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Spinoza and Mechanical Infinities

The Mechanically Bound Infinite

I want to respond to Corry Shores’ wonderful incorporation of my Spinoza Foci  research into his philosophical project (which has a declaimed Deleuzian/Bergsonian direction). It feels good to have one’s own ideas put in the service of another’s productive thoughts. You come to realize something more about what you were thinking. And to wade back through one’s arguments re-ordered is something like coming to your own house in a dream.

This being said, Corry’s reading of my material thrills, for he is, at least in evidentary fashion, one of the first persons to actually read it all closely. And the way that he fits it in with his own appreciation for Spinoza’s concepts of Infinity certainly open up new possibility for the Spinoza-as-lens-grinder, Spinoza-as-microscope-maker, Spinoza-as-technician interpretations of his thinking.

There is much to take up here, but I would like to begin at least with the way in which certain parallels Corry draws that change the way that I see what Spin0za was saying (or more exactly, what Spinoza was thinking of, and perhaps associating on), when talking about infinities. Key, as always, is coming to understand just what Spinoza had in mind when drawing his Bound Infinities diagram:

Corry points out in his analysis/summation of Letter 12, grafting from Gueroult’s commentary, that in order to understand the epistemic point (the status of mathematical figures, and what they can describe), one has to see that what Spinoza as in mind in writing to Meyer is a very similiar diagram found in his Principles of Cartesian Philosophy, of which Meyer was the active editor. There the diagram is not so Euclidean, but rather is mechanical, or, hydro-dynamical:

The diagram illustrates water moving at a constant rate (a “fixed ratio” one might say), but due to the nature of the tube it must be moving at point B, four time faster than at AC, and a full differential of speeds between. There you can see that any section of the intervening space between the two circles composed of “inequalities of distance”  in the Letter 12 diagram (AB/CD) is not really meant as an abstraction of lines and points as it would seem at first blush (the imaginary of mathematics), but rather real, mechanical differentials of speed and material change. The well-known passage

As, for instance, in the case of two circles, non-concentric, whereof one encloses the other, no number can express the inequalities of distance which exist between the two circles, nor all the variations which matter in motion in the intervening space may undergo. This conclusion is not based on the excessive size of the intervening space. However small a portion of it we take, the inequalities of this small portion will surpass all numerical expression. Nor, again, is the conclusion based on the fact, as in other cases, that we do not know the maximum and the minimum of the said space. It springs simply from the fact, that the nature of the space between two non-concentric circles cannot be expressed in number.

Letter 12

The Lathe Buried Under the Euclidean Figure

But, and this is where Corry Shores alerted me to something I did not formerly see, the relationship between the two diagrams is even further brought forth when we consider Spinoza’s daily preoccupation with lens-grinding and instrument making. It has been my intuition, in particular, that Spinoza’s work at the grinding lathe which required hours of patient and attentive toil, MUST have had a causal effect upon his conceptualizations; and the internal dynamics of the lathe (which fundamentally involve the frictioned interactions of two spherical forms under pressure – not to mention the knowing human eye and hand), must have been expressed by (or at least served as an experiential confirmation of) his resultant philosophy. If there was this heretofore under-evaluated structuring of his thought, it would see that it would make itself most known in his Natural Philosophy areas of concern, that is to say, where he most particularly engaged Descartes’s mechanics (and most explicitly where he refused aspects of his optics, in letters 39 and 40). And as we understand from Spinoza’s philosophy, Natural Philosophy and metaphysics necessarily coincide.

What Shores shows me is that Spinoza’s Bound infinities diagram (letter 12), his very conception of the circle, is intimately and “genetically”  linked to the kinds of motions that produce them. It is with great likelihood that Spinoza is thinking of his off-center circles, not only in terms of the hydrodynamics that circulate around them, but also in terms of Descartes’ tangents of Centrifugal force.

There is a tendency in Spinoza to conflate diagrams, and I cannot tell if this is unconscious (and thus a flaw in his reasoning process) or if he in his consummatephilosophy feels that all of these circular diagrams are describing the very same thing simply on different orders of description. But the connection between a tangential tendency to motion conception of the circle (which Corry makes beautifully explicit in terms of optics) and Spinoza’s consideration of bound Infinities in the letter 12 (which remains implicit in Corry’s organization of thoughts), unfolds the very picture of what Spinozahas in mind when he imagines two circles off-center to each other. Spinoza is thinking of is lens-grinding blank, and the spinning grinding form.

One can see the fundamental dynamic of the lathe from Van Gutschoven’s 1663 letter to Christiaan Huygens, illustrating techniques for grinding and polishing small lenses,

And it is my presumption that Spinoza worked at a Springpole lathe, much like one used by Hevelius, Spinoza’s Grinding Lathe: An Extended Hypothesis, the dynamics of which are shown here:

In any case, when one considers Spinoza’s Bound Infinity diagram, under the auspices of tangential motion tendencies, and the hydrodynamic model of concentric motions, I believe one cannot help but also see that the inner circle BC which is off-center from the first, is representationallythe lens-blank, and the larger circle AD, is potentially the grinding form. And the reason why Spinoza is so interested in the differenitals of speed (and inequalities of distance) between two, is that daily, in his hand he felt the lived, craftsman consequence of these off-center disequilibria. To put it one way sympathetic to Corry’s thinking, one could feel them analogically, with the hand, though one could not know them digitally, with math. The human body’s material (extensional) engagements with those differentials (that ratio, to those ratios), is what produced the near perfectly spherical lens; and the Intellect intuitionally – and not mathematically – understands the relationship, in a clear and distinct fashion, a fashion aided by mathematics and figure illustration, which are products of the imagination.

What is compelling about this view is that what at first stands as a cold, abstract figure of simply Euclidean relationships, suddenly takes on a certain flesh when considering Spinoza’s own physical experiences at lens-grinding. Coming to the fore in such a juxtaposition is not only a richer understanding of the associations that helped produce it, but also the very nature of Spinoza’s objection to the sufficiency of mathematical knowledge itself. For him the magnitudes of size, speed and intensity that are buried between any two limits are not just abstract divisions of line and figure, or number to number. They are felt  differentials of real material force and powers of interaction, in which, of which, the body itself necessarily participates. The infinities within (and determinatively outside of) any bound limits, are mechanical, analogical, felt and rational.

Corry raises some very interesting relationship question between the Spinoza Bound Infinities Diagram and the Diagram of the Ideal Eye from letter 39. They are things I might have to think on. The image of the ideal eye is most interesting because it represents (as it did for Descartes) a difficult body/world shore that duplicates itself in the experiential/mathematical dichotomy. Much as our reading of the duplicity of the Bound Infinity Diagram which shows mathematical knowledge to be a product of the imaginary, the diagram of the ideal eye, also exposes a vital nexus point between maths, world and experience.

From Mechanics to Optics (to Perception)

It should be worthy to note that Spinoza’s take on the impossibility of maths to distinguish any of the bound infinities (aside from imposing the bounds themselves), bears some homology to Spinoza’s pragmatic dismissal of the problem of spherical aberration which drove Descartes to champion the hyperbolic lens. When one considers Spinoza’s ideal eye and sees the focusing of pencils of light upon the back at the retina (focusingswhich as drawn do not include the spherical aberration which Spinoza was well-aware of), one understands Spinoza’s appreciation of the approximate nature of perceptual and even mathematical knowledge. This is to say, as these rays gather in soft focus near the back of the eye (an effect over-stated, as Spinoza found it to be via Hudde’s Specilla circularia), we encounter once again that infinite grade of differential relations, something to be traced mathematically, but resultantlyexperienced under the pragmatic effects of the body itself. “The eye is not so perfectly constructed” Spinoza says, knowing as well that even if it were a perfect sphere there as yet would be gradations of focus from the continumof rays of light so refracted by the circular lens. What Spinoza has in mind, one strongly suspects, and that I have argued at length, is that the Intellect, with its comprehensive rational in-struction from the whole, ultimately Substance/God, in intuitional and almost anagogic fashion, is the very best instrument for grasping and acting through the nature of Nature, something that neither bodily perception, or mathematical analysis may grasp. Indeed, as Corry Shores suggests in his piece, it is the very continuum of expressional variability of Substance (real infinities within infinities) which defies the sufficiency of mathematical description, but it is the holistic, rational cohesion of expression which defies experiential clusterings of the imagination: the two, mathematics and imaginary perception, forming a related pair.

In the end I suspect that there is much more to mine from the interelationship between Spinoza’s various circular diagrams, in particular these three: that of the relationship of the modes to Substance (EIIps), that of the the hydrodynamics of circulating water (PCP, implicit in the Letter 12 diagram of Bound Infinites), and the Ideal eye (letter 39), each of these to be seen in the light of the fundamental dynamics of the lens-grinding lathe to which Spinoza applied himself for so many years, and at which he achieved European renown expertise.



The Infinities Beneath the Microscope

I would like to leave, if only for Corry Shores’ consideration, one more element to this story about Real Infinities (and I have mentioned it in passing before on my blog). There is an extraordinary historical invocation of something very much like Spinoza’s Bound Infinities in the annals of anatomical debates that were occurring in last decade of Spinoza’s life. I would like to treat this in a separate post and analysis, but it is enough to say that with the coming of the microscope what was revealed about the nature of the human body actually produced more confusion than understandings in what it revealed, at least for several decades. Only recently was even the basic fact of the circulation of blood in the body, something we take for granted, grasped. And in the 1670s the overall structure or system of human anatomy was quite contested, contradictory evidence from the microscope being called in support one theory or another. Among these debators was Theodore Kerckring, who was weighing in against the theory that the human body was primarily a system of “glands” (and not ducts). Kerckring’s  connection to Spinoza is most interesting, much of it brought to light in Wim Klever’s inferential and quite compelling treatment of the relationship of Van den Enden  and Spinoza. In any case Kerckring  is in possession of a microscope made by Spinoza (the only record of its kind), and by virtue of its powers of clarity he is exploring the structure of ducts and lymph nodes. Yet he has skepticism for what is found in the still oft-clouded microscope glass leads him to muse about the very nature of perception and magnification, after he tells of the swarming of tiny animals he has seen covering the viscera of the cadaver, (what might be the first human sighting of bacteria). He writes of the way in which even if we see things clearly, unless we understand all the relationships between things, from the greatest breadth to the smallest, we simply cannot fully know what is happening, if it is destruction or preservation:

On this account by my wondrous instrument’s clear power I detected something seen that is even more wondrous: the intestines plainly, the liver, and other organs of the viscera to swarm with infinitely minute animalcules, which whether by their perpetual motion they corrupt or preserve one would be in doubt, for something is considered to flourish and shine as a home while it is lived in, just the same, a habitation is exhausted by continuous cultivation. Marvelous is nature in her arts, and more marvelous still is Nature’s Lord, how as he brought forth bodies, thus to the infinite itself one after another by magnitude they having withdrawn so that no intellect is able to follow whether it is, which it is, or where is the end of their magnitude; thus if in diminishments you would descend, never will you discover where you would be able to stand.

Spicilegium Anatomicum 1670

Several things are going on here (and in the surrounding context), but what seems most striking given our topic, we once again get a glimpse into the material, and indeed historical matterings of what bound, mechanical infinities might be. (As a point of reference, at the time of Kerckring’s  publishing Spinoza had just moved to the Hague and published his Theological-Political Treatise, having taken a respite from his Ethics approximately half done, and he will have died seven years later.) Kerckring  in a remarkable sense of historical conflation looks on real retreating infinities with Spinoza’s own microscope, and exacts much of the same ultimate skepticism toward human scientific knowledge, as per these infinities, as Spinoza  does in his letter to Meyer. This does not mean that we cannot know things through observation, or that imaginary products are not of use to us, but only that there is ultimately for Spinoza and Kerckring  a higher, rational power of interpretation, the comprehensiveness of what abounds. Neither measurement or calculation is disqualified, in fact Spinoza in his letters and experiments and instrument making showed himself to be quite attentive to each. It is rather that the very nature of human engagement requires both attention to the bodily interaction with devices and the measured thing, and also a sensitivity to anagogic, rational clarity, something found in the very unbroken nature of Substance’s Infinity. What Kerckring’s description does is perform the very consequence of conception in scientific observation itself, almost in Spinoza’s stead (expressing very simililar  sentiments as Spinoza does in Letter 32 to Oldenburg on lymph and blood, and the figure of the worm in blood,

Let us imagine, with your permission, a little worm, living in the blood¹, able to distinguish by sight the particles of blood, lymph, &c., and to reflect on the manner in which each particle, on meeting with another particle, either is repulsed, or communicates a portion of its own motion. This little worm would live in the blood, in the same way as we live in a part of the universe, and would consider each particle of blood, not as a part, but as a whole. He would be unable to determine, how all the parts are modified by the general nature of blood, and are compelled by it to adapt themselves, so as to stand in a fixed relation to one another.

There is great conceptual proximity in these two descriptions, suggesting I imagine that Spinoza used his microscopes as well, for observation, not to mention that Kerckring and Spinoza come from a kind of school of thought on scientific observation of human anatomy, perhaps inspired by or orchestrated by Van den Enden, as argued by Klever. Just the same, at the very least, Kerckring  presents greater context of just what kinds of retreating infinities Spinoza  had in mind in his letter 12 diagram, not simply a differential of motions, but also a differential of microscopic magnitudes, each of which were an expression of an ultimate destruction/preservation analysis, something that falls to the very nature of what is body is. Spinoza not only ground lenses, but also made both telescopes and microscopes, gazing through each at the world, this at a time when the microcosmic and macrocosmic, nested infinities were just presenting themselves to human beings. And as such his critique of scientific observation and mathematical calculation preserves a valuable potentiality for our (postish) modern distancings and embrace of the sciences.

Evidence toward the nature of Spinoza’s Lathe(s)

Writing an email today to an interested party I found myself running over the evidence that Spinoza used either a hand driven lens-grinding lathe, or one of the springpole variety, such as the Hevelius lathe (Selenographia, 1647). It seemed best to briefly summarize them hear, as though the evidence is scant, it is not non-existent. I have already written briefly on these two lathes here: Spinoza’s Grinding Lathe: An Extended Hypothesis ; Spinoza’s “Spring Pole” Lathe: Experience to Metaphysics and Back

1. The auction of Spinoza’s estate held nine months after his death (4 Nov 1677), accounts for more than one “mill” (mollens). If such mollens are taken to be grinding lathes, it shows that he had more than one, likely for more than one purpose (telescope/microscope; grinding/polishing). It is also very possible that the estate had already lost a number of its items by the time of the auction.

2. Spinoza is generally assumed to have been tubercular. While in remission the disease may not inhibit the stenuousness of activity, when manifest any grinding lathe that would greatly reduce exertion would seem almost necessary. A springpole lathe frees the hands, and allows the larger leg muscles to bear the burden.

3. There is some evidence that Spinoza did work on larger telescope objective lenses, ones that would require heavier iron grinding forms, less conducive to a hand-driven lathe. For instance, Huygens writes his brother in reference to calculations Spinoza had done for a 40 ft. lens (in collaboration with J. Hudde), and ten years after Spinoza’s death, Constantijn Huygens writes of using a 42 ft. Spinoza grinding/polishing form (I have not checked the primary source on this yet, OC IX p. 732) which worked so well that he did not have to lift the lens from the glass to check it for blemishes even after an hour straight of use (suggesting a fixed-glass, hands free devise).

4.Christiaan Huygens at several points in his letters to his brother refers to Spinoza’s championing of small spherical lenses for microscopes. If these are not unground spherical bead-drop lenses, then these would be the kind that required very precise grinding and polishing. One can certainly imagine that hand-driven grinding lathes would be more suitable for this kind of work.

This rough sketch seems to suggest a combination of grinding and polishing lathes were used. Spinoza in his criticism of Huygens’ semi-automated grinding lathes, and artisan concern for basic tried techniques, does strongly advise that whatever Spinoza’s lathe designs, they were of a simple, efficient design. He did not appreciate speculative mechanical experimentation, at least not for its own sake. One imagines that his springpole- and/or hand- lathe was of a tried and true fundamental design, though from Huygens’s comments on Spinoza’s polishing techniques, it does appear that he possessed distinctive techniques which were either discovered by himself as a inventive craftsman, or were from a source not commonly available to others.

Spinoza’s Brilliant Neighbor, The Huygens Estate: Hofwijck

Christiaan Huygens drawing of Hofwijck, where Spinoza would have visited

To add to the picture of the quiet Huygens family estate at Voorburg we have a drawing of it by the hand of Christiaan Huygens himself (undated).

This brings a certain vividness to any imagined visit by Spinoza in the summer of 1665 [thought about here: What Spinoza and Huygens Would Have Seen that Summer Night ]. It would be likely that at one of these top floor windows Huygens would have placed his telescope, and through which he and Spinoza would have gazed at Saturn. An interesting sidenote: Spinoza is said by Colerus to have drawn portraits of some quality (and he lived in the house of a master painter), so in this the two men may have shared some small interest.

What Spinoza and Huygens Would Have Seen that Summer Night

Telescoping with Spinoza and Christiaan Huygens

The night sky July 13th 1665 near Voorburg

The night sky, 11:31 p.m. July 13th 1665 near Voorburg

 …with which they have been able to observe the eclipses of Jupiter caused by the interposition of satellites, and also a kind of shadow on Saturn as if made by a ring. – Spinoza to Oldenburg, May 1665

It is tempting to imagine that having seemingly met and discussed with some enthusiasm issues of astronomy and microscopy in late April of 1665, Spinoza may have visited often with Huygens at his country estate which was some 10 minutes walk from Spinoza’s rooms on Kerkstraat. Aside from issues of social standing, Spinoza as a maker of telescopes and microscopes, surely would have wanted to talk with Huygens on the state of the art of the day, and further, Christiaanin esteem and sharing may have wanted to share the facts of his experiences of discovery with his most compelling neighbor. But we must ask, what could Spinoza and Huygens have seen, if they had looked through a telescope together?

It is probably without doubt that Huygens had set up one of his long telescopes permanently on the estate, for though he had not made astronomical discoveries for nearly decade, issues of significance were happening in the sky. In the Winter of 1664-5 a brilliant comet had showed itself, and then another in March 27thof ’65, the last being visible to the naked eye for a month. These were occasions not only for religious fervor, and signs of the end of Times, but also windows into the structure of the universe, events to observe closely so to feed the growing theories on the nature of cosmic bodies and their travel. Plague and war was rife, and yet and imperative of knowledge was blooming. As a general note, everyone’s eyes were on the sky, and Huygens’s telescope most surely was trained there. 

from Lubinetski’s 1667 treatise Theatrum Cometicum

from Lubinetski’s 1667 treatise Theatrum Cometicum

Aside from the striking sky pyrotechnics of comets, there is further in evidence that the sky was still much on Huygens’ mind in the summer of 1665. As recently as 1660-1661 Huygens was busy defending the power and accuracy of his telescope to the accusations of fraud come from the famed Italian telescopist Divini. (Huygens had controversallydiscovered the rings of Saturn in 1656, lead to them by his discovery the moon of Saturn later to be named Titan, in 1655, which he regarded as “my moon”.) Withthe existence of the Saturn’s rings still in dispute, evidence for them resting solely on the strength of his telescope, the prestigious Prince Leopold of Tuscany had twice proposed a paragone, a face-off field test between Huygens’s telescope and Divini’s, before persons of high social status, an offer that Huygens each time refused. In this vien of concerns, Roger Hahn in “Huygens and France” suggests as quite likely that Huygens continued interest in the telescope in April of 1663 lead him to the house of Adrien Auzout in Paris, and to a group that included Pierre Petit who were working on a 80 to 100 ft. telescope under the promise of seeing Huygens’ rings of Saturn more clearly. Then, what must have been a great relief to Huygens, in April of 1664 the rising Italian telescopist Campani himself faced the arrogant and well-connected Divini in a paragone, and definitively defeated him, soon after publishing a confirmation of Huygens’s Saturn findings through the report of the shadow of the questionable rings (look closely at the wording of Spinoza’s letter 26, where this shadow is mentioned). Following this history of observation and dispute, Spinoza writes of his early meeting of Huygens in May of 1665, and their talk of issues of astronomy. He mentions in his letter their discussion of the rings of Saturn, as well as the eclipses of Jupiter. With Saturn, comets and Huygens’s telescope in the forefront of the last years of European astronomy, and fresh to their friendship, one can easily imagine Spinoza having walked the ten minutes to the Huygens estate (pictured below), as the sun was lowering into the late evening of a long summer day, in order to look through the long-contested and now vindicated device. The sky would not have completely lost the sun’s light until just after 11:30.

If we imagine the night to be something like that of July 13th, there would be no moon. The canal’s lapping could be heard perhaps from the upper story, and somehow too the breadth of the property, the rush of the breeze across the rows of orchard and bush, so symmetically laid forth. Dark shadow-lines set out in geometry, ringing faintly as if strings. Here, would not Christiaan Huygens have trained his telescope on Saturn, the home of his distantly reached sight of rings, and a moon he had discovered? How many times had he looked at it? Saturn happened to be at its zenith on this night, due South, low on the horizon as the sky blackened. Christiaan had carved into the lens withwhich he had seen Saturn’s moon and rings with a line from Ovid: Admovere oculis distantia sidera nostris : They carried distant stars to ours eyes. This would have been a remarkable moment for Spinoza as he contemplated the Infinite.

East on the ecliptic there was Jupiter. Would they not have focused then on that great planet, having discussed the discovery of its eclipses only a few months earilier? Would not the glass telescope have brought to two great, but quite distinct minds into intersecting conversation. Neighbors of such diversity, such disjunction, living a short walk from each other, stretched thin across the solar system by means of a glass and metal? Would Huygens have mentioned, tipping that lens to its precise point, that he believes that light moves in spherical waves?

The Huygens Estate at Voorburg

The Huygens Estate in Voorburg

Constantijn Huygens Uses Spinoza’s Grinding Dish (1687)

In correspondence Wim Klever directs my attention to evidence that the Huygenses used Spinoza’s grinding equipment as late as 10 years after his death. The citation is here, thus translated from the OCCH:

[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. XXII, p. 732, footnote).

If I have the details here correct, it seems either that indeed the Huygenses had purchased Spinoza’s lens grinding equipment at auction in November of 1677 and maintained the use of that equipment, or that Spinoza may have made a grinding-dish for the brothers under their specification before he died. What is revealed is that Spinoza’s skill had been directed toward not only microscope instruments, but also towards telescopes of a rather large magnitude. This lens appears to have a focal length of 42 ft. And secondly of course, here Constantijn jr., a rather experienced lens-grinder himself, seems to have marveled at the confidence in the lumininocity of the lens produced.

(This reported Spinoza lens is much shorter in focal length than three known to have been made in 1686 by Constantijn: w/ diameters 195, 210 and 230 mm, and w/ focal lengths of 122, 170 and 210 ft.; each “made from the same very poor glass – a heterogeneous and discoloured potash-rich, but essentially lead-free `forest glass’.”)


Descartes’ Dioptrics 7th Discourse and Spinoza’s Letters 39 and 40

[For a fuller treatment of the topic read “Deciphering Spinoza’s Optical Letters”]

Telescopes and Turning a Flea into a Elephant

To offer context to the question that Jelles poses in a letter we have lost, regarding the size of objects on the retina, I post here the likely text that Jelles has in mind, and to which Spinoza is responding. Spinoza writes in answer:

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

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 (letter 39)

It is the 7th discourse that Spinoza and Jelles are discussing. Here is a portion of the relevant passage:

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).

At this point in the explanation Descartes seems to have touched on the factor of the “the size of the angle which the rays make when they cross one another at the surface of the eye” for this would seem implicit in a discussion of the refraction of rays. But Spinoza seems to have focused on what follows, which leaves off any concern for this factor:

Using this diagram, Descartes continues:

“Thus it is evident that the image RST would be greater than it is, if the object VXY were nearer to the place K, where the rays VKR and YKT intersect, or rather to the surface BCD, which is properly speaking the place where they begin to intersect, you will see below; or, if we were able to arrange it so that the body of the eye were longer, in such a way that there were more distance than there is from its surface BCD, which causes the rays to intersect, to the back of the eye RST; or finally, if the refraction did not curve them so much inward toward the middle point S, but rather, if it were possible, outward. And whatever we conceive besides these three things, there is nothing which can make this image larger.”

Here, Descartes has claimed to total all possible means of enlarging an image. He indeed has talk about the surface of the lens (BCD), but perhaps in keeping to Spinoza point, has not talked about the “size of the angle” that the rays make at the surface of a lens. (An issue Spinoza would like to make regarding the powers and functions of a telescope, it would seem.)

Descartes continues, detailing the kinds of improvements of magnfication that are possible:

“Even the last of these [the refraction curving outward from point S] is scarcely to be considered at all, because by means of it we can augment the image no more than a little bit, Read more of this post

Some Personal Thoughts on a Possible Spinoza Lathe

Some discussion has been going on over at the Practical Machinist forum, where I have sought any views about the real world workings of any of the devices Spinoza may have used at grind lenses. I have come to the thought that it might very well be a rather simple device that Spinoza used, not much differnt than the one Manzini depicts for the start of the 17th century:

In response to my query someone was kind enough to relate some of his own, unique experiences with a machine not unlike the one illustrated. I post them here because they serve to vivify the elementary nature of these technical movements, in the manner of which a 20th century workman and a 17th century philosopher might share an experience of material and design effects.

Joe writes:

When I was in my 20s I worked for a couple of years at the Peerless Optical Co in Providence, Rhode Island, making lenses for glasses. While much of the work was automated to a degree there was still a little corner of the shop where very special lenses were ground. Because I was actually interested in the work, that became my department.

The lenses were ground against iron forms, called “laps” (either convex or concave) using a variety of progressively finer abrasives. The final polish was achieved by gluing a thick disc of felt to the lap and using a much finer polishing media. The lap spun in a bucket-like contraption that worked very much like a potters wheel. The lens was kept in contact with the lap by means of a hinged arm with an adjustable pin. The arm was held in place with the left hand, the pin pushing against the lens, while you added abrasive to the lap with the right hand. To secure the lens without damaging it, a small flat piece of metal with a center hole was “glued” to it using thick green pitch, exactly like the “sealing wax” used before the invention of gummed envelopes. We melted the pitch onto the lens with a bunsen burner. It was removed by chilling the whole piece, at which point the pitch would harden and fall off the glass.
Other than the motor that spun the lap, there isn’t a thing about this whole process that any 17th century mechanic would find surprising. Also, with particularly difficult lenses, I would have to forgo the hinged arm and hold the lens against the lap with my hand.

In our case, a special purpose-built machine re-cut the laps when they wore…I had a beautiful engraved set of brass gauges which I used to check them (by holding the gauge and lap up to a window) and which must have been 100 years old or more when I was using them. I can see where a lathe of some sort would be essential for making the laps, a primative lathe would suffice, but I can’t see it being used to actually make the lens itself.

The machine illustrated in the post above this one is very much like what I am describing. In fact, other than the hand operation it would be instantly recognizable to anyone who was making lenses in the manner I was. I actually made a couple of lenses for an antique telescope on this equipment…they worked perfectly.

In coincidence to this, Rijk-Jan Koppejan sent me a photograph of a reproduction of just this illustrated device, built by his team and part of their exposition on the invention of the telescope, organized around the 400th year Middelburg anniversary. There is to be a symposium of speakers in September, which I just may have to find a way of attending. He says he may be able to take new, more revealing photographs and send them. I will post them as he might.

Joe mentions that the curvature of this grinding “dish” may be too extreme, but that Manzini’s illustrator may not have thought this a significant factor (also, we cannot see the internal curavature of the reproduction). I don’t know enough about the optics of the time to comment.

He mentions as few more interesting details of his memory of lens grinding with such a lathe, in particular the method he had to use to correct the wear on the “laps” (as he calls them) – Spinoza calls them patinas or scutellae, plates or dishes – and thoughts about processes by which a spherical lens is checked for its optical quality:

I suspect that the drawback to using male/female laps against each other is that both pieces will wear. I am guessing that if the lens maker had a set of gages like I used, which are simply used to check the curve, the lap could be spun in any lathe-like machine and the surface selectively filed or ground to return it to true. As I’ve said, I held the lap and the gage up to a window and looked for a streak of light between them…a very accurate way of measuring once you have some practice and know what to look for.

…Another memory just came back…I think that the felt was attached to the lap with fish or hide glue. The lens was checked by holding it up to a light bulb with a single filiment. You held it in such a way that the light from the filiment reflected off the surface. If there were no breaks or nicks in the reflection, the lens was perfectly true. This could also be done by stretching a hair across a window and picking up the shadow. You could never see the imperfections with the naked eye..

…The lens was finished in what we called an “edger” which was nothing more than a lathe-like spindle that gripped the little metal piece glued to the lens and spun it against a grinding wheel. These were not the modern clay-based wheels but slow turning natural stone wheels that ran in water, the grinding wheel turning one way and the lens in the opposite direction. In this way the outer edge was gradually reduced in a manner perfectly concentric with the optical center. Even if the metal attachment was slightly off center on the original lump of glass, this process insured that it would be perfectly concentric when finished. You could only remove the metal piece after this was done and you could not replace it perfectly so it was a once-chance-only affair.

Althought at this point it is only a collective intution that Spinoza did not use a large, spring-pole lathe such as the one shown at the Rijnsburg, there are some facts that lead to me this thought. First is that when Huygens writes of the superior polish of Spinoza’s lenses, he describes them as “little lenses”:

“the Jew of Voorburg finishes his little lenses by means of the instrument and this renders them very excellent” (Complete Works, 6:155).

I do not have the original word from which “instrument” is translated, but at least at this point it strikes me that this is a small device. And these lenses are small. I am unsure if Huygens is talking about telescope lenses or microscope lenses, but there is the implication of very fine work. This also coincides with Spinoza’s own light criticism of Huygens’ very complex machine, in letter 32 to Oldenburg. (See some of my thoughts on this here.) It is of course possible that Spinoza had a spring-pole lathe much like the Rijnsburg and Hevelius lathes, but the contrast between his own approaches and Huygens’s seems more at home with a simpler device. There are other factors that cause me to think that this is so, but for now this is enough to discuss.

How Much were Spinoza’s Lenses and Microscopes?

In the interest of making Spinoza’s lens-grinding, polishing, and telescope and microscope building more vivid to those considering his metaphysics, this evidence is posted as to the kinds of prices for those services one would expect.

Lueken and Lueken (1694)

E. G. Ruestow writes:

At a date I read to be late 1670’s: “…Johan van Musschenbroek in Leiden [sold micro-beaded lenses] forty for a gilder – roughly a day’s wages for skilled manual labor in the Netherlands. Musschenbroek otherwise advertised his cheapest simple microscope for 7½ gilders and his most elaborate, with nine seperate and interchangeable lenses, for nearly ten times as much” (The Microscope and the Dutch Republic, 28).

And the footnote reads: “Johan van Musschenbroek advertised six beads – “Glaze dropjes, en bolletjes” – for three stuivers, which, there being twenty stuivers to the guilder, was the price equivalent to forty for a guilder…Earlier in the century, Constanijn Huygens, Sr., had paid forty guilders for one of Drebbel’s microscopes.”

If indeed Spinoza made simple bead lenses, provided a buyer was available – which for these type lenses would be likely be infrequently – he could make a laborer’s day’s wages in about an hour (Ruestow points out that Swammerdam said he could make them at this rapid rate, 40 and hour). The prices of any primary grinding of lenses to specific focal lengths or uses for other salesmen or instrument makers of course are not reflected here. But perhaps a week’s wages could be made for his simplest microscopes.

Lenses not Rare

One can see from the depiction of a spectacle makers’ storefront, strewn with glimmering lenses and spy glasses, that by the late 17th century such devices are quite common. In fact, when Descartes writes his Dioptrics in 1637, when Spinoza is five years old, he mentions how common “flea glasses” have become. It is good to remember both the commonality of Spinoza’s trade, its brute, craftsman standing, but also the elite circulation of ideas which came about in applying these somewhat widespread devices, both in terms of theories about the nature of what was seen, but also changing techniques and optical conceptions on how to see it. Spinoza stood in both worlds.

Spinoza and Hooke’s Micrographia: The minascule made Large

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


As wiki writes of it:

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

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

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

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

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

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

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

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

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




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