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Tag Archives: Christiaan Huygens

To Understand Spinoza’s Letter 32 to Oldenburg

It is November of 1665, and Spinoza has just that summer likely spent much time in communication and possible visitation with the esteemed Christiaan Huygens, whose estate is a mere 5 minutes walk from where he lives. The two of them are ensconced in the quiet village of Voorburg, but it was a summer in which plague was ravaging London at a rate of nearly 6000 a week, and the secretary of the Royal Society of England, Oldenburg, has begged Spinoza for an update on the discoveries and devices of Huygens, as if upon such innovations the figurative health of society depends.

Spinoza responds with some telling remarks, upon which I have already registered some thoughts: Spinoza’s Comments on Huygens’s Progress. Here though, I want to post some relevant illustrations from Huygens’s notebooks, which make much more clear just what Spinoza may find objection to in Huygens’s fabrica. What Spinoza writes is this:

The said Huygens has been a totally occupied man, and so he is, with polishing glass dioptrics; to that end a workshop he has outfitted, and in it he is able to “turn” pans – as is said, it’s certainly polished – what tho’ thusly he will have accomplished I don’t know, nor, to admit a truth, strongly do I desire to know. For me, as is said, experience has taught that with spherical pans, being polished by a free hand is safer and better than any machine.

(This was the summer that Huygens will have solved the issue of spherical aberration using a solely combination of spherical lenses. But Spinoza does not know this.) We can assume that Spinoza has seen the machine that Huygens is fast at using. In order to see with Spinoza what this machine likely entailed, one must turn to several illustrations. Since the 1650s Huygens had experimented with (and likely used) an assisted means of steadying the glass blank against the spinning metal grinding form. The nature of this technical strategy was a long “bâton” which would restrict the kinds of movements the blank was capable of:

This is a detail of the device, followed by the wide view:

Oeuvres Complètes, XVII (p.300)

Oeuvres Complètes, XVII (p.299)

As one might see, the glass blank can toggle to a degree. This is what professor Graham Burnett writes of it in his Descartes and the Hyperbolic Quest: Lens Making Machines and Their Significance in the Seventeenth Century:

In the late 1650s, [Huygens] outlined the improved “bâton” technique for handling the lens in the forming pan [above illustrations cited]. Previously, the lens blank had been afixed by means of pitch or rosin to a short wooden or stone handle called a mollette. This short handle and wide distribution could lead to a rocking of the blank as it guided over the form, resulting in distortions of shape. Huygens’s improvement made use of an iron pin which acted as a bearing in the center of a piece of wood sitting over the glass. The pin was affixed to a wooden shaft that was suspended from above. This arrangement did not necessitate the use of pitch to attach the lens, and thus avoided fouling the abrasive with fragments of rosin. The technique must have worked well, because Huygens referred to using it into the early 1660s and even dedicated to it several pages in his extensive De Vitris Figurandis…representing work done in the 1670s and 1680s (97 – 98 )

Whatever the fabrica that Spinoza saw and commented on, it most surely employed something of the bâton mechanism. And it is likely that it is at least in part this that Spinoza is commenting on when he says: “experience has taught that with spherical pans, being polished by a free hand is safer and better than any machine”. But the automated potential of Huygens’s machine exceeds this semi-assisted mechanism, for there is a long history of Huygens’s conceptual experimentation with a fully automated device which would both hold the glass blank, but also turn and grind it. In these the glass blank and the forming pan apparently spun against each other in opposing directions. Here are several of these prospective machines:

Oeuvres Complètes, XVII (p.303)

 Oeuvres Complètes, XVII (p.304)

As Graham Burnett describes:

They are gear- and belt- driven, imparting both rotary and epicyclic motion to the glass blank, and they are all represented as self-contained boxes out of which lenses would emerge more or less by the turn of the crank. In fact in [the figure from page 303 of OC], it appears that the crank itself was forgotten and had to be added as an afterthought – a pentimento that speaks volumes concerning the preoccupation with excessive of the process (98 )

Burnett’s global point, if I read him right, is that Huygens’ plans for a completely mechanized production of mathematically exact lenses, purged from the human errors of the craftsman, is in the heritage of Descartes own, highly unrealistic schemata for a hyperbolic lens-grinding machine, symptomatic perhaps of a tendency to divorce body from mind. Burnett is quick to point out that Huygens, unlike Descartes, had extensive experience both in grinding lenses, and using them for discovery (for instance his discovery of the moon and rings of Saturn in 1656 is epic), yet the overall point of this tendency in conception holds. And likely it is to this that Spinoza is in some degree responding.

To better conceive of the contrast between whatever state the Huygens machine may have exhibited (in this spectrum of automatizations), and the simple lathe Spinoza may have employed, a juxtaposition of one of Huygens’s drawings a reproduction of a possible Spinoza lathe will serve:


 Oeuvres Complètes, XVII (p.302)

From the Middelburg 400th Anniversary of the Telescope Exhibit, design from Manzini’s “L’occhiale all’occhio, dioptrica pratica”  (1660), circa 1614.

From Manzini

One can immediately see the kind of condensed block mechanism that Huygens would like to have built, and to some degree had built, and the kind of traditional lathe that Spinoza may have used. In fact I have come to strongly suspect that in addition to the simple hand driven lathe depicted above, he likely also used a spring-pole lathe (such as the one in the Rijnsburg museum [here], though this museum piece is not of the period, nor a lens grinding lathe), most likely of the kind Hevelius used (pictured below) the hypothesis discussed here:

Spinoza’s lathe emphasized personal skill, the sensitive hand-eye-machine interface that drew not only on experience and a patient, attentive eye, but also on the particular passed on abrasive recipes and techniques of individual masters. Huygens’s ambition, as was Descartes’ was to transcend the event of crafting, mathematically. That is, with a mathematics that was embodied by the mechanism itself he hoped to simply machine the accuracy. Spinoza’s doubts to whatever fabrica he saw at the Huygens Estate were doubts about removing the “free hand” from the technology. And there is something to this that goes beyond whether this machine or that is at any one moment in history the better machine.

Speculatively: What Spinoza has in mind with the “free hand” is that the human element must be included in any epistemological assemblage. He would no more refuse the mechanized advances in contemporary technology than he would refuse more and more adequate ideas, but he would still look for the “free hand”, the touching point that circulates that knowledge back down to the user, and other men. Technical knowledge still must be human knowledge. The causes of things related to the causes of men. This is what I believe he meant by the fourth stimulation of the “means necessary to attain our end” in the Emendation of the Intellect:

4. To compare this result [the extent to which things can and cannot be acted upon] with the nature and power of man.

There is no doubt that Huygens was on the right track. His mentality was to lead him to a wave theory of light to complement Newton’s spectrum discoveries of the same. In fact, Huygens’s scientific discoveries and inventions are prodigious for the age, but it is good to note that Spinoza in the year of 1665 was fairly close to Huygens, and in many ways Spinoza’s optical and practical knowledge circulated with that of Huygens. That latter would affirm as late as 1668 that Spinoza was in fact right all along about the superiority of small objectives in microscopes, and had marveled at the lens polish that Spinoza was able to achieve through rather craftsman-like means. In reading the objection that Spinoza makes to Huygens’ machine one should understand it at two levels. The first is simply the pragmatic matter of an experienced craftsman who is not intoxicated by technical marvels in their own right. The turning of shiny gears does not make his heart sing. Taking his hand off the lens seems to him one of the last things one would want to do, and it would take a striking result to convince him otherwise, a result which Huygens would not be able to provide. The second level is as vast as the first is earthbound. Spinoza’s notion is that no matter how intricate the device (or the mathematical figure), the meaning of its products, the degrees to which their ideas set us free or not, must relate back to the human being itself, as it finds itself in history. In a sense, Spinoza is looking microscopically beneath, and macroscopically beyond Huygens’s improvements in his letter 32, as a craftsman and a metaphysican.


Spherical Aberration: Descartes’ Solution

Philosophical Context: Optics

For those unclear about what spherical aberration is, and attempting to follow Spinoza’s comments about Descartes below, it is the tendency of rays at the edge of a spherical lens to refract at a focal length shorter than those near the axis:

This tendency is corrected by a hyperbolic lens, as theorized by Rene Descartes, which in its progressive sloping, sends the rays to one mathematical point.

It was Spinoza’s thought, probably working from the theories of mathematician Johannes Hudde, that light rays gathered at what he called a “mechanical point” and not a mathematical point. So spherical aberration was considered inconsequential, a fact of working with lenses and light. (A practical mid-century address of spherical aberration in telescopes was the use of much larger, flatter lenses, with longer focal lengths, making telescopes sometimes reach lengths over 30 ft.)

Unfortunately, for the greater part of the 17th century this spherical aberration effect was confused with, and assumed to be the cause of, chromatic aberration which produced a bluish halo at the edges of images, and which is, in most cases, more disruptive than spherical blurring. Not until Newton discovered the spectrum components of light from 1670 – 72 was it theoretically realized that the one was not the other. Even Christiaan Huygens, as he worked to resolve spherical aberration solely with spherical lenses, probably thought that he was resolving what was latter understood to be chromatic, as he was somewhat deflated by Newton’s discovery:

Spinoza and Hooke’s Micrographia: The minascule made Large

Look at Robert Hooke’s incredible, and conception-changing Micrographia (1664). And see it as if you are looking at the very book with excellent viewing software, at Turning the Pages Online (click “Turn The Pages”). This book must have struck one as if from another planet. See the overleafs open up into the most extraordinary illustrations of the smallest of things. It was the 3D, Surround Sound, epic film of its time.


As wiki writes of it:

Hooke most famously describes a fly’s eye and a plant cell (where he coined that term because plant cells, which are walled, reminded him of a monk’s quarters). Known for its spectacular copperplate engravings of the miniature world, particularly its fold-out plates of insects, the text itself reinforces the tremendous power of the new microscope. The plates of insects fold out to be larger than the large folio itself, the engraving of the louse in particular folding out to four times the size of the book. Although the book is most known for foregrounding the power of the microscope, Micrographia also describes distant planetary bodies, the wave theory of light, the organic origin of fossils, and various other philosophical and scientific interests of its author

Published under the aegis of The Royal Society, the popularity of the book helped further the society’s image and mission of being “the” scientifically progressive organization of London. Micrographia also focused attention on the miniature world, capturing the public’s imagination in a radically new way. This impact is illustrated by Samuel Pepys’ reaction upon completing the tome: “the most ingenious book that I ever read in my life.

This is the book that Oldenburg speaks of when he writes Spinoza about his hope that English booksellers will soon be able to send copies of various important books (the Second Anglo-Dutch War had interrupted commerse):

There has appeared a notable treatise on sixty microscopic observations, where there are many bold but philosophical assertions, that is, in accordance with mechanical principles (April 28th, 1665)

Spinoza answers in May, regarding his new talks with Christiaan Huygens, (it is unclear if he has just met him, or if Spinoza is answering in a condensed fashion, having not mentioned to Oldenburg this relationship before – he may even have known him since the summer of ’64 [letter 30A:”…I know that about a year ago he told me”]):

The book on microscopic observations is also in Huygens’ possession, but, unless I am mistaken, it is in English. He has told me wonderful things about these microscopes, and also about certain telescopes in Italy (letter 26).

Several prospective questions arise here. As mentioned, we are unsure if Spinoza has just met Huygens (Nadler brings up a counter argument beside the one that I suggest), so we cannot say if Spinoza was able to look at the book at the Huygens estate. At this point he seems only to have heard of it, but this may even be a polite deferment. If Spinoza did visit the estate and looked at its pages one certainly can imagine its impact upon the lens-grinder. He must have been mystified and pleased. There is a very good chance that its viewing set off a change in Spinoza’s thinking about his lenses, optics and the world, for by letter 32 in late November Spinoza is making metaphysical analogies that seem to appeal to microscopic observations, the “tiny worm living in blood”¹. This seems to suggest that somewhere in the summer of 1665 Spinoza became more focused on both the telescopic and the microscopic uses of lenses. Years later, after much experimentation, Huygens would finally admit that Spinoza was right that the smallest of lenses were best for microscopic viewing. Nadler suggests, via Klever, that Spinoza had a reputation for his telescopes and lenses as early as 1661, (“Borch’s Diary”, from A Life 182). Whether Spinoza in this summer decided to make new observations, or had already been making them, or if there were microscopes at Huygens’ estate, we cannot know.

Whether Spinoza was working with microscopes or not, the presence of the Micrographia at the Huygens estate, the likelihood that Spinoza would have seen its breath-taking layout (not to mention the possibility that Hooke’s generous and detailed description of how he made his lenses by a thread of glass was relayed to him), combined with Huygens own experiments with lens designs, lens lathes, and spherical aberration at the time, the summer of 1665 must have had a concerted conceptual and imaginative impact on Spinoza’s thinking and practices.

1. There is scientific context for the imagine of worms in blood: The “dust” on old cheese was found to be not dust at all but little animals, and swarms of minute worms were discovered tumbling about in vinager (Fontana 1646, Borel 1656, Kircher 1646). Kircher announced that the blood of fever victims also teemed with worms, and there was talk that they infested sores and lurked in the pustules of smallpox and scabies. (Ruestow, 38).




Also featured at Turning The Pages On-line: Ambroise Paré’s Oeuvres; Conrad Gesner’s Historiae Animalium and Andreas Vesalius’s De Humani Corporis Fabrica

Spinoza’s Blunder and the Spherical Lens

Did Spinoza Understand the Law of Refraction?


In seeking to uncover the nature of Spinoza’s lens-grinding practices, and the competence of his optical knowledge, I believe I may have uncovered another small, but perhaps significant mis-translation of the text. The text is letter 39, written to Jelles (March 1667)), wherein Spinoza explains the insufficiency of Descartes treatment of the cause by which objects appear smaller or larger on the back of the eye. Spinoza points out that in La dioptrique Descartes fails to consider the angle of incidence of the rays at the surface of the lens, and only considers “the crossing of the rays proceeding from different points of the object” at various distances from the eye. I believe that there are subtle epistemological issues that come with this point, and which Spinoza has in mind, but for now it is best to pay attention to what follows this complaint.

Spinoza suspects that Descartes has fallen intentionally silent on this factor, in part because Descartes has it within his plan to promote the importance of the hyperbolic lens, a reported improvement upon spherical lenses. Hyperbolic lenses, which Descartes championed and tried at great length to produce in an automated way, by the time of Spinoza’s writing had proven to be impossible to manufacture. Yet here Spinoza objects to them at the geometrical level. Spinoza argues that their means of collecting light rays, due to their aspherical nature, was less proficient than that of spherical lenses. I shall have to leave this immediate comparison of lens shapes aside, as well, to narrow our attention to a reported blunder Spinoza makes in his explanation, a blunder that has lead some to conclude that Spinoza really could not have had much comprehension of theoretical optics at all. He might have known how to make lenses, but was unclear as to how they worked. The charge is that he didn’t even understand the law of refraction: the principle relation between the angle of incidence and the refractive properties of a lens medium, derived by Descartes to explain the main insufficiency of spherical lenses: spherical aberration. 

Spinoza draws a diagram to help explain to Jelles just what the advantages of spherical lenses are. It depicts what Spinoza regards as a natural product of its symmetry, the capacity to focus rays that are parallel along its infinite number of axes, across its diameter, to a point opposite:

This is how Spinoza describes the relation, as translated by Shirley, (a wonderful translator):

For the circle, being everywhere the same, has everywhere the same properties. For example, the circle ABCD has the property that all the rays coming from the direction A and parallel to the axis AB are refracted at its surface in such a manner that they all thereafter come together at point B. Likewise, all the rays coming from the direction C and parallel to the axis CD are refracted at the surface in such a way that they all come together at point D. This can be said of no other figure, although hyperbolae and ellipses have infinite diameters

Now this description has lead notable, modern critics to conclude that Spinoza seems to have failed to understand the Law of refraction altogether, for in seeking to raise the value of the figure of the circle Spinoza has attributed properties to it that appear to ignore just that law. Principle here is the apparent, unqualified claim, “the circle ABCD has the property that all [parallel] rays coming from direction A…are refracted [at] point B”:

This is how Alan Gabbey articulates the failure on Spinoza’s part, and the reasonable conclusion to be drawn from it:

One’s immediate suspicions of error is readily confirmed by a straightforward application of Descartes Law of refraction. For the circle to have to the dioptrical property Spinoza claims, the refractive index of the glass would have to be a function of the angle of incidence [Gabbey secures his point with a footnote of the formula], a condition of which there is not the slightest hint in the letter. In his next letter [March 25, 1667] to Jelles, who asked for clarification, Spinoza explained that light rays from a relatively distant object are in fact only approximently parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintains the same property of the circle in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous “Ovals of Descartes”] that Descartes had constructed in Book 2 of La Géométrie to provide a general solution to the problem of spherical aberration [Ep 40]. I suggest on these grounds alone that though Spinoza may well have written a treatise on the rainbow [which he allegedly burned shortly before his death: Spinoza 1985b: 8], it is very unlikely that he wrote Stelkonstige Reekening van den Regenboog (“Spinoza’s Natural Science and Methodology” in The Cambridge Companion to Spinoza, 154)

For me what is at issue is not Gabbey’s concern as to whether Spinoza wrote the particular treatise on rainbows or not; it is the general appraisal of Spinoza’s aptitude in the theoretical optics of his day. Just what did Spinoza understand? Gabbey’s point rests on the slender idea that Spinoza does not seem to be able to even conduct “a straightforward application of Descartes’ Law of refraction”.

The problem arises I think from a slight neglect in the translation, the loss of the word “if”. I do not know what translation Gabbey was using, but if he was working from the Latin I believe he made a similar mistake. This is the Latin taken from his Opera:

Nam quia circulus idem ubique est, easdem ubique proprietates habet. Si, exempli causa, circulus A B C D hanc possideat proprietatem, ut radii omnes, axi A B paralleli, a parte A venientes, ad eum modum in ejus superficie refringantur, ut postea omnes simul in puncto B coeant; omnes quoque radii, axi C D paralleli, a parte C venientes, ita in superficie refringentur, ut simul in puncto D conveniant; id quod de nulla alia figura affirmare licet, licet Hyperbolae ac Ellipses infinitas habent diametros (the conditionals emphasized)

Key is to note that Spinoza has shifted into a hypothetical subjunctive. It is not as Shirley translates: “For example, the circle ABCD has the property that…” (though Shirley’s translation does not wholly obscure my point). The line reads “If, for example, circle ABCD should have the property that…”. It is not an assertion. This small difference has lasting effects on the nature of the point that Spinoza is trying to make. He is not attempting to say that circle ABCD, and therefore all circles, has the general property that ALL rays that are parallel to ANY of its axes, would be refracted to a point opposite. This would be absurd, and as a lens-maker and user he would know this. He is saying that given that this particular circle, as it represents a lens, can be said to have this property that all rays [all rays so depicted] come from direction A and do refract to a point opposite B, then another set of rays, at the same angle of incidence to a different axis would be refracted to another point in just the same manner [C to D].

Thus, what Gabbey mistakes as a general definition of the capacities of a spherical lens, ANY spherical lens, is actually for Spinoza a description that has two levels of comprehension. The first level is manifest and actual, the second is abstract and geometrical. At the manifest level circle ABCD is understood to be a description of a hypothetical, though real lens, with specific properties and focal points (here the Law of refraction is simply assumed); at the second level, given the acceptance of the first level, there a general property of ALL spherical lenses due to their geometry. This property is: if the index of refraction allows the focusing of a set of rays at a range of angles of incidence to a particular axis (as drawn), so that they meet at one point, the same will be said of other such axes of the arc. The loss of the hypothetical “If” in translation makes it appear that Spinoza is making a much broader claim about sphericals which simply ignores the Law of refraction. He is not.

Here is the text in a translation more sensitive to the conditional (wording suggested by S. Nadler):

For because a circle is everywhere the same, it has the same properties everywhere. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D, this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters.

The construction of his thought is: “if this can be said of this particular figure, it can be said of no other figure, hyperbola and ellipse included”. We can see that part of the problem of translation lies with the very condensed way in which Spinoza is employing the subjunctive. His hypothetical has two levels of comprehension. To repeat, the figure ABCD is being treated as a real world manifestation, and so has particular properties such as the limitations granted to it by its index of refraction as being made of glass, but also it is taken to be an illustration of an abstract property of spherical shapes in general. It is both general and particular, and, as such, it is quite easy to miss both aspects, as Alan Gabbey and others seem to have done.

To really make clear Spinoza’s point, as he sees it, one just has to overlay his circle with the figure of a hyperbolic, and assume, as Spinoza does, that both sets of parallel rays (A and C) would be refracted to a single point (B and D). The hyperbolic lens simply would not have the same property:


There are several issues at stake here, some of them optical, some epistemological, some pragmatic, most of which must be left aside for the moment. It is my purpose to expose that Spinoza is making a point about the capacities of spherical lenses. We are to understand, under his proposed description, that these are lenses that are capable of focusing rays at a mutuality of angles of incidence by virtue of the geometrical properties of their class.

Although a surface reading of Spinoza’s diagram and description seems to present an elementary blunder, and may lead to the conclusion that he did not even understand how the Law of refraction was to be applied, a closer reading shows, at the very least, that he was intimately aware of Descartes’ law, and how Descartes attempted to solve the problem of spherical aberration. Key to understanding Spinoza’s approach was that he considered spherical aberration itself to be incidental to the limitations of lens use. Most likely he thought of lenses as devices, and the slight blurring at the edges of focus to be endemic to them. In this he followed closely his friend in correspondence, mathematician Johannes Hudde, who in his now lost treatise, Specilla circularia, argued just this conclusion. Instead of accepting Descartes’ attempt to define a mathematical point of focus, and construct a lens to achieve it, Hudde claimed that the point of focus of a lens is what he called a “mechanical point”. And it is precisely this “mechanical point” notion of focus that Spinoza has in mind when he speaks of rays meeting up at points B or D in his figure, (he uses the phrase in the following letter of clarification to Jelles, letter 40). Spinoza may have been on the wrong side of the argument, but to decide that he was unfamiliar with the argument itself in a fundamental way would be incorrect.

There is an additional piece of evidence of Spinoza’s theoretical familiarity with the application of the Law of refraction. We have the historical fact that he was in conversation and likely visitation with Christiaan Huygens in the summer of 1665, just as Huygens was working out his solution to spherical aberration using solely spherical lenses in composite. Huygens was bent on solving the aberration problem in a non-Cartesian way, as he meant to publish in his Dioptrics. In letter 30A to Oldenburg, as Spinoza responds to the Royal Society secretary’s urgings for an update on Huygens’s progress, despite Spinoza’s love of the spherical lens he expresses his doubts that Huygens would solve the problem that Descartes thought he had solved with hyperbolics:

The problem which he says he is trying to solve in Dioptics is as follows: Is it possible to arrange the lenses in telescopes in such a way that the deficiency in one will correct the deficiency in the other, and thus bring it about that all parallel rays passing through the objective lens will reach the eye as if they converge on a mathematical point? As yet this seems to me impossible. Further, throughout his Dioptrics, as I have both seen and gathered from him (unless I am mistaken), he treats only spherical figures. (quoted on October 7, 1665).

By the time that Oldenburg had read this Huygens indeed had solved much of problem of spherical aberration using two spherical lenses. (He would improve upon his solution later.) While it is likely Spinoza did not know of Huygens’s success – as he kept his discoveries close to the vest – Spinoza most certainly had discussed with Huygens, and seemingly read in the draft of the Dioptrics itself, the theoretical nature of the pursuit. He likely understood the issue that was at hand, most particularly the failure of Descartes to provide a practical solution to the problem of aberration, and how the Law of refraction itself might be used to overcome this aberration. It is noteworthy that Spinoza denies the possibility of Huygens’s aim in the context of his working solely spherical figures.

Given these factors of a theoretical correspondence with Hudde, the reading of Hudde’s treatise, and also Spinoza’s personal exchange with Huygens, the closer translation of letter 39 attests to a proficiency of optical knowledge far greater than what Alan Gabbey allows. Spinoza did seem to be more than familiar with the “importance” of Descartes figures, in particular, with the disappointments of that importance, at least as far as they were understood in his day. Central though to the issue of the “if” and the subjunctives that follow, is that the conditional itself assumes Spinoza’s own “mechanical point” (Hudde) notion of the focus of a lens. When Spinoza says, if circle ABCD has the property that certain parallel rays would meet at point B, he means at “mechanical point” B. Granting that they meet at such a mechanical point, other such rays must meet in a similar fashion at other points, as due to the nature of a circle, and this could not be said of Descartes’ other figures.

[For a Full Treatment of Spinoza’s Letters 39 and 40: Deciphering Spinoza’s Optical Letters ]

Huygens’s Lens

This struck me as such a remarkable image, one that I have found in my just nascent research into Spinoza’s lens-grinding techniques (Huygens was a companion and nearby neighbor of Spinoza’s in the years 1664 and ’65). This is the lens with which Huygens discovered the first moon of Saturn (in March of 1655), and then the rings, as unlike Descartes before him he embodied his studies on the nature of optics.

There are several things that are eye-catching. First is its thinness and size. Likely ground by hand, it is merely 5.7 centimeters across and yet has a focal length of over 3 meters. It is only 3.4 millimeters thick.

Then there is the inscription at the edges, found scratched into the wax-yellow material:

A line from Ovid:

Admovere oculis distantia sidera nostris : They carried distant stars to ours eyes.

In anagram:

Saturno luna sua circunducitur diebus sexdecim horis quatuor : With Saturn, his moon circles itself around in 16 days, 4 hours”

It is this artifact of discovery, the materiality of the lens, emphasized by inscription, the presence of the poetry and the observation, that we glimpse something of the physicality of sciences, and the conceptions that enlighten it.

Later, when other moons besides this one were discovered, Huygens simply called this moon, “my moon”. One gets a sense of the personal, material and historic conception of invention.


Spinoza’s Comments on Huygens’s Progress

The Particularities of Spinoza’s Questions of Technique 

In Spinoza’s letter (15/32) to Royal Society secretary, Henry Oldenburg, after a summary of the reasons why Spinoza believes that “each part of nature agrees with its whole,” in which our knowledge position is compared to that of a worm living in our blood, the letter finishes with a few seemingly mundane topics, one of which is on Oldenburg’s interest in scientist Christiaan Huygens’s work. It is a passing note, but telling, both in its tone and substance:

The said Huygens has been a totally occupied man, and so he is, with polishing glass dioptrics; to that end a workshop he has outfitted, and in it he is able to “turn” pans – as is said, it’s certainly polished – what tho’ thusly he will have accomplished I don’t know, nor, to admit a truth, strongly do I desire to know. For me, as is said, experience has taught that with spherical pans, being polished by a free hand is safer and better than any machine.

Dictus Hugenius totus occupatus fuit, et adhuc est, in expolientis vitris dioptricis; in quem finem fabricam adornavit, in qua et patinas tornare potest, fatis quidem nitidam; quid autem ea promoverit adhuc nescio, nec, ut verum fateor, valde scire desidero. Nam me experientia fatis docuit, in patinis sphaericis libera manu tutius et melius expoliri, quam quavis machina.


In terms of tone, we get a sense of what Spinoza thinks of the wealthy Huygens’s fabrica. The shop has been fully furnished [ adornavit ] and perhaps Spinoza’s humorous word-play is evident as he calls it nitidam, a shiny, glossy or polished thing. It is a workshop for polishing lens, and itself is quite polished: spiffy. We can feel a contrast to Spinoza’s much more humble abode and hand-polishing buisness. He is the still the merchant thinker, the “Jew of Voorburg” (a village outside The Hague) or at times the “Isrealite” whose small lenses have a “remarkable polish”, in the mind of Huygens. Huygens, at the cusp of a mechanical age when the machines still have the aura of the divine about them, is pursuing a mechanized way of producing lenses, one that Spinoza cannot embrace at quite a few levels.


Also interesting is that there seems some lexical ambiguity which obscures just what is being turned. Is it the patinae (templates), or is it the patinae (understood as “tools”)? There are various descriptions of the full lathing process. By Cherubin’s report (1671), which may be rather late, and more complex than usual, there are three stages to lens-making: turning, grinding, and polishing, all of which though can be called “turning”. Turning first involves the making of templates proably made of iron, but in Huygens’s case may even be made of the superior material brass. Two are made in a convex/concave pair, and these are ground against each other to insure sphericality. These turned metal templates are then used to grind and polish a pair of brass or iron “tools” which are then used to grind and polish the glass blank. It seems the case that what Huygens’s workshop is capable of is not only polishing lenses, but also of grinding tools, and even perhaps turning the templates themselves. There is some evidence that the turning of the template was in the case of spectacle making done by a guilded turner (17).¹ So it is not perfectly clear enough if the patinae are templates or tools. But what does seem clear is that Huygens’s is a kind of impressive all-in-one machine, or workshop, one that does more than the usual. This is also suggested from his notebook drawings from the period. Possibly, everything from the work of the turner on to the final glass product can be achieved. [Spinoza, by the evidence of his letter to Hudde shown below, at least at the time of the writing of that letter, may have not only had his patinae (templates or tools) made for him, but perhaps even both, as he there uses the term scutellae (dishes), which are to be fashioned by someone else. Whether this term is synonymous with patinae is undecided.

In the latter half of our considered passage the image becomes potentially more complex. It is commonly translated and understood that Spinoza is talking about his experience with polishing of the patinae (templates/tools) themselves. Perhaps. Yet, in patinis sphaericis really gives the perspective of the polishing being carried out in respect to the use of patinae – perhaps even “within pans”, as in: “being polished by a free hand in spherical pans, is safer and better”. A free interpretation of the Latin does not easily produce the idea that the polishing is solely being done to the pans themselves. This is further supported, I believe, by the passive form of the infinitive expoliri. The sense is that being polished by the means of a free hand (the technique for hand-holding a glass blank, or any polishing device onto the pan) is both safer and better, with the infinitive operatating as the subject of the clause.

These are of course tentative thoughts about this passage, but it seems that instead of reading Spinoza’s comments as pertaining only to his long-time experience of polishing metal pans – patinae – Spinoza seems to be talking instead about his preference for using a free hand for glass itself, in metal forms. This matches up with the known fact that Huygens’s machines were ones that held the glass blank fixed in some mechanically guided way, put against the form as part of the final process. It makes more sense for Spinoza to be responding to this semi-automated, glass-grinding aspect of Huygens’ machine, and not just a form-polishing technique. The entire mechanism is organized in a way that defies Spinoza’s experienced wisdom of grinding and polishing.

Lastly, by specifying the sphericality of the patinae, he is also setting himself against any of the much-pursued quests for a way to mechanically produce Hyperbolic Lenses, (initiated by Descartes own discovery of a law of refraction, his own belief that the Hyperbola was a revealing form). Not only is Spinoza commenting upon Huygens’s social affluence, in an off-hand way, but also upon any non-spherical lens aims, and the idea that an insensate hand could create the fineness of results needed.

Here is Elwes’ translation of the passage:

The above-mentioned Huyghens is entirely occupied in polishing lenses. He has fitted up for the purpose a handsome workshop, in which he can also construct moulds. What will be the result I know not, nor, to speak the truth, do I greatly care. Experience has sufficiently taught me, that the free hand is better and more sure than any machine for polishing spherical moulds. I can tell you nothing certain as yet about the success of the clocks or the date of Huyghens journey to France.

And Shirley:

The said Huygens has been, and still is, fully occupied with polishing dioptical glasses. For this purpose he has devised a machine in which he can turn plates and a very neat affair it is. I don’t yet know what success he has had with it, and, to tel the truth, I don’t particularly want to know. For experience has taught me that in polishing spherical plates a free hand yield safer and better results than any machine.

I hesitate of course to re-translate such esteemed translators, but it seems that there are good arguments for reading Spinoza’s meaning another way. At the very least, the possibility of a second meaning seems present.


For evidence that Spinoza himself did not fashion his own templates (and perhaps not even his own “tools”), from the letter to the mathematician Hudde (41/36):

“Hisce finirem; verum, quia, ut mihi novae ad polienda vitra scutellae fabricentur, animus est, tuum hac in re consilium audire exoptem. Non video, quid vitris convexo-concavis tornandis proficiamus”

“With these I may have ended, in truth, but because for me new dishes for glasses being polished may be fashioned, such is the spirit, your council in this matter I would be eager to hear. I do not see what we may profit in ‘turning’ convex-concave glasses.”

“I might have ended here, but since I am minded to get new plates made for me for polishing glasses, I should very much like to have your advice on this matter. I cannot see what we gain by polishing convex-concave glasses” (trans. Shirley: likely June 1666; page 142 Opera).

It does not seem likely that Spinoza had very much experienced, first hand, the safety of fashioning metal plates (it may of course be the case that though not fashioning them, he did polish them, but then the safety of the process – mentioned in the letter to Oldenburg – would seem to be less of an issue).  To sum up, it seems more the case that Spinoza is, first, noting the furnished and “spiffy” nature of Huygens’s fabrica, as it polishes lenses (and even patinae ), and then secondly, that Spinoza is referring to and judging a more particular aspect of Huygens’s machine – one well-known, since Huygens was working on a semi-mechanized process for polishing lenses for a decade – that the glass is not held in a free hand.



1. D. J. Bryden and D. L. Simms. “Spectacles Improved to Perfection and Approved by the Royal Society.” Annals of Science 50 (1993) 1-32.