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The “Corporeal Equation” of 1:3: What Makes A Body for Spinoza?

If a Body Catch a Body Comin’ Through the Rye

I have always been fascinated by Spinoza’s defintion of a body as found in the Second Part of the Ethics. Not because it reflected some proto-physics, but because it allowed a radical revisioning of what defined boundaries between persons, and between persons and things. What seems implicit in such a definition is that something of a cybernetic recusivity surrounds and defines any isolated “part” of the Universe, yet, a recursivity that only comes clear by taking a perspective. One understands that really for Spinoza the entire Universe composes a single such body.

Here is Spinoza’s famous Ethics  defintion, and an even more elementary and bold one from his much earlier Short Treatise on God, Man and His Well-Being (KV)

Ethics: When a number of bodies of the same or different magnitude form close contact with one another through the pressure of other bodies upon them, or if they are moving at the same or different rates of speed so as to preserve an unvarying relation of movement among themselves, these bodies are said to be united with one another and all together to form one body or individual thing, which is distinguished from other things through this union of bodies (E2p13a2d)

KV: Every particular corporeal thing [lichaamelijk ding] is nothing other than a certain ratio [zeekere proportie] of motion and rest.

Yet, such a vision for Spinoza is more than an instructive imaginary relation, it indeed is a proto-physics, a concrete real which must be accepted as such. There is a certain sense in which Spinoza’s conception of a body must be reconciled with the “facts” of contempory physics if we are to geta stronger impression of the truth of his metaphysics and psychology. As Spinoza wrote to Blyenbergh, “Ethics, … as everyone knows, ought to be based on metaphysics and physics” (Ep 38). At a general level, in Spinoza’s own terms, if his physics is radically wrong this may pose serious doubts as to his Ethics (an entirely rationalist reading of his philosophy notwithstanding). And concordantly, one might assume, new information in physics could have a rippling effect across his philosophy and Ethics.

It is not my aim here to explore these wider meta-questions, but rather to for a moment pause upon a change in my own thinking. I had always taken Spinoza’s above defintions just as I explained, fantastic frameworks for revisioning the world as it common-sensically and historically has come down to us, intellectual opportunities for instance to see the connections between bodies in a Batesonian or an Autopoietic sense. This still remains. But I came to realize that when Spinoza is thinking about a “certain ratio” (as Shirley translates) or a “fixed manner” (Curley), he is thinking of something quite quantifiable, something numeric. I had of course loosely thought that this was the case, but until recently I had never strictly thought about it.

Spinoza’s Objection

There is an interesting, rather provocative point in Spinoza’s letters to Oldenburg, as he is reporting back to this Secretary of the Royal Society on the progress of his brilliant neighbor Christiaan Huygens. It seems apparent from what Spinoza reports that he has had intermittent, but somewhat substantive discussions on not only optics and lens-grinding, but also on physics. Huygens, by what history tells, had corrected Descartes’ rules of motion, and done so through experiment. Huygens was quite interested in the rules of motion for he had invented the pendulum clock way back in 1656 (the same year he had discovered the rings and a moon of Saturn), and for a decade was focused on improving it. Spinoza reports back to Oldenburg Huygens’ disagreement with Descartes, but tantalizingly also speaks of his own disagreement, in particular, with the sixth rule of motion:

Spinoza: “It is quite a long time since he [Huygens] began to boast that his calculations had shown that the rules of motion and the laws of nature are very different from those given by Descartes, and that those of Descartes are almost all wrong…I know that about a year ago he told me that all his discoveries made by calculation regarding motion he had since found verified by experiment in England. This I can hardly believe, and I think that regarding the sixth rule of Motion in Descartes, both he and Descartes are quite in error.” (Letter 30A)

Oldenburg: “When you speak of Huygens’ Treatise on Motion, you imply that Descartes’ Rules of motion are nearly all wrong. I do not have to hand the little book which you published some time ago on ‘Descartes’ Principia demonstrated in geometrical fashion’. I cannot remember whether you there point out that error, or whether you followed Descartes closely to gratify others.” (Letter 31)

Spinoza: “As to what you say about my hinting that the Cartesian Rules of motion are nearly all wrong, if I remember correctly I said that Mr. Huygens thinks so, and I did not assert that any of the Rules were wrong accept the sixth, regarding which I said I thought that Mr. Huygens too was in error.” (Letter 32)

Many commentators have not been able to make much headway when interpreting Spinoza’s objection to Descartes sixth rule of motion, for at the very least, it seems woven to his other rules, and the objection should have spread far wider than this, as in the case with Huygens. Alan Gabbey (The Cambridge Companion ) for instance simply finds it nonsensical. And Lachterman in “The Physics of Spinoza’s Ethics”, really almost avoids the issue altogether. (Wim Klever has taken the question directly on in “Spinoza and Huyges: A Diversified Relationship Between Two Physicists”, tying it to a Cartesian difficulty in explaining cohension, while Rivaud finds what seems to be an untenable conceptual connection between speed and essence in his “La physique de Spinoza”.)

I certainly am not one here to solve the question, but it did get me thinking about how Spinoza conceived of a body, and what a “certain ratio” meant to him.

Descartes’ Sixth Rule of Motion and Spinoza’s Defintion of a Body in the Short Treatise

Below is the sixth rule of motion to which Spinoza found objection. It essentially describes what would ideally happen if two bodies of the same size, one in motion and one at rest, struck. Descartes suggests that if the moving body had four (4) degrees of speed before impact, after impact the ratio would be 1:3, with the body at rest taking on one (1) degree of speed, the bodies rebounding:

Descartes:51. Sixth rule.
Sixthly, if body C at rest were most accurately equal to body B moved toward it, it would be partly impelled by B and would partly repel it in the contrary direction. That is, if B were to approach C with four degrees of speed, it would communicate to C one degree and with the three remaining would be reflected in the opposite direction.

Huygens reportedly showed through experiments at the Royal Society that instead all the degrees of speed would be imparted to the body at rest, and the intially moving body would then be stopped, and it was to this, as well as to Descartes’ rule that Spinoza expressed an unspecified objection. But this is not the ultimate point here for me. I was rather struck by an early note on Spinoza’s defintion of a body found in the Short Treatise , which proposes the same ratio of 1:3 that Descartes used to illustrate his sixth rule, here below stated as the ratio of motion to rest, and not as “degrees of speed”:

Spinoza: Short Treatise, notes to the Preface to Part II:

12. As soon, then, as a body has and retains this proportion [a proportion of rest and motion which our body has], say e.g., of 1 to 3, then that soul and that body will be like ours now are, being indeed constantly subject to change, but none so great that it will exceed the limits of 1 to 3; though as much as it changes, so much does the soul always change….

…14. But when other bodies act so violently upon ours that the proportion of motion [to rest] cannot remain 1 to 3, that means death, and the annihilation of the Soul, since this is only an Idea, Knowledge, etc., of this body having this proportion of motion and rest.

What is striking to me is that such an elementary numerical value for the definition of a body would occur to Spinoza in this context. Alan Gabbey wants us to point out that this ratio of 1:3 is found in editorial notes, and my not even be of Spinoza’s hand, though I am unsure if Spinoza would have allowed such a strong example to slip through if it was alien to his thinking. Provocative is that the context for this proposed illustration of a “corporeal equation” (as Matheron has named it), of 1 to 3, is that it is the human body that is being discussed and not abstract solids such as those Descartes discusses in his physics. Even if Spinoza does not imagine that the human body might actually retain such an elementary 1:3 ratio of motion to rest, somewhere in his conception of the human body there is an affinity to such an simple math. One for instance would not be describing a super computer whose mark would be its complexity, and turn to such a number. It would appear that at least figuratively Spinoza at the time of the Short Treatise  thought of the human body as elementarily composed such that its conatus expressed a homeostasis that was comprehesible and simple. The numerical value of 1 to 3 held perhaps a rhetorical attraction.

By the time of Spinoza’s geometrical treatment of Descartes’ philosophy, the proposed illustrative values that Descartes included in his rules for motion are no longer there. Spinoza generalizes them apart from any particular equation. One could see in this perhaps already a distancing from some of Descartes’ assertions, and Oldenburg tells Spinoza that he looked over Spinoza’s exposition of Descartes to see signs of his disagreement, finding none.

What the sixth rule Meant for Spinoza

For my part, if we take Descartes’ sixth rule at face value, and imagine the interaction between two bodies of the same size, one at rest, one in motion, we get a glimpse into the kind of change Spinoza thinks makes a body. For once the supposed transfer of a degree of speed occurs, the two bodies are now in communication. As long as they are not interacted with by other bodies their ratio will remain 1:3, and they would be considered an “individual”. And if one of those bodies interacted with another body so as to change its speed, immediately one realizes that if the idea of a single body is to be preserved the definition of parts needs to be expanded so that the ratio is to be expanded across a host of interactions. One sees how the definition of a body as a body is entirely contingent upon how you calculate.

Wim Klever finds in Spinoza’s 1665 objection to Descartes’ sixth rule (made almost 4 years after the writing of the Short Treatise ) a testament to Spinoza’s thorough-going commitment to a physics of immanence. This could be. But one could also imagine the case that Spinoza had been caught up in a conversation with Huygens at the Hofwijck estate and was entirely caught off guard by Huygens’ sweeping dismissal of Cartesian physics, which up to that point had been a touchstone for most scientific thinking in Europe. Spinoza’s objection to the sixth rule may have only been a reaction, one that prudently and instinctively placed himself between Descartes and Huygens, on a single point, a point he could not elaborate on.

But what was it about Huygens’ correction to Descartes which may have also given Spinoza pause, especially if Descartes’ rule for the transfer of motion between two equal bodies, one moving, one at rest helped frame Spinoza’s general notion of what makes a body? Would it not be that there was a complete tranfer of motion from one to the other, that one stopped and the other started? Because Spinoza envisioned bodies moving together in community, and integrated communication of impinging interactions that could be analyzed either in terms of their recursive cohensions (for instance how the human body can be studied solely in terms of its own internal events, as one might say, immanent to their essence), or in terms of extrinsic interactions which “through the pressure of other bodies” cause these internal events, the intuitional notion that a body in motion would deliver all of its motion to another body at rest, and not be rebounded simply defied the over all picture of what Spinoza imagined was happening.

I suggest that somewhere in the genealogy of Spinoza’s thought about what defines a body he found Descartes sixth rule quite suggestive. The idea that two bodies which do not seem to be in communication, one moving, one unmoving, (an essential perceptual differential which allows us to distinguish one thing from another in the world), suddenly can appear in communication from the change they bring about in each other in collision, now departing at a ratio of speeds, helped Spinoza psychologically and causally define the concrete yet contingent composition of an individual. The corporeal equation of 1 to 3 standing in for the possibility of mathematical determination which could conceptually unite any two parts in a single body, given the right analysis.

But when Spinoza encountered Huygens’ thorough dispatch of Cartesian mechanics we can suspect that Spinoza came in contact with his own theoretical disatisfactions with Descartes. As we know, Spinoza was part of a small cadre of mathematicians and thinkers which found dissatisfaction with Descartes idealized optics, something that no doubt formed part of his discussions with fellow-lense grinding and instrument maker Christiaan Huygens. And too, Spinoza likely felt that though Descartes’ mechanics provided an excellent causal framework for rational explanations of the world, his determinations lacked experimental ground. It would seem to me that Spinoza’s objection to the sixth rule of motion poses something of a revelation into the indeterminancy of Spinoza’s physics. The sixth rule may have played a constructive role in his imagination of what a body must be, but in particular in view of Huygens’ confirmed rejection of the rule, it became simply insufficient. Spinoza’s physical conception of a body stands poised between a Cartesian rational framework of causal interaction and mechanism, which proves lacking in specifics, and the coming Newtonian mechanics of force. However, in such a fissure, one does have to place Spinoza’s notion of immanence.

Autopoiesis Comes?

Signficantly, and something which should not be missed, is that the definition from axiom 2 of proposition 13 of Part 2 above is not the only conclusive one that Spinoza provides in the Ethics. Lemma 4 under axiom 3 actually provides a view of the body which does not require that the parts themselves remain in a fixed ratio to each other. Rather, it is only the ratio itself that must be preserved:

If from a body, or an individual thing composed of a number of bodies, certain bodies are separated, and at the same time a like number of other bodies of the same nature take their place, the individual thing will retain its nature as before, without any change in its form [forma].

This allows us to see that by the time of his writing of the Ethics, Spinoza’s notion of ratio, the aim of his mechanics, is far from what Newton would develop. The causal histories traceable through interactions between bodies certainly were signficantly important for Spinoza, but it was the preservation of a mode of interaction which really concerned Spinoza’s focus. That all the bodies that compose and individual could conceivably be replaced, without that individual being considered as changed (as for instance we know of nearly every cell of the human body), is something that Newtonian physics would not enumerate. It is within this conception of preservation that I think Spinoza’s mechanical conceptions have to be framed, in the entirety of an effect between bodies, the cohesiveness of the modal expression.

One need only turn to something like Autopoietic theory (both those of life by Maturana and Varela, and suggestively of social forms by Luhmann) to see a lineage given from Spinoza’s Lemma 4 description:

The defintion of a living thing understood to be a self-producing machine:  “An autopoietic machine is a machine organized (defined as a unity) as a network of processes of production (transformation and destruction) of components which: (i) through their interactions and transformations continuously regenerate and realize the network of processes (relations) that produced them; and (ii) constitute it (the machine) as a concrete unity in space in which they (the components) exist by specifying the topological domain of its realization as such a network.” (Maturana, Varela, 1980, p. 78)

On the difference between “organization” and “structure”:  “…[I]n a toilet the organization of the system of water-level regulation consists in the relations between an apparatus capable of detecting the water level and another apparatus capable of stopping the inflow of water. The toilet unit embodies a mixed system of plastic and metal comprising a float and a bypass valve. This specific structure, however, could be modified by replacing the plastic with wood, without changing the fact that there would still be a toilet organization.”
(Maturana & Varela, 1987, p. 47)

Where Lies Spinoza’s Physics?

Spinoza’s immanent connection between physics and metaphysics in a turn toward a decisive ethics, is one in which any outright mechanics must be understood beyond simply A causes B, and the appropriately precise mathematical calculation of what results. If Spinoza’s physics (and even its relationship to Descartes who preceded him, and Newton who followed him) is to be understood, it is this recursive relationship between parts that has to be grasped, the way in which parts in communication can be analyzed in two ways, along a differential of events internal to a horizon, and events external to that horizon, interior and exterior, even with a view to the conceived totality. It seems that it is this replaceable nature of body-parts in composite that qualifies Spinoza’s physics as interpretively distinct, and what allows it to place within the domain of cause not only questions of material interaction, but also psychology and belief, and ultimately social values of good and bad. 

What it seems that Spinoza was most concerned with in his assessment of a physics is the kinds of concrete reactions which ground our selective ability to usefully distinguish one thing from another, a usefulness that ever trades on the community of rational explanations with share with others. The result of this physics is an ultimate ground upon which we can and do build our own mutual body of social wholes, our own physics of decisions and distinctions. Physics both ground and distinguish us for Spinoza, always suggesting an anatomy of joined, contiguous parts; it is an anatomy that guides the effortless butcher’s knife that ideally, knowingly, seldom would need sharpening.

Govert Bidloo, A Spinoza Microscopist?

[Addendum, September 10th: in looking at the full text of the letter referenced below, indeed Bidlow did NOT use a Spinoza microscope, but was only referencing Kerckring's use as well as his observations on the limitations of the microscope. I keep the post up though, to preserve the thought process of a deadend of research, for whatever that may be worth, as well as for the value of Bidloo's citation of Spinoza at a near near the death of his friend Eric Walten: Govert Bidloo’s 1698 Refference to a Spinoza Microscope ]

Physician to the King and Another Spinoza Microscope?

[The arguments below I present prospectively, waiting for a confirmation of the source]

I stumbled upon some evidence that there is a second Spinoza microscope in the historical record, and it is my hope that this glass may bring to view more of the details for which I have been straining. Thus far, the only first hand report we have is from Spinoza’s fellow Latin student, and possible van den Enden disciple, Theodore Kerckring, who in his Spicilegium anatomicum  (1670), describes how with Spinoza’s glass he had seen a “infinitely minute animalcules” teeming upon the viscera. This description is to be questioned, firstly, because Kerckring himself warns us a few sentences before, that all observations of microscopes have to be doubted; but also because Kerckring reported elsewhere some microscopic observations which plainly come from the imposition of fantasy upon sight.

In this case the account may be more sobering and exact, though I have yet been able to actually assess the content of the claim. The report comes apparently from Govert Bidloo, and man of fairly high standing, and apparently connections to Spinozist political movements of his day. In 1694 Bidloo was appointed professor of anatomy and medicine at the university of Leiden, a post to which he was not able to well-attend due to also becoming the personal physician to stadholder William III, who would die in his arms in 1702. If indeed Govert Bidloo did use and favor a Spinoza microscope, he was a well-connected anatomist and physician, and public champion of microscopic investigation.

Collaboration with van Leeuwenhoek: Parasitic Protozoon

The fact of Bidloo’s use of a Spinoza microscope is at this point circumspect, as for the moment I have only a summary of the mention of praise for a Spinoza microscope-glass (vergrootglass), in a memoir-letter written to the famed microscopist Antony van Leeuwenhoek, subsequently published in the same year, 1698. I do not read Dutch, so I had to rely upon the summation of a website owner to understand its content.

“Passage from a letter of Govard Bidloo (Henrik van Kroonevelt Ed., 1698, page 27) a memoir to Antonie van Leeuwenhoek, about the animals which are sometimes found in the liver of sheep, on the etiology of diseases (the Plague) and referring to remarks of scientists abroad on his work, and quoting the quality of the Magnifying glass made by Benedict de Spinoza.”

This is found here. The citation given, aside from the letter itself, is not traceable. Perhaps it is a television production: [52] “Cells of Spinoza”: Tetsuro Onuma, Representative of Yone Production Co.Ltd. (2002).

The phase “quoting the quality of the Magnifying glass” I assume probably means “citing the quality”. Because the context is missing for me, there is no way to affirm what I would suspect, that Bidloo is writing to van Leeuwenhoek about his observations of small parasites and their eggs, as found in the liver of sheep, and it is by virtue of the excellence of Spinoza’s glass that his observations are assured. This is somewhat also how Kerckring references his Spinoza microscope.

Historical Context For Bidloo’s Letter to Van Leeuwenhoek 

Two decades before Bidloo presented his findings to van Leeuwenhoek, in 1674 van Leeuwenhoek was startling the world as he peeled away the curtain of the microscopic, revealing to a new level of exact description and illustration, a world of minute animals and structures. Under his tiny, spherical lenses the first bacteria and protozoans were coming to life, and he began letting the world know about in through letters written to leading scientists in London. And in October ’74, he wrote to the Royal Society about his discoveries of “globules” and “corpuscles” in the bile of domesticated animals, the first Sporozoa and parasitic protozoon. It would be as an expansion upon these observations that Bidloo would conduct his own microscopic examnations. I quote here from Dobell’s excellent book in van Leeuwenhoek to give a sense of the early material and Bidloo’s connection to it, first from the letter, and then from Dobell’s commentary:

…in the bile of suckling lambs there are very little globules, and some, though very few, bright particles. which are a bit bigger; besides irregular particles, of divers figures, and also composed of globules clumped together.

The bile of yearling sheep I find to be like that of suckling lambs, only with this difference, that in this bile there are also oval corpuscles of the bigness and figure of those I remarked in ox-bile. (Letter 7 to the Royal Society, October 19th 1674).

I think there can be no doubt that the “oval corpuscles” – called eijronde deeltgensin the original – which Leeuwenhoek discovered in the gall-bladder of one of his “three old rabbits,” were the oocysts of the coccidian Eimeria stiedae; while the comparable structures which he found in the bile of sheep and oxen were, equally certainly, the eggs of trematodes [Dobbell notes: Fasciola hepatica- the worm itself -was well known to L.; for the Dutch anatomist Bidloo (1649-1713) dedicated a little memoir to him, in 1698, in which it was described and figured. If my interpretations be correct, the foregoing extract records the first observations ever made upon the Sporozoa or upon any parasitic protozoon (200)

Antony van Leeuwenhoek and his 'Little Animals'

Eggs and the Source of Disease

It is regarding these Fasiola hepatica that Bidloo is writing to van Leeuwenhoek in 1698, apparently part of a collaboration of observations between the two microscopists. This is how Frank Egerton sums up the correspondence in his article for the Bulletin for the Ecological Society of America : 

Leeuwenhoek examined flatworms (flukes) from the livers of diseased sheep under a microscope and suspected that the sheep got the worms from drinking rainwater that collected in fields (21 February 1679, Leeuwenhoek 1939-1999, II:417-419). He pursued the subject no further until 1698, when he and Professor of Medicine Goderfridus Govard Bidloo (1649-1713) of Leiden University (van der Pas 1978) discussed liver flukes in sheep. Boththen wrote up their observations for publication, with Leeuwenhoek sending his to the Royal Society and Bidloo sending his to Leeuwenhoek, who had them published in Delft. Bidloo sent with his letter an overly precise drawing of a fluke, which shows two eyes, a heart, a circulatory system, and intestines that existed only in his imagination. Nevertheless, Bidloo did recognize the eggs and concluded correctly that the species is hermaphroditic. He also generalized from his observations that these worms seem to cause disease in sheep and that worms probably also cause disease in humans (Bidloo 1698, 1972). Leeuwenhoek went out and attempted to find fluke eggs in fields and ditches, where they might have been deposited in sheep feces (2 January 1700, 1939-1999, ?), but he had no way to identify them if he had found them. The fluke life cycle is so complex that it was not fully understood until the mid-1800s (Reinhard 1957). (53)

"A History of the Ecological Sciences, Part 19"

Indeed, the lifecycle of F. hepatica is quite complex, as it relies upon a symbiont aquatic snail, something no microscope would reveal to these men, but it is good to note that Bidloo's microscope and analysis did properly identify the eggs of F. hepatica, something which may give clue to the magnification of his glass. It would appear that the two men were operating under at least remotely similar powers of glass, and at this point van Leeuwenhoek had achieved magnification really beyond compare for the century.  

Bidloo's illustration of the flatworm F. hepatica

The size of the eggs in question may be in order. They come in the thousands, so together are visible to the naked eye, but the eggs themselves are microscopic, measuring approximately 130-160 µm, or 130/1000th of a millimeter:

According to their optical appearance and approximate measurements, we isolated about 1,300-1,500 ‘large’ eggs from a fairly large quantity of sheep faeces. Of these, 300 were measured and their average size was found to be 154 (143-180) x84 (75-102) µm. Fasciola eggs of normal size found in the faeces of the same sheep measured 129 (107-162)x 71 (61-79) µm.

“Unusually Large Eggs of a Fasciola hepatica Strain” (1982) D. Duwel

As I have not read Bidloo’s account, I as yet cannot tell if his glass resolved such detail, but van Leeuwenhoek’s description of “oval corpuscles” must have. And we should keep our mind open to this possibilities.

If we are to speculate, having identified what Bidloo saw and concluded, and assumed that he used a Spinoza made glass, what was the nature of Spinoza’s “vergrootglas”? Literally, this word means “magnifying glass”, something distinct from the word for microscope. It is the same word used to describe the instruments sold from Spinoza’s estate at auction on November 4th, 1677. (It is even conceivable that this was one of those instruments.) A vergrootglas could be anything from a swivel-armed spectacle glass used for dissection and study, to the very powerful simple, single-lens microscopes that Swammerdam and van Leeuwenhoek used. Aside from the more famous Leiden anatomists who used a simple microscope, we are told that Bidloo’s successor to the university position, Boerhaave, used a lens as small as a grain of sand (Ruestow 95). But the story is unclear. Bidloo was a student and friend to Ruysch, a fellow student and associate of Kerckring from ’61 onwards, who used magnification quite sparingly, and would have had no need of such an intense and difficult lens.

Devils and Parasites

There is another interesting point of about Bidloo’s biography which makes his 1698 reference to Spinoza’s lens more than a point of curiosity. It is twenty-one years after Spinoza’s death, but something more than simply the persistence of the efficacy of Spinoza’ instrument forces his name into consciousness. Bidloo, the physician of William III, was apparently a political activist of a sort, a champion of republican values. And just the year before his rather vociferousfriend Eric Walden had died in prison, perhaps by suicide following a series of failed suits for his freedom, under the general accusation of being a Spinozist-atheist. Walten’s escalating pamphleted attacks against the Dutch Reformed Church, in defense of Bathasar Berkker’s “The World Bewitch’d”, were fierce and reminiscent of Spinoza’s friend Koerbagh, who also died as a political prisoner. Berkker had maddened the religious in his Cartesian-like argument that because their could be not causal interaction between Spirit and Matter, devils and angels could have no effect on this world. This denial of both the miraculous and the diabolical enraged the pious, and when Walten wrote on Berkker’s behalf, the ire came to be directed towards him, eventually with legal consequence. This connection between Bidloo and Walten I find, thinly, but indicatively here:   

In 1688 he took up the cause of William III against James II and showed himself to be a staunch defender of popular sovereignty and the elective nature of monarchy. Next, he turned to the question of the civil rights of governments over the church, and two local disputes, one concerning the privileges of the regents of Amsterdam, and other Rotterdam tax upheavals. [note, after "regents of Amsterdam": It is unclear which pamphlets in this particular row were written by Walten and which by his friends Govert Bidloo and Romeijn de Hooghe, the famous engraver. See Knuttle, "Ericus Walten", p. 359-383.] (44)

“Eric Walten (1663-1697): An Early Enlightenment Radical In the Dutch Republic”, by Wiep van Bunge, in Disguised and Overt Spinozism In and Around 1770

Whether Govert Bidloo used Spinoza’s microscope in his observations on the hepatica or not, I cannot say for certain now, but his reference in the published memoir, in the context of his observations on parasites of the body and a suspicion that they lead to human illness no doubt reflected to some degree the events that of the years previous, and the sourness of the death of Walten in prison. What comes to mind is Spinoza’s reflection to Oldenburg so many years before, that we are like a worm in the blood, how our perceptions are only most often local to what jostles us, itself a reflection on Kircher’s microscopic discovery of worms in the blood of plague victims. (Some thoughts here:  A Worm in Cheese ). One must remember that this was not only a time of political and religious upheaval, but also a time of plague. The clearness of Spinoza’s glass no doubt, in the minds of his admirers, expressed the clarity with which the political body must be examined. Bidloo’s study of the bile of sheep, in search of parasites with Spinoza’s glass either in hand or in mind, surely struck him as fitting.

To Understand Spinoza’s Letter 32 to Oldenburg

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

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

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

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

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

Oeuvres Complètes, XVII (p.300)

Oeuvres Complètes, XVII (p.299)

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

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

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

Oeuvres Complètes, XVII (p.303)

 Oeuvres Complètes, XVII (p.304)

As Graham Burnett describes:

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

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

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


 Oeuvres Complètes, XVII (p.302)

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

From Manzini

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

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

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

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

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

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 ]


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