Seeing Machines and Modes of Slavery

The Human Machine

Corry Shores puts up a wonderful response to some of my optical research on Spinoza, drawing on some threads and putting them together in a way that I just had not yet: Seeing Machines. There he picks out what for me are several vital issues that are found in not only Spinoza’s, but also Descartes’ preoccupation with optical matters, both theoretical and practical, and really touches the primary concern. How do these modes of production (ideas, machines) reflect, express and criticize the very rise of instrumentality and really Capitalized labor (and merchant class related freedoms) in which they arose?

Consider how Descartes proposed his own notions of a transcendent God and free will. His sharp division between mind and body was essential for his project. Spinoza, however, reconciled the two [by means of his parallelism]. He was not so narrowly focused on abstract rational conceptions. He did not just design lenses for seeing things with greater focus. As well, he ground and polished them with his own hands. Ideas and their material instantiations cannot be divorced. In fact, kvond writes, “a calculation, for Spinoza, must be seen as an act, the mathematical point, as a relation and expression, and an instantiation, a persistence.” We do not just see, we see from a certain conceptual perspective. [Descartes saw the world mechanically. This perspective might view slaves as machines and not people.] Kvond puts it that we are always seeing-with.

In my view this connection is exactly right. Descartes’ preoccupation with the narrow focus of optical (and mental) clarity, and the attendant vision of machinic Instrumentality, is precisely related, ultimately, to the question of human slavery. It is no mere metaphor that Spinoza uses in his Ethics when he devotes his fourth part to the subject matter of “Human Slavery”. He is speaking of the emotions, but for Spinoza ideational over-focus was material over-focus. Emotional Slavery expressed itself in physical slavery. And he is not only thinking of individuals. It would seem out of place to give Descartes responsibility for 17th century slavery (why not, so much else gets laid at his feet!), but there are valid, thematic, if not arguments, parallels to be drawn between Descartes’ pursuit of a machinic world vision (paired from Mind), his attempt build automated devices that would not be stained by human hand interference, the attempt to mentally isolate clarity in terms of a point of focus, and the general colonial trend towards labor efficiency that would eventually replace indentured servitude (practical slavery) with outright slavery itself (the evaporation of the “human” in the name of production). I see in the very “object” oriented, optical preoccupation with central clarity – the hallmark of much, if not all of Idealism that followed – the conceptual cornerstone for Instrumentality itself, the mode of thought that regards a clarity and sureness of an intentional part as the grounds for what human beings should know, and what they do.

Additionally, it is precisely how we eroticize the boundary (that which lies outside our view of clarity, the “object” of our orientation), that fuels – both literally and imaginarily – our very Instrumentalities.

This is no mere theoretical question, but a large scale question of concept and human action. Much, if not all of the value of philosophy is that at the widest level in a certain register, what hu/man is capable of thinking becomes reconfigured, and I cannot help thinking that the preoccupations with optics and lenses that distinguished many of the great, newly affluent minds of the mid-17th century, bears a conceptual connection to the real human and institutional relationships that constituted the nature of their wealth. Optics, Instrument and slavery are not divorced, or at least Spinoza would refuse to divorce them. Corry did not realize it, but in the time of my optical study of Spinoza I also found compelling the likelihood that Spinoza, and the Spinoza family had at the very least tangential ties to the slave trade enhanced sugar buisness, leaving me with the suspicion that slavery and its connection to commerce lead in part to Spinoza’s decision to leave the occupation of family merchant behind, and devote himself both to philosophy and lenses.

Most of these are conjectural sketches, but because it seems that no one in Spinoza scholarship has much brought up the matter, they perhaps form a sketch of what is worth thinking about: Spinoza the Merchant, Caliban and the Prophetic Imagination, The London Question, Spinoza and the Ethiopian, The Sephardim and the Slave Trade, Spinoza Family Sugar Trade Timetable, Gabriel Spinoza and Barbados.

In this way it is possible perhaps to address the knot of questions behind recent talk about Ontology and Politics. The relationship between the two is I think best expressed by Spinoza’s political expression of ontologies, achieved through the erasure of the human/natural-world divide, descriptively turning Man into a force of nature, which is likely what it always was. But, as Corry helps me remember, this is not just a conceptual position, but also a part of the very intimacy philosophy bears to its time.

Spinoza and Mechanical Infinities

The Mechanically Bound Infinite

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

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

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

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

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

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

Letter 12

The Lathe Buried Under the Euclidean Figure

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

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

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

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

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

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

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

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

From Mechanics to Optics (to Perception)

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

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

The Infinities Beneath the Microscope

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

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

Spicilegium Anatomicum 1670

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

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

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

Spinoza’s Optical Letters: Redux

As some know, primarily last summer I spent my time researching and theorizing on Spinoza’s lensgrinding and optical concepts, a largely underdeveloped field in Spinoza studies. The greater portions of my findings are listed here on this site under the sub-heading Spinoza’s Foci. A spearpoint of this research was uncovering the substantive arguments and conceptions that lay behind Spinoza’s rejection of Descartes’ optics, as found in his two letters 39 and 40, letters that have be nearly completely ignored by commentators on Spinoza, or if address, addressed in what seems a delinquent, or dismissive fashion. Spinoza is mostly thought to not know what he is talking about. On the other hand, Spinoza’s objections if carefully examined reveal both technically an alternate position on the problem of “spherical aberration,” but more deeply, a radically distinct conception of what vision is, in particular how it works as an insufficient analogy for consciousness. While Descartes wanted to emphasize the power of the central clarity powers of hyperbolic vision (both in the human eye, and in his proposed lenses), Spinoza understood vision and conciousness both as holistic events, ones best approached with the pragamatic appreciation of our limitations. I provide very little philosophical extrapolation here, though the implications are vast, perhaps running through down to the root of Idealism and Phenomenology. This epistolary commentary also does not touch on such other important factors such as the kind of lathe Spinoza likely used, nor much on his likely technique, and kinds of instruments he made and calculated for, which form a significant secondary branch of my research. Yet as these letters remain nearly the only first hand statement Spinoza made on optical matters, they are the anchorage point for anything else that is likely to be asserted.

For the convience of interested readers I here post a Word document version of my line-by-line explication of these rarely read and rather under-interpreted letters. I realized that the previous weblog versions were very difficult to read and browse through, hopefully something this version will correct. The two entries that can be found in this document are: Deciphering Spinoza’s Optical Letters and  Spinoza: Letter 40 and Letter 39. These are both the English translation of the two letters by Spinoza, and then my explication. This version is not footnoted (though there are citations), and it retains some of the idiocyncratic paragraphing and color coding. It is a 14,000 word document (48 pages), though Spinoza’s letters are only 900 words or so.

[click download]: Deciphering Spinoza’s Optical Letters Line by Line

Downunder: Central Clarity Consciousness (CCC)

The Series of Objections to “Object” Consciousness

Larval Subjects posted a round-up of the latest OOP (Object-Oriented Philosophy) weblog discussion, as everyone seems to be seeking the “New” metaphysics (one if reminded of Nietzsche’s clever, and unfortunately philosophical reminder that one should hate the “Good” that is in the mouth of one’s neighbor [BGE 43]). Cordiality all around, but metaphysics (the logical necessities) in the mouth of another invites objection. These were my three posted objections, in their order:

1. The “Picture” behind Intention: What Lies at the Center of Perception

2. The Bounce of the Being of Beings

3. Harman Brings Central Clarity to the Issue (wink, nod)

Larval Subjects though finds my main objection to Graham Harman’s ontology of necessarily hidden objects as they are caught in a Husserlian/Heideggerian Four Fold, not even remotely applicable, (how can hidden objects be a product of optical metaphorization?). This is surely a clue that I have done a very poor job of making the nature of my point clear for each time I see it summarized, the nature of my point seems to not be there in the mouths of my summarizers. This is my fault. The purpose of this post is to set some of the history backdrop for the nature of my critique, a backdrop which may make my critique more plain, at least in concept. Then, when plain, it can be debated whether it applies.

Part of the problem with my critique is that it cuts so broadly across the swathe of Philosophy, hitting at all the fibers of terms that have developed from Descartes on, that arguing about it within the vocabularies of those derived terms is counter-productive. It touches such elementary philosophical analytics dyads Self/World, or subject/object, and even Being/Non-Being, overturning something of the each of these terms even when separated out from their traditional dyad partner.

And part of the problem is that my point is a historical one, that is, I am tracing the genaology of a thought back to a moment in theory, and perhaps it requires that one understand this moment in theory in order to understand the subtle perniciousness of the continuity. It is for this reason that I’ll summarize something about what was going on when the exact anatomical nature of the eye meet the Cartesian idea of the ideal shape of a lens, all brought in metaphor to a concept of consciousness itself, the Cartesian concept of Will organized around the idea of a Central Clarity of Object.

The Hyperboliod Lens, Bending Rays Toward the Center of the Eye

Come out of the Perspectivist Tradition of a Theory of Vision which for centuries thought that vision consisted of a point by point correspondence of rays coming from a place in the object directly to the back of the eye, like so many uninterrupted strings, Kepler was the first to make rigorous discovery of how rays radiated out in all directions from all the points of an object, and were generally refracted and refocused on the back of the eye (to over-simplify the history). Part of this discovery was his proposal/discovery that the lens of the eye was hyperbolic in shape, a shape which focused dispersing rays from a central point toward a perpendicular which was the lens’s axis, as the illustration shows:

This meant that by the very nature of the hyperboloid lens of the eye (it turns out it is not a hyperbola), rays coming from a central point are most perfectly focused, and those coming from cones at angles which are not on the axis of the lens, are confusedly focused to the edges of vision, as he explains in his Paralipomena:

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

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

As you can see, he goes further than simply stating this relative confusion of the border, but tells us that the sides of the border “carry over” their measure of sense to the perfection of the central vision, a central object. And this physics of sense is seen in the very satisfactions of the soul. That is, the soul and Nature are in agreement.

Descartes though was the first scientist/philosopher to actually do the mathematical work which shows the refracting powers of hyperbolic lenses (having discovered the Law of Refraction, a fact of discovery still under dispute). The precise gradation of hyperbolic lenses were impossible to make the time, and Descartes became fairly obsessed with making an automated machine which would be capable of grinding them, a doomed endeavor. Spherical lenses suffered from aberrations, which after Descartes were generally presumed to be solved with the Ideal Hyperbolic Lens. But Descartes did something more to Kepler’s notion of natural, hyperbolic center of object focus. He made the benefit of the periphery a quantitative measure (one sees more things via the periphery), but not a qualitative measure. He attaches to a concept of qualitative central clarity his most important concept of Will. The relatively confused border only serves help us direct our free Will in the right direction (oddly, this contradicts the sheerly quantitative measure of the periphery, for numericity alone is not helpful); the clarity in a way seems to speak for itself:

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

One can see that Descartes rather easily transfers this notion of central object clarity to central idea clarity. The quelling of confusion comes from separating out the lone clear thing (object/idea) from the many things you might be confusedly taking in. Interestingly, he appeals to the precision vision of the craftsman whose “delicate operations” are supposed to reflect this intentionality principle of consciousness clarity. Ignored is that the craftsman is able to narrow his/her attention due to the very clarity of perceptions of the entire border of the process (the process borders are not confused, but stable); and even more so, it should be pointed out that if one really notices what a craftsman is attending to, it is not a central object, or a central idea, but rather the potentiating dissonance at the center of the project. At the center of the craftsman’s gaze is not a clarity, but an eruption:

Rule 9: We must concentrate our mind’s eye totally upon the most insignificant and easiest matters, and dwell upon them for long enough to acquire the habit of intuiting the truth distinctly and clearly.

…We can best learn how mental intuition is to be employed by comparing it with ordinary vision. If one tries to look at many objects at one glance, one sees none of them distinctly. Likewise, if one is inclined to attend to many things at the same time in a single act of thought, one does so with a confused mind. Yet craftsmen who engage in delicate operations, and are used to fixing their eyes on a single point, acquire through practice the ability to make perfect distinctions between things, however minute and delicate. The same is true of those who never let their thinking be distracted by many different objects at the same time, but always devote their whole attention to the simplest and easiest of matters: they become perspicacious.

– Descartes, The Regulae, Rule 9

Narrow or Broad?

That is the historical stage. From here I will move faster. While many did embrace both the physical theory that hyperbolic lenses would solves all the aberrations of spherical lenses, and the metaphysical theory that central clarity of an object/idea is constitutive of a directed, Willing consciousness, and key to truth pursuits, Spinoza who was a lens-grinder, and designer of optical instruments and as he will become known as, a metaphysican, did not.

In two letters Spinoza discusses the insufficiency of Descartes’ hyperbolic lenses (which again, did not exist). From his objections we get a firm clue into the nature of his metaphysical objections to Descartes, objections which cut the very nature of how consciousness, the mind, and the Will is conceived. Spinoza’s objection to the hyperbolic lenses is simple:

“Moreover, it is certain that, in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points” (Letter 40).

He draws a diagram to illustrate the problem, and its solution. A hyperbolic lens can concentrate rays only come perpendicular to its axis, a spherical lens has an infinity of axes, so despite a small amount of aberration, clarity is achieved broadly, by bringing as much relational clarity together across the full breadth of the retina. (I superimpose a red-line hyperbola to illustrate Spinoza’s point.)

Now the facts of optics are not ultimately important here. Indeed Descartes and Spinoza were both wrong for a variety of reasons. What is significant is the way that opticality and focus as they are theoretically conceived work to ground fundamental conceptions of how consciousness is. As we can see, Descartes notion of an object/idea centered free Will and consciousness was foregrounded upon first Kepler’s idea that the eye itself had a lens of the shape that Descartes had theorized would be best for the improvement of magnified vision. The question is, once this anatomical, naturalizing foundation is kicked away, and once Descartes’ theological imperative for an essentialized Freedom of the Will is let go of, what has come of the picture of consciousness itself as essentially a Central Clarity Consciousness. Spinoza, as he does away with the Freedom of the Will, does so by dismissing the notion of a pictorial Mind, in which Ideas of varying clarity play like objects of varying clarity:

E2p48 – In the mind there is no absolute, or free, will, but the Mind is determined to will this or that by a cause which is also determined by another, and this again by another, as so to infinity.

Scholium – We must investigate, I say, whether there is any other affirmation or negation in the Mind except that which the idea involves, insofar as it is an idea – on this see the following Proposition [49] and also D3 – so that our thought does not fall into pictures. For by ideas I understand, not the images that are formed at the back of the eye (and, if you like, in the middle of the brain), but concepts of Thought [NS: or the objective Being of a thing insofar as it consists only in Thought]; – trans. Curley

Ultimately for Spinoza as he minimizes the human in the universe, an idea of the mind is only an expression of Substance, and has for its object a state of its body. It is not composed of something that is Intentionality, an object of thought. But more than this, Spinoza’s panoramic understanding of clarity, that object clarity comes from the relative clarity of the entire field of vision, and not from some core surety, reveals the very comprehensiveness of his metaphysical approach. It is not that a single, central proposition “Cogito ergo sum” or any other that could provide real clarity, but rather the entire interlocking breadth of coherence (it is for this reason that some contemporary philosophers have found Spinoza to be a Coherence Theorist of Truth). The clarity comes from abroad, so to speak. It is for this reason that he calls his propositions of the Ethics “the eyes of the mind” (in a very rare metaphor). No one proposition could function as a clarity, just as no central rays from an object could. It is the width of them that brings clarity (and it is for this reason that he argues that one in reasoning must pass as fast as possible to the Idea of God, Substance, and Idea of comprehensivity of cause and explanation, a field upon which any proposition then considered may receive its own sharpness).

How Can Not-Seeing the “object” Come from CCC?

Now briefly Larval Subjects (and probably Graham Harman and many others) does not see how Cartesian Central Clarity Consciousness pervades Graham’s hidden object ontology. As a brief answer, one must not just look at Graham’s conclusions, but also the constituent parts of his conclusion. As Graham himself put forth, he relies upon Husserl’s Cartesian “intentional object” to make up his Four Fold, from which his “hidden objects” are derived:

My own concept of objects is of a fourfold tension that erupts into view most clearly when you look at the two domains of intentional objects (Husserl) and real objects (Heidegger, in my reading at least).

Perhaps this is a misreading on my part, but I do not consider Husserl’s split of the object within consciousness to have removed him from the Intentionality of Consciousness composed of a Central Clarity picture which Descartes put forth. By my accounting, to rely upon this conceptual designation, even in part, is to partake in a mistaken and oversimplified view of the nature of Mind. (And to no small point, in partaking of it, one brings the human-centricist aspects which brought it into view in the first place.) So, for those confused as to why I could claim that Graham is offering up a wrongful optical metaphor when he is asserting objects which cannot be seen at all, one has to keep track of the foundations of the “hidden object” (its place in the Intentionality reduction of Consciousness). It is not just that Graham’s ontology is caught up in an opticality (a general metaphorization of mental clarity with optical clarity), but rather is caught up in a specific model of opticality and consciousness, that clarity is CENTRALIZED (sorry for the caps, but his point seems to be slipping out summaries). This problematic central object of consciousness be it a phenomenological object, or an idea, composes several again essentialized dyads. Upon it is grafted a primary (clear) figure/(confused) ground distinction, a Being/Non-Being binary, and ultimately even a Self/World problematic (to name a few). Once Consciousness is assumed to be CCC, much binary machination can be performed. By simply crossing out the object, through the reflections of Non-Being, pointing to its insufficiency, saying that the object is forever in retreat from this constitutive relations (and thus “hidden”) becomes simply a product of the this mistaken object-centered characterization of consciousness itself.

It is to the periphery that philosophy must look, I suggest, to the connectives. Much as in Spinoza’s diagram of the optics of the human eye, it is the field of (mental) vision which provides any such clarity experienced to be towards the center. And beyond this, if one is to embrace a post-human, or pre-post-Kantian philosophy of mind and object, one really should look right towards the center of vision, where a functioning dissonance, a living eruptive line forces itself up, in the midst of the object and between them. Attentiveness to the phenomenology of consciousness, the nature of the center, is what displaces the very essentialized notion of object (as privileged ontological unit), and as Spinoza would tell us, this living line is constituted by the very thought to thought affirmations of our Body under fluctuating degree of Being and Power…In a sense, Graham retreats from the human, but stops at the object because of a mistaken (and historically human-centricist) conception of consciousness. He simply does not go far enough. There is no reason to stop at the object as the final ontological stop. The fullness, the breadth of what we can do under that trajectory toward the ultimately unknown X, is not the hidden object, but Spinoza’s Substance.

Graham though now suggests that he is coming closer to embracing panpsychism (I believe he restricts his position to an essential dyad endo-panpsychism), so it may be that we will eventually come to find ourselves in agreement, perhaps as I come to better understand him.

[Addendum: Graham responds:

“Kevin thinks the periphery is where the action is. But I’m not sure what to say in response to this aside from what has already been written… Within the intentional sphere, it is simply the case that we do perceive objects and not fuzzy horizons. And within the real sphere, it can be logically deduced that the world is object-oriented, since the alternative is a slippery slope through fields of pre-individual intensities all the way to outright monism.”

My thoughts: The problem is not whether we do perceive objects, of course we do. The question is what is the means by which we do so, and what is their metaphysical grounding. That is, not only do we perceive objects, but our cognition is composed of many other aspects, and reliant upon a wide coherence of beliefs (and ideas are not clear in the same way that objects are clear).  And as I point out, unless you want to pre-posit the “intentional sphere” as defined by its object, a category of reduction that I challenge, a close attention to the phenomenology of what we are aware of is not SIMPLY an object, but a dissonance within and across objects, not to mention aspects of the entire field. My entire point that the concept of the “intentional spheres”  is miscontructed, both in terms of concept and in terms of phenomenal observation. By invoking the “intensional sphere” as a self-evident, and self-defined essentialization of consciousness, Graham simply performs the point I am making, that is, his “hidden objects” ontology is based upon the CCC conception, inherited from Descartes, and cannot operate without it.

As to the other horn of the dilemma that Graham proposes, “Either accept my hidden objects or slide into lump intensities (or monism)” the general conflation of lump intensities, what he elsewhere calls a “molten slag”, and monism is unwarrented, for Spinoza’s monism is not molten and not much of a slag. In fact it contains a vertical dimension carefully constructed along a degree of Being analysis along a vector of power. I don’t know if “Outright monism” is supposed to carry the rhetorical weight of something like “Outright Anarchy” or worse “Outright Balderdash,” but I would think that outright monism is preferable to implicit or only evocative monism. I have yet to read his objection to Spinoza’s notion to Substance, so I cannot really respond to this deadend. There are objects in Spinoza, and they even have essences. It is just that they have no ontological priority, as objects, over the expression of Substance, which they are.]

Here is Spinoza’s much neglected optical letters with commentary, if interested:

Deciphering Spinoza’s Optical Letters

And a Word.doc version: Spinoza’s Optical Letters: Redux

A Look Back for a Moment, The Hole of Spinoza’s Vision

Right now I’m busy composing my Cabinet article, a result of this width of research I have done. Part of this process is looking back at my various conclusive essays to see where I have gotten. There is one that really struck me as a signficant reduction of the kinds of philosophical conclusions that can be drawn from my study of Spinoza’s optical endeavours, in particular pointing out how deeply he diverged from Descartes who preceded him. I repost it here for anyone else’s pleasure, for I read it again this morning and was really moved by its import (sometimes it is like that, one forgets what one wrote):

here: “The Hole at the Center of Vision”

Comments are of course appreciated kvdi@earthlink.net

 

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.

Deciphering Spinoza’s Optical Letters

Line By Line

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

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

Spinoza Answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Soft Focus: Spherical Aberration

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

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

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

diagram letter 39

“[referring to the above] For because a circle is everywhere the same, it has the same properties everywhere. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D…,”

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

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

One’s immediate suspicions of error is readily confirmed by a straight forward application of Descartes Law of refraction. For the circle to have to the dioptrical property Spinoza claims, the refractive index of the glass would have to be a function of the angle of incidence, a condition of which there is not the slightest hint in the letter…[he is] apparently unaware of the importance the “[other] figures”…that Descartes had constructed (154).

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

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

diagram from the Ethics 2, prop 8, scholia

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

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

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

Kepler's diagram from proposition 14

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

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

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

 

 

figure 5, Robert Hooke's Micrographia

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

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

Man on the Moon

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

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

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

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

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

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

Hevelius' 150 ft. arial telescope

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

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

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

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

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

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

 

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

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

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

As Michael John Petry writes:

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

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

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

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

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

figure 14 from the Fifth Discourse of the Dioptrics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Seeing More, or Seeing Narrowly

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

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

Kepler's drawing the hyperbolic crystalline humor, 167

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

figures 1 and 2 from the Optima Promota

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

“the language of mechanics”

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

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

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

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

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

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

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

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

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

Until this day it is believed that spherical surfaces are…less apt for this use [of making telescopes]. Nobody has suspected that the defects of convex lenses can be corrected by means of concave lenses. (OC13-1, 318-319).

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

In the optimal lens the radius of the convex objective side is to the radius of the convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.

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

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

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

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

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

“With concave and convex spherical lenses, to make telescopes that are better than the one made according to what we know now, and that emulate the perfection of those that are made withellipticor hyperbolic lenses” (OC13, 318-319).

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

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

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

 

 

Descartes' diagram 17 of the eye, Sixth Discourse

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

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

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

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

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

Spinoza’s diagram from Letter 40

 

 

 

Descartes diagram 30, Seventh Discourse

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

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

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

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

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

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

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

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

 

These English selections and links to the Latin text: here

A Diversity of Sight: Descartes vs. Spinoza

Different Foci of “Clear and Distinct”

for more on Spinoza, Optics and His Lens Grinding here

This may be premature in my process, but I would like to set down some simple correspondences that have arisen in my reading of Kepler’s Paralipomena to Witelo (1604), correspondences that elicit strong tidal differences between Descartes’ approach to vision – with its attendant metaphysical counterpart, the clear and distinct Idea – and Spinoza’s. That this is found in some of the historically least interesting of Spinoza texts, what has been taken by many to be Spinoza’s blundering into optical theory [ letters 39 and 40 to Jelles, full text ], is suggestive of Spinoza’s critical brilliance. For within Spinoza’s conception of optics and his close-cut rejection of Descartes reasoning seem to be important radical divergences, insofar as vision and light are understood to be more than analogous of metaphysical truths. In these letters Spinoza questions the very vision-philosophy that privileges a central-line of axis, one that fundamentally connects a viewer within to an external and opposing point: a hidden, underpinning assumption of Descartes’ mathematization of the experiential self, Will and the world. Though there is much in Descartes’ philosophy that substantially departs from the Perspectiva epistemology tradition, his enthused embrace of the powers of the hyperbola embody a grounding of the eye, (and therefore the mind), in a centralized perception and knowledge upon which Spinoza places his criticism.  

I have not encountered any analysis which shows that Descartes’ enthusiasm for the hyperbola derives directly from Kepler’s Paralipomena, yet as I am not fully familiar with the literature perhaps this is a commonplace understanding. Gaukroger, for instance, gives evidence that Descartes read this text around 1620, and allows three possible points of influence, but does not speak to the potential influence of Kepler’s offer of the hyperbola as a natural normative of vision. Yet in the process of reconstructing Spinoza’s conceptions in these two letters of objection to Cartesian optics this connection with Kepler owes be fleshed out. One will see that in the Paralipomena Kepler not only sets out the virtues of the hyperbola’s anaclastic line in terms of its resolution of spherical aberration from focus – the primary and crowning demonstration of Descartes’s 8th discourse of La Dioptrique – but he also naturalizes the hyperbola, claiming that the hyperbolic shape of the posterior of the crystalline humor is the very thing that helps establish the soul’s satisfaction with centered-object viewing. In fact, the wide sweep of Descartes’ metaphysics and optics in brief seems signaled by Kepler in his treatment of the hyperbola. And, because I suspect that Spinoza has Kepler’s descriptions in mind when responding to Jelles, making an argument for the importance of a radii of axes of perception, and emphasizing the importance of the angle of incidence in the measure of magnification (see letters linked above), his alternate response to Kepler provides a valuable clue to the stake Spinoza is making against Descartes, both in philosophy and science. In short, Descartes makes a virtue out of the Kepler’s description of the hyperbola and the eye, and Spinoza makes it something of a flaw or limitation. 

Setting Forth Kepler’s Hyperbola

Kepler’s illustration exploring the properties of refraction, page 106

First, perhaps it is best to set out what Kepler has to say about the hyperbola so that a comparison can be made. Initially, he embraces the figure in an attempt to assess a law of refraction, which he hopes to find through the virtue of its special properties. He will not attain the law of refraction, but what he does make clear is that the hyperbola alone possesses the capacity to focus parallel rays to a single mathematical point, or vice versa. Thus, he writes of the figure above:

What is required is to say what sort of surface it is upon which these radiations in this position coming forth from α, so that they strike just as do here the lines βλ, μγ, and so on, so that these lines are either tangents to that surface, or lines equidistant to the tangents…These, moreover, are found only in the hyperbola, not in the parabola, which tends toward a straight line parallel to the axis, not one meeting the axis, as X A here. (107-108).

Not to get lost, were are simply establishing that Kepler points out the solitary characteristic of the hyperbola which will make a lens of this shape central to Descartes’ ambitions for providing aberration-free vision through hyperbolic lenses.

The next appearance of the hyperbola in Kepler’s text occurs in chapter 5, where the geometry and anatomy of the human eye is discussed. Here Kepler presents a small, simple drawing of the “crystalline humor” which behind the cornea in the eye is the primary means of focusing light. I enlarge it here for clarity:

Kepler’s drawing of the crystalline humor, page 167

Kepler writes of the shape of the humor:

 On the posterior side…[the] figure is a hyperbolic cononoid, a hyperbola rotated on its axis. For [Johannes] Jessenius thus relates, that it is not spherical as [Felix] Platter said but that it protrudes markedly, and is made oblong stretching up almost into a cone; and that on its anterior face it is of a flattened roundness…(167).

Kepler has emphasized a correction. The humor is not spherical as Platter claimed, but hyperbolic, and made nearly like a cone. It is this particularity of the eye, coupled with his earlier hyperbolic observations that will create a certain naturalization of hyperbolic vision, something that speaks to the very nature of the human soul. The aspherical effect of the crystalline humor on vision is made clear a few pages later where Kepler discusses how the various cones of light coming from each of the points of an object to be seen, interact with and refract upon the cornea and the crystalline humor. What will be shown is that the humor is biased towards the production of clarity oriented towards its central axis, and that light cones that come obliquely to this angle, will appear less clear to the human eye, as they are focused to the borders of the image:

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

Kepler then takes these facts of refraction and uses them to explain the experience we have of satisfactory vision. What is most notable is that Kepler wants us to understand how the oblique “more confused” images at the borders of an image actually complete and serve the focus of vision around the central axis:

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

The “surroundings of the hemisphere” invite our eye from to this or that. Oblique vision proves satisfying to the soul only to the degree that it inspires the eye to turn its hyperbolically-aided central axis across its field. The concept is that the deprivation of clarity somehow drives the soul to complete its picture. This is an important point when considering the influence of vision as a primary analogy for Cartesian metaphysics.

Kepler Completes the Synthesis of Eye and Lens

Lastly, Kepler will place the facts and inclinations of vision within the context of the powers of hyperbolic focus, and he does this in such a way that it would seem sure that when Descartes looks to solve the problem of spherical aberration he would see in the hyperbola a natural bias towards centralized focus.

Kepler’s illustration of the spherical aberration of rays, page 194

After a protracted investigation of the behavior of light rays in refraction through a crystal globe filled with water, and an explanation of our visual experiences of the images and confusions that result, Kepler will conclude that the hyperbola alone resolves the need for gathering a cone of rays into a single point:

Proposition 24 – Rays converging towards some single point within a denser medium are gathered by the hyperbolic conoidal surface bounding the medium to one single point, closer than the former point…These two things [two requirements], however, are accomplished, not by one or another circle but by conic sections…Further, of the conic sections, only the hyperbola or some line very close to it, is the measure of refractions, as was shown in Sect. 5 of chapter 4. Indeed, this very thing was demonstrated there: that the surface making all the rays outside the denser medium parallel is a conoid that does not differ from hyperbolic (198).

Given this remarkable property, Kepler then concludes that “nature’s plan” has endorsed the priorty of the hyperbola. And it seems that there can be little doubt that Descartes worked to synthesize this priorty which his own metaphysical priorty for “clear and distinct” ideas:

Hence it is evident nature’s plan concerning the posterior surface of the crystalline humor in the eye. That is, she wished to gather all the radiation of any visible object entering the opening of the uvea [pupil] into a single point of the retina in order both that the point of the picture might be all the more evident, and that the rest of of the points of the picture might not be confused by extraneous rays whether stray or gathered together. – Chapter 5, proposition 24, corollary (199).

Spinoza: “the eye is not so exactly constructed..” (letter 40)

To pull back for a moment. If this analysis is correct, that a decisive rift in Cartesian philosophy could be seen in Spinoza’s letter to a friend discussing what at most appears to be a trivial oversight on Descartes’ part is striking. Many scholars seem to have struggled over the meaning of Spinoza’s words in these letters (39 and 40) sensing that there is an elementary blunder in Spinoza’s thinking, but, as seen in their lack of a careful examination of it, there is an inability to locate just what this blunder would be. Instead, because Spinoza has been read as solely a metaphysician, his foray into optical matters in these two letters was largely dismissed as Spinoza simply wading in too deep a water. With very few exceptions, most only assumed that Spinoza was unfamiliar with the issues at stake, and his in-concordant use of terms appeared to prove this; yet a change in scholarship is occurring. Perhaps our examination of these two letters can add to this shift of perspective.

What does Spinoza mean in his letter 40 talk about the inexactness of the human eye?:

Moreover, it is certain that, in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points. And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci. And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others (letter 40, full text)

We have seen from Kepler that the likely reference is to the distortions of “more confused images” at the borders of vision, in part due to the aspherical, single axis nature of the eye’s lenses. This stands in contrast to Spinoza’s diagram, which he takes to be an ideal spherical refraction: 

Spinoza’s diagram of the virtue of spherical refraction, Letter 39

Spinoza is extolling the comprehensive capacities of spherical refraction, and his embrace of this concept marks out a distinct divergence from Descartes’ naturalized endorsement of a center-focused vision.  Descartes will develop a theory of clear and distinct thinking which narrows the field of mental vision. This vision is imagined to be in concert with what Nature had planned in having given the eye its own hyperbolic lens; conversely, Spinoza will take from Descartes the notion of “clear and distinct”, but the concept of vision in which it is to be deployed is dramatically different. Spinoza emphasizes the clarity of a connective, hemispherical scope; Descartes is aimed at the close focus on the surest of things.

Looking With the “Mind’s Eye”

This difference perhaps can be made more distinct by considering the use of the phrase “the Mind’s eye” by both thinkers. For instance, in his Regulae, after his very influential Rule 8 [the full text of which I post and briefly engage here ], Descartes tells us in Rule 9 how we must compare improvements in thinking by attending to how we naturally see things. Human vision provides the exemplar of how of mental vision is, and the issue is one of central focus:

Rule 9: We must concentrate our mind’s eye totally upon the most insignificant and easiest matters, and dwell upon them for long enough to acquire the habit of intuiting the truth distinctly and clearly.

…We can best learn how mental intuition is to be employed by comparing it with ordinary vision. If one tries to look at many objects at one glance, one sees none of them distinctly. Likewise, if one is inclined to attend to many things at the same time in a single act of thought, one does so with a confused mind. Yet craftsmen who engage in delicate operations, and are used to fixing their eyes on a single point, acquire through practice the ability to make perfect distinctions between things, however minute and delicate. The same is true of those who never let their thinking be distracted by many different objects at the same time, but always devote their whole attention to the simplest and easiest of matters: they become perspicacious.

– Descartes, The Regulae, Rule 9

It is not without significance that Descartes in his metaphor of close focus appeals to artisans and craftsmen to praise the powers of concentrated vision. These are not a class of persons that he would embrace in society- his attitude toward de Beaune and Ferrier is well known – but their analogous use in a hierarchy of powers is in keeping with his concept of knowledge being like an ascent of mechanized complexity, from primitive mental tools to those most intricate (Rule 8, Regulae). The same artisan trope is found again in his La Dioptrique, where the limits of the eye are by craftsmen strained and improved by the exercise of ocular muscles. I quote at length below because it is an important passage. For one, it is found at the end of the 7th Discourse which is the text that Spinoza and Jelles are commenting on in their letters, and is at the cusp of his presentation of the 8th Discourse praise of the hyperbola. Secondly, here the virtue of a dispersive vision (which Kepler describes as limited by the hyperbolic shape of the lens) is denied: seeing more is not seeing better, as is testified by experience; and thirdly, Descartes treats the possible alteration of the limits of crystalline humor and pupil’s capacities as being achievable, not in the direction that Spinoza would like (toward a more-than-human breadth of clear vision), but in terms of an exceedingly close narrowing, achieved by trained specialists:

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

I have still to warn you that the faults of the eye, which consist in our inability to change sufficiently the shape of the crystalline humor or size of the pupil, can bit by bit diminish or be corrected through practice: for since this crystalline humor and the membrane which contains this pupil are true muscles, their functions become easier and greater as we exercise them, just like those of other muscles of our body. And it is in this way that hunters and sailors train themselves to look at very distant objects, and engravers or other artisans who do very subtle work to look at very close ones. 

– Descartes, Seventh Discourse

In the highlighted passage Descartes repeats Kepler’s summation of the benefits of a hemisphere of vision, that it simply leads to a field of vision which helps serve our central axis: Kepler: “On this account, oblique vision is least satisfying to the soul, but only invites one to turn the eyes thither so that they may be seen directly “. Yet Descartes has changed the emphasis some and placed the notion of “willing” at this central axis (a literal conflation). The widening of a view only provides the occasion for free choice, which will express itself in the turning of the single axis of the eye. (It is thus clear that Descartes and Spinoza mean different things by “the faults of the eye” and “the eye is not so exactly constructed”.)

Spinoza: “so that we must not fall into pictures”

What is compellingly consistent is that Spinoza’s own use of Descartes’ “Mind’s eye” phrasing will directly address the Cartesian issue of the freedom of the will, not to mention the pictorial conception of clarity and distinctness. The phrase is most pointedly found in his Ethics, just where the distinction between willing and understanding is by Spinoza denied (E2p48 and 49):

E2p48 – In the mind there is no absolute, or free, will, but the Mind is determined to will this or that by a cause which is also determined by another, and this again by another, as so to infinity.

Scholium – We must investigate, I say, whether there is any other affirmation or negation in the Mind except that which the idea involves, insofar as it is an idea – on this see the following Proposition [49] and also D3 – so that our thought does not fall into pictures. For by ideas I understand, not the images that are formed at the back of the eye (and, if you like, in the middle of the brain), but concepts of Thought [NS: or the objective Being of a thing insofar as it consists only in Thought]; – trans. Curley

Descartes’ notion of looking with the “Mind’s Eye” requires learning what distinctness is in terms of our experience of human vision, a lesson that requires that we focus closer and closer upon obvious things, training our eye to become more and more exact, a lesson which in turn gives us to understand the centrality of the focus of a single axis, and the use of a breadth of vision as merely the field for a freedom of choices; yet in Spinoza the human eye itself is seen as “inexact” in its singular axis of focus. And looking with the Mind’s eye is for Spinoza not so much a process of learning to see clearer and clearer pictures, (or even holding one clear idea or another in mental vision), but rather learning to look in a way quite unlike the way of the human eye, within a matrix of conceptual understandings; and this matrix is one which decenters the central axis of vision (and one could say the “self”), strives to achieve something akin to an infinity of axes of vision, (nothing more than a breadth understanding of the order of Adequate Ideas and thus the causes of the phenomena we witness, and that we affectively experience). While Descartes would say, following closely our analogous experiences of human vision, that seeing more is not seeing better, Spinoza would say that one is only seeing better if one is seeing more: hence his thought moves very, very quickly to the intuition of the Adequate Idea of God.

A Difference in Method

One grasps this if one compares Descartes’ notion of method with Spinoza’s own early Emendation notion of method, which is in response to it. The below passage is important because it follows several points which bear 8th Rule influence (a focus upon human powers, viewing knowledge building like blacksmithing, for instance). While Descartes is interested in focusing the mind on simple truths which may serve ultimately to connect one to a transcendent God, Spinoza’s method is not one of focus upon this or that truth, but upon the standard of truth itself as it immediately directs one’s attention to a maximalization of thought: a most perfect Being. It is the distinction between one kind of perception and all others, which throws the vision wide:

That is, the most perfect method will be one which shows how the mind should be directed according to the standard of a given idea of the most perfect Being…From this one can readily understand how the mind, as it understands more things, at the same time acquires other tools which facilitate its further understanding. For, as my gathered from what has been said, there must first of all exist in us a true idea as an innate tool, and together with the understanding of this idea there would likewise be an understanding of the difference between this perception and all other perceptions. Herein consists on part of our method. And since it is self-evident that the more the mind understands Nature, the better it understands itself, it clearly follows that this part of our method will become that much more perfect as the mind understands more things, and will become then most perfect when the mind attends to, or reflects upon, the knowledge of the most perfect Being. (trans. Shirley 39)

The web of truths that “Mind’s eye” vision focuses on is a breadth of vision, governed by a comprehension of determined causes. And in a sense, this begins with God, God as a totalizing reality of Being from which we are not separate. A field of vision, which for Descartes provides an array of choices which an axis-eye then willfully judges and picks its way through, for Spinoza is an incandescent weave of causes and effects, any adequate understanding of which leads to all others. It is a spherical conception of a refraction along an infinity of axes, in which the Will plays no part.

Descartes’s Hyperbolic Doubt and Hyperbolic Lens

In considering Kepler’s introduction of hyperbolic lenses and his Nature’s single-axis plan for the eye, and then Descartes synthesis of the two, there is the happy result of support found for a contested interpretation of Descartes offered by Graham Burnett, in his book Descartes and the Hyperbolic Quest . Professor Burnett offers that Descartes’ obsessive, mechanized pursuit of the grinding of a hyperbolic lens, and his project of legitimatizing his Natural Philosophy through “hyperbolic doubt” are something more than a mere conflation of uses of the word “hyperbolic”. Burnett tells us, citing Gaukroger’sbiography, that the two may correspond to a single conception of mind, (quoting at length):

What configuration of mind of mind allows natural light to coalesce into a clear and distinct idea? The answer…is hyperbolic doubt. If once we saw, as in a glass darkly, and if at some (beatific) point we will see face to face, for the time being the best we can seem to do is to see through the right kind of glass that one that does not distort or obscure: and this just might be, at least least initially, the focusing glass of hyperbolic doubt. To play out the suggestion then: Descartes greatest scientific success lay from his perspective, in his systematic investigation of optics and the perfection of human vision those investigations promised; his optics presented an instanteous light that could be focused into clear and distinct images by means of the imposition of a hyperbolic form. Descartes’ greatest philosophical success lay, from his perspective, in a systematic investigation of the human mind and the perfection of cognitive operations those investigations promised; that the human mind received, via natural light of reason, an instantaneous, clear and distinct illumination, but only by means of the interposition of another hyperbolic focusing device – the hyperbolic doubt.

I do not wish to overemphasize the signification of the parallelism, tantalizing as it is. Following Gaukroger’s reconstruction of Descartes’ psychology [Descartes, an intellectual biography ], a quite elaborate extension of the hyperbolic (lens)/hyperbolic (doubt) analogy would be possible. In Gaukroger’s reading, the imagination mediates between the pure intellect and the realm of the senses, and the experience of cognition inheres in this intermediate faculty, which represents the content of the intellect and the content of the senses both as “imagination.” Where these two map onto each other the experience is that of “perceptual cognition.” As the project of hyperbolic doubt is abundantly imaginative, and as Descartes has insisted that the natural light of reason does not stream down from God but is within our intellects, it would be possible to argue that the imagination plays the role of the focusing of the hyperbolic lens, and receives the light emanating from the intellect, which normally enters the imagination confusedly, quickly distorted by the “blinding” profusion of imagery from the senses (126-127).

Indeed, if we follow professor Burnett’s conclusion, and allow it to have a substance greater than mere lexical coincidence, we find that when armed witha knowledge of Kepler’s antecedent approach to the hyperbola such a reading begins to cohere. Descartes’ embrace of Kepler’s hyperbola of the human eye shows that “extreme doubt” and the focus of the Will that it accomplishes is for Descartes truly both a mechanism for focusing the mind upon simple truths and a naturalized legitimization of a will-centered, single axis organized perception. What professor Burnett intuited through his study of Descartes’ life-long pursuit of an automated, hyperbolic lens-grinding machine, is given traction when the genealogy of Descartes’ conception of the importance of the hyperbola is traced back to Kepler, its orgin. And this is exposed in two almost-ignored letters written by Spinoza on a subject he long had been considered deficient in, as the force of Spinoza’s attack upon both Descartes’ metaphysics andhis optics is to be considered as being of one cloth. Spinoza has Kepler in mind because Descartes had Kepler in mind. This is suggestive of the power of Spinoza’s critique, and the level at which he carried it forth. It touched not only the abstraction of Descartes’ metaphysics, but also the optical-theory origins of Descartes preoccupation, that human vision was somehow naturally hyperbolic and therefore offering a guide toward the perfection of the mind. Because our inheritance of the optical trope of human vision is so rooted in our conceptions of the world, and our acceptance of Descartes’ approach to thought and mechanism is so pervasive, Spinoza’s optical critique proves promising of a radical importance at the very least.

Lasting Questions

None of this goes any distance toward proving whether Spinoza’s critique of Descartes in letters 39 and 40 was correct in the terms he meant it by, speaking of how light and lenses behave. Clearly Spinoza was well informed about the nature of Descartes’ claim as to the importance of the hyperbola. He had read and followed Johannes Hudde’s Specilla circularia, which dismissed the importance of spherical aberration in a mathematically exact way, minimizing Descartes’ impractical solution; and he was likely familiar with Christiaan Huygens’ own complaints about Descartes’ failures in treating telescope magnification accurately. Additionally, it seems quite likely that Spinoza was familiar with the Ur-source of Descartes’ own embrace of the hyperbola, Kepler’s Paralipomena to Witelo, since he addresses specific terms of its explanation, and the argument he presents in brief cuts to the quick of the virtues of the hyperbola presented there: the idea that the human eye’s hyperbola somehow expresses Nature’s plan which in Cartesian hands would naturalize a priorty of a single-axis, will-driven priority of focus and choice. By arguing for the “inexactness” of the eye, Spinoza is undermining a primary vision/knowledge metaphor which helps form part (but most certainly not all !) of Descartes’ metaphysics of clarity.

There are therefore a few questions that remain. For one, Kepler does not merely serve as a source for a negative critique of Descartes, insofar as he has followed Kepler. For instance Kepler’s conception of light in many ways diverges from Descartes’, and could be said to have concepts which presage Christiaan Huygens’ own wave theory, which would eclipse Descartes’ optics. It remains to be seen if Spinoza’s optical understanding stems directly from Kepler in a positive sense, that is, if Spinoza’s holds optical concepts which were superior to Descartes’ theories due to Kepler’s influence. Spinoza has used Kepler to undercut Descartes’ metaphysics, but where does Spinoza stand in terms of contemporary optics? For this to be answered, Spinoza’s praise of the versatility of an infinity of axes has to be set up against the contemporary science of telescope construction. For though Descartes’ hyperbolic lenses were nearly impossible to make at that time, in theory at least they would have offered an advantage. Spinoza’s objection is that Descartes is incomplete in his analysis of magnification, and that the capacity of a lens to handle a variety of axes is important in compound telescope magnification. Such a possible importance remains unaddressed, though all existing telescopes obviously achieved their magnification without hyperbolic lenses (and notably, Christiaan Huygens had privately solved the issue of spherical aberration using only spherical lenses in the summer of 1665, when Spinoza and he were closest).

The other question that remains is to determine the large scale consequences of Spinoza’s rejection of a naturalized, single-axis concept of hyperbolic vision, upon his own preoccupation with lens-grinding and instrument making. The grinding of a lens, after all, is exactly the kind of “craftsman” or “artisan” practice that Descartes lauded in his 9th Rule, one that lead to an acuity of vision. The purpose of a lens is most often to achieve a magnification which concentrates the vision at a local point. And this is the mode of narrow focus to which Spinoza seems to making his objection. But if we allow the analogy between craftwork and mental tools found in the works of both Descartes and Spinoza, the careful refinement of a proposition, such as those found in Spinoza’s Ethics, would be read as a kind of development of perspicuity. Permitting the Ethics to stand as our model for complex, intricate knowledge, by analogy any grinding of glass into a polished shape must be seen as part of an interlocking of all other actions, ideas and material states; for just as there is extensive cross-reference in Spinoza’s Ethics, Spinoza’s own daily preoccupation with lens-grinding and instrument building must be seen as cross-referenced to an infinity of other causes and actions, all leading to an theorized increase in freedom. As Spinoza clarified a piece of glass and made it capable of magnification, was it that he was concerned not just withthe the lens, but how this magnification fit with other lenses, in devices, with phenomena to be discerned, and the Ideas we hold as we use them? It would seem at that this is so, but this question has to be answered more fully.

for more on Spinoza, Optics and His Lens Grinding here

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

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

Telescopes and Turning a Flea into a Elephant

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

I have looked at and read over what you noted regarding the Dioptica of Descartes. On the question as to why the images at the back of the eye become larger or smaller, he takes account of no other cause than the crossing of the rays proceeding from the different points of the object, according as they begin to cross one another nearer to or further from to eye, and so he does not consider the size of the angle which the rays make when they cross one another at the surface of the eye. Although this last cause would be principle (sit praecipua ) to be noted in telescopes, nonetheless, he seems deliberately to have passed over it in silence, because, I imagine, he knew of no other means of gathering rays proceeding in parallel from different points onto as many other points, and therefore he could not determine this angle mathematically.

Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered (letter 39)

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

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

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

Using this diagram, Descartes continues:

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

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

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

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

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.