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Category Archives: Specilla circularia

An origin of Spinoza’s “cones of rays” explanation, Letter 40

[addendum: in addition to these thoughts, the influence of a more recent source, James Gregory's Optica Promota (1663) has to be considered]

Kepler and How Spinoza Viewed the Eye and Light

As a point of reference it is important to locate the origin of Spinoza’s phrase “cones of rays” found in his letter 40, since implicit in this phrase is likely the conception of light and refraction which would help us make sense of his objection to Descartes. This phrase has a history of what seems a bit of interpretive confusion, for instance, that expressed by Alan Gabbey in his Cambridge Companion to Spinoza article, “Spinoza’s Natural Science and Methodology”. Here professor Gabbey quotes the phrase as if it embodies the locus of Spinoza’s befuddlement:

Spinoza explained that light rays from a relatively distant object are in fact only approximently parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintains the same property of the circle in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous "Ovals of Descartes"] that Descartes had constructed in Book 2 of La Géométrie to provide a general solution to the problem of spherical aberration [Ep 40].

I have already pointed out that Spinoza indeed was not “unaware” of the “importance” of Descartes’ figures (since he was intimate with the debate over that importance), and that part of Gabbey’s difficulty may stem from a weakness in translation, or not taking into account Spinoza’s familiarity with Hudde’s Specilla circularia: here. Spinoza, all the same, is constructing an argument that seems to shift parameters. In Letter 39 he speaks of the capacity of spherical lenses to focus parallel rays to an (approximate) point of focus opposite, along an infinity of axes, and now he tells Jelles that this capacity is to be understood not for parallel rays, but for “cones of rays”, which is more accurate to what is actually occurring. Where does Spinoza get his conception of “cones of rays”?

I believe it is found in Kepler’s Paralipomena to  Witelo (1604), a work I am beginning to suspect holds some of Spinoza’s resistance to Descartes. Descartes called Kepler his “first teacher” in optics, so when there is a divergence between the two, Kepler and Descartes, one may perhaps look to Kepler as a source for other resistance to Descartes’ conclusions. (It is a mistake to assume that solely in terms of temporal advancement, all of Descartes deviations from Kepler are corrections, for in some ways Kepler held views antecedent to our better conceptions on the nature of light.) In letters 39 and 40 Spinoza is critiquing Descartes explanation of how image size is produced in telescopes, and he finds in Descartes’ explanation some delinquencies which give undue favor to the hyperbola. Where Spinoza likely draws his conception of “cones of rays” is where Kepler is discussing the manner in which images are formed in the human eye:

Now in order to approach closer to the way this picturing happens, and to prepare myself gradually for the demonstration, I say that this picturing consists of as many pairs of cones as there are points in an object seen, the pairs always being on the same base, the breadth of the crystalline humor, or making use of a small part of it, so that one of the cones is set up with its vertex at the point seen and its base at the crystalline (though it is altered somewhat by refraction in entering the cornea), the other, with the base at the crystalline, common with the former, the vertex at some point of the picture, reaches to the surface of the retina, this too undergoing refraction in departing from the crystalline. And all the outside cones come together at the opening of the uvea [pupil], at which space the intersection of the cones takes place, and right becomes left..

…[now speaking of a single cone of those cones of rays] Thus those rays which previously were spreading out in their progress through the air, are gathered together now that they have encountered in to the cornea, so much so that any great circle described by those rays upon the cornea, which in their decent touch the edges of the opening is wider than the circle of the opening of the uvea; however, these rays, all the way to the opening of the uvea, are so strongly gathered together through such a small depth of the aqueous humor, that now the edges of that opening are trimmed of by the extremes, and by that decent they have made illuminous a portion on the surface of the crystalline humor, if indeed they all have first arisen at a point at a certain and proportionate distance (which is pecular to each eye, and not just the same for all), they fall approximately perpendicularly, because of the similar convexity of the cornea and the crystalline humor. (trans. Donahoe, 170)  

Included in this reference is also the obvious fact that for an object to be seen, light from all its points must be gathered. It is part of Kepler’s picture:

Spinoza writes: “…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.”

Because Spinoza is arguing that the hyperbolic lens – designed to receive rays solely parallel to its one axis - is insufficient for the variety of angles at which light arrives, the question of parallel or coned rays does not seem germane to his argument. His emphasis in the original description seems meant to be in terms of axes, assuming a “mechanical point” of focus definition. (Whether it is ultimately germane to contemporary telescope construction is another question.)

It must be noted, though here is both a most significant implication of the cone of light having a spherical (wave?)front, something ungrasped by Descartes but captured later by Huygens, in the text that follows as Kepler closely describes this action of cones of rays in the eye, he emphasizes the “hyperbolic posterior surface of the crystalline” (171), possibly disturbing the cohesion of Spinoza’s purely spherical ideal of light refraction. If indeed Spinoza is taking Kepler’s description as his source, this gives us to consider how Spinoza might mean the inexactness of the construction of the eye (letter 40). In what way can the eye be considered imperfect, and is there a Kepler source for this notion?

Spinoza writes: “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.”

There is an antecedent to this in Keplers’ description of the action of rays as they come from cones at angles oblique to the axis of the cornea:

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 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

…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). 

Here Kepler seems to be making the exact same point as Spinoza, with an additional hint towards the necessity of the oblique in Spinoza’s concern. The construction of the eye, in so far as its lenses are aspherical, it is retarded its capacity to handle the focus of cones of rays oblique to its single axis. This first calls our attention to the limits of human vision (in individuals and in plan), and then suggests that Spinoza’s point is one of practical application in terms of lenses: that in aiding human vision and constructing telescopes, the symmetry of spherical lenses is preferred for magnification, handling a greater variety of angles of incidence through its infinity of axes.

This does not of course establish the veracity of Spinoza’s argument, but in locating a likely origin for Spinoza’s conception, we at least place Spinoza’s argument within the context of a larger view, to be weighed with all other anti-hyperbolic (Cartesian) positions  of his day (Hudde, Huygens). As I have said, it is my sense that Spinoza derives more than this from Kepler’s account of light. More posts to follow.

Spinoza’s Blunder and the Spherical Lens

Did Spinoza Understand the Law of Refraction?


In seeking to uncover the nature of Spinoza’s lens-grinding practices, and the competence of his optical knowledge, I believe I may have uncovered another small, but perhaps significant mis-translation of the text. The text is letter 39, written to Jelles (March 1667)), wherein Spinoza explains the insufficiency of Descartes treatment of the cause by which objects appear smaller or larger on the back of the eye. Spinoza points out that in La dioptrique Descartes fails to consider the angle of incidence of the rays at the surface of the lens, and only considers “the crossing of the rays proceeding from different points of the object” at various distances from the eye. I believe that there are subtle epistemological issues that come with this point, and which Spinoza has in mind, but for now it is best to pay attention to what follows this complaint.

Spinoza suspects that Descartes has fallen intentionally silent on this factor, in part because Descartes has it within his plan to promote the importance of the hyperbolic lens, a reported improvement upon spherical lenses. Hyperbolic lenses, which Descartes championed and tried at great length to produce in an automated way, by the time of Spinoza’s writing had proven to be impossible to manufacture. Yet here Spinoza objects to them at the geometrical level. Spinoza argues that their means of collecting light rays, due to their aspherical nature, was less proficient than that of spherical lenses. I shall have to leave this immediate comparison of lens shapes aside, as well, to narrow our attention to a reported blunder Spinoza makes in his explanation, a blunder that has lead some to conclude that Spinoza really could not have had much comprehension of theoretical optics at all. He might have known how to make lenses, but was unclear as to how they worked. The charge is that he didn’t even understand the law of refraction: the principle relation between the angle of incidence and the refractive properties of a lens medium, derived by Descartes to explain the main insufficiency of spherical lenses: spherical aberration. 

Spinoza draws a diagram to help explain to Jelles just what the advantages of spherical lenses are. It depicts what Spinoza regards as a natural product of its symmetry, the capacity to focus rays that are parallel along its infinite number of axes, across its diameter, to a point opposite:

This is how Spinoza describes the relation, as translated by Shirley, (a wonderful translator):

For the circle, being everywhere the same, has everywhere the same properties. For example, the circle ABCD has the property that all the rays coming from the direction A and parallel to the axis AB are refracted at its surface in such a manner that they all thereafter come together at point B. Likewise, all the rays coming from the direction C and parallel to the axis CD are refracted at the surface in such a way that they all come together at point D. This can be said of no other figure, although hyperbolae and ellipses have infinite diameters

Now this description has lead notable, modern critics to conclude that Spinoza seems to have failed to understand the Law of refraction altogether, for in seeking to raise the value of the figure of the circle Spinoza has attributed properties to it that appear to ignore just that law. Principle here is the apparent, unqualified claim, “the circle ABCD has the property that all [parallel] rays coming from direction A…are refracted [at] point B”:

This is how Alan Gabbey articulates the failure on Spinoza’s part, and the reasonable conclusion to be drawn from it:

One’s immediate suspicions of error is readily confirmed by a straightforward application of Descartes Law of refraction. For the circle to have to the dioptrical property Spinoza claims, the refractive index of the glass would have to be a function of the angle of incidence [Gabbey secures his point with a footnote of the formula], a condition of which there is not the slightest hint in the letter. In his next letter [March 25, 1667] to Jelles, who asked for clarification, Spinoza explained that light rays from a relatively distant object are in fact only approximently parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintains the same property of the circle in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous "Ovals of Descartes"] that Descartes had constructed in Book 2 of La Géométrie to provide a general solution to the problem of spherical aberration [Ep 40]. I suggest on these grounds alone that though Spinoza may well have written a treatise on the rainbow [which he allegedly burned shortly before his death: Spinoza 1985b: 8], it is very unlikely that he wrote Stelkonstige Reekening van den Regenboog (“Spinoza’s Natural Science and Methodology” in The Cambridge Companion to Spinoza, 154)

For me what is at issue is not Gabbey’s concern as to whether Spinoza wrote the particular treatise on rainbows or not; it is the general appraisal of Spinoza’s aptitude in the theoretical optics of his day. Just what did Spinoza understand? Gabbey’s point rests on the slender idea that Spinoza does not seem to be able to even conduct “a straightforward application of Descartes’ Law of refraction”.

The problem arises I think from a slight neglect in the translation, the loss of the word “if”. I do not know what translation Gabbey was using, but if he was working from the Latin I believe he made a similar mistake. This is the Latin taken from his Opera:

Nam quia circulus idem ubique est, easdem ubique proprietates habet. Si, exempli causa, circulus A B C D hanc possideat proprietatem, ut radii omnes, axi A B paralleli, a parte A venientes, ad eum modum in ejus superficie refringantur, ut postea omnes simul in puncto B coeant; omnes quoque radii, axi C D paralleli, a parte C venientes, ita in superficie refringentur, ut simul in puncto D conveniant; id quod de nulla alia figura affirmare licet, licet Hyperbolae ac Ellipses infinitas habent diametros (the conditionals emphasized)

Key is to note that Spinoza has shifted into a hypothetical subjunctive. It is not as Shirley translates: “For example, the circle ABCD has the property that…” (though Shirley’s translation does not wholly obscure my point). The line reads “If, for example, circle ABCD should have the property that…”. It is not an assertion. This small difference has lasting effects on the nature of the point that Spinoza is trying to make. He is not attempting to say that circle ABCD, and therefore all circles, has the general property that ALL rays that are parallel to ANY of its axes, would be refracted to a point opposite. This would be absurd, and as a lens-maker and user he would know this. He is saying that given that this particular circle, as it represents a lens, can be said to have this property that all rays [all rays so depicted] come from direction A and do refract to a point opposite B, then another set of rays, at the same angle of incidence to a different axis would be refracted to another point in just the same manner [C to D].

Thus, what Gabbey mistakes as a general definition of the capacities of a spherical lens, ANY spherical lens, is actually for Spinoza a description that has two levels of comprehension. The first level is manifest and actual, the second is abstract and geometrical. At the manifest level circle ABCD is understood to be a description of a hypothetical, though real lens, with specific properties and focal points (here the Law of refraction is simply assumed); at the second level, given the acceptance of the first level, there a general property of ALL spherical lenses due to their geometry. This property is: if the index of refraction allows the focusing of a set of rays at a range of angles of incidence to a particular axis (as drawn), so that they meet at one point, the same will be said of other such axes of the arc. The loss of the hypothetical “If” in translation makes it appear that Spinoza is making a much broader claim about sphericals which simply ignores the Law of refraction. He is not.

Here is the text in a translation more sensitive to the conditional (wording suggested by S. Nadler):

For because a circle is everywhere the same, it has the same properties everywhere. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D, this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters.

The construction of his thought is: “if this can be said of this particular figure, it can be said of no other figure, hyperbola and ellipse included”. We can see that part of the problem of translation lies with the very condensed way in which Spinoza is employing the subjunctive. His hypothetical has two levels of comprehension. To repeat, the figure ABCD is being treated as a real world manifestation, and so has particular properties such as the limitations granted to it by its index of refraction as being made of glass, but also it is taken to be an illustration of an abstract property of spherical shapes in general. It is both general and particular, and, as such, it is quite easy to miss both aspects, as Alan Gabbey and others seem to have done.

To really make clear Spinoza’s point, as he sees it, one just has to overlay his circle with the figure of a hyperbolic, and assume, as Spinoza does, that both sets of parallel rays (A and C) would be refracted to a single point (B and D). The hyperbolic lens simply would not have the same property:


There are several issues at stake here, some of them optical, some epistemological, some pragmatic, most of which must be left aside for the moment. It is my purpose to expose that Spinoza is making a point about the capacities of spherical lenses. We are to understand, under his proposed description, that these are lenses that are capable of focusing rays at a mutuality of angles of incidence by virtue of the geometrical properties of their class.

Although a surface reading of Spinoza’s diagram and description seems to present an elementary blunder, and may lead to the conclusion that he did not even understand how the Law of refraction was to be applied, a closer reading shows, at the very least, that he was intimately aware of Descartes’ law, and how Descartes attempted to solve the problem of spherical aberration. Key to understanding Spinoza’s approach was that he considered spherical aberration itself to be incidental to the limitations of lens use. Most likely he thought of lenses as devices, and the slight blurring at the edges of focus to be endemic to them. In this he followed closely his friend in correspondence, mathematician Johannes Hudde, who in his now lost treatise, Specilla circularia, argued just this conclusion. Instead of accepting Descartes’ attempt to define a mathematical point of focus, and construct a lens to achieve it, Hudde claimed that the point of focus of a lens is what he called a “mechanical point”. And it is precisely this “mechanical point” notion of focus that Spinoza has in mind when he speaks of rays meeting up at points B or D in his figure, (he uses the phrase in the following letter of clarification to Jelles, letter 40). Spinoza may have been on the wrong side of the argument, but to decide that he was unfamiliar with the argument itself in a fundamental way would be incorrect.

There is an additional piece of evidence of Spinoza’s theoretical familiarity with the application of the Law of refraction. We have the historical fact that he was in conversation and likely visitation with Christiaan Huygens in the summer of 1665, just as Huygens was working out his solution to spherical aberration using solely spherical lenses in composite. Huygens was bent on solving the aberration problem in a non-Cartesian way, as he meant to publish in his Dioptrics. In letter 30A to Oldenburg, as Spinoza responds to the Royal Society secretary’s urgings for an update on Huygens’s progress, despite Spinoza’s love of the spherical lens he expresses his doubts that Huygens would solve the problem that Descartes thought he had solved with hyperbolics:

The problem which he says he is trying to solve in Dioptics is as follows: Is it possible to arrange the lenses in telescopes in such a way that the deficiency in one will correct the deficiency in the other, and thus bring it about that all parallel rays passing through the objective lens will reach the eye as if they converge on a mathematical point? As yet this seems to me impossible. Further, throughout his Dioptrics, as I have both seen and gathered from him (unless I am mistaken), he treats only spherical figures. (quoted on October 7, 1665).

By the time that Oldenburg had read this Huygens indeed had solved much of problem of spherical aberration using two spherical lenses. (He would improve upon his solution later.) While it is likely Spinoza did not know of Huygens’s success - as he kept his discoveries close to the vest – Spinoza most certainly had discussed with Huygens, and seemingly read in the draft of the Dioptrics itself, the theoretical nature of the pursuit. He likely understood the issue that was at hand, most particularly the failure of Descartes to provide a practical solution to the problem of aberration, and how the Law of refraction itself might be used to overcome this aberration. It is noteworthy that Spinoza denies the possibility of Huygens’s aim in the context of his working solely spherical figures.

Given these factors of a theoretical correspondence with Hudde, the reading of Hudde’s treatise, and also Spinoza’s personal exchange with Huygens, the closer translation of letter 39 attests to a proficiency of optical knowledge far greater than what Alan Gabbey allows. Spinoza did seem to be more than familiar with the “importance” of Descartes figures, in particular, with the disappointments of that importance, at least as far as they were understood in his day. Central though to the issue of the “if” and the subjunctives that follow, is that the conditional itself assumes Spinoza’s own “mechanical point” (Hudde) notion of the focus of a lens. When Spinoza says, if circle ABCD has the property that certain parallel rays would meet at point B, he means at “mechanical point” B. Granting that they meet at such a mechanical point, other such rays must meet in a similar fashion at other points, as due to the nature of a circle, and this could not be said of Descartes’ other figures.

[For a Full Treatment of Spinoza's Letters 39 and 40: Deciphering Spinoza’s Optical Letters ]


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