Refraction, 4-12.
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.
Alhacen, Optics I, 81 (book 1, section 61).
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.
Alhacen, Optics I, 82 (book 1, section 62)
Alhacen, Optics I, li (Sabra’s introduction). See also Sabra, “Physical and mathematical”, 7-9.
Lindberg, Theories, 120-121 and 109-116.
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
Alhacen, Optics I, lxxiii-lxxix (Sabra’s introduction).
Lindberg, “Cause”, 30-31.
Risner, Optica thesaurus, 112-113. Witelo relied the argument of the shortest path: Risner, Optica thesaurus,
Lindberg, “Cause”, 25-29. Sabra, “Explanation”, 551-552.
Risner, Optica thesaurus, 112-113. “Huius aút rei simile in naturalibus motibus videre possumus, & etiá in
accidentalibus.” Translation: Lindberg, in Grant, Source book, 418.
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
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.
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.
Alhacen, Optics I, lxi (Sabra’s introduction); Risner, Optica thesaurus, XVII-XIX (Lindberg’s introduction).
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
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
Lindberg, Theories, 122-132.
Dupré, Galileo, 17-19.
Lindberg, “Roger Bacon”, 249-250.
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.
Lindberg, “Incorporeality”, 240-243.
Kepler, Paralipomena, 13 (KGW2, 24)
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
Kepler, Paralipomena, 8 (KGW2, 20). “Nam diximus affectari à luce figurationem Sphaerici.”
Propositions XII-XIV: Kepler, Paralipomena, 10-11 (KGW2, 22-23)
Kepler, Paralipomena, 13 (KGW2, 25). “Lux in superficiem illapsa repercutitur in plagam oppositam,
unde advenit.”
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.
Proposition XX: Kepler, Paralipomena, 15-21 (KGW2, 26-31).
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
Kepler, Paralipomena, 16 (KGW2, 27). “Liceat enim hîc mihi verba Opticorum contra mentem ipsorum
usurpare, et in meliorem sensum traducere.”
Kepler, Paralipomena, 85-87 (KGW2, 85-86)
This is equivalent with sini : tanr = constant. Lohne, “Kepler und Harriot”, 197. Compare Buchdahl,
“Methodological aspects”, 283.
experience.53 He tried some more ideas flowing
from this geometry, including some ways of
evaluating the ‘refractaria’ - the locus of images
where the points of a line are percieved, D for
point L, I for point M, etcetera.54 All ideas were
refuted by experience and Kepler abandoned his
attempt of finding a measure of refraction on the
basis of an analysis of the physics of light rays.
In the next three sections, Kepler temporarily
ignored the causes of refraction and focused on
finding mathematical regularities in the given
angles of incidence and refraction. Building on the
known properties of reflection, he tried certain
analogies between reflection and refraction. Kepler
argued that in the case of refraction in a medium
with infinite density, all rays must be refracted into
the perpendicular. He then correlated this case to
reflection by a parabolic mirror with rays coming
from its focus. This led him to consider the Figure 36 The first stage of
relationship of conic sections with refraction. He Kepler’s
constructed a diagram of angles of incidence and
refraction and considered the intersection of the accompanying rays. The
resulting curve is similar to a hyperbola, but points from where the rays
come are not the matching foci, so Kepler dismissed this attempt as well.
This and other trials with conic sections – including the effort to construct
an anaclastic curve – still did not give Kepler a correct ‘measure of
refractions’ and he abandoned this line of thought as well.
Finally, Kepler returned to his causal analysis of refraction of chapter 1 to
query whether - “may God look kindly upon us” - this would yield the
measure of refraction.55 As contrasted to the ray analysis of the first stage, he
now considered the interaction of the surface of light with the surface of the
refracting medium.56 Kepler warned beforehand he would perhaps stray
somewhat from his goal of finding the measure of refractions in its causes,
and halfway through his exercise he would acknowledge “In demonstrating
the true cause of this directly and a priori, I am stuck.”57 He did not formally
deduce a ‘measure’ from the causes of refraction, but rather had employed
(in Buchdahl’s words) “physical considerations to guide the intuitive search
for responsible factors relevant to the result.”58
Kepler, Paralipomena, 86 (KGW2, 86). “Hic modus refutatur experientiâ: ...”.
Kepler, Paralipomena, 88-89 (KGW2, 87-88).
Kepler, Paralipomena, 110 (KGW2, 104). “quod Deus benè vertat”
Kepler, Paralipomena, 110-114 (KGW2, 104-108).
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.”
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 Figure 37 The final stage of Kepler’s
Kepler the angle of deviation must be analysis of refraction
proportional to the angle of incidence.
This condition is met when only the (surface)density of the refracting
medium is assumed to be effective. With increasing obliquity, BM increases
and therefore the resistance met by the light increases. Now “… there is
more density in BM than in LM ...” so that the proportion LM to BM must be
added as a factor of refraction onto the proportionality of angles of
incidence and deviation.59 However, the proportion LM to BM – or sec i –
implied a paradox. Horizontal rays would be refracted at an infinitely large
angle. Kepler therefore changed his perspective and now considered BR of
the refracted ray. He concluded that the secans of the angle at the upper
surface of the denser medium ‘plays a part’ in refraction.60 Refraction was
thus a composite of two factors: the proportionality of i-r to i and the
proportionality of i-r to sec r – in other words: i-r = c· i· sec r, where c is some
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
Kepler, Paralipomena, 111 (KGW2, 105). “Plùs igitur densitatis est in BM, quàm in LM.”
Kepler, Paralipomena, 113 (KGW2, 107). “..., sciendum igitur, eorum angulorum incidentiae secantes concurrere ad
mensuram refractionum, qui constituuntur ad superficiem in medio densiori.”
Kepler, Paralipomena, 116 (KGW2, 109). “Neque te moveat tantilla discrepantia, credas mihi, infra tantam
subtilitatem, experientiam in hac minus apt materia non descendere.”
Kepler, Paralipomena, 116 (KGW2, 109). “Ergò in Vitellionis refractionibus culpa haeret.”
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
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
See for example: Buchdahl, “Methodological aspects”, 291.
True measures
Even without a true measure of refraction, an inventive mathematician like
Kepler could solve problems in the behavior of refracted rays. In Dioptrice, he
determined properties of spherical lenses in a less rigorous way, pragmatically
applying a rule that had only limited validity. Likewise, his predecessors had
used their limited knowledge to discuss isolated problems regarding
refraction. In order to turn ‘dioptrics’ into a genuine part of the
mathematical science of optics, a true measure of refraction was still needed.
However impressive Kepler’s persistence to find a true measure of
refraction, his efforts will always have a tragic side. Around the same time he
was struggling with the phenomenon, across the Channel the exact law had
already been found by the very man Kepler had been corresponding with:
Thomas Harriot.
Harriot had done so by traditional means that were accessible to Kepler
too: analysis of the observed propagation of light rays. The difference was
that Harriot made new observations and had a lucky hand in this. Harriot’s
success shows that, in the case of refraction, traditional methods could yield
the required result. Around 1597, Harriot had begun looking for a law of
refraction. Initially, he also tried to find a law on the basis of Witelo’s tables.
As these efforts were unsuccessful, he decided that Witelo was unreliable and
started to measure angles of refractions anew.64 After some fruitless attempts,
he chose a way of measurement that proved very lucky.
In 1601, he measured refraction
by means of an astrolabe suspended
in water (Figure 38). Viewing along
the center R of the astrolabe, he
determined the positions O where a
point was seen when moved along
the lower edge of the astrolabe. Then
he determined the image points B.
The cathetus rule (see page 33) said
that the image point is the
intersection of the normal to the
refracting surface and the incident
ray. All image points were on a circle. Figure 38 Harriot’s measurements (Lohne).
This meant that RO and RB were in constant proportion, and likewise were
the sines of i and r. In a table Harriot compared angles of deviation as he had
measured them with calculated ones, but he did not reveal how he had used
the figure with two concentric circles – which he called ‘Regium’ – for his
calculations. The calculated values give reason to believe that it was the sine
relation he used.65
Lohne, “Geschichte des Brechungsgesetzes”, 159-160.
Lohne, “Kepler und Harriot”, 202-203.
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.
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.
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.
Hentschel, “Das Brechungsgesetz”, 302-308.
Hentschel, “Das Brechungsgesetz”, 334 note 22.
Smith, “Saving the appearances”, 86-89.
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
Smith, “Big picture”, 587-589.
Shapiro, “The Optical lectures”, 137.
Smith, “Ptolemy’s search”, 239-240.
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.
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
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 Figure 39 Mydorge's rule
of the radii of the two semi-circles.
Second, the constant length of the parallel components FO and OI before
and after refraction. In the diagram of Mydorge these assumptions are
represented directly by the lengths of the respective lines. However, instead
of a distance diagram, in La Dioptrique Descartes used a time diagram where
the lengths of lines represent duration (Figure 40). Instead of the two semicircles representing the constant ratio of the effect of the media, it shows a
single circle. As a consequence the constancy of the parallel component was
represented by lines of differing length (AH and HF).
I have begun with Mydorge’s
rule because it somewhat bridges
conceptualization of refraction and
our understanding of the sine law.
It gives the modern reader a clear
derivation as well as the way he
adapted it to his own line of
thinking. As I will argue below, the
diagrams Descartes used fitted Figure 40 Descartes’ analysis of refraction
perspectivist analysis of refraction rather than his own account, and he chose
them deliberately. The account of La Dioptrique, with all its complicating
facets, was how seventeenth-century readers got to know the sine law and
the mechanistic interpretation of refraction. It formed the starting-point of
all subsequent accounts of the causes of refraction, although few adopted
Descartes’ conceptual and methodological notions in full.
Mersenne, Correspondence I, 404-415. This letter and its import for Descartes’ optics is discussed
thoroughly in Schuster, “Descartes opticien”, 272-277 and Schuster, Descartes and the Scientific Revolution, 304308.
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
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
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)
“… en quelle sorte se doivent mesurer les refractions”, AT6, 101-102.
“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.
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
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
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.
“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.
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
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 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
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.
Sabra, Theories, 97-107; Schuster, Descartes, 333-334.
Schuster, “Descartes opticien”, 279-285; Schuster, Descartes, 334-336.
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)
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
“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.
Except Clerselier who expressly defended Descartes’ mechanistic models; Sabra, Theories, 116-135.
Sabra, Theories, 44.
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
Schuster, “Descartes opticien”, 261-272 Schuster, Descartes, 273; Mahoney, Fermat, 387-393; Sabra,
Theories, 78-89.
Descartes, AT6, 94-95.
Schuster, “Descartes opticien”, 258-261; Schuster, Descartes, 293
Schuster, Descartes, 288.
Schuster, “Descartes opticien” 261-265.
Sabra, Theories, 112-116.
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
Schuster, Descartes, 321-326. See also: Costabel, “Refraction et La Dioptrique”.
Schuster, Descartes, 343-346.
Shea, Magic, 156-157.
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
Newton, Optical lectures, 170-171 & 310-313.
A similar conclusion can be drawn with respect to Descartes’ theory of hydrostatics on which his
concept of ‘conatus’ was based. Shapiro, “Light, pressure”, 260-266.
“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.
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
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.
Sabra, Theories, 12.
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
Barrow, Lectiones, [10].
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
Barrow, Lectiones, [26].
Barrow, Lectiones, [15-16].
Barrow, Lectiones, [15]; (emphasis in original).
Barrow, Lectiones, [8, 15].
Barrow, Lectiones, [26].
Shapiro, “Kinematic optics”, 177-181. Hobbes’ optics is discussed in the next chapter, section 5.2.1.
Barrow, Lectiones, [17]
refraction.112 For reflection, he
considered BD – a line of light in the
most realist sense of the word –
colliding obliquely with a reflecting
surface EF (Figure 42). He argued
that, after B hits the surface, this end
of the line of light rebounds while end
D continues its way, resulting in the
rotation of BD around its center Z. Figure 42 Barrow’s explanation of reflection.
This rotation lasts until D hits the
surface and the line of light is in position ƢƤ. The line of light then continues
towards ơƪ. From the symmetry of the situation the equivalence of the
angles of incidence and reflection follows directly.
To substantiate his claim, Barrow invoked a general ‘law’ of motion:
“… that it is constantly found in nature, when a straight movement degenerates into a
circular one, that it is the extreme parts of the moving objects that direct and control all
He applied the same law to derive
the sine law (Figure 43). On
entering the more resisting
medium below EF, point B of the
line of light BD will be slowed
down while D continues with the
original speed. As a consequence
DB will be rotated around a point
Z until D also reaches the ‘denser’
medium. Then the line of light ƢƤ
will continue along a straight path.
Now, the proportion between ZD
and ZB is constant for any angle
of incidence and depends upon
the particular difference of the
densities. From this it easily
Figure 43 Explanation of refraction.
follows that for i =‘GBM and
r =‘NƤƪ, sin i : sin r = ZD : ZB. After thus explaining refraction into a rarer
medium and total reflection, Barrow was ready to elaborate the ‘Optic
Science’ of his lectures in the common manner:
“…considering rays as one-dimensional (seeing that the other dimensions, in which
physicists delight, have no importance for the calculations here undertaken).”115
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.
Barrow, Lectiones, [28].
Barrow, Lectiones, [29-31]
Barrow, Lectiones, [39-41]
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
Tacquet, Opera mathematica (Antwerp, 1669), Catoptricae libri tres, 217-218
Shapiro, Fits, 31-36.
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
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.
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,
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.
Hug2, 173v. One half of Hug2, 174r is torn away; the page contains a remark that seems of a later date.
Hug2, 173v; OC19, 407. “Perpendiculariter incidens refringitur Non facit duplicem reflexionem.” The
editors combine this with a remark written on Hug2, 175v.
Figure 44 Sketch of refracted rays in Iceland crystal: r (‘regelmatig’) for
ordinary refraction; o (‘onregelmatig’) for strange refraction.
On 8 July 1672, Pardies wrote Huygens about strange refraction. He had
visited Picard and taken a look at a piece of Iceland crystal brought from
Denmark. Pardies did not believe the phenomenon contradicted the sine
law, as he thought it could be explained from the composition of the crystal.
“… it seems to me that it is not as troublesome as I had imagined to explain this effect.
… I am very much mistaken if one cannot demonstrate that, if one were to cut various
pieces of glass in rhomboid shape and simply put one on the other to make a total
rhomboid out of them, two refractions would present themselves.”123
Some sketches Huygens made in his notebook
around the same time are reminiscent of Pardies’
view (Figure 45). They seem to explore how the
composition of the crystal may explain strange
refraction. The surface is drawn indented, so that
part of the perpendicularly incident light actually
falls on an oblique surface. A perpendicular ray
falls upon the indented surface so that part of the Figure 45 A refracted
wave is divided into many small wavelets, that perpendicular caused by the
composition of the crystal.
proceed obliquely to the surface.124
Apparently, Huygens did not accept Pardies’ idea, for he did not
elaborate it beyond these sketches. Moreover, he ended his first study with
the conclusion that the refracted perpendicular contradicted the wave
explanation of refraction.125 A sketch on the next page of his notebook makes
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.”
Hug2, 178v; OC19, 415.
See below page 151 footnote 148.
it clear what kind of problem strange refraction constituted
for the wave theory (Figure 46). It shows the strange
refraction of a perpendicular ray along with, what seems to
be, the propagation of waves.126 After having passed the
refracting surface, the waves proceed obliquely to their
direction of propagation, which contradicts the assumptions
of Pardies' theory. Thus this tiny sketch illustrates what
Huygens called the ‘difficulté’ of strange refraction.
Strange refraction posed a problem for the explanation of
ordinary refraction that Huygens intended to adopt and he
first wanted to solve it. His first attempt is recorded in those
Figure 46
ten notebook pages. The notes are revealing. Despite the fact
Waves through
that strange refraction was a problem of waves, this tiny
the crystal.
sketch is the only place where waves enter his investigation.
Instead, Huygens approached strange refraction in a rather traditional way.
He tried to find out what mathematical regularities rays refracted in Iceland
crystal might display. This was the same way Bartholinus had approached the
phenomenon, namely by trying to find a law describing the behavior of
strangely refracted rays.
Huygens began by recording the
main characteristics of the crystal
(Figure 47).127 With explicit
reference, he reproduced the
Bartholinus. The crystal has the
form of a parallelepiped, of which
the obtuse angles of each
parallelogram like ACB are 101q.
Consequently, the angle AXB
between faces GOCA and FOCB is
103q40' and those between lines
OC and CI (bisecting angle BCA) is
72q34'. A ray of light falling on a
Figure 47 Shape and main angles of the crystal.
face of the crystal is double
refracted. One of the refractions conforms to the sine law, whereby the
index of refraction is approximately 5 to 3, a value Bartholinus had
determined empirically. The other refraction does not follow the sine law
and is therefor called extraordinary or strange. Huygens observed some
physical characteristics of the crystal as well, in particular the fact that the
I experienced some difficulty seeing this sketch as a two-dimensional section, as most historians have
done. I once thought it was meant to be drawn in perspective, a ray refracted out of the paper towards the
reader. Despite this ambivalence, I think after all that the two-dimensional interpretation is correct.
Hug2, 173v and 175r. OC19, 407-408; Bartholinus, Experimenta, 8-11 and 40.
crystal is easily cleft but only along surfaces parallel to its faces. He noted
that other ways of cutting it had not been successful yet. He evidently
alluded to Bartholinus’ discussion of refraction in planes that are not parallel
to the natural faces of the crystal.
These data were written mainly in French. The investigation continues
with an analysis, written in Latin, of the way rays are refracted by the crystal.
Huygens did not adopt the conclusion Bartholinus had drawn from his
experiments with the crystal, in the form of a ‘law’ of strange refraction.
Huygens proposed an alternative ‘law’.
Bartholinus’ experimenta
To see where he took off in his study of strange refraction and to compare
Huygens’ analysis with Bartholinus’, we first turn to the main argument of
Experimenta crystalli islandici disdiaclastici. As regards its methodological
structure, Experimenta is an interesting work in the history of optics.
Bartholinus explicitly discussed how the main principle for the mathematical
analysis of the phenomenon under consideration - his law of strange
refraction - was derived from and subsequently founded upon empirical
investigation. He did not report literally on his experiments, but stylized and
ordered them into a mathematical argument.128 This way of integrating
experimental inquiry into mathematical inference is akin to the approach of
men like Pascal and Mariotte, and seems to have been a common strategy in
Figure 48 Double refraction according to Bartholinus.
that period of (continental) mathematicians to appropriate the new
philosophies.129 Experimenta is divided in two parts: seventeen ‘experimenta’
and ten ‘propositiones’. These parts pivot around a section ‘Observationes
ad demonstrationem præcedentium’, in which he put forward his law of
strange refraction. The ‘experimenta’ describe Bartholinus’ observations of
the crystal and the behavior of rays refracted by it. From these experiments
he inferred his law. The content of the ‘propositiones’ is identical with the
findings of the ‘experimenta’ in the first part. The difference is that in these
propositions Bartholinus used his law of strange refraction to derive the
results of his observations. In this way, Bartholinus demonstrated that the
law agreed with observation.
In the twelfth experiment, Bartholinus introduced the line ED which
bisects the obtuse angle of the upper face (Figure 48).130 This line has
particular properties. In the first place, when the eye is in a plane
perpendicular to the face of the crystal and through ED, both images of an
object A will appear on this line. In the second place, the separation of the
images is maximal on this line. In the following, Bartholinus confined his
account to refractions of rays in the plane perpendicular to the upper face of
the crystal through ED and the edge EM. Huygens later introduced the term
principal section for this plane.
In the next, thirteenth experiment, Bartholinus explained what is strange
about strange refraction. When the crystal is rotated on the table, the image C
of object A is fixed, as it ought to. The second image B, however, moves
around. Therefore, B must be produced by some strange, extraordinary
refraction.131 Throughout Experimenta, Bartholinus distinguished between
ordinary and strange refraction by referring to the ‘fixed’ and the ‘mobile’
images.132 Only in the sixteenth experiment, after he had ruled out reflection
as a cause for the second image, did he introduce the epithet ‘extraordinary
refraction’.133 Huygens did not speak of a mobile image, but only talked of
strange or extraordinary refraction.
Lohne has published and translated an earlier draft of Experimenta, which shows that Bartholinus
reordered the propositions for the final version and apparently rewrote the context of discovery. Lohne,
“Nova experimenta”, 106-107.
See Dear, Discipline and experience, 201-209.
Bartholinus, Experimenta, 19-20. The figure is erroneous, as Bartholinus pointed out too, for ED bisects
the acute angle of the upper surface.
Bartholinus, Experimenta, 20-22. In the text the thirteenth experiment is also numbered XII.
In his fourteenth experiment, he explained how the mobile image might be rendered fixed – and viceversa – by considering alternative surfaces for observing the images. (Bartholinus, Experimenta, 22) It is
questionable whether Bartholinus had actually made the observations he described here. Lohne has
pointed out that, had it been carried out, it would have appeared to yield trivial or erroneous results.
Lohne, “Nova Experimenta”, 135 note 29. Buchwald and Pedersen point out, however, that these
observations are quite difficult to perform. Bartholinus, Experiments, 19-20 (Introduction).
Bartholinus, Experimenta, 29-30. Bartholinus attributed his discovery of strange refraction - that the
duplicate image is caused by refraction instead of reflection - to this experiment, which however the
earlier draft of his treatise does not include. Lohne, “Nova experimenta”, 106-107.
Figure 49 Refraction in two positions of the crystal.
The mobile image contradicts the ordinary laws of refraction, but it does
not “seem to vary with uncertain laws”, so Bartholinus began his fifteenth
experiment.134 To deduce the ‘laws’ governing strange refraction, he recorded
some empirical properties. The mobile image rotates around the fixed image
and does not describe a perfect circle. When the eye is in O, the separations
DC and CB of the images for two positions of the crystal are unequal. When
the eye is in N, on the other hand, the separations QC and CP are the same.
The properties Bartholinus employed to define strange refraction consisted
of qualitative observations where the behavior of strangely refracted rays is
linked to the crystallographic data. This also applies to the observation
crucial to his law of strange refraction: a ray parallel to the edge of the crystal
is not refracted. This probably was a naked eye observation and, as we will
see in the next chapter, it was inaccurate. He did not perform direct optical
measurements of strange refraction. From these rather meager data,
Bartholinus concluded that the mobile image does describe a perfect circle
when the eye is in N.
After seventeen experiments, Bartholinus was ready to formulate a law of
strange refraction: a mathematical construction explaining how to construct
the strange refraction of a ray in the principal section. Bartholinus’ solution is
ingenious; we may surmise his line of thought like the following. As the sine
law does not apply to strange refraction, refractions cannot be measured with
reference to the normal of the refracting plane. Some other instance of
reference should be found, and this was the oblique ray that passed the
surface unrefracted: “… the extraordinary refraction took for its normal a
parallel to the sides of the birefringent crystal, while the ordinary refraction is
Bartholinus, Experimenta, 24. Translation by Archibald.
directed to the perpendicular to
reproduced the Cartesian diagram
of the sine law and added the lines
governing strange refraction
(Figure 50). QGS is the face of the
crystal, DN the normal governing
ordinary refraction. FGL is an
ordinarily refracted ray, so FK : LN
is constant. Bartholinus had
determined empirically the index
of (ordinary) refraction for the
crysal FK : LN = 5 : 3. CP is the
unrefracted oblique ray parallel to Figure 50 Bartholinus’ law of strange refraction.
the edge of the crystal. It governs
strange refraction in the same way as DN does its ordinary counterpart.
Consequently, when FGM is an extraordinarily refracted ray, the sines FI and
PM are in constant ratio, namely 5 : 3.136
It is clear that Bartholinus’ law of strange refraction was an extension of
Descartes’ law of ordinary refraction. According to Pedersen and Buchwald,
the leading idea behind Bartholinus’ law of strange refraction was to preserve
Descartes’ sine law of refraction, changing only its frame of reference.
Strange refraction is strange because its ‘normal’ is oblique to the refracting
surface rather than perpendicular. In one sentence, Bartholinus suggested a
physical explanation of the law, which resembled Descartes’ explanation of
ordinary refraction:
“For it appears that this birefringent crystal has pores running along the faces and
parallel to them, since we may observe that the fracture and separation of fragments
follows this disposition of the sides; and [further] one image, namely the mobile one,
passes through these same [pores].”137
He seems to have adopted Descartes’ theory of light, but he did not
elaborate a causal analysis of strange refraction. Buchwald and Pedersen
point out that Bartholinus expressly distinguished the mathematical law and
the physical structure regarding strange refraction.138 His sole objective was to
establish the law governing the behavior of strangely refracted rays. In his
view he had succeeded in formulating a law from which its observed
properties could be derived. He had also suggested an experiment further to
substantiate it. In the fourteenth experiment, he discussed refraction in
planes that are not parallel to the natural faces of the crystal. He claimed that
Bartholinus, Experimenta, 32. Translation by Archibald.
Bartholinus, Experimenta, 46-48. Modern notation: sin(i – 17):sin(r – 17) = 5:3; Lohne, “Nova
experimenta”, 142.
Bartholinus, Experimenta, 54. Translation by Archibald.
Bartholinus, Experiments, 18-19 (Introduction).
the fixed and the mobile image would swap place, but had not been able to
substantiate this, as he could not cut the crystal appropriately.139
Huygens’ alternatives
Huygens followed Bartholinus’ approach to consider only the observed
properties of strangely refracted rays. He adopted the Dane’s data and he
even seems to follow him in his line of thinking: to extend Descartes’
account of ordinary refraction. Nevertheless, Huygens’ analysis differs in two
respects. In the first place he changed perspective by focusing on the
refracted perpendicular ray instead of the unrefracted oblique ray. Which is
not unexpected, for the refraction of the perpendicular ray formed the heart
of the ‘difficulté’ of strange refraction. Accordingly, the one original datum
Huygens supplied was the angle of the refracted perpendicular: slightly
smaller than 7o.140 Secondly, he went beyond Bartholinus by considering rays
outside the principal section. The outcome was a new law of strange
Having described the crystal,
Huygens began his analysis by
drawing the principal section and
some rays (Figure 51). This plane is
formed by the edge of the crystal
and the line AB that bisects the
obtuse angle of the upper face of
the crystal. The perpendicular ray
GH is refracted to HE. This meant,
according to Huygens, that each ray
in plane GH – the plane through
GH, perpendicular to the paper – is
refracted into the plane HE. KLE is
the unrefracted oblique ray (parallel
to the edge of the crystal through
B). Now, Huygens writes, the Figure 51 Rays in the principal section.
refraction of rays in plane KL (the plane through KL, perpendicular to the
paper) that are not parallel to ray KL, do not lie in plane LE. These rays
outside the principal section are refracted towards the perpendicular, and the
more oblique they are to KL, the closer their refractions are to the plane
through LS, the refracted perpendicular.141
By considering rays outside the principal section, Huygens surpassed
Bartholinus’ account. The ‘oblique’ sine law applied only to rays in the
principal section and therefore was not a general law. By considering rays
Bartholinus, Experimenta, 22-24.
Hug2, 175v; OC19, 410 “Angulus FBC refractionis radii perpendicularis est paulo minor 7 grad. cum ad
solis radios inquiritur.” The reference is to Figure 51.
Hug2, 175v; OC19, 410 “… introrsum versus perpendicularem refringuntur ut in LS, idque tanto magis
quanto erunt ad KL radium obliquiores.”
Figure 52 Construction for strangely refracted rays in the principal section
outside the principal section the refracted perpendicular NLS came into view
as an important line of reference. Bartholinus’ analysis had been fully based
on the unrefracted oblique ray. He had not assigned special significance to
the refracted perpendicular. The refracted perpendicular promptly suggested
an alternative law, which Huygens elaborated on the next, third page of his
Huygens began with a new drawing of
the principal section (Figure 52). In DBCF are
given the normal to the refracting surface
ABF and the refracted perpendicular ABC,
with FBC slightly smaller than 7q. TTC is the
unrefracted oblique ray, parallel to the edge
of the crystal. In the case of ordinary
refraction, Huygens applied the sine law as
he was used to do in his dioptrics. To find
the refraction DD of a ray DF, draw S on BF
so that DS is to DF as the ratio of sines 5 to 3
(or slightly smaller than 8 to 5). Next, he
simply stated the following: in strange Figure 53 Main lines of Figure 52
refraction rays DD are refracted towards C,
where the refracted perpendicular reaches the bottom of the crystal.142 In
other words, strange refraction adds the line FC to the ordinarily refracted
ray. So, to find the strange refraction DC of a ray DD, draw its ordinary
refraction DF and add the line FC. The fact that a parallel component is
added precisely marks the strangeness of strange refraction: the line FC is the
component Iceland crystal ‘strangely’ adds to the perpendicular ray.
Hug2, 176r; OC19, 411.
Figure 54 Huygens’ alternative for Bartholinus’ law.
The previous construction applied to rays in the principal section only, but
Huygens went on to extend it to arbitrary rays (Figure 54). PD is an arbitrary
ray in the plane PDE perpendicular to the surface of the crystal. At the point
of incidence D, the perpendicular ray is refracted towards DG. Now draw
through DB and DG plane DGCB, in which the refracted ray will lie. In order
to find the strange refraction DC of PD, determine its ordinary refraction DF
and then add the ‘strange component’ KG, by drawing FC equal and parallel
to KG.
“So that the motion of the refracted ray within the crystal is composed, as it were, of
the motion it would regularly have, and of a lateral motion whose quantity in the whole
descent through the crystal is equal to the line FC.”143
This, then, was Huygens’ alternative to Bartholinus. He gave no justification
for his construction. He did not reveal whether it accorded with the
observed properties of strange refraction, nor why it was better. Huygens
evidently did not accept Bartholinus’ law, but he nowhere says so explicitly,
nor does he give reasons for his non-acceptance. To be sure, Huygens’
alternative was more general, not being confined to rays in the principal
section. But it was likewise a mathematical construction applicable to rays. It
is comparable to Bartholinus’ law in another respect as well, as it is also
based on Descartes’ account of refraction. Huygens’ construction was a
rather straightforward extension of Descartes’ derivation. It added a ‘strange
parallel component’ to the components of a regularly refracted ray, equal to
the one added by strange refraction to the perpendicular ray. Huygens seized
Hug2, 176v; OC19, 412. “Adeo ut motus radij refracti intra crystallum sit veluti compositus ex motu
quem regulariter haberet, et ex motu laterali cujus quantitas in toto descensu per crystallum est æqualis
rectæ FC.”
the refracted perpendicular in order to
understand the behavior of strangely
refracted rays, whereas Bartholinus started
from the unrefracted oblique ray.
Huygens was not done yet. He
directed a ray AB through two pieces of
crystal GKVH and LNM, aligned with all
their faces parallel (Figure 55).144 By
strange refraction one ray BC continues
unrefracted while the other BD is refracted
ordinarily. Upon leaving the crystal the
rays CE and DF become parallel, as
expected. Yet, a curious thing happened
as they entered the second crystal. Ray CE
was not split up into EO and EP, nor was
DF split up into FP and FQ, as ought to be
(unrefracted) along EO and OS, and DF
was refracted (ordinarily) to FQ and QR.
When, on the other hand, AB was not
parallel to the edge of the crystal, or when
both pieces were not parallel to each
other, the rays were split up by the second
piece. A sketch on the preceding page of
his notes shows how the phenomenon
puzzled Huygens.145 Even if a pair of rays
EP-FP would join, three instead of two
Figure 55 Description of polarization.
images should have been visible.
Huygens’ notes do not reveal whether he just happened to align two
pieces of crystal in this way. The experiment may also have been induced by
his alternative law of strange refraction. The sketches make it clear that he
was considering the unrefracted oblique ray and he may have wondered what
happened if the strange component added to it was equal to the distance EF
between two rays. Huygens may also have been considering the explanation
of strange refraction Pardies had given. If, as Pardies would have it, the
crystal was composed of tiny congruent pieces of crystal, some idea of the
effects might be got by aligning two pieces of the crystal. In that case the
observation would confirm Pardies’ idea as the incident ray would not be
infinitely split up. Whatever induced Huygens to perform the experiment, in
all probability he was the first to observe the phenomenon, polarization as it
is called nowadays.
It puzzled him. In an effort to ‘make sense’ of it, waves returned:
Hug2, 177v; OC19, 412-413.
Hug2, 177r. Not reproduced in the Oeuvres Complètes.
“I have imagined that in the crystal there are two different matters, and that there are
likewise two different ones in the air or ether where the motion happens that we call
light. And that the two motions of undulation of these two matters of the ether have
power to move each its analogous matter of the two that compose the crystal, and
reciprocally, that these different matters of the crystal being stirred, these would be able
to impress this motion of light only upon its analogous matter of the ether.”146
In this case ray AB contains both motions, whereas CE and DF contain only
the motions belonging to strange and ordinary refraction respectively. But
then, Huygens said, it remains to be seen why CE and DF are split up for
other positions of the crystals.
“Which is very difficult, because for this it is required, that these rays CE, DF, although
not composed when hitting the surface LN of the crystal in some direction, are able to
move the two different matters that compose it, and in other directions not.”147
Besides this puzzling phenomenon, the actual problem of strange refraction
was not really near its solution. Huygens concluded his first study of strange
refraction with a formulation of the true problem with the anomalous
refraction of the perpendicular, accompanied by a sketch of a Pardies-like
explanation (Figure 45 on page 142):
“How can the perpendicular ray become oblique by the refraction, for it happens that
the waves will not be at right angles to the line of their extension or emanation,
contrary to what our hypothesis of light demands.”148
Huygens had now found a means to construct a strangely refracted ray,
which he apparently preferred over Bartholinus’ law. It was more general but
Huygens still did not have a clue towards an explanation. The core of his
alternative ‘law’ was the refracted perpendicular, which indeed was the core
of the problem of strange refraction. Maybe Huygens thought that
explaining strange refraction might benefit from the mathematical regularity
expressed by his law, or the other way around. As of yet, he had no idea
what might happen to waves so as to explain strange refraction, nor do his
notes suggest that he had considered the matter in any detail. He had
extended ordinary refraction by adding a strange component. What this
component – or any of the components in Descartes’ derivation – might
mean in terms of waves he had not considered. At this point Huygens may
have brought some clarity to the behavior of strangely refracted rays, but the
Hug2, 178r; OC19, 413-414. “Pour rendre raison du phenomene de la page precedente, je me suis
imaginè que dans ce crystal il y a deux matieres differentes, et qu’il y en a pareillement deux differentes en
l’air ou ether dont le mouvement fait ce que nous appellons lumiere. Et que les deux mouvements
d’undulation de ces deux matieres de l’ether ont pouvoir d’emouvoir chacun sa matiere analogue des deux
qui composent le crystal, et que reciproquement, ces matieres differentes du crystal estant esbranlees, ne
sçavroient imprimer ce mouvement de lumiere qu’a leur matiere analogue de l’ether.”
Hug2, 178r; OC19, 414. “Ce qui est tres difficile, car il faut pour cela, que ces rayons CE, DF quoyque
non composez en frappant en certain sens la surface du crystal LN, puissent esbransler les 2 differentes
matieres qui le composent, et en d’autres sens point.”
Hug2, 178v; OC19, 414-415. “Comment le rayon perpendiculaire peut il devenir oblique par la
refraction, car il arrivera que les ondes ne seront pas a angles droits a la ligne de leur extension ou
emanation, contre ce que demande notre hypothese de la lumiere.”
behavior of strangely propagated waves was just as obscure as when he
began his investigation.
In the view of the way he had formulated the problem of strange refraction,
Huygens’ first investigation of strange refraction may strike us as odd. He
clearly considered it to be a problem of waves, but these do not enter his
analysis. He approached strange refraction in a rather traditional way as a
problem regarding the behavior of rays. Moreover, the fact that it was an
extension of Descartes’ derivation of the sine law may strike us as odd. Such
a line of thought seems contradictory to his adoption of a Pardies-like
explanation of refraction, and to his express rejection of Descartes’
explanation. Of course, Huygens may just have been trying to see how far he
could get. We need not make too much of the alternative Huygens devised in
1672, for we never hear of it again. Still, I will briefly discuss the possible
conflict between a Cartesian analysis of strange refraction and a Pardies
conception of ordinary refraction, as this will be illuminating for our
understanding of the physical conceptualization of light involved.
The ‘Projet’ reveals that Huygens rejected Descartes’ view of light as a
tendency that propagates instantaneously. In addition, he noticed difficulties
in Descartes’ derivation of the sine law, particularly the assumed increase
when a ray is refracted towards the normal. Does this rule out an analysis in
terms of components of motion as Huygens employed in his ‘law’ of strange
refraction? I believe it does; Descartes’ derivation of the sine law conflicts
with Pardies’. Yet, Huygens need not have been aware of such a conflict. He
might have accepted the mathematical structure of Descartes’ derivation in a
general sense – refraction adds a component to the motion of light –
regardless of the question how the motion of rays and its components ought
to be interpreted physically. He may have assumed – deliberately or not –
that an interpretation of the derivation in terms of waves would also be
possible. It was not, I will argue, as the assumptions of Descartes’ derivation
are meaningless with a wave conception of light.
In order to substantiate the last point, I now sketch Ango’s derivation of
the sine law, assuming it to be similar to Pardies’ account.149 The derivation
depends on two premises: rays are always normal to waves and waves
propagate with a definite speed in different media.150 With these premises a
ray refracted according to the sine law can be constructed. Consider spherical
waves passing the surface BED between two media (Figure 56). Ray ccC is the
direction of propagation of the wave and normal to the tangent Cm. In order
to construct the propagated wave in the second medium, its tangent Cn is
constructed. Draw an arbitrary circle that cuts BED in C and K and Cm in m.
Draw Kn so that Km : Kn = v1 : v2 , where v1 and v2 are the speeds of
propagation in the respective media. Now Cn must be tangent to the
Ango, Optique, 61-66.
Shapiro, “Kinematic optics”, 209-218.
refracted wave and thus
perpendicular to its direction
of propagation Cee. Thus
Cee is the refracted ray for
incident ray ccC. It is easily
shown that the sine law
Now compare Descartes’
derivation and Huygens’
extension of it to strange
assumed that the parallel
component of the ray was
conserved. He did not say
perpendicular component.
This accords with Huygens’ Figure 56 Ango’s explanation of refraction.
construction, which adds a ‘lateral’ component to an ordinarily refracted ray.
The second assumption of Descartes’ derivation was a constant proportion
of the motions of the ray before and after refraction. Pardies also assumed
such a constant proportion, but exactly the other way around. Waves move
faster in air than in glass, whereas in Descartes’ derivation rays necessarily
move fastest in glass. Consequently, a Cartesian derivation contradicts a
Pardies-like explanation of refraction. Moreover, in Pardies’ derivation of the
sine law, both components of the ray have changed after refraction,
rendering the Cartesian analysis meaningless.151
How could so gifted a man as Huygens
overlook such an obvious inconsistency?
We should bear in mind that, in Dioptrica,
Huygens never used the Cartesian circle
diagram (Figure 40 on page 127). He always
visualized the sine law as a ‘cathetus’
construction, where CG and CD are
constructed according to the ratio of sines
(Figure 57). A similar approach is also
suggested by DS and DF of the ordinarily
refracted ray (Figure 54 on page 149). The
Figure 57 The sine law in Tractatus.
details of Descartes’ derivation need not
have been on top of his head when Huygens added his strange component.
Although he stayed closer to the drift of Descartes’ derivation as compared
to Bartholinus – who merely extended the circle diagram – he nevertheless
In Ango’s diagram the proportion of speeds is directly represented by the ratio Kn : Km and
subsequently by the distances cc and ee traversed by the waves of light. These distances can, in its turn, be
analyzed in parallel and perpendicular components, but both change according to the figure. The
assumption visini = vrsinr is meaningless.
left aside the possible physical implications of his Cartesian analysis of
strange refraction. It remains to be seen how seriously he took his proposal.
In view of his commitment to a wave conception one might say that
Huygens was just taking considerable liberty of reasoning in order to see
how far he could get at fathoming the oddities of strange refraction.
Huygens also afforded himself liberty in another respect. In his notes, he
did not explain his motives for proposing an alternative to Bartholinus’ law.
Like Bartholinus, he extended the mathematical structure of the sine law
through rational analysis. He did not question Bartholinus’ data and confined
his study to mathematical analysis.152 Also, irrespective of the virtues of
Bartholinus’ verification, Huygens made no effort to justify his conclusions
empirically. If his alternative was more general, he did not check whether it
was anywhere near the truth. Huygens was familiar with such an approach of
mathematical reasoning and it had been successful several times. Although in
his optical studies in De Aberratione it had led him somewhat astray, in his
studies of motion this strategy had been very rewarding. As mentioned in
chapter three, an empirical study of gravitational acceleration had got him
nowhere. The breakthrough had been effected by mathematical analysis of
circular motion.
In his correction of Descartes’ rules of impact, Huygens likewise relied on
rational analysis. He built upon the established, Galilean laws of motion to
find the true laws of impact by means of rational analysis, a strategy he also
chose in his analysis of strange refraction where he built upon the established
law of ordinary refraction. Huygens carried out his study of impact between
1652 and 1656. It has been discussed in full detail by Westfall.153 He had
found out that Descartes’ rules of impact - a crucial topic in mechanistic
philosophy - where wrong save for the first. In particular in the case of
unequal bodies or speeds, the rules proved inconsistent and failed to obey
Galileo’s principle of relativity. Huygens’ solution lay in rigorously applying
this principle, in combination with the principle of inertia. In so doing, he
converted the study of impact into an extension of Galileo’s theory of
uniform motion, namely, the inertial motion of the center of gravity of two
colliding bodies.154 As Westfall shows elaborately, the main thread in
Huygens’ study of impact was an increasing desire to treat impact in terms of
velocities instead of forces, which in his view defied mathematical clarity. As
we shall see in the next chapter, the concept of velocity would be crucial to
Huygens’ understanding of the mechanistic causes of natural phenomena. In
his study of impact Huygens repeatedly solved problems by transforming
them into problems subject to known principles. The principle of relativity
made possible the treatment of all collisions of equal bodies. Collisions of
There is no way Ziggelaar’s observation that “Huygens repeats carefully the experiments of Bartholin
on the crystal, measures more exactly, …” can be substantiated. Ziggelaar, “How”, 182.
Westfall, Force, 149-155.
Westfall, Force, 152-153.
unequal bodies could be solved with the principle of inertia combined with
the hypothesis stating that one body conserves its original motion if the
other does so.
Huygens approached strange refraction in a way similar to his analysis of
impact: by trying to reduce it to established principles, in this case Descartes’
derivation of the sine law. His study of strange refraction consisted of a
mathematical analysis of the behavior of strangely refracted rays, trying to
transform the new phenomenon into a phenomenon whose mathematical
properties were already known. Measuring or experimenting was not the way
in which Huygens explored strange refraction. He first tried to fathom the
mathematical structure of the phenomenon. As we have seen in the case of
chromatic aberration, he ruled out the possibility that such understanding
could be attained beforehand, consequently he did not set up an empirical
investigation of it. The precision he achieved in measuring the constant of
gravitational acceleration was made possible by his mathematical
understanding of the matter. Huygens approached his subject first of all
theoretically, interpreting concepts geometrically and analyzing phenomena
by means of his mathematical mastery.
The particular problem of strange refraction: waves versus masses
Huygens’ initial approach to strange refraction does not suggest that his
account would stand out much as compared to Bartholinus’. It does not give
reason to expect that it would lead to a memorable contribution to the
history of optics. Had Huygens realized his ‘Projet’, this study of mine would
probably not have been written. Huygens’ ‘Dioptrique’ would have been a
treatise in geometrical optics, an analysis of the behavior of rays where the
nature of light played an elucidative role at the most. Its most distinguishing
feature would have been the way it combined rigorous analysis with a focus
on the telescope. The dioptrical theory would have been preceded by some
mechanistic justification of the sine law – one that would probably be
counted nowadays as a repetition or a variant of Pardies. Maybe it would also
have contained a discussion of strange refraction in terms of waves. What
this would have looked like, we cannot tell.
Huygens had tried to make sense of strange refraction by reducing it to a
strange component added to ordinary refraction. From this point of view, his
first investigation seems successful. He had found a law that was more
general than Bartholinus’ law, and was also more in line with Descartes’
analysis of refraction. Yet, his alternative law did not solve the actual
problem of strange refraction, which was a problem of waves. Whatever the
virtues of his law, it did not explain how waves could become oblique to
rays. And this is the most remarkable thing of all about Huygens’ study.
Strange refraction was a problem of waves but they did not enter his analysis.
He was thinking in terms of rays. He tried to establish a general law by
analyzing the mathematical regularities of rays refracted in Iceland crystal. In
this sense his approach was traditional. His ideas about the nature of light
and his analysis of strangely refracted rays remained separate. As yet,
Huygens had not considered the interaction of waves with Iceland crystal
beyond the observation that it constituted a problem.
This, then, is where the strategy that had been so successful with motion
broke down. Whereas motion only involves observable entities like balls and
pendulums, refraction also entails a consideration of unobservable entities.
As contrasted to the laws of motion, mechanistic causes are involved in the
laws of optics. Huygens believed that a Pardies-like theory of light was a
plausible way to explain refraction, but he had not figured out how it related
to strange refraction. Still, he had committed himself to this theory, and this
was why his earliest study of strange refraction ended inconclusively. In view
of geometrical optics, it may appear strange that Huygens would reject his
law on the basis of mechanistic considerations. Why did he not drop Pardies’
Looking at his writings prior to 1672, one would not expect Huygens to
feel so strongly about a mechanistic theory. Prior to the ‘Projet’, Huygens
never considered the nature of refraction more than incidentally. During his
trip to Paris, on 3 January 1661 he discussed refractions – “contre des
Cartes” – with Clerselier and others.155 In a letter to Moray of 9 June 1662 he
ridiculed Vossius’ ideas on light and said that he had a totally different “…
doctrine concerning refraction …”156 What it was he did not say. The same
year his brother Lodewijk copied him a letter containing Fermat’s derivation
of the sine law, based on the assumption that light follows the quickest path.
On 8 March 1662 he answered that he admired the ingenuity of Fermat’s
proof, but he found his fundamental assumption “pitoyable” and considered
his doctrine unsatisfactory. According to him, neither Descartes nor Fermat
were capable of proving the law of refraction, “the fundamental theorem of
refractions” – and only experience renders it certain.157 Apparently Huygens
saw no point in going more deeply into the matter of the cause of refraction.
Despite his deep involvement in dioptrics, Huygens had never elaborated or
adopted a theory of light or refraction. Whether he was simply not really
interested in the matter or had ideas of his own on the back of his head, is
hard to tell.158
In general, his attitude towards Cartesians and mechanistic philosophizing
had been ambivalent during the previous years. In Paris, as he had already
OC22, 544.
OC4, 149. “… doctrine touchant la refraction …”.
OC4, 71. “Pour faire donc l’accord entre luy [Fermat] et Monsieur des Cartes je dirois que ny l’un ny
l’autre a prouvè la theorem fondamental des refractions, et qu’il n’y a que la seule experience qui nous en
rende certains.”
There is one reference to the nature of light in a planned introduction, of 1656, to his treatise on
impact, which appears, however, to be in contradiction with his later views, as Huygens seems to adopt
instantaneous propagation of light: “… if nature as a whole consists of certain corpuscules from the
motion of which every diversity arises, and from the fastest impulse of which light is propagated in a
moment of time and flows throughout the immense expanse of the sky, …” (OC16, 150; translation:
Gabbey, “Huygens and mechanics”, 189)
experienced during his earlier travels, savants displayed much more concern
for philosophical and metaphysical topics than he did. During his trip to
Paris and London in 1660-1661, he expressed his appreciation for the downto-earth attitude of the Londoners as compared to the more esoteric bent of
the Parisians. The particular group of Parisians he started to cultivate during
that visit consisted of like-minded ‘mathématiciens’ like Auzout and Petit.159
In this light, his commitment to a mechanistic theory of light seems
There is reason to believe that Huygens’ move to Paris brought about a
change in his interests. After 1666, at the Académie, he was confronted with
many discussions about natural philosophical topics. Huygens took part, in
particular, in a discussion on gravity in August 1669.160 Van Berkel has
suggested that Huygens began to emphasize mechanistic explanations
because he was dissatisfied with the many theories put forward at the
Académie that were not (properly) mechanistic.161 His opponents in the
discussion on gravity assumed, for example, attractive forces. Huygens’
paper on gravity of 1669 may have had the effect that he saw the value of
discussing natural philosophical questions. In the discussion about gravity,
allusions to the nature of light also come out for the first time. In the notes
he took during this discussion the question is asked how light can be
understood when perfectly hard bodies do not rebound. And he added that
if the corpuscles explaining light were elastic and composed this would
accord with “…l’hypothese du P. Pardies …”.162 It is reasonable to suggest
that the Parisian scene compelled Huygens to think more and more actively
on questions of mechanistic philosophy than he had done in The Hague.
Pardies himself may have been a decisive factor in this regard. It is not
inconceivable that Pardies’ wave theory showed Huygens that it was possible
to pursue mechanistic philosophizing in a satisfactory manner. Although we
do not know in what manner Huygens intended to treat the mechanistic
causes of refraction, the Pardies-like theory may have offered the kind of
middle course between a non-committal, Barrovian account and a Cartesian
derivation that suited him. Statements in the ‘Projet’ seem to rule out a
Cartesian view, whereas his investigation of strange refraction suggests that
Huygens took questions regarding the nature of light more seriously than
Barrow did. In the next chapter, I discuss what may have attracted Huygens
in Pardies’ theory. I think the example set by Pardies made Huygens realize
that it was possible to treat the nature of light in a ‘comprehensible’ way.163
He saw no alternative for Pardies’ waves and, strange as it may seem in view
Hahn, ”Huygens and France”, 58-59.
See below, section 6.3.1.
Van Berkel, “Legacy”, 55-59.
OC16, 184.
I owe this suggestion to Alan Shapiro. It is elaborated in section 5.2.2.
of his earlier lack of interest for mechanistic explanation, he did not want to
drop they idea of giving an explanation of refraction.
I suspect few contemporaries would have objected if Huygens had passed
over strange refraction in a treatise on dioptrics. Few would realize that it
contradicted his explanation of refraction; even Pardies himself thought that
strange refraction could easily be resolved with his theory. It looks like
Huygens made things difficult for himself by choosing to bring up the
‘difficulté’ of strange refraction. Not only did he decide to include an
account of strange refraction, but he also wanted to reconcile this with his
mechanistic conception of ordinary refraction. According to Huygens, the
causes of the various properties of light could only be plausible when they
were consistent with one another. Therefore, his study of strange refraction
had not yet come to an end.
The question Huygens had posed in the ‘Projet’ – the ‘difficulté’ of
strange refraction – carried the seed of a turn towards a new way to consider
the problem of finding a new law of optics. His attitude towards the
justification of established laws and the way he had searched for a law of
strange refraction had been traditional. Had he been satisfied with the
results, ‘Dioptrique’ would have been a traditional treatise in geometrical
optics, unaffected by the changes initiated by Kepler and Descartes. The
problem of strange refraction turned out to be not traditional, as it was a
problem of waves instead of rays. In the next chapter we shall see how
Huygens took the remaining step, that of analyzing strange refraction on the
level of waves of light.
Chapter 5
1677-1679 - Waves of Light
The road to the wave theory and the transformation of geometrical optics
At the end of his first attack on strange refraction in 1672, Huygens was
literally back at his original question: how can strange refraction be
reconciled with Pardies-like waves? The refracted perpendicular contradicted
the principal assumption of wave propagation, that rays are normal to waves.
He left the problem unsolved and left his ‘Projet’ for what is was, a project.
Five years later, he returned to the problem. This time he went straight to the
heart of the matter: what happens to waves when they enter Iceland crystal?
This was the question he had left aside in 1672. Now, in the summer of
1677, he found an answer that solved the problem of strange refraction in
one stroke. “EUPHKA”, he exclaimed on 6 August 1677.
This solution was preceded by a study of some topics of refraction in
which Huygens was reconsidering, so we can say with hindsight, the question
of how exactly waves propagate. In the course of these investigations, he
implicitly formulated his principle of wave propagation and subsequently
applied it to waves in Iceland crystal. He did not elaborate the principle until
he presented his theory of light at the Académie in the summer of 1679. This
theory makes it clear that something special had happened in the summer of
1677. He had developed a rigorously mathematized definition of light waves,
that is: of the mechanistic nature of light. Huygens’ principle of wave
propagation enabled him to derive the laws of optics from a theory
explicating the mechanistic nature of light. It enabled him to give mutually
consistent explanations of the rectilinearity of rays, of reflection and
refraction, and of strange refraction. It enabled him, in other words, to
elaborate the ‘Projet’. But then the question is whether his explanations still
fitted the methodological scheme laid out in the ‘Projet’. It did not, as
Huygens was really doing a new kind of optics that went beyond traditional
geometrical optics. In his new theory the mechanistic nature of light was no
longer an additional elucidation of the laws of optics but the very heart of
the theory.
The question is to what extent Huygens realized that he was breaking
new ground, beyond the mere solving of the puzzle of strange refraction.
The text of the eventual Traité de la Lumière presents the epistemic novelties
of his theory of light in a rather matter-of-fact way. Yet, to objections raised
at the Académie in 1679 to his explanation of strange refraction, Huygens
had replied in a way that brought its innovative character into sharp relief.
His new theory did not allow of direct empirical proof, but required an
indirect empirical confirmation by means of an experimental test of an
hypothesis derived from it. The test succeeded, giving rise to a second
“EUPHKA” on 6 August 1679. This event underscores the remarkable fact
that until this late point, Huygens had largely proceeded by rational analysis
without much thought for the empirical foundation of his ideas.
This chapter discusses the development of Huygens’ wave theory of light
from 1677 until the events of 1679. It is marked by a twofold “EUPHKA”
written out in Huygens’ notebook, and that may be said to signify the
context of discovery and of justification respectively. Unlike the “EUPHKA”
that had hailed Huygens design for a perfect telescope in 1669, these two
would stand the test of time. Traité de la Lumière of 1690 contains the larger
part of the text he read at the Académie in 1679. It remained unchanged
until 1689, when Huygens made some corrections and additions as he finally
prepared his theory for publication.1
Times were still eventful for Huygens. After his return, upon his
recovery, to Paris in June 1671, years of productivity followed. His encounter
with Newton and his reflective telescope has been discussed in chapter 3, the
‘Projet’ and the first attack on strange refraction in the previous chapter. A
discussion of Alhacen’s problem with de Sluse had started early 1671 and
reached its peak late 1672.2 On 9 August 1673 he wrote a letter for Colbert,
in which he summarized the state of the art in contemporary dioptrics. He
explained the value of dioptrical theory: with “… the rules of refraction …,
one could predict in advance the effect of telescopes”.3 He described the
most powerful telescopes available and the problems with grinding good
lenses, especially in Paris. A design for grinding non-spherical lenses by
Smethwick had caught his attention in 1671. In 1675, he discussed it, without
becoming fully convinced of the validity of the method.4 In addition, he was
engaged in various activities, like trials of his pendulum clock at sea and the
invention of the spring balance and its subsequent priority disputes, early
1675, with Hooke and Hautefeuille. Several papers in Journal des Sçavans and
Philosophical Transactions appeared and in 1673 Huygens published his master
piece Horologium Oscillatorium. He dedicated it to Louis XIV, in spite of the
fact that his patron had invaded his fatherland the previous year and
occasioning the Republic’s ‘disaster year’ and the lynching of his
mathematics soul mate Johan de Witt .
The text in which these changes are made is preserved in two manuscript copies. OC19,
“Avertissement”, 383.
The problem is, to find the point on a spherical mirror were a light ray is reflected when the position of
the light source and the eye of the observer are given. In June 1669, Huygens sent his initial solution to
Oldenburg, who began sending de Sluse’s work to Huygens in August 1670. Extracts of ensuing letters
were printed in Philosophical Transactions of October and November 1673 after the discussion had
ended in January. It is primarily a mathematical problem and less relevant for my account of Huygens’
optics, so I will not discuss it. For a detailed account of the problem and its solution: Bruins, “Problema”.
OC7, 350-351. “… les regles de refraction …, l’on pouvoit predire par avance l’effect des lunettes
d’approche” It is not clear on what occasion he wrote this.
OC7, 111 (October 1671); 117 (November 1671); 487 (July 1675); 511-513 (October 1675)
1677-1679 – WAVES OF LIGHT
Amidst these various activities there is no sign of further work on the
issues raised in the ‘Projet’. After the inconclusive end of his investigation of
strange refraction, Huygens seems to have let the matter rest. Then, in the
winter of 1675/6 his ‘melancholie’ reared its head again. Huygens went
home to The Hague the following summer, returning to Paris two years later
in June 1678. But he came back with valuable stuff. He had discovered Van
Leeuwenhoek and his microscopes, adding several innovations as well as a
new topic for his dioptrics. And he had a new insight in the nature of light
and the solution to the puzzle of 1672: how can Iceland crystal refract a
perpendicular ray?
5.1 A new theory of waves
On 15 September 1676, Constantijn Sr. wrote to Oldenburg that ‘his
Archimedes’ had brought a piece of that remarkable Iceland crystal with
him.5 In the Hague, sometime during the next year, Christiaan returned to
the problem of strange refraction. On 14 October 1677 he wrote to Colbert
that he had recently demonstrated the properties of Iceland crystal “…,
which is not a small wonder of nature, nor easy to fathom”.6 The solution is
found in Huygens’ notebook, right after an investigation of caustics in which
he first formulated his principle of wave propagation. This principle then
turned out to provide the basis for solving the problem strange refraction
posed for waves.
The study of caustics and the solution of strange refraction together take
up 11 pages of the notebook, which in my view form one continuous whole.
However, in their customary manner, the editors of the Oeuvres Complètes
have split up the contents into five paragraphs of the section ‘La Lumière’ in
OC19.7 They blended material from different pages into what they
considered coherent issues, to the point of inserting material that dates from
years later.8 This not just disturbs chronology but even Huygens’ actual line
of thinking. I shall now offer my reconstruction of how his conception of
the propagation of waves developed hand in hand with the study of
OC8, 19.
OC8, 36-37. “… demontrè … depuis peu celle [les proprietez] du Cristal d’Islande, qui n’est pas une
petite merveille de la nature, ni aisée a aprofondir.”
Hug9, 38r-48v; OC19, 416-431. With considerable effort, the original order may be reproduced on the
basis of information given in the editors’ annotations. In order to give an idea of the way the manuscript
material has been mixed up in the Oeuvres Complètes, I will list the way the order in which the illustrations
are given (page number in Hug9, number of the illustration in OC19 - section number in OC19. ‘nu’: an
illustration not used) 38r, 137-3, nu (shortest path); 38v, 148-6 (ovals); 39r, 149-6, nu (ovals); 39v, nu nu
nu (ovals); 40r, nu nu (ovals); 40v, 138-3, 139-3, nu nu (shortest path); 41r, 141-4, 144-5, 140-4, nu
(caustics); 41v, 145-5, nu (caustics); 42r, 146-5, nu (caustics); 42v, 150-6 (caustics, wave propagation); 43r,
142-5, 143-5, nu nu (wave propagation, principle); 43v, calculations; 44r, nu nu (telescope and a
wavefront); 44v, nu nu nu (idea of spheroidal wave?); 45r, nu nu nu (idem); 45v, nu (spheroidal waves,
sketch related to Eureka); 46r, ...; 46v, calculations; 47r, 151-7, 152-7, 156-7 (eureka); 47v, nu nu (waves?);
48r, 154-7, 153-7 (shape of crystal); 48v, 147-5, nu nu (athm refraction, (faulty?) propagation (?) of
spheroidal waves)
§4 on OC19, 430-431 is of a much later date than the insertion in the section on the explanation of
August 1677 suggests. With respect to content and place in the notebook it must be closer to the
experiment of August 1679.
Cartesian ovals and caustics also conducted on these pages. In my view, the
solution to the problem of strange refraction came directly out of this line of
thought, although it may be argued that at some point some break occurred.9
The editors failed to reproduce the material on the five pages preceding the
EUPHKA. It is indisputably related to the solution and, although somewhat
obscure, it clarifies the way the solution may have taken shape in Huygens’
Both Huygens’ account of caustics and his
explanation of strange refraction depend upon
Huygens new conception of wave propagation.
Yet, it remains implicit throughout the notes
involved. Only one or two tiny sketches reveal
that an adjustment of Pardies’ wave theory had
taken shape in his mind. The crucial insight is
first found in the sketch reproduced in Figure
58. All points on a wave are centers of a
multitude of wavelets spreading in all directions;
the tangent to these wavelets is the propagated
wave. Only in Traité de la Lumière did Huygens
elaborate his principle of wave propagation and Figure 58 Huygens’ principle.
its application to the behavior of light rays.
To my knowledge, only two previous historians have consulted parts of
the manuscripts reproduced in volume 19 of the Oeuvres Complètes. They were
puzzled. According to Shapiro, they contain “the most subtle refinement of
Huygens’ optics” which cannot have been its starting point.10 Ziggelaar has
suggested that they reflect the “flash of genius” in which Huygens found his
principle, that he then applied to the sophisticated problem of caustics.11 In
my view, rather than a matter of application, the principle gradually emerged
from the analysis of caustics. By following the manuscript material, we may
find out what the “flash of genius” was – if it was one at all – and what
sparked it.
The pages begin with a drawing of a ray refracted by a plane surface,
accompanied by a formulation of Fermat’s principle of least time: a ray is
refracted in such a way that light travels between two points in different
media in minimal time.12 On the next pages Huygens used the ensuing
equation for the lengths of the two parts of the refracted ray to construct a
Cartesian oval, the curve that refracts rays from one point to exactly a
Hug9, 42v is clearly written later, as it is dated 24 March 1678. The following three pages contain some
scattered sketches and calculations.
Shapiro, “Kinematic optics”, 241.
Ziggelaar, “How”, 187.
Hug9, 39r; §1on OC19, 416.
1677-1679 – WAVES OF LIGHT
second point.13 (In a way this brought him back to the very beginning of his
dioptrical studies.) Then he returned to the principle of least time, now to
derive it from the sine law (Fermat had worked the other way around).14 All
of this dealt with rays refracted by plane and curved surfaces.
Figure 59 Two rays refracted by a plane
surface. Lettering added.
Figure 60 Wave VK refracted by a plane surface
VM forming a caustic VHN.
On the next page, Huygens moved on to the case when rays do not
intersect in one point after refraction.15 In such cases, the intersections of
refracted rays form a caustic. Huygens first considered two rays refracted
from glass (top) to air (bottom) at a plane surface (Figure 59). Before
refraction, the rays AP and DP intersect in P. AP : AE = DP : DF = 3 : 2,
according to the sine law. The refracted rays AE and DF intersect in H.
Huygens continued: “the difference of the two PA, PD must be 23 of the
difference of the two EA, FD.”16 These differences are AB and AC
respectively; AC equals 23 AB. They indicate the paths traversed by rays in air
and glass in equal times: in the time light covers the distance AC in air, its
covers AB in glass. With this, Huygens derived an expression for the position
of H on AE in terms of the position of S (OS perpendicular to AE).
Hug9, 38v-40r; partly reproduced in §1and §2 on OC19, 424-425.
Hug9, 40v; §2 on OC19, 416-417.
Hug9, 41r. This page (reproduced in OC19, 419 §2) begins with an analysis, invoking his theory of
spherical aberration, of the point of intersection of two near parallel rays refracted by a sphere.
OC19, 418. “differentia duarum PA, PD debet esse 3/2 differentiae duarum EA, DF.”
In a drawing plus text right above this, the same case is considered, but
now in terms of a wave.17 (Figure 60) In other words, all rays intersecting in P
and all the points H of intersection of refracted rays are considered. A wave
VK propagates from the glass above VM into the air below it. It propagates in
such a way that all incident rays VP, CP and KP – like the rays AP and DP
above – intersect in P. These rays are refracted towards VM, CG and MP.18 The
intersections of the refracted rays – like point H above – form a curve VHN
tangent to all refracted rays. Huygens now wanted to prove VHN = NM
+ 23 MK. That is, the time for light to cover VHN in air, is equal to the time
required to cover NM in air and MK in glass. He explained the meaning of
this statement by considering the moment when point K of the wave has
reached the refracting surface. In the time K moves to M through glass, point
V moves to Q through the air (VQ = 23 KM). At this moment, Huygens said
without explanation, a wave is formed consisting of parts RM and RQ, which
are the involutes of parts VR and NR of curve VHN. Ergo, NR + RV equals
NM + VQ = NM + 23 MK. It is still not clear what Huygens exactly was after.
He was thinking in terms of rays being paths covered by light in a certain
time, but the point of considering the unfolding wave QR-RM after refraction
is unclear.
Figure 61 A wave refracted at the plane surface of a glass
medium. (Letters ABCDE added by editors, Xxxxx by me)
OC19, 421 §2.
KP is perpendicular and not refracted; VP is incident at about 48o, the critical angle, and is refracted to
the parallel VM. It is an odd case, a wave propagating to a single point instead of away from it. It might be
connected to the preceding discussion of ovals, in that Huygens is considering what happens when the
wave has passed the aplanatic surface and crosses a plane surface.
1677-1679 – WAVES OF LIGHT
On the next two pages, the issue at stake does become clear.19 Here he
considered the same wave, but now propagating from air to the glass
between E and D. The incident rays intersect in C (Figure 61). When the
whole wave has passed the refracting surface, the wave XxxxxE is formed in
the glass. The accompanying text reads:
“The common tangent curve of all the particular waves will be the propagation of the
principal wave in glass. Therefore, the straight lines which cut this tangent curve at right
angles will be the refracted rays. These, however, are given otherwise. Therefore these
will cut that curve at right angles. Therefore the curve is the involute of the other curve
which is the common tangent of these rays. It is sufficient to know that the waves are
propagated along these straight lines. But since the lines must cut the waves at right
angles, it can appear surprising how the lines, not tending to one center, can always cut
the waves at right angles. But this is now explained by the involute.”20
This text makes two things clear. In the first place, Huygens finally says what
problem had been involved in the preceding exercises. In both cases
discussed, rays do not intersect in one point after refraction. The question
therefore is how the accompanying wave ought to be imagined. Apparently,
as we shall soon see, caustics (or aberration in general) also raised questions
with Pardies’ theory, as it is not immediately clear whether or how rays are
normal to waves. Huygens had settled the matter by means of involutes. The
refracted rays form a curve AB, like the curve VHN in Figure 60. The wave
XxxxxE is the involute of this curve.
In the second place, Huygens was applying a new conception of wave
propagation. The particular waves he talks about are not drawn, but are
thought to be the various spherical waves spreading in all directions through
the glass around the points of incidence. At the time the wave in air reaches
E, these wavelets have covered the distance to points x. Their tangent is
XxxxxE, the propagation of the principal wave. Huygens leaves out this step
and immediately goes on to draw the ‘straight lines’ along which it is
propagated, the rays that is. He can do so because these are ‘given otherwise’,
namely by the sine law. So, instead of determining the refracted rays by
constructing the propagated wave, he determines the propagated wave by
constructing the refracted rays. As I see it, instead of applying his principle to
construct the propagated ray, Huygens was using it to justify the
construction by means of refracted rays.
The insight underlying this justification does not go beyond a small
sketch one page further down (Figure 58 on page 162).21 We cannot be
certain to what extent the insight was already in his mind. I believe it was
beginning to take shape when he was analyzing caustics in figures 59 to 61. I
find the preceding study of Fermat’s principle and of aplanatic surfaces
telling. Huygens was beginning to consider rays in terms of a path covered in
Hug9, 41v and 42r. OC19, 421 §3 and 422 respectively. I only discuss 42r.
OC19, 422; translation from Shapiro, “Kinematic optics”, 236.
Hug9, 43r. On the intermediate page 42v he applies it to the propagation of a wave crossing an aplanatic
surface (OC19, 425-426 §2), but this apparently is much later as Huygens dated it 24 March 1678.
a certain amount of time, that is, as optical paths. He then applied this to
caustics. In this case he considered a number of rays. He measured out equal
times covered along different rays and recognized the relationship between
caustics and involutes, curves he was fully acquainted with. Somewhere along
the line, Huygens realized that in constructing waves this way he was
assuming that rays are always normal to waves. If that is so, the notion of
wavelets spreading in all directions from the points of incidence and
subsequently forming a propagated wave by their enveloping curve must
have come up as a justification.22
Huygens’ account of caustics resolved two ambiguities in Pardies’
derivation of refraction. Pardies had determined the direction into the
refracted wave propagates by constructing the refracted ray (Figure 56 on
page 153). The line sections Ce, ee, etc. on the refracted rays are regarded as
the intervals by which the wave proceeds in a given time. The resulting wave
is not spherical anymore, but this is not accounted for any further. The
meaning of the curve in terms of light waves thus remains vague.
Furthermore, as Shapiro has pointed out: “Pardies has not explained why at
the point of refraction the wave should be refracted in one direction, … He has
simply assumed that refraction occurs. Therefore, he has demonstrated only
that if the wave is refracted, then it must be propagated in the new medium
in a direction such that the rays are always normal to the wave fronts.”23
Huygens now stated that light, in the form of ‘particular waves’, spreads in all
directions from the points of refraction. The propagated wave can be
constructed by drawing the envelope of the wavelets, even if the wave is not
spherical. Implicitly, Huygens stated that the wave is a surface of constant
phase, i.e. the locus of points where wavelets unite. It remains to be seen
whether Huygens was explicitly reconsidering Pardies’ theory of light at this
moment. It is unclear, for example, whether the ‘surprising’ observation that
‘rays not tending to one center, can always cut the waves at right angles’ had
given rise to the study of ovals and caustics.24
Irrespective of the question of whether he was explicitly tackling
problems with Pardies’ theory, Huygens had begun to focus on distances
covered by light in a specific time. And irrespective of the question of
whether, and if so how, his principle of wave propagation arose from the
ensuing analysis of caustics, he realized that both waves and rays are
In other words, I tend to disagree with Shapiro’s view that this ‘most subtle refinement of Huygens’
optics’ cannot have been the starting point for the formulation of Huygens’ principle. (241) I do not
consider the equality of optical paths to have been derived from his theory of light (232), but rather to be
Huygens’ starting point in these studies, which he subsequently, and rather implicitly, justified by a vague
notion of his principle. I suspect that Shapiro has been misled by following the text of Traité de la Lumière,
that is, the analysis of the enveloping wave refracted by a spherical surface, which is indeed the most
subtle refinement and application of Huygens’ theory of light. It must, however, be of a much later date
as this case is not found in the 1667 notes. Shapiro, “Kinematic optics”, 231-236.
Shapiro, “Kinematic optics”, 215-217. Emphasis in the original.
Ziggelaar states, without argument, that caustics posed a crucial objection to Pardies’ theory and that
this induced Huygens to formulate his own theory: Ziggelaar, “How”, 186-187.
1677-1679 – WAVES OF LIGHT
subordinate to the speed of propagation of light. What distinguishes the
notion implicit in this analysis from Pardies is the idea that light spreads in all
directions. The flash of genius would then consist in the insight that Pardies’
premise, rays are normal to waves, still holds as it directly follows from the
fact that the wave is tangent to all wavelets. What Huygens had done was to reduce
waves to their speed of propagation.
Another question is what stimulated Huygens to resume the riddles of
light waves. It had, after all, been 5 years since he put aside the unsolved
questions raised by strange refraction. May an article in the Philosophical
Transactions of 25 June 1677 (6 July N.S.) have been the occasion for
Huygens’ study? The article, by Ole Rømer, contained a proof of the finitude
of the speed of light.25 It was based on observed irregularities in the motions
of Jupiter's satellites. The idea had already been proposed by Cassini in 1675,
but he had withdrawn it shortly afterwards.26 Cassini would remain one of
the main opponents of Rømer’s assertions.27 Understandably, Huygens was
to welcome this observational confirmation of the main premise of his
understanding of light. In Traité de la Lumière he would repeat “the ingenious
demonstration of Mr. Romer.”28 In his view, the wave theory and the finite
speed of light were necessary linked.29 He wrote Rømer on 16 September
1677 to express his gratitude.30 In the following months they exchanged
various letters on the subject.
Rømer’s article would not have affected Huygens’ conviction that the
speed of light is finite, but it is likely that reading it induced him to look at
the propagation of light anew. The centrality of the speed of propagation in
the notes we have just examined – from Fermat’s principle to the analysis of
caustics – makes this plausible. This would mean that this study took place
between the middle of July and 6 August 1677.31 It would imply that he
moved on to the problem of strange refraction almost at once upon solving
the problem of caustics. In view of the content and appearance of the
manuscript material I find such a short span of time quite plausible.
OC8, 30n1. It was the translation of an article that had been published in French on 7 December 1676:
“A demonstration concerning the Motion of Light, communicated from Paris, in the Journal des Sçavans,
and here made English.” Apparently, Huygens had not seen the issue of Journal des Sçavans.
Sabra, Theories of Light, 205. Not all historians agree on this point; compare: Cohen, “First
determination”, 345-346; Van Helden, “Roemer’s speed of light”, 140n1 and Wroblewski, “De Mora
Luminis”, 629.
Other opponents to the view that the speed of light is finite were Hooke and Fontenelle. Wroblewski
argues that the controversy came to an end in 1729.
Traité, 7. “l’ingenieuse demonstration de Mr Romer.”
In the ‘Projet’ he had made one reservation: “Light extends circularly and not in an instant, at least in
the bodies down here, because for the light of stars it is not without difficulty to say that it would not be
instantaneous.” Such a reservation was now needless, for Rømer’s argument was based on astronomical
observations. OC19, 742. “lumiere s’estend circulairement et non dans l’instant, au moins dans les corps
icy bas, car pour la lumiere des astres il n’est pas sans difficulté de dire qu’elle ne seroit pas instantanee.”
OC8, 30-31.
The editors of Oeuvres Complètes think that the study of caustics may have taken place as early as 1676.
The solution of the ‘difficulté’ of Iceland Crystal
In August 1677, what we see as Huygens’ principle was nothing but an
implicit application in the analysis of caustics and a tiny sketch reflecting this
idea. Huygens had not yet elaborated his principle nor explained how it
ought to be applied. This did not refrain him from passing on to the puzzle
that still remained: what happens to waves when they enter Iceland crystal?
In his analysis of caustics, Huygens had confirmed Pardies’ premise that
waves are normal to rays. Yet, it had become secondary to his understanding
of wave propagation. But strange refraction still contradicted it.
As we cannot tell with certainty to what extent Huygens was tackling
problems in Pardies’ theory with his study of caustics, it is not clear in what
way the problem of strange refraction was on his mind. In his notebook the
study of caustics is preceded by a single leaf filled with sketches relating to
strange refraction.32 All relevant matters are reviewed, Bartholinus’ law and
the supposed pores of the crystal, Pardies’ explanation and the propagation
of a ray through successive layers of crystal. No progress, in comparison with
the 1672 investigation, is apparent. Dating the page is hazardous. On the
basis of its place in the notebook it can be prior to December 1674, but it
might as well have been a vacant page Huygens scribbled on later during his
stay in The Hague.
The next allusion to strange refraction follows almost immediately upon
the analysis of caustics and the sketch of his principle. After two pages with
calculations, a diagram of a telescope and another sketch (apparently) of
wavelets, a page follows with intriguing content.33 From left to right we see
three sketches: a horizontal ellipse with two rays parallel to its axis refracted
to its focus; an ellipse plus axis, yet drawn obliquely; rays refracted with some
wavelets indicated. The last sketch is remarkable, as the refracted ray seems
to be drawn obliquely to its accompanying wave. The problem of strange
refraction had returned, and the following pages make it clear that Huygens
had found the solution: an oblique ellipse. On the mirror page the ellipse
returns, now with some small wavelets and – this was the solution – a
horizontal tangent.34 The line connecting the point of tangency and the
center of the ellipse makes an angle with the tangent. In other words: ray and
wave intersect obliquely. The speed of propagation of light in Iceland crystal
is not equal in all directions, light spreads spheroidally instead of spherically.
Consequently, the wave is not normal to its direction of propagation.
We are able to understand these sketches in this way, as we have the
hindsight knowledge of Huygens’ elaborating in Traité de la Lumière. In his
notebook he did not explain what precisely the oblique ellipse was and how
it explained strange refraction. On the next three pages, he only offered
Hug9, 7r. None of it is reproduced in the Oeuvres Complètes.
Hug9, 44v. Not reproduced in the Oeuvres Complètes.
Hug9, 45r. Not reproduced in the Oeuvres Complètes.
1677-1679 – WAVES OF LIGHT
some more sketches and numbers calculations. Then a glorious EUPHKA
“EUPHKA. 6 August 1677. The cause of strange refraction in Iceland crystal.”35
The page contains a large drawing of this ‘cause’, surrounded by explanatory
texts and additional drawings and calculations. In the central figure the
oblique ellipse returns, now abundantly rigged up with geometry. The editors
of the Oeuvres Complètes have proposed a plausible order in which the texts
around it have been written. This means that Huygens started with the figure
reproduced in Figure 62, then wrote the ‘Eureka’ at the top-right corner and
started the explanation of the drawing, continuing in the top-left corner with
a smaller sketch and further notes.
Figure 62 “Causam mirae refractionis in Crystallo Islandica”.
The explanation begins with a description of the figure. The crystal has a
principal section for each dimension and AS, the axis of the obtuse solid
angle of the crystal, is the intersection of these (from the center diagonally
down to the right). The plane of the paper is one of the principal sections.
The upper face of the crystal is KA, the lower face DƦ, and AC (from the
center down second to the left) is parallel to the edge of the crystal. AC is
also the unrefracted oblique ray (i.e. parallel to the edge of the crystal). Also
drawn is AB, the refracted perpendicular (from the center down first to the
left). BPSHN is an ellipse, with a circle drawn inside it with center A and
radius AS. The ellipse and circle therefore touch in S. The axis of the ellipse
AP (from the center diagonally down to the left) is normal to AS.
Huygens showed that DƦ is tangent to the ellipse in point B, its
intersection with the refracted perpendicular AB. The smaller ellipse ưƧƪ is
constructed by drawing the quadrangle µƪLA (ƪµ parallel to the tangent of
Hug9, 47r. OC19, 427, it is preceded by a facsimile of this manuscript page. “EUPHKA. 6 Aug 1677.
Causam miræ refractionis in Crystallo Islandica.”
the large ellipse in C, µL parallel to AC and normal to LA). Huygens then
explained the meaning of all this:
“In the time light forms a sphere with radius µL in the air, it forms a spheroid ưƧƪ,
congruent with PCH, within the crystal.”36
The ellipse signifies a wave of light and presumably this explains strange
refraction. How it did Huygens did not explain in detail, but – again with
hindsight – we can tell. The horizontal line BD is tangent to the wave at the
point where the refracted perpendicular AB cuts it. In other words, the wave
is oblique to its direction of propagation.
Figure 63 Strange refraction of an arbitrary ray.
Huygens went on to explicate how the ellipse can be used to determine the
strange refraction of an arbitrary ray. In a sketch at the top-left of the page
he discusses the incident ray MA (Figure 63). Again, Cµ is tangent to the
ellipse, µL parallel to CA (the unrefracted oblique ray) and perpendicular to
LA. Now draw AV perpendicular to MA, VX perpendicular to AV (where
VX = µL), and XF tangent to the ellipse. Then AF is the refracted ray.
Huygens only proved that XF is the tangent. In terms of waves, the following
happens. A plane wave AV propagates along MA reaching the surface of the
crystal obliquely in A. In the time light covers the distance VX in the air, a
spheroidal wave HFBC is formed inside the crystal. The tangent XF is the
refracted wave propagating along AF, which therefore is the refracted ray.
Again, the meaning of the construction in terms of waves remained largely
implicit in Huygens’ proof. He had defined a construction to determine the
refraction of an arbitrary ray. In other words, he had formulated a new law
of strange refraction, the kind of thing he had been looking for in 1672.
The new law differed considerably from the proposal of 1672, however.
The latter depended upon the refracted perpendicular – as contrasted to
Bartholinus’ law that depended upon the unrefracted oblique ray. The
refracted perpendicular offered a straightforward extension of Descartes’
derivation of the sine law in terms of components added to rays. The new
law depended upon both the refracted perpendicular and the unrefracted
oblique ray: AB is the basic parameter of the ellipse and AC is used to draw
VX (via µL), the propagation of light in air proportional to the propagation
OC19, 427. “Quo tempore lux in aere facit sphaeram cujus radius µL, eadem intra crystallum facit
spheroides ưƪƧ simile PCH.”
1677-1679 – WAVES OF LIGHT
of the ellipse in the crystal. Yet, the most important difference between both
laws is their nature. Whereas the proposal of 1672 was fully phrased in terms
of rays and their components, the new law utilizes waves, unobservable and
hypothetical entities expressing the mechanistic nature of light. The refracted
ray is constructed by means of the tangent to the ellipse, by means of waves
and their properties. The unrefracted oblique ray and the refracted
perpendicular have become secondary to the speeds with which light
propagates through the crystal. The core of the new law was the strange
mode of wave propagation in Iceland crystal. With this Huygens had fully
departed from both his own line of thinking of 1672 and from Bartholinus’.
He had returned to his ideas regarding the nature of light to understand the
peculiar phenomena displayed by Iceland crystal.
The EUPHKA of 6 August 1677 signaled the solution to what Huygens
saw as the problem of strange refraction. As with his proposal of 1672 – and
also with Bartholinus’ law – no empirical confirmation is given. Huygens
had, by the way, improved his empirical data. He introduced an accurate
method of determining crystallographic angles that required only one
measurement.37 Yet, he had not improved or added optical data, nor did he
explicitly verify his new law empirically. Huygens seems to have been fully
convinced that the ‘cause’ he had found was valid. What was the ‘cause of
strange refraction’ Huygens had found? The EUPHKA did not hail the
discovery of spheroidal waves. Instead, it hailed the invention of the way a
strangely refracted ray could be constructed by means of the ellipse.
Although Huygens said what the ellipse was – which in retrospect was
sufficient – he was relatively silent on the question how the construction
should be interpreted in terms of waves. Drawing on a distinction made by
Shapiro, we can say that Huygens was more concerned with the question how
an spheroidal wave could explain strange refraction than with the question
whether it could.38 He did not explain the idea that light produces a spheroid
wave in the crystal. Only implicitly did he assume that the speed of
propagation or the action of the crystal differs in each direction. As with his
principle of wave propagation, the physical concepts underlying the
mathematical construction were at the back of his head, but he did not take
the trouble to elaborate them.
The proof of the pudding was in the eating: before spelling out the idea
of his principle of wave propagation, Huygens first saw to it that it could be
successfully applied. He found out that, with his new idea, he could
understand both caustics and strange refraction in terms of waves. He
focused on the new mathematical structure of wave propagation, and even
this he did not elaborate in detail. In his notes, he did not explain if and how
the construction conformed to his ideas on the propagation of light waves.
He only brooded a little on the composition of the crystal. Scattered around
Discussed in Buchwald, “Experimental investigations”, 313-314..
Shapiro, “Kinematic optics”, 238-239.
the central figure are sketches of piles of round and elliptical balls.
Apparently, he was figuring out how these could make up a rhomboid and
maybe also how elliptical particles could pass on a movement asymmetrically.
This was a question that still stood open regarding his theory of light in
general. He had thought out a principle describing its wavelike propagation,
but he had not answered the question what light is and why these waves
propagate this way. For answers to all these questions we have to turn to the
theory as he presented it at the Académie in 1679.
Announced directly after his return to Paris in the summer of 1678, the
reading of “the treatise of Mr. Huygens on dioptrics” began on 13 May
1679.39 Huygens expounded his ideas on the nature of light and its wavelike
propagation, concluding with an explanation of the rectilinearity of light rays
on this basis. This account is found in chapter one of Traité de la Lumière.
Chapters two to six deal successively with reflection, refraction, atmospheric
refraction, strange refraction, and finally aplanatic surfaces and caustics.
Hereafter, I follow the text of Traité de la Lumière except where otherwise
Huygens began with an exposition of the mechanistic nature of light. The
whole argument is aimed at establishing the one defining characteristic of
waves: their finite and constant speed of propagation. Besides reproducing
Rømer’s proof (“But what I used only as a hypothesis has recently received
every appearance of a definite truth, by the ingenious demonstration of Mr.
Rømer …”40) he explained how an action is propagated, with finite speed,
through imperceptibly small particles. Light, according to Huygens,
originates from the agitation of particles in luminous objects, colliding with
the particles of the surrounding, all-pervading ether. These collisions are
communicated in all directions through the ether, without particles being
displaced. The waves thus produced constitute light. On the basis of the
mechanical properties of hard, elastic balls, Huygens argued that this impact
spreads in all directions with a finite and uniform velocity. The velocity
depends only on the degree of elasticity of the particles and is independent
of the strength of the impact. Moreover, any particle can communicate
different impulses simultaneously.
At the bottom, Huygens’ account of the mechanisms explaining the
propagation of light waves had only one purpose: to show that the ether
consists of elastic particles. If this be the case, the basic premise of his theory
is valid: the collisions propagate with a finite and uniform velocity that
OC19, 441. (The dates when the reading was continued are given on 441-443) “… le traitté de Mr.
Hugens de la Dioptrique.”
Traité de la Lumière, 7. “Mais ce que je n’employois que comme une hypothese, a recue depuis peu grande
apparence d’une verité constante, par l’ingenieuse demonstration de Mr. Romer …”
1677-1679 – WAVES OF LIGHT
depends only on the medium.41 In other words, Huygens mechanistically
justified his reduction of light propagation to velocity. Consequently, light
consists of waves:
“If light thus takes time for its passage (which we now will examine) it will follow that
this movement impressed on matter is successive; and consequently it spreads, …, by
spherical surfaces and waves: …”42
After Huygens had shown how light can be
thought to consist of spherical waves propagating
with a considerable yet finite velocity, he moved on
to consider their propagation in more detail.43 This
is where he finally elaborated his principle of wave
propagation. First of all, each point of a luminous
source produces spherical waves (Figure 64). The
circles represent the propagation of a single wave,
so he added, and should not suggest any regular
succession of particular waves. Although Huygens’
theory is in fact a pulse theory, for sake of
convenience I will speak of waves. While a wave
moves away from its origin, its speed is maintained
although it gradually loses its strength. In the long Figure 64 Waves around a
run, the waves will become imperceptible to our source of light
eyes. Still, light produced by such small actions can be perceptible over long
distances, because innumerable waves
“… unite in such a way that to the senses they make up only one single wave, that
consequently must have enough force to make itself felt.”44
In a ‘Particular remark on the extension of light’, Huygens went on to
describe this uniting more precisely. It was an elaboration of the premise he
had formulated earlier in his discussion of caustics: “The common tangent
curve of all the particular waves will be the propagation of the principal wave
…”. Although he introduced this principle of wave propagation as a
‘remark’, it was the core of Huygens’ theory. Waves of light were not his
idea; his contribution consisted of this principle:
“This is what was not known to those who previously began to consider the waves of
light, among whom are Mr. Hooke in his Micrographia, and father Pardies, who in a
treatise of which he has shown me a part, …, undertook to prove by these waves the
The notes on caustics already reveal that the speed of propagation had become central for Huygens.
Implicitly, he constructed the refracted wave by considering rays as isochronous paths. He explicitly
considered the caustic as a path travelled in a specific time, when he showed that its length is equal to a
rectilinear distance traveled by light.
Traité, 4. “Que si avec cela la lumiere employe du temps à son passage; ce que nous allons examiner
maintenant; il s’ensuivra que ce mouvement imprimé à la matiere est successif, & que par consequent il
s’etend, … , par des surfaces & des ondes spheriques: …”
Traité, 15-17.
Traité, 17. “… s’unissent en sorte que sensiblement elles ne composent qu’une onde seule, qui par
consequent doit avoir assez de force pour se faire sentir.”
effects of reflection and refraction. But the principal foundation, that consists of the
remark I have made, was lacking in his demonstrations, …”45
This principal foundation – Huygens’ principle – explained how a
propagated wave can be constructed mathematically:
consideration of these waves,
that each particle of matter in
which a wave spreads, ought not
to communicate its motion only
to the next particle which is in
the straight line drawn from the
luminous point, but that it also
imparts some of it necessarily to
all the others which touch it and
which oppose themselves to its
movement. So it arises that
around each particle there is
made a wave of which that
particle is the center. Thus
(Figure 65) if DCF is a wave
emanating from the luminous
Figure 65 Huygens’ principle.
point A, which is its center, the
particle B, one of those comprised within the sphere DVF, will have made its particular
wave KCL, which will touch the wave DCF at C at the same moment that the principal
wave emanating from the point A has arrived at DCF; and it is clear that it will be only
the region C of the wave KCL which will touch the wave DCF, to wit, that which is in the
straight line drawn from AB. Similarly the other particles of the sphere DCF, such as bb,
dd, etc., will each make its own wave. But each of these waves can be infinitely feeble
only as compared with the wave DCF, to the composition of which all others contribute
by the part of their surface which is most distant from the center A.
One sees, in addition, that the wave DCF is determined by the distance attained in a
certain space of time by the movement which started from the point A; there being no
movement beyond this wave, though there will be in the space which it encloses,
namely in parts of the particular waves, those parts which do not touch the sphere DCF.
And all this ought not to seem fraught with too much minuteness or subtlety, since we
shall see in the sequel that all the properties of Light, and everything pertaining to its
reflection and its refraction, can be explained principally by this means.”46
Huygens had drawn the ultimate consequence of the notion that in a fluid
medium an action must spread in all directions: each particle in the medium
is the source of a new wave. These wavelets are too feeble to be perceptible,
but when they unite at certain loci they form a principal wave. The action of
this principal wave constitutes visible light. Why this laborious exposition of
waves producing wavelets forming a new wave? Because the laws of optics
can be explained properly only with this principle. Huygens explicitly warned
Traité, 18. “C’est ce qui n’a point esté connu à ceux qui cy-devant ont commencé à considerer les ondes
de lumiere, parmy lesquels sont Mr. Hook dans sa Micrographie, & le P. Pardies. qui dans un traitté dont il
me fit voir une partie, …, avoit entrepris de prouver par ces ondes les effets de la reflexion & de la
refraction. Mais le principal fondement, qui consiste dans la remarque que je viens de faire, manquoit à ses
demonstrations, …”
Traité, 17-18; translation: Shapiro, “Kinematic Optics”, 222-223.
1677-1679 – WAVES OF LIGHT
his readers not to judge this principle foundation at face value, but to wait
and see how the behavior of rays was derived from it.
The subsequent explanation of the laws of optics is indeed the key to
Huygens’ wave theory. The ‘remark’ was not a mere explication of his idea
how feeble actions may produce visible light over tremendous distances. It
was the mathematical representation of his understanding of wave
propagation. Waves are represented by circles and defined as the “distance
attained in a certain space of time by the movement which started from
point A”. These express his premise that waves propagate with a uniform
and finite velocity. Huygens’ principle explains how to construct a principal
wave that has traveled for a certain time: construct the secondary waves and
draw their tangent. It did so, as we shall see, for all situations where the
speed of propagation changes when the medium changes. The only
assumption to be made is the distance covered by waves in a certain time.
The first thing to be explained was the rectilinearity of light rays. Light
propagates rectilinearly when the medium is homogeneous and the speed of
propagation does not change. In his explanation, Huygens stated the mutual
relationship between the original wavelets and the principal wave more
explicitly. Huygens argued that a wave spreads in such a way (in Figure 65,
from BG on to CE) that it is always between the same straight lines (ABC and
AGE) drawn from the luminous points.
“… For although the particular waves produced by the particles comprised within the
space CAE spread also outside this space, they yet do not concur at the same time
instant to compose a wave which terminates the movement, as they do precisely at the
circumference CE, which is their common tangent.”47
The wavelets outside the region CAE are “… too feeble to produce light
there.” Because this applies to any portion of the principal wave, the opening
BG can be made arbitrary small. “Thus then we may take the rays of light as
if they were straight lines.”48
Refraction now is
explained by the change
of the speed of waves
propagating from one
medium to another.
During the time a wave
covers LL in the medium
above the refracting
surface AB, it will cover
a smaller distance OO in
the medium below
(Figure 66). The points
the Figure 66 Huygens’ explanation of refraction.
Traité, 19; translation: Shapiro, “Kinematic optics”, 223.
Traité, 19-20; translation: Shapiro, “Kinematic optics”, 223. For an extensive discussion of the validity of
his argument, see Shapiro, “Kinematic optics”, 225-227; De Lang, “Originator”.
surface in points AKKKB successively, producing wavelets spreading in all
directions through the refracting medium around these points. When the
whole wave has reached the surface – when C arrives in B – around A a
wavelet will have propagated over the distance AN. The common tangent NB
of all wavelets around the points of incidence is the propagated principal
The sine law of refraction easily follows. CB represents the speed of light
in the upper medium, but also the sine of the angle BAC, equal to the angle
of incidence DAE. Likewise, AN is the speed in the lower medium and the
sine of angle ABN, equal to the angle of refraction FAN. The assumption that
CB and AN are in constant proportion directly yields the sine law. In the same
way, Huygens could derive the law of reflection by considering only the
propagation of waves. Assuming that the motion of light rebounds at a
reflecting surface, the tangent of wavelets spreading around the points of
reflection is constructed and the equality of the angles of incidence and
reflection readily follows.
Huygens’ theory was simpler and contained less ambiguities than Pardies’.
He had reduced waves to a single property of light, its speed of propagation.
Waves are the effect of an action spreading with a certain speed. In his
derivation of the sine law, Huygens did not have to presume that a wave
refracts. He only had to consider the consequence of an alteration in the
speed of propagation. The curve (or line) resulting from his construction has
an unambiguous meaning, established by his principle of wave propagation.
He preserved the premise that rays are normal to waves, at least in the case
of spherical waves. As a ray is the path traveled by a point of a wave in a
specific time, it is a direct consequence of the fact that the principal wave is
the tangent of secondary waves.
Explaining strange refraction
Precisely by this reduction of waves to speed of propagation, the puzzle of
strange refraction had been solved. In Iceland crystal light propagates with
differing speeds in differing directions. In his notes, Huygens had not
explained why spheroidal waves account for the refracted perpendicular, nor
why light propagates spheroidally in Iceland crystal in the first place. In the
fifth chapter of Traité de la Lumière, Huygens elaborated his discovery of 6
August 1677. He began with a description of the crystal and its peculiar
properties. It displays double refraction, so supposedly light propagates
through the crystal in two different ways. The first one was regular and
produced ordinary refraction in agreement with the sine law. The other one
was irregular, as a perpendicular ray was refracted. To account for this
strange phenomenon, Huygens “wanted to try what elliptical, or better
speaking spheroidal, waves would do”.49 In other words: try and see what
would happen when light propagates in this direction faster than that.
Traité, 58. “Quant à l’autre émanation qui devoit produire la refraction irreguliere, je voulus essaier ce
que feroient des ondes Elliptiques, ou pour mieux dire spheroïdes; …”
1677-1679 – WAVES OF LIGHT
He began with a qualitative
account of spheroidal waves
produced by a perpendicularly
incident wave (Figure 67).50 RC is
part of a plane wave incident
perpendicularly on the surface AB
of the crystal, so that all points
RHhhC arrive in AKkkB at the
same time. Suppose spheroidal
wavelets SVT spread around these
points, as the speed of
propagation in the direction AV is
larger than in the direction AZ.
According to Huygens’ principle Figure 67 Refraction of the perpendicular.
the common tangent of these spheroidal waves is the refracted wave.
“And it is thus that I have comprehended, what had seemed to me very difficult, how a
ray perpendicular to a surface could suffer refraction on entering the transparent body;
seeing that the wave RC, having come to the aperture AB, continued forward thence,
extending between the parallels AN, BQ yet itself remaining always parallel to AB, such
that here the light does not extend along lines perpendicular to its waves, as in ordinary
refraction, but these lines cut the waves obliquely.”51
Thus the application of Huygens’ principle applied to spheroidal waves
showed that these could explain the refracted perpendicular. Huygens had
solved the original problem of strange refraction, as the refracted
perpendicular implied that waves would not be at right angles to the line of
their extension. Strange refraction was strange precisely because of this: the
propagation of light in Iceland crystal is extraordinary and produces waves
that are not normal to rays.
At this point in the Traité de la Lumière the problem was solved, but only
in principle. For Huygens the most important question still had to be
answered: how spheroidal waves could explain strange refraction. The answer,
the exact properties of the spheroid, was that of 6 August 1677.
In Traité de la Lumière, he began with observing that in the principal
sections of each face of the crystal (the dotted lines in Figure 68) strange
refraction behaves the same. Consequently, the spheroid must have the same
section in all three planes. This is so when the axis of the spheroid is also the
axis of the obtuse solid angle C of the crystal. Choosing plane GCF, Huygens
drew the ellipse PSG, the cross-section of the spheroid around center C
(Figure 69). CS is the axis of the obtuse solid angle C as well as the axis of
Traité, 60-62.
Traité, 61-62. “Et c’est ainsi que j’ay compris, ce qui m’avoit paru fort difficile, comment un rayon
perpendiculaire à une surface pouvoit souffrir refraction en entrant dans le corps transparent; voyant que
l’onde RC, estant venue à l’ouverture AB, continuoit de là en avant à s’étendre entre les paralleles AN, BQ
demeurant pourtant elle mesme tousiours parallele à AB, de sorte qu’icy la lumiere ne s’étend pas par des
lignes perpendiculaires à ses ondes, comme dans la refraction ordinaire, mais ces lignes coupent les ondes
Figure 68 Orientation of spheroid in
the crystal.
Figure 69 Shape of the spheroidal wave.
revolution of the spheroid, with angle GCS = 45º20´. CM is the refracted
perpendicular and thus – on account of the preceding – the ellipse must be
tangent in M to the lower surface FH of the crystal, with angle MCL = 6º40´.
Choosing CM = 100,000 yields CP = 105,032, CS = 93410 and CG = 98779.52
Except for differences in the specific values (due to later measurements) this
what his new method of August 1677 gave.53
In Traité de la Lumière Huygens went on to explain how the strange
refraction of an arbitrary ray can be found. He applied his principle of wave
propagation to spheroidal waves, which had been implicit in the notes of
August 1677 (Figure 70):
“Coming now to a search for the refractions that the obliquely incident rays must make,
following the hypothesis of these spheroidal waves, I saw that these refractions
depended upon the proportion of the speed that is between the movement of the light
outside the crystal in the ether, and the movement inside the same. For supposing for
example that this proportion was such that, while the light in the crystal makes the
spheroid GSP, as I have just said, outside it makes a sphere of which the semi-diameter
is equal to the line N, which will be determined further down; then this is the manner to
find the refraction of the incident rays.”54
Except for the line N, the construction is the same as on 6 August 1677: RC
is incident on surface kCK of the crystal. To find the (strangely) refracted ray
CI, draw CO normal to RC and OK normal to CO, with KO = N. Drawing the
tangent through K to the ellipse GSP yields point I, and CI is the refracted ray.
The proof, using spheroidal wavelets, proceeds along the same lines as in the
case of ordinary refraction. By the time O arrives in K, points H have arrived
at the points x on the surface and spheroidal wavelets have spread in the
Traité, 62-63.
CM = 100,000 yields CP = 105,022, CS = 93095 and CG = 98473. In 1677 angle FCL = 70º57´. As a result
of the measurement of August 1679 (see section 5.3.1) FCL = 73º20´ in Traité de la Lumière.
Traité, 63-64. “Or passant à la recherche des refractions que les rayons incidens obliques devoient faire,
suivant l’hypothese de ces ondes spheroides, je vis que ces refractions dependoient de la proportion de la
vitesse qui est entre le mouvement de la lumiere hors du cristal dans l’éther, & le mouvement au dedans
du mesme. Car supposant par exemple que cette proportion fût telle que, pendant que la lumiere dans le
cristal fait le spheroide GSP, tel que je viens de dire, elle fasse au dehors une sphere dont le demidiametre
soit égal à la ligne N, laquelle sera determinée cy apres; voicy la maniere de trouver la refraction des rayons
1677-1679 – WAVES OF LIGHT
Figure 70 Construction of the refraction of an arbitrary ray in Traité de la Lumière.
crystal. IK is the common tangent of these wavelets and therefore IK is the
propagated wave and IC the refracted ray.55
In this construction the line N replaces the unrefracted oblique ray in the
construction of August 1677. It represents the proportion of the speeds of
propagation in the air and in the crystal. As Buchwald explains, it provided
an absolute parameter for the construction.56 Originally, this proportion
followed from the unrefracted oblique ray. In 1679 he found out, as we will
see in section 5.3.1, that this ray was not parallel to the edge of the crystal.
Although this did not change his construction, he could not use it as a
parameter anymore. Instead he used the line N, introduced as an
observational value:
“To find the length of the line N, proportional to CP, CS, CG, it is through the
observations of the irregular refraction that occurs in this section of the crystal, that it
must be determined; and I find in this way that the ratio of N to GC is a little less than 8
to 5.”57
With the ellipse construction thus quantitatively determined, Huygens
derived several properties of the strange refraction. He showed which ray
passed without refraction; considered rays outside the principal section, and
discussed the apparent position of images. Finally, Huygens could conclude:
Traité, 65.
Buchwald, Rise, 315-316. However, Buchwald only discusses the final text of Traité de la Lumière and
therefor does not take into account the historical background of this choice of parameters. On page 317
he raises the possibility that Huygens did not determine the value of N directly - as the next quote
suggests - but deduced it from the angle of the unrefracted oblique ray and subsequently reversed the
calculation to confirm is theory. In the manuscripts he could have caught Huygens more of less in
flagrante delicto.
Traité, 66. “Pour trouver la longueur de la ligne N, à proportion des CP, CS, CG, c’est par les observations
de la refraction irreguliere qui se fait dans cette section du cristal, qu’elle se doit determiner; & je trouve
par là que la raison de N à GC est tant soit peu moindre que de 8 à 5.”
“In this way, I have searched in every detail the properties of the irregular refraction of
this crystal, to see whether each phenomenon that is derived from our hypothesis
agrees with what is actually observed. This being so, it is not a light proof of the truth
of our suppositions and principles.”58
In glaring contrast with the acuity with which he thus derived the behavior
of strangely refracted rays from his hypothesis stands the vagueness with
which Huygens dealt with the question why light produces spheroidal waves
in Iceland crystal. When he proposed the idea, he said that he only needed to
assume that the speed of light differed for various directions of the crystal:
“As for the other emanation that must produce the irregular refraction, I wanted to try
what elliptical or, speaking better, spheroidal waves would do; and these I supposed
would spread indifferently both in the ethereal matter diffused throughout the crystal
and in the particles of which it is composed; …It seemed to me that the disposition, or
regular arrangement, of these particles could contribute to forming the spheroidal
waves (nothing more being required for this than that the successive movement of light
should spread a little more quickly in one direction than in the other) and I hardly
doubted that there is in this crystal such an arrangement of equal and similar particles,
due to its shape and its angles of definite and invariable measure.”59
The crucial assumption that light propagates somewhat faster in one
direction of the crystal is being introduced here rather incidentally, in
parentheses. Huygens only vaguely suggested how this assumption in its turn
could be explained mechanistically: by the disposition of the particles of the
crystal. That was about all Huygens said about it, and it was quite meagre
compared to the work he put in accounting for the finite and uniform speed
of light.
At the end of the chapter, Huygens discussed the composition of the
crystal in some detail. He pictured a pile of balls and explained how it would
produce a body with a specific shape. According to this line of reasoning
Iceland crystal would be composed of spheroidally shaped particles.
“… these little spheroids might very well contribute to forming the spheroids of the
light waves assumed above; both being situated the same, and with their axis parallel.”60
Except for this suggestion, Huygens said nothing about the mechanistic
explanation of his hypothesis. I figure it is quite difficult indeed to explain
Traité, 85. “J’ay recherché ainsi par le menu les proprietez de la refraction irreguliere de ce Cristal, pour
voir si chaque phenomene, que se deduit de nostre hypothese, conviendroit avec ce qui s’observe en effet.
Ce qui estant ainsi, ce n’est pas une legere preuve de la verité de nos suppositions & principes.”
Traité, 58. “Quant à l’autre émanation qui devoit produire la refraction irreguliere, je voulus essaier ce
que feroient des ondes Elliptiques, ou pour mieux dire spheroïdes; lesquelles je supposay qu’elles
s’estendoient indifferement, tant dans la matiere étherée repandue dans le cristal, que dans les particules
dont il est composé; … Il me sembloit que la disposition, ou arrangement regulier de ces particles,
pouvoit contribuer à former les ondes spheroïdes, (n’estant requis pour cela si non que le mouvement
successif de la lumiere s’étendit un peu plus viste en un sens qu’en l’autre,) & je ne doutay presque point
qu’il n’y eust dans ce cristal un tel arrangement de particules égales & semblables, à cause de sa figure &
ses angles d’une mesure certaine & invariable.”
Traité, 96. “J’ajouteray seulement que ces petits spheroides pourroient bien contribuer à former les
spheroides des ondes de lumiere, cy dessus supposez; les uns & les autres estant situez de mesme, & avec
leur axes paralleles.”
1677-1679 – WAVES OF LIGHT
differing speeds of propagation in terms of successive impact. At any rate, in
Traité de la Lumière Huygens simply avoided the problem.
With the solution of the problem of strange refraction the last obstacle for
elaborating the ‘Projet’ was out of the way. Huygens had begun reading his
treatise on dioptrics in May 1679. At the end of June 1679 the reading began
of “the first part of his treatise that contains the physical causes of refraction
and the phenomena of Iceland crystal.”61 Compared to the ‘Projet’ he had
altered the organization of its contents. The chapters on the causes of
refraction and on strange refraction had become a separate part. However, as
the opening lines of Traité de la Lumière reveal, his views on the place and
function of an explanatory theory had not altered:
“The demonstrations that concern optics, as is the case in all sciences where geometry
is applied to matter, are founded upon truths derived from experience; such as that the
rays of light extend in right lines; that the angles of reflection and incidence are equal;
and that in refraction the ray is broken according to the rule of sines, nowadays so wellknown, and no less certain than the preceding ones.
The majority of those who have written concerning the different part of optics
have contented themselves with presupposing these truths. But some more curious
have wanted to investigate their origin and causes, considering them as admirable
effects of nature themselves. Having advanced ingenious things in this, but not to the
extent that the most intelligent would not want explications that satisfy them better, I
want to propose here what I have considered on this subject, to contribute as much as I
can to the clarification of that part of natural science that is not without reason reputed
to be one of the most difficult.”62
Although Huygens now spoke less in disparaging fashion about those
curious minds that want to know the reason of everything, he had not
changed his mind about the necessity of explanations. The laws of optics
were empirical laws, whose causes could be investigated, in a supplementary
way, as effects of nature. However plausible, explanation does not add to
their validity. In wording akin to the ‘Projet’, he continued:
“In this book I will therefore try, by the principles accepted in the philosophy of today,
to give clearer and more probable reasons, firstly of these properties of light directly
extended; secondly of that which is reflected by the encounter with other bodies.
OC19, 440. “… la premiere partie de son traitté qui contient les raisons physiques de la réfraction et des
phenomenes du cristal d’Islande”
Traité, 1-2. “Les demonstrations qui concernent l’Optique, ainsi qu’il arrive dans toutes les sciences où la
Geometrie est appliquée à la matiere, sont fondées sur des veritez tirées de l’experience; telles que sont
que les rayons de lumiere s’etendent en droite lignes; que les angles de reflexion & d’incidence sont egaux;
& que dans les refractions le rayon est rompu suivant la regle des Sinus, desormais si connue, & qui n’est
pas moins certaine que les precedentes.
La pluspart de ceux qui ont écrit touchant les differentes parties de l’Optique se sont contentés de
presupposer ces veritez. Mais quelques uns plus curieux en ont voulu rechercher l’origine, & les causes, les
considerant elles mesmes comme des effets admirables de la Nature. En quoy ayant avancé des chose
ingenieuses, mais non pas telles pourtant que les plus intelligens ne souhaittent des explications qui leur
satisfassent d’avantage; je veux proposer icy ce que j’ay medité sur ce sujet, pour contribuer autant que je
puis à l’éclaircissement de cette partie de la Science naturelle, qui non sans raison en est reputée une des
plus difficiles.”
Further I will explicate the symptoms of rays that, are said to, suffer refraction when
passing through transparent bodies of a different kind. …
Then I will examine the causes of the strange refraction of a certain crystal that is
brought from Iceland.”63
The reconciliation of strange refraction with his wave theory had cleared the
way to propose waves as a plausible cause of refraction. Yet, how did strange
refraction fit in this scheme of experiential truths additionally explained by
the principles of accepted philosophy? Huygens did not mention the
properties of strange refraction among the common ‘truths derived from
experience’. He did not explicate what law strange refraction was subject to.
He would examine its causes – spheroidal waves as we know – but what
were these to explain?
In view of Huygens’ epistemological statement, strange refraction would
be properly accounted for if he could give a ‘law’ of strange refraction that
was empirically valid. Similar to the sine law, such a law should prescribe in
exact fashion how rays are refracted in Iceland crystal. This is what Huygens
had been looking for in 1672. Rejecting Bartholinus’ law, he searched for a
‘law’ of strange refraction that was general. The proposal of 1672 provided a
general construction for the strangely refracted ray, but did not solve the
underlying problem that strange refraction could not be reconciled with the
wave explanation of ordinary refraction. In August 1677 Huygens found
what he had been looking for: a general construction that could be
accounted for in terms of waves. But was this construction a law of optics in the
same sense as the sine law?
In chapter five of Traité de la Lumière, Huygens presented an account of
strange refraction that, at first sight, had been structured like the previous
chapters on rectilinear propagation, reflection and refraction. In the case of
refraction, he first laid down ‘the principal properties of refraction’: the sine
law and the reciprocity of refraction.64 Then he went on to explain these by
means of his wave theory.65 In the case of strange refraction, he also began
with a description of the (mathematical) properties of strange refraction.
Huygens carefully described the observable properties of strange refraction –
the perpendicular is refracted, strange refraction contradicts the sine law, the
unrefracted oblique ray is not parallel to the edge of the crystal. He even
Traité de la Lumière, 2. “J’essaieray donc dans ce livre, par des principes recues dans la Philosophie
d’aujourd’huy, de donner des raisons plus claires & plus vraysemblables, premierement de ces proprietés
de la lumiere directement etenduë; secondement de celle qui se reflechit par la rencontre d’autres corps.
Puis j’expliqueray les symptomes des rayons qui sont dits souffrir refraction en passant par des corps
diaphanes de differente espece: …
Ensuite j’examineray les causes de l’étrange refraction de certain Cristal qu’on apporte d’Islande.”
Note by the way as was explained above on page 112 that Huygens seldom spoke of ‘laws’ of optics
(and few did in the seventeenth century). In Traité de la Lumière he called the sine law the principle
property of refraction, elsewhere he spoke of ‘ratio of sines’. Sometimes he used ‘laws’ to indicate the
various properties of refraction. He neither called his construction for strange refraction a ‘law’. I will use
the modern terminology of ‘law of refraction’ and in analogy call his construction a law. Later readers
conceived of it as such.
Traité, 32-33.
1677-1679 – WAVES OF LIGHT
formulated a ‘remarkable rule’: if two rays from opposing directions are
incident with equal angles, the distance between the refracted rays and the
refracted perpendicular was also equal.66 This was, however, not a general law
prescribing how an arbitrary ray was refracted by the crystal. Besides, it
probably was a residue from the line of thinking guiding his 1672 attack on
strange refraction. Unlike with the common properties of light, in the case of
strange refraction he did not give an experiential law before turning to its
causes. Instead he first proposed the hypothesis of spheroidal waves, and
then showed how all observable properties could be derived from it.
The hypothesis did yield a law of strange refraction, a general procedure
to construct strangely refracted rays. It was, however, a law of a different
kind than the laws of reflection and refraction. The law of Traité de la Lumière
was a law of waves. His proposal of 1672 had been a law that described strange
refraction in terms of rays and their components and therefore in principle
an experiential law. The ellipse construction also described the behavior of
strangely refracted rays, but it was based on an analysis of the propagation of
waves. Moreover in its implementation it employed spheroidal waves, that is,
unobservable, hypothetical entities. This ‘manner of finding the refraction of
incident rays’ at the same time incorporated its probable cause. In the ellipse
construction waves and rays were inseparably tied, being explanans and
explanandum at the same time. In this sense, strange refraction did not fit
Huygens’ scheme of providing explanations for empirically established
optical laws.
Huygens had found his law of strange refraction by exploring the
propagation of light waves. In a way similar to Kepler’s analysis of ordinary
refraction, he had figured out the mathematical properties of strangely
refracted rays by liberally applying the idea of spheroidal waves. It was not a
particularly new thing to speculate upon causes in optics, not even to use this
in order to find new laws. Yet, by presupposing them in his eventual
construction this law was not a traditional law of optics, mathematically
describing the behavior of rays. By deriving the observed properties of
strange refraction from it, he gave ample proof for the empirical soundness
of his assumptions. Still, the ellipse construction was not a ‘truth drawn from
experience’ like the sine law. It was an application of Huygens’ principle,
which expressed his conception of the propagation of waves. Huygens could
derive the properties of reflected and refracted rays from his principle, by
assuming that the speed of propagation of these waves depended upon the
medium traversed. Unlike the laws of reflection and ordinary refraction, in
his ‘law’ of strange refraction he could not separate the properties of rays
from the properties of waves after his derivation.
Huygens did offer an abridged version of the ellipse construction, similar
to the circle diagram for the sine law.67 A refracted ray was constructed by
Traité, 57.
Buchwald, Rise, 316 calls it the law of proportions.
means of a single ellipse, without secondary waves being used. The ellipse
can be seen as a mere geometrical tool without a meaning in terms of waves,
but Huygens called it an abridged repetition of the preceding ‘maniere’. One
can read the abridged version of the ellipse construction as a purely
mathematical construction in which the ellipse lacks physical meaning. Yet,
the order of his presentation contradicts this interpretation. He put forward
the abridged version only after his explanation of strange refraction. He did
not present it as an empirically founded law applying solely to rays. The
observed properties of strange refraction could be derived from the ellipse
construction, but an empirically founded law of strange refraction was not
among them.
The solution of the problem of strange refraction thus produced an
unanticipated result. The goal of his study had been to attain consistency in
the causes of the various forms of refractions. The original problem of
strange refraction had been the refracted perpendicular. This did not just
contradict the sine law, it constituted a problem for his Pardies-like
explanation of refraction. It was a problem of waves. In order for waves to
be a plausible cause of ordinary refraction, the explanation ought not be
contradicted by strange refraction. After his initial attempt to solve it in
terms of rays, the solution had come from a reconsideration of the
microphysics involved. The opening lines of chapter five of Traité de la
Lumière, then, sum up what had constituted the problem of strange refraction
and wherein consisted its solution:
“From Iceland, …, is brought a kind of crystal, or transparent stone, very remarkable
for its shape, and other qualities, but above all for its strange refractions. The causes of
which seemed to me all the more worthy to be investigated curiously, as among
diaphanous bodies only this one, in respect of the rays of light, does not follow the
ordinary rules. I even had some necessity to make this investigation, because the
refractions of this crystal seemed to overthrow our preceding explication of regular
refraction; which, on the contrary, it will be seen they confirm a good deal, upon being
reduced to the same principle.”68
The problem had turned into its opposite. Whereas strange refraction had
first constituted a problem for his explanation of ordinary refraction, it now
confirmed it. The solution did not, however, fit the original scheme
anymore. There was no empirical law of strange refraction that could be
reduced to the principle of wave propagation. The law that described the
behavior of strangely refracted rays was a law of waves. Solving the problem
of strange refraction had yielded a novelty in the mathematical study of
optics. And Huygens’ principle was the key.
Traité, 48-49. “L’on apporte d’Islande, …, une espece de Cristal, ou pierre transparente, fort
remarquable par sa figure, & autre qualitez, mais sur tout par celle de ses estranges refractions. Dont les
causes m’ont semblé d’autant plus dignes d’estre curieusement recherchées, que parmy les corps
diaphanes celuy cy seul, à l’egard des rayons de la lumiere, ne suit pas les regles ordinaires. J’ay mesme eu
quelque necessité de faire cette recherche, parce que les refractions de ce Cristal sembloient renverser
nostre explication precedente de la refraction reguliere; laquelle, au contraire, l’on verra qu’elles
confirment beaucoup, apres reduites au mesme principe.”
1677-1679 – WAVES OF LIGHT
The ellipse construction broadened the traditional meaning of what
constituted a law of optics. As contrasted to the sine law, Huygens’ law of
strange refraction was not independent of the underlying conception of the
nature of light. It was derived from his wave theory, as an application of his
principle of wave propagation, in the same way as he had managed to derive
the laws of rectilinear propagation, of reflection and refraction from his wave
theory. The novelty of Traité de la Lumière went deeper than this peculiar
law of strange refraction. Huygens’ principle of wave propagation
constituted the means to reduce the laws of optics to one and the same
‘principle foundation’.
5.2 Comprehensible explanations
In Traité de la Lumière, Huygens ‘wanted to propose’ what he had considered
on the subject of the origin and causes of the laws of optics. After the
passage quoted at the beginning of section 5.1.3, he continued:
“I acknowledge to be much indebted to the first ones who have commenced to dispel
the strange obscurity in which these matters were shrouded, and to give hope that they
could be explicated by intelligible reasoning. But on the other hand, I am also amazed
how the same have quite often wanted to make pass little evident arguments for very
certain and demonstrative: not finding anyone who has yet explicated in a probable way
these first, notable phenomena of light, namely, why it extends only along right lines,
and how the visual rays, coming from an infinity of diverse places, cross without
impeding each other in any way.”69
Huygens would do a better job. He would give ‘clearer and more probable
reasons’ of the laws of optics than his predecessors had. By means of the
‘principles accepted in the philosophy of today’, he added. One page further
down he explained what these principles were. Light consists of the
movement of a certain matter. Both the origin of light – flames and such –
as well as its effects – heat and burning – indicate motion,
“… at least in the true philosophy, in which one comprehends the cause of all natural
effects by reasons of mechanics. That is what must be done in my view, or give up all
hope ever to comprehend anything in physics.”70
Huygens was going to give better explanations by means of ‘raisons de
Although these words about the proper conduct in physical explanation
are clear, Huygens put his opinion in a fairly general way. What did he
Traité, 1-2. “Je reconnois estre beaucoup redevable à ceux qui ont commencé les premiers à dissiper
l’obscurité estrange ou ces choses estoient enveloppées, & à donner esperance qu’elles se pouvoient
expliquer par des raisons intelligibles. Mais je m’étonne aussi d’un autre costé comment ceux là mesme,
bien souvent ont voulu faire passer des raisonnements peu evidents, comme tres certains & demonstratifs:
ne trouvant pas que personne ait encore expliqué probablement ces premiers, & notables phenomenes de
la lumiere, sçavoir pourquoy elle ne s’étend que suivant des lignes droites, & comment les rayons visuels,
venant d’une infinité de divers endroits, se croisent sans s’empêcher en rien les uns et les autres.”
Traité, 3. “… la vraye Philosophie, dans laquelle on conçoit la cause de tous les effets naturels par des
raisons de mechanique. Ce qu’il faut faire à mon avis, ou bien renoncer à toute esperance de jamais rien
comprendre dans la Physique.”
Since I do not want to prejudge my ensuing discussion of what ‘raisons de mechanique’ are meant by
Huygens to be I here leave the phrase untranslated.
understood by ‘raisons de mechanique’ and what did it entail to comprehend
phenomena by them? In Traité de la Lumière he mentioned some others,
Descartes, Hooke and Pardies, but criticized their theories in only general
terms.72 The best way to get answers to questions like these is to compare
Huygens approach with that of others. After all, he did not operate in an
intellectual vacuum but against the rise of mechanistic philosophy in which
Descartes’ trailblazing work in optics was critically assessed. This section
discusses relevant texts and see how Huygens predecessors and
contemporaries realized and employed the so-called true philosophy in
optics, that is: how they gave shape to the mechanistic explanation of the
properties of light. In this way I intend to expose the contours of Huygens’
implementation of mechanistic philosophy. The discussion of the texts of
others is instrumental to this, they are judge from the point of view of Traité
de la Lumière. Huygens had a very clear goal, I will argue, to acquire in
mechanistic philosophy the same level of comprehensibility as in
mathematical science. He did so by rigorously treating the matter the world
was thought to be made of in the same way as the observable phenomena
arising from it. Others were to be judged by this same standard.
Descartes was, of course, the one who had set the tone in the ‘true
philosophy’. For Huygens too, he was a major point of reference, but it was
not all euphony he heard. In the previous chapter we saw the difficulties with
Descartes’ optics revealed in the ‘Projet’. In Traité de la Lumière, Huygens
expressed his critique in no uncertain terms:
“Because it has always seemed to me, and to many others with me, that even Mr.
Descartes, who had the goal of treating intelligibly of all subjects of physics, and who
certainly succeeded in this much better than any person before him, has said nothing
that was not full of difficulties, or even inconceivable, regarding light and its
Like the remarks in the ‘Projet’, these lines were probably aimed at the
account of refraction in La Dioptrique. In the previous chapter, we have seen
that as regards its mechanistic underpinnings, this account left open many
questions. Descartes had put the nature of light between brackets and the
physical foundations of the proposed mechanisms were only intimated. To
consider the details of the ‘raisons de mechanique’ of light propagation we
have to turn to the natural philosophical works in which Descartes
elaborated them.
That is: with respect to the nature of light. He mentions Bartholinus of course with respect to his
(faulty) ideas about strange refraction. In addition he discusses Rømer’s proof of the speed of light and
Fermat’s principle of least time. Leibniz (aplanatic surfaces), Barrow (caustics), Newton (aplanatic surfaces
and dispersion) he only mentions in the passing.
Traité, 6-7. “Car il m’a tousjours semblé, & à beaucoup d’autres avec moy, que mesme Mr. Des Cartes,
qui a eu pour but de traitter intelligiblement de tous les sujets de Physique, & qui assurément y a beaucoup
mieux reussi que personne devant luy, n’a rien dit qui ne soit plein de difficultez, ou mesme inconcevable,
en ce qui est de la Lumiere & de ses proprietez.”
1677-1679 – WAVES OF LIGHT
Descartes presented his tendency theory of light in Le Monde (written
1630-1632, published posthumously in 1664) and Principia Philosophiae
(written 1641-1644, published 1644). Light was central in his system of
natural philosophy, the nature of light being ultimately connected with the
essence of the cosmos. The full title of Le Monde was Le Monde ou Traité de la
Lumière. Descartes envisaged a system of natural philosophy founded solely
on mathematical principles.74 Quantity was the only thing to be investigated
about material substance and it was subject to the laws of motion. According
to Descartes, the cosmos is completely filled with matter, which is manifest
in three kinds or elements. The third element, composed of the bulkiest
parts, constitutes the visible objects around us, like the earth, the planets and
comets. The first and finest element makes the Sun and the stars; the second
consists of spherical particles and makes the heavens.75 All three elements
have their share in the explanation of light: the bodies of the first element
produce light, the second element makes up the medium that propagates it,
and reflecting and refracting bodies are made of the third element. Descartes
argued that the Sun and Stars, in rotating about their axes, exert a radial
tendency (‘conatus’) upon the Heavens which instantaneously spreads
outward along straight lines. This tendency is light, and Descartes explained
its properties in a discussion of circular motion.76
In Traité de la Lumière Huygens did not waste his breath on the tendency
theory. He only mentions it once, to reject it on the basis of just one
“… [Descartes] has light consist in a continual pressure, that only tends to movement.
As this pressure cannot act at once from two opposing sides, against bodies that have
no inclination whatsoever to approach, it is impossible to comprehend what I have just
said of two persons who mutually see each others’ eyes, nor how two flambeaus can
illuminate each other.”77
Huygens did not bother to criticize the mechanistic underpinnings of the
tendency theory in detail. Although we can figure that Huygens would reject
the ‘raisons de mechanique’ employed, he regarded instantaneous
propagation as the decisive problem of the theory. It returns unremittingly
whenever mention is made of Descartes. To Huygens the speed of light was
necessarily finite.
Descartes, Principles, [76].
Descartes, Principles, [110].
Descartes, Principles, [111-118]. For a detailed discussion see: Shapiro, “Light, pressure”, 243-266.
Traité, 20. “… Descartes, qui fait consister la lumiere dans une pression continuelle, qui ne fait que
tendre au mouvement. Car cette pression ne pouvant agir tout à la fois des deux costez opposez, contre
des corps qui n’ont aucune inclination à s’approcher; il est impossible de comprendre ce que je viens de
dire de deux personnes qui se voyent les yeux mutuellement, ni comment deux flambeaux se puissent
éclairer l’un l’autre.”
“I have therefore not had any difficulty, …, in supposing that the emanation of light
occurs with time, seeing that therewith all its phenomena could be explained, and that
following the contrary opinion all was incomprehensible.”78
An instantaneously propagated action flew in the face of everything his
Galilean understanding of motion entailed. This understanding was
geometrical at the same time. As in his studies of impact and circular motion,
Huygens reduced light to velocity, its speed of propagation. It is a concept
that lends itself to geometrical representation in an obvious way, by means of
a line segment. Instantaneity, on the other hand, is hard to picture.
In addition to the incomprehensibility - in Huygens’ view - of its ‘raisons
de mechanique’, there was another problem with Descartes’ optics. Where
Huygens said the main objective for treating the nature of light is to explain
its properties - the laws of optics - Descartes never thoroughly connected
these two parts. Le Monde and Principia did not give an explanation of the
laws of optics, they only treated the mechanistic nature of light. Instead,
Descartes referred his readers to La Dioptrique.79 There the nature of light
remained below the surface. The only more or less explicit connection
between the nature and the properties of light consisted of his claim that
tendency obeys the same laws as motion proper. Descartes’ derivation of the
sine law was problematic, to say the least. The analogies implied that light
was of a different nature than he himself proclaimed. The claim that moving
balls and tendencies were compatible may have been acceptable to himself,
but he did not – whether in Le Monde or in La Dioptrique – explain why and
how a tendency refracted.
Huygens wanted to explain the laws of optics. Le Monde and Principia were
probably hardly relevant to Huygens. Descartes derived the laws of optics in
La Dioptrique, and this is where Huygens found his difficulties. In my view,
the argument about the unimpeded crossing of rays should be read with the
sticks and wine barrels of La Dioptrique in mind, rather than the various kinds
of elements of Le Monde. Although Huygens may initially have considered
the mathematics of the derivation a useful way to look at strange refraction,
he never accepted Descartes’ physical interpretation of it. He could not see
how diagrams invoking distances could be understood in terms of an
instantaneously propagating action. From 1672 on he remained completely
silent on Descartes’ derivation. Whatever the merits of Descartes’ theory of
light, we can fairly say that Huygens did not accept the way it should have
explained the laws of optics.
Although Huygens unrelentingly rejected Descartes’ theory of light, he
openly acknowledged his indebtedness in a more general sense. Descartes
had been the first to show how physics can be treated in a comprehensible
way. That is: ‘par des raisons de mechanique’. In optics Descartes’ project
Traité, 6. “Je n’ay donc pas fait difficulté, …, de supposer que l’emanation de la lumiere se faisoit avec le
temps, voyant que par là tous ses phenomenes se pouvoient expliquer, & qu’en suivant l’opinion contraire
tout estoit incomprehensible.”
Descartes, Principles, [159-164].
1677-1679 – WAVES OF LIGHT
may not have materialized satisfactory, but Huygens would show how it
ought to be done.
Despite all the critique, Huygens’ theory of light shared one fundamental
element with Descartes’: the idea that light must be an action propagated
without transport of matter. In Descartes’ case this conception was dictated
by his conception of the cosmos as a plenum. In Huygens’ case it was the
only viable alternative. Emission conceptions of light - where light is thought
to consist of corpuscles emitted by its source – he rejected on basic grounds:
it conflicted with the unimpeded crossing of light rays.80 He did not waste
more words on the matter. Wave theories - or more generally: the idea of
successive motion without transport of matter - formed Huygens’ frame of
reference. He knew and credited two predecessors: Hooke and Pardies. The
progenitor of wave theories was Thomas Hobbes, who formulated a wave
theory in direct response to La Dioptrique. It displays a sustained effort to
mathematize the mechanistic nature of light. His ideas were passed on by
Maignan and Barrow, to whom they are often attributed.81 Huygens was
familiar with Barrow’s Lectiones - discussed in the previous chapter - but he
nowhere referred to its physical parts. Although Huygens reacted to none of
these wave theorists, a discussion of Hobbes’ theory is illuminating with
respect to the pitfalls of corpuscular reasoning. He did respond to Hooke’s
rendering of the wave conception, which enlarged the deficiencies in
Hobbes, Hooke and the pitfalls of mechanistic philosophy: rigid waves
Descartes had provided the most elaborated attempt to explain the nature
and properties of light in mechanistic terms.82 Despite all its flaws, Descartes’
optics was of overriding importance for the development of seventeenthcentury thinking on light. Hobbes, a fierce defender of mechanistic thinking,
was the first of many to react to the derivation of the sine law. In the course
of a dispute over La Dioptrique he devised an alternative derivation of the
sine law.83 It was published by Mersenne in 1644 – probably against Hobbes’
intentions – under the title of Tractatus opticus as the seventh book of Universae
geometriae mixtaque mathematicae synopsis. It is rather sketchy and confusing. In
the unpublished Tractatus Opticus II (1640) and in ‘A Minute or First Draught
of the Optiques’ (around 1646) the theory is elaborated in more detail.84
According to Hobbes, a source of light dilates and contracts like a heart,
thus producing an action that is propagated as pulses in the surrounding
Traité, 3.
Shapiro, “Kinematic optics”, 143-145. Hobbes is regarded here as the onset of seventeenth-century
continuum theories. As the paper focusses on the so-called kinematic tradition, Shapiro does not include
Descartes as he did not contribute to the the concepts of waves and rays.
Gaukroger, Descartes, 269.
Prins, “Hobbes on light and refraction”, 132.
Stroud, First draught, 18-20. The dating of Tractus Opticus II at 1640 is derived from Horstmann, “Hobbes
und das Sinusgesetz”, who also argues that the published Tractatus Opticus, commonly referred to as ‘I’, is
in fact of later date.
medium. He called such a pulse a ‘line of light’. A ‘line of light’ traces out a
rectilinear path, and the resulting parallelogram is a ray of light.
In Hobbes’ view the rectilinearity of a ray
followed from the nature of this action (Figure 71):
“… the luminous object acts with its entire force; for if
the sides of the ray would leave obliquely from line of
light AB, as AE and BF, the luminous object would not act
with its entire force, but diminished in the ratio of AG to
Hobbes justified this statement by drawing a
comparison with a cylinder pushed with equal force
at each end.86 In the ‘First draught’, this argument is Figure 71 Hobbes’ rays.
used to explain that a perpendicular incident ray is not refracted.
When both ends of the cylinder are not pushed
with equal force, its motion can be compared to
that of a cone with bases AE and BF tracing out a
curved path AH, BR (Figure 72). If combined with
Hobbes’ assumption that rarer media like air are
less resistant to motion than denser media like glass
and water, the sine law can be derived simply from
this cone model. Barrow was to employ this
Figure 72 Refraction.
reasoning (see above, page 138). Hobbes used this
comparison only to discuss refraction in qualitative terms.
Hobbes’ derivation of the sine law was a formal proof based on two
assumptions: a line of light has a constant width and there is a constant ratio
with which a ray submerges into a refracting medium. Hobbes demonstrated
that CD follows the curved path to GH when D submerges into the denser
medium but C has not yet reached its surface ED (Figure 73). His
demonstration rests upon the constancy of the proportion between LF (the
‘quantity of submergence’) and CQ. G is an arbitrary point on ED, and GH is
found by GH = AB and MH = LF.87
Hobbes did not discuss the mechanics of a line of light passing the
boundary of two media of differing resistance. In the published Tractatus
opticus, Hobbes did not prove the equivalence of this derivation with his the
cone model.88 A probable reason why Hobbes left out this explicit linking of
his derivation and his physical model is that he claimed to be considering
infinitesimally narrow rays of light. The very reason G can be drawn
arbitrarily on ED lies in the fact that “… all of ED, just as AB or GH, must be
Shapiro, “Kinematic optics”, 151.
Shapiro, “Kinematic optics”, 152; Stroud, First draught, 122-125.
For a detailed discussion of the demonstration see Shapiro, “Kinematic optics”, 261-262.
Shapiro, “Kinematic optics”, 260n410; Stroud, First draught, 126a-126m. This equivalence rests, Shapiro
explains, on the equality of LF and MH. In the unpublished Tractatus Opticus II, Hobbes proved this by
considering the curved path CG, DH the line of light traces in refraction. The extension of lines CD and EF
intersect in N and CN : DN = CE : DF, the proportionality of the resistance of the media. Now MH = DP
can be constructed and by the proportionality CG : DH = CN : DN it follows that CQ : MH = CE : FD.
1677-1679 – WAVES OF LIGHT
Figure 73 Hobbes’ derivation of the sine law.
understood as insensible.”89 With this assumed insensibility the equivalence
of Hobbes’ physical rays and mathematical rays was secured. In his physical
model such a transition to infinitesimal lines was problematic because the
width of a line of light was crucial in this account. Despite its flaws, Hobbes’
account of refraction was an advance over Descartes in one respect. It
directly invoked his understanding of the nature of light by phrasing his
derivation in terms of the behavior of lines of light.
Yet, a more fundamental problem emerges in Hobbes’ understanding of
the nature of light. By his ‘lines of light’ he explained light and refraction in
terms of bodies in motion, rather than of particles in motion. He implicitly
regarded a line of light as a coherent entity. Refraction therefore involved the
mechanics of a rigid body, rather than the quasi point-masses of Descartes.
The problem is that no laws governing the motion of such rigid bodies were
known. Although they soon became superseded, Descartes had at least
worked out a set of laws to which the matter constituting light in his view
was subject. Hobbes had not, he simply presumed that a rigid rod behaved
like he claimed. The mechanisms he employed to explain the phenomena
are speculative and qualitative because he did not provide a theoretical or
empirical justification of the supposed motion. He did not come out with
observational evidence in terms of macroscopic rods, nor did he produce
some theory of motion to substantiate it. We see here a fundamental
problem of corpuscular reasoning and quite common in the work of
seventeenth-century savants.
Shapiro, “Kinematic optics”, 260.
In Maignan’s and Barrow’s elaborations of the sketchy theory of Tractatus
opticus, this problem only got worse. Both transformed Hobbes’ theory into
an emission theory by interpreting a line of light explicitly as a moving body.
I have discussed Barrow’s explanations of reflection and refraction in section
4.1.3. He did invoke a general law of motion to justify his claim that a rod
‘gyrates’ in the way he claimed, but the status of this law was unclear, to say
the least. As a matter of fact, around 1680 the physical pendulum was the
only rigid body whose motion was understood mathematically.90 The
difference with Huygens’ wave theory is clear. His waves were not some
coherent body but the effect of motions of ethereal particles.91 In the guise of
a macroscopic model he advanced empirical evidence for his basic claim –
that this action propagates with finite speed and does not displace the
ethereal particles. Moreover, he knew the laws governing impact.92 We may
infer that this is what Huygens meant with ‘raisons de mechanique’. The
mechanisms assumed to be at work on the microscopic level ought to be understood on the
macroscopic level. Imperceptible matter should be subject to the laws of an
established science of motion.
Huygens did not mention Hobbes, Barrow, or Maignan, but it is not
difficult to see why he would not accept their theories. They employed the
method of transduction, which extends the properties of macroscopic bodies
to the unobservable motion of corpuscules, in a deficient way.93 Deficient, to
wit, from the perspective of Traité de la Lumiére. On a qualitative level of
everyday observation they may have thought rigid bodies to behave the way
they claimed, as no laws describing these motion were available,
mathematically the extension was incomplete. By reducing the propagation
of light to the one property of velocity, Huygens steered clear of this pitfall.
He possessed a theory of motion and impact that covered his claims about
the waves propagated in ether, given that ether corpuscles were indeed hard.
In his view he thus had succeeded in applying his beloved rigor of
mathematics to the mechanistic nature of things.
These pitfalls of corpuscular reasoning are accentuated in the theory of
waves Hooke included in his Micrographia (1665). Hooke was one of the
precursors Huygens mentioned in Traité de la Lumière and elsewhere he
severely criticized his theory. From the point of view of the mathematician
Huygens this is to be expected, for exactness was precisely the weakness in
Hooke’s account. Hooke did, however, provide the most detailed and
complete account of colors, experimental and theoretical, of the time, the
very subject Huygens had avoided and would remain silent on. As a
theoretical exposition instead of description of microscopic observations,
Shapiro, “Kinematic optics”, 177.
Newton also understood that a wave cannot be conceived as some kind of coherent entity without
smuggling in unproven assumptions. Shapiro, “Definition”, 195-196.
Although gaps in the transition from the behavior of single particles to waves in the sea of ether may be
pointed out, as Burch has done. Burch, “Huygens’ pulse models”, 56-60.
For an exposition of this concept and references to the literature see Shapiro, Fits, 40-48.
1677-1679 – WAVES OF LIGHT
Hooke’s causal account of colors is somewhat the odd man out of
Micrographia. The book is a sustained effort to show that the world is made
up of seemingly imperceptible particles and structures and that the newly
invented microscope has enabled manhood to open up these uncharted
world plus ultra the visible surface of things.
Hooke’s theory of waves is found in ‘Observation XI’ of Micrographia
(1665). This section contains an experimental investigation of the colors
produced in thin films of transparent material, like the lamina of ‘Muscovy
glass’, or the space between two lenses pressed together, or soap-bubbles.
According to Hooke, these colors meant a refutation of the explanation of
prismatic colors Descartes had given in Les Météores (1637). The latter had
argued that no colors are produced when there is no net refraction. Hooke
argued that in thin films there is no net refraction, so according to Descartes’
theory no colors should be produced. Yet, observation shows they are.94 In
addition he argued that according to Descartes’ own theory no colors would
be produced in rain-drops.95 In both cases, Hooke gave an alternative
explanation, thus demonstrating the superiority of his theory of colors over
Descartes’. These explanations were based on his own pulse theory of light,
according to which the short, vibrating motions of luminous objects produce
pulses that propagate rectilinearly through a transparent, homogenous
When such a light pulse falls obliquely on the surface of a denser
medium, the following happens.97 Adopting Descartes’ viewpoint, Hooke
assumed that the pulses propagate faster in the denser medium.98 The end of
the pulse that first reaches the surface will therefore come to move ahead of
the other end. As a result, the pulse will become oblique to its direction of
propagation, the refracted ray as found by means of the sine law. According
to Hooke the preceding end of the pulse is resisted most by the medium and
thus becomes weaker than its other end, whose passage has been prepared
by the first. This difference accounts for the primary colors red and blue.
With this ‘hypothesis’ Hooke explains the production of colors when light
passes a drop of water in a succession of refraction, reflection and another
refraction.99 It may be clear that in this account a pulse is necessarily a
coherent whole, otherwise no difference can be made between its acute and
obtuse ends. Moreover, Hooke made no effort to mathematize the
mechanism presumed in the passage of the pulse to the denser medium, nor
did he suggest a macroscopic phenomenon comparable to it.
Hooke, Micrographia, 54.
Hooke, Micrographia, 59.
Hooke, Micrographia, 54-56.
Hooke, Micrographia, 56-59.
Compare Shapiro, “Kinematic optics”, 194-196.
Hooke, Micrographia, 61-62.
Things became even more problematic when Hooke turned to the colors
produced by thin films.100 On the basis of several experiments, he came to
the conclusion that the various colors depended upon the thickness of the
film.101 In his view, part of the incident light is reflected at the upper surface
of the film, part at the lower surface. Consequently, two pulses following
shortly upon each other are produced of which the second, having traveled a
longer distance, is weaker. The amount of retardation depends upon the
thickness of the layer, thus explaining the variety of colors. In this way,
Hooke had formulated two different, even inconsistent theories of color.
This did not keep him from a generalization:
“That Blue is an impression on the Retina of an oblique and confus’d pulse of light,
whose weakest part precedes, and whose strongest follows. And, that Red is an
impression on the Retina of an oblique and confus’d pulse of light, whose strongest
part precedes, and whose weakest follows.”102
This explanation ought to encompass both the oblique pulse (with the acute
end being the weakest) and the retarded pulse (with the pulse reflected at the
lower surface of the film being the weakest). It is, however, stated in most
general terms and Hooke did not explicitly consider the question whether his
two mechanisms explaining colors could be made consistent. The
formulation rules out any possibility of mathematization the two individual
theories may have had. Several other inconsistencies and obscurities in
Hooke’s theory can be pointed out.103
Huygens’ verdict was merciless. The annotations in his copy of
Micrographia make it clear that he did not think much of Hooke. “This must
not be presumed but ought to be demonstrated, …” Huygens wrote in the
margin of the page where Hooke introduced the sine law.104 He was quick to
point out the sloppiness of Hooke’s reasoning, in particular his use of
‘pulses’ and ‘rays’.105 In Traité de la Lumière, he mentioned Hooke without
comment, but elsewhere he made no secret of his dissatisfaction. In optics
Hooke had only made ‘shameful blunders’, he wrote to Leibniz in 1694.106
If Micrographia did not come up to the standards of Traité de la Lumière,
these were not, after all, Hooke’s standards. His goal was not to elaborate a
mathematical, but an experimental theory of colors derived from and
founded on exhaustive empirical evidence. He accepted the sine law as an
empirically founded truth that needed no further mathematical or other
Hooke, Micrographia, 65-67.
Hooke, Micrographia, 50.
Hooke, Micrographia, 64.
In his subsequent analyses of refracted pulses, of the refraction of a beam (by a drop of water), his
interpretation of ‘ray’ and ‘pulse’ continuously changed, switching without notice from a microscopic
point of view to a macroscopic and back. Huygens noted several gaps, and some vagueness as well: Barth,
“Huygens at work”, 612-613. See also: Shapiro, “Kinematic optics”, 198-199.
Barth, “Huygens at work”, 612.
Barth, “Huygens at work”, 612 (in particular 57 II & III).
OC10, 612. “… bevues honteuses …”
1677-1679 – WAVES OF LIGHT
proof.107 He set great store by his experimental refutation of Descartes’
theory. As an experimental philosopher he proceeded by minutely recording
observations and experiments of colors in order to infer their causes. His
observations were largely qualitative. As regards the colors in thin films,
Hooke admitted that he had not been able to “… determine the greatest or
least thickness requisite for these effects, …” 108 He suggested that the colors
in thin films were periodical in some way, without attempting to state this in
more exact terms. Upon reading Micrographia both Huygens and Newton
readily determined the thickness of the film and the kind of periodicity
involved.109 In this way, Micrographia is typical of the experimental philosophy
in which observations and explanations were qualitative and theories never
rose to an exact level.
From the perspective of Traité de la Lumière, Pardies’ theory of waves met the
standards of proper ‘raisons de mechanique’, even if it could be improved
somewhat. In the first place, it employed mechanistic concepts Huygens
accepted. It was based on the idea that light consists of motion without
transport of matter, and crystallized in a conception of waves produced by
successive collisions of ethereal particles. The basic corpuscular entity was
the particle and the basic motion was impact, instead of some kind of body
whose exact motions were obscure. The combination of these constituted an
action governed by established mathematical laws. Waves produced by
impacts of ethereal particles were, in other words, proper ‘raisons de
Secondly, the form of Pardies’ theory met Huygens’ demands. It was cast
in the form of a geometrical construction. In this way, the mechanistic
consideration of refracting waves reduced to geometrical manipulation on
the basis of some mathematical premises. As we have seen in section 4.2.2,
this was not wholly unproblematic in Pardies’ explanation of refraction. The
curve resulting from the construction had an ambiguous meaning in terms of
waves. In Ango’s rendition Pardies’ waves remain entities whose existence is
presupposed in the derivation of the sine law.
In his own theory, Huygens rigorously defined waves as an effect of
colliding ethereal particles. At all times, a wave is the resultant of the way this
action has spread indifferently through a sea of disconnected particles.
Waves are reduced to the one property that was central to his understanding
of motion: velocity. The idea that each part of a wave is the source of a new
wave, combined with the assumption that visible light is produced only
where secondary waves coincide, enabled Huygens to consider speeds of
propagation only. In his ‘principal foundation’ this was cast in mathematical
form, thus reducing the consideration of wave propagation to geometrical
Hooke, Micrographia, 57.
Hooke, Micrographia, 67.
Westfall, “Rings”, 64-65.
construction.110 It improved the mechanistic conceptualization and its
mathematical representation of Pardies’ theory. These two aspects of
Huygens’ wave theory are two sides of the same coin: comprehensibility. To
Huygens, sound mechanistic concepts were those that could be represented mathematically.
In Huygens’ principle the mechanics of wave propagation was absorbed into
geometry.111 However, in Traité de la Lumière Huygens did not explicate what
precisely he meant with ‘raisons de mechanique’. The strict definition of a
mathematized model of actions that are governed by established laws of
motion remains implicit in his treatment.
Against the background of the problems in Descartes’ optics of not
completely integrating mathematical science and natural philosophy,
Huygens got further than his contemporaries in turning the principles of
mechanistic philosophy in true ‘raisons de mechanique’. In this sense I tend
to disagree when Buchwald says Huygens’ principle of wave propagation
does not explain the rectilinear propagation of rays or the sine law any better
than Pardies/Ango. Conceptually it was clearer and it was a more successful
exercise in transduction. Yet, these are subtle differences and it is indeed
little surprising that the fact that only Huygens’ theory could account for
strange refraction was not enough to gain him many adherents.112
Newton’s speculations on the nature of light
Given Huygens’ (implicit) conception of ‘raisons de mechanique’, an
emission conception of light may seem a viable alternative but this he had
ruled out in advance. Only one seventeenth-century adherent took the
trouble of mathematically elaborating an emission conception of light:
Newton. In Principia and Opticks he presented an analysis of the dynamics of
a particle passing into a denser medium. Newton understood ‘raisons de
mechanique’ in a way similar to Huygens: the established laws of motion
applied to unobservable particles. Before continuing my discussion of the
achievements of Traité de la Lumière, I discuss Newton’s analysis of refraction
in some detail.
Although Newton had a similar idea of the proper ‘raisons de
mechanique’, he dealt with them quite differently. Only with the greatest
circumspection did he reveal his ideas on the nature of light in public. He
had adopted the idea that light consists of moving particles since his earliest
studies of prismatic colors.113 Yet, according to Newton this was irrelevant
for his theory of colors. An experimentally founded theory explicating the
behavior and properties of rays could and should be separated from
speculative ideas regarding their causes and the nature of light. Different
Compare Shapiro, “Kinematic optics”, 208.
Shapiro (“Kinematic optics”, 244) puts it as follows: “The key to Huygens’ success in optics was his
continual ability to rise above mechanism and to treat the continuum theory of light purely kinematically
and, thereby, mathematically.”
Buchwald, Rise, 5.
Hall, Unpublished, 403; Newton, Certain, 432-435; Westfall, Never at rest, 159-163, 170-172.
1677-1679 – WAVES OF LIGHT
refrangibility was an observational property derived from experiment. In his
paper on colors of 1672 he remained silent on the mechanistic nature of light
and colors, and in his lectures on optics he likewise kept his mouth shut. In a
paper Newton sent Oldenburg a few years later, on 7 December 1675, he
disclosed his views in public for the first time. “An Hypothesis explaining
the Properties of Light discoursed of in my severall Papers” it was called. It
gave qualitative explanations for reflection, refraction and the diversity of
Twelve years later, in Principia, Newton mathematized the explanation of
the sine law sketched in the ‘Hypothesis’. However, he did not present it
explicitly as a derivation of the sine law. Section XIV of Book I discussed
“The motion of minimally small bodies that are acted on by centripetal
forces tending towards each of the individual parts of some great body”.114
Only in a scholium after the exposition did Newton point out the
resemblance of the results here obtained with the behavior of light rays. He
added cautiously:
“Therefor because of the analogy that exists between the propagation of rays of light
and the motion of bodies, I have decided to subjoin the following propositions for
optical uses, meanwhile not arguing at all about the nature of rays (that is, whether they
are bodies or not), but only determining the trajectories of bodies, which are very
similar to the trajectories of rays.”115
In proposition 94, Newton pictured (Figure 74) a space AabB between two
similar media, bounded by parallel planes, through which a body passes that
is attracted or impelled perpendicularly towards either of those media and
showed “… that the sine of the angle of incidence onto either plane will be
to the sine of the angle of emergence from the other plane in a given
A body moves along GH
and is attracted upwards
between Aa and Bb. During
its passage it follows the
curve HI, then leaves the
layer along IK. The curve is
constructed by producing
GH to M and IK to L,
drawing IM perpendicular to
Bb, and a semi-circle PNIQ
with center L and radius LI.
Newton then showed that
when the attraction is
uniform, HI will be part of a
Figure 74 Refraction in Principia.
parabola with the following
Newton, Principia, 622.
Newton, Principia, 622- 626. The following propositions were on anaclastics.
Newton, Principia, 622.
properties: HM2 = MI· p (where p is the given latus rectum of the parabola),
HM bisected in L and IR = MN. Consequently “… the ratio of the sine of the
angle of incidence LMR to the sine of the angle of emergence LIR is given.”117
Next, he showed that if AabB is divided into several parallel planes the same
holds for the angle of incidence on the first plane and the angle of
emergence from the last plane.
Newton did not explain the physical meaning of parameter p. Bechler has
shown that it depends upon the properties of the material in the layer AabB
and is constant for all angles of incidence. Furthermore, he argues
convincingly that this leads to seeing the ratio of sines – that is, the index of
refraction – as dependent upon the initial velocity of the body. So for the
sine law to hold, all incident rays must have the same velocity.118 It may be
clear that this derivation of the sine law employed proper ‘raisons de
mechanique’ as Huygens understood them implicitly. In the preceding
sections of Principia Newton had established his theory of accelerated motion
which he now applied to ‘very small bodies’. It goes without saying that
Huygens’ mechanistic vocabulary lacked the forces employed (see chapter
In Book 1, Part 1 of Opticks, Newton presented an adjusted and
abbreviated version of the demonstration in Principia. He carefully eliminated
all traces of unobservable particles and couched the derivation solely in
terms of rays and their properties. He did not explicitly say that the ray is
curved at the refracting surface and referred to an ordinary ray diagram,
considering the parallel and perpendicular components of the ray’s motion.
Although he said the sine law to be adequately founded empirically – which
had been sufficient in his optical lectures of 1670 – he now added a
demonstration in order to show that it is ‘accurately true’.119 He was, after all,
a mathematician rather than a mere experimentalist. He did so by the
supposition “That Bodies refract Light by acting upon its Rays in Lines
perpendicular to their Surfaces.”120 As Sabra has shown, the supposition
comes down to saying that only the perpendicular component of the rays is
affected and its parallel component remains constant.121 This means that
Newton adopted the mathematical assumptions of Descartes’ demonstration
of the sine law, while giving them a new physical interpretation.
Newton did not, like Descartes, simply assume the second assumption –
the velocities in the respective media are in constant proportion, and larger
in denser media. He derived it by stating, without proof, a consequence of
the demonstration in Principia: the perpendicular velocity at emergence is
equal to the square root of the sum of the square of the perpendicular
Newton, Principia, 623.
Bechler, “Newton’s search”, 16-17.
Newton, Opticks, 79. Newton, Optical papers 1, 169-171; 311-313.
Newton, Opticks, 79.
Sabra, Theories, 300-301.
1677-1679 – WAVES OF LIGHT
velocity at incidence and
of the square of the
perpendicular velocity at
emergence of a ray at
grazing incidence (Figure
relationship holds between
rays MCN and ACE:
CF = ¥(DC2 + CG2), where
medium. Combined with
AD = DH, as the diagram
should be read, the
proposition yields the sine
law of refraction.
Figure 75 The sine law in Opticks.
“And this Demonstration being general, without determining what Light is, or by what
kind of Force it is refracted, or assuming any thing farther than that the refracting Body
acts upon Rays in Lines perpendicular to its Surface; I take it to be a very convincing
Argument of the full truth of this Proposition.”123
Nevertheless, assumptions on the level of unobservable entities are implicit:
AC and MC have equal velocity and the constancy of the index of refraction
depends upon the velocity of the incident ray.124 Bechler argues that Newton
chose to use the confusing one-circle diagram – instead of a diagram
representing the relative ‘velocities’ in both media – precisely to obscure the
meaning and physical implications of the fact that MC : NG is given. In this
way, Newton could refer to the components of the rays without explicitly
referring to the velocities.
There may have been an extra reason for Newton to obscure the central
role velocity played in his derivation of the sine law, besides his effort to
convey the demonstration in terms of experimentally founded entities. The
derivation had unfortunate consequences for his understanding of colors, the
very ‘raison d’être’ of Newton’s optics. The derivation implied that the index
of refraction depends upon the velocity of rays. In this way, different
refrangibility might be identified with the different velocities of rays of
various colors. This in turn might lead to the derivation of a law of
dispersion correlating the refractive indices of various colors. In fact,
Newton had already formulated the law that would result from such an
extension of the sine law. In his optical lectures of the 1670s he had laid
down a law of dispersion without proof – be it experimental or
Newton, Opticks, 79-80.
Newton, Opticks, 81-82.
Bechler, “Newton’s search”, 28-31.
mathematical.125 It assumed a constant difference between the parallel
components of the colored rays. The law can be seen as a natural extension
to dispersion of Descartes’ derivation of the sine law, identifying the variety
of colors with various sizes of the parallel component.126
In Opticks this ‘Cartesian’ dispersion law has disappeared. Supposedly,
Newton had realized that his dispersion law implied that color depended
upon velocity. This he could not accept, as the immutability of colors was
the core of his theory. As velocity changes in even the most elementary
mechanisms, it seemed a unlikely candidate for an original and conservable
property of light rays.127 Therefore, the ‘Cartesian’ dispersion law of the
optical lectures was unacceptable.128 This left a model for dispersion based on
size or mass, but Newton never articulated a mechanism through which
different refrangibility might be explained this way. What is more, the only
mechanism he elaborated for refraction – based on perpendicular forces
acting upon particles – was at odds with such a model, for acceleration is
independent of mass. In Opticks, Newton put forward an alternative law of
dispersion with dubious empirical evidence and whose mechanistic causes he
had never elaborated – not even in private.129
The status of ‘raisons de mechanique’
In the seclusion of his private quarters, Newton allowed himself a far greater
liberty of reasoning than in his publications. From the very start, his
experimental inquiries had been accompanied by speculations on the
corpuscular nature of light and colors. Shapiro has analyzed the way in which
Newton employed his vibration model to develop his theory of periodicity of
colors.130 The derivation of the sine law shows that Newton was on a par
with Huygens as regards the mathematization of mechanistic causes. He
employed the method of transduction with a comparable meticulousness
mathematizing the physics of unobservable particles by founding them upon
the established laws of motion. Unfortunately, Newton’s consideration of
‘raison de mechanique’ turned problematic because it produced discrepancies
that left considerable gaps in his mathematical science of colors. It did not
affect his experimental theory of color, though. Although he did not have an
exact law of dispersion, the experimentally disclosed and secured theory of
different refrangibility stood unshaken.
Newton, Optical papers 1, 199 & 335-337.
See Shapiro, “Dispersion law”, 99-104 & 126-127; Bechler, “Newton’s search”, 4-5. I discuss this
matter in more detail in my “Once Snel breaks down”.
Bechler, “Newton’s search”, 32-33.
In 1691, Newton figured out a test for the assumption that color differs with velocity: when a moon of
Jupiter disappears behind the planet the slowest color – red – should be seen last. In February 1692,
Flamsteed reported that such a difference could not be observed. This empirical evidence definitely ruled
out velocity. Shapiro, Fits, 144-146.
Newton, Opticks, 128-130. Shapiro, “Dispersion law”, 97-99; 126-127. Shapiro suggests that it might be
based upon a small angle approximation of refracting angles.
Shapiro, Fits, 200-201.
1677-1679 – WAVES OF LIGHT
If Newton and Huygens had comparable opinions about the nature of
mechanistic principles, the took opposite positions regarding their status and
their relationship to the laws of optics. The way Huygens derived and
presented his law of strange refraction would have been unacceptable for
Newton, who had gone to such great lengths to erase all traces of
speculation from his experimentally established theory of light and colors.
Huygens deliberately used hypotheses to explain the observed properties of
light. True, this implied speculation, but to do so was inevitable when it came
to explanations.
“One will see here demonstrations of the kind as not to produce a certainty as great as
those of geometry, and that even differ much from it, for whereas the geometers prove
their propositions by secure and incontestable principles, here the principles are verified
by the conclusions drawn from them; as the nature of these matters does not allow that
this is done otherwise.”131
The wave theory belonged to ‘la Physique’, offering a plausible explanation
of the empirical truths on which ‘l’Optique’ was founded. Yet, explanation
was not arbitrary. In order to be plausible an explanation ought to employ
proper ‘raisons de mechanique’. His principle of wave propagation satisfied
this demand. But still, it was probable at best. It derived no proof value from
being plausible. Huygens did not claim a priori truths. Consequently, the
burden of proof for the wave theory did not rest with its merits as a
mechanistic theory. The raison d’être of Huygens’ principle was that ‘all
properties of light, and everything pertaining to its reflexion and its
refraction, can be explained principally by this means’.132 His ‘principal
foundation’ should not be judged at face value but on its adequacy to derive
the laws of optics.
Pardies’ theory had the same focus on explaining the laws of optics.
Huygens pointed out the advantage of his principle of wave propagation
precisely in this context. It filled a gap in Pardies’ demonstrations of the laws
of reflection and refraction. What this gap was, and why his own
demonstrations were better, Huygens did not explicate. He had found a
better ‘foundation’ for deriving the laws of optics. Huygens was rather
awkward in spelling out the mechanics of wave propagation in full and did
not always – as in the case of strange refraction – elaborate it in full detail.
His focus was on his principle of wave propagation. First of all, he assured
himself that all properties of reflection and refraction could be reduced to it.
His wave theory had developed accordingly. In his notes on caustics and the
explanation of strange refraction, Huygens satisfied himself that he could
construct a propagated wave in the troublesome situation of caustics, and a
refracted ray in the equally troublesome situation of strange refraction.
Traité, “Preface”, [2-3]. “On y verra de ces sortes de demonstrations, qui ne produisent pas une
certitude aussi grande que celles de Geometrie, & qui mesme en different beaucoup, puisque au lieu que
les Geometres prouvent leurs Propositions par des Principes certains & incontestables, icy les Principes se
verifient par les conclusions qu’on en tire; la nature de ces choses ne souffrant pas que cela se fasse
See the passage quoted above on page 174.
With the full elaboration of his theory in Traité de la Lumière, Huygens
showed that with his principle propagated waves could be constructed in any
situation. Only the speed of propagation, as it depended upon the medium
traversed, needed to be varied. In this way he derived all observable
properties of light rays from one and the same principle in a mutually
consistent way. This reduction was what he understood by explanation.
Reducing the properties of light rays to Huygens’ principle was explaining
these properties mechanistically, because the principle explicated the
essentials of successive impact in ethereal particles. The validity of this
‘principal foundation’ rested upon the fact that the laws of optics could be
reduced to it. In other words, it did not rest upon the appropriateness of
‘raisons de mechanique’, but on the plausibility of mathematical inference.
In Huygens’ wave theory three levels can be distinguished: a mechanistic
model of colliding particles, the laws of optics and – in between – Huygens’
principle. As the mathematical representation of the mechanistic nature of
wave propagation, Huygens’ principle serves as a intermediary of a special
kind between the nature of light and the laws of optics. It was the
indispensable link between Huygens’ mechanistic picture of collisions of
ethereal particles and the mathematical laws of light rays. In the light of
seventeenth-century geometrical optics, where the laws of optics functioned
as the postulates or principles of mathematical science, Huygens’ principle of
wave propagation can be called a law of optics. Not in the modern sense of a
law of nature in physical science, but in a then traditional sense. Remember
that the sine law and the like were rarely called laws then, but rules, measures
or properties. In the mathematical science of optics Huygens had disclosed a
new law, a more fundamental one to which the various properties of light
propagation were subordinated to. However, this new ‘law’ was of an entirely
different nature than the traditional principles of optics. Huygens’ principle
did not describe the behavior of rays but the behavior of waves; it was a
mathematical law describing the behavior of unobservable entities.
Comprehended in this way, Huygens’ principle was a novel element in the
mathematical science of optics. Huygens’ principle not only unified ordinary
and strange refraction, it unified all properties of light rays. It was a more
general law and a law of different character at the same time, describing the
behavior of unobservable waves mathematically.
One might say that Huygens had brought geometrical optics to a new
level, that of microphysics. He focused on the geometrical constructions
with his principle and did not spell out its mechanistic underpinning. In
Traité de la Lumière, waves have taken the place of rays. Waves are entities
with well-defined mathematical properties, the causes of which are explained
rather informally, like in Barrow’s elucidations. Huygens switched to the
mathematical consideration of waves in a matter-of-course way. He applied
geometry to these unobservable entities with the same ease as he applied it to
observable balls and pendulums. In his wave theory he extended Galileo’s
mathematical physics of observables to that of unobservables. As one
1677-1679 – WAVES OF LIGHT
applied geometry to matter in optics or any other branch of mathematics,
one could apply it to the ether. To Huygens, it went without saying that the
‘vraye Philosophie’ meant applying the ‘raisons de mechanique’ to
unobservables just as one applied them to pendulums. He gave no sign of an
awareness of the possibility that mathematizing the ether might raise
philosophical or ontological problems.
The only difference between applying geometry to observable matter and
applying it to ethereal particles was the level of certainty that could be
attained. The nature of these things does not allow other than speculations,
but – so the text quoted on page 201 continued:
“It is still possible in this to arrive at a degree of probability that quite often yields
hardly to full evidence. This is so when the things that are demonstrated by the
supposed principles correspond perfectly with the phenomena that experience has
drawn attention to; particularly so when there is a great number of them, and moreover
principally when one conceives and foresees new phenomena that must follow from
the hypotheses one employs, and when one finds that in this the effect answers our
expectation. When all these proofs of probability converge in what I have designed to
treat of, as it seems to me they are, this must be quite a great confirmation of the
success of my research, and it is only with difficulty possible that things would not be
more or less as I represent them.”133.
The wave theory occupied the twilight zone of probability that lies between
the truth of the laws of optics and the arbitrariness of mere speculations.
This was an area where Huygens, unlike Newton, dared to tread because in
his own view he could still cling here to the reliability of mathematical
inference. The reduction of the laws of optics to the principle of wave
propagation did not add to their validity, it only proved the probability of the
explanation. These laws were empirically founded, as contrasted to the
principle of wave propagation that described unobservables. His principle of
wave propagation was probable at most, precisely because it could not be
demonstrated by direct observation. By means of sound and mutually
consistent derivations, Huygens demonstrated that his principle was more
plausible and more probable than other explanatory theories. This is, of
course, what we call hypothetico-deductive inference.
The explanation of strange refraction had been decisive. It singled out
Huygens’ theory because it was the only one that accounted for the
phenomenon in a way that could be reconciled with the other properties of
light rays. In Huygens’ view, this matter had been settled. But it would not be
that easy. This had everything to do with the particular nature of the ellipse
construction. As we have seen in the previous section, it differed essentially
Traité, “Preface”,[3]. “Il est possible toutefois d’y arriver à un degré de vraisemblance, qui bien souvent
ne cede guere à une evidence entiere. Sçavoir lors que les choses, qu’on a demontrées par ces Principes
supposez, se raportent parfaitement aux phenomenes que l’experience a fait remarquer; sur tout quand il y
en a grand nombre, & encore principalement quand on se forme & prevoit des phenomenes nouveaux,
qui doivent suivre des hypotheses qu’on employe, & qu’on trouve qu’en cela l’effet repond à nostre
attente. Que si toutes ces preuves de la vraisemblance se rencontrent dans ce que je me suis proposé de
traiter, comme il me semble qu’elles sont, ce doit estre une bien grande confirmation de succês de ma
recherche, & il se peut malaisement que les choses ne soient à peu pres comme je les represente.”
from the established laws of optics. It was no ‘truth drawn from experience’,
for it mixed the properties of rays with the properties of waves. The ellipse
construction was inextricably bound up with the principle of wave
5.3 A second EUPHKA
With his theory explaining the established laws of refraction as well as the
strange refraction of Iceland crystal, such as we have set it forth in the
preceding section, Huygens addressed the Académie in the summer of 1679,
nearly two years after the EUPHKA of 6 August 1677. We can imagine his
expectations. He would present to his colleagues a truly mechanistic
explanation of the properties of light, firmly founded upon the laws of
motion. Only with his principle could the laws of optics be derived in a
sound and coherent way. In addition, he would present a wonderful
confirmation by explaining the baffling phenomenon of strange refraction
with it. Things could hardly be otherwise. His theory of waves was the only
comprehensible explanation conceivable. It would show what rigorous
thinking could yield. Thinking that was not easily satisfied, but aimed at
rendering matters intelligible without compromise.
At least one member of the Académie was not convinced immediately. It
was Rømer, the same who in 1677 had provided Huygens with observational
proof of the finite speed of light. Rømer’s intervention forced Huygens out
of the safe domain of rational analysis, where the properties of light are
derived from clear and distinct concepts by means of rigorous deduction, to
the empirical domain of tinkering with the unpolished reality of
measurement and experimentation. Huygens managed to counter Rømer’s
objections by measurements acquired by a precise and powerful observing
technique and a ingenious experiment that reveals a remarkable command.
Remarkably, as up to this point Huygens had repeatedly steered clear of
empirical grounds. These measurements of 1679 provided the data of the
eventual Traité de la Lumière which, in other words, date from Huygens’ third
go at strange refraction.
In a letter of 11 November 1677 Huygens had informed Rømer of a
letter he had written to Colbert on October 14.134 He had praised Rømer’s
“belle invention”, and now added that he had always assumed the same in
order to explain the properties of light.135 He added further that his
hypothesis to explain strange refraction was so simple and so accurate and
agreed so well with observation that he did not doubt that everyone would
accept it.136 Replying on December 3, Rømer expressed the opinion that
optical principles that could not account for strange refraction were useless.137
He was curious after Huygens’ ideas and added that he himself had also done
OC8, 41.
OC8, 36-37. This letter is quoted on page 161.
OC8, 41.
OC8, 45.
1677-1679 – WAVES OF LIGHT
some thinking on refraction, in particular on Descartes’ account of it. He
threw doubts of the validity of Descartes’ proof. It was unclear to him what
kind of impact Descartes had in mind when comparing light rays with balls
struck into water. And, according to him, contrary conclusions could be
derived from Descartes’ assumptions. Rømer himself had read a paper to the
Académie in which he rejected the assumption that the speed of light is
larger in denser mediums and derived the sine law in a way similar to
Fermat’s derivation.138 Huygens’ letter had made him consider strange
refraction and he had promising ideas, he said. But without a piece of crystal
and precise data he could not pursue his thinking in a satisfactory manner.139
Rømer had to wait for one and a half year before hearing the details of
Huygens’ explanation. And when the time came, in the summer of 1679, he
raised serious objections. Huygens was to recall what happened when he sent
him a copy of Traité de la Lumière in 1690.140 From this letter it is also clear
that some notes Huygens wrote in July and August 1679 were directed at
Rømer’s objections. From all this we can infer that Rømer had advanced the
theory of Bartholinus – his father in law – as a viable alternative to Huygens’
ellipse construction.141 Bartholinus had argued that strange refraction is
governed by an ‘oblique perpendicular’ that is parallel to the edge of the
crystal. Evidence for this he found in the fact that the unrefracted oblique
ray is parallel to the edge of the crystal. Therefore, so Bartholinus had
concluded, the unrefracted oblique ray must have a function similar to the
perpendicular ray in ordinary refraction. According to Bartholinus’ ‘oblique’
sine law, the sines of incident and strangely refracted rays are in constant
proportion when measured with respect to the unrefracted oblique ray.
Bartholinus had suggested that pores in the crystal could explain the
unrefracted passage of the ray parallel to the edge of the crystal.
According to the registers of the Académie, Huygens read from his
‘Dioptrique’ on 1 July 1679.142 This may well have been the session at which
Rømer pointed out that Bartholinus’ explanation was equally plausible and
had not been refuted by Huygens’ explanation. On 3 July, Huygens in his
turn could refute a central assumption of Bartholinus’ explanation:
“Observation made on 3 July 1679. which proves manifestly that it is not the ray
parallel to the sides of the crystal that passes without refraction as I thought until
Cohen, “Roemer”, 344.
OC8, 45-46.
OC9, 489. No direct evidence from the late 1670s of Rømer’s objections is available.
As Ziggelaar also assumes; “How”, 185.
OC19, 440n2.
OC19, 440. “Observation faite le 3 juillet 1679. qui prouve manifestement que ce n’est pas le rayon
parallele aux costez du cristal qui passe sans refraction comme j’avois creu jusqu’icy.”
Huygens described a simple but precise
method for comparing the angle of the
edge of the crystal and the angle of the
unrefracted oblique ray (Figure 76).144
reminiscent of the one employed by
Bartholinus to determine the index of
(ordinary) refraction.145 The manuscripts
suggest that Huygens went back to his
earliest notes on strange refraction for Figure 76 The new measurement.
the observations are recorded on a remaining part of the last page of the
1672 investigation.146 On the upper surface of the crystal he marked a point D
and determined the point B on the opposite surface so that BD is parallel to
the edges AH and KL. Then he positioned a ruler EF under the crystal,
perpendicular to the diameter AC and through point B. On the diameter AC
he then marked the place were B is seen and found that this was not D. The
angle between the edge of the crystal and the refracted oblique ray was about
2½ degrees. On the next session of the Académie – 8, 15 and 22 July –
Huygens continued to read his ‘Dioptrique’ and presumably presented this
new result.
Huygens’ observation still did not suffice to counter the Dane’s
objections. Rømer was not convinced that the ‘oblique’ sine law had been
refuted. Apparently, he had referred to Bartholinus’ suggestions about
refraction in surfaces of non-natural sections of the crystal. According to
Bartholinus, strange refraction was related to the orientation of the refracting
surface with respect to the crystal. He had predicted that ordinary and
strange refraction would swap place – so that the ‘fixed’ image becomes
mobile and vice versa – in alternative, non-natural sections of the crystal.
On 6 August 1679, Huygens
found out that this was not the
case. He figured the crystal cut
along plane MN, which makes an
angle of 45º20' with the axis of
the spheroid governing strange
refraction (Figure 77). According
to the ellipse construction, this
plane should produce the same
refractions as the natural plane
gG. The verification of this
Figure 77 The EUPHKA of August 1679.
assumption was troublesome
because it was not easy to cut and polish the crystal, but Huygens claimed to
See the assessment of Buchwald, Rise, 312-313.
Bartholinus, Experimenta, 34-41 (experimentum XVII).
Hug2, 178r.
1677-1679 – WAVES OF LIGHT
have succeeded: “In this way I have made section MNO, and I have found
that the surfaces it makes have the same refractions as the surface gG, …”147
After considering various ways of cutting the crystal, he concluded: “It
appears that it is not the disposition of the layers of the crystal that
contributes to the irregular refraction.”148 Bartholin’s explanation and
Rømer’s objections thus lacked a foundation. A second EUPHKA followed:
“EUPHKA. The confirmation of my theory of light and of refractions.”149
Huygens had proven that his was the only acceptable explanation of strange
refraction. Unexpectedly, Rømer had made it clear that Bartholinus’ law
could not be dismissed forthwith and that Huygens should produce decisive
evidence against it. In doing so, Huygens showed that his was the only
theory that could also explain strange refraction. Ergo, to take Rømer by his
own words, his principle of wave propagation was the only useful principle
in optics. On 12 August he continued the reading of his ‘Dioptrique’ at the
Forced innovation
In the summer of 1679 Huygens showed himself an able measurer and
inventive experimenter. In a two-stage reaction to Rømer’s objections he
refuted Bartholinus’ law and confirmed his own ellipse construction. The
measurement and the experiment added a new, empirical element to his
study of strange refraction. It is remarkable that Huygens had not questioned
Bartholinus’ data previously. In 1672 he had improved Bartholinus’
measurements of the angles of the crystal by means of a more reliable
technique.150 Yet, apart from the angle of the refracted perpendicular ray –
which Bartholinus had not provided – he had not measured any angle of
refraction. He had never measured the unrefracted oblique ray or any other
rays. That he developed the technique to measure the refraction of a ray only
in 1679 appears from the fact that this section of Traité de la Lumière was
inserted into the original manuscript.151 Until that time his theory had
developed in an empirical void. He discovered the ‘law’ of strange refraction
by mathematical reasoning, not from precise observations as Buchwald
concluded from his study of Traité de la Lumière.152 The empirical solidity of
the finalized theory was acquired only at the third stage of Huygens’ studies
of strange refraction, when he was forced by Rømer’s objections to take a
closer look.
OC19, 442. “De cette maniere j’ay fait la section MNO, et j’ay trouvè que les surfaces qu’elle a faites
avoient les mesmes refractions que la surface gG, …” On 3 November 1679 he wrote his brother: “I have
found means to grind and polish this crystal which was thought impossible, …” OC8, 241. “J’ay trouvè
moyen de tailler et de polir ce cristal ce qu’on croioit impossible,…”
OC19, 443. “Il paroit que ce n’est point la disposition des feuilles du cristal qui contribue a la refraction
OC19, 441. “EUPHKA. La confirmation de ma theorie de la lumiere et des refractions.”
Buchwald, “Experimental investigations”, 313-314.
OC19, “Avertissement”, 385.
Buchwald, “Experimental investigations”, 313 & 316-317 and Buchwald, Rise, 313.
The Eureka of 6 August 1679 provided an additional confirmation of
Huygens’ explanation of strange refraction. It was presented in Traité de la
Lumière accordingly. Having derived in detail the properties of strange
refraction from his ellipse construction, he concluded:
“This being so, it is not a light proof of the truth of our suppositions and principles.
But what I am going to add here confirms them marvelously once more. These are the
different cuts of this crystal, of which the surfaces produced by them bring about
refractions precisely such as they must be and as I have foreseen them, following the
preceding theory.”153
Huygens inserted the results of his measurements and the experiment in the
original manuscript. He had to adjust his earlier ellipse construction, as it had
originally employed Bartholinus’ observation that the unrefracted oblique ray
runs parallel to the edge of the crystal. This did not, as we have seen, affect
the ellipse construction as such. At the beginning of the chapter, when he
gave the angle of the unrefracted oblique ray, he had warned his readers:
“This is to be noted, so that one does not search in vain the cause of the
singular property of this ray, in its parallelism to said sides.”154 He did not
bother to tell his readers that he had long assumed the same.
Rømer’s objections had not shaken Huygens’ confidence in the validity of
the ellipse construction and in his wave theory as a whole. He could reduce
all properties of light to a single principle by assuming only that the speed of
propagation varied with a specific medium. The explanation of strange
refraction singled out his theory, as it was the only one that could account
for the phenomenon in a consistent way. He had dismissed Bartholinus’ law
without further notice right in 1672 and never mentioned it. Only when
Rømer pointed out that it was a viable alternative did he put some work into
refuting it. He decided upon a remarkable way to counter Rømer’s
objections. What makes it remarkable is that his study of strange refraction
had consisted of rational analysis. He could have argued that Bartholinus’ law
was not general because it applied only to rays in the principal section. He
could have questioned the effect of the suggested pores – why do they affect
rays only partially? Huygens did not take this line of approach. Instead, he
called upon nature to refute Bartholinus’ law.
He did so in a particular way. The improved measurement of the
unrefracted oblique ray undermined the logic of Bartholinus’ law. If the
unrefracted ray was not parallel to the edge of the crystal, its connection with
the pores of the crystal became dubious. The result did not take away
Rømer’s objections, though. The ‘oblique perpendicular’ governing strange
refraction need not be parallel to the edge of the crystal for an ‘oblique’ sine
law to be plausible. It might well be that in strange refraction the sines
Traité, 85. “Ce qui estant ainsi, ce n’est pas une legere preuve de la verité de nos suppositions &
principes. Mais ce que je vais adjouter icy les confirme encore merveilleusement. Ce sont les coupes
differentes de ce Cristal, dont les surfaces, qu’elles produisent, font naistre des refractions precisement
telles qu’elles doivent estre, & que je les avois prevuës, suivant la Theorie precedente.”
Traité, 57. “Ce qui est à noter, afin qu’on ne cherche pas en vain la cause de le proprieté singuliere de ce
rayon, dans son parallelisme ausdits costez.”
1677-1679 – WAVES OF LIGHT
should be measured with respect to an arbitrary line, empirically determined.
So, even after the measurement, Bartolinus’ law remained a viable alternative
to Huygens’ ellipse construction. It was an exact law, that could account for
the basic observable properties of strange refraction.
Huygens had to find a more substantial refutation. He succeeded, as the
experiment of 6 August 1679 went right to the heart of Bartholinus’ law. He
could have challenged its empirical accuracy directly, by offering a set of
measurements that refuted the constancy of sines. Instead, he devised an
experiment of a special kind. He set Bartholinus’ law and his ellipse
construction side by side and compared them on their merits as exact laws.
He thought up a situation in which the two constructions yielded opposite
results and let nature decide. Compared to the way Huygens rejected other
theories the refutation of Bartholinus’ law takes a special place in Traité de la
Lumière. He refuted Descartes’ theory and emission conceptions by means of
straightforward, qualitative observations. Bartholinus’ law was now refuted
by means of a crucial experiment of the sort Hooke had employed to refute
Descartes’ theory of colors. The colors of Muscovy glass, however, were a
qualitative observation and in this sense they decided between Descartes’ and
Hooke’s theories. Huygens’ experiment, on the other hand, produced a
situation in which exact predictions of two mathematical laws were
compared. This was a new kind of experiment, a ‘mathematized’ crucial
experiment and Huygens had been forced into this innovation by Rømer’s
unanticipated objections.
Huygens’ experiment undeniably refuted Bartholinus’ law in favor of the
ellipse construction. That law was inherently connected with the particular
shape of the crystal and the measurement and the experiment were aimed at
undermining this connection. First, by casting doubts on the relationship
between the angles of the crystal and the line governing strange refraction.
Then, by questioning the relationship between strange refraction and the
orientation of the refracting surface. Huygens, too, explained strange
refraction by properties of the crystal. In his case, however, it was a property
of the material the crystal was made of instead of its (macroscopic) shape.
Strange refraction was caused by a property of the medium (affecting the
speed of propagation) just as any optical phenomenon was caused by various
properties of the media. The experiment did not so much confirm the ellipse
construction, it confirmed Huygens’ explanation of strange refraction by
spheroidal waves and thereby his wave theory as a whole.
The experiment of 1679 made the hypothetico-deductive structure of
Huygens’ wave theory manifest. Prior to it, the evidence for Huygens’
principle consisted of the reduction of the common properties of light to
one and the same principle. In addition, the successful explanation of strange
refraction by the same means reinforced its probability. It singled out his
principle of wave propagation as the only useful – as Rømer would put it –
principle in optics. Still, Huygens’ waves were hypothetical entities. Unlike
Descartes, Huygens did not intend to prove the laws of optics by means of
his theory of waves. It was the other way around: the verities of optics
proved the probability of the way he imagined waves to propagate. The laws
of optics were the ultimate foundation of Huygens’ theory of light. However,
the ‘law’ of strange refraction did not really fit this scheme as it mixed up
waves and rays and was not an empirical truth. Indirectly, via the successful
derivation of the ellipse construction, Huygens’ principle was founded upon
singular – but important – observations of strange refraction. Rømer’s
objections made it clear that the ellipse construction was not the only law
that was consistent with those observations. In order to counter these
objections, Huygens chose to employ the keystone of hypothetico-deductive
inference: experimental verification.
Although he could have refuted Bartholinus’ law otherwise, Huygens
went to the heart of the matter. He devised an experiment with the suggested
relationship between the crystal and strange refraction in mind. An unnatural
section of the crystal would reveal whether a law of strange refraction should
be related to the shape and structure of the crystal or to its material. He put
the very foundation of his wave theory at stake: waves are defined by their
speed of propagation, which depends solely on the medium traversed. This
could not be verified directly, but only by comparing consequences drawn
from the alternatives. The crux of Huygens’ employment of hypotheticodeductive inference was that he had the laws predict what would happen. He
derived exact predictions to be put to the test. The drawing accompanying
the experiment can be regarded as the essence of Traité de la Lumière: a wave
with respect to an unnatural section. The mathematical representation of the
mechanistic nature of light is here being experimentally verified.
The Eureka of 6 August 1679 was the ultimate consequence of Huygens’
mathematico-mechanistic thinking. Unexpectedly drawn into the problem of
the nature of light, our dioptrical geometer had set up a search for the
mechanistic causes of the properties of light. He had found waves caused by
collisions of ethereal particles and fitted out with mathematically defined
properties. Huygens’ principle was the plausible cause he needed, a law of
waves. It was a new kind of law, unifying the observable properties of light
rays by reference to unobservable waves. It also was a hypothetical law, as it
was not drawn from experience. The ellipse construction derived from it was
likewise hypothetical, although less explicitly so. It described strangely
refracted rays while presupposing spheroidal waves. When forced to test it,
Huygens chose to put to test this assumption of a medium-dependent
propagation of waves. The experiment was not a necessary step, but it was
the obvious choice. Waves were not just a plausible cause of the properties
of light, ultimately they were their true cause. Things could not reasonably be
otherwise than Huygens imagined. Therefore one could deduce phenomena
from this hypothesis, which experiment should show to be real.
1677-1679 – WAVES OF LIGHT
Huygens’ waves, while hypothetical and probable, were nonetheless
thought to be real things. This marked his explanations off from the
analogies invoked to elucidate the laws of optics. The balls and swords of
Alhacen clarified the assumptions of his mathematical account of reflection
and refraction, but did not prove them. Light was compared to cleaving
swords and bouncing balls only with respect to its ability to be deflected, not
because its nature was sword- or ball-like. Descartes’ moving balls were
likewise analogies not meant to represent the true nature of light. Only
Kepler had tried to derive the measure of refraction from its proper cause,
although he did not have a corpuscular conception of light. Barrow’s pulses
may have given a reasonable idea of the nature of light, they did no more
than that. Barrow did not explore this idea, to see what new consequences it
might reveal. The waves of Huygens were meant to reflect the true nature of
light, so that its properties could be derived from it. Huygens did, and thus
set the wheel of hypothetico-deductive inference in motion. One might say
that he did what Galileo had done in Discorsi, except that Huygens applied
mathematics to the motions of unobservable objects.
Between 1672 and 1679, a new way of doing the mathematical science of
optics developed. The wave theory contains three elements typical of early
modern science: mathematical description, mechanistic explanation,
experiment. By mathematically formulating his principle of wave
propagation, Huygens brought these three elements to a fruitful synthesis
that made possible the discovery and establishment of the nature of strange
refraction. In the attack upon strange refraction, a methodical process had
got going that (as Hakfoort tentatively concluded earlier) is typical of modern
mathematical physics, in which theory is extended by mathematically derived
consequences that are experimentally verified.155 Within the limited scope of
reflected and refracted rays, Traité de la Lumière constitutes the birth of
physical optics. Not the whole of it, but at least it contained a new,
mathematical science of optics in which the nature of light and its observed
behavior was fruitfully integrated for the first time.156
Hakfoort, Optics in the age of Euler, 183-184.
In his theory of colors Newton had invented another kind of physical optics, in which experiment was
used as a heuristic tool for finding new, mathematical properties of light. Newton at the same time
refrained from integrating explanatory hypotheses into his mathematico-experimental theories. I return to
this in section 6.2.
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Chapter 6
1690 - Traité de la Lumière
Retrospection upon the coming about of the wave theory
in the context of Huygens’ oeuvre
and the mathematical sciences in the seventeenth century
Huygens’ new science of optics developed in a markedly contingent way. If
he had not conceived of a plan for the elusive publication of his dioptrics; if
he had not fallen in with the custom of providing some explanation for
refraction; if he had not recognized the problem strange refraction posed for
his Pardies-like explanation of refraction; if he had not decided to include it
in his treatise; if he had not pressed ahead after his investigations of strange
refraction of 1672; and if Rømer had not compelled him to devise the special
experiment of 1679. If all this had been otherwise, then his celebrated wave
theory had not come about. In that case, some time, some kind of
‘Dioptrique’ may have been published. As a treatise in geometrical optics it
would hardly have marked itself off from – say – Barrow’s lectures, except
for its practical outlook. But Traité de la Lumière was something different.
What had begun as a fairly conventional, natural philosophical introduction
to a treatise on dioptrics had become a new way of treating light
mathematically that went beyond traditional geometrical optics. We are now
ready to look back and ask how the wave theory related to the seventeenthcentury development of optics and of mathematical science in general and,
second, what light it sheds on Huygens’ oeuvre.
Traité de la Lumière was not presented by its author as a revolutionary new
way of doing optics. Hypotheses were simply inevitable in these matters,
Huygens said as a matter of fact, and he did not draw attention in any way to
the special character of his principle of wave propagation and his account of
strange refraction. Did he realize he was breaking new ground? We cannot
read his mind, of course, but there is reason to think that he did not value his
findings in the same vein as we do, as some kind of methodological
innovation, that is. He never abandoned the original plan of 1672, in which
his theory of light would be a preparatory part to his dioptrics. Only at the
very last moment did he abandon his plan to publish a ‘dioptrica’ and created
a ‘traité de la lumière’. He published his wave theory in 1690 as Traité de la
Lumière, a title he had chosen at the very last moment. He did so after many
hesitations over the best way to present his wave theory, which suggests that
Huygens himself was also not sure about its exact status.
The publication of the wave theory took no less than twelve years. The
years after its presentation at the Académie witnessed Huygens’ step by step
departure from Paris. In 1681, he fell ill again and he returned to The Hague
in September. In Paris, the climate for Protestants was growing less tolerant
in Paris, and when in 1683 Colbert died, Huygens decided to remain in
Holland. He spent some happy years enjoying the reunion with his brother
until duty in the form of Stadtholder William III (= King William) called
upon Constantijn to go to London in 1688. Their correspondence in these
final years of separation gives once again proof of their intimate
comradeship. In 1687, at the ripe old age of ninety, their father Constantijn
sr. passed away. As the second son, Huygens inherited Hofwijck, the county
house in Voorburg, and the title ‘Lord of Zeelhem’, an estate of the family in
what is now Belgian Limburg.1 The last years of his life he spent much time
at the seclusion of Hofwijck, where his science experienced somewhat of a
prime with, among other things, contributions to the recent developments in
This chapter begins with the publication history of Traité de la Lumière and
a short outline of his later dioptrics. It continues with a review of
seventeenth-century optics from the perspective of Triaté de la Lumière. The
development of Huygens’ wave theory has revealed some themes that in my
view are important for our understanding of the development of
seventeenth-century optics. I will not offer a worked-out history but rather
sketch the lines of a re-interpretation. In the final part of this chapter, I turn
to Huygens’ science as a whole, in particular his alleged Cartesianism. Read
as a textbook example of Cartesian science, Traité de la Lumière is often seen
as exemplary for Huygens’ science. The eventual Traité de la Lumière should
not, however, be taken at face value. When assessing Huygens’ scholarly
goals and conceptions the winding road of its creation needs to be taken into
account. And it particularly casts doubts on its reputed cartesianist essence.
6.1 Creating Traité de la Lumière
With the solution of the problem of strange refraction, nothing stood in the
way of elaborating the ‘Projet’ of 1672. Yet, it lasted more than ten years
before Huygens put his wave theory to print. In the preface of the eventual
Traité de la Lumière, he mentioned three reasons for the delay:
“One may ask why I have tarried so much with publishing this work. The reason is that
I had written it rather negligently in the language in which one sees it, with the intention
to translate it into Latin, doing so in order to have more attention to things. Upon
which I planned to give it together with another treatise on dioptrics, where I explain
the effects of telescopes, and the other things that also belong to that science. But the
pleasure of the novelty being gone, I have gone on postponing the execution of this
Father Constantijn had given the estate Zeelhem - and probably the title too - to his son Constantijn in
1651. Christiaan did not have a title, but he bore ‘Lord of Zuylichem’, for example on the title page of
Horologium Oscillatorium. After their father’s death, Constantijn inherited the house at the ‘Plein’ in The
Hague and the title ‘Lord of Zuylichem’, while Christiaan now became a ‘real’ lord, of Zeelhem. Keesing,
“Wanneer”, 63 and Keesing, Constantijn en Christiaan, 112-113.
plan, and I do not know when I would have been able to put this to order, being often
diverted either by things to do or by some new study. ”2
It was true, many things had distracted him since his ‘Eureka’s’. In the
summer of 1678, Hartsoeker and Leeuwenhoek had kindled his interest in
microscopes and microscopical observation. He devised several
Leeuwenhoek-style simple microscopes himself and made some technical
improvements and additions.3 In 1680, he published a design for a telescopeenhanced level, followed by disputes which lasted for years.4 Around the
same time he studied the properties and nature of magnetism. Back in
Holland, Huygens designed his planetarium and continued working on his
pendulum clock and its application at sea. With Constantijn he
recommenced the grinding of lenses and manufacture of telescopes, earning
fame for their skills and creating demand for their products.5 They built a
grinding lathe and Huygens published the description of an aerial telescope
of his design, Astroscopia Compendaria (1684). Around the same time the
brothers were also discussing a treatise on the grinding of lenses.6 So, even if
the list is confined to optical matters there was enough to divert his attention
from his treatise in dioptrics. Moreover, Huygens somewhat lost interest in
his theory, witness his response to Leibniz’ inquiries after his wave theory.
During the 1680s Leibniz repeatedly asked him his opinion on the nature of
light and refraction, but Huygens proved rather reluctant to discuss the
details of his wave theory. He replied Leibniz’ questions were often much
later and when he did he was rather succinct and shallow.7
The delay was not only effected because other things caught Huygens’
eye. His indecisiveness as regards the final format of his optics, to which he
alluded in the first sentences quoted above, also played a part; a substantial
part in my view. Although the preface does not say so, Traité de la Lumière
had been intended as part of a treatise on dioptrics, rather than an
accompanying discours. Until Traité de la Lumière went into print, Huygens
had maintained his original plan of a ‘Dioptrique’ of which his theory of light
was an integral part. When he presented it to the Académie, he had upgraded
his wave theory to form a separate part, but still the ‘first part’ of his
‘Dioptrique’. Now, in 1690, Huygens mentioned ‘Dioptrique’ as a separate
Traité, ‘Preface’, [1]. “On pourra demander pourquoy j’ay tant tardé à mettre au jour cet Ouvrage. La
raison est que je l’avois escrit assez negligement en la Langue où on le voit, avec intention de le traduire en
Latin, faisant ainsi pour avoir plus d’attention aux choses. Apres quoy je me proposois de le donner
ensemble avec un autre Traité de Dioptrique, ou j’explique les effets des Telescopes, & ce qui apartient de
plus à cette Science. Mais le plaisir de la nouveauté ayant cessé, j’ay differé de temps à autre d’executer ce
dessein, & je ne sçay pas quand j’aurois encore pû en venir à bout, estant souvent diverti, ou par des
affaires, ou par quelque nouvelle étude.”
For example: OC8, 112-113. For details on his observations: Fournier, “Huygens’ observations”.
OC8, 263-266; 273-276, and further.
The quality and distribution of their lenses is recorded in Van Helden & van Gent, The Huygenscollection
and Van Helden and van Gent, “Lens production”.
For example: OC8, 432-435; OC9, 8; 25.
See OC8, 244-245; 250-251; 256-257; 267; OC9, 259.
work not inherently connected to Traité de la Lumière. Still, the two parts of
his ‘Dioptrique’ had only gradually drifted apart between 1672 and 1690.8
Below the surface the two parts were of a fundamentally different nature.
The innovative character of his eventual theory of waves had broken it loose
from the geometrical optics of lenses and Huygens only gradually came
aware of the gulf that had formed between both parts. His allusion to a Latin
translation of Traité de la Lumière underscores the close tie he remained to see
between the two parts of his ‘Dioptrique’.
The text of the eventual Traité de la Lumière had been ready for by far the
greater part in 1678. The events of August 1679 necessitated some
corrections and additions, but these could be inserted into the existing text,
as indeed they were. Neither the theory as such, nor the main line of his
argument was affected by these changes. What, then, needed in view of the
‘Projet’ to be done about the second part of ‘Dioptrique’? Huygens decided
to consider the content of the dioptrical part anew. The content of Tractatus
still had to be rearranged to begin with and a final text elaborated. On and
off, Huygens worked on this during the 1680s, making some tables of
content, writing some prefaces, and composing parts of the theory.
Huygens’ dioptrics in the 1680s
In the meantime, Huygens added new dioptrical studies. The recent
developments, and his own involvement, in microscopy made Huygens
decide to include a discussion of the dioptrical properties of these
instruments, too. To this particular topic he had assigned only historical
importance in 1672.9 The application of his theory to microscopes is similar
to his treatment of telescopes, so I will not discuss much detail.10 The most
interesting, and somewhat unexpected, dioptrical addition was his
investigation of lens aberrations. Although it had been discarded with the
‘Projet’, Huygens made a fresh start with his ill-fated theory of spherical
aberration and combined it with a mathematical analysis of chromatic
Due to the 1672 debacle, Huygens now set the aims of his dioptrical
theory considerably lower. He had given up hope of neutralizing aberrations.
He set full focus on the mathematical understanding of the quality of images
and on deriving guidelines for improving it. This was a continuation of the
table that had concluded De Aberratione, in which he listed the optimal
configurations of Keplerian telescopes. Probably by early 1684 Huygens
began to examine these anew. He remarked that his earlier studies were
useless as they presupposed that spherical aberration was the key factor in
Yoder has suggested that, even after the publication of Traité, Huygens intended to attach it to the
second part of ‘Dioptrique’. Yoder, “Archives”, 106..
OC13, 748-749.
Most of it can be found in OC13, 512-585.
the limited quality of telescope images. He now realized that the effect of
spherical aberration was small as compared to the aberration “… that arises
from the Newtonian dispersion of rays.”11 He set out to make a new table of
optimal configurations, which would also take the ‘Newtonian’ aberration
into account.12
Huygens explained the difficulty with telescopes in the same way he had
done in 1665: increasing the magnification renders images fuzzy and
obscure.13 The problem was how to increase the power of a telescope whilst
maintaining the clarity and distinctness of images. This came down to
determining the aperture of the objective lens in proportion to the aperture
of a given telescope of good optical qualities.14 Huygens first determined the
amount of chromatic aberration of a lens relative to its focal distance. He
more or less repeated what he had written to Newton in 1672. Huygens had
argued that the ratio between the aberration and the focal distance was
1 : 25, whereas Newton had used 1 : 50.15 This meant that chromatic
aberration exceeded spherical aberration 39 times and would imply that it
was superfluous to take spherical aberration into account when dealing with
the quality of images.16 Huygens explained that in reality things were not as
bad as these figures suggested. Repeating further arguments from his dispute
with Newton, he said that many of the dispersed rays were not perceptible.
Therefore lenses did produce images that were sufficiently distinct, although
they might be surrounded by a faint ‘nebula’.17
First, Huygens considered the chromatic aberration NM, produced on the
retina by a telescope consisting of two convex lenses AC and DP (Figure 78).
The axis of the system is TPC, F is the focus of the red rays refracted by the
objective lens AC and B the focus of the violet rays. In this type of telescope
the foci of objective and ocular lens should coincide, but the focal distances
of the various colors differ. Huygens assumed the foci of the red rays to
coincide. F is also the focus of the red rays for the ocular PD, G is the focus
of the violet rays. Consequently, red rays will be refracted along AFO,
towards LK parallel to the axis, and point N, where the axis intersects with
the retina. Next, Huygens considered the path of the violet rays. These are
refracted by the objective lens towards ABD. As G is the focus of the violet
rays for the ocular, a ray GD will be refracted towards DE and N on the
retina. Ray ABD is not refracted towards DE, however, but towards DK and
thus reaches the retina in M. Consequently NM is the aberration produced by
the system. Because – by a small angle approximation – angle NKM is equal
OC13, 621. “… ex dispersione radij Newtoniana.”
OC13, 496-499.
OC13, 480.
OC13, 482. Compare section 3.2.1 on De aberratione below.
OC13, 484-487 and 485 note 8. The manuscript is confusing as Huygens first derived his own figure of
1 : 25 but used Newton’s figure of 1 : 50 when he later inserted the numbers into the text.
According to his own figure of 1 : 25 this should be 79.
OC13, 486-487.
to angle DKL and EDK equal to BDG,
it follows that BDG fixes the
aberration at the back of the eye.18
Next, Huygens queried how the
length of the telescope can be
increased while maintaining the
degree of aberration, that is: keeping
the angle BDG constant. A second
telescope consists of objective lens ac,
ocular pd and focus f. A
straightforward derivation yields the
conditions for the aberration to be
equal in both telescopes. The
apertures need to be in proportion to
the square roots of the focal distance
of the objective: ac : AC
cf : CF .
Consequently, the focal distances of
the ocular are in proportion to the
apertures: fp : FP = ac : AC.19 From this
he derived a rule to determine the
aperture and the ocular when the
objective lens is given. Given a
satisfactory 30-foot telescope the rule
is as follows. Multiply the focal
distance of the objective lens
(measured in inches) by 3000; the
square root of this figure gives the
diameter of the aperture (in
hundredths of inches); adding one
tenth yields the focal distance of the
ocular (in hundredths of inches).20 It
magnification by the system is in
proportion to the aperture. The table
listing the appropriate configurations
found in the manuscript was,
however, calculated by a simpler rule
where the focal distance is equal to
the diameter of the aperture.21 Thus
Huygens revised and updated De
Figure 78 Chromatic aberration
OC13, 488-491.
OC13, 492-493.
OC13, 494-495.
It was this simpler rule he communicated to his brother on 23 April 1685. OC9, 6-7.
Aberratione, now taking into account the ‘newtonian’ aberration as well.
Huygens applied his account of chromatic aberration to microscopes. In
the first place he considered simple microscopes in which only the aperture
could be adjusted but loss of clarity could not be prevented.22 With
compound microscopes the application of the theorem had the surprising
outcome that the aperture could be increased without loss of clarity and
distinctness.23 This compelled Huygens to take spherical aberration into
account as well. Calculating the degree of both aberrations in a specific
microscope, he came to the conclusion that in this case the spherical
aberration exceeds the chromatic.24 This being established, he could apply his
theory of spherical aberration in order to prescribe the optimal
configurations in a compound microscope.25 Although of limited use for
telescopes, Huygens’ theory of spherical aberration was still of some avail for
the improvement of microscopes. It does not seem that he put his findings
to practice. His own microscopic observations date back to the late 1670s,
when he had used simple microscopes like Leeuwenhoek’s.
Without professing to do full justice to Huygens’ dioptrical studies of the
1680s, I leave it at this. It suffices to make clear that they remained in line
with his work prior to 1672, both qua content and qua character. Huygens
maintained his focus on instruments, which now included microscopes as
well. Huygens seems to have learned from the debacle of De Aberratione. In
his discussion of the effect of aberrations on the quality of images he was
more perceptive of the relative nature of mathematical results. Having
demonstrated how clarity of images relates to the dimensions of telescopes,
he remarked that his results did not fully agree with experience, in particular
when it came to observing faint objects.26 He then came to a discussion of
dimensions actually employed, elaborating on their justification in a fairly
qualitative way and appealing to his personal experience.27
From time to time during the 1680s, Huygens made sketches for a revised
‘Dioptrique’. The earliest can be dated in or after 1682.28 It contains, among
other things, two sketches of a transitional paragraph between part one “…
on the physical causes of the rules that light observes, …” and part two
containing “… the explanation of the effects of glass lenses …”29 The
historical sketch originally planned as the first chapter in the ‘Projet’ would
OC13, 530-535.
OC13, 542-543.
OC13, 564-565.
OC13, 576-585.
OC13, 502-503.
OC13, 502-509.
OC13, 745n11.
OC13, 747. “… des causes physiques des regles qu’observe la lumiere, …”; “… l’explication des effects
des lentilles de verre …”
now open the second part of ‘Dioptrique’. In the adjusted plan, this part
would be a trimmed down version of Tractatus. Huygens remarked that he
had written his theory of lenses a long time ago and that some of its content
had been dealt with since by others. He probably had Barrow’s lectures in
mind. Matters like these he would sketch only briefly and then present the
most important theorems of Tractatus. The principal topic of the second part
of ‘Dioptrique’ was magnification, a topic that according to Huygens still
passed without proper treatment in existing literature.30
Huygens’ dioptrical studies of the 1680s have been gathered by the
editors of the Oeuvres Complètes in a separate section in OC13, called
“Dioptrica. Pars Tertia”. For the large part it consists of an essay labeled ‘De
Telescopiis’. Dating it is hazardous, but it appears that the latest version is of
1692. Material with varying dates – ranging from the 1660s to the 1690s – is
collected and alternative versions of several parts can be found in
appendices. Huygens probably started assembling and elaborating this in
1684 or 1685.31 In or after 1684, he made another outline for the second part
of ‘Dioptrique’, listing the order of subjects treated.32 This outline
corresponds for the most part with ‘De Telescopiis’. The essay cannot have
been intended as a definite second part of ‘Dioptrique’, though. At several
places Huygens referred to theorems of Tractatus. If ‘De Telescopiis’ was
eventually intended to replace Tractatus entirely, Huygens still would have to
find a way to integrate these theorems.
At any rate, Huygens did not round off his new plans. In 1687 he
considered the state of his ‘Dioptrique’ once again. He still considered his
theory of waves and his theory of dioptrics as two parts of a single work. He
doubted, however, whether ‘Dioptrique’ was still an appropriate title. A note
gives a new title: ‘Optique. I partie.’33 In addition, the problem of languages
remained. He made a start with a Latin translation of the first part under the
title ‘Versio Diatribæ de Luce’ but got no further than some 10 pages.34 At
the beginning of 1690, Traité de la Lumière was published as a autonomous
treatise. It did not, as we have seen, reveal that it had been intended as the
first part of a larger work. Still, it was not fully separated from the
‘Dioptrique’. At the time the Traité de la Lumière was being printed, Huygens
began a French translation of his dioptrics: “Beginning of the treatise on my
dioptrics in French that I planned to join with the treatise on light, …”35 The
decision to treat them as separate treatises had been made just before. The
opening had at first read: “Beginning of my second part of the ‘Dioptrique’
OC13, 746-748.
OC13, 434-511. See notes 1 and 2 of pages 434-435 on the dating.
OC13, 750-752.
OC13, 754.
OC19, 458-470. He probably did not start translation before May 1687: OC9, 133.
OC13, 754; 755-770. “Commencement du Traitè de ma Dioptrique en François que j’avois dessein de
joindre au Traitè de la Lumière, …”
in French to join it with the first which is in this same language. This plan
has changed for it will remain in Latin.”36
After the publication of Traité de la Lumière, Huygens continued to work
on the dioptrical treatise. In 1692, his plans changed once again after he had
read Molyneux’ Dioptrica Nova. Huygens’ notes on Dioptrica Nova survive.37
Molyneux had treated telescopes better than anyone before, but had “little of
the things my treatise contains on this matter”38 Huygens knew that he was a
better mathematician than the Irishman, and realized that he had still
something valuable to present in dioptrics. Some time after his reading of
Dioptrica Nova, he made a new (also the last) outline for his dioptrics: “De
Ordine in Dioptricis nostris servando”39 He would leave out what Molyneux
had treated and emphasize his own strong points: the theories of spherical
aberration and magnification. He gave particular attention to his theorem on
the magnification produced by a given system of two lenses, the lens-formula
as it is called nowadays.40 Huygens would prove it (instead of just stating it, as
Molyneux had done) and extend it to more complex systems.
‘De Ordine’ brought together scattered material ranging from the 1650s
to the early 1690s. In accordance with this scheme, Huygens ordered his
manuscripts and numbered the pages in red.41 Roughly the set runs as
follows: the first part of Tractatus, with De Aberratione (without the rejected
parts) inserted after proposition twenty, propositions one and two of part
three of Tractatus, part two of Tractatus, various fragments of ‘De Telescopiis’.
It makes it clear that in the course of 40 years Huygens’ views on what
dioptrics was about had not changed: it was about telescopes. It had to
account for the working and improvement of telescopes mathematically.
Huygens did not live to see his dioptrical treatise through the press. In his
will he instructed De Volder and Fullenius to look over his “mathematical
writings” and “to edit as best they can whatever in it might be fit to
publish”42 He explicitly named the ‘Dioptrika’ and three other treatises. De
Volder and Fullenius followed Huygens’ ordering pretty closely, so that the
1704 edition gives a good indication of his final idea of Dioptrica.43
OC13, 754n4. “Commencement de ma seconde partie de la Dioptrique en francois pour la joindre a la
Première qui est en cette mesme langue. Ce dessein est changè car elle demeurera en Latin.”
OC13, 826-844.
OC10, 279. “… peu de ce que contient mon Traitè sur cette matiere.”
OC13, 770-778.
OC13, 773-774. The original version was: OC13, 186-197 and is treated in section 0.
Hug29, 101bis-205.
OC22, 775-776. “schriften van Mathematique” and “… ‘tgeen daerin soude mogen weesen bequaem
om gepubliceert te werden, hetselve willen besorgen ten besten sij sullen connen, …”
The editors of the Oeuvres Complètes chose not to follow the final ordering, but have attempted to make a
chronological reconstruction of Huygens’ dioptrical papers.
The publication of Traité de la Lumière
Why did Huygens not publish his dioptrics himself, given the quite
publishable state he left it in?44 After the unceasing postponements of the
previous decades, with none of his plans – beginning with that of 1652 –
completely executed, it does not really come as a surprize he did not finish
‘De Ordine’. Still, this leaves the question why he never went public with his
dear dioptrics. Huygens’ general tardiness of publishing is often pointed out,
suggesting some psychological factor for failing to publis his important
discoveries and inventions. However, there may well be cultural factors in
play as well. I will not elaborate this, except for advancing some questions.
Most important, in my view, is reversing the question that opened this
paragraph. Rather than asking why Huygens did not publish what from our
modern point of view was well worth publishing, we should ask why he
published what he did? What credit could a savant like Huygens have gained
by publishing? To answer questions like this, his social position needs to be
taken into account. Was an ‘aristocrat’ like Huygens in the position to
publish whatever he liked at the moment that suited him, or did he need to
observe specific forms? Were books and articles the obvious means for
disseminate one’s ideas, or would someone like him prefer letters to wellchosen peers? In order to explain Huygens’ tardiness in publishing Dioptrica
(and other texts), his publication pattern ought to be surveyed. This would
account for the swiftness he published his early mathematical works with
around 1650, as well as the haste with which he usually applied for patents.
So, we let the question why Huygens did not publish his dioptrics rest and
turn to the more important question why, in the end, he did publish the first
part of his ‘Dioptrique’, the wave theory and his explanation of strange
A direct incentive to publish Traité de la Lumière may have been plans at
the Académie to publish papers of its (former) members.45 On 8 September
1686 Huygens received a letter from De la Hire asking his permission to
publish some of his manuscripts kept in Paris.46 Huygens did not hesitate to
list some interesting treatises, but at first no mention was made of his theory
of light.47 In his letter of 20 April 1687 De la Hire started to inquire about the
state of affairs concerning Huygens’ treatise of dioptrics.48 Huygens answered
that it was almost ready; at least the part on “… physics, Iceland crystal
etc.”.49 The following letter makes it clear that Huygens’ explanation of
strange refraction was well remembered in Paris but also that it was not fully
Yoder, “Archives”, 91-92.
Divers Ouvrages de Mathématique et de Physique. Par Messieurs de l’Academie Royale des Sciences was published in
1693, containing eight papers by Huygens.
OC9, 91.
OC9, 95-95.
OC9, 129.
OC9, 133. “… la Physique, le Cristal d’Islande &c.”
understood.50 This interest displayed in his theory, combined with an
apparent ignorance of its contents, may have influenced the decision to
publish Traité de la Lumière.
In his preface to Traité de la Lumière, Huygens alluded to another reason.
During the 1680s, Leibniz and Newton had published on anaclastic curves.
Huygens had first heard of Newton’s derivation of anaclastic surfaces from
Fatio de Duillier in June 1687.51 Fatio, who had visited Huygens at the end of
the preceding year, wrote him about the excitement among the members of
the Royal Society over Newton’s forthcoming Principia. According to Fatio,
Newton’s method for finding anaclastic surfaces accorded with Huygens’, in
that he assumed that each ray travels in the same time from one focus to the
other, although he employed a different principle.52 Huygens, in a letter of 11
July 1687, responded that he did not see how the same conclusion could be
reached from a different assumption.53 He did not return to the matter in his
correspondence. When Principia was published, he would discover that
Newton assumed bodies instead of waves to travel in equal times. Leibniz, in
the meantime, had also heard of the Principia (he claimed only to have read
the review in Acta eruditorum at that point) and acted appropriately. For
reasons of priority, he sent three articles to the Acta, including one on
anaclastic surfaces. In chapter 6 of Traité de la Lumière, Huygens presented his
own derivation on the basis of his wave theory. In addition, this chapter
contained his determination of caustics by means of his wave theory. It was
an elaboration of the notes of 1677 discussed in section 5.1.1.
Some hold that Newton’s Principia was the main incentive for Huygens to
publish Traité de la Lumière together with his theory of gravity, Discours de la
Cause de la Pesanteur. After he had read the Principia, Huygens wrote his
brother Constantijn in London that he was impressed and would like to
come to Cambridge only to meet Newton.54 In the summer of 1689, he went
to England and met Newton at the Royal Society. Huygens spoke about his
theories of light and gravity; Newton is reported to have discussed, out of all
possible subjects, strange refraction.55 Unfortunately, no records remain but
it is likely that the two men did not fully agree. In Principia, Newton had
asserted that a wave-like motion diverts after passing through an aperture.
This implied that waves could not explain the rectilinearity of rays.56 In
Discours de la Cause de la Pesanteur – a treatise on gravity published together
with Traité de la Lumière – Huygens retarded that waves do spread around
OC9, 164.
OC9, 167-171.
Newton, Principia, 626-628 (Propositions 97 and 98 of section XIV of book I). I do not precisely know
what and how Fatio knew of Huygens’ ideas.
OC9, 190.
OC9, 305.
Westfall, Never at rest, 488; Shapiro, “Pursuing and eschewing hypotheses”, 223.
Newton, Principia, 762-767. For an extensive discussion see: Shapiro, “Light, pressure”, 284-291.
corners but that these dispersing waves are too weak to produce light.57 This
argument was, of course, elaborated in the first chapter of Traité de la
Lumière.58 At the beginning of 1690, Traité de la Lumière was printed and he
sent some copies to England.59 Huygens wrote Fatio on 7 February 1690 that
he was anxious to know what Newton thought of his “… explanations of
refraction and of the phenomena of Iceland crystal, but I am not quite sure
whether he understands French, …”60 He would not live to see Newton
entirely reject the idea of light waves, and ignore his explanation of strange
refraction, in the queries appended to Opticks.
With Traité de la Lumière now published, we may ask what Huygens
thought he had published. The hesitations about titles and languages suggests
that Huygens was not sure anymore of the status of his wave theory. Did it
still belong to his ‘Dioptrique’? He had written it in French, the language
used at the Académie. But French was also the language of the particular
topics discoursed of there, issues in physics and so on. Latin, on the other
hand, was the language he used for mathematical topics.61 His doubts about
translating it might also indicate that he was doubting whether it still
belonged to the mathematical science of optics. At the very last moment
Huygens decided to publish it as an independent treatise in French, under
the title Traité de la Lumière. In so doing, he cut through the umbilical cord of
his wave theory, thus withdrawing it from the realm of mathematics.
Huygens had put, so to speak, his theory of light in the milieu of the Paris
Académie and the post-Cartesian debates proliferating there. Through the
decoupling of Traité de la Lumière from ‘Dioptrique’ he focused attention on
the wave theory, that is, on the explanation of the laws of optics instead of
their application to the behavior of rays. The title he chose, Traité de la
Lumière instead of Dioptrique, suggests that the treatise had switched from a
correction of La Dioptrique to a rebuttal of Le Monde. However, its stated aim
was much more modest:
OC21, 475.
Cohen argues that one of the reasons Huygens was urged to publish Traité de la Lumière was to guarantee
his priority regarding the theory of wave propagation. Cohen, “Missing author”, 32.
Cohen explains that a series of Traité de la Lumière exist that was issued by the publisher Vander Aa with
the author’s name spelled out on the title page: “Par Monsieur Christian Huygens, seigneur de Zeelhem.”
The other copies of Traité de la Lumière, including the large-size edition Huygens also printed to distribute
among acquaintances, only have the author’s initials on the title page: “Par C.H.D.Z.”. Cohen assumes
that the title page was altered just after printing had begun. The reason may have been that Huygens had
used the title ‘Lord of Zuylichem’ in earlier publications and wanted to prevent confusion by used a
neutral ‘D.Z.’. Since the death of their father, Constantijn formally was ‘Lord of Zuylichem’. Cohen,
“Missing author”, 33-35.
OC9, 358. “… Explications de la Refraction et des phenomènes du cristal d’Islande, mais je ne suis pas
bien assurè s’il entend le François, …”
After his move to Paris in 1666 Huygens began writing more and more in French, with the exception
however of his works on mathematics. Illustrative are his 1672 notes on strange refraction, where – as we
have seen above – he switched between both languages accordingly. Huygens’ use of languages
corresponds with what seems to be a general pattern. Halfway the seventeenth century, the vernacular had
begun to replace Latin in scholarly writings, especially in France and England. The notable exception are
mathematical texts, which remained to be written in Latin well into the nineteenth century. A systematic
study of the use of languages in scholarly writings would be well worth pursuing.
“Where are explained the causes of what happens to it in reflection and in refraction.
And particularly in the strange refraction of Iceland crystal.”62
Huygens did not intend to expound the principles of natural philosophy nor
the methodology of mathematical science. The subtitle of Traité was not
something like ‘Treatise of light, where the nature and all properties of light
are wonderfully explained in a clear and most probable way according to the
true philosophy’. He expounded natural philosophical principles insofar as
these want to explain the mathematical laws of optics. He considered the
principle of wave propagation – this ‘principal foundation’ – his main
achievement. The validity of his principle was based upon, and implicitly
confined to, the successful derivation of those laws. Traité de la Lumière
offered an example of the proper use of mechanistic philosophy.
Traité de la Lumière did indeed offer better explanation – more plausible,
more probable – but we value it for its epistemic innovations. The wave
theory had originally been planned as a non-committal, explanatory
introduction to his mathematical theory of dioptrics. The eventual outcome
really exceeded geometrical optics. The few methodological issues Huygens
raised, were passed over as a matter of course. In other words, he does not
seem to have been aware of the epistemically innovative character of the
wave theory. Too modestly, from our point of view, Huygens presented his
wave theory as a better explanation of the laws of optics, instead of a new
way of doing the mathematical science of optics.
6.2 Traité de la Lumière and the advent of physical optics
With Traité de la Lumière, Huygens created a new kind of optics, an instance
of what we would call physical optics. I have argued that his actual interest in
optics was the dioptrics of telescopes but that the phenomenon of strange
refraction rather coincidentally directed him to questions pertaining to the
mechanistic nature of light, which he subsequently subjected to the rigorous
mathematical treatment of his dioptrics. This account of its historical
development sheds new light upon our understanding of Huygens’ science,
as I will argue in the next section. This section deals with lines of
interpretation Huygens’ case suggests for the history of seventeenth-century
optics. I will therefore broaden the outlook of my discussion and see how
the themes in my account of Huygens’ optics may be generalized. I do not
profess to offer a new history of seventeenth-century, rather I want to
suggest some lines of interpretation that I consider important for our
historical understanding of the origin physical optics. A major point of
reference will be, of course, Newton, who created his own particular instance
of a physical optics. Despite fundamental differences in their outlook,
intentions and activities, there are important parallels between the optics of
Huygens and Newton.
Traité de la Lumière, title-page. “Où sont expliquées les causes de ce qui luy arrive dans la reflexion, &
dans la refraction. Et particulierement dans l’etrange refraction du cristal d’Islande.”
Historical studies of seventeenth-century optics tend to focus on the
development of theories of light. They do so with good reason, for it is in
the changing thinking over the nature of light and its properties that the
development of optics exemplifies the transformation of science known as
the scientific revolution. However, such a focus presupposes some kind of
coherence and continuity in the pursuits of thinkers on matters optical. In
my view the study of phenomena of light in the seventeenth century was a
rather heteroneous affair. More specifically, I think that historians have
tended to overlook the fact that something akin to physical optics did not yet
exist before Newton’s and Huygens’s work.63 The part of exact science that is
organized around the question ‘what is light and how does this explain its
properties?’ did not come about until they created and established it. The
one synthetic study of seventeenth-century optics, Sabra's incomparable
Theories of Light from Descartes to Newton, is telling in this respect.64 It is
structured around the conceptualizations of the physical nature of light and
the way seventeenth-century students of optics employed these to explain
the properties of light. Likewise, most monographic studies treat
seventeenth-century optics qua physical optics. Yet, the issues basic to
modern physical optics did not guide the development of seventeenthcentury optics but were gradually recognized as fundamental to the science
of optics through the pursuits of Newton and Huygens.
For a historical understanding of their optics, and of the development of
seventeenth-century optics in general, its origins need to be taken into
account. The historian should turn his eyes away from their future
achievements and look back to the mathematical science of optics as they
had encountered it. In this regard, traditional geometrical optics is historically
significant. It formed a prominent disciplinary, conceptual, and
methodological context for the pursuits of seventeenth-century students of
optics, and Huygens and Newton in particular. In contrast to later physical
optics, this science was organized around the questions ‘what are the
properties of rays in their interaction with various media and how can their
behavior be deduced geometrically?’, questions that were directional in
seventeenth-century optics and, as such, have left their mark on Huygens’
and Newton’s pursuits as well.
That does not mean that geometrical optics was the sole root of early
modern optics. The emerging new philosophies of nature were equally
significant, interacting with the established mathematical sciences and
regauging their basic principles. Yet, existing literature considers
seventeenth-century optics primarily within the context of these philosophies
Cantor points this out but does not elaborate this theme. Cantor, “Physical optics”, 627-628 in
particular. Cohen does point out the novelty of the content of Traité de la Lumière and Opticks compared to
traditional geometrical optics, but he neither elaborates it further; Cohen, “Missing author”, 30-32.
A.I. Sabra, Theories of Light, 11-15 in particular. I call it incomparable in the way Leibniz called Huygens
an ‘incomparable man’ at the news of his death (Acta eruditorum August 1695, 369): although he regretted
Huygens’ lack of interest in metaphysical issues, he greatly admired his scientific abilities.
and underrate, in my view, the historical significance of geometrical optics.
Shapiro’ studies of the development of Newton’s optics offer a favorable
exception. He has analysed Newton’s theory of colors as a failed effort to
establish a mathematical science of colors after the model of geometrical
optics.65 His work has been very inspiring and instructive in developing my
ideas on the historical significance of geometrical optics. Taking into account
the influence of geometrical optics does not, to be sure, yield a radically new
historical picture, but I do think that it can further illuminate the
developments of seventeenth-century theories of light.
Mathematization by extending mathematics
Unlike Huygens, Newton was explicit about the new ground he was
breaking. The lectures on optics he delivered from 1670 on as newly
appointed professor at Cambridge, were the first occasion where he
presented his new ideas in optics. The core of it is formed by the theory of
different refrangibility which is integrated into an elaborate discourse in
geometrical optics. Laboriously, he accounted for his digression into the
origins of colors, a topic that traditionally belonged to philosophy rather than
“But lest I seem to have exceeded the bounds of my position while I undertake to treat
the nature of colors, which are thought not to pertain to mathematics, it will not be
useless if I again recall the reason for this pursuit. The relation between the properties
of refractions and those of colors is certainly so great that they cannot be explained
separately. Whoever wishes to investigate either one properly must necessarily
investigate the other. Moreover, if I were not discussing refractions, my investigation of
them would not then be responsible for my undertaking to explain colors; nevertheless
the generation of colors includes so much geometry, and the understanding of colors is
supported by so much evidence, that for their sake I can thus attempt to extend the
bounds of mathematics somewhat, just as astronomy, geography, navigation, optics,
and mechanics are truly considered mathematical sciences even if they deal with
physical things: the heavens, earth, seas, light, and local motion. Thus although colors
may belong to physics, the science of them must nevertheless be considered
mathematical, insofar as they are treated by mathematical reasoning.”66
Newton here calls his mathematization of prismatic colors an extension of
the bounds of mathematics. He justifies it by comparing it to the fields of
mixed mathematics where physical things were traditionally treated
From a historical point of view these are telling words. They provide
contemporary reflections on what it meant to subdue new domains of
natural inquiry to mathematical treatment in early modern science. Newton’s
words indicate that mathematization is not a mere application of cut-anddried concepts and methods plucked from some mathematical air, but that
existing fields of mathematical sciences provided the context for it and
formed a steppingstone to treat new domains mathematically. The explicit
methodological awareness Newton displays here, is vainly searched for in
In particular his “Experiment and mathematics” and “Dispersion law”
Newton, Optical papers 1, 88-87 & 438-439.
Traité de la Lumière. For Huygens it went without saying to apply mathematics
to matter in ‘Physique’ just like it was done ‘Optique’, in spite of the very
different nature of this was matter.
Besides the difference in tone in the way Huygens and Newton made
public their achievements, there is, of course, a big difference in the
character of their pursuits. Each created a form of physical optics by
mathematizing new domains of light, but of an essentially different nature.
Newton mathematized a new range of optical phenomena (which Huygens
did not); Huygens extended mathematics into the new kind of realm of the
unobservable, hypothetical nature of light (what Newton could do equally
well but only did so privately). In each case, traditional geometrical optics
was a major starting point, yet embedded in different natural philosophical
contexts and problem definition. Huygens primarily responded to the issues
raised by mechanistic philosophy, continuing along the lines of mathematical
optics that run from Alhacen over Kepler to Descartes. The roots of
Newton’s optics were more diverse. As much, if not more, as his optics was
guided by his proficiency in and commitment to mathematical science, it was
informed by his quest for the true nature of matter. In addition to
mathematical optics, it built on the teachings of experimental philosophy and
questions of matter theory articulated by Aristotle, Descartes and Gassendi,
and Boyle.
Despite these differences, the development of their pursuits show
conspicuous similarities. Huygens and Newton came to their novel ways of
doing optics only after they had moved beyond the confines of traditional
geometrical optics. At an early stage, each worked much closer to the tenets
of traditional geometrical optics than there final theories suggest. In section
4.2, I have shown that, in first attack on strange refraction in 1672, Huygens
approached the phenomenon in a traditional way aimed at establishing the
properties of rays interacting with Iceland crystal. Only at a later stage did he
focus his attention on the mechanics of waves involved. Newton likewise
formulated a law of dispersion in his Optical Lectures that defined additional
properties of rays to mathematically account for the amount of dispersion.
In these lectures he also allowed himself an epistemological freedom of
solely providing a rational foundation that can only be understood in the
context of geometrical optics.67 In addition, both employed a similar strategy
in their early efforts to fathom the mathematical regularities of the two
phenomena of strange refraction and color dispersion that challenged the
universality of the newly discovered law of refraction. Both extended on
Descartes’ analysis of ordinary refraction adding some extra component to
the (now irregularly) refracted ray. These examples, which I have elaborated
in detail elsewhere, serve to show that mathematization also involves
transferring to new domains ideas and strategies from established fields of
See Shapiro, Fits, 24-26.
mathematics.68 However, by that time establishing the properties of rays was
not a straightforward a matter as it had used to be in traditional geometrical
optics. With the rise of corpuscular thinking the ray no longer was a selfevident physical concept. Properties of rays now needed further justification,
beyond the realm of visible phenomena or everyday experience. Huygens
and Newton recognized the full import of these new questions and were the
first to directly face them.
The matter of rays
Kepler can be said to have sharpened the question after the nature of rays
and their properties. In perspectivist accounts these were answered only
loosely, by an appeal to analogies between the motion of rays and that of
bodies. With his rigorously realist reading of mathematical description,
Kepler thought that the causes of rectilinearity, reflection and refraction
ought to be contained in their measure. In his theory, this implied
considering the interaction of incorporeal surfaces with the surfaces of
diverse media. In the case of refraction this indeed led to a quasi-physical
analysis of refraction on a microscopic level, as we have seen in section 4.1.2.
In Paralipomena, Kepler explicitly distinguished the mathematical ray from the
physical ray and, in a note on what he calls the ‘fourth kind of light’ meaning
light communicated by the interaction with bodies, he can be said to have
put the question after the physical nature of light propagation on the
agenda.69 He did so in the first place, however, by the general reorientation of
perspectiva into optics: from a theory of vision to a theory of the behavior
and properties of light.
As contrasted to his achievements in geometrical optics proper, however,
Kepler’s ideas on the physics of light were little referred to later. Besides the
Renaissance idiom of his thinking, the conduct of Descartes seems to have
blocked the view on Kepler. Not only did he conceal the inspiration he had
drawn from him, more importantly, he gave a radical twist to the
perspectivist-cum-keplerian understanding of the behavior of light rays.
Descartes’ mechanistic interpretation of perspectivist causal analyses of the
laws of optics, turned these into material interactions. By the same token a
good deal of traditional conceptualization was channeled into seventeenthcentury theories of light. Of old, geometrical optics had been geometry
applied to matter, the matter of light rays. Descartes now raised the question
of what matter these rays were and how this could explain their behavior.
Still, the question of the nature of the light ray no longer was a simple one.
Hobbes’ concept of a line of light indicates that the once natural
identification with a geometrical line no longer was valid. Questions arose
concerning the relationship between light and the geometrical line, whether it
somehow expressed the nature of light, or whether it was the route of the its
propagation, or merely an abstraction of some kind. Depending on one’s
Dijksterhuis, “Once Snel breaks down”.
Kepler, Paralipomena, 37 note (KGW2, 46) in particular; Kepler, Paralipomena, 35 and note (KGW2, 31)
specific conception of the corpuscular nature of light, questions like these
could be answered in a more or in a less straight-forward manner.
Broadly speaking, corpuscular conceptions of light can be divided in two:
emission and medium conceptions. That is, light is either propagated matter
or an action propagated through matter. The first maintained the primacy of
the light ray in a rather natural way. The rectilinearity of light rays seems to
follow directly from the view of a moving particle. The most prominent
exponent of the emission conception was Newton, who considered its
purport more thoroughly than anybody else. Apart from his explicit
refutation of medium conceptions - in particular waves - he carefully
considered his own understanding of light. At least in his early years he
thought of light in terms of atoms, but soon developed a precise definition
of a light ray that covered his emission conception without being dependent
on it, as well as carefully determining the relationship between a geometrical
ray and a physical ray, where a physical ray not necessarily is the
mathematical line of geometrical optics.70 When he later reconsidered the
papers and letters he had published in 1672 in the Philosophical Transactions, he
added a footnote where he once again went into the details of the question
whether light is a “body” or “the action of a body”.71
Medium conceptions of light marked a more decisive break with
traditional geometrical optics. A light ray came to be seen as the effect of
the propagation of light, not as its essence. This implied abandoning the idea
that a ray has much intrinsic physical significance, as Buchwald explains, and
he adds that few at that time were willing to do so.72 In a medium conception
rectilinearity requires explanation. Significantly, Descartes, who originally
formulated the idea that light consists of an action propagated without
transport of matter, evaded the question.73 The same can be said for Hooke
and Barrow who turned Hobbes’ pulse theory into what in essence was an
emission conception of moving rods. Huygens, who adopted his medium
conception in an almost a priori manner, took the problem seriously. In the
first chapter of Traité de la Lumière, he put great stock in demonstrating that,
and how, waves are capable of producing rectilinear light rays.
The choice for an emission or a medium conception determined the way
in which refraction could be conceptualized. In an emission conception, one
must account for the fact that a transition to a different medium results in an
instantaneous change of direction. The conception of refraction as a surface
phenomenon rooted in perspectivist analyses. Kepler (himself neither a
medium nor an emission theorist, to be sure) had explicated this by his
notion of surface density. In his final analysis of refraction, he tried to
analyze the interaction between the two-dimensional surfaces of light and
Shapiro, “Definition”, 206-208.
Cohen, “Missing author”, 23-26.
Buchwald, Rise, 5.
See also: Shapiro, “Light, pressure”, 254-260.
medium. The conceptualization of refraction as a surface phenomenon was
to culminate in proposition 14 of Principia, where it is articulated as an event
occurring at the boundary layer between two media. It subsequently to be
reformulated entirely in terms of rays and their properties in the proof in
In medium conceptions, refraction could be explained in a much more
straightforward way. The explanation reduced to accounting for the fact that
a change of velocity results in a change in the direction of propagation. In his
explanation of refraction, Descartes inserted this notion into a perspectivist
analysis of refraction. The result was a rather ambiguous account, as he
blended his medium conception of light with a surface conception of
refraction. He was the first to state the propagation of light in terms of
properties of the refracting medium. In the first assumption of his
derivation, he mathematized this insight. Yet, in his second assumption he
maintained the conception of refraction as a surface phenomenon by
attributing the constancy of action to the surface of the refracting medium.
In terms of the corpuscular nature of light, Descartes’ derivation thus
raised more problems than it solved, which his successors did not refrain
from pointing out. The mathematics of Descartes’ derivation, however,
made an indelible impression on seventeenth-century savants. By stating the
interaction between light and media in terms of rays and their actions, the
derivation gained a significance that went beyond refraction per se. It provided
a promising clue to seize all phenomena in which refraction was involved. Extending
Descartes’ diagrams is a strategy that recurred many times in course of the
seventeenth century. Bartholinus took this lead to fathom the behavior of
strangely refracted rays, and Huygens initially did so, too.
Newton in particular would always remain impressed with the cogency of
Descartes’ elegant proof.74 In Opticks, he preserved it while putting it on a
firmer (emission) foundation than La Dioptrique had done. In his search for a
law of dispersion Newton’s first proposal was an extension of Descartes’
derivation. Naturally, so one would say, as this would preserve the harmony
with monochromatic refraction and preserve the analysis of the
phenomenon in terms of rays. Likewise, when Newton turned his mind to
strange refraction in Opticks, he proposed – without justification – a
construction that added the irregular component of the refracted
perpendicular to the ordinary refraction of each ray.75 Indeed, the same
construction Huygens had proposed in 1672. So, even after he had dismissed
his ‘Cartesian’ dispersion law (see above 5.2.2), Newton remained confident
that a Cartesian analysis had broader significance for phenomena of
Shapiro, “Light, pressure”, 239-241.
Newton, Opticks, 356-357.
The mathematics of light
In their early optics, both Newton and Huygens employed strategies and
methods common in traditional geometrical optics, i.e. determining
properties of rays in order to account for phenomena of light mathematically
and thus establishing principles for the mathematical science of optics. Both
realized, however, that the new philosophies of nature had by then imposed
limitations to such an affair, that the ray was not a natural physical entity and
its properties needed supplemental accounting for. Besides the question what
precisely was the (corpuscular) nature of light – discussed in the previous
section – epistemic issues had arisen: how can the corpuscular nature of light
be understood mathematically, how do conceptions regarding the nature of
light relate to the laws of optics, what part do they have in the mathematical
science of optics?
Such questions were already shining through in Kepler’s account, but
Descartes made them manifest without, however, fully answering them. The
status of his models was ambiguous and the explanation problematic.
Moreover, La Dioptrique lacked an explicitly empirical foundation of the sine
law, and the derivation suggested that it was founded upon a priori
mechanistic principles. In this sense he put the mathematical science of
optics upside-down. Later students of geometrical optics were quick to
restore the primacy of empirically founded laws, among which the sine law
could now be counted as well. The validity of the laws of optics was
independent of the mechanisms underlying them. Few directly addressed
such questions regarding the physical nature of rays and light, be it in terms
of empirical validation or corpuscular explanations. The explanations of a
mathematician like Barrow were non-committal elucidations reminiscent of
the way analogies had functioned in perspectivist theory. In their further
optical studies, Huygens and Newton were the first to fully confront the
issues involved. The content of their pursuits differed greatly, but in each
case can be interpreted as dealing with the question what a new physical
basis of the mathematical science of optics ought to look like.
In his study of strange refraction Huygens moved, more or less unseen,
from a consideration of the properties of rays to the actual problem of the
properties of waves. In this domain of the hypothetical corpuscles and their
motions, his basic problem was to ascertain the mechanistic causes of the
laws of optics in a properly mathematical way. Newton expressly rejected
this kind of ‘Hypothetical Philosophy’. His goal was to establish a
mathematical science of colors on the precepts of ‘Experimental
Philosophy’, to propose and prove the properties of light by reason and
experiments.76 Newton’s use of the two pairs of ‘proposing and proving’ and
‘reason and experiments’ show the consciousness with which he pursued his
Newton used the terms ‘hypothetical’ and ‘experimental philosophy’ in a letter to Cotes in 1713, cited in
Shapiro, Fits, 16. The phrase ‘propose and prove the properties of light by reason and experiment’ is
derived from the opening lines of Opticks, Book I: Newton, Opticks, 1.
project. They express his unique combining of mathematical science and
experimental philosophy. In mathematical science, including Galileo’s ‘nuova
scienza’, experiment was used as a tool of verification of hypotheses and
theories. From the experimental philosophers, Newton adopted the heuristic
use of the experiment, to discover and explore new phenomena and
properties. Yet Newton looked upon his experiments with the eye of a
mathematician. He saw rays and he looked for laws and did so by measuring
and analyzing mathematically the outcomes of his experiments. In so doing
he extended geometrical optics to new properties of rays: colors.
In the Optical Lectures, Newton still confidently proposed properties of
light by reason, but he soon qualified his statements. In particular after the
disputes over the ‘New Theory’, he distinguished the certainty of
mathematical demonstration from the conditional certainty of experimental
conclusions.77 Explaining his view on the certainty of mathematical science to
Hooke in 1672, he wrote “… the absolute certainty of a Science cannot
exceed the certainty of its Principles”. And in optics these principles were
physical.78 In his view different refrangibility was an experimentally proven
property of rays and the true cause of the appearance of colors, and he was
trying to convince Hooke of it. Newton maintained the conceptual primacy
of the light ray in optics, thus ensuring the connection of his theory of colors
with the mathematical science of optics. He did not take the physical
significance for granted, like traditional geometrical optics, but carefully
defined its mathematical and physical meaning respectively, and carefully
determined its properties experimentally. The theory of fits of Book 2 of
Opticks, in which he attributed periodicity to rays and that should account for
the colors in thin films, can be seen as the culmination of this project of
establishing a mathematical science of colors.
As such, the project had foundered, however. Newton had not been able
to establish the theory of unequal refrangibility as a mathematical science of
colors. The first step in finding the laws of colored rays was to establish a
one-to-one correspondence between the color of a ray and its index of
refraction. The next, crucial step for the science of colors to become
mathematical was to determine the regularity of the various indices by means
of a law of dispersion. Having dropped the ‘Cartesian’ dispersion law of the
Optical Lectures, in Opticks he resorted to a law whose validity, both by reason
and experiment, was unclear. This is only a symptom of the fact that Newton
never succeeded in elaborating the mathematical science of colors projected
in the lectures.79 The first book of Opticks is the direct descendant of the
Optical Lectures, but Newton had transformed his theory of unequal
refrangibility from a mathematical deduction into an experimental
exposition. In this he consolidated the presentation of the ‘New theory’.
Shapiro, Fits, 12-14.
Newton, Corrspondence 1, 187-8. Cited and discussed in Shapiro, Fits, 36-38.
See the quote above on page 227
Books 2 and 3 concern his later experimental investigations of the colors in
thin films and of diffraction. Although Newton’s mathematical perspective is
unmistakable in the concepts employed and the quantification effected, the
mathematical reasoning at the heart of his understanding of colors is implicit.
Opticks presented the new science of colors as an experimental theory.
The core of Newton’s optical investigations consisted of the search of
phenomenological laws and properties of light. He did speculate on the
corpuscular nature of light and color and their properties, but from the onset
he barred these from his established theories. In his view, experimental
philosophy should not be contaminated by hypotheses or other ill-founded
assumptions, as Descartes had done. This level of causation, distinguished
from the level of experimentally demonstrated properties, implied so-called
hypothetical philosophizing which Newton pursued publicly only once, in
his 1675 paper for the Royal Society ‘Hypothesis explaining the properties of
light’. Speculations on unobservable matter in motion, did play a part in
Newton’s optics however. One of the reasons he dropped the ‘Cartesian’
dispersion law seems to have been that it conflicted with his most private
thought on the corpuscular nature of colors. Yet, this remained concealed
from its readers.
To stress the lucidity of mathematics against the obscurity of natural
philosophy was rather a ‘topos’ in the seventeenth-century. Like Newton,
Huygens thought that Descartes had gone astray in presuming that the laws
of optics could be derived from a priori truths. A mechanistic hypothesis
could not by itself prove anything. Unlike Newton however, Huygens did
not banish hypotheses from his optics. On the contrary, the question how to
establish proper ‘raisons de mechanique’ and built a mathematical science of
optics from them, was his main concern. According to Huygens, the causes
of the behavior of light were ultimately hypothetical. His problem was how
to find the right hypotheses. In the first place, this meant to establish
veritably mechanistic causes of the properties of light. As contrasted to the
speculations of Hooke and Descartes, Huygens wanted his mechanistic
explanations to be comprehensible. That is, a hypothetical mechanism had to
be exact and to conform to the established laws of motion. Huygens’
principle defined ethereal waves mathematically and prescribed how a
propagated wave could be constructed geometrically. In this way it explained
the laws of optics accurately, by means of mathematical derivation.
From the perspective of seventeenth-century geometrical optics,
Huygens’ principle can be seen as a law. But it was a new kind of law: a law
of unobservable waves instead of rays. Light consisted of waves and these
could be treated in the same manner as the rays of traditional optics. He
defined the properties of these waves in the same law-like manner. By
mathematizing the mechanistic causes of the laws of optics, Huygens
extended geometrical optics into the realm of the unobservable. For
methodological reasons Newton would not allow such a thing, although he
was quite capable of mathematizing mechanistic causes. As compared to the
derivation in Principia, Huygens’ model of colliding particles was fairly
economic. Huygens’ focus was on the construction and its applications to
variously deflected rays not on the subtleties of ethereal collisions.
Causes in optics ought to be comprehensible, that is, be mathematical in
the first place. But what about their status? Of course waves were real, they
must be, but were they true? By admitting hypotheses Huygens sacrificed the
full, indisputable certainty Newton wanted to preserve at all costs. According
to him, one could and should conjure up a picture of light propagation, as
long as one showed that the established rules of motion did not leave room
for alternatives. Such a plausible cause could be used subsequently to derive
a possible law of strange refraction. Huygens conjured up a principle to
which all laws of optics could be reduced. The principle was demonstrated
by confirming experimentally conclusions drawn from it. Such a proof was
necessarily indirect and less than fully conclusive. Huygens’ ultimate goal was
not the mechanistic theory per se, but a theory that properly explained the
laws of optics. These conditioned his explanatory theory and were its
ultimate foundation.
Generalizing my findings regarding Huygens’ optics, I have tried to show
how the traditional mathematical science of optics offers useful clues for the
historical understanding for the development of physical optics in the
seventeenth century. Geometrical optics is not the only root, in particular
not in the case of Newton, but I think it is an important one that has been
relatively neglected. To interpret Huygens’ and Newton’s pursuits in optics
as extensions of the mathematical science of optics, reveals some noticeable
similarities that tend to be overshadowed by the vast differences between
them. Let me conclude by setting their eventual publications side by side.
There are many resemblances, as Cohen has amply shown. In particular the
novelty of the subject combined with the imperfectness of both works, has
led him to suggest that Newton may have used Traité de la Lumière as a model
for Opticks.80 For my argument, the most conspicuous parallel between
Opticks and Traité de la Lumière is the way their mathematical roots are
obscured. Although Opticks preserved the deductive structure of definitions,
axioms, and propositions, the line of inference was clearly experimental. The
deductive structure of Traité de la Lumière is not visualized as such. Moreover,
both publications only established the principles of their new mathematical
sciences of optics. Traité de la Lumière had been intended as the prelude for a
dioptrics in which the mathematical theory would be elaborated by applying
the principles to lenses and their configurations. Newton applied his
principles to a few problems like the rainbow, but did not elaborate his
theory of colors in the way he had done in his lectures. In this way, Traité de
la Lumière and Opticks can be said to spotlight their new ways of doing the
mathematical science of optics.
Cohen, “Missing author”, 30-33.
6.3 Traité de la Lumière and Huygens’ oeuvre
Traité de la Lumière looks like a complete and purposively elaborated whole,
an exemplar of a seventeenth-century mathematical physics in which the
principles of optics are derived from a mathematized theory of the
corpuscular nature of light. Traité de la Lumière is often regarded as exemplary
for Huygens’ science as well. In particular, historians has regarded it as a
proof of the fundamental role mechanistic philosophy played in his science.
Yet, it is risky to base an interpretation on the eventual text, as it barely hints
at the winding road towards the final result. In my view, the historicization
of Traité de la Lumière of the preceding chapters sheds new light on Huygens’
science in general.
In the historical literature, Traité de la Lumière is often seen as a direct
response to Descartes’ optics. Sabra first characterizes Huygens’
Cartesianism and then shows how it produced the wave theory.81 Along
similar lines, E.J. Dijksterhuis calls Traité de la Lumière the high-point of
seventeenth-century mechanistic science and its author the first perfect
Cartesian.82 In Traité de la Lumière the mechanistic conception of nature was
indeed perfected, but mechanistic science had not given the momentum to
its materialization. Seeing it as a response to Descartes fails to account for
the fact that mechanistic thinking is virtually absent in his optics prior to the
1670s. At a relatively late stage, Huygens began considering the causes of the
laws of optics and only while developing the wave theory did Huygens
become a ‘mechanistic thinker’. The new form Traité de la Lumière gave to
mechanistic explanation was the outcome of questions pertaining to
geometrical optics, rather than some preconceived plan or mechanistic
I think that Huygens’ commitment to mechanistic philosophy was not as
decisive as is often assumed. His ideas on the nature of light were, of course,
based on prevailing mechanistic conceptions, and Huygens was well aware of
the problems in Descartes’ optics. Yet, questions of mechanistic theory were
not the impetus or drive of his consideration of causes in optics. The
problem of strange refraction set it going and its persistence gave it an
unexpected twist, eventually resulting in the wave theory as we know it.
Huygens’ mechanistic conceptions regarding the nature of light only played a
limited role in the development and establishment of the wave theory. He
displayed a particular lack of interest in elaborating the mechanistic finesses
of his theory, for example by avoiding the question why waves propagate
asymmetrically in Iceland crystal. The strict definition of what counts as a
‘raison de mechanique’ is largely my interpretation, as Huygens himself did
not explicate it in Traité de la Lumière.
“There is in fact evidence to show that Huygens first arrived at his views regarding the nature of light
and the mode of its propagation through an examination of Descartes’ ideas.” Sabra, Theories of Light, 198.
Dijksterhuis, Mechanization, 503-507.
The question now is, to what extent this interpretation of Traité de la
Lumière stands up with regard to his science in general. Historians’ view of
Huygens as a Cartesian at heart has been fostered by his confrontations with
Newton over colors and gravity. They have created the impression that
Huygens refused to accept Newton’s discoveries because they did not fit his
mechanistic conception of nature. I believe that this view of Huygens’
Cartesianism can, and must, be qualified by drawing upon my interpretation
of the development of Traité de la Lumière.
Huygens himself has given ample reason for seeing him as a Cartesian. In
1693 wrote a commentary of Baillet’s biography of Descartes which
developed into a reflection upon the virtues of his teachings. He went back
to his earliest encounter with Cartesian philosophy:
“When I read this book of Principles for the first time it seemed to me that everything
in the world went as well as it could, and I believed that, when I found some difficulty
in it, it was my fault for not grasping his thought well enough. I was only 15 or 16 years
old. But having since then discovered in it from time to time things visibly false, and
others very little probable I have well returned from the preoccupation where I had
been, and right now I find almost nothing that I can approve of as true in his entire
physics, nor in his metaphysics, nor in his meteors.
What was very pleasant in the beginning when this philosophy began to appear, is
that one understood what Mr. Descartes said, instead of the other philosophers who
gave us words that made nothing comprehensible, such as those qualities, substantial
forms, intentional species, etc. He rejected more universally than anyone before this
impertinent ragbag. But what above all recommended his philosophy, is that he did not
confine himself to instilling distaste for what is old, but that he dared to substitute for it
causes which one can comprehend of all there is in nature.”83
Concluding his comment with:
“Notwithstanding this little amount of truth I find in the book of Principles of Mr. des
Cartes, I do not deny that he displayed quite a good deal of wit in fabricating, the way
he did, this whole new system, and in giving it such a twist of truth-likeneness as to
make infinitely many people satisfied with it and pleased with it. One may also say that
by presenting those dogmas with much assurance, and becoming a very celebrated
OC10, 403. “Il me sembloit lorsque je lus ce livre des Principes la premiere fois que tout alloit le mieux
du monde, et je croiois, quand j’y trouvois quelque difficultè, que c’etoit ma faute de ne pas bien
comprendre sa pensée. Je n’avois que 15 à 16 ans. Mais y ayant du depuis decouvert de temps en temps
des choses visiblement fausses, et d’autres tres peu vraisemblables je fuis fort revenu de la preoccupation
ou j’avois estè, et à l’heure qu’il est je ne trouve presque rien que je puisse approuver comme vray dans
Ce qui a fort plu dans le commencement quand cette philosophie à commencè de paroitre, c’est qu’on
entendoit que disoit M. des Cartes, au lieu que les autres philosophes nous donnoient des paroles que ne
faisoient rien comprendre, comme ces qualitez, formes substantielles, especes intentionnelles, etc. Il a
rejettè plus universellement que personne auparavant cet impertinent fatras. Mais ce qui a surtout
recommandé sa philosophie, c’est qu’il n’est pas demeurè à donner du degout pour l’ancienne, mais qu’il a
osè substituer des causes qu’on peut comprendre du tout ce qu’il y a dans la nature.” Also quoted and
translated in Westman, “Problem” 95-96 and 99.
author, he has excited all the more those who wrote after him to resume it and to try to
find something better.”84
This can hardly be read otherwise than as the outline of a program of
Cartesian physics. On the basis of the clear and comprehensible
philosophical foundation laid by Descartes’ followers would erect the
mechanistic science he himself failed to realize. At this moment, Huygens
does not tell that not long after his introduction to Cartesian philosophy
another protagonist of the new sciences would make an even deeper an more
decisive impression on him: Galileo. But first his reception Descartes’
Undeniably, Huygens’ ideas about the ultimate nature of things have
always been uncompromisingly mechanistic. The question is, however, what
relevance these philosophical ideas for his science and when, and how, this
mechanistic framework was mobilized in his actual investigations. At what
moments did Huygens become a mechanistic philosopher, philosophizing
about the mechanistic causes of the phenomena? In Traité de la Lumière his
mechanistic thinking had a designate yet limited role of providing plausibility
of his hypotheses. His focus was on the derivation of the laws of optics. This
aspect of Huygens’ optics developed late in his career, probably not before
the late 1669s during his sojourn in Paris. In Huygens’ optics in general, La
Dioptrique formed the main point of reference, first as regards the dioptrics of
telescopes, then the mathematics and mechanics of refraction.
Responding to Descartes does indeed form a thread in Huygens’ optics,
but one that needs qualification. In the early decades of his career he
primarily responded to Descartes’ ideas and achievements insofar as they
pertained to the mathematical sciences. ‘Physique’, the consideration of
topics in mechanistic philosophy and the response to Descartes’ natural
philosophical conceptions, enters his oeuvre at a later stage, in the context of
his Académie activities. These qualifications of Huygens’ Cartesianism put
his famous confrontations with Newton in 1672 and 1690 in a different
perspective. The first, in 1672, concerned Newton’s theory of colors and has
been discussed in chapter 3. The second, around 1690, was Huygens’
reaction to Newton’s theory of universal gravity, and I will now discuss it
briefly. To do so, we have to go back to the late 1660s, when Huygens laid
the foundation of his ideas about the mechanistic cause of gravity.
The subtle matter of 1669
On one page in his notebook, probably between September 1667 and
February 1668, Huygens listed a series of statements about “… a matter very
OC10, 406. “Nonobstant ce peu de veritè que je trouve dans le livre des Principes de Mr. des Cartes, je
ne disconviens pas qu’il ait fait paroitre bien de l’esprit à fabriquer, comme il a fait, tout ce systeme
nouveau, et a luy donner ce tour de vraisemblance qu’une infinitè de gens s’en contentent et s’y plaisent.
On peut encore dire qu’en donnant ces dogmes avec beaucoup d’assurance, et estant devenu autheur tres
celebre, il a excitè d’autant plus ceux qui escrivoient apres luy a le reprendre et tacher de trouver quelque
chose meilleur.”
subtle and lithe and that is agitated by an extremely swift movement.”85 The
behavior of this subtle matter may explain gravity, the effect of gunpowder,
flames, magnetism, elasticity of solid bodies and of air. Huygens did not
make clear what united these phenomena in his view. Was it because they all
arise from vortices, streams of subtle matter as Descartes had proposed
them? This might explain why light was not mentioned among the
phenomena subtle matter would explain, as according to his (later) views it
was not a vortex but an action propagated through the ether. Or did the
phenomena Huygens mentioned constitute a different class? The ensuing
discussion at the Académie might suggest that they did, namely, a group of
hidden forces in nature that called for mechanistic explanation. It might also
suggest that his recent move to Paris somehow occasioned Huygens to make
these notes on a subject he had not considered before. He may have been
reacting – as was often the case with him – to some discussions going on at
the Académie. For the cases of gravity and magnetism Huygens elaborated
the idea of vortices in papers he read before the Académie in 1669 and 1680
respectively. The “Traité de l’aimant” of 1680 is mainly an effort to
determine the exact way in which turbulences of subtle matter stream on the
basis of observations of patterns of filings of iron.86 I will not discuss this
Huygens considered his ideas on gravity further in some notes he made
probably between February and May 1668.87 Besides remarks about various
materials like iron, lead and water, he laid down what he considered its main
properties. Gravity works towards the center of the earth; the subtle matter
causing it easily penetrates all bodies; weight is proportional to the quantity
of matter in a body. In these notes he discussed some of Descartes’ claims,
including an experiment to simulate the effect of vortices. It described a
vessel containing rotating water in which pieces of lead pushed pieces of
wood towards the center. Because lead is heavier than wood, Huygens
thought that “this experiment does not serve to show the cause of gravity,
…”.88 Gravity ought to be explained by movement alone so there should not
be a difference in weight between the various materials in the vessel.
In the paper he read at the Académie on 29 August 1669, Huygens
described an alternative experiment. In a vessel filled with water, two strings
allow a small sphere to move along the diameter. Turning the vessel around
its axis will make the water rotate. If it is suddenly made to stop, the water
OC19, 553. “Qu’il y a une matiere tres subtile et deliée et qui est agitée d’un mouvement extremement
OC19, 575-581. Huygens appears to have modified his interpretations afterwards, laying particular stress
on the pores of the magnet and the direction of the streams: OC19, 591-603.
OC19, 625-637. See also: Westfall, Force, 185-186.
OC19, 626. “Cette experience ne sert point a faire voir la cause de la pesanteur, …” Descartes had
described the experiment in a letter to Mersenne of 16 October 1629, AT2, 593-594.
will continue to rotate, thereby pushing the sphere towards the center.89 What
did this experiment demonstrate?
“Well then, having found in nature an effect equal to that of gravity and of which the
cause is known it remains to be seen whether one can suppose that something similar
happens as regards the earth, namely some movement of matter that constrains bodies
to tend towards the center and that matches at the same time with all the other
phenomena of gravity.”90
The emphasis should be on the phrase ‘of which the cause is known’. In
Huygens’ view the experiment showed that vortices are a plausible cause of
gravity because a comparable motion exists in nature. As in Traité de la
Lumière, in his 1669 paper on gravity Huygens intended to derive the
observable properties of the phenomena with a hypothesis employing proper
‘raisons de mechanique’.
There are, however, marked differences. In Traité de la Lumière, Huygens
founded his hypothesis on a discussion of the law-like behavior of single
particles. The action causing gravity he could not specify in such terms. He
did not explain the circularity of vortices by picturing the motions of single
ethereal particles. The cause of the push exerted by rotating water was
known to Huygens only on the basis of the observation of a macroscopic
instance of such a motion, not in terms of established laws describing it.
Moreover, he did not mathematize the picture. In other words, his theory of
gravity could hardly count as providing proper ‘raisons de mechanique’.
Consequently, the explanation of the properties of gravity was not a
geometrical derivation as in Traité de la Lumière, but rather an account of the
behavior of vortices with respect to heavy bodies. These properties, in their
turn, lacked mathematical specification. To the ones mentioned above,
Huygens had added Galileo’s law of fall: bodies are continually accelerated
proportional to the time.91 It was the only mathematical one, but he
explained it in just one paragraph. The first and the third can be said to be
potentially mathematical, but in his explanation he did not go beyond a
broad, qualitative formulation.92
Huygens’ improvements and adjustments of Descartes’ explanation were
rather marginal. With his alternative experiment, he could argue more
convincingly that a vortex-like action exists in nature. He brought in some
quantity by calculating the speed of the subtle matter at the surface of the
earth at 17 times the speed of rotation of the earth. He defined more
precisely what needed to be explained (whereby the proportionality of weight
and quantity of matter was a new and important insight). All in all, the 1669
OC19, 633.
OC19, 634. “Or ayant trouvé dans la nature un effect semblable a celuy de la pesanteur et dont la cause
est connue il reste a voir si l’ont peut supposer qu’il arrive quelque chose de pareil à l’esgard de la terre,
sçavoir quelque mouvement de matiere qui contraigne les corps a tendre au centre et qui convienne en
mesme temps a tous les autres phoenomenes de la pesanteur.”
OC19, 640.
Westfall, Force, 186-187 discusses some of these problems.
paper contained a mechanistic theory that remained essentially qualitative. In
the light of the precepts of Traité de la Lumière it could not be called an
instance of ‘true philosophy’.
The message of the paper could not, however, be mistaken. It made clear
what, in Huygens’ view, mechanistic explanation ought to be about. His was
the fourth and last of a series of papers on gravity read at the Académie in
August 1669. On the 7th, Roberval had opened the debate with a paper that
seems to be an express denial of everything Huygens stood for.93 He chose to
account for gravity by proposing a mutual attraction between bodies of the
same kind. He did specify how such an attraction explained the properties of
gravity. He explicitly rejected efforts to explain it by the movement of a
subtle matter because he had never seen anything that was not problematic.
In his reaction to Huygens’ paper, four weeks later, Roberval said:
“… he excludes from nature without proof attractive and expulsive qualities and he
wants to introduce without foundation solely sizes, shapes and movement.”94
In reply, Huygens said that he excluded those qualities because
“… I search for an intelligible cause of gravity, as it seems to me that it would be saying
as much as nothing when attributing the cause why heavy bodies descend to the earth
to some attractive quality of the earth or of these bodies themselves, but for the
movement, the shape and the sizes of bodies I do not see how one can say that I
introduce them without foundation since the senses make use know that these things
are in nature.”95
This, then, was the raison d’être of the paper on gravity. In Roberval’s use of
attractive qualities he saw a relapse into the incomprehensible thinking
Descartes had dispensed with. He would not accept active principles in
nature that were not intelligible in terms of matter in motion.
But did Huygens regard attractive qualities as incomprehensible only
because of his Cartesian leanings? I think not. The 1669 paper on gravity had
been preceded by years of mathematical study of motions. In these he had
persistently reduced all phenomena of impact and acceleration to the
Galilean science of motion. Westfall describes in full detail his “… constant
effort to eliminate dynamic concepts and to treat mechanics as kinematics,
…”96 Huygens found that whatever could not be expressed in terms of
velocities escapes mathematical treatment. In my view, Huygens rejected
attractive forces not only because his mechanistic convictions forbade them,
but also because such concepts were problematic from a mathematical point
OC19, 628-630. Frenicle and Buot followed on 14 and 21 August respectively, summarised in OC19,
630-631. Huygens read his paper – the longest – on 28 August. On 4 September, Roberval and Mariotte
gave their comments to which Huygens replied on 23 October; OC19, 640-644.
OC19, 640. “… il exclud de la nature sans preuve les qualitez attractives et expulsives et il veut
introduire sans fondement les seules grandeurs, les figures et le mouvement.”
OC19, 642. “… parce que je cherche une cause intelligible de la pesanteur, car il me semble que ce seroit
dire autant que rien que d’attribuer la cause pourquoy les corps pesants descendent vers la terre, a quelque
qualité attractive de la terre ou des corps mesmes, mais pour le mouvement la figure et les grandeurs des
corps je ne vois pas comment on peut dire que je les introduicts sans fondement puisque les sens nous
font connoistre que ces choses sont dans la nature.”
Westfall, Force, 177.
of view. This does not mean that mechanistic philosophy did not influence
Huygens’ studies of motion. Mechanistic views were a source of inspiration,
for example when he equated gravity and circular motion in 1659.97 Yet, their
role was limited and they were virtually absent in his subsequent analysis of
circular motion.98 Forces were gradually pushed back and velocity became
Huygens’ ultimate concept of motion. Not just because of the dictates of
mechanistic philosophy, it was the only way he could deal with motion
Huygens lack of interest in the mechanistic nature of light, at least the
absence of any recorded consideration up to the late 1660s, was not singular.
In this sense, his optics is exemplary for his science in general. Before his
move to Paris in 1666, philosophizing about the mechanistic nature of things
is virtually absent in his writings. As I see it, with his brief note on subtle
matter Huygens returned for the first time to the realm of thought that had
made such an impression on his youthful mind. By that time his thinking on
matter and motion had ripened into a thoroughly Galilean conception that
he subsequently injected into mechanistic philosophy. He understood
mechanistic philosophy as an ultimately mathematical idiom. Not the
ontological idiom of Principia Philosophiae, but the mathematical idiom of the
laws of motion of the Discorsi.
The 1669 paper on gravity makes clear that the conception of
mechanistic philosophy that underlies Traité de la Lumière had already taken
shape. In Huygens’ view ‘raisons de mechanique’ invoked matter moving
according to established rules and they were intended to formulate
hypotheses that explained the observable properties of phenomena. As
regards gravity the doctrine had not been realized in full, and in optics it had
not yet taken form. The mechanistic nature of light first enters Huygens’
writings with his approving reference to Pardies’ theory in course of the 1669
debate on gravity.
Huygens versus Newton
In his dispute with Newton on colors Huygens took a more strict position
regarding the need for mechanistic explanation than he did in the ‘Projet’
jotted down at the same time. When in September 1672 Huygens finally
realized the full import of the new theory of light and colors he (grudgingly)
accepted different refrangibility but he did not accept the compound nature
of white light.99 According to him, Newton first had to solve the difficulty of
explaining it mechanistically. Apparently, Huygens thought that sometimes
mechanistic explanation was more than just an optional subject to satisfy ‘the
mind that loves to know the reason of everything’ as he phrased it in the
See above, page 96.
Yoder, Unrolling time, 17-19.
See above, page 88.
What difficulties Huygens was thinking of, is not clear. I suspect the
letters of Pardies had given him the idea that Newton’s theory presupposed
an emission conception of light. This he could not accept, as it conflicted
with his basic notions about the nature of light. Did he fear a diversity of
‘lights’ would undermine his understanding of the nature of light? His (later)
view of light being an action propagated with a velocity depending only upon
the nature of the medium, cannot be reconciled with such a diversity. On the
other hand, around the same time he had suggested the idea that double
refraction, including its odd absence in a second crystal, is caused by two
undulations linked to two kinds of particles in Iceland crystal. I think there
was a good deal of rhetoric in Huygens’ remark, and I do not believe he had
considered the issue in any great detail at this point. It appears that he
considered the mere semblance of being unclear in terms of the mechanistic
nature of light sufficient to request clarification on this point. As long as
Newton had not done so Huygens could not accept his conclusions about
the nature of light on the basis of the ‘accident’ of different refrangibility.
The dispute, 15 years later, over Principia followed a similar course.
Huygens accepted the ‘accident’ of the inverse-square law, but rejected
Newton’s conclusions regarding universal gravity. Principia did what Huygens
had not been able to do: to unify all forms of accelerated motion.100 He did
so by means of a new, mathematized concept of force. Newton considered
circular motion in terms of a force that seeks the center and coined the term
centripetal force. With this force he could treat all frictionless motions of
point masses. Book one of Prinicipia laid down ‘the science of motions that
result from any forces whatever’ as they can be investigated from the
The aim of Principia went beyond a mere science of motion. The laws and
conditions of motions and forces established in book one were the principles
of a philosophy from which the phenomena of nature were to be derived.
“For many things lead me to have a suspicion that all phenomena may depend on
certain forces by which the particles of bodies, by causes not yet known, either are
impelled toward one another and cohere in regular figures, or are repelled from one
another and recede.”102
In book three, Newton gave an example of this by unfolding a system of the
world founded upon the force of gravity. He showed that the force that
holds satellites in their orbit is the same as the force that causes an apple to
fall on earth. The centripetal force established mathematically in book one
formed the basis. He correlated the centripetal acceleration of the moon and
the acceleration of gravity and showed that both are instances of a force that
varies inversely as the square of the distance.
Westfall (Force, 178-179) discusses a paper from about 1675 (OC18, 496-498) that was the start of a
generalised theory of accelerated motion, but in which Huygens failed to see the dynamical equivalence of
change of direction and change of linear velocity.
Newton, Principia, 382.
Newton, Principia, 382-383.
From the viewpoint of mechanistic orthodoxy the introduction of
attractive forces posed, to say it mildly, a problem. In being causes external
to matter they seemed to attribute an active quality to it. Newton admitted
that he had not yet been able to deduce the cause of gravity from the
phenomena, but this did not refrain him from setting forth its properties.
“And it is enough that gravity really exists and acts according to the laws that we have
set forth and is sufficient to explain all the motions of the heavenly bodies and of our
In definition four, Newton defined ‘impressed force’ descriptively:
“Impressed force is the action exerted on a body to change its state either of
resting or of moving uniformly straightforward.”104 Forces could thus be
measured by changes in the motion of a body, as expressed in the second
law: “A change of motion is proportional to the motive force impressed and
takes place along the straight line in which that force is impressed.” 105 The
mathematical correlation of the change of motion of a satellite and of an
apple convinced Newton that the inverse square law defined gravity as a
really existing force of which the cause was yet to be discovered.
When Huygens heard of the forthcoming Principia, his response was as
might be expected. Fatio informed him in 1687 of the forthcoming
publication of a book that would change all of physics. Huygens replied that
he was curious to see the demonstrations that Fatio had sketched. “I do not
mind that he is no Cartesian as long as he does not give us suppositions like
that of attraction.”106 Seemingly, a new Roberval had stood up across the
Channel. Upon reading Principia Huygens must have realized, however, that
this one was of different stature. Newton did derive the properties of gravity.
What is more, he derived a whole science of motion from the concept of
gravitational attraction. Here was an able mathematician treading on ground
that Huygens himself had explored earlier and on which he had acquired
What was Huygens’ reaction upon reading Principia? “Vortices destroyed
by Newton. Vortices of spherical movement in their place.”107 In 1690, he
published – together with Traité de la Lumière – Discours de la Cause de la
Pesanteur, his 1669 paper on gravity extended with a critique of Principia.
However, in the second book Newton had demolished Descartes’ notion of
vortices. In a discussion of the motion of bodies through fluids, he had
demonstrated that a system of vortices could not explain Kepler’s laws of
planetary motion and, moreover, could not exist without some external
agent. How could Huygens believe that Principia did not refute his
explanation of gravity?
Newton, Principia, 943.
Newton, Principia, 405.
Newton, Principia, 416.
OC9, 190. “Je veux bien qu’il ne soit pas Cartesien pourveu qu’il ne nous fasse pas des suppositions
comme celle de l’attraction.”
OC21, 437. “Tourbillons destruits par Newton. Tourbillons de mouvement spherique a la place.”
First of all because he did not believe that Newton intended to explain
gravity by attraction:
“It would be a different matter if one supposes that gravity is a quality inherent to
corporeal matter. But that is what I do not believe Mr. Newton agrees with, because
such an hypothesis would lead us far from mathematical or mechanical principles”.108
Secondly, Huygens had reason to believe that Newton’s concept of gravity
was not universal. His calculations yielded a degree of oblateness of the earth
that stood in sharp contrast to the value Newton had derived on the basis of
the supposition of gravitational attraction between its parts.109 In addition,
these calculations agreed with the data gathered on the most recent test of
his clocks at sea.110 Huygens had not examined Newton’s derivation in detail
“… because I likewise do not agree with a principle he supposes in that calculation and
elsewhere; which is that all the little parts, …, mutually attract or tend to approach one
another. That I would not know how to admit, because I believe to see clearly that the
cause of such an attraction is not explicable by any principle of mechanics, nor by rules
of motion.”111
Huygens had no objections to Newton’s mathematical demonstrations
regarding the inverse-square law per se:
“Thus I have nothing against the Vis Centripeta, as Mr. Newton calls it, by which he
makes the planets be heavy towards the sun, and the moon towards the earth, but I
rather remain in agreement with that without difficulty: … I had not thought either of
that regular diminution of gravity, namely, that it is in reciprocal ratio to the squares of
the distances from the center: which is a new and very remarkable property of gravity,
of which it is well worth to seek the cause.”112
In explicitly refraining from explaining the properties of gravity
mechanistically, Newton did not offer an alternative for spherical vortices.
Huygens would be happy to fill this gap by means of his alternative to
Descartes’ theory. Only, what did vortices explain? He could argue that his
spherical vortices were consistent with an inverse-square law. But could he
derive the properties of gravitational acceleration in the way he had derived
the properties of light rays? No, for he did not show that this kind of
movement follows from the behavior of single particles as he did for waves
in Traité de la Lumière. Moreover, in the light of book two of Principia,
OC21, 474 (Discours, 163). “Ce seroit autre chose si on supposoit que la pesanteur fust une qualité
inherente de la matiere corporelle. Mais c’est à quoy je ne crois pas que Mr. Newton consente, parce
qu’un telle hypothese nous eloignoit fort des principes Mathematiques ou Mechaniques.”
Smith, “Huygens’ empirical challenge”, 2-3.
Mahoney, “Determination”, 258-260.
OC21, 471 (Discours, 159). “… parce qu’aussi bien je ne suis pas d’accord d’un Principe qu’il suppose
dans ce calcul & ailleurs; qui est, que toutes les petites parties, …, s’attirent ou tendent à s’approcher
mutuellement. Ce que je ne sçavrois admettre, par ce que je crois voir clairement, que la cause d’une telle
attraction n’est point explicable par aucun principe de Mechanique, ni des regles du mouvement.”
OC21, 472 (Discours, 160). “Je n’ay donc rien contre la Vis Centripeta, comme Mr. Newton l’appele, par
la quelle il fait peser les Planetes vers le Soleil, & la Lune vers la Terre, mais j’en demeure d’accord sans
difficulté: … Je n’avois pensé non plus à cette diminution reglée de la pesanteur, sçavoir qu’elle estoit en
raison reciproque des quarrez des distances du centre: qui est une nouvelle & fort remarquable proprieté
de la pesanteur, dont il vaut bien la peine de chercher la raison.”
Huygens should have been more cautious regarding claims about such
Why did Huygens publish Discours? Huygens may have felt that Newton
challenged him on his own territory, having seen things – the inverse square
law – he had admittedly missed. With Principia Newton threatened to eclipse
the stature of Horologium Oscillatorium and of its author as one of Europe’s
most eminent scholars. Fortunately, Huygens still had this old paper lying
around that treated of just the thing Newton refused to do: the mechanistic
nature of gravity. So, by publishing Discours, Huygens could show that he
could not yet be written off and, in one stroke, how such occult qualities
ought to be properly dealt with. Besides, the 1669 paper on gravity was
already going to be published anyway. It had been among the papers he had,
in 1686, allowed De la Hire to publish.113 Publishing an extended version
offered Huygens the opportunity to make some corrections and additions
with a view to Principia.114
The confrontations between Huygens and Newton ran a most
unfortunate course. Their philosophical differences prevented a fruitful
exchange of ideas on the mathematical ground they shared. Discours offered
the very kind of explanation Newton refused to give. Not only did it employ
hypothetical objects like the particles and the streams of subtle matter, but it
also assumed laws applicable to them that lacked foundation. It was the kind
of reasoning Newton would not even do in private and one we would not
expect from the author of Traité de la Lumière. For Newton this made no
difference, Huygens’ waves were as speculative as his vortices and neither
had a place in good science.
Still, Huygens and Newton ultimately wanted the same: to do better
where Descartes had been led astray. This meant that the ‘raisons de
mechanique’ ought to be based on the established laws of motion and were
exempt from a priori truth. Descartes, Huygens observed in his comment on
Baillet’s biography,
“… should have proposed to us his system of physics as an essay of what one can say
with probability in this science whilst admitting nothing but principles of
Newton would agree, but he gave a different interpretation. To the
‘mechanical principles’ he added a new, mathematized concept of force. This
was something Huygens could not do. In the end, he remained faithful to
mechanistic philosophy by restricting its principles to their mathematical
core: Galileo’s science of motion. At the same time, however, in his wave
theory he was the first to realize Descartes’ ideal of a mathematical physics.
He freely applied the Galilean laws of motion to the imperceptible particles
See above, page 222.
Mahoney argues that Discours offered Huygens an opportunity to publish corrected values for the
Earth’s rotation, resulting from the V.O.C. trials of his clock. Mahoney, “Determination”, 259.
OC10, 405. “Il devoit nous proposer son systeme de physique comme un essay de ce qu’on pouvoit
dire de vraisemblable dans cette science en n’admettant que les principes de mechanique …”
of the ether. Thus he mathematized the mechanistic foundations of natural
science, which Newton had not dared to. As regards the essential probability
of mechanistic explanations, Newton had drawn a conclusion opposite to
Huygens’. If speculations lead you astray, then stop speculating (at least in
public), Newton said. If they do so, speculate better, Huygens would say.
Huygens’ self-image
We have seen, in his comments on Baillet, how Huygens saw the merits and
faults of Descartes’ science. According to Huygens, Descartes had shown
what the principles of natural philosophy ought to be like, but at the same
time he had failed to elaborate them properly in explaining particular
phenomena. Still, if he had been too confident of the truth of his ideas, this
may have been a stimulus for so many to do better. The picture returns in
“I confess that his essays, and his views, although false, have served to open to me the
road to what I have found on this very subject.”116
Yet the clear and distinct ideas that had made such an impression on the
youthful Huygens were activated only occasionally during his scholarly
career. His science consisted primarily of mathematical inquiry, sometimes
inspired by his mechanistic thinking. He turned mechanistic philosopher
only rarely. He did so, in my view, mainly when he saw ghosts of the occult
thinking he thought Descartes had disposed of.117 And he did so in Traité de la
Lumière, establishing the ‘raisons de mechanique’ of the laws of optics. Or
should we say that, with his waves, he went beyond mechanistic philosophy
by applying Galileo’s science of motion to the imperceptible particles of the
In his comments on Baillet’s biography, Huygens moved from a
reflection upon Descartes’ merits to a discussion of the virtues of other
natural inquirers. This is illuminating regarding the way he saw himself. The
ancient atomists had had the right categories, but could not explain natural
phenomena. Moderns like Telesio, Campanella and Gilbert were no better
than Aristotelians in that they maintained occult qualities. Gassendi and
Bacon exposed the inadequacies of Peripatetic philosophy but, like so many
moderns, lacked understanding in mathematics. Only one man could stand
the test of Huygens’ critique:
“For wit and knowledge of Mathematics Galileo had all that is necessary to make
progress in physics, and one must acknowledge that he has been the first to make
beautiful discoveries concerning the nature of motion although he has left very
considerable ones still to be made. He had neither the audacity nor the conceit to want
to take it upon himself to explain all natural causes, nor the vanity to wish to be head of
a sect. He was modest and loved the truth too much; he believed moreover to have
OC21, 446 (Discours, 127). “Et cependant j’avoue que ses essais, & ses vuës, quoyque fausses, ont servi
à m’ouvrir le chemin à ce que j’ay trouvé sur ce mesme sujet.”
Probably, in the eyes of new age thinkers like Lewis Mumford, this would turn Huygens into a criminal
of the stature of Galileo and Descartes. See Vanheste, Copernicus is ziek, 40-45.
acquired sufficient reputation – one which would endure forever by his new
But Mr. Descartes, who seems to me to have been very jealous of Galileo’s renown,
[and who] had this great urge to pass for the author of a new philosophy, as appears
from his efforts and his hopes to have it taught at the academies instead of Aristotle’s,
or from his wish that the Society of Jesus embrace it: but in the end because he stuck at
all costs to things once he had put them forward, even though often very wrong ones.”
It is not difficult to understand where Huygens placed himself: as the heir of
the unpretentious mathematician rather than the cocky philosopher.
Westman has pointed out he was actually describing himself in his picture of
Galileo as a humble, moderate lover of the truth.119 With mathematics rather
than philosophy he had made progress in physics.
In Traité de la Lumière he had succeeded to extend it to the causes of
natural phenomena. In it, Huygens took the modest stance he ascribed to
Galileo. His theory was not all-compassing, as he had nothing to say about
colors.120 After his explanation of strange refraction he described a new
phenomenon (polarization) he had discovered adding that his theory could
not explain it:
“For although until now I have not been able to find its cause, I do not want to refrain
from indicating it, in order to give occasion to others to seek it. It seems that still other
suppositions would have to be made besides the ones I have made; which nevertheless
will not fail to preserve all their probability, after having been confirmed by so many
The limited reach of his theory did not alter the fact that only waves could
explain the reflections and refractions of light rays properly.
The focus and the foundation of Traité de la Lumière were on the
mathematical science of optics. In its opening lines Huygens was explicit
about his conception of optics. Optics was one of the sciences where
geometry is applied to matter. He did not explicitly say that he extended this
to the unobservable matter of mechanistic philosophy. He applied geometry
OC10, 404. “Galilee avoit du costè de l’esprit, et de la connoissance des Mathematiques tout ce qu’il
faut pour faire des progres dans la Physique, et il faut avouer qu’il a estè le premier à faire de belles
decouvertes touchant la nature du mouvement, quoy qu’il en ait laissè de tres considerables à faire. Il n’a
pas eu tant de hardiesse ni de presomption que de vouloir entrepretendre d’expliquer toutes les causes
naturelles, ni la vanitè de vouloir estre chef de secte. Il estoit modeste et aimoit trop la veritè; il croioit
d’ailleurs avoir acquis assez de reputation et qui devoit durer à jamais par ses nouvelles decouvertes.
Mais M. des Cartes qui me paroit avoir estè fort jaloux de la renommee de Galilee avoit cette grande envie
de passer pour autheur d’une nouvelle philosophie. Ce qui paroit par ses efforts et ses esperances de la
faire enseigner aux academies à la place de celle d’Aristote; de ce qu’il souhaitoit que la societè des Jesuites
l’embrassast: et en fin parce qu’il soutenoit a tort et a travers les choses qu’il avoit une fois avancees,
quoyque souvent tres fausses.”
Westman, “Problem”, 97. Yet, in what seems to be a slip of the eye, Westman misses the phrase: “… ni
de presomption que de vouloir entrepretendre d’expliquer toutes les causes naturelles …”
Traité, “Preface”.
Traité, 88-89. “Car bien que je n’en aie pas pû trouver jusqu’icy la cause, je ne veux pas laisser pour cela
de l’indiquer, afin de donner occasion à d’autres de la chercher. Il semble qu’il faudroit faire encore
d’autres suppositions outre celles que j’ay faites; qui ne laisseront pas pour cela de garder toute leur vraisemblance, apres avoir esté confirmées par tant de preuves.”
to matter, observable and unobservable alike, without racking his brain over
philosophical issues involved. Mechanistic explanations raised a
methodological problem – as compared to the ordinary laws of mathematical
science – and as a matter of course Huygens explained that these required a
different mode of inference. Huygens had a clear conception of proper
method, without putting this into a prescriptive doctrine. This does not
mean that he did not know methodological constraints. He relied on
mathematics as he had learned to pursue it and submitted to it
unconditionally. In this way he had discovered a new property of light,
namely the exact way in which waves of light propagate.
Huygens did not realize that his principle of wave propagation and its
application to strange refraction went beyond traditional ways of studying
light mathematically. He looked upon himself as a mathematician who had
advanced seventeenth-century science by new discoveries and better
theories. He perfected the science of motions and the measure of time, the
telescope and telescopic observation, and so on. He believed that he had
discovered the true causes of refractions and that he had perfected the study
of mechanistic causes in optics. He did not realize that, with the latter, he
had pursued optics in a new, unwittingly and inadvertently revolutionary
kind of way.
Contemporaries looked upon Huygens as a prominent mathematician. He
made his name with his astronomical and mechanical inventions and
discoveries. In optics he had raised high expectations since the 1660s.
During the late 1670s, many thought he would finally publish his long
awaited dioptrics. Huygens’ father had begun to spread word that his son’s
treatise was ready for publication.122 On 15 may 1679, Leeuwenhoek wrote
“Sir, your father writes me …, that the main part of your dioptrica is almost
ready to be printed from a good copy, …”123 Because Hooke had begun
publishing papers on similar subjects, Leeuwenhoek urged him to publish his
own.124 On 8 September 1679, Leibniz wrote to Huygens to breathe new life
into their relation. He told him about his recent work in mathematics and
said he had heard “… from Mr. de Mariotte that you will soon give the
Dioptrique so long desired.”125 Still, he would have to wait another decade.
When finally published, Traité was praised by several men. Leibniz was
impressed. He was surprised by the ease with which the properties of light
could be explained by waves, but when he proceeded to the explanation of
For example the letter of Susanna of 1 February 1680: OC8, 272.
OC8, 166-167. “U.Edele hr. vader schrijft mij …, dat het voornaemste deel van U.Edele dioptica (sic)
bij na in staet is, om uijt een goede copie gedruct te connen werden, …”
OC8, 166-167. Early 1678, Hooke had commenced publishing Lectures and collections made by Robert Hooke
which were continued in 1679 under the title Philosophical Collections (the successor of Philosophical
OC8, 214. “… de Mr. de Mariotte que vous donnerés bien tost la Dioptrique si longtemps souhaittée.”
strange refraction, his esteem turned into admiration.126 He valued Huygens’
theory more highly than Pardies’, let alone Ango’s, but he still wanted to hear
Huygens’ opinion on colors and on diffraction. Papin’s response was similar,
but he had doubts whether the hypothesis explaining strange refraction
could be true.127 From Paris, too, Huygens received appreciation.128 From
London, Fatio called it a pleasure to read it but also had some comments.129
Amidst profuse apologies, he confessed he did not quite understand
Huygens’ explanation of strange refraction. How could the wave strike the
eye along rays not perpendicular to it? Huygens’ reply is lost, but Fatio
withdrew his doubts completely and apologized for making objections
without having studied Huygens’ explanation in a satisfactory manner.130
The correspondence with Fatio is revealing in another regard. The main
part of Fatio’s comments did not concern Traité de la Lumière but Discours de la
Cause de la Pesanteur. In this Fatio was no exception: Principia dominated even
Huygens’ own correspondence. It therefor does not come as a surprise that
the reactions to his own publication were dominated by Discours, his
response to Principia. In this sense, Traité de la Lumière fell between two stools
of gravity. After Huygens had gathered some compliments, it more or less
disappeared from his correspondence. Huygens took no trouble to change
this; apparently he had lost interest again. Early 1690, unbeknown to
Huygens, Traité de la Lumière was discussed at the Royal Society. Hooke raised
objections, mainly pointing out its failure to account for colors. Halley
responded with a paper in which he discussed the virtues of wave theories,
preferring Huygens’ over Hooke’s.131 Newton, who owned and dog-eared
two copies of Traité de la Lumière, first referred to Huygens in the second
English edition of Opticks in 1717.132
The subject returned once more in a letter Leibniz wrote on 26 April
1694. Huygens’ theory, he reported, had been expounded by Martin Knorre
at the University of Wittenberg.133 Along with his own letter, Leibniz sent a
copy of a letter he had received from Fatio, who reported on his and
Newton’s opinions concerning the nature of light and gravity. Fatio and
Newton, Leibniz wrote, still upheld an emission conception of light and
explained different refrangibility with it. Leibniz still had problems with such
OC9, 522.
OC9, 559-560. It seemed to imply that Iceland crystal was not a homogeneous substance, which
Huygens presupposed in order to explain refraction. Huygens responded that this was a question
concerning the structure of the crystal, on which he had only some speculations: OC10, 177-179.
For example from La Hire and Huet: OC10, 5-6; 53.
OC9, 381.
OC9, 410. On Fatio’s letter, he made a note: “IC is the light ray, but it affects the eye as if coming along
the perpendicular of the wave IK”: OC9, 388. “IC est le rayon de lumiere, mais il agira sur l’oeil comme
venant suivant la perpendiculaire de l’onde IK.” (See Figure 70)
Albury, “Halley and Traité de la Lumière”, 449-454. Albury seems to miss the point that Halley was
discussing wave theories only, by claiming that the paper displays his rejection of Newton’s optics.
Cohen, “Missing author”, 32.
OC10, 601.
a conception, and noted that Mariotte had not been able to verify the
invariability of colored rays. He thought it was difficult to explain refraction
with light conceived as particles. He still preferred Huygens’ hypothesis but
also wanted his opinion on the matters discussed by Fatio.134
In reply to Leibniz, Huygens wrote he was glad that his theory was being
approved of, although he was not pleased to see it equated with those of
Hooke and Pardies as the Wittenberg professor did.135 For one thing, Pardies
and Hooke had not been able to explain the “…bizarreries du cristal
d’Islande, …” The explanation of strange refraction was the ‘Experimentum
Crucis’ of his theory and as long as they could not explain ordinary refraction
satisfactorily – let alone strange refraction – their views lacked a solid
foundation. As regards Newton’s nice and interesting observations on
different refrangibility, Huygens was of the same opinion as of universal
gravitation: “… he does not explain what color in those rays is, and it is
because of this that I, too, have not been fully satisfied until now.”136 What
Huygens’ own thoughts on colors were, he once again did not tell.
Leibniz went so far as to place Traité de la Lumière at the same level as
Principia: these were in his view the two most important works in
contemporary science.137 In this, Leibniz was the exception. Huygens
believed that he had surpassed all his predecessors in establishing a plausible
cause for the laws of optics. It was not hailed by his contemporaries as the
success he saw in it. Newton, for one, was not convinced by its argument as
he rejected a wave conception of light altogether. And he probably was not
impressed by Huygens’ arguments against an emission conception. But he
did take the trouble to discuss the difficulties of a wave conception in detail.
In addition, he gave his own account of strange refraction, apparently in
order to undo the uniqueness of Huygens’ explanation. After all, one
strength of Huygens’ wave theory was that it was the only one that could
explain this phenomenon. As Fatio put it:
“You, Monsieur, always have the advantage, that one cannot claim to have something
better until one has explained the phenomena of Iceland crystal so successfully …”138
This was true for the few – Leibniz, Papin, de la Hire – who accepted
Huygens’ theory, but not for the majority in the eighteenth century who did
not.139 Even ’s Gravesande, who published Opera Varia and Opera Reliqua of
Huygens, in optics followed Newton in his widespread textbook Physices
elementa mathematica of 1720.
OC10, 602. According to Fatio, Newton’s view that space is empty posed a serious but not
insurmountable problem for Huygens’ theory: OC10, 606.
OC10, 611-612.
OC10, 613. “… il n’explique pas ce que c’est que la couleur dans ces raions, et c’est en quoy je ne me
suis pas pleinement satisfait non plus jusqu’à present.”
Heinekamp, “Huygens vu par Leibniz”, 108.
OC9, 381; translation: Shapiro, “Kinematic optics”, 244.
Shapiro, “Kinematic optics”, 245. Shapiro offers a concise discussion of the way several scholars
understood and reacted upon the physical concepts of Huygens’ wave theory: 245-252.
A combination of several factors caused the almost complete rejection of
Huygens’ theory in the eighteenth century.140 His account of rectilinear
propagation was generally thought to be inadequate, even by those who did
not follow Newton’s emission conception of light. The scope of the theory
was limited. In particular Huygens’ omission of colors was a drawback in
comparison with Opticks. In addition, the phenomenon on which Huygens’
theory was founded – strange refraction – was largely ignored during the
eighteenth century. Students of crystallography consulted Traité de la Lumière,
but only for his description of the crystal.141 Only with the studies of Haüy,
Malus, and Wollaston the optical theory of strange refraction was
rediscovered. In some German textbooks Huygens’ theory was appealed to,
but adopted only in broad outline.142 Huygens’ explanations of specific
phenomena such as rectilinear propagation were passed over. Hakfoort
explains this by the mathematical content of Huygens’ theory, that exceeded
the goals of books on natural philosophy, in which the nature of light was
customarily being discussed.
Hakfoort broadens his explanation of the neglect of Traité de la Lumière by
pointing out a disciplinary factor. Huygens’ mathematical treatment of
mechanistic causes eluded the capacities and interests of scholars dealing
with such explanatory theories. As had been the case previous to Huygens,
they resorted to qualitative explanations. Mathematicians, on the other hand,
continued to confine themselves to the behavior of light rays irrespective of
its underlying causes. According to Hakfoort, the eighteenth-century
disciplinary barriers between mathematics and natural philosophy, and the
lack of savants who successfully overcame them, caused Traité de la Lumière
to fall in neglect.143 Opticks, on the other hand, had the advantage that it could
be read as an experimental theory of colors. Newton’s queries offered a
qualitative account of the nature of light and were adopted accordingly. This
would mean that the true ‘raisons de mechanique’ Huygens prided himself to
have established were an important factor in the neglect of his theory. What
Huygens considered as a comprehensible explanation eluded the savants of
the eighteenth century.
If Traité de la Lumière fell into oblivion, such was not the fate of Huygens’
oeuvre as a whole. Smith’s A Compleat System of Opticks is exemplary in this
regard. Whereas he ignored Traité de la Lumière – even for strange refraction
he adopted Newton’s account – his praise for Huygens’ accomplishments in
both practical and theoretical dioptrics was high. If Huygens was forgotten
as a mechanistic philosopher, he remained renowned as a mathematician. In
particular Horologium Oscillatorium earned him fame. In perfecting Galileo’s
science of motion, he was perceived as preparing the ground for Newton.
Hakfoort, Euler, 53.
Shapiro, “Kinematic optics”, 257.
Hakfoort, Euler, 119-126.
Hakfoort, Euler, 183-185.
Some ambivalence resounds in the judgements of men like Laplace and
Lagrange. Huygens brought the science of motion to a new level, but did not
develop this into a new science like Newton.144
By the end of the eighteenth century Huygens’ investigation of strange
refraction underwent a more positive valuation. However, his natural
philosophical ideas met with doubts. In his Exposition du système du Monde of
1796 Laplace observed:
“ … [the insufficient explanation of spheroidal waves and polarization] combined with
the difficulties the theory of luminous waves presents is the cause why Newton and the
majority of geometers who have followed him failed to appreciate with justice the law
Huygens attached to it.”145
Malus praised Huygens for finding an accurate law of strange refraction, but
lamented his troublesome ‘system of undulations’:
“That law, considered in itself and cleared of the explanation to which Huygens had
attached it is one of the finest discoveries of that celebrated geometer.”146
He did not realize that this troublesome system was essential to the discovery
of the law and was inherently connected to it.147 He did not recognize the
new way of studying light mathematically that was being pursued in Traité de
la Lumière. His physical optics still had to be rediscovered. Nineteenthcentury students like Fresnel developed a Huygens-like way of doing optics
in which microphysical hypotheses were the starting point of the
investigation – a way Huygens himself had barely recognized as a kind of
‘Optique’ that went beyond geometrical optics.
Bachelard, “Influence”, 244-247.
Laplace, Oeuvres Complètes 6, 353-354. “… joint aux difficultés que présente la théorie des ondes
lumineuses est la cause pour laquelle Newton et la plupart des géomètres qui l’ont suivi n’ont pas
justement apprécié la loi qu’Huygens y avait attachée.”
Malus, Theorie de la double réfraction…, 289-290. “Cette loi, considérée en elle-même et débarrassée de
l’explication à laquelle Huygens l’avait attachée, est une des plus belles découvertes de ce célèbre
The eighteenth-century development of mechanics, in which mathematical science and natural
philosophy tacitly drifted apart, is illuminating in this regard. See Boudri, Het mechanische van de mechanica, in
particular 257-265.
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Chapter 7
Conclusion: Lenses & Waves
A sketch of Huygens in the light of his optics
This study has been aimed at finding out the coming into being of Traité de la
Lumière. How did Huygens’ work in optics develop into the wave theory of
light, a new way of doing optics in which the laws of optics are derived from
an experimentally confirmed, mathematized theory of the mechanistic nature
of light?
Huygens’ work in optics comprises in the first place his dioptrical studies,
but these have hardly been taken into account by historians. Dioptrica has
very little been studied historically and its potential relevance for
understanding Traité de la Lumière has not been considered previously.
Huygens’ optics tends to be identified with his wave theory and the
mechanistic reasoning in it is often taken as a natural part of his science. Yet,
little in Huygens’ work in optics prior to 1672 gives reason to suspect that he
would have given a new form to mechanistic science by 1679. Taking
Dioptrica into account while discussing the development of Huygens’ optics
raises a historical problem. The kind of theorizing pursued in Traité de la
Lumière is completely absent from Dioptrica. Generally speaking, Huygens
does not appear to have had a particular interest in mechanistic topics prior
to the 1670s. The Huygens of Dioptrica was a seventeenth-century
mathematician who does not at all resemble the alleged, ‘first thoroughgoing
Cartesian’ of Traité de la Lumière.
What I wish to do now, is to forget about Huygens’ Cartesianism for a
while and focus on the Huygens of Dioptrica. By comparing his pursuits in
dioptrics to those of his precursors and contemporaries a picture has arisen
of a mathematician with an idiosyncratic approach to questions of
mathematical theory. I shall generalize this picture to include his pursuits in
other fields and make a sketch of his scientific persona. Only then shall I ask
how Traité de la Lumière may fit in and how we should assess his alleged
A seventeenth-century Archimedes
The Huygens who went to Paris in 1666 pursued the various branches of the
mathematical sciences: geometry, arithmetic, statics, optics, harmonics, some
astronomy, and the study of motions. He pursued these brilliantly, and
marked himself off by a particular sense of practical possibilities. Huygens’
orientation on instruments in Dioptrica was unique for its day, having his
theoretical investigations guided by questions of practical relevance.
Huygens was the first (and for a long time the only one) to pursue the
question raised by Kepler right after the invention of the telescope: how can
we understand its working in a mathematical way? Students of dioptrics like
Descartes, Barrow, and Newton focused on solving sophisticated
mathematical problems like determining aplanatic surfaces and analyzing
optical imagery. They did not elaborate their findings to explain the
dioptrical properties of ordinary lenses and their configurations.
Astronomical observers, starting with Galileo, did not elaborate a dioptrical
theory of telescopes either. Only when the telescope was turned, towards
1670, into an instrument of precision did its users like Flamsteed and Picard
begin to bother about questions of dioptrics. Without Huygens’
mathematical proficiency they could not, however, obtain the rigor and
generality of Tractatus and De Aberratione. But Huygens was not of help, he
never came to publish his dioptrics.
Despite the fact that he always had an open eye for practical implications,
the orientation on instruments characteristic of Dioptrica cannot be directly
generalized. The organ did not direct his studies of consonance, as clocks did
not direct his studies of motion.1 Still, in a broader sense Dioptrica does reveal
a particular feature of Huygens’ science. Whereas Descartes contented
himself with establishing the principles of refraction and perfect vision,
Huygens applied the sine law to establish the properties of actual lenses and
telescopes. Whereas others analyzed the mathematics of lenses in order to
find perfectly focusing surfaces, he did so in order to understand the
properties of real lenses and fathom their imperfections mathematically.
What Huygens did in Dioptrica – apart from the practical relevance of his
pursuits – was elaborating mathematical theory by applying general principles
to specific problems of real objects.
Huygens ‘applied geometry to matter’, to use his phrase in Traité de la
Lumière. Real, rather than ideal matter. Application in the sense of elaborating
established mathematical theory for particular cases, rather than
mathematization of new phenomena in the sense Newton did with colors.
Even in his theories of impact and circular motion he substantially built
upon mathematical foundations already laid by Galileo. As contrasted to
Newton, he mathematized no phenomena that had not already latently been
mathematized. Rather than establishing an investigation into the physics of
consonance, he elaborated the mathematics of the coincidence theory. In
dioptrics, he confined himself to the analysis of the properties of refracted
rays and left colors for what they were. Brilliantly pursuing mathematical
reasoning, he rarely went beyond the established boundaries of the
mathematical sciences.
One cannot escape the impression that the elaboration of mathematical
theory for particular problems interested him more than laying new
Although these have never, to my knowledge, been studied from the viewpoint of the relationship
between theory and practice.
foundations. Bos’ aptly wrote: “Huygens as a mathematician was not a man
of abstract theories and methods, his preference lay towards the use of these
to solve problems, preferably problems in physics.”2 In analytical geometry
he did not develop new methods but applied existing methods to new
curves.3 Having established a theory of circular motion, Huygens quickly
moved on to apply it to a physical pendulum. In the broad sense of a
propensity to application, his dioptrics seems to me typical of Huygens’
mathematics in general.
Such elaborating of theories gives rise to a different kind of science than
the development of new principles. Lowbrow science so to say, that
distinguishes itself by the complexity of problems and the elegance of
solutions. It is mathematization in the reverse direction of the process we
usually associate with the term; it is exploring the richness of theory by
deriving specific theorems from its primary principles, rather than unfolding
the mathematical nature of new phenomena. In the hands of Huygens,
elaborating mathematical theory took on a special form. His theorems where
not solutions to problems merely arising from theory, but applications to
concrete, physical objects like lenses and pendulums. As contrasted to the
mathematical mirrors of perspectiva, he applied the principles of optics to
real lenses and telescopes. This idiosyncrasy of Huygens stands out clearest
in comparison with Barrow, an equally gifted mathematician who stuck to
the abstractions of optical theory. Huygens’ mathematics was applied in a
more modern sense of the word, to real objects rather than abstract puzzles.
With this, Huygens stood out among his contemporaries, dealing with
problems others left aside, took for granted, or simply did not notice.
Huygens applied geometry to things, real things, we can paraphrase the
line from Traité de la Lumière. The application of mathematical theory to real
(rather than ideal) objects was not very common in the seventeenth century,
all the more so because for Huygns it seems to have been an end in itself.
Newton was capable of this kind of elaboration, too, and he did so in his
application of his theory of universal gravity to the system of the world. Yet,
this exercise had the higher goal of giving an experimental proof of its
principles.4 Similar epistemological aims are absent with Huygens. In his case
application seems to be a natural part of mathematics, a challenge in its own
right. This challenge then consists of tackling problems with ever greater
complexity. Getting a grip on an increasing number of parameters, from the
foci of spherical lenses to their aberrations; from ideal pendulums to physical
pendulums. It is as if for Huygens the fun only started when the principles of
a phenomenon had already been established. Unlike Newton or Galileo, he
did not bother too much about epistemological, methodological, or
ontological difficulties the application of mathematics to nature might give
Bos, “Huygens and mathematics”, 126.
Bos, “Huygens and mathematics”, 143-144.
Cohen, Newtonian revolution, 62-64 and 100.
rise to. In this sense Huygens’ mathematics is indeed lowbrow. He was more
fascinated by the frayed fringes of mathematics when it came down to
penetrating the behavior of concrete objects like brass pendulums and glass
lenses, than in developing indisputable foundations or in gaining access to
the truth about nature.
However tempting, it would be too easy to explain the factual character
of his mathematics by some sort of artisanal attitude. True, one can discern
interesting parallels between artisanal practice and Huygens’ mathematics.
The visual reasoning by which his mathematics has been characterized has
also been pointed out to typify the way of thinking of the craftsman and the
engineer.5 Like a craftsman wants things to work, Huygens first of all wanted
to get the mathematics right. By his propensity for application he had to
reckon with parameters eluding mathematical theory, as a craftsman has to
cope with the imperfections of concrete materials standing in the way of a
perfectly functioning apparatus. In either case this yields a different view of
knowledge as never final and always open to improvement. As said, an
explanation of the character of Huygens’ mathematics along such lines
would be too easy. De Aberratione suggests that he did not fully grasp the
mismatch between mathematical and artisanal knowledge. Yet, he probably
had a clearer few of the gap between science and technology than anybody
else in the seventeenth century, and he came closest to bridging it. In the
end, however, Huygens could not integrate his theoretical and practical
Huygens always had a keen eye for useful applications, but these did not
drive his theoretical studies. Moreover, his interest in mathematical theory
went beyond mere instrumentalism, in view of the way he sought to establish
general theories that made possible a rigorous mathematical analysis of the
subject at hand. Even in dioptrics, where the tie between theory and practice
was closest, his practical approach cannot be said to originate in practice.
Tractatus was written prior to his work on lens grinding and his telescopic
observations. At the end of chapter three I have pointed out that, however
strange it may seem, his orientation on telescopes emanated from itself. The
same applies, I believe, to his approach in general. The Huygens I see was
fascinated by figuring out the mathematics of real, tangible things and this
concrete puzzling was to him of intrinsic value.6
All in all, I see quite a lot of coherence in Huygens’ activities before his
move to Paris. First of all, most formed a part of the mathematical sciences.7
Secondly, he displayed a marked predilection for the elaboration of
mathematical theory, including its application to concrete objects and
Bos, “Huygens and mathematics”, 132 and Ferguson, Engineering and the mind’s eye, 1-12.
6 In a way, his tutor Henricus Bruno foresaw this with the fourteen-year old Christiaan, when he wrote
Constantijn sr. that they would have to fear that he might turn into an engineer, given the his fascination
and skill with taking apart clocks. OC1, 552.
His work on pumps in 1661 being the most notable exception, but in this his principal interest was in
apparatus rather than vacuum. For an overview see Sparnaay, Adventures in vacuums.
phenomena. The coherence I see is therefore not one of content of theories,
but one of a common approach and of disciplinary-connected fields of
study. He believed in the power and fertility of rigorous mathematical
reasoning, as opposed to the mere empiricism of ordinary craftsmen. Our
pre-Parisian Huygens was a mathematician, a seventeenth-century
mathematician with an idiosyncratic approach. A new Archimedes, Mersenne
concluded when he was confronted with the youthful Huygens.8
From mathematics to mechanisms
What happens, we may now ask, when a mathematician of this Archemidean
inclination meddles with questions of the mechanistic nature of light?
Nothing special needs to happen and nothing did at first. The nature of light
became of interest to Huygens when he needed a preparatory chapter for his
‘Dioptrique’ that would explain the laws of optics. No problem, Pardies had
shown how refraction could be explained by waves of light. Only an exotic
phenomenon displayed by Iceland crystal posed a bit of a problem. A
refracted perpendicular ray negated the perpendicularity of rays and waves
that was crucial to Pardies’ explanation. Although the problem of strange
refraction thus pertained to the wavelike nature of light, Huygens first
approached the phenomenon in the meanwhile traditional, mathematician’s
way. He sought a law of strange refraction in terms of the properties of rays.
Not surprisingly, the law he found did not solve the problem of strange
Five years later, Huygens returned to the problem. And this time
something special did happen. Following on his analysis of waves refracted
by curved surfaces, he considered the question what happened to waves
when they traverse Iceland crystal. The special thing is that he now took the
propagation of waves mathematically. He defined a wave as the result of a
disturbance propagated with a specific velocity in all directions, which he
could then apply by geometrical construction only. At the background was
the conviction that the mechanics of wave propagation ought to follow from
the laws of motion. But the mechanistic picture was explicated – and maybe
also recognized – only afterwards. In the notes of 1677 we see Huygens less
concerned about the broad ideas of his principle and of spheroidal waves
than about their mathematical elaboration. The same line of reasoning that
explained refraction should also explain the other properties of light rays,
including strange refraction. The result was a law of wave propagation
yielding an indissoluble tie between the mechanistic nature of light and the
laws governing the behavior of rays.
What Huygens did not realize, was that he had not just solved another
problem in optics. It was a problem regarding the physical foundations of
geometrical optics, but Huygens had phrased it in a particular way.
Reconciling strange refraction with waves was a problem of reconciling
Yoder, Unrolling time, 179. OC1, 47. “Je ne croy pas s’il continue, qu’il ne surpasse quelque jour
mathematical description and physical explanation. That had not been
Bartholinus’ problem, who sought only a correct mathematical description of
the phenomenon. As a matter of fact, few students of optics made a problem
of the coherence of physical explanation and mathematical laws. Kepler,
Descartes, Newton, and possibly Pardies were likely to recognize the
problem. The first two, who did not know strange refraction, did not solve
the problem of ordinary refraction in a satisfactorily coherent manner.
Pardies saw strange refraction as a problem of the crystal, not as a problem
of waves. And Newton … With respect to strange refraction he avoided the
problem by proposing a law of strangely refracted rays without even
suggesting, whether in print or in private, a possible explanation in terms of
light particles. This law happened to be identical with the first stab Huygens
had made at the problem of strange refraction but had failed to solve it. With
a surefootedness that can only be called astonishing Huygens had taken
precisely this problem of reconciling mathematical description and physical
explanation seriously and brought it to a fortunate conclusion. Posing the
problem in this way was, however, not a matter of course, nor was the
eventual solution.
Huygens did not realize that he was doing something new in a broader
sense. In his view, he had merely solved the problem of strange refraction.
And in a sense he was right. He had set his teeth in another challenging
mathematical problem: reconciling waves and strange refraction. Just as he
did not content himself with rough answers about pendulums and lenses, he
wanted to get the mathematics of wave propagation right. Given the
conscientiousness with which he handled all problems, it is quite natural that
Huygens ended up with a coherent and thoroughly mathematical answer. In
his attack on the problem we see the same versatility in applying his
mathematical skills to concrete objects. In this case, however, these concrete
objects were invisible waves of light. As if it went without saying, he had
approached unobservable particles in the same Archimedean way he
approached the tangible objects of his earlier mathematical studies. He had
mathematized the mechanistic causes of the behavior of light rays. Huygens
was therefore wrong as well. He had not just solved the problem of strange
refraction. In effect he had invented a new way of doing optics.
Within the limited scope of reflected and refracted rays, Huygens had
invented that part of physical optics in which mathematics fruitfully
integrated the nature of light and its observed behavior. Kepler had realized
that the mathematical description of light rays also ought to reflect its
physical nature, but had not succeeded in deriving the ‘measure’ of refraction
from its ‘cause’. Descartes had proclaimed the mechanistic nature of light
but, by seeking mathematics in the ontology of matter rather than its
motions, had not succeeded in mathematizing his picture. Newton could
mathematize the motions of particles of light, but he would not allow this to
be integrated with his experimentally established theory of the mathematical
behavior of colored rays. Parallel to Huygens, Newton had developed that
other part of physical optics in which experiment was used as a heuristic tool
for exploring new phenomena of light and establishing their mathematical
properties. In this way he had extended the mathematical science of optics to
the quality of color. Huygens had extended it to the mechanistic causes of
the laws of optics. By applying Galileo’s science of motion to the motions of
ethereal particles, he had invented the most complete form of mathematical
physics in the seventeenth century.
Huygens and Descartes
Traité de la Lumière gave a new form to mechanistic science, the first
‘thoroughly Cartesian’ theory of light. Yet – and this is the gist of my
argument – it was not the outcome of some program in mechanistic, or even
Cartesian, science. A careful reconstruction of what exactly were the leading
questions for Huygens, juxtaposed with comparable pursuits of other
protagonists of seventeenth-century optics, reveals that Huygens’ wave
theory was the outcome of his typically rigorous and tenacious approach to a
problem raised in the context of geometrical optics. As was his wont, he first
of all wanted to get the mathematics of his solution right. He wanted the
explanations of the various laws to be mathematical derivations that were
mutually consistent. As a result of the particular character of the ‘matter’ of
geometrical optics – light rays, which had come to be seen as being of a
mechanistic nature – he got involved in mechanistic questions. He did so in a
deliberately mathematical way, intending to stick to the rigor of mathematics
he missed in the reasonings of his fellows at the Académie. He believed in
the power of mathematical reasoning and did not content himself with illdefined mechanisms.
This reaction to the Parisian Cartesians can be seen as a continuation of
what I regard as Huygens’ lifelong reaction to Descartes. Much of his oeuvre
was a direct response to what Descartes had said on impact, circular motion,
curves, lenses, light, halos, etc. He did so in a clearly mechanistic context,
accepting fundamental concepts and drawing inspiration from some of
Descartes’ ideas. In his theories of gravity and light he also considered the
conceptualization of the mechanistic nature of things. Discours was induced
by the – in his view – obscurities vented on the Parisian scene. Pardies may
have inspired his thinking on the nature of light and the intellectual climate
at the Académie, but strange refraction – together with the problem of
caustics – may well have been the sole occasion for Huygens’ consideration
of the mechanics of light propagation.
As an adolescent, Huygens had soaked up Principia Philosophiae and its
clarity of reasoning had made an indelible impression on him. The idea that
nature ultimately consists of passive matter in motion was always at the back
of his mind. But this does not turn Huygens into a Cartesian. Mechanistic
philosophy was merely a tacitly assumed background of his thinking.
Huygens quite consistently confined himself to the mathematics of these
matters – or, more properly speaking, to Descartes’ dealings with
Descartes set Huygens’ agenda as he did for seventeenth-century science
in general, but in this case it was a mathematical agenda instead of a natural
philosophical one. He was no builder of a system of natural philosophy,
neither in the Cartesian sense, nor in the way Newton was. Leibniz criticized
him for failing to draw philosophical conclusions from his laws of impact.
“He had no taste for metaphysics”.9 Huygens did not pursue questions raised
by Descartes’ natural philosophical program, he responded to his
contributions to the various branches of mathematics. He firmly criticized
these and even his whole approach. In Huygens’ view, Descartes had
corrupted mathematical science and he would do better.
The point I want to make here is that the historical significance of Traité
de la Lumière has blown up Huygens’ alleged Cartesianism and distorted our
view of the whole of his optics – and his science in general. Sabra offers an
example of the pitfall created by presuming Huygens to be pursuing
Cartesian science. He discusses the wave theory prior to the dispute on
colors with Newton, which is historically incorrect, thus making him more
mechanistic than he actually was at that time.10 Huygens did not have some
kind of research program aimed at unraveling the mechanistic nature of
things, not even a program aimed at establishing the mathematical nature of
The small Archimedes
For Huygens, applying mathematics to real things (large and small) went
without saying. He investigated the mathematical aspects of phenomena and
one suspects that he did not have explicit ideas about the ultimate
mathematical nature of nature like Galileo had. Huygens seems to have
lacked “…a personal conviction about access to deep secrets of nature.”11 I
regard his revolutionary conception of the probable nature of explanatory
knowledge not as an outcome of some philosophical or epistemological
conviction, but rather as a reflection of the cumulative character of his
Archimedean mathematics. I do not think that the preface of Traité de la
Lumière reflects some scepticist attitude. Colors or polarization were simply
additional parameters not yet fathomed, but for Huygens this did not detract
from the validity of his wave theory.
Huygens has been called a problem solver, and this he was, marking
himself off by solving problems his contemporaries passed over. He was
perfectly happy with brilliantly solving sophisticated problems of
Heinekamp, “Huygens vu par Leibniz”, 106. Leibniz, Philosophische Schriften III, 611. “Il n’avoit point de
goust pour la Metaphysique.”
Sabra, Theories of Light: chapter VI “Huygens’ Cartesianism and his theory of conjectural explanation”,
chapter VIII “Huygens’ wave theory”, chapter X “Three critics of Newton’s theory: Hooke, Pardies,
Hall, “Summary”, 307.
mathematical physics without bothering too much about their mutual
connections or their theoretical and philosophical background. As a result of
Huygens’ pragmatism or eclecticism, his oeuvre may sometimes seem like a
mishmash of isolated problems, brilliantly yet pragmatically solved. If we
expect a brilliant savant to be searching for new or better foundations,
Huygens may indeed pose a problem. Elaborating and applying theories does
not yield new foundations. This makes it difficult to situate him among the
Galileos and Newtons.
Yet, Huygens’ pre-1670s oeuvre has historical significance in several
other respects. It reveals another aspect of modern science, the application
of mathematics to concrete things. Huygens’ mathematics was the kind of
the rational mechanics that was to develop in the course of the eighteenth
century.12 I have the impression that this kind of science tends to be
overlooked by historians of science. When it comes to the scientific
revolution, they primarily look at the development of new conceptual and
methodological foundations.13 As regards seventeenth-century optics this is
evident: whereas the discovery of the sine law and of dispersion as well as
the mathematization of the corpuscular nature of light have amply been
studied, the development of geometrical optics as cultivated by Huygens in
Dioptrica has received hardly any attention.
Historians seem to feel a bit awkward about the fact that someone of the
stature of Huygens was a mere problem solver. Some have tried to distill
some kind of underlying philosophical scheme from his activities.14 I do not
expect that seeing him as some kind of neo-Cartesian will shed more light on
the character of his oeuvre.15 I am more taken by the fact that a seventeenthcentury savant of his stature was so unprogrammatic and displayed such a
lack of interest in epistemological, methodological and philosophical issues.
Those historians tend to overlook that this problem solving for the pleasure
of problem solving has historical significance in its own right. It made
Huygens into one of the first – perhaps the very first – of a new kind of
scientist, an investigator of nature desiring to do things better than others,
Mulder, “Pure, mixed and applied mathematics”, 37-39.
This has also been suggested by Gabbey, at the 1979 symposium, in trying to understand why Huygens’
mechanics has received relatively little attention from historians: “…, I would suggest that historians have
clustered around Galileo, Descartes, or Newton because one of their central aims was to describe the
fundamental nature and workings of the physical world, and since this is the primordial purpose of
physical science, the concomitant difficulties and inconsistencies, the associated philosophical and
mathematical problems, the false starts, the anomalies, the blind allies, all the unfinished business, are
irresistible to the historian. By contrast, Huygens has been visited relatively infrequently by historians
because he solves problems, and does so magnificently, by an appeal to principles and hypotheses his
intuition and empirical sense tell him are right, rather than erect an explanatory system of the world that
has its roots in an original analysis of the nature of things.” Gabbey, “Huygens and mechanics” 175-176.
Fortunately, since 1979 the situation around Huygens’ mechanics has changed to the better with, first of
all, Yoder’s Unrolling time and, more recently, with Mormino’s ‘Penetralia motus’ .
Elzinga does so on the basis of Traité de la Lumière. Elzinga, On a research program.
Hall, “Summary”, 309-310.
rather than seeking to fathom the deep secrets of nature. A ‘contemplator of
nature’ he did not become until he composed Kosmotheoros.16
Huygens has obtained his historic stature as a pioneer in mathematical
physics because he solved a special problem: the problem strange refraction
appeared to him to pose to waves. In Traité de la Lumière, the small
Archimedes – as his father used to call him – made new science. Galileo had
mathematized motion, Newton mathematized qualities, Descartes had
ontologized mathematics. Huygens mathematized the invisible nature of
things. He had become the Archimedes of the small.
See Harting, Christiaan Huygens, 45-50.
List of figures
Figure 1 Huygens: sketch of 6 August 1679
Figure 2 Spherical aberration
Figure 3 Cartesian oval.
Figure 4 Huygens: focal distance of a bi-convex lens
Figure 5 Huygens: punctum concursus
Figure 6 Huygens: refraction at the anterior side of a bi-convex lens
Figure 7 Huygens: refraction at the posterior side of a bi-convex lens.
Figure 8 Huygens: focal distance of a bi-convex lens
Figure 9 Huygens: extended image.
Figure 10 Huygens: magnification by a convex lens.
Figure 11 Huygens: four of the cases of magnification by telescopes.
Figure 12 Huygens: analysis of Keplerian telescope with erector lens.
Figure 13 Diagram for Keplerian telescope with erector lens.
Figure 14 Kepler’s solution to the pinhole problem
Figure 15 Kepler: focal distance of a plano-convex lens
Figure 16 Kepler: image formation by a lens
Figure 17 Della Porta: image of a near object
Figure 18 Della Porta: image of distant object
Figure 19 Della Porta: image by a telescope
Figure 20 Barrow’s analysis of image formation in refraction.
Figure 21 Huygens: observations of Saturn with the 12- and a 23-foot telescope.
Figure 22 Huygens: beam to facilitate lens grinding.
Figure 23 Daza’s scale
Figure 24 Huygens’ eyepiece.
Figure 25 Diagram for Huygens’ eyepiece.
Figure 26 Huygens: spherical aberration of a plano-convex lens.
Figure 27 Huygens: spherical aberration of a bi-convex lens
Figure 28 Hudde’s calculation of spherical aberration
Figure 29 Huygens: Galilean configuration in which spherical aberration is neutralized.
Figure 30 Huygens: ‘Circle’ of aberration.
Figure 31 Huygens: Aberration produced by a Keplerian configuration.
Figure 32 Rendering of Huygens’ sketch of chromatic aberration.
Figure 33 Huygens’ invention of 1669
Figure 34 Huygens’ crossed out EUREKA.
Figure 35 Newton’s determination of chromatic aberration.
Figure 36 The first stage of Kepler’s attack of refraction.
Figure 37 The final stage of Kepler’s analysis of refraction
Figure 38 Harriot’s measurements.
Figure 39 Mydorge’s rule
Figure 40 Descartes’ analysis of refraction
Figure 41 Descartes’ analysis of reflection
Figure 42 Barrow’s explanation of reflection.
Figure 43 Barrow’s explanation of refraction.
Figure 44 Huygens: sketch of refracted rays in Iceland crystal.
Figure 45 Huygens: a refracted perpendicular caused by the composition of the crystal.
Figure 46 Huygens: waves through Iceland crystal.
Figure 47 Huygens: shape and main angles of the crystal.
Figure 48 Bartholinus: double refraction.
Figure 49 Bartholinus: refraction in two positions of the crystal.
Figure 50 Bartholinus’ law of strange refraction.
Figure 51 Huygens: rays in the principal section.
Figure 52 Huygens: construction for strangely refracted rays in the principal section
Figure 53 Diagram of Huygens’ construction for strange refraction.
Figure 54 Huygens’ alternative for Bartholinus’ law.
Figure 55 Huygens: description of polarization.
Figure 56 Ango’s explanation of refraction.
Figure 57 The sine law in Tractatus.
Figure 58 Huygens’ principle.
Figure 59 Huygens: two rays refracted by a plane surface.
Figure 60 Huygens: wave refracted by a plane surface forming a caustic.
Figure 61 Huygens: wave refracted at the plane surface of a glass medium.
Figure 62 Huygens: “Causam mirae refractionis in Crystallo Islandica”.
Figure 63 Huygens: strange refraction of an arbitrary ray.
Figure 64 Huygens: waves around a source of light
Figure 65 Huygens’ principle.
Figure 66 Huygens’ explanation of refraction.
Figure 67 Huygens: refraction of the perpendicular.
Figure 68 Huygens: orientation of spheroid in the crystal.
Figure 69 Huygens: shape of the spheroidal wave.
Figure 70 Construction of the refraction of an arbitrary ray in Traité de la Lumière.
Figure 71 Hobbes’ rays.
Figure 72 Hobbes: refraction.
Figure 73 Hobbes’ derivation of the sine law.
Figure 74 Refraction in Principia.
Figure 75 The sine law in Opticks.
Figure 76 Huygens: new measurement of strange refraction.
Figure 77 Huygens’ EUPHKA of August 1679.
Figure 78 Huygens: chromatic aberration of lenses.
References are made by author and short title. References to the twenty-two
volumes of the Oeuvres Complètes are given by OC followed by the volume and
page numbers, as in OC10, 153. References to the manuscripts, all of which
belong to the collection Codices Hugeniorum housed at the University of
Leiden, are given by manuscript number and folio, as in Hug2, 45v. Huygens
is left out as author.
I do not explicitly refer to all seventeenth-century works listed in the
bibliography, but I have drawn upon these in general conclusions regarding
optics in the seventeenth century.
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Académie Royale des Sciences 44, 49, 53,
105, 107, 110, 157, 159, 160, 172,
204, 205, 206, 207, 213, 215, 222,
224, 238, 239, 241, 261
Aguilón, François d’ (1567-1617) 35, 42,
Alhacen, or Ibn al-Haytham (ca.9651039) 27-28, 38, 83, 108, 112-116,
124, 131, 160, 211, 228
Andriesse, C............................................ 107
Ango, Pierre (1640-1694) 110, 152, 153,
195-196, 250
Auzout, Adrien (1622-1691) .......... 44, 157
Bacon, Francis (1561-1626) ...99, 105, 247
Bacon, Roger (1214-1292).............. 27, 114
Baillet, Adrien (1649-1706) . 237, 246, 247
Barrow, Isaac (1630-1677) 38-41, 50, 95,
102, 108, 112, 125, 136, 138-140, 157,
189, 190, 192, 213, 220, 230, 232,
256, 257
cause of refraction ........137-138, 202, 211
image formation......................................39
Lectiones XVIII (1669) 39, 82, 83, 136,
Bartholinus, Erasmus (1625-1692)..... 111,
135, 142-147, 149, 151, 153, 154, 155,
168, 170, 171, 182, 205-210, 231, 260
Beeckman, Isaac (1588-1637) ....42, 57, 58
Berkel, K. van......................................... 157
Bolantio, Giovanni Christophoro () ... 6162, 76
Bos, H.J. .......................................................3
Boyle, Robert (1627-1691) .....94, 135, 228
Brahe, Tycho (1546-1601)... 26, 43, 44, 45
Buchdahl, G............................................ 120
Buchwald, J.Z....... 146, 179, 196, 207, 230
Campani, Guiseppe (1635-1715)..... 59, 62
Cassini, Gian Domenico (1625-1712) . 4445, 107, 167
Cavalieri, Bonaventura (ca.1598-1647) .38,
Cherubin d’Orleans (1613-1697)............62
Colbert, Jean-Baptiste (1619-1683) .......53,
160-161, 204, 214
Daza de Valdez, Benito (1591-1634) .....61
Descartes, René (1596-1650) 2, 5-9, 15,
25, 36, 37, 40-42, 50, 55-57, 67, 71,
72, 82, 94, 96, 101, 108-109, 112-113,
116-117, 126-127, 129-136, 139-140,
146-147, 149, 151-156, 158, 170, 186189, 191, 193, 195-196, 198, 200, 205,
209-211, 228-232, 234, 236-241, 244247, 256, 260-262, 264
cause of refraction .................126-130, 187
Discours de la Methode (1637) ..............24
La Dioptrique (1637) 8, 11, 13-14, 16,
24, 36, 37, 38, 41, 49, 56, 62, 71,
82, 109, 111, 125-128, 131-134,
136, 155, 186, 188-189, 206, 213,
224, 231-232, 238
La Géométrie (1637)...........11, 13-14, 41
Le Monde, ou Traité de la Lumière (1664)
...................133, 187, 188, 224, 253
lenses ............................................... 36-37
Les Météores (1637)...............13, 24, 193
Principia Philosophiae (1644) 133, 187,
242, 261
Dijksterhuis, E.J............................. 2, 6, 236
Divini, Eustachio (1610-1685) 53-54, 59,
with H. Fabri, Brevis Annotatio in
Systema Saturnium (1660).............54
Domini, Marko Antonij (1560-1624) ....33
Dupré, S. ............................................. 29, 35
Fatio de Duillier, Nicolas (1664-1753)
...........................223-224, 244, 250-251
Fermat, Pierre de (1601-1665) 135, 156,
162, 165, 167, 205
Ferrier, Jean (fl. 1620-1640) ............. 36, 56
Flamsteed, John (1646-1719) 45-50, 72,
Fontana, Francesco (ca.1585-1656) 43, 59
Fresnel, Augustin (1788-1827)....... 42, 253
Fullenius, Bernardus (1640-1707) ........221
Galileï, Galileo (1564-1642) 9, 41, 53-55,
59, 63, 96, 97, 98, 117, 126, 135, 154,
202, 211, 233, 238, 240, 246-248, 252,
256, 257, 261-262, 264
Discorsi e Dimostrazioni Matematiche
Intorno a Due Nuove Scienze (1638)
............................................ 211, 242
Sidereus Nuncius (1610) ..........25, 35, 55
Gascoigne, William (ca.1610-1644). 43, 47
Gassendi, Pierre (1592-1655)....... 228, 247
Gravesande, Willem Jacob ‘s (1688-1742)
Gregory, James (1638-1675) ..38, 125, 135
Grimaldi, Francesco Maria (1618-1663)
Grosseteste, Robert (ca.1170-1253).....114
Gutschoven, Gerard van (1615-1668) 1418, 57
Hakfoort, Casper .......................9, 211, 252
Hall, A.R. .....................................................8
Halley, Edmond (1656-1743)...40, 45, 250
Harriot, Thomas (ca.1560-1621) 35, 123124
Harting, Pieter (1812-1885).......................7
Hartsoeker, Nicolaas (1656-1725)....... 215
Hérigone, Pierre (-ca.1643) .....................36
Hevelius, Johannes (1611-1689) 41, 43,
45-46, 49, 62
Hire, Philippe de la (1640-1718) ...... 222,
246, 251
Hobbes, Thomas (1588-1679) 37, 137,
189-192, 229, 230
cause of refraction .........................190-191
Hooke, Robert (1635-1703) 43, 45, 78,
89, 91, 94, 110, 135, 160, 186, 189,
192-194, 209, 230, 233-234, 249-251
Hudde, Johannes (1628-1704) 11, 71-72,
Huygens, Christiaan (1629-1695) 1, 2, 53,
107, 135
‘Adversaria ad Dioptricen’ (1665) 68, 70,
72, 81, 83
‘De Aberratione Radiorum a Foco’ (1666)
5, 72, 74-77, 83, 91-93, 95, 97,
100-102, 105, 154, 216, 219, 221,
256, 258
‘De Ordine in Dioptricis nostris servando’
(ca. 1692).............................221-222
‘De Telescopiis’ (1680s)...............220-221
‘mon Archimède’.......... 10, 161, 259, 264
‘Projet du Contenu de la Dioptrique’
(1672) 108-109, 111-112, 135-136,
140, 152, 155-161, 181, 186, 214,
216, 219, 242
‘Tractatus de refractione et telescopiis’
(1653) 4, 12, 15-20, 22-24, 30, 32,
41, 49-50, 53, 58, 63-68, 76, 82,
93, 108-109, 153, 216, 220-221,
256, 258
accelerated motion ........................... 95-100
Astroscopia Compendaria (1684) ....... 215
De Saturni Luna Observatio Nova (1656)
Dioptrica 6, 8, 91-91, 95, 100-101, 104105, 112, 153, 220-221, 255-256,
Discours de la Cause de la Pesanteur
(1690)........................ 223, 244, 250
Horologium Oscillatorium (1673) 53, 101,
107-108, 160, 246, 252
impact................................................. 154
Memorien aangaande het slijpen van glasen
tot verrekijckers (1703)...................62
music ............................................. 97-100
nature of gravity ...................239, 244-245
Systema Saturnium (1659) 44, 50, 54,
63, 105-106
Traité de la Lumière (1690) 2-4, 6, 9, 95,
108, 112, 159, 160-162, 167-168,
172, 176-179, 181-187, 192, 194196, 202, 204-205, 207-211, 213216, 220-225, 228, 230, 235-238,
240-242, 246-257, 261-262, 264
Huygens, Constantijn jr. (1628-1697)...11,
33, 53, 58, 63, 64, 77-80, 107, 214,
Huygens, Constantijn sr. (1596-1687) ..11,
56, 77, 88, 135, 214, 249, 264
Huygens, Lodewijk (1631-1699) 64, 80,
Kepler, Johannes (1571-1630)4-7, 12, 23,
27, 34, 36, 38, 40-43, 48, 50, 66-67,
104, 108, 112-113, 117, 120-125, 130131, 134, 140, 158, 183, 228-230, 232,
244, 256, 260
Ad Vitellionem Paralipomena (1604) ..2631, 35, 38, 41, 47,