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Tag Archives: Huygens

Spinoza’s Circle and the Interior

Some Ruminations on Spinoza’s “Simplest” Thought

A single figure keeps returning to me as I contemplate Spinoza’s optical concepts and metaphysics, in view of the prevailing Cartesian science of the age, that elementary figure that Spinoza provides to help explain how he conceives of the reality of the modes, even the modes of ideas for non-existent things, in relationship to the ultimate reality of Substance, God or Nature. As Spinoza explains, the figure stands for something that has not parallel, not example, because it is a fundamental relationship which is unique and totalizing.

Ethics part 2, prop 8 schol. — The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles; wherefore they are distinguished from the remaining ideas of the remaining rectangles. (Elwes trans)

This has always been a convenient image for the explanation how the modes dependently are an expression of the totality which is all of what there is. One practically sees how the modes, while causally interacting with each other to produce current states of being, immanently rely upon a larger comprehesivity. And if one wants to even leave off from the narrowness of Spinoza’s explicit thought, the chords D and E could be read as vectors which produce an intersecting point (unlabeled), the modal expression of two lines of force, a point of focus, this seems at least arguable when considering much of the rest of Spinoza’s view.

But two other realizations come to me as well. The first is that the rectilinearitythat Spinoza presents within the borders of the circle very well may for Spinoza have represented the linear analyses of motion provided by Descartes. Part of the challenge presented to Spinoza, as he tried to heal a perceived breech between Body and Soul, Thought and Extension, was how to bridge mathematical descriptions with affective realities. In a certain sense the rectilinearity with the circle of Substance for Spinoza represents the ideational crispness of mathematical description understood as a partial, yet sufficient expression of substance. The mathematics is to some degree subsumed. Just as Huygens and others struggled to assume the advantages of the Perspectiva  tradition in optics, yet still fully explain the properties of light’s propagation (which Huygens would show to be that of a wave or a pulse), Spinoza too sought to preserve the mechanized sureties of Descartes mathematizations, yet with a view that could account for the affective propagations living bodies (as early as Kepler’s Paralipomena light is described as an “affectus”).

Secondly though is the realization that the circle does not only represent Substance as a whole, but also any conatus expression of Body (defined by Spinoza as a preservation of a ratio of motion and rest over time). The circle above can also be read as the recursive coherence of a body, and the internal causal relationships of which it is composed, immanent to its essence. 

In other words, a body, in the ordinary as well as in the Cartesian sense, preserves its physical integrity in just the same way the whole universe preserves it: they differ only with regard to the degree of probability that each can ‘survive’ externally caused modifications. Only the unique individual of level-ω can be modified in infinite ways without giving up its identity.

The Physics of Spinoza’s Ethics, David Lachterman

What level-ω is in Lachterman’s nicely conceived reduction is the last border of an entire body. Level-2 is the most elementary kind of body for Spinoza, a hypothetical composition of what one supposes is at least two of the most simple bodies “corpora simplicissima”. The human body, and all such bodies we regularly recognize as such are what Lachterman regards as level-k bodies (2 < k < ω). Following these descriptions, the circle above from the Ethics could be read not only as the schema of the level-ω Body, but also of any typical level-k Body. It is only the degree of power, freedom and being that preserves such a circle (ratio) that distinguishes it from the ultimate whole. Spinoza world can be seen as one of circles within such circles. The modal causal interactions are immanent expressions not only the totality of Substance or Nature, but also are expressions of the ratio of motion and rest of which a said body (level-k) is composed. The existential modality of ideas and extensions which is interior to the circle (level-k) can be read as contingent to the essence of that Body taken as a whole, as it is determined to exist.

While it is fair and best to understand the diagram exactly in the way that Spinoza meant it, I think it is important to realize that in a figure such as this, representing something for which Spinoza says there is no parallel, the image is likely overdetermined by several other Spinoza concerns and conceptions. The circle and its interior floats through Spinoza’s thinking process (additionally). The diagram can be read across at least these two layers: the affective subsumption of Cartesian mathematicization of Nature in terms of propogating, interacting bodies; and, the immanence of constituent modes of bodies under gradations of Being and power. And I would end that the suggestive recursivity of this body (level-k) opens itself perhaps to Autopoietic descriptions, at least for those bodies considered to be alive, and Autopoiesis itself to Spinozist re-description.

Spinoza’s “Spring Pole” Lathe: Experience to Metaphysics and Back

Spinoza’s Practiced Knowing

I mentioned in my recent post on the likely design of Spinoza’s grinding lathe that the dynamics of the Hevelius’ spring pole lathe may be tied to Spinoza’s ideas of Substance and the modes, such that one would be able to see how the epistemo-kinetic experiences Spinoza had during his many hours, days and years of lens grinding on such a lathe may have bore influence upon his metaphysical conceptions.

Here I want to take up this intuition, and perform the appropriate visualizations that would allow us, if for a moment, to picture what Spinoza’s body went through in communications with his device. In this way we might place ourselves, materially and affectively, in a relationship to his ideas, such that reading them alone in text would not allow (even if this goes against what one could argue is Spinoza’s rationalist program of understanding). That is, in short, one hopes to understand through body and affect what ideas Spinoza thought at the most abstract of levels, through their causal origins; for we can follow what Spinoza wrote, “experience can determine our mind to think…of certain essences of things” (Ep. 10), and assume that a similiarity of experiences may determine us to think of a similarity of essences. If we attain the experiences Spinoza underwent which determined him to think of certain essences (in his terms), this I believe can provide clarification to the same thoughts reached through his geometric pedogogy alone. 

The Horizontal and the Vertical: An Initial Philosophical Platform

Further below is an illustration of the tension dynamic of a spring pole lathe, as taken from Hevelius (second diagram). But first in order to follow my thinking here I want to insert a most revealing point made by Gatens and Lloyd in their critique of Hegel’s critique of Spinoza. As they point out, Hegel, while in great praise of Spinoza, felt that he did not embrace the full reality of the “negation”, Hegel’s personal contribution to the progress of philosophy and mankind. Spinoza’s description from Hegel’s point of view simply collapses into an acosmism, an undifferentitated whole of Substance, leaving no specific reality for either rocks, lakes or most importantly Man. In examining Hegel’s objection, Gatens and Lloyd introduce the vectoral notion of verticality and horizontality. It is suggested that Hegel’s problem is that he is only thinking of the “vertical” relationship of the modes to Substance, their individual, expressive relationship to the Totalizing whole. I quote at length here from the two authors because it is a very good paragraph, as they make an extremely important point:

Hegel’s critique of Spinoza thus focuses on the relation between individual mode and Substance. His complaint is that Spinoza cannot coherently articulate that relation without collapsing the infinite mode back into Substance; Substance remains undetermined, undifferentiated, while the individual mode is merely negative. But this is to miss the other dimension of Spinoza’s treatment of the finite modes -their mutual interaction, in which the determining force of Substance is mediated through the whole interconnected network of modes. Hegel’s critique of Spinoza is oriented, as it were, to the vertical relation between Substance and individual mode, rather than to the horizontal relation in which finite modes act on and are acted on by one another. Here, along the horizontal axis of finite modes, the claim that determination involves negation can be seen not as a repudiation of finite individuals but as an insight into their interdependence (71)

– Moira Gatens and Genevieve Lloyd, Collective Imaginings, chapter 3 “Re-imagining Responsibility”

Gatens and Lloyd are making clear the a key emphasis Spinoza places on modal expressions, as real, when included in our own constructive and immanent projects of freedom. I would graph out the differentiation that they find in Spinoza in this way:

I would parenthetically add to Gatens and Lloyd’s point that the reason why Hegel is thinking solely about the verticality of the modes is that the binary of individual/God, (or individual/world) is one of the primary schemas of the intersubjective investigations and inculcations of European Christianity. The salvation of the soul, through its relationship to an all-encompassing God, (via various institutional mediations), was the essential dogmatic concern, and this lengthy heritage necessarily brings the philosophical focus back to this binary difficulty: how does the individual soul return to God. Hegel, because he sees man at the apex of history, primarily must read the problematic in terms of a vertical dynamic, the individual vs. the whole. But Spinoza, because he is not burdened by the primacy of man or his reflective powers, allows another dimension of analysis which Hegel cannot see: that of the horizontal. The institutional mediations of the State, Church, Family and their imaginary relationswhich simply interpose themselves as aids to an essential verticality, are by Spinoza exploded out to the ultimate horizontal limit: the infinite expression of the modes. And the imaginary status of their mediating ideas become proliferate and constitutive vectors of power, degrees of freedom, across the field of Being (if one could put it that way without too much obfuscation).

Heveliuss Selenographia (1647) Spring Pole, Foot Pedal Lathe, illustrated

Spinoza’s Lathe

I want to return to the assumed spring pole, foot pedal lathe (pictured above), to get a much more concrete, affective sense of the vertical and the horizontal in Spinoza, and how the practices of Spinoza’a lens grinding may have helped construct his metaphysical conceptions. One must recall that Spinoza spent hours upon hours at such work lens work. To grind, polish and re-polish a lens could take several days, much of it in non-stop and highly repetitive, one might even say meditative, action (Auzout in 1664 records a time of 15 days for a single objective lens). So let our attention be called to the internal dynamics depicted above, found in focus of the dotted yellow frame. Here one should picture Spinoza seated or standing at length, his foot rhythmically pushing down a vertical tension from the spring pole at the ceiling. The vertical rising and falling motion forms a kinetic warp which not only works to orient the body spatially to the height of the room, connecting consciousness from the floor boards to the ceiling, but also creates and punctuated temporality, a timed ratio to the work. Transverse to this warp is the weft of horizontal action. The oscillations of the grinding form are pulled across the body by the push and pull connectionsof the foot the ceiling pole, and distinctly lateral to the focus of concentration. Again and again for hours these up-down, left-right actions literally weave a room of fluctuating conscious attentions, in which the craftsman necessarily embraces the lived experiences of the space, aware that whatever precise spotlight of focus he may have, it is merely a part of much larger, wider degrees of perception. Against Hegel’s fear of an individual’s collapse into the undifferentiated, the agentized craftsman becomes the draw-string of every quarter of the room, a focal point of at least two vectors action, from which and to which he is a differentiated, yet interconnected and expressive part. The melding of the craftsman and his tool is more than a metaphor. It is an experiential and metaphysical certainty.

Within this dotted frame of loomed space, at its center is a rotating circle. It whips at varying speeds in response to both the intensities of the leg, and the limits of the spring pole above, in a concentric motion. It is no secret that Spinoza had great love for the circle as a diagramed exemplar of the relationship between the modes and Substance (Ethics 2p8s, pictured below), but also as an ideal of vision and the actions of optical focus (Letter 39 to Jelles, March 3rd 1667). Here, for hours on end Spinoza would stare determinitively as a rotating circular form which remained both fixed (stable in its ratio), and changing, expressing in both the consummation of the vectors of the room’s actions. One cannot help but think that such concentrated attention upon the spinning form would leave at least a conceptual imprint upon the philosophical craftsman, especially as he considered the modal expressions to be causal interactions immanent to the whole, just as internal rectangles can be considered immanent to the properties of a circle (his diagram below). It is most suggestive to see that the rotating circular form becomes a bed of friction and idea, producing realized changes in the material of glass held in Spinoza’s sensing hand.

from Ethics 2p8s
The Turning Lap

To carry our instructive analogizing further, one must look closer at the actions immediate to Spinoza’s attention as he worked his glass into the required shape.

In the scalloped metal form likely an abrasive would be applied to aid in the grinding, the light blue arrow above represents the hand’s actions upon the circular rotation. The horizontal and vertical tensions are vortexed into an oscillating circularity. There a recipe of frictions and intelligenced experiences interact to bring about an ideal result. What Antonio Negri calls the “concrete…unique terrain of reality, [the] fruit of the paradoxical determination [a metaphysical dilation of unity and multiplicity]” (The Savage Anomaly, 127), the modal “surface of the sea”, occurs here, in the turning scalloped dish, a product of the cybernetic expressions of a room, a mechanism, and man, in which the craftsman’s hand performs a living shore of perceptual action.

The Sublime Tool

If this notional leap from mechanism to conceptual metaphysics seems too great, too fantastic, I believe that this is because we do not have a strong enough sense of the bodily, affective, imaginary foundational means of immanent abstract thought, something that Spinoza’s own metaphysics works to make more clear. Further I believe we must adjust ourselves from thinking of Spinoza merely in terms of propositions and proofs, though the rhetorical form of his work certainly at first or even second glance invites us to think of him in this way. Richard Sennett for instance in his recent pragmatic and near-poetic book, The Craftsman (2008), perhaps gives us a bridge for thinking about craft and abstraction as part of one constitutive process. He invites us to understand how human progress and freedom comes by thinking through one’s tools, how tools help frame our questions and solutions. In fact this is very much how Spinoza has conceived of abstract thinking itself, as he followed Descartes’ analogy found in the 8th rule of the Regulae : just as how a blacksmith’s tools had to be originally made by simpler tools themselves, so too simple tools of the intellect are needed to make other, more complex tools of the intellect (On the Emendation of the Intellect ). In a certain sense, one needs something that hammers in order to make a hammer:

The matter stands on the same footing as the making of material tools, which might be argued about in a similar way. For, in order to work iron, a hammer is needed, and the hammer cannot be forthcoming unless it has been made; but, in order to make it, there was need of another hammer and other tools, and so on to infinity. We might thus vainly endeavor to prove that men have no power of working iron. But as men at first made use of the instruments supplied by nature to accomplish very easy pieces of workmanship, laboriously and imperfectly, and then, when these were finished, wrought other things more difficult with less labor and greater perfection; and so gradually mounted from the simplest operations to the making of tools, and from the making of tools to the making of more complex tools, and fresh feats of workmanship, till they arrived at making, with small expenditure of labor, the vast number of complicated mechanisms which they now possess. So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations gets again fresh instruments, or the power of pushing its investigations further, and thus gradually proceeds till it reaches the summit of wisdom.

But is our reading of Spinoza’s metaphysics anything more than simply a coincidence of horizontal and vertical vectors in the lens-grinding lathe, and Gatens and Lloyd’s horizontal and vertical spatialization of his metaphysics in defense against Hegel? Is this conflation merely accidentally bolstered by the analogy of ideas to be taken read tools in Spinoza’s very early work? I think there is much more to this than that, and that aside from the notion of the horizontal and the vertical there is a multiplicity of core principles that seem to stem from Spinoza’s unique, and one must say classed, artisan experiences. In particular we must understand that Spinoza, unlike Descartes, was through and through a practiced craftsman, an artisan by trade and value, who repetitious and refining practices which he took rather seriously must have influenced his guiding conceptions of Mind, Body, Idea and Power. In fact, it seems that it is a tooled notion of idea and body that I believe informs his vital definition of the power of the body, a defintion which will reconceptualize any of our instrumental approaches to material augment of the human body or mind:

Whatever so disposes the human Body that it can be affected in a great many ways, or renders it capable of affecting external Bodies in a great many ways, is useful to man; the more it renders the Body capable of being affected in a great many ways, or of affecting other Bodies, the more useful it is; on the other hand, what renders the Body less capable of these things is harmful.

– E4p38

Think on how this expressive yet instrumental numerical notion of power can be found within the most elementary experiences of tool use, as Richard Sennett tells us about the wonders of the flat-edged screwdriver:

…in its sheer variety this all-purpose tool admits all manner of unfathomed possibilities: it, too, can expand our skills if only our imagination rises to the occasion. Without hesitation, the flat-edged screwdriver can be described as sublime – the word sublime standing, as it does in philosophy and the arts, for the potentially strange. In craftwork, that sentiment focuses especially on objects very simple in form that seemingly can do anything (195)

The Craftsman, Chapter Six “Arousing Tools”

I believe that Spinoza’s lifelong craft experiences with the lens-grinding lathe (among so many other simple tools) had a lasting effect on his conceptions of Mind and Body, and their necessary unification. The grinding lathe, with its intimate, indeed cybernetic, interweave of body, mind and material construction, its concentric use of the spinning semi-sphere, must have struck Spinoza as sublime in the sense that Sennett tells us. There is the evidence of Spinoza’s resistance to the sophisticated, semi-automated designs of his brilliant and wealthy neighbors the brothers Huygens ( EP 15/32 ) which tells us that Spinoza was quite hesitant to leave behind the interface of the machine with the understanding and felt hand. But it is more than this. It seems that the grinding lathe leaves its conceptual, kinetic trace all the way up through to the most abstract, and most radical of conceptions. In fact there is the very real sense in which we may read Spinoza’s Ethics (as it exists both in idea and extension) as a tool which can affect and be affected in the greatest number of ways. 

Picturing Work and the Work of Picturing

Once we have a vivid sense of the kinds of material engagements Spinoza had concerned himself with, his bodily practices of concentrated creation and refinement, we get a better sense of how Spinoza conceived of his own Rationalist, propositioned philosophical aims. From there we can place ourselves with the lived historical space of the man who lived at the cusp of our modernity, and feel something of the material and pragmatic focus of his articulations of freedom. Below is Spinoza’s rented room in Rijnsburg where he lived roughly from age 29 to 31 having fled the upheavals of Amsterdam, perhaps with concern for the return of his tuberculosis from remission. It is today’s Spinozahuis museum. As mentioned before the wood turners lathe depicted there is NOT the kind of lathe Spinoza would have used, but if you look to the upper center left of the photo you can see a hypothesized spring pole, the vertical vector of his practice. It is not known if Spinoza’s later rooms in the village of Voorburg, where it is thought that he did his most concentrated grinding work, were of this size, but the combination of the Rijnsburg room and Hevelius’s illustration gives us I believe some determinative sense of the internal dynamics of Spinoza’s lived experiences as a craftsman and thus as a thinker; they directs us to the material and conceptual causes that may have privileged Spinoza treatment of the Mind and Body over his predecessor, Descartes. There is much that divided these two thinkers from each other, but perhaps even more than joined them. Each was concerned with lens-grinding, optics and the improvement of the telescope, but only one of them was a practiced maker of lenses and instruments. Only one of them touched the glass.

The Rijnsburg Wood-turners Lathe, Spinozahuis

Hevelius's Spring Pole Lathe, from the Selenographia (1647)


The Rijnsburg Wood-turners Lathe, another angle

The Rijnsburg Wood-turners Lathe, another angle

The Text of van Gutschoven’s Letter to Huygens No. 1148

[Posted here is the full original text of Huygens letter 1147, for which there are some comments here A Method of Grinding Small, Spherical Lenses: Spinoza ]

No. 1147.

[G. van Gutschoven] à [Christiaan Huygens].


Apendice au No. 1146.

Instrumentum ad vitra minoris sphaerae terenda.

Laminam eream AB schamno tornatorio affixam excavabis secundum circulum CD cuneo parato in formam circuli aequalis circulo vitri formandi. dice aequalis circulo maximae sphaerulae cuius lens particulam referat. in hoc excabato canali CD, atteres vitra capulae affixa et canali appressa : modo capulam in manu continuo dum vitrum attertur vertas, ut ex omni parte vitrum aequaliter atteratur : consultissimum autem erit laminam horizontalier circumagatur, nam hoc modo non tam facile arena decidet, vltimam autem polituran vitro addes simili lamina sed stannea terra tripolitana inserta : vel lamina simili lignea ex lingo aliquot molliori, quale est salicis, vel populi.

Lentes cavas formabis sphaerula stannea vel plombea EF schamno tornatorio affixa ut vides, polies vero eadem sphaerula vel lignea ut superius dictum terra tripolitana inserta.

Hoc concavo cono schamno tornatorio affixo, scabies vitrorum limbos polies, ne dum ultimam in charta inducimus vitro polituram, particulae tenuiores vitri exfilientes et in poros chartae sese infinuantes vitrum deturpent.

Spinoza’s Lens-Grinding Equipment


Door to the Hofwijck Estate where Spinoza likely strode

Door to the Hofwijck Estate where Spinoza likely strode


Spinoza Purchases Lens Grinding Laps

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

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

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

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

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

View from the Huygens Estate, the Hofwijck

View from the Huygens Estate, the Hofwijck


Christiaan Huygens’ Marbler and Chimney-Sweep

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

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


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

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

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

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

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

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

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

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

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

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

Approaching Huygens

Approaching Huygens

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.

A History of Early Spherical Microscope Lenses – Spinoza’s Place in Optics

Spinoza’s Microscope

In view of organizing the possible context for Spinoza’s lense-grinding and instrument making, here is a brief timetable of spherical lens use, such that surrounds Spinoza. Spherical lens making was the use of tiny spheres of melted glass, some of them ground and polished. The smaller the sphere, the greater the magnfication, surpassing all compound microscopes at the time, making at a times very minute magnifying glass to which one pressed one’s eye [see here for Van Leeuwenhoek’s example]. This history is quite approximent.

1628 Harvey sees the heart of a flea.
1629 painter de Gheyn II dies at age 64, before Constantijn Huygens, Sr. can convince him to illustrate microscopic views for the proposed “The New World”.
1644 Odierna uses a chickpea sized globule, Toricelli as well.
1646 Kircher sees flea’s leg and hair with spheres no bigger than the smallest of pearls; (Cardinal Medici’s gift).
1646+ Bettini uses a sphere smaller than a millet seed.
1654-1663 Hudde studies under van Schooten, working on Descartes’ Geometry, with J. de Witt, at Leiden.
1656 Borel sees the nerves, feet and eyes of a mite.
1657 Schott uses “transparent atoms”
1657 Hudde writes of plans to study generation microscopically, letter to Van Velthuysen
1658 Kircher sees worms in the blood of fever victims.
1659 Van Leeuwenhoek claims (40 years later) to have produced bead glass lenses, in Delft.
1659 Spinoza begins periodic studies at the University at Leiden.
1660 Blaes influential speech at the Athenaenum of Amsterdam, addresses the microscopes use in anatomy.
1661 Spinoza said to be a maker of telescopes and microscopes, Borch’s Diary
1661-1662 Enrolled at Leiden University studying medicine Swammerdam, Steno, Kerckring, de Graf, Ruysch
1661-1663 Spinoza lives in Rijnsburg grinding lenses and making instruments, a couple of miles from the University at Leiden and the young anatomists.
1663 Spinoza moves to Voorburg, five minute walk from Huygens’ estate, a few miles from The Hague.
1663 At The Hague Vossius shows the French diplomat Monconys his simple microscope of a little hemispherical lens mounted in woodframe behind black board (Journal).
1663 Hudde in Amsterdam shows to visiting Monconys his technique of forming lenses by melting beads of glass, and polishing them with salt (Journal).
1664-5 Hooke‘s influential Micrographia, wherein he describes his beads from threads of glass method. Likely seen by Spinoza in visits to the Huygens’ estate in the summer of 1665 when he also saw Huygens’ machine for grinding lenses.
1666 Vossius, Spinoza’s associate, writes that smaller sphericals of glass best. De Nihi et aliorum…
1666 Spinoza in correspondence with Hudde over the nature of God and technical questions of optics and lens grinding (Letters 34 to 36).
1667 Spinoza speaks to Vossius about an alchemical matter.
1667 Huygens twice writes his brother, complimenting the polish that Spinoza is able to achieve by “the instrument” in small lenses for his microscopes. (Parenthetically, also writes to his brother about Spinoza‘s collaboration with Hudde on calculations on a40ft lens which would have been among the largest in Europe.)
1668 Swammerdam declares sphericals best lenses.
1668 May 11, Christiaan Huygens admits to his brother in letter that he agrees that Spinoza was right, smaller objectives in microscopes represent objects much finer fashion.
1670 Spinoza moves to The Hague.
1670 Kerckring (Spinoza’s former classmate), declares he has seen vascular bundles with Spinoza’s microscope, and organs teeming with minute, possbily symbiotic animals [post-mortum microorganisms or “still active cilliated surfaces”?, Ruestow 265]: yet declares that things viewed solely by microscope should be held suspect. Spicilegium anatomicum.
1671 Kerckring claims to have seen, and draws a tiny, tiny human inside a 3-day old fertilized human egg [seen with Spinoza’s microscope?], Anthropogeniae ichnographia.
1674 Hartsoeker makes beads of glass lenses after his father visits Van Leeuwenhoek.
1677 Johan Ham, a student a Leiden, possibly with a bead-lens, first sees spermatozoa and brings them to Van Leeuwenhoek.
1677 Hartsoeker, a student at Leiden with a bead-lens, later claims to be the first to see spermatozoa.
November 1677 The Huygenses purchase Spinoza’s lens-grinding equipment at auction.
March 1678 Huygens makes his first bead-glass lens microscope aided by Hartsoeker.


The importance of this time table is that it builds a net of associations with the bead-like, small spherical lens techniques which made up some of the most powerful yet simply microscopes of the day, in part through the person of Hudde around Spinoza. Hudde taught at least two persons his technique of microscope lens-making, and possibly many more. It also places Spinoza in the context of the anatomist debates of the early 1660’s, as a microscope maker and a Cartesian living in Rijnsburg. The role of Kerckring in this debate (glands vs. vessels and preexistence) under the authority of Spinoza’s microscope also is suggestive of Spinoza’s connection to this group, and the possible powers of his instruments.

Monconys’ Visit: Six Degrees of Separation for Spinoza


The Distance from Spinoza’s Tydeman home, just 5 minutes walk from the Huygens Estate, and The Hague is just over 2 miles. The distance from the Huygens Estate and Vermeer’s House, just about 4 miles.

Veermer, Van Leeuwenhoek, Huygens, Hudde, Monconys, Oldenburg

A selection from Philp Steadman’s Vermeer’s Camera, which works to establish a fine net of painterly and optical relations to which Spinoza was connected in 1664-5. :

Six years previously, in 1663, Vermeer had received another distinguished visitor, the French diplomat and traveller Balthasar de Monconys. Monconys records in the Journal of his travels how he went to Delft briefly on August 3 where he admired the tombs of Admiral Tromp, Piet Hein, and William of Orange. Eight days later, on the 11th, he was back again with the sole purpose of visiting Vermeer. The meeting was not, by Monconys’ account, a great success. He was disappointed in his hopes to buy a painting…

…Broos notes that how, before going to Delft,  Monconys had been to see the Huygens family in The Hague, and had passed by again after this meeting with Veermer. “One gains the strong impression,” says Broos, “that it was thanks to his contacts in The Hague that the French diplomat had been able to take note of the most famous Dutch artists of that era, such as van Mieris and Dou in Leiden, and Johannes Veermer in Delft…”

…[and then on Monconys’s optical concerns, in addition to his pursuit of high art] Indeed, while he was with the Huygens family in The Hague, Monconyswas comparing their designs of telescopes with his own, and admiring the clarity and sharpness of the images produced by their lenses. On this same trip he went on to call on the mathematician Johan Hudde in Amsterdam and on the scholar Vossius in The Hague to see their microscopes…Both men’s instruments had single lenses. Hudde demonstrated to Monconys his methods for melting glass bead lenses and polishing them with salt. He also described his techniques for illuminating specimens (55-56).

Steadman makes this chain of associations part of a story of the possible optical resources for Vermeer, concluding that Constantijn Huygens is a candidate [earlier in the text he strongly considers a van Leeuwenhoek influence], but it also works as a roadmap for the web of relations which characterize Spinoza’s own artisitic and lense oriented connections. Monconys’ visit to Delft and Voorburg occurred only a few months after Spinoza had moved to that quite village in the Spring. Spinoza’s Voorburg landlord was a “Master painter”, [as would be his next one] and If indeed Spinoza had some reputation for either lenses or instruments, as is possible, his coming to the neighborhood must have been noted with special interest by the Huygens Three. Spinoza may have meet mathmatician and lens-grinder Johannes Hudde several years before while living at Rijnsburg, through the Leiden Cartesian Circle, as suggested by Klever to Nadler, or he might have met him later through Christiaan Huygens himself (although F. J. Dijksterhuis attempts to minimize the connection between the two, Ruestow tells of how Huygens actually translated some of the Micrographia for Hudde); in any case Spinoza was in correspondence with him in early 1666, and took his optical theories largely to be right ones. This net of contacts and friendships both in the arts and optics circles rather tightly around Spinoza’s person. If we allow Philip Steadman’s groupings of influences, we can see that they indeed are more those for Spinoza, than for Vermeer. The Guild of Luke tensions between Delft and The Hague artists notwithstanding, if the Huygens House was a confluence of paintings and optical knowledge, we can assume to some degree Spinoza too was touched.

 As Steadman concludes, if it was not Van Leeuwenhoek who introduced Vermeer to lenses and cameras, it likely was Huygens senior, quoting Heinrich Schwartz,

…the evidence makes it rather possible that one of Monconys Dutch scientist friends may have called his attention to a painter in Delft who used with some amazing results an optical contrivance and that, therefore, his paint may have had a particular interest and appeal for him (58 )

Just as the conflation of paint and optics occurs in the figure of Vermeer, so too it does with Spinoza, who taught himself to draw (camera obscura? reflecting concave mirror?) and associated with painters of the Guild.

A Worm in Cheese

Campanella and Spinoza On Perspective

It has long been my suspicion that Spinoza had read, and has a debt to the much forgotten Tommaso Campanella. There is evidence that Descartes’ most central contributions have in some way be under this influence, and Spinoza’s position indeed is in many ways a response to Cartesianism. But it more than this. There is in the thought of Campanella a particular panpsychic, and cybernetic understanding of what is epistemological, and power, which reads open, and corrects, the overly rationalistic reductive reading of Spinoza’s many propositions.

Here I take up only a tangential connection, one that moves from the sharing of a single, (if common), trope. It does not prove the influence of the one on the other, but at the very least it operates as a happy coincidence, or even an expression of a continuity of Geist, (however delineated), between thinkers and times. This is the figure of the “worm in cheese”.

It was a prevalent image Headly tells us, a part of the “folk culture” of the Late Renaissance, the cosmological idea that we are all severely limited in our perceptions and knowledge, locked within a localization of understanding, surrounded by our food source, on which we blindly feed. But Campanella lifts this picture of the world up to larger purposes, sharpening it to both epistemological and political effect, drawing forth the consequences of such an instructive image.

First, there is his famous sent letter to the Emperor of China. It should be known at first that Campanella really was in a worm-in-cheese predicament. He had endured acute torture at the hands of the Spanish authorities, in which he had to feign, and therefore prove his “madness”, and thus avoid an inquisitional burning at the stake as heretic. And then he had suffered multiple years locked away in the worst dungeon of Italy, chained and manacled in a nearly lightless cell of the formidable Castel San Elmo, where he worked would work free sonnets on bits of paper, and access his prodigious, certainly photographic memory, sketching out his metaphysics. It is a story I really do have write, but it is the not purpose here. Rather, only, if anyone knew what a worm in cheese was, Campanella had a sort of privileged view of the condition.

But back to the letter to the Emperor. Campanella was fast on the idea of restoring in the papacy some sort of universal, indeed Catholic, governance, one that put the whole world into communication with itself, so that there were to be a free flowing of knowledge and sciences that would help liberate men from their ignorance. Ming China, by Campanella’s understanding, had cut itself off from outside influences, and turned itself inward. This is something extraordinary that gives us a piece of the so-called “volcanic” mind of Campanella, that a monk of no importance, under years of imprisonment, would even concern himself with the notion of a world history, and take it upon himself to compose such a letter.

Campanella writes to the Emperor:

Those men [your subjects] are lacking in aspiration; they seem like men but like worms born inside a cheese, who reckon nothing more or better there to be in the world beyond their own cheese from which they are nourished, sustained, hidden, or as worms born in a man’s stomach who know nothing of man, nor his mind, but cocooned away, complacent, not wanting to be disturbed, jealous of their remove. So, oh King, [the monarch of China], you seem to us… Stick your head out beyond your cheese, beyond the stomach of your land (TC-QR, 221).

It is a powerful call somehow, extending far beyond the prison walls, and the walls of history, into history. Campanellla makes use of a similar worm-image, to a different effect in his utopian vision “The City of the Sun”. Here, describing the conceptions of the people of La Cità del Sole, ones who live in a kind of blissful perfection of knowledge, in a city that is shaped with concentric ramparts so that it is the shape of the Solar System:

They [the Solarians] assert two principles of the physics of things below, namely, that the sun is the father, and the earth the mother; the air is an impure part of the heavens; all fire is derived from the sun. The sea is the sweat of earth, or the fluid of earth combusted, and fused within its bowels, but is the bond of union between air and earth, as the blood is of the spirit and flesh of animals. The world is a great animal, and we live within it as worms live within us. Therefore we do not belong to the system of stars, sun, and earth, but to God only; for in respect to them which seek only to amplify themselves, we are born and live by chance; but in respect to God, whose instruments we are, we are formed by prescience and design, and for a high end.

Here we have the more benign, and perfecting simile of the world as an immense and sensate animal, with we but like parasitic worms, feeding on it, but also part of its expression, its system. And lastly, in concert with Campanella’s notion the importance of empirical knowledge, direct experience, the testings of science and observation, come from his study of Telsio, we have another use of the worm-in-cheese metaphor, that expressing the historical linking of observations, the importance of communicability:

Just as namely, through individual perceptions, the mind adds to truth, so too with what belongs to others. Otherwise one would be like a worm in cheese, knowing nothing, except those parts of cheese that touch it. Every narrator, whether by letter, or in mouth stretched, or in movements, a historian is.

And lastly, we come to the worm of Spinoza. Memorably, after writing to the founding secretary of the Royal Society, Henry Oldenburg, about his arguments and notions of God, as being both a totality and expression of parts, all of which fit together. If we, in our experience on thing or part as a complete entity, this is a kind of selecting out of the whole, from a perspective of ignorance. This he compares to the kind of knowledge that a worm in the blood of the body has, as it goes about bumping into something so vast it has no possibility of understanding. One has to keep mind that Spinoza was an early maker of microscope lenses (attested to be of rather high quality), and it is perhaps likely that he had stared into lenses, looking at blood and the what must have seemed infinitesimally small forms therein.

This was an extraordinary time, when the smallest and the most distant were coming into view (Spinoza would become a companion of Christiaan Huygens, the discoverer of the rings of Saturn, and it is imagined to be likely that he would then look through his telescope too, when they spent time together at Huygens’ country estate). It is said by Colerus, his first biographer, than he would lift his magnifying glass and stare at mosquitoes and flies. All the vastness was opening itself up, and closing in.

Let us imagine, with your permission, a little worm, living in the blood¹, able to distinguish by sight the particles of blood, lymph, &c., and to reflect on the manner in which each particle, on meeting with another particle, either is repulsed, or communicates a portion of its own motion. This little worm would live in the blood, in the same way as we live in a part of the universe, and would consider each particle of blood, not as a part, but as a whole. He would be unable to determine, how all the parts are modified by the general nature of blood, and are compelled by it to adapt themselves, so as to stand in a fixed relation to one another. For, if we imagine that there are no causes external to the blood, which could communicate fresh movements to it, nor any space beyond the blood, nor any bodies whereto the particles of blood could communicate their motion, it is certain that the blood would always remain in the same state, and its particles would undergo no modifications, save those which may be conceived as arising from the relations of motion existing between the lymph, the chyle, &c. The blood would then always have to be considered as a whole, not as a part. But, as there exist, as a matter of fact, very many causes which modify, in a given manner, the nature of the blood, and are, in turn, modified thereby, it follows that other motions and other relations arise in the blood, springing not from the mutual relations of its parts only, but from the mutual relations between the blood as a whole and external causes. Thus the blood comes to be regarded as a part, not as a whole. So much for the whole and the part.

Letter 15 (32), 1662

There are obvious wide-sweeping parallels, none of which create an argument of influence: comparison to worms living in the body, and the locality of perceptions which seal each person off from the rest of existence, and contingency of our immediate sense knowledge. And there is the political character of communications itself, the sharing of descriptions across countries and the globe, and the kind of epistemic building (albeit from a difference in emphasis or even process: Spinoza looked for a rational grasp of “common notions” which joined bodies and minds, Campanella appraised observation and a species of synthetic becoming what one observed), by grasping the larger and larger wholes, of which one is participating. At the very least there is something shared; it is that animate sense that one is within a psychic, sensate thing, when one is in the world, and that knowledge consists in identifying with, and constructing epistemic conjoinings, as part of an over-arching, and yet un-understood entirety. And in this service, a catholic freedom of exchange becomes the nexus for that building of communications, a Renaissance notion of political and ideal creation.

1. A notable annex to this comparison of worms in cheese and worms in blood is Kircher’s microscopic observation that the blood of fever victims was worm-filled:

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).

This is likely the main triggering thought in Spinoza’s mind – though I have never seen it noted by scholars – as Oldenburg mentions the very same Kircher’s later work Subterranean World in the previous letter which Spinoza is answering.