A Diversity of Sight: Descartes vs. Spinoza

Different Foci of “Clear and Distinct”

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 demostration 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 apparence 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 conoidalsurfacebounding 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 emdorsement 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 hyperobolic 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 emphasises 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 (tpward 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 breadthof 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 maximalizationof 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 ahyperbolic 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 instanteous, 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 critque 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 ChristiaanHuygens’ 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 magnfication, 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.