Spinoza: Not As Abused As Is Said

Two Kinds of Disparagment Found In the Huygens Letters

I am looking at the references to Spinoza made by Christiaan Huygens, coming to them with the expectation that they would reveal a general disparagement of the man, either in terms of his optical knowledge, or in terms of his person, for these letters have been characterized as proof of a certain diminishment Spinoza had suffered in the minds of those who came to know him.

I quote below two sources that typify this kind of conclusion.

It is however the letters which Christiaan Huygens wrote to his brother Constantijn between 9 September and 11 May 1668, which provide us with the clearest evidence that by then, those engaged on actual research into dioptrics had begun to take a somewhat patronizing attitude to Spinoza’s theorizing on the subject. They make it perfectly clear that although Huygens valued “our Israelite’s” practical skill in producing first-rate lenses, he thought it very unlikely that he was capable of adding anything of value to the understanding of optical phenomena (97)

– Spinoza’s Algebraic Calculation of the Rainbow & Calculation of Chances, by Michael John Petry  

Over time, he earned praise from some notable experts for his expertise in lens and instrument construction. Huygens, writing to his brother from Paris in 1667 (when Spinoza was living in Voorburg) noted that “the [lenses] that the Jew of Voorburghas in his microscopes have an admirable polish.” A month later, still using the somewhat contemptuous epithetet – occasionally replaced in his letters by “our Israelite” – he wrote that “the Jew of Voorburg finishes [achevoit] his little lenses by means of the instrument and this renders them very excellent (183) 

– Spinoza: A Life, by Steven Nadler

The Optical Israelite

While Petry finds in these letters clear evidence for an accumulation of doubt as to Spinoza’s capacities as an optical thinker, the relegation of him to simply that of an excellent craftsman, Nadler acknowledging that although Spinoza’s instrument achievements were much respected, strongly suggests that he was seen merely as a “Jew”, or perjoratively as “our Israelite”. The picture that is left by these writers and others is that somehow Spinoza was seen in a poor intellectual and ethnic light by the Huygnses.

In looking at these letters, this simply does not seem to be the case. Firstly, Nadler’s implied characterization that in these letters Spinoza is ONLY the Jew or Israelite does not hold. He is also “Le Sieur Spinoza” , “Sir Spinoza” (September 9, 1667, May 11 1668), and just “Spinoza” several times. He is also addressed in combination of “Spinoza et Monsieur Hudde” (Semptember 23, 1667); whether this is a sign of his diminishment in contrast to Mr. Hudde, or one of familiarity is hard to weigh. In fact it is hard to measure the full texture of the Jewish nomenclatures, some of which Nadler finds distinctly “contemptuous”. There very well may be social contempt in these, but the title “the Jew of Voorborg” may be a title Spinoza had somehow informally gained in circles, and not simply one of Christiaan’s invention, and though “our Israelite” may strike our eyes in a jarring fashion, it is difficult to parse out the affection from the diminishment, if indeed there is such. (To understand what Huygens means by “our Israelite” one for instance may have to anachronistically ask, Is Spinoza diminishing others when he refers to the “Brazilian” in his waking dream, as an “Ethiopian” [Ep. 17].)  Because of these telescopic difficulties across centuries, at the very least I want to present the picture of the Huygenes social relationship to Spinoza as more complex and varied than what I assumed by reading the tale of these references without looking at them. And I wish to open the possibility that there was more social respect there, against the tremendous currents of the prejudice of the times, than otherwise would be assumed possible in a less nuanced reading, a respect that Spinoza had personally earned across social barriers.

Petry’s point I am unclear on, for in the letters Spinoza’s optical (vs. craft) acumen does not seem to be in question. There seems to me to be clear evidence rather that Spinoza rather had collaborated with the well-respected mathematician Johannes Hudde on calculations for a 40 ft. lens (Sept 23, 1667), and that these calculations had perhaps influenced Huygens’ own calculations for even longer lenses. Perhaps Petry has in mind Huygens’ thoughts in his May 11, 1668 letter, where Huygens discusses his new eyepiece with Constantijn. Spinoza certainly had no knowledge of the optics of this eyepiece, or its principles, but if I am reading Huygens correctly, this is his proposed solution to spherical aberration using only spherical lenses (against a hyperbolic solution). Not only would Spinoza have no knowledge of these principles, neither would any other man in Europe, Johannes Hudde included. I am unsure if we could say that this was a “patronizing attitude”. I am certainly open to evidence to the contrary.

Others have suggested Christiaan’s warnings to Constantijn should keep quiet about his invented lenses, and not reveal them to Spinoza, proves that he regarded Spinoza to be a “competitor” in lens-making. I find this an odd, or perhaps incomplete conclusion. Christiaan’s invention simply was not ready to be made public, and he knew Spinoza to be at times in close contact with Oldenburg, the secretary of the Royal Society of London. Spinoza had kept Oldenburg abreast of the details of Huygens’s progress. There is a sense though in which Spinoza may have been a competitor to Christiaan. The Huygens brothers may have had an intimate relationship to lens-grinding, and there are signs that Constantijn grew cold to Christiaan’s instructions when Christiaan had gone to Paris. The lenses ground during the time of their separation are thought by Anne van Helden to have been entirely farmed out to craftsmen. If though Constantijn continued his conversations about optics and lenses with his neighbor Spinoza, having lost his brother partner to fame in Paris, indeed Spinoza may have represented, however slightly, an emotional threat to Christiaan. It seems, by several accounts, that Spinoza was an engaging man to talk with. Any disparagement we may find in these letters from Christiaan, insofar as we find it, I think should be understood within this context as well, that the brothers were extremely close on the subject and practice of lens-grinding.

A Method of Grinding Small, Spherical Lenses: Spinoza

Van Gutschoven’s Design for Grinding Small Lenses: Letter No. 1147

We have in a letter written to Christiaan Huygens by G. van Gutshoven, descriptions and diagrams of the essential processes for grinding small spherical lenses, as they were likely shared by most contemporaries of the age. The letter is surely a response to a request from Christiaan who may have been in need of smaller lens grinding techniques, either due to his future interest in compound eye-pieces for telescopes, or in regard to the question of the best lenses for microscopes which would later arise in discussions between himself and Johannes Hudde and Spinoza as well. In any case, van Guschoven an Antwerp mathematician, was Huygens’ initial teacher of the complete essentials of lens grinding in the first place, which he gave to him ten years before in a letter dated Feb. 10, 1653. It was by the aid of these instructions, among others,  that Huygens was able to grind one of the most powerful lenses in Europe, and discover the rings of Saturn in 1656.

This letter is dated only as 1663 by the editors of Huygens’s Oeuvres. 1663 was the year that Spinoza had moved to Voorburg, where the Huygens family kept their Hofwijck country estate. That spring Spinoza rented rooms in the home of master painter Daniel Tydeman, but a five minute walk from the Hofwijck. But Christiaanwas not yet there. He was living in Paris with his father who was attempting to curry the diplomatic favor of King Louis XIV, an effort which would result in Constantijn Sr.’s son becoming the secretary to Louis’ Royal Academy of the Sciences, in 1666.

None of this has occurred yet. Christiaan and Spinoza have not yet met (unless they crossed paths momentarily in the summer of 1663, when a traveling Christiaan took leave of Paris to go to London in the off-season). Huygens would not arrive in Holland and develop his relationship to Spinoza until after May of 1664.

What this letter reveals to us though is the basic mechanism and techniques used in the grinding of small lenses. We know that Spinoza made microscopes (and telescopes) at least since the year of 1661, and in his coming debates over techniques and optics with Christiaan he would champion much smaller, more highly curved lenses for microscopes, against Huygens’ designs of lesser magnification. One would think that from van Gutschoven’s descriptions we can receive a sense of the physical practices that preoccupied Spinoza for many of his daylight hours, specializing at times in these smaller lenses.

It should be noted that the Huygens brothers by this time are among the best lens-grinders in Europe, and Christiaan had already worked on several sophistocated semi-automated designs of grinding machines. These instructions must have been experienced as extremely rudimentary to Christiaan (or perhaps, it is from another date).

The letter has three figures, pictured below. The first of these shows a vertical grinding form that is likely of iron or copper. One can see the core movement of a lathe, as foot petal likely drove the strap that turns the shaft, spinning the form concentrically. For larger lenses the form would be hollow, holding the concavity of a curvature that one would want the glass to have. Here though, the small lens is to be ground in the “canal” near its lip:

“Now in this hollowed out canal C D you will grind glasses affixed to a handle and pressed into the canal, with the handle in the hand continuously; while grinding the glass you would turn it until all parts of the glass are equally ground.”

After this equanimity is roughly achieved, attention is turned to the “laminate” or layered strip A B, which turns so the top of it is horizontal to the turner’s bench. By van Gutschoven’sdirection, the laminate is of a soft wood, polar or willow. (Other techniques of the day call for paper.) The roughness of dimples are by hand ground away, and Tripoli, which is a chalky substance made of the remains of microscopic marine life, is added to the laminate to smooth the way.

After this, there is a third process recommended which can either be done in a concave wooden form G H, it too aided by Tripoli, pictured here:

Or, what seems to be a pillow (plombae), affixed to a lathe shaft EG:

There are several things of interest here. The date of the letter makes this description contemporaneous with Spinoza’s own practices, so one might assume a basic correspondence. The grooved canal method strikes one as similar to those a-centric grinding techniques discs used by diamond polishers which Spinoza may have come in contact with either briefly as a merchant of gems, or simply by growing up in a community where gem polishing. The process remained unchanged for several hundred years as late 19th century illustration below shows:

Like the diamond polisher, it is quite possible that Spinoza’s form was oriented horizontally, and not as van Gutschoven suggests, vertically. This was part of a gradual change in lens grinding techniques, much of it initiated by gem polishing influences. The horizontal mould simply made the glass easier to control, and the variable polishes to be administered more cleanly. For this reason, any polishing with Tripoli also occurred on a horizontal, turning wheel. The grinding forms designs that I have seen that the Huygens were using now all had a horizontally oriented lap. 

The second thing to note that in 1667 and 1668, after Christiaan had come to know Spinoza and become familiar with Spinoza’s techniques, he clearly did not still feel comfortable with the limits of van Gutschoven’s design, whenever he had received it, as he in repeated letters urged his brother about the fineness of Spinoza’s small lens polishing. Spinoza’s technique was not that of van Gutschoven. It is my feeling that he had developed, either though his associations in the community he grew up in, those influenced by the practices of gem polishing, means of polishing that were not common to the rest of Europe. Whether these be methods of grit application, the use of diamond dust, particular designs of a simple but effective lathe, one can only surmise. But it seems that Spinoza’s glasses were of a quality and luminosity that made them distinct.

Here is the Latin Text of van Gutschoven’s letter: The Text of van Gutschoven’s Letter to Huygens No. 1148

Conclusion

Aside from this I would want our investigative imagination to extend itself to the physical understanding of these practices, and the conceptual impression they would leave upon a thinking man who engaged in them repeatedly. This has been a theme of my thinking, that if Spinoza had been a potter we may do well to think about his metaphysics and arguments in terms of the potter’s wheel with which he was familiar. The grinding lathe is not so different from the potter’s wheel, and van Gutshoven’s diagrams give us a visual vocabulary for the kinds of effects and exertions that Spinoza produced in perfecting his craft. What in particular these diagrams allow, apart from the general understanding of the grinding lathe, is the picture of a grooved grinding practice, the canal, which varies from the greater method of placing a glass blank within a concave metal form. If indeed Spinoza used this method for his small objectives for microscopes, we can think along with him in the craft of it, and see him bent over the lip of the spinning canal.

As pictured here before, here is an example of a foot petaled lathe from the year 1647, that used by Hevelius. It may give us a dynamic sense of the physical engagement:

Here is a closer look at the Hevelius Lathe: Spinoza’s Grinding Lathe: An Extended Hypothesis

Leibniz’ “optical” Response to the Theologico-Political Treatise

Letter 45, Leibniz to Spinoza…

Leibniz wrote a short, almost entirely ignored by scholarship letter to Spinoza whose subject seems to be a lens invention of Leibniz’s, a “pandochal” (all receiving) lens which may have been something of a fish eye. What is of interest is the nature of the optical conflation Leibniz seems to be performing, and how this letter is sent right in the middle of the brewing tempest of the Spinoza’s blasphemous and anonymous Theological-Political Treatise. Leibniz appears to be offering, as he slandering Spinoza on the side, an optical Ideal world of pure perception, one which Spinoza ultimately shrugs off.

The Problem of the TTP

Leibniz’ letter to Spinoza on an issue of optics occurs just as he is positioning himself in correspondence with others who are outraged by Spinoza’s recently published Theologico-Political Treatise. Some of exchange:

Graevius writes on April 12, 1671, concerning TTP,

Last year there appeared this most pernicious book, whose title is Discursus Theologico-Politicus, a book which, having pursued a Hobbesian path, nevertheless quite often deviates rather far even from that, sets up the height of injustice as natural law, and having undermined the authority of sacred scripture, has opened the window very wide to ungodliness. Its author is said to be a Jew, named Spinoza, who was previously excommunicated from the synagogue because of his wicked opinions, but his book has also been proscribed for the same reason by the authorities. I think that you have seen it, but if you haven’t, I shall make it a point to have a copy sent to you. (A I, i, 142)

Leibniz’s replies, May 5 1671:

I have read Spinoza’s book. I grieve that a man of his evident learning should have fallen so far into error. Hobbes’ Leviathan has laid the foundations of the critique he carries out against the sacred books, but that critique can be shown to often be defective. These things tend to overturn the Christian Religion, which has been established by the precious blood of the martyrs and by such great labors and vigilance. If only someone could be stirred to activity who was equal to Spinoza in erudition, but [dedicated?] to the Christian cause, who might refute his frequent paralogisms and abuse of oriental letters. (A I, i, 148)

And then after his Letter 45 and 46 exchange with Spinoza, he writes to Gottlieb Spitzel, urging an erudite refutation,

Doubtless you have seen the book published in Holland, called The Liberty of Philosophizing. They say the author is a Jew. He employs a judgment which, while indeed erudite, is at the same time interspersed with much poison against the antiquity, genuineness, and authority of the sacred scripture of the Old Testament. In the interests of piety he should be refuted by some man solidly learned in Oriental studies, such as yourself or someone like you. (A I, i, 193)

Leibniz’s optical letter to Spinoza, given the epistolary machinations – and it is interesting that Leibniz hides from Spitzel the fact that he already knows Spinoza to be the author of TTP as Spinoza had incriminatingly offered to send Leibniz a copy of the text – reads as a scientico-political entreaty to the author of the Theologico-Political Treatise, an engagement of radical politics through science. This is supported by the very nature of the optical work that Leibniz includes. If one reads Leibniz’s very short A Note on Advanced Optics (“Notitia opticae promotae”) one sees that the intent of the work is to see in the perfection of optics a unification of all people under a rational perception of the world, framed as a distinctly political ambition of drawing heroic men together on a single path. Leibniz’s newly invented “Pandochal” (all-receiving [of rays]) lens, seems to manifest for him the rational and political power of his thought.

“Notitia opticae promotae” and J. Hudde

Also of significance is that Leibniz requests that his “Notitia opticae promotae” be forwarded to Johannes Hudde, who is on the verge of being appointed as Burgomaster of Amsterdam (a position he would hold for 30 years). Spinoza writes back that Hudde tells him that he is quite busy, but will look at the text in a week or two. This shows that Hudde and Spinoza are still in contact (despite the climbing rancor over his TTP); but also, it is from Hudde’s optical treatise, “Specilla circularia” that Spinoza composes much of his anti-Cartesian, or at least anti-hyperbolic, arguments. Leibniz, having himself studied Hudde’s Specilla, seems to be aware of this connection between the two men, and his conflation of the political and the optical in the Notitia, in part as a response to Spinoza’s Theologico-Political Treatise, marks out what is at stake in the literalization of optical metaphors for some at the time.

Given this, it is most interesting how Spinoza responds to Leibniz’s Notitia and letter. He takes up, not in the least, the invitation of an optical-political conflation, but simply asks for a clarification how Leibniz conceives of spherical aberration, and thus how his Pandochal lenses might allow an aperture any size. And, then a month later offers to send Leibniz a copy of the Theologico-Political Treatise, if he had not read it.

Spinoza’s Refusal

Whether this separation out of the questions of optics from the questions of politics by Spinoza represents extreme circumspection on his part, or a genuine difference in concept with Leibniz, we cannot say with certainty. It makes sense that at this time Spinoza must be very sure of the motives of all who respond to his TTP, as his friend Koerbagh has only recently died in prison over published texts (August . But I suspect that optical theory does not represent for Spinoza what it does for Leibniz. It does not hold “secrets” which will put all of man into much more rational communication. I suspect that this is because Spinoza’s path to freedom is quiet divorced from metaphors of light, pictures or imagery, and that he viewed the products of observations accomplished through telescopes and microscopes with as informing, but not revealing the nature of things.

Leibniz’s exactly timed letter and its implicit optical-political conflation makes a very good case study for Spinoza. For Spinoza would like to treat even something as fluid as human emotion as if it were the lines and planes of Euclidean geometry. His resistance to Leibniz’ enthusiasm for his pandochal lens, and the rhetoric of illustrious men marching together on the rational path, marks out I think, a certain sobriety toward questions of science; or perhaps greater finesse in understanding the totality of causes at play at that very volatile crossroads in history, the full and ballasting weight of the joined, imaginary perception of the social, something not to be solved by, or even addressed by the capacities of a paricular kind of lens.

Spinoza’s Lens-Grinding Equipment

 

Door to the Hofwijck Estate where Spinoza likely strode

 

Spinoza Purchases Lens Grinding Laps

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…

…A further reason why convex-concave glasses are less satisfactory, apart from the fact that they require twice the labour and expense, is that the rays, being not all directed to one and the same point, never fall perpendicular on the concave surface. – Letter 36, June 1666

This is what Spinoza writes to mathematician and microscopist Johannes Hudde, part of a correspondence that had begun before the end of the previous year, a correspondence which may have had its impetus in another lettered exchange: the on-going discussions on probability and actuary models between Hudde and Christiaan Huygens of that same year. Spinoza was getting to know his neighbor Huygens, and ends up writing to Hudde, someone he may have known since Rijnsburg and Leiden in the early 60’s.

The value of this letter for those investigating any potential lens-grinding practices Spinoza may have had is that here Spinoza cues that he had his metal shapes or laps made for him, at least at this time in the summer of 1666. The context of these remarks is Spinoza’s argument for the superiority of convex-plano lenses, using the same mathematical analysis of refraction that Hudde uses in his brief “Specilla circularia” (1655). Huygens, the previous summer, had personally calculated to a new degree of precision the phenomena of spherical aberration using convex-plano lenses (something Spinoza may have been privy to), and as Huygens has just left for Paris in April, Spinoza asks Hudde for both practical and theoretical optical advice.

I only wish to present to this context information about the kinds of workmen the Huygens brothers used over the years for their own telescopic lens-grinding projects:

View from the Huygens Estate, the Hofwijck

 

Christiaan Huygens’ Marbler and Chimney-Sweep

There is no doubt that at first their work consisted solely in grinding and polishing the glass. Even the metal shapes, on which the lenses were ground, were obtained from the outside. Their first ones were of iron (in 1656 Caspar Kalthof supplied one of these, [OC1, 380-81]), later they used copper shapes but for many years they did not make any themselves. In 1662 Christiaan still stated quite emphatically that he had never bothered with making shapes, although he did correct and finish them (OC4 53), and in 1666 in Paris he had a copper shape supplied to him (OC6, 87). But by 1668 we hear that Constantijn makes small shapes (for eyepieces) himself, on a lathe (OC6, 209), and it would appear that later he learned to make larger ones as well, for the instructions for making Telescope-lens (written in 1685 by the two brothers together) contains  detailed instructions about this part of the work (OC21, 251).

“Christiaan Huygens and his instrument makers” (1979), J. H. Leopold

And,

It is not entirely clear if the brothers made their own eyepieces around 1660, but they did not do so later on. Occasionally their correspondence contains references to local craftsman who prepared glass or ground eyepieces; the brothers focused on the delicate work of grinding object lenses.

In 1667 and 1668, Constantijn employed Cornelis Langedelf for polishing glass and grinding eyepieces, and in 1683 this same man delivered the tubes for one of Constantijn’s telescopes. From 1682 the brothers preferred the services of Dirk van der Hoeven, who lived nearby in The Hague. At the same time the brothers also did business with a marbler van der Burgh, who supplied them with grinding laps and glass. The relationships the brothers had with these two craftsmen were not identical. In the case of van der Hoeven – he was often simply referred to as Dirk or the chimney-sweeper – it was only his labour was hired. The brothers supplied the materials and tools, including the grinding laps. Van der Burgh, on the other hand, had a workshop of his own, and the Huygens[es] were were not his only clients. Moreover, one might expect this marbler to have been a more skillful grinder than his chimney-sweeping fellow citizen. So, it was probably was not the routine preparatory work that Constantijn asked of van der Burgh in April 1686, when he sent him two pieces of glass to be flattened 15.

“The Lens Production by Christiaan and Constantijn Huygens” (1999), Anne C. van Helden and Rob H. van Gent

What seems evident from both the Helden and Gent account, and the interpretation of Leopold is that it was not uncommon at all to hire-out for the production of the metal laps in which lenses would be ground. It seems clear from Spinoza’s letter 32 to Oldenburg in November of 1665 that the Huygenses were at least in the possession of a lathe that not only could grind lenses, but also laps or pans, for it is regarding this very (semi-automated) turning of the pans that Spinoza had his greatest doubts:

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. [See: Spinoza’s Comments on Huygens’s Progress .]

Whether anything good came of this Huygens lathe we cannot know. What is significant though in this combination of evidence, is that Spinoza seems to have made use of someone like the marbler Dirk van der Hoeven, at The Hague, just as the Huygens did, but also that Spinoza maintained a priority using the free had to either polish these purchased laps, or to polish lenses in them. That a chimney-sweep and a marbler would both be hired by someone as wealthy as the Huygens family, suggests a rather wide-spread and eclectic economic foundation for the procurement of these services and other related grinding services, something that did not require a specialist.

It is interesting to place Spinoza somewhere between the handyman Chimney-sweep and the savant Christiaan Huygens. Perhaps, if we take a more refined glance back through history, he seems to be between holding the straight-forward lathe experience of marbler-turner van der Hoeven and the specialized knowledge of Christiaan’s brother Constantijn, who spared no expense in carrying out Christiaan’s designs and theories.

As I have mentioned several times on this weblog, the picture is a bit more complicated than that. Christiaan Huygens had to bow to Spinoza’s assertion that the smaller objective lens makes a better microscope, and marveled at the polish that Spinoza was able to achieve in his microscopes, a polish achieved by “by means of the instrument” in a method that Christiaan did not seem to know. The speciality of knowledge did not restrict itself to just microscopes, but to telescope lenses as well. It is reasonable that the Huygenses purchased telescopes, microscopes and lathes from the Spinoza estate upon his death, and there does seem to have been something special about Spinoza’s laps (one’s he likely polished), as Constantijn used one in 1687, ten years after Spinoza’s death:

[I] have ground a glass of 42 feet at one side in the dish of Spinoza’s clear and bright in 1 hour, without once taking it from the dish in order to inspect it, so that I had no scratches on that side ” (Oeuvres completes vol. XI, p. 732, footnote). [cited by Wim Klever]

Spinoza, it would seem, used a man like van der Hoeven, but held at least to some particular degree both theoretical and craft advantages over Christiaan Huygens.

Approaching Huygens

Deciphering Spinoza’s Optical Letters

Line By Line

Below is my reading of Spinoza’s Optical Letters (39 and 40) as best as I have been able to extract interpretations from them. They are letters that are in general ignored, or when brushed over, taken to be evidence for Spinoza’s incompetence in optical matters. It seems that few have thought to examine in detail Spinoza’s point, or the texts he likely had in mind when formulating his opinion and drawing his diagrams. It should be said right from the start that I am at a disadvantage in this, as I have no formal knowledge of optics, either in a contemporary sense, nor in terms of 17th century theory, other than my investigation into Spinoza lens-grinding and its influence upon his metaphysics. In this research, the reading of this letter has proved integral, for it is one of the very few sources of confirmed scientific description offered by Spinoza. That being said, ALL of my facts and inferences need to be checked and double checked, due to my formal lack of familiarity with the subject. It is my hope that the forays in this commentary reading, the citations of likely texts of influence and conceptual conclusions would be the beginning of a much closer look at the matter, very likely resulting in the improvements upon, if not outright disagreement with, what is offered here.

[The Below English selections and links to the Latin text: here ]

Spinoza Answers

“I have looked at and read over what you noted regarding the Dioptica of Descartes.”

Spinoza is responding to a question we do not know, as we have lost Jelles’s letter. We can conclude from several points of correspondence that it is a section of Descartes’Dioptrics that Jelles’ question seems to have focused on, the Seventh Discourse titled “Of The Means of Perfecting Vision”. There, Descartes describes the interactions between light rays, lenses and the eye for purposes of magnification, preparing for the Eighth Discourse where he will present the importance of hyperbolic lenses for telescopes, and also onto the Ninth, “The Description of Telescopes”, where that hyperbola is put to use in a specific proposed construction.

“On the question as to why the images at the back of the eye become larger or smaller, he takes account of no other cause than the crossing of the rays proceeding from the different points of the object, according as they begin to cross one another nearer to or further from to eye…”

This is the beginning of Spinoza’s attack on Descartes’ rendition of how light refracts through lenses to form images of various sizes at the back of the eye. In the Seventh Discourse Descartes claims to have exhausted all factors that can influence the size of the image, which he numbers at three:

“As to the size of images, it is to be noted that this depends solely on three things, namely, on the distance between the object and the place where the rays that it sends from its different points towards the back of the eye intersect; next on the distance between this same place and the base of the eye; and finally, on the refraction of these rays (trans. Olscamp).”

His descriptions that follow are varied. Among his either trite or fanciful augmentations he considers moving the object closer to the eye, then the impossibility of lengthening the eye itself, and lastly musing that if the refraction of the crystaline humor would spread rays more outward, then so too should magnification be achieved. This seems to be the extent to which Descartes will treat the factor of refraction in this discourse (hence perhaps Spinoza’s claim of the repression of a very important factor); but what Spinoza has cast his critical eye upon, I believe, is Descartes characterization of the solution to questions of magnification achieved by fundamentally extending of the distance of the intersection of rays:

There remains but one other means for augmenting the size of images, namely, by causing the rays that come from diverse points of the object to intersect as far as possible from the back of the eye; but this is incomparably the most important and the most significant of all. For it is the only means which can be used for inaccessable objects as well as for accessable ones, and its effect has no limitations; thus we can, by making use of it, increase the size of images indefinitely.

It is good to note that in his description of the strategies of telescope magnification, Descartes is operating under an extended analogy, that the telescope can work like a prosthetic lengthening of the human eye, causing the refraction that would regularly occur at the eye’s surface to happen much farther out, as if the retina were being placed at the end of a very long eye. This is his mechanical concept.

Descartes distance-analysis of magnification (and an assertion of the significance of the hyperbola) is then carried forth in the Ninth Discourse, where again Descartes will treat magnification in terms of the proximity to the eye of the crossing of rays, which here he will call the “burning point” of the lens. The descriptions occur both in the context of solutions to far and near sightedness as well as in proposals to the proper construction of telescopes, and generally follow this idea that one is primarily lengthening the eye. 

“…and so he does not consider the size of the angle which the rays make when they cross one another at the surface of the eye. Although this last cause would be principle (sit praecipua ) to be noted in telescopes…”

What Spinoza is pointing out is that when constructing telescopes, as he understands it, the aim is to increase the magnitude of the angle of rays upon the surface of the eye (the cornea), something not solely achievable merely through the adjustment of the distance of the “burning point” or the crossing of the rays of the lens from the eye. Attention to the angle of intersection is for Spinoza a more accurate discriminator probably because it leads to calculations of refraction which include the angle of incidence upon the lens, giving emphasis upon the varying refractive properties of different shapes and thicknesses of lenses in combination, some of which can increase magnification without lengthening the telescope. Descartes conceived of the objective and eyepiece lenses as mimicking the shape and powers of the eye’s lens(es), just further out in space. Though he states at several points that we do not know the exact shape of the human eye, under this homological view, he still sees a correspondence between his proposed hyperbolic-shaped lenses and those of the eye, likely drawing upon Kepler’s observation that the human crystalline humor was of a hyperbolic shape.

The fuller aspects of the factor of refraction – the third factor listed in Descartes three – are left out in such a distance calculation, Spinoza wants us to see. As mentioned, in the combination of lenses, depending upon their shape and powers, the required lengthening of the telescope can be shortened (Spinoza presents just this sort of argument to Hudde in Letter 36, arguing for the efficacy of convex-planolenses). One can also say that this same emphasis on the powers of refraction was also at play in Spinoza’s debate with Huygens over the kinds of objective lenses which were best for microscopes. Huygens finally had to privately admit in a letter written to his brother a year after these two letters, that Spinoza was right, smaller objective lenses with much greater powers of refraction and requiring much shorter tubes indeed made better microscopes (we do not know if Spinoza had in mind the smallest of lenses, the ground drop-lenses that Hudde, Vossius and van Leeuwenhoek used, but he may have). It should be said that Huygens’ admission goes a long way toward qualifying Spinoza’s optical competence, for Spinoza’s claim could not have simply been a blind assertion for Huygens to have taken it seriously. Descartes to his pardon is writing only three decades after the invention of the telescope, and Spinoza three decades after that. Be that as it may, Descartes’ measure is simply too imprecise a measure in Spinoza’s mind, certainly not a factor significant enough to be called “incomparably the most important and the most significant of all”.

Because Jelles’ question seems to have been about the length of telescopes that would be required to achieve magnification of details of the surface of the moon (the source of this discussed below), it is to some degree fitting for Spinoza to draw his attention away from the analysis of the distance of the “burning point”, toward the more pertinent factor of the angle of rays as they occur at the surface of the eye and calculations of refraction, but it is suspected that he wants to express something beyond Jelles’ question, for focal and telescope length indeed remained a dominant pursuit of most refractive telescope improvements. And Spinoza indeed comes to additional conclusions, aside from Descartes imprecision. Spinoza suspects that Descartes is obscuring an important factor of lens refraction by moving the point of analysis away from the angle of rays at the surface of the eye. This factors is, I believe, the question of the capacity to focus rays coming at angles oblique to the central axis of the lens, (that is, come from parts of an object off-center to the central line of gaze). Spinoza feels that Descartes is hiding a weakness in his much treasured hyperbola.

“…nonetheless, he seems deliberately to have passed over it in silence, because, I imagine, he knew of no other means of gathering rays proceeding in parallel from different points onto as many other points, and therefore he could not determine this angle mathematically.”

Descartes, in Spinoza’s view, wants to talk only of the crossing of rays closer to or farther from the surface of the eye, under a conception of physically lengthening the eye, and not the magnitude of the angle they make at the surface of the eye because he lacks the mathematical capacity to deal with calculations of refraction which involved rays coming obliquely to the lens. For simplicity’s sake, Descartes was only precise when dealing with rays coming parallel to the center axis of the lens, and so are cleanly refracted to a central point of focus, and it is this analysis that grants the hyperbola its essential value. In considering this reason Spinoza likely has in mind Descartes’ admission of the difficulty of calculation when describing the best shapes of lenses for clear vision. As well as the admitted problem of complexity, Descartes also addresses the merely approximate capacties of the hyperbola to focus oblique rays.

[regarding the focusing of rays that come off-center from the main axis]…and second, that through their means the rays which come from other points of the object, such as E, E, enter into the eye in approximately the same manner as F, F [E and F representing extreme ends of an object viewed under lenses which adjust for far and near sightedness]. And note that I say here only, “approximately” not “as much as possible.” For aside from the fact that it would be difficult to determine through Geometry, among an infinity of shapes which can be used for the same purpose, those which are exactly the most suitable, this would be utterly useless; for since the eye itself does not cause all the rays coming from diverse points to converge in exactly as many other diverse points, because of this the lenses would doubtless not be the best suited to render the vision quite distinct, and it is impossible in this matter to choose otherwise than approximately, because the precise shape of the eye cannot be known to us. – Seventh Discourse

This is an important passage for several reasons, but first because it comes the closest to the question of the focus of rays para-axial to the center. Again, one must keep in mind that Descartes is thinking about trying to make lenses of a shape that are exact to the shape (or powers) of the eye. Here he is thinking about ever more exotic geometrical shapes which may achieve this, and insists upon the fruitlessness of such a pursuit; it is significant that in contrast to this, Spinoza imagines rather a very simple solution to the question of aberration: the acceptance of spherical aberration and the embrace of the advantage of spherical omni-axial focus. The quoted passage directly precedes Descartes’ summation of the three factors in magnification, with which I began my citations. And I will return to the latter parts of this passage later when we investigate Spinoza’s critique of the hyperbola and the eye. (Note: Aside from this direct reference to Descartes on the issue of calculation, perhaps Spinoza considers also James Gregory, who had some difficulty calculating paraxial rays for his hyperbolae and parabolae in his Optica Promota, though writing an entire treatise devoted to their value.)

Nonetheless, Spinoza suspects that Descartes has shifted the analysis of magnification not simply because it is not amenable to calculation, but more so because, had Descartes engaged the proper investigation, he would have had to face an essential advantage of spherical lense, lessening to some degree his hyperbolic panacea to the problems of the telescope. Again, we will leave aside for the moment Descartes’ justification of this approximation on the basis of the human eye and Nature.

Soft Focus: Spherical Aberration

“Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered.”

Spinoza goes on to expound for Jelles the virtues of the simple circle, as it expresses itself in spherical lenses. One has to keep in mind that since the publishing of Descartes’ Dioptrics  (1637), there had been a near obsessional pursuit of the grinding of hyperbolic lenses, a lens of such necessary precision that no human hand was able to achieve it. The hyperbolic lens promised – falsely, but for reasons no one would understand until Newton’s discovery of the spectrum character of light in 1672 – a solution to the problem of spherical aberration. Spherical aberration is simply the soft focus of parallel rays that occurs when refracted by a spherical lens. Kepler in his Paralipomena provides a diagram which illustrates this property:

As one can see, rays that are incident to the edges of the lens (α, β) cross higher up from the point of focus, which lies upon the axis (ω). It was thought that this deviation was a severe limitation on the powers of magnification. With the clearing away of the bluish, obscuring ring that haloed all telescopic vision, the hope was for new, immensely powerful telescopes. And it was to this mad chase for the hyperbola that Spinoza was opposed, on several levels, one of which was the idea that spherical lense shapes actually had a theoretical advantage over hyperbolics: the capacity to focus rays along an infinity of axis:

diagram letter 39

“[referring to the above] 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 a very important point in the letter, for I believe it has been misread by some. At the same time that Spinoza seems to be asserting something painfully obvious in terms of the geometry of a circle, he, at first blush, in bringing this geometry to real lenses appears to be making a serious blunder. And, as I hope to show later, beneath both of these facts there is a subtle and deeper phenomenal-epistemic philosophical point being made, one that echoes through to the roots of Cartesian, and perhaps even Western, metaphysics. Let me treat the first two in turns, and then the third in parts.

The first point is obvious. As we can see from the diagram Spinoza provides, each of the refractive relationships of rays parallel to one axis are symmetical to the same relationships of other parallel rays to another axis. The trick comes in Spinoza’s second sentence, where he seems to be asserting an optical property of actual spherical lenses. As one email correspondent to me concluded, (paraphrased) “Spinoza thinks that the focal point of such a lens lies on the diameter, and this only occurs in rare cases.” The index of refraction of glass simply is not 2 in most cases. Spinoza seems to be making an enormous optical blunder in leaving the refractive index of the glass out, opening himself to a modern objection that he simply does not know the significance of the all important Law of Refraction, put forth by Descartes. This is a similiar prima facie reading done by Alan Gabbey in his widely read essay “Spinoza’s natural science and methodology”, found in The Cambridge Companion to Spinoza,

One’s immediate suspicions of error is readily confirmed by a straight forward 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, a condition of which there is not the slightest hint in the letter…[he is] apparently unaware of the importance the “[other] figures”…that Descartes had constructed (154).

The problem with these readings, among many, is that Spinoza is not at all asserting that there exists such a lens which would have this refractive property (Gabbey’s concerns about Spinoza’s awareness of the Law of Refraction should be answered by looking his familiarity with Johannes Huddes “Specilla circularia”, in letter 36, which will be taken up later). I have corrected a weakness in the prominent English translation of the text which helps to bring out the distinction I am making. If one looks at the sentence closely, Spinoza is presenting an if-then assertion (he uses the subjective in the intitial clause). IF, and only if, a circular lens can be said to have the focusing property along axis AB, THEN it would have the same property along axis CD. To repeat, he is not asserting such a property in real glass and therefore he remits any refractive index reference because it is not germane to his point; he is only at this point emphasizing the property of an infinity of axes of focus, and he is using a hypothetical sphere for several reasons.

The first reason I suspect is that he is trying to draw out the remarkable resonance of spherical forms, making his diagram evocative of notions of completeness and internal consistency. This is of course not an optical concern, but we have to consider it as an influence. We have a similiar looking diagram presented by Spinoza in the Ethics, showing an argued relationship between Substance and the modes that express it. As Spinoza writes:

diagram from the Ethics 2, prop 8, scholia

“The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle.” E2p8s

There is perhaps much speculation to be made as to Spinoza’s feelings about the the interweave of causes that express themselves in modes and the apparitions of focus generated by hypothetical spherical lenses (are modal expressions seen in some way like a confluence of rays?), but at this point I only want to point out Spinoza’s affinity for the sphere, and thus this one possible reason for using a full sphere to illustrate an optical property of spherical lenses. (Remember, this is just an informal letter written to a friend, and not meant as a treatise.)

The second reason is that Spinoza very likely is thinking of a real sphere, that is, the “aqueous globe” that Kepler used to investigate refraction in his Paralipomena, a work in which he was the first to articulate with mathematical precision the dynamics of spherical aberration (before there was a telescope, in 1604), and also was the first to suggest the hyperbola as the resolving figure for such aberration. Here is Kepler’s diagram of his sphere through which he gazed at various distances, illustrating his Proposition 14: “Problem: In an aqueous globe, to determine the places of intersection of any radiations parallel to an axis”.

Kepler's diagram from proposition 14

Thus, Spinoza’s use of a sphere in his diagram has at least two readings that have heretofore not been noticed. The first is that his description is operating at solely the hypothetical level, asserting the abstract properties of spherical symmetry, but secondly, he is referencing, or at least has in mind, a primary historical optical text, in all likelihood the text which spurred Descartes’ enthusiasm for the hyperbola in the first place (likely read by Descartes around 1620). It is precisely in this parallel fashion, between the geometrical and the manifest, that Spinoza seems to work his optical understanding.

The third reason that Spinoza is using a full sphere to illustrate his principle of omni-axial refraction is that Descartes’ treatise deals not only with lenses, but also with the human (and ox) eye. And this eye in diagrams is represented as a sphere. I will return to this point a little later, because as he encounters Descartes, he is making an argument, however loosely, against not only his optics, but his essential concepts of clear perception. By taking up a full sphere in his objection, he also poses a relation to Descartes schemas of the eye.

Aside from Descartes’ pseudo-spherical diagram of the eye, we have to consider as an additional influence Hooke’s spherical depiction of the eye with two pencils of rays focused along different axes, used to illustrate the reception of color (pictured below left). The reason why I mention this diagram is not only because it bears some resemblance to Spinoza’s, but also because Hooke’s extraordinary Micrographia might have been the source of Jelles’ question, as I will soon address, and so may have been a text Spinoza thought of in his answer, though we are not sure if he ever read it, or even looked at it, as it was in published in English. Christiaan Huygens owned a copy of it and it was the subject of a conversation between the two. If Spionoza indeed visited the Hofwijck several times, it is hard to believe that he would not have looked closely at this page of diagrams.

 

 

figure 5, Robert Hooke's Micrographia

“…this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters.”

Spinoza here declares the exclusivity of a property that only spheres and their portions possess. It is hard to tell exactly at what level Spinoza is making his objection. Is it entirely at the theoretical level of optics that Spinoza believes hyperbolic lenses to be impaired, such that even if people could manufacture them with ease, they still wouldn’t be desired. If so, he would be guilty of a fairly fundamental blindness to potential advantages in telescope construction that such a lens would grant, rather universally understood. If indeed he was an accomplished builder of telescopes – and we have some evidence that he may have been – this would be a difficult thing to reconcile, forcing us to adopt an estimation of a much more craftsman level understanding of his trade. But it is possible that Spinoza is asserting a combine critique of hyperbolic lenses, one that takes into account the difficulty in making them. There are signs that spherical aberration after Descartes was taken to be a much greater problem than it calculably was, and Spinoza brings out a drawback to hyperbolic focus that adds one more demerit to an already impossible-to-make lens. Thus, as a pragmatic instrument maker he may not be assessing such lenses only in the abstract, but in reality. It may be that Spinoza sees the ideal of the hyperbolic lenses as simply unnecessary, given the serviceability of spheres, and the perceived advantage of oblique focus. This question needs to be answered at the level of optical soundness alone, but such an answer has to take in account the great variety of understandings in Spinoza’s day and age, even among those that supposedly “got it right”. For instance, such an elementary and widely accepted phenomena as “spherical aberration” was neither defined, nor labeled in the same way, by any two thinkers; nor were its empirical effects on lensed vision grasped. We often project our understanding backwards upon those that seem most proximate to our truths. Spinoza’s opinions on aberration seem to reside exactly in that fog of optical understandings that were just beginning to clear.

Man on the Moon

“So the case is as you describe; that is, if no account is taken of anything except the focal lenth of the eye or of the telescope, we should be obliged to manufacture very long telescopes before we could see objects on the moon as distinctly as those on earth.”

Here we possibly get a sense of Jelles’ question. It must have come from a reflection upon Descartes’ comments on crossing of rays at various distances from the eye, posed as a question to whether we might be able to view the Moon with such clarity as we see things here – remember, Descartes’ promised infinite powers of magnification. I mentioned already that Jelles’ question may have come in reference to Hooke’s work. We must first overcome the problem of language of course, for I do know that Jelles read English, though it is possible that he read a personal translation of a passage, as Huygens had translated a passage for Hudde. But given these barriers, I believe there is enough correspondence to make a hypothesis that is not too extravagant: Jelles had recently read a portion of Hooke’s Micrographia. The reason that I suspect this, is that the Micrographia published with extraordinarily vivid plates of magnified insects and materials, concludes with a speculative/visual account of what may be on the moon, seen through his 30-foot telescope (and a suggested 60 ft. telescope), coupled with a close up illustration of a moon’s “Vale” crater, he writes of an earthly lunar realm:

…for through these it appears a very spacious Vale, incompassed with a ridge of Hills, not very high in comparison of many other in the Moon, nor yet very steep…and from several appearances of it, seems to be some fruitful place, that is, to have its surface all covered over with some kinds of vegatable substances; for in all portions of the light on it, it seems to give a fainter reflection then the more barren tops of the incompassing Hills, and those a much fainter then divers other cragged, chalky, or rocky Mountains of the Moon. So that I am not unapt to think that the Vale may have Vegetables analogus to our Grass, Shrubs, and Trees; and most of these incompassing Hills may be covered with so thin a vegetable Coat, as we may observe the Hills with us to be, such as the Short Sheep pasture which covers the Hills of Salisbury Plains.

As one can see from this marvelous, evocative passage, the suggestion that the moon’s vales are pastorially covered with rich meadows, calling up even flocks of sheep before the mind, one can easily see that Jelles has something like this in mind when he asks what it would take to see objects on the moon, as we can see objects on the Earth. One might speculate that, having read such a passage, Jelles had a spiritual or theological concern in mind and excitment over the possibility of other people on the moon, but this would be perhaps only wistful supposition on our part. But it is too much to suppose that it was likely Hooke’s description of the moon Jelles was thinking of when he wrote his question to Spinoza, for not only are the details of an Earth-like moon present, but also Hooke’s urging of the reader to use a more power and much longer telescope than he used. Spinoza is responding directly to this aspect of telescope length.

(An alternate thought may be that Jelles had come upon Hevelius’s Selenographia, sive, Lunae descriptio 1647, filled with richly engraved plates of the moon’s surface. It did not have the same fanciful description of moon meadows, and was not circulated with the acclaim of Hooke’s Micrographia, but it did name features of the moon after Earth landmarks, giving it an Alps, a Caucasus and an Island of Sicily.)

If we allow this supposition of a posed question on Jelles’s part, we might be able to construct something of Spinoza’s thinking in his response. It would seem, in our mind’s-eye, that Jelles had read Hooke’s description of the moon and his urge for a longer telescope and set about checking Descartes’ Dioptrics if it were the case that we really would have to build an extraordinarily long telescope to see the details that Hooke invoked (indeed Huygens built a 123 ft. arial telescope; and Hevelius one of 150 ft., pictured below).

Hevelius' 150 ft. arial telescope

Following this evolution of the question, it would seem that Jelles came to Descartes’ treatment of magnification in the Seventh (and related) Discourses, one that defined the power of magnification by the all important distance of the crossing of rays from the surface of the eye, treating the telescope as an extended eye. If indeed Jelles was not familiar with optical theory he may have taken this increase of distance for an explanation why telescopes had to be so very long to see the moon with desired detail. It would seem natural for Jelles to pose this question to Spinoza, who not only was regarded as the expert on Descartes in the Collegiant group, but also was a grinder of lenses and a designer of telescopes.

If this hypothetical narrative of the question is correct, Spinoza responded in a slightly misdirected way, taking the opportunity to vent an objection to Descartes thinking which did not have acute bearing upon Jelles’s question. For Descartes’ description of a “burning point” distance and Spinoza’s emphasis on the angle of incidence of rays oblique to the center axis, makes no major difference in the conclusion that Jelles came to, that indeed it would take a very long telescope to do what Jelles imagined, and Spinoza admits as much, above. Yet, when Spinoza qualifies his answer “if no account is taken of anything except the focal lenth of the eye or of the telescope” he is pointing to, one imagines, factors of refraction, for instance in compound telescopes and lenses of different combinations, which do not obviate the contemporary need for very long telescopes, but may affect the length.

Aside from this admission, Spinoza has taken the opportunity to express his displeasure over a perceived Cartesian obscurance, one that has lead to an over-enthused pursuit of an impossible lens, and as we have seen, in this context Spinoza puts forward his own esteem for the spherical lens, and the sphere in general. But this is no triffling matter, for out of Spinoza’s close-cropped critique of Descartes’ Dioptrics run several working metaphors between vision and knowledge, and a history of thinking about the optics of the hyperbola that originates in Kepler (made manifest, I contend, in a full-blown metaphysics in Descartes). Though Spinoza’s objection is small, it touches a fracture in thinking about the Body and Perception, a deep-running crack which might not have direct factual bearing on optical theory, but does have bearing on its founding conceptions. As I have already suggested, we have to keep in mind here that though we are used to thinking of a field of science as a closed set of tested truths oriented to that discipline, at this point in history, just when the (metaphysically) mechanical conception of the world was taking hold, it is not easy, or even advisable, to separate out optical theories from much broader categories of thought, such as metaphysics and the rhetorics of philosophy. For example, how one imagined light to move (was it a firery corpuscula, or like waves in a pond?), refract and focus was in part an expression of one’s overall world picture of how causes and effects related, and of what bodies and motions were composed: and such theories ever involved concepts of perception.

“But as I have said, the chief consideration is the size of the angle made by the rays issuing from different points when they cross one another at the surface of the eye. And this angle also becomes greater or less as the foci of the glasses fitted in the telescope differ to a greater or lesser degree.”

Spinoza reiterates his point that it is the intersecting angles of incidence at the surface of the eye which determined the size of the image seen through a telescope. He finally connects the factor of the angle of incidence and intersection to the foci of lenses themselves. It is tempting to think that Spinoza in his mention of lenses is also thinking of compound forms such as the three-lens eyepiece invented by Rheita in 1645, or as he was already familiar through visits to Christiaan Huygens’s home in 1665, proposed resolutions of spherical aberration by a complex of spherical lenses. Such combinations would be based upon angle of incident calculations.

“If you wish to see the demonstration of this I am ready to send it to you whenever you wish.”

 

Spinoza will send this evidence in his next letter (pictured at bottom).

Letter 40 “…I now proceed to answer your other letter dated 9 March, in which you ask for a further explanation of what I wrote in my previous letter concerning the figure of a circle. This you will easily be able to understand if you will please note that all the rays that are supposed to fall in parallel on the anterior of the glass of the telescope are not really parallel because they all come from one and the same point.”

Jelles has apparently had some difficulty with understanding Spinoza’s explanation. It is interesting because this confusion on Jelles’ part has actually been taken as evidence that Spinoza not only is impaired in his understanding of optics (this may be the case, but Jelles’ confusion, I don’t believe, is worthy of being evidence of it), but that those close to Spinoza around this time became aware that Spinoza’s optical knowledge was superficial at best, something not to be questioned too deeply.

As Michael John Petry writes:

“There is evidence that after 1666 Spinoza’s ideas on theoretical optics were less sought after by his friends and acquaintences…Even JarigJelleswasquiteevidently dissatisfied with the way in which Spinoza explained the apparent anomaly in Descartes’ Dioptrics” (Spinoza’s Algebraic Calculation of the Rainbow & Calculation of Chances, 96)

Petry cites other evidence which needs to be addressed (primarily Huygens’ letters), but a close reading of the nature of Jelles implied question does not seem to support in any way the notion that Spinoza’s optical knowledge had been exposed as a fraud of some sort. Alan Gabbey as well, who maintains serious doubts about Spinoza’s optical proficiency, seems to focus on Spinoza’s need to explain himself to Jelles as a sign that he is somewhat confused:

In his next letter…to Jelles, who has asked for a clarification, Spinoza explained that light rays from a relatively distant object are in fact only approximately parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintained the same property of the cirlce in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous “Ovals of Descartes”] (154).

It seems quite clear that Spinoza was aware of the “importance” of these figures, at least he was aware of Hudde’s and Huygens’ attempt to minimize that importance. But Gabbey here seems to suggest that Spinoza is evading a point of confusion by simply changing descriptions, instead of parallel rays of light, Spinoza now uses “cones of rays”. For these reasons of suspicion it is better to go slow here.

The question that Jelles raised apparently has to do with the reading of Spinoza’s circular diagram and its focus of two pencils of light rays, for Spinoza imagines that if Jelles understands these pencils as cones of rays his confusion will be cleared up. To take the simplest tact, it may very well be that Jelles, upon seeing Spinoza’s diagram, turned back to Descartes’ text in order to apply it, and found there a diagram which was quite different. What Jelles may have seen was Descartes’ figure 14 from the Fifth Discourse (pictured below, left), or really any of his diagrams which depict the interaction of rays with the eye:

figure 14 from the Fifth Discourse of the Dioptrics

One can see how in this context Jelles may have been confused by Spinoza’s diagram of the focus of two pencils of rays, and even by the accusation that Descartes is being somehow imprecise, for the illustration seems to depict rays as something like cones of rays, not rays flowing parallel to an axis, as they are in Spinoza’s drawing. Aside from this plain confusion, Jelles’ question may have dealt with some other more detailed aspect, for instance, a question about the importance of a lens’s ability to focus rays oblique to its center. If so, Spinoza would require not only that Jelles understand that rays come in cones, but also have a fuller sense of how those rays refract upon the eye, perhaps provided by the diagram that will follow. In either case, rather than understand Spinoza’s change in descriptive terminology as an attempt to dodge his incomprehension, Spinoza simply appears to be guiding Jelles in the reconsilation of both kinds of diagrams, or preparing ground for a more complete explanation.

Note: Regarding the analytical descriptions of a pencil of parallel of rays or “cones of rays” there is no standing confusion between them. They exhibit two different ways of analyzing the refractive properties of light. But there is more than this, the use of the phrase “cones of rays” by Spinoza gives a clue to what texts he has in mind in his answer. The orgin of this phrase for Spinoza likely comes from Kepler’s Paralipomena  (1604), in a very significant passage. As mentioned, Kepler has already provided a description of the phenomena of spherical aberration (shown in diagrams including the one I first cited here), and forwarded the hyperbola as a figure that would solve this difficulty. Further, he has claimed that the crystalline humor of the human eye has a hyperbolic shape. Here Kepler describes how light, having proceded from each point of an object in a cone of rays (truly radiating in a sphere), intersects the eye’s lens at varying degrees of clarity. The cone that radiates directly along the axis of the lens is the most accurately refracted:

All the lines of the direct cone [a cone whose axis is the same as the axis of the cornea and crystalline] are approximately perpendicular to the crystalline, none of those of the oblique cones are, The direct cone is cut equally by the anterior surface of the crystalline; the oblique cones are are cut very unequally, because where the anterior surface of the crystalline is more inclined [aspherical], it cuts the oblique cone more deeply. The direct cone cuts the hyberbolic surface of the crystalline, or the boss, circularly and equally; the oblique cone cuts its unequally. All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side…(Paralipomena 174)

This passage has multiple points of importance, in part because I suspect that it is the orgin passage of Descartes’ enthusiasm for the hyperbola, but also, as I will show later, for a naturalized justification for hyperbolic vision, something which will play to Spinoza’s optical critique. But at this point it is just sufficient to register the citation as a reference point for Spinoza’s phrase. We have already pointed out that Spinoza may have Kepler’s aquaeous globe in mind for his intial diagram, so there is something distinctly Keplerian in Spinoza’s approach.

Another reference point for Spinoza’s phrase is James Gregory’s 1663 Optical Promota, a treatise written without the aid of Descartes’ Dioptrics, but which all the same proposed parabolic and hyperbolic solutions to refraction aberrations and proposed reflective mirror telescopes to avoid the problem altogether. This text we know Spinoza had in his personal library, and he seems to be reasoning from it in part. Gregory regularly uses both “pencils of rays” and “cones of rays” as modes of analysis.

As a point of reference for us, he offers these defintions to begin his work:

6. Parallel rays are those which are always equally distant each to the other amonst themselves.

7. Diverging rays are those which concur in a point when produced in both directions: those rays produced in the opposite direction to the motion from the ray-bearing cone – the apex of the cone is the point of concurrence of the rays.

8. Converging rays are those rays are those which concur in a point in the direction of the motion when produced in both directions; these rays are called a pencil, and the point of concurrence the apex of the pencil…

10. An image before the eye [i.e. a real image], arises from the apices of the light bearing cones from single radiating points of matter brought together in a single surface.

Pencils of parallel rays feature in many of the diagrams, within the understanding that rays proceed as cones. So seems to me that Spinoza is operating with both Kepler and Gregory in mind as he answers Jelles’ question.

“But they are considered to be so because the object is so far from us that the aperture of the telescope, in comparison with its distance, can be considered as no more than a point.”

Spinoza follows Gregory’s Fourth Postulate: “The rays coming from remote visible objects are considered parallel.”

“Moreover, it is certain that, in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points.”

Spinoza may be still addressing the nature of Jelles’ request for clarification. He follows the reasoning of Gregory’s Tenth defintion (above). Whether the rays be treated as parallel pencils, or cones does not make a strict difference to Spinoza’s point, though understanding that they are coming to the lense as cones does something to express their spherical nature (one must recall that Kepler asserted that light radiates as a sphere as it can, and even that Hooke proposed that it moves in waves; Spinoza’s attachment to the sphere may be in regards to this). It is the lens’ capacity to gather together these rays come from diverse points of the object, and not just rays parallel to its central axis, that Spinoza emphasizes. In other words, though considered no more than a point, it is a point that must gather rays from a variety of angles.

“And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci.”

It should be noted that Spinoza is talking about glass lenses here, and not the eye’s lens. Spinoza has taken his ideal model of a spherical refraction from the first letter, and has applied it to actual lenses (there is no requirement to the index of refraction of the glass). As Spinoza envisions it, because a glass has to focus rays coming obliquely, the foci along those alternate axes are significant factors in clarity.

Seeing More, or Seeing Narrowly

“And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others.”

This is the big sentence, the one that opens up the place from which Spinoza is coming from. What does Spinoza mean “the eye is not so exactly constructed”? How odd. Descartes’ comments on optics indeed are often made in the service of correcting far- and near-sightedness, so there is context for a notion of the “inexactness” of the eye, and for his own uses Descartes picks up on the notion that the eye is limited or flawed: …”in as much as Nature has not given us the means…”, “I still have to warn you as to the faults of the eye”. But this is not what Spinoza has in mind. What I believe Spinoza is thinking about is the hidden heritage behind a naturalizing justification of hyperbolic vision itself. This is not strictly an optical point, as we have come to understand optical theory, but an analogical point. And this distinction organizes itself around the failure that a hyperbolic lens to handle rays oblique to its axis, with clarity, and whether this failure is something to be concerned with.

Kepler's drawing the hyperbolic crystalline humor, 167

Kepler begins the justification. The passage continues on from the conclusion of the one cited above, which ended with an explanation of why the image of the eye is blurred at its borders,

All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side…

…so the sides of the retina use their measure of sense not for its own sake, but whatever they can do they carry over to the perfection of the direct vision. That is we see an object perfectly when at last we perceive it with all the surroundings of the hemisphere. On this account, oblique vision is least satisfying to the soul, but only invites one to turn the eyes thither so that they may be seen directly (174).

This is a striking passage in that we know the history of the hyperbolic lens, and Descartes’ fascination with it. Due to the hyperbolically shaped crystalline humor (as Kepler reasons it), the image at the border, projected at the edges of the retina, is said to be more confused due to the inability of the lens to focus oblique rays. This is what Spinoza has in mind when he says that the eye is not so exactly constructed. But there is more to this passage. Not only is the image more confused, but Kepler goes so far was to qualify this confused quality as an explanation for why the soul is dissatisfied with oblique vision. At the margins of blurred vision, according to Kepler, the sides of the retina do not “sense” for their own sake, but for the sake of central axis perfection, in effect serving the center. Kepler has provided the hyperbola as the solution for spherical aberration, but has also couched that shape within a larger context of human perception and the nature of what experience satisfies the soul or not.

This theme of the hyperbola’s justifcation through Nature continues. I will leap forward to Gregory’s Optica Promota, a writer who, as I have said, had no access to Descartes’ treatise but did read Kepler closely. At the end of a thorough and brilliant work on the value of hyperbolic and parabolic forms for use in telescopes, Gregory as well evokes Kepler’s notion of the weakness of the hyperbola, along with its naturalization. This is how he ends his Optica :

But against hyperbolic lenses, it is only objected that nothing will be able to be most clearly seen, except a visible point arising on the axis of the instrument. But this weakness [ infirmitas ] (if it would be worthwhile to call it that) is sufficiently manifested in the eye itself, though not to be impuning Nature, for whom nothing is in vain, but how much all things most appropriately she carries out [ peragit]. Nevertheless, withconicallenses and mirrors not granted, it shall be rather with spherical portions used in place of spheriods and paraboloids in catoptrics; as with hyperboloids in dioptrics, in which portions of spheres are less appropriate.

With these we go to the stars – His itur ad astra

Just as Kepler justifies hyperbolic vision by appeal to the eye’s own weakness, redeemed by the roles of the retina and the satisfactions of the soul, so here too Nature herself is the justification of central axis priority. This is a curious naturalization, given that so much of optics addresses the failings or the limitations of Nature. Such a self-contradiction deserves attention, especially with a focus upon the foundations of valuations that make one adjustment to Nature desired, and another not. But here I would like to continue the line of justifications of the hyperbola through the construction of the eye that Spinoza likely has in mind.

Descartes, if you recall from a passage cited above, also justifies the shape of the hyperbolic lens through appeal to the shape of the human eye. After he admits that the foci of rays that come obliquely to the axis of the hyperbola can only approximate a point of focus,

…for since the eye itself does not cause all the rays coming from diverse points to converge in exactly as many other diverse points, because of this the lenses would doubtless not be the best suited to render the vision quite distinct, and it is impossible in this matter to choose otherwise than approximately, because the precise shape of the eye cannot be known to us…

Descartes has not strictly forwarded Kepler’s claim that the crystalline humor has a hyperbolic shape, perhaps because his own anatomical investigations caused him to doubt the accuracy of this, but he maintains Kepler’s reasoning to some degree. While Descartes has long let go of any notion that spherical lenses may be preferred due to their omni-axial focus, he shrugs off the necessity for anything more than approximate foci along these oblique axes. The reason he provides for this is unclear. Either it is proposed that because the eye does not focus oblique rays, the benefits of any lens that does so would simply be lost – yet, if this were the reason, it would not result in the conclusion that such shapes are not best for precise vision, for they would be no worse than his hyperbola; or, he means to say that hyperbolic lenses are simply preferred because their weaknesses are natural weaknesses of the eye, with Nature not to be improved upon. This is emphasized in conclusion of the passage:

…Moreover we will always have to take care, when we thus place some body before our eyes, that we imitate Nature as much as possible, in all things that we see she has observed in constructing them; and that we lose none of the advantages that she has given us, unless it be to gain another more important one. – Seventh Discourse

There is additional evidence for the naturalized justification of the hyperbolic “weakness” (notice the question of valuation in the phrase “important one”). Firstly, when he proposes his notion that the telescope is simply an extension of the eye, Descartes imagines that all the refraction would occur in one lens, thus, “…there will be no more refraction at the entrance of that eye” (120). In this analogical conception of the extended length of the eye Descartes imagines his hyperbola as supplimenting and even supplanting the eye’s refractions. Secondly, when Descartes addresses the possibility that seeing at the borders may be an improvement of vision, he denies this, by virtue of how Nature has endowed our sight. Seeing more is not seeing better.

There is only one other condition which is desirable on the part of the exterior organs, which is that they cause us to perceive as many objects as possible at the same time. And it is to be noted that this condition is not in any way requisite for the improvement for seeing better, but only for the convenience of seeing more; and it should be noted that it is impossible to see more than one object distinctly at the same time, so that this convenience, of seeing many others confusedly, at the same time, is principally useful only in order to ascertain toward what direction we must subsequently turn our eyes in order to look at the one among them which we will wish to consider better. And for this, Nature has so provided that it is impossible for art to add anything to it. – Seventh Discourse

What Kepler has stated as simply the role of the borders of the retina to serve the perfection of the center, Descartes has made an occasion to assert the virtue of the human Will (a cornerstone of his metaphysics, and a cornerstone which Spinoza rejects, which makes the two philosophers quite opposed in their philosophy of ideal perception). For Kepler the edges serve the center, as is shown in the satisfactions of the soul. For Descartes the width of blurred vision becomes only a field upon which the Will manifests itself in making judgements of good and bad. Not only is the hyperbola’s condensed vision naturalized, it is key to how the Individual Will functions. Nature herself has foreclosed the possibility of improving the capacity to see more in a better way. Spinoza’s philsophy of mind’s-eye perception is based on the principle that one sees clearly as one sees more – more at once. (It is interesting that immediately following this assertion Descartes uses the examples of sailors and hunters who are able to improve on Nature’s provisions, but only in the direction of further sharpening their eyes to a more narrow focus. Descartes valuation is both implicit and naturalized.)

It suffices to say that in this long digression what Spinoza means by “the eye is not so exactly constructed” is that the non-spherical shapes of the eye (and our tendencies of vision that come from it) provides a focus that is not optimal. Spinoza here likely conflates his metaphysics and his optics, as perhaps does Descartes. His critique, right down to the root of centralized conceptions of a naturalization of hyperbolic vision, opens to Post-modern and Post-structuralist critiques of marginalization and philosophies of Presence, locating his objection not in the glorification of the human eye, but in the understanding of its limitations. Descartes’ philosophy of “clear and distinct” and its parasitic conceptions of Human Will are cut at in a very essential way. But the question remains, is there an optical advantage to spherical lenses, as they exhibit the flexibility of omni-axial foci? The obvious objection to hyperbolics is that they proved impossible to grind, either by hand, or in the kinds of automated machines that Descartes proposed. As a practiced lens-grinder Spinoza better than most would surely know this. But aside from this serious detraction Spinoza finds one more, and it is one that Kepler, Descartes and Gregory all admit, as they justify it not in optical terms, but in terms of naturalized conceptions of the eye and perception. Perhaps we can assume that Spinoza, out of his love for the sphere, coupled with the Keplerian sense of the spherical radiation of light, the practical considerations of lens grinding, and a epistemological conception of Comprensive Vision, saw in the admitted weakness of the hyperbola (and the eye) something that outweighed the moderate weakness of spherical aberration. In a sense, Spinoza may have seen spherical aberration in terms of his acceptance that almost all of our ideas are Inadequate Ideas. [More of this line of thought written about here: A Diversity of Sight: Descartes vs. Spinoza ]

“Now since a definite segment of a circle can bring it about that all the rays coming from one point are (using the language of Mechanics) brought together at another point on its diameter, it will also bring together all the other rays which come from other points of the object, at so many other points.”

A modified version of the letter 39 diagram, showing what Spinoza believed to be the failings of the hyperbola

Spinoza repeats his insistence upon the virtues of spherical lenses. As the modified diagram here shows, the capacity to refract rays along an infinity of axes is in Spinoza’s mind an ideal which hyperbolic forms cannot achieve. He does not accept the notion that an assumed narrow focus of human vision, nor the supposed shape of the crystalline humor (Kepler) determines that “hyperbolic abberation” is negligable to what should be most esteemed. This insistance upon the importance of the sphere calls to mind James Gregory’s description of refraction on sphere of the “densest medium” presented in his first proposition of the Optica:

If truly, everything is examined carefully, then it will seem – on account of the aforementioned reasons -that all the rays, either parallel or non-parallel, which are incident on the circular surface of the densest medium for refraction, are concurrent in the centre of the circle. Now we ask: how does this come about? The answer is: – Well, however a line is drawn incident on the circle, (provided they are co-planar) an axis can be drawn parallel to it and without doubt the circle can be considered a kind of ellipse so that any diameter can be called the axis, from which it appears that the special line sought is the axis of a conic section. – Optica Promota

figures 1 and 2 from the Optima Promota

One feels that there seems something of this ideal conception of the densest medium floating behind Spinoza’s conception of the spherical lens. Material glass somehow manifests for Spinoza, in its particularities of modal expression, these geometric powers of unified focus, and peripheral focus is a part of what Spinoza conceives of as ideal clarity.

“the language of mechanics”

But there is another very important clue in this section of the letter: the phrase “using the language of Mechanics”); for now I believe we get direct reference to Johannes Hudde’s optical treatise “Specilla circularia” (1655), an essential text for understanding Spinoza’s approach to spherical aberration.

Rienk Vermij and Eisso Atzema provided a most valuable, but perhaps sometimes overlooked insight into the 17th century reaction to Descartes resolution to spherical aberration in their article “Specilla circularia: an Unknown Work by Johannes Hudde”. They present Hudde’s small tract (it is not quite nine typed journal pages) which offers a mathematical treatment of the problem of spherical aberration. Interestingly, as it was published anonymously, Hudde’s teacher at Leiden, Van Schooten, actually thought that the work belonged to his star student Christiaan Huygens. Presumably this was because of the closeness it bore to Huygens’ 1653 calculations of aberration, and he wrote him to say as much, and he likely sent him a copy of it as Christiaan requested. Hudde’s approach is a kind of applied mathematics to problems he considered to be pragmatic mechanical issues. In a sense he simply took spherical aberration to be a fact of life when using lenses, and thought it best to precisely measure the phenomena so as to work with it effectively. The hyperbolic quest was likely in his mind a kind of abstract unicorn chasing. He wanted a mechanical solution which he could treat mathematically, hence his ultimate distinction between a “mathematical point” of focus and a “mechanical point”. As Vermij and Atzema write describing this distinction and its use in analysis:

At the basis of Hudde’s solution to the problem is his distinction between mathematical exactness and mechanical exactness. Whereas the first is exactness according the laws of mathematics, the second is exactness as far as can be verified by practical means. After having made this distinction, Hudde claims that parallel incident rays that are refracted in a sphere unite into a mechanically exact point (“puntum mechanicum”). In order to substantiate his claim Huddethen proceeds to the explicit determination of the position of a number of rays after refraction.

Restricting his investigation to the plane, Huddeconsiderstherefraction of seven parallel rays by explicitly computingthepoint of intersection of these rays with the diameter of the circle parallel to the incident rays for given indices of refraction. The closer these rays get to the diameter, the closer these points get to one another until they finally merge into one point. Today, we would call this point the focal point of the circle; Hudde does not use this term.

Returning to spheres, Hudde erects a plane perpendicular to the diameter introduced above and considers the disc illuminated by the rays close to this diameter. He refers to this disc as the “focal plane”. On the basis of the same rays he used earlier, Hudde concludes that the radius of the focal plane is very small compared to the distance of the rays to the diameter. Therefore this disc could be considered as one, mechanically exact point. In other words, parallel rays refracted in a sphere unite into one point (111-112).

From this description one can immediately see a conceptual influence upon Spinoza’s initial diagram of spherical foci, and far from it being the case that Spinoza knew nothing about spherical aberration and the Law of refraction, instead, it would seem that he was working within Hudde’s understanding of a point of focus as “mechanical”. We know that Spinoza had read and reasoned with Hudde’s tract, as he writes to Hudde about its calculations, and proposes his own argument for the superiority of the convex-plano lens. And the reference to “the language of mechanics” seems surely derived straight from Hudde’s thinking. What these considerations suggest is that Spinoza’s objection to the hyperbola to some degree came from his agreement with Hudde that spherical aberration was not a profound problem. As it turns out, given the diameters of telescope apertures that were being used, this was in fact generally correct. Spinoza joined Hudde in thinking that the approximation of the point of focus was the working point of mechanical operations, and the aim of shrinking it down to a mathematical exactness was not worth pursuing (perhaps with some homology in thought to Descartes’ own dismissal of the approximations of focus of rays oblique to the axis of the hyperbola).

F. J. Dijksterhuis summarizes the import of Hudde’s tract, in the context of Descartes’ findings in this way:

The main goal of Specilla circularia was to demonstrate that there was no point in striving after the manufacture of Descartes’ asphericallenses. In practice one legitimately makes do with spherical lenses, because spherical aberrations are sufficiently small. (Lenses and Waves. Diss. 72)

Spinoza has a connection to the other main attempt to resolve the difficulty of aberration from focus using only spherical lenses, that which was conducted by Christiaan Huygens. Spinoza in the summer of 1665 seemed to have visited Huygens’ nearby estate several times, just as Huygens was working on developing a theory of spherical aberration and devising a strategy for counteracting it which did not include hyperbolas. In that summer as Spinoza got to know Huygens, he was busy calculating the the precise measure of the phenomena. In 1653 he had already made calculations on the effects in a convex-plano lens, an effort he now renewed under a new idea: that the combination of defects in glasses may cancel them out, as he wrote:

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. (OC13-1, 318-319).

What followed was a mathematical finding which not only gave Huygens the least aberrant proportions of a convex-plano lens, but also the confirmation of its proper orientation. In addition he found the same for convex-convex lenses. In August of that summer Huygens wrote in celebration:

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.

During this time the secretary of the Royal Society was writing Spinoza, trying to get updates on the much anticipated work of Spinoza’s illustrious neighbor (he was about to become the founding Secretary for the Académie Royale des Sciences for Louis XIV. Spinoza writes to Oldenburg:

When I asked Huygens about his Dioptricsandabout another treatise dealing with Parhelia he replied that he was still seeking the answer to a problem in Dioptrics, and that soon as he found the solution he would set that book to print together with his treatise on Parhelia. However for my part I believe he is more concerned withhisjourneytoto France (he is getting ready to to live in France as soon as his father has returned) than with anything else. The problem which he says he is trying to solve in the Dioptrics is as follows: It is possible to arrange the lenses in telescopes in such a way that the deficiency in the one will correct the deficiency of the other and thus bring it about that all parallel rays passing through the objective rays will reach the eye as if they converged 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.

This letter is dated October 7, 1665, two months after Huygens had scribed his Eureka optimalization of the lens shape. Significantly, Huygen found that lenses of this optimal shape actually were not the best for his project of combining lens weaknesses (302-303), rather lenses with greater “weaknesses” were better combined. Several facts can be gleaned from Spinoza’s letter, and perhaps a few others guessed at. Spinoza had both looked at and discussed with Huygens his contemporary work. So the sometimes guarded Huygens was not shy about the details of his project with Spinoza. It may well have been Huygens’ treatment of the convex-plano lens here that caused Spinoza to write to Hudde less than a year later with his own calculations in argument for the superiority of the convex-plano lens, using Hudde’s own Specilla as a model. (Hudde seemed quite interested in Spinoza’s proofs of the unity of God, and the correspondence seems to have begun as early as late 1665.) What cannot be lost is that with a joint awareness of both Hudde’s and Huygens’ attempts to resolve spherical aberration, Spinoza was in a very tight loop of contemporary optical solutions to the problem. Not only is his scientific comprehension trusted by both Huygens and Oldenburg at this point, but perhaps also by Hudde.

What is striking though is Spinoza’s pessimism toward Huygens’ project. Given Spinoza’s optical embrace of spherical lenses (in the letters 39 and 40 we are studying), what would lead Spinoza to such a view he qualifies as “As yet this seems to me impossible.” Is this due to a familiarity with Huygens’ mathematics, and thus comes from his own notable objections? Has Huygens actually shared the frustrations of his experiments? Or is he doubtful because Spinoza has only a vague notion of what Huygens is doing? He seems to deny the very possibility of achieving a mathematical point of focus, though his mind remains tentatively open. His added on thought, Further, throughout his Dioptrics, as I have both seen and gathered from him (unless I am mistaken), he treats only spherical figures” , is also curious. He seems privy to the central idea that Huygens is using spherical lenses to achieve this – what other figure would it be? – but it is possible that Spinoza here qualifies his doubt as a general doubt about sphericals which he only believes Huygens is using in his calculations, showing only a cursory knowledge. Perhaps it is only an addendum of information for Oldenburg.

Huygens indeed would soon find such a solution to aberration writing,

“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 withellipticor hyperbolic lenses” (OC13, 318-319).

I am unsure if he had come to this solution before he left for Paris in mid 1666, or if he would even have shared this discovery with Spinoza, but he also came to the same pessimistic conclusion as Spinoza held, at least for Keplerian telescopes, for his design only worked for those of the Gallelian designs which had fairly low powers of magnification. By combining convex lenses the aberration was only increased. This would be the case until February of 1669, when Huygens finally came up with right combinations of lenses.

“For from any point on an object a line can be drawn passing through the center of a circle, although for that purpose the aperture of the telescope must be made much smaller that it would otherwise be made if there were no need of more than one focus, as you may easily see.”

Again Spinoza returns to his initial point, now putting it in context of real telescopes. Such telescopes required the stopping down of the aperture, something that reduced the impact of spherical aberration; but restricting the aperture reduced the amount of light entering the tube, hence making the image less distinct. I am unsure what Spinoza refers to in “as you may easily see”, for neither of his diagrams seem to distinctly address this aspect. Perhaps Spinoza has in mind two diagrams of the eye that Descartes provides, contrasting the angles of rays entering the eye with a narrow and a wide pupil aperture. Was this a diagram which Jelles had mentioned in his response (below, left)?

 

 

Descartes' diagram 17 of the eye, Sixth Discourse

“What I here say of the circle cannot be said of the ellipse or the hyperbola, and far less of other more complex figures, since from one single point of the object only one line can be drawn passing through both the foci. This what I intended to say in my first letter regarding this matter.”

I am unsure what Spinoza means by “both the foci”, but it appears that he asserts again that because there is only one axis of either hyperbolics or ellipse available to any rays of light arriving for refraction, and that spherical lenses, again, have the advantage that rays come from any particular point of an object then can be focused to a single “mechanical point” along an available axis. Under Spinoza’s conception, this is an advantage that cannot be ignored.

Below I post Spinoza’s last diagram to which he refers with his final remarks. I place it beside Descartes diagram to which it most likely refers. This may be the most telling aspect of Spinoza’s letter, for we have to identify just what Spinoza is making clear as distinct from what Descartes was asserting.

Descartes’ diagram is a variation of as similar diagram which illustrated his prototype idea of forming a single lens made of an objective lens and a tube of water which was imagined to be placed directly upon the eye, making a long prosthetic lens, physically extending the eye. In this version he proposes that because such a watery tube is difficult to use, the tube may be filled with one large glass lens, with surfaces A and B acting as the anterior and posterior surfaces. And yet again acknowledging that the making of such a lens is unlikely, the same diagram is meant to serve as a model of an elementary telescope:

…because there would again be some inconvenience…we will be able to leave the whole inside of this tube empty, and merely place, at its two ends, two lenses which have the same effect as I have just said that the two surfaces GHI and KLMshouldcause. And on this alone is founded the entire invention of these telescopes composed of two lenses placed in the two ends of the tube, which gave me occasion to write this Treatise. – Eighth Discourse

Spinoza’s diagram from Letter 40

 

 

 

Descartes diagram 30, Seventh Discourse

“From the attached diagram you will be able to see the proof that the angle formed at the surface of the eye by rays coming from different points becomes greater or less according to the difference of the foci is greater or less.”

There are several ways to look at Spinoza’s diagram, but it is best to take note of where it diverges from Descartes’ (for Jelles would have had the latter to compare it to). The virtual image of the arrow appearing to be much closer to the eye is eliminated, presumably because the appearance of magnification is not in Spinoza’s point. The refraction of the centerpoint of the arrow remains, and is put in relation to refractions of rays coming from the extreme ends of the arrow. The refractions within the eye have been completely collapsed into an odd, artfully drawn eye, (the touch of lid and lashes actually seem to speak to Colerus’ claim that Spinoza was quite a draftsman, drawing life-like portraits of himself and visitors). Behind this collapse of the eye perhaps we could conclude either a lack of effort to portray his version of refractions into the mechanisms of the eye, or even a failure of understanding, but since this is just a letter to a friend, it probably marks Spinoza’s urge just to get a single optical point of across, and he took more pleasure in drawing an eye than he did tracing out his lines of focus. An additional piece of curiousness, which may be a sign of a very casual approach is that the last arrow in the succession, which to my eye appears to be one supposed to be in the imagination of the mind, Spinoza fails to properly reverse again so that it faces the same direction as the “real” one, although perhaps this is an indication that Spinoza thought of the image as somehow arrived within the nervous system at a point, on its way to be inverted by the imagination (though in the Ethics he scoffs at Descartes’ pituitary concept of projective perception). There is of course the possiblity that I am misreading the diagram, and the the final arrow somehow represents the image as it lies on the retina at the back of the eye. At any rate, it is a confusing addition and one wonders if it is just a part of Spinoza’s musings.

As best I can read, below is an altered version of the diagram designed to emphasize the differences between Descartes’ drawing the Spinoza’s:

The first thing to be addressed, which is not labeled here, is what C is. There is the possibility that it is a crude approximation of the crystalline humor, acknowledged as a refractive surface. If so, the upper arc of the eye and the figure C would form some kind of compound refractive mechanism approximate to what Descartes shows in his eye, here compressed and only signified. But I strongly suspect that C is the pupil of the eye, as the aperture of the telescope has been recently has been referred to in terms of its effect on the requirements of refraction, and in Descartes text there is a definite relationship between the telescope aperture and the pupil of the eye (it has also been proposed to me that C is the eyepiece of the telescope).

The primary difference though is the additional emphasis on the cones of rays that come from either end of the object to be seen (here shaded light blue and magenta). This really seems the entire point of Spinoza’s assertion, that spherical lenses are needed for the non-aberrant focus of oblique cones for a object to be seen clearly. In addition to this, the angle that these rays make at the surface of the eye (indicated) points to Spinoza’s original objection to Descartes incomplete description of what is the most significant factor the construction of a telescope.

What remains is to fully assess this conception of refraction that Spinoza holds. While it is made in the context of historic discussions of the blurred nature of the borders of an image’s perception, it is also true that such an oblique focusing must occur, however slightly, at any point exactly off from the center axis of a hyperbolic lens. It may well be that Spinoza is balancing this aberration of focus in hyperbolic lenses with the found-to-be overstated aberration of spherical focus. Given his comprehensive conception of clear mental vision -seeing more is seeing better – and its attendant critique of the Cartesian Will, given his love for the sphere, perhaps aided by a spherical conception of the propagation light come from Kepler, with Spinoza being much sensitized to the absolute impracticality of ground hyperbolic glasses through his own experiences of glass grinding, it may have been quite natural for Spinoza to hold this optical opinion…though it is beyond my understanding to say definitively so. 

“So, after sending you my cordial greetings, it remains only for me to say that I am, etc.”

This is a curious ending for such a wonderful letter. Perhaps we can assume that once again the editors of his Opera suppressed important personal details.

 

These English selections and links to the Latin text: here

An origin of Spinoza’s “cones of rays” explanation, Letter 40

[addendum: in addition to these thoughts, the influence of a more recent source, James Gregory’s Optica Promota (1663) has to be considered]

Kepler and How Spinoza Viewed the Eye and Light

As a point of reference it is important to locate the origin of Spinoza’s phrase “cones of rays” found in his letter 40, since implicit in this phrase is likely the conception of light and refraction which would help us make sense of his objection to Descartes. This phrase has a history of what seems a bit of interpretive confusion, for instance, that expressed by Alan Gabbey in his Cambridge Companion to Spinoza article, “Spinoza’s Natural Science and Methodology”. Here professor Gabbey quotes the phrase as if it embodies the locus of Spinoza’s befuddlement:

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 have already pointed out that Spinoza indeed was not “unaware” of the “importance” of Descartes’ figures (since he was intimate with the debate over that importance), and that part of Gabbey’s difficulty may stem from a weakness in translation, or not taking into account Spinoza’s familiarity with Hudde’s Specilla circularia: here. Spinoza, all the same, is constructing an argument that seems to shift parameters. In Letter 39 he speaks of the capacity of spherical lenses to focus parallel rays to an (approximate) point of focus opposite, along an infinity of axes, and now he tells Jelles that this capacity is to be understood not for parallel rays, but for “cones of rays”, which is more accurate to what is actually occurring. Where does Spinoza get his conception of “cones of rays”?

I believe it is found in Kepler’s Paralipomena to  Witelo (1604), a work I am beginning to suspect holds some of Spinoza’s resistance to Descartes. Descartes called Kepler his “first teacher” in optics, so when there is a divergence between the two, Kepler and Descartes, one may perhaps look to Kepler as a source for other resistance to Descartes’ conclusions. (It is a mistake to assume that solely in terms of temporal advancement, all of Descartes deviations from Kepler are corrections, for in some ways Kepler held views antecedent to our better conceptions on the nature of light.) In letters 39 and 40 Spinoza is critiquing Descartes explanation of how image size is produced in telescopes, and he finds in Descartes’ explanation some delinquencies which give undue favor to the hyperbola. Where Spinoza likely draws his conception of “cones of rays” is where Kepler is discussing the manner in which images are formed in the human eye:

Now in order to approach closer to the way this picturing happens, and to prepare myself gradually for the demonstration, I say that this picturing consists of as many pairs of cones as there are points in an object seen, the pairs always being on the same base, the breadth of the crystalline humor, or making use of a small part of it, so that one of the cones is set up with its vertex at the point seen and its base at the crystalline (though it is altered somewhat by refraction in entering the cornea), the other, with the base at the crystalline, common with the former, the vertex at some point of the picture, reaches to the surface of the retina, this too undergoing refraction in departing from the crystalline. And all the outside cones come together at the opening of the uvea [pupil], at which space the intersection of the cones takes place, and right becomes left..

…[now speaking of a single cone of those cones of rays] Thus those rays which previously were spreading out in their progress through the air, are gathered together now that they have encountered in to the cornea, so much so that any great circle described by those rays upon the cornea, which in their decent touch the edges of the opening is wider than the circle of the opening of the uvea; however, these rays, all the way to the opening of the uvea, are so strongly gathered together through such a small depth of the aqueous humor, that now the edges of that opening are trimmed of by the extremes, and by that decent they have made illuminous a portion on the surface of the crystalline humor, if indeed they all have first arisen at a point at a certain and proportionate distance (which is pecular to each eye, and not just the same for all), they fall approximately perpendicularly, because of the similar convexity of the cornea and the crystalline humor. (trans. Donahoe, 170)  

Included in this reference is also the obvious fact that for an object to be seen, light from all its points must be gathered. It is part of Kepler’s picture:

Spinoza writes: “…in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points. And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci.”

Because Spinoza is arguing that the hyperbolic lens – designed to receive rays solely parallel to its one axis – is insufficient for the variety of angles at which light arrives, the question of parallel or coned rays does not seem germane to his argument. His emphasis in the original description seems meant to be in terms of axes, assuming a “mechanical point” of focus definition. (Whether it is ultimately germane to contemporary telescope construction is another question.)

It must be noted, though here is both a most significant implication of the cone of light having a spherical (wave?)front, something ungrasped by Descartes but captured later by Huygens, in the text that follows as Kepler closely describes this action of cones of rays in the eye, he emphasizes the “hyperbolic posterior surface of the crystalline” (171), possibly disturbing the cohesion of Spinoza’s purely spherical ideal of light refraction. If indeed Spinoza is taking Kepler’s description as his source, this gives us to consider how Spinoza might mean the inexactness of the construction of the eye (letter 40). In what way can the eye be considered imperfect, and is there a Kepler source for this notion?

Spinoza writes: “And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others.”

There is an antecedent to this in Keplers’ description of the action of rays as they come from cones at angles oblique to the axis of the cornea:

All the lines of the direct cone [a cone whose axis is the same as the axis of the cornea and crystalline] are approximately perpendicular to the crystalline, none of those of the oblique cones are, The direct cone is cut equally by the anterior surface of the crystalline; the oblique cones are cut very unequally, because where the anterior surface of the crystalline is more inclined [aspherical], it cuts the oblique cone more deeply. The direct cone cuts the hyberbolic surface of the crystalline, or the boss, circularly and equally; the oblique cone cuts its unequally. All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side…

…so the sides of the retina use their measure of sense not for its own sake, but whatever they can do they carry over to the perfection of the direct vision. That is we see an object perfectly when at last we perceive it with all the surroundings of the hemisphere. On this account, oblique vision is least satisfying to the soul, but only invites one to turn the eyes thither so that they may be seen directly (174). 

Here Kepler seems to be making the exact same point as Spinoza, with an additional hint towards the necessity of the oblique in Spinoza’s concern. The construction of the eye, in so far as its lenses are aspherical, it is retarded its capacity to handle the focus of cones of rays oblique to its single axis. This first calls our attention to the limits of human vision (in individuals and in plan), and then suggests that Spinoza’s point is one of practical application in terms of lenses: that in aiding human vision and constructing telescopes, the symmetry of spherical lenses is preferred for magnification, handling a greater variety of angles of incidence through its infinity of axes.

This does not of course establish the veracity of Spinoza’s argument, but in locating a likely origin for Spinoza’s conception, we at least place Spinoza’s argument within the context of a larger view, to be weighed with all other anti-hyperbolic (Cartesian) positions  of his day (Hudde, Huygens). As I have said, it is my sense that Spinoza derives more than this from Kepler’s account of light. More posts to follow.

Did the Huygenses “buy” Spinoza’s lens polishing technique?

The Meteoric Rise of Huygens’s Microscope

The following is an exercise in historical imagination, only meant to elicit what is possible from what we know. Perhaps a fiction bent towards fact.

Wim Klever has brought to my attention a detail which sheds some light upon the possible lens polishing techniques Spinoza employed. Admittedly the connective tissue for a conclusion is not there, but the inference remains.

Professor Klever tells me that in his “Insignis opticus: Spinoza in de geschiedenis van de optica” he cites Freundenthal’s publication of the advertisement of the auction of the Spinoza’s estate in the Haarlemse Courant. The advertisement was printed on November 2nd, and occurred on November 4th (almost 9 months after Spinoza’s death). It seems likely that Constantijn Huygens jr., and/or his brother the famed scientist Christiaan,  bid at and purchased what remained of Spinoza’s estate. This is how Wim Klever roughly translates some of the items:

books, manuscripts, telescopes (‘verrekyckers, mind the plural!), microscopes (‘vergrootglazen’, also plural), glasses so grinded (‘glazen soo geslepen’), and various instruments for grinding (‘en verscheidene slypgereedschap’) like mills (‘molens’, also plural!) and great and small metal dishes serving for them (‘groote en kleine metale schotels daartoe dienende’) and so on” (en so voort).

It is the number of devices and equipment that is Klever’spoint. Spinoza is not a dabbler in optics. He does not grind a few spectacle glasses for the near-sighted, but rather is interested in full-blown optical instrument production. There are multiple telescopes and microscopes to be had, as well as perhaps something more important, his grinding dishes, and at least two lathes or mills not to mention other small details of his process. Certainly the bill of sale attests to a rather thorough industrial investment on Spinoza’s part, making of his optical enterprises something quite substantial, but what I am most interested in here is the timing of this auction, in the view of the events that immediately are set to follow, events which may give clue to the nature of just what it is that Constantijn Huygens purchased for his brother.

Spinoza’s death, and auction occurs right at the doorstep of a very important moment in history: the official discovery of protozoa, bacteria, and then spermatozoa by Van Leeuwenhoek in nearby Delft. And it is this discovery which will eventually catapult the single lens simple microscope into European renown. But there is, I suggest, a good chance that Spinoza had been making, using, giving to others and possibly selling this kind of microscope for a very long while (Klever translates “vergrootglazen” as “microscope” as one should, but there is another word for microscope, and this word means “glass that magnifies” perhaps more suitable for a single lens microscope.)  

 

First, I should point out that Christiaan Huygens had been a neighbor to Spinoza since 1663 when Spinoza moved to Voorburg, a sleepy village just outside ofThe Hague. He is a profound experimenter and scientist, having, among other remarkably brilliant things, invented the pendulum clock and discovered the rings of Saturn in the very same year of 1656. Spinoza had, most agree, become a conversational friendinthe summer of 1665, when the two of them discussed optical theory it seems with some regularity and detail. The Huygenses lived about a 5 minutes walk from Spinoza’s room at the house of master painter Daniel Tydeman, just down the road. Christiaan moved to Paris in 1666 to take the prestigious position of founding Secretary to Académie Royale des Sciences established by the Sun King Louis XIV to rival the Royal Society of London. There was no doubt extreme pressure to counter and surpass the great flow of knowledge that was collecting at the Royal Society under the supervision of Oldenburg. 

During the intervening years, as Huygens attempted to bolster his Academy, in letters written to his brother back in Voorburg he expressed interest in Spinoza’s lens polishing technique. As early as 1667, he writes Constantijn “the [lenses] that the Jew of Voorburg has in his microscopes [I don’t have the original word here] have an admirable polish” and a month later again, “the Jew of Voorburg finishes his little lenses by means of the instrument and this renders them very excellent”. Here we have an attestation to both the mystery of the quality of Spinoza’s polish, (it was a technique which Spinoza apparently kept to himself); and also there is the hint that the instrument used was meant for very fine work, on the smaller of lenses. (In general, the difficulty in acquiring a fine polish on lenses was a significant aspect of lens-crafting technique, as polishing away the pitting of the glass brought in the grinding often would change the spherical shape of the lens.) In 1668 Christiaan then writes to his brother a concession over a debate that he must have been having with Spinoza, that Spinoza is right that the smallest objective lenses make the very best microscopes.

These references by Christiaan establish that the Huygens brothers’ had interest in techniques which Spinoza was not free with, and that Spinoza was on the side of the debate that theoretically would favor the use of single lens microscopes; this, at the very least, confirms their acquisition of his equipment and lenses to be something of a notable event. If there was anything to Spinoza’s technical capabilities which resided in the equipment he used (small grinding dishes, the nature of his lathe, an abrasive recipe, a polishing material), this fact might be evidenced by a sudden change in the capacities of either brother in making microscope lenses.

And remarkably, such a change was to come.

Now the issue of timing. Here is a timetable of events that led up to Christiaan Huygens presenting a “new microscope” to the Académie Royale des Sciences, one that perhaps reflects something of Spinoza’s technique in crafting lenses.

9 Oct. 1676  Van Leeuwenhoek sends his letter regarding the discovery of protozoa and bacteria.

21 Feb. 1677  Spinoza dies at the The Hague.

22 Feb. 1677  Van Leeuwenhoek’s letter 18 to the Royal Society is read aloud, the “first ever written account of bacteria” (Dobell).

August 1677 Van Leeuwenhoek discovers the animalcules in semen, spermatozoa

4 Nov. 1677 Spinoza’s auction, the Huygenses seem to have acquired some of Spinoza’s equipment.
@ 4 Nov. 1677 Van Leewenhoek writes to the president of the Royal Society, William Brouncker, about his observation of the spermatozoa in semen. This sample was brought to him by Leiden medical student Johan Ham (who also might have had a single lens microscope).
Late 1677 Christiaan expresses interest in the Van Leeuwenhoek/Ham discovery (OCCH 8:77; and 62-3, 65).

March 1678  Hartsoeker explains to Christiaan how he makes lenses from beads of glass.

16 July 1678  Christiaan presents to the Académie Royale des Sciences the “new microscope” that differs from others in Holland and England only in the very small size of the lens.

Aug. 1678  Christiaan writes “my microscopes” have made a “great noise” in Paris.

One must know that single lens microscopes had already been in use in the Netherlands for some time before these dates. It had been used, but its capacity for magnification had not been regularly harnessed to make scientific discovery. Part of this was due to a difficulty in using it, for it must be pressed very closely to the eye, requiring great patience, and lighting techiques for the specimen in contrast had to be developed. And part of this dearth of scientific discovery was due to simply the lack of a conceptual framework for the microscopic world. This was a new world. Few as yet would even know where and why to point such a small and powerful viewing glass. Be that as it may, the microscope technique of forming tiny bead lenses from threads of melted glass was certainly known and talked about in a close scientific circle of experimenting savants (a short history of the spherical glass here). Among those notables were Spinoza’s correspondent Johannes Hudde who made them at least since 1663 when he showed his design to the French diplomat Monconys, and possibly used it as early as 1659 when he youthfully writes in a letter how he will uncover the secrets of generation through its powers. The scholar Vossius has one in 1663 which he also shows to Monconys, and in 1666 publishes the claim that the smaller the lens the stronger the magnification. And then to greatest attention Hooke describes his own bead microscope in the Micrographia in 1665 (some comments here), complaining though that it is too difficult to regularly use, fearing the loss of his eyesight.

 

Hooke's Fly's Eye, from the Micrographia

And of course, it is the king of all microscopists, Van Leeuwenhoek, who exclusively employed this kind of microscope, making over 500 of them almost all for his personal use (some comments here). When he began using them is of much debate. He makes a claim late in life that had had made bead microscopes as early as 1659 (so simple are they to make!), yet some scholars find him to have been directly informed by the description left by Hooke in the Micrographia. We do not hear of his use until 1774, and the nature of his microscope he keeps secret for sometime. It is Van Leeuwenhoek’s microscope – upon the reading of his 18thletterto the Royal Society, the day after Spinoza’s death – that will suddenly take center stage through its discoveries (although its nature at this time remains largely unknown). The single lens microscope is the strongest microscope in the world, but only now will Christiaan Huygens be coming to realize it.

For many years it seems Johannes Hudde had to defend his tiny spherical lenses against Huygens’ intution that larger, compound scopes would do a better job. It seems quite likely that Spinoza found himself mostly on the Hudde side of the argument, even I think it likely that it was Hudde himself, or one in his circle who disseminated the technique to him, either in Amsterdam or at Leiden. To this possibility, the famed Leiden anatomist Swammerdam attributes Van Leeuwenhoek’s technique to Hudde, as he does his own’ and Borch in his diary mentions the heavy influence of Hudde upon these Cartesians. Apart from this debate, Christiaan as a user of the compound scope as late as January 1675 to Oldenburg expresses an outright pessimism towards Van Leeuwenhoek discoveries already relayed to the Royal Society. These may be founded on his own frustrations when attempting to repeat the experiments, as he simply did not have enough magnification power, or they may even be a product of Van Leeuwenhoek’s low social standing as a mere draper in Delft (while Christiaan does not strictly know what kind of microscope Van Leeuwenhoek possesses, he may have guessed. There may be a class issue that folds into the conception of the microscope. Bead lenses are simply, too simple. They are not the shiny, gearing tubes of an upper machinery):

I should greatly like to know how much credice Mr. Leeuwenhoek’s observations obtain among you. He resolves everything into little globules; but for my own part, after vainly trying to see some of the things which he sees, I much misdoubt me whether they be not illusions of his sight…(Dobell 172)

Christiaan Huygens Makes His Turn

But back to the excitment. Something has turned Christiaan Huygens’ pessimism of the simple microscope into an enthusiasm. Most certainly some of this can be attributed to the sudden notability of Van Leeuwenhoek’s discovery of the protozoa and bacteria in marshy and boggy water. In November he will have discovered what male semen looks like under high magnification. At stake were arguments over just how Life itself was generated. (Did it arise spontaneously as it seemed to do in moulds, or was there some “mechanism” to it?) One can imagine the primacy of such a question. Secondly though, it is thought that Christiaan Huygens’s sudden leap towards the simple microscope was nearly entirely triggered and faciliated by the young microscopist Hartsoeker, who not long too before had discovered this technique for himself. The two were in correspondence and in March 1678 Hartsoeker reveals to him his secret. As Edward Ruestow narrates in his wonderful history The Microscope and the Dutch Republic:

The announcement of the discovery of spermatozoa in the fall of 1677 arouses the particular interest of Christiaan Huygens and, through the young Hartsoeker, drew him belatedly to the bead microscope…but having heard of a young man in Rotterdam whose microscopes could reveal the recently discovered spermazoa, Christiaangot in touch with Hartsoeker.

The essential account of their first contact, which is Hartsoeker’s, is tainted by its entanglement with his later claim that he had in fact been the first to discover spermatozoa. The surviving correspondence begins with a reply from Hartsoeker in March 1678 in which he explained how he made the bead with which he observed the “animalcules” found in semen. He presented Christiaan with a number of these sphericals, as well as some wood and brass devices to hold them in place, and by the endofthe month had himself come to The Hague to show Christiaan the spermatozoa of a dog. Hartsoekercontinued to correspond with Christiaan about the employment and improvement of these instruments, all of which Christiaan meanwhile shared with his brother Constantijn. The following year Constantijn spoke of Hartsoeker as “the inventor of our microscopes,” and years later Christiaan recalled Harksoeker having taught them to make little spheres that served as lenses (24-25)

This is all very convincing. Christiaan, after many years of resistance to the idea of tiny spherical lenses, debating with Hudde and possibily Spinoza, spurred on by the need for more powerful magnfication due to the discovery of protozoa, bacteria and then the most importantly, the elusive key to life, spermatozoa, collaborates with a savantish, largely unknown young man from Rotterdam who even claims that had discovered the technique himself when he was a young boy, and suddenly is applying his own rather vast device-making knowledge to craft the best microscopes in Europe, presenting them to the Paris academy, confirming Van Leeuwenhoek’s discoveries only three and a half months after having learned how to bead lenses himself. Huygens is shopping his microscope across the continent, while Van Leeuwenhoek refuses to allow anyone to look into or even see his.

But the problems with this quick reversal narrative is subtle. For one the lens-bead techique is extremely simple. Hartsoeker himself said he discovered it while toying with a thread of glass and a candle. Swarmmerdam says that he could make 40 more or less servicable bead lenses in an hour. It also, as I have said, was rather ubiquitous. To recount: Huddehadbeen in possession of it at least since 1663, was willing to depart with it for at least Swammerdam and Monconys, andin fact had discussed its advantages with Huygens in April 1665. As M. Founeir describes Huygens’ objection to Hudde:

Hudde discussed the merits of these lense with Huygens [OCV, 308-9, 318, 330-1], who declined their use. He particularly deplored their very limited lack of depthof field. He foundit inconvenient that with such a small lens one could not see the upper and underside of an object, a hair for instance, at the same time (“Huygens’ Design…” 579).

Vossius, Huygens’s friend seems to be in possession of it then, and it is no doubt related to the “flea glasses” that Descartes speaks of in 1637, “whose use is quite common everywhere”.  Further, of course, when Hooke describes it in brief in his 1665 Micrographia, he exposes the method to the whole English reading world. This text Huygens remarkably had in his possession very soon after its publication, one of the few copies in Europe despite the Anglo-Dutch war of that year; and we have that copy, a section of which is annotated with Huygens’ hand.  Huygens had even been so kind to actually translate some of the English for Johannes Hudde.

Further in evidence that Christiaan Huygens was well-familliar with this lens, in November 1673 Hooke demonstrates to the Royal Society “microscope with only one globule of glass, fastened to an instrument with many joints” likely made in wide production by the Dutch instrument maker Musschenbroek. And even more conclusively, Christiaan’s own father Constantijn Sr. a few months later writes of a powerful “machine microscopique” used by both Swammerdam and Leiden professor of Botany Arnold Seyn (Ruestow, 24 n.96); and we know that Swammerdam later favored a single lens scope. Given their prevalence, simplicity andthe extent of Huygens’ likely intercourse with these lenses, it could not be that Christiaan Huygens and his brother were somehow deprived, waiting to be told how to bead glass by the 22 year old [Leiden student?] Hartsoeker? It may be imagined that perhaps Hudde kept his personal means of grinding tiny lenses secret from Huygens due to some competitive antagonism and Huygens’ obstinancyover the larger, compoundlens microscope design. Perhaps. But it could not be that all of educated Europe keep it a secret from one of the foremost scientific minds of the time. Something does not sit right. Was it simply Huygens’s disinterest in such a low-depth of field, simple lens, andhis proclivities for certain other types of lens formations (compound, like his telescopes) that kept him from wanting to know? Was Hartsoeker simply the expedient when Christiaan needed to catch up quickly? The way that Edward Ruestow tells it we get the sense that it merely took the interest of Huygens, the timely injection of technique, and then the application of the Huygens’ brothers marvelous technical sense. Perhaps.

But I suggest that one piece is missing from this puzzle. It may be not until the Huygenses acquired the lens-grinding equipment and lens examples from Spinoza’s estate that they possessed the technical means of polishing these small spherical bead lenses: a talent for minute polish which Spinoza had showed early on. Could it be that this was the link, the technical means which accelerated the rapid development of the Huygens microscope from concept to actuality?

The Huygens droplet design, as it ended up in late 1678

Ruestow cites the kinds of changes that the Huygens brothers made to the Hartsoeker lens technique, such as “removing the molten globule from the thread of glass withametal wire, or, with one end of the wire moistened, picking up small fragments of glass to fuse them into globules over the flame” (25). All these seem aimed trying to make the sphere smaller and smaller, increasing its magnification. In the endChristiaan would proclaim to his French audience that his microscope is not much different than those in Holland and England, other than the size of its smaller lens, supposedly something which he alone had achieved.

He also produced a casing that was built around this tiny lens, “mounting their own beads in small squares of thin, folded brass; with the bead trapped between the opposing holes pierced with a needle through the two sides of the folded brass, those sides were pinched together with hammered pieces of wire. The microscope would go through several revisions.

As Ruestow writes of its appearance in Paris:

“on July 16th he presented to the assembly the ‘new microscope’ he had brought back withhim from Holland – one that, according the the academy minutes, was ‘extraordinarily small like a grain of sand’ and magnified incredibly…before July was out, Christiaanusedthe instrument to show the members of the academy the microscopic life Leeuwenhoek had found in pepper water, soon after publishing the first public announcement of their discovery in the Journal des Sçavans, Christiaanalsoidentified it with the discovery of the spermatozoa.” 

By August his microscope had caused the “great noise” all over Paris, so much so that John Locke at Blois had heard of it. Through the next year he had “cultivated the impression” that Van Leeuwenhoek’s observations were made with a microscope like his own. French instrument makers set to copying his invention. The response was not altogether gleeful. In London Hooke was somewhat put out that so much excitment surrounded what for him was a well-known device, one that he himself had fashioned, used and written of. And Hartsoeker, having finished his third year at the University at Leiden, all the while had been left in the shadows, not something that sat well with his rather conceitful temperment. Traveling to Paris Hartsoeker sought in some way to unmask his role in the creation of this remarkable device, exposing Huygens to be something of a plagerist. As Ruestow reports, knowing wisely Christiaan steered him from that course,

but [Christiaan] quickly took his younger compatriot under tow and wrote a brief report for him, published in the influential Journal des Sçavans, that asserted Hartsoeker’s active role in making new bead microscopes (27).

We have here evidence of Christiaan’s tendency to obscure the origins of his microscope. Yet was there more to the development than simply Hartsoeker’s revelation of the thread melting techique? Was it that in the purchase of Spinoza’s lens-polishing equipment they acquired something of the techiques long appreciated by the brothers? Does this technique prove essential to Christiaan’s implementation of a rather simple bead-glass lense? Was Hartsoekersimply solicited for the one remaining aspect of the technique that Spinoza’s equipment would not provide, that of simply melting the glass into a lens? We do know that the grinding of the already quite spherical bead was common among its users. For instance Van Leeuwenhoek ground and polished almost all of his tiny bead lenses, (and modern assayers do not quite know why). Further, Johannes Huddealsopolished his bead lenses, reportedly with salt. Was there something to Spinoza’s knowledge of small lens-crafting that facilitated Huygens’ suddenly powerful microscope design? Something even that Hartsoeker was privy to? And lastly, if Spinoza’s equipment and techniques are implimented in this sudden rise of the simple microscope, what does this say about Spinoza’s own microscope making practices.

All this fantastic story is just speculation of course

It could merely be a coincidence that, with Spinoza having died just as protozoa and bacteria were being discovered; and with his equipment coming into the hands of the brilliant Huygenses almost 9 months later, they they then just happen to be aided by a young microscopist that gives the means needed to suddenly develop a microscope that will sweep across Europe in merely a few months. Christiaan Huygens and his brother were brilliant enough for that. Perhaps Spinoza’s ginding dishes and recipes simply sat in the dust, having been acquired. But it should be noted that many years before this, the physcian Theodor Kerckring, a friend of Spinoza’s and a member of the inner, Cartesian circle, son-in-law to its central member Franciscus Van den Enden, writes of his use of Spinoza’s microscope:

“I have to my disposal a very excellent (praestantissimum) microscope, which is fabricated by that noble Benedictus Spinosa, mathematician andphilosopher…What I in this way discovered with the help of this admirable instrument…[are] endless many extremely small animalcula….”

This is found in his Spicilegium anatomicum published in 1670, seven years before Van Leeuwenhoek’s acclaimed description of the protozoa and bacteria in letter 18. It is not clear at all what “animalcula” Kerckring saw (some offer that they are post-mortum microbes, or mistaken ciliated action), but there is the possibility that these were the earliest microorganisms to be described, or at the very least, Spinoza had perfected an advanced form of the single lens, bead-microscope whose powers of magnfication approached many of those of Van Leeuwenhoek, and even that of its copist Christiaan Huygens. The timing remains. In November of 1677 the Huygenses lmay have acquired Spinoza’s lens grinding equipment, and in 8 months they have a microscope of remarkable powers.

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 ]

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.

 

Footnote:

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.