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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.


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

Letter 45, Leibniz to Spinoza…

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

The Problem of the TTP

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

Graevius writes on April 12, 1671, concerning TTP,

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

Leibniz’s replies, May 5 1671:

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

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

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

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

“Notitia opticae promotae” and J. Hudde

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

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

Spinoza’s Refusal

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

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

A Sum on Spinoza Sugar

Spinoza at Sea

I’m waiting for Wolf’s book on the Canary Island Inquistion, so for now that should probably be all on the possible connections between the Spinozas and sugar production in Brazil and Barbados. It is my instinct that there is something there, that the bonds between the Amsterdam community and Recife, and also the wide-spread opportunity for short-term turn around would surely place some of Michael Spinoza’s investments in Brazil. It strikes me that the collapse of the Spinoza buisness upon Michael’s death is too immediate to not be due to either an erosion brought on by a decade of English harassment (as Jonathan Israel seems to suggest), or by Baruch’s incompetence. Rather, it would seem that as English naval attacks on Dutch shipping began in ’51, Michael had already secured himself a fall-back within London in the person of Antonio Carvajal who petitioned many times on his behalf. The confiscated Brazilian sugar seized by the English ship George, already consigned by the Spinozas to de Morais in Rouen, suggests a substantial connection to Brazilian sugar, London and Rouen. Remember, Carvajal had strong connections to the Rouen community. When the Portgugeuse would retake Recife and send the Amsterdam community into a chaos of exiled immigrants, Michael Spinoza died. Significant would be his debt to the same Rouen merchant de Morais, to whom he had consigned sugar shipments. It would seem that the collapse of Recife, Sephardic sugar signaled the collapse of the Spinoza firm, and that Michael had leveraged himself too far. Baruch’s charitable donation of 5 guilders to the Brazilian poor in Brazil, at a time of personal financial difficulty, suggests a family connection to that community which may very well have been an economic one. 

The relevance of this for anyone looking into the motivations and principles going through Spinoza’s mind at the time of his break with the community, and his subsequent stand in politics and arguments for freedom, is that there may have been a substantial experience of colonial collapse, with an attendant association of messianic Judaism (in the roles of the Kabbalist Aboab da Fonseca and the political envoy Menasseh ben Israel), which sprang Spinoza forwards. Sugar, with its highly problematic ethical question of slave labor, perhaps lies on a fault line in the fortunes of the Spinoza family, not to mention his community. It is interesting that Spinoza would continue this Jewish connection to the English in his correspondence with Oldenburg the Royal Society Secretary, a philosophical and scientific continuation of the economic and cultural advantages his father and Menasseh were carrying out at the time of his expulsion. As his brother Gabriel seems to have followed firm connections to London and trade in his immigration to Barbados, Spinoza was seeking another kind of sugar.

Some general thoughts on the matter.

Descartes and Spinoza: Craft and Reason and The Hand of De Beaune

Some Reflections on Letter 32

Descartes in 1640 reports to Constantijn Huygens, “You might think that I am saddened by this, but in fact I am proud that the hands of the best craftsman do not extend as far as my reasoning” (trans. Gaukroger). And as Graham Burnett translates, “Do you think I am sad? I swear to you that on the contrary, I discern, in the very failure of the hands of the best workers, just how far my reasoning has reached” (Descartes and the Hyperbolic Quest, 70).

The occasion is the wounding of the young, brilliant craftsman Florimond De Beaune on a sharp piece of glass, as he was working to accomplish the automated grinding of a lens in a hyperbolic shape on a machine approximating Descartes’ design from La Dioptrique. This at the behest of Descartes himself:

His wound to the hand was so severe that nearly a year later De Beaune could not continue with the project, a project he would not take up again. Descartes’ craftsmanless, all-turning machine could not be achieved. It is as if its “reason” had chewed up even the best of earth’s craftsman.

Compare this to Spinoza’s comment on Christiaan Huygen’s own semi-automated machine, in letter 32 to Oldenburg. (One wonders if he may even had had a now infamous injury to De Beaune in mind.) Descartes seems to write callously to Christiaan’s father in 1640 [following Gaukroger’s citation], 25 years later Spinoza writes soberly about the machine of the son:

…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 more sure [tutius] and better than any machine.

Issues of class play heavily into any attempt to synthesize the rationality of a mechanism with the physical hands [and technical expertise] of the required craftsman to build it. What comes to mind for me is the same Constantijn’s Huygens enthused reaction to the baseness of the youths Rembrandt and Lievens in 1629, when he discovered their genius. As Charles Mee relates and quotes:

Unable to have Rubens, Huygens evidently decided to make his own Rubens, and he saw the raw material in Leivens and Rembrandt. He loved the fact that this “noble pair of Leiden youths” came from such lowly parentage (a rich miller was still a miller after all): “no stronger argument can be given against nobility being a matter of blood” (Huygens himself had no noble blood). And the fact of their birth made the two young men all the more claylike, so much more likely to be shaped by a skilled hand. “When I look at the teachers these boys had, I discover that these men are barely above the good repute of common people. They were the sort that were available for a low fee; namely with the slender means of their parents” (Rembrandt’s Portrait ). 

The standing of the rising Regent riche had to position itself between any essentialist noble quality of blood, and the now stirring lower merchant and artisan classes, whose currencied freedoms in trade and mobility were testing ideological Calvinist limits. Leveraging itself as best it could on rational and natural philosophy, a philosopher-scientist-statesman was pursuing a stake in freedom and power, one that rested on the accuracy of his products. In this way it seems that Descartes’ – feigned? – glee over De Beaune’s injury, insofar as it embodied a superhuman outstripping of remedial others, manifests this political distancing to a sure degree. De Beaune was no ignorant worker, for his high knowledge of mathematics made him much more “technician” than craftsman, (in fact de Beaune had proposed the mathematical problem of inverse tangents which Descartes would not be clear on how to solve (letter, Feb 20 1639), and it was his Notes brièves and algebraic essays which would make Latin editions of Descartes Géométrie much more understandable to readers). Reason and rationality could in the abstract certainly in some sense free even the most economically and culturally base kinds (at least those with a disposition to genius), but in fact savants likely imagined that their lone feats of Reason actually distanced themselves from the “hands and limbs” on which they often relied.

Seen in this way, Spinoza’s sober view of Christiaan Huygens machine perhaps embodies something more than a pessimism of design, but rather more is a reading of the very process of liberation which technological development represented for a class of thinkers such as Leibniz or the Huygenses. The liberation of accuracy and clarity was indeed a cherished path, but perhaps because Spinoza was a Jewish merchant’s son, excommunicated, because Spinoza understood personally the position of an elite [his father had standing], within a community itself ostracized though growing with wealth, a double bind which he relinquished purposively, any clarity was necessarily a clarity which connected and liberated all that it touched. It was inconceivable to have dreamed a rationality so clear that it would distance itself from the the hands that were to manifest it. Perhaps Spinoza keeps in his mind the hand of De Beaune.

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.

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.

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