Frames /sing


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.

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

  1. josé Médina August 14, 2012 at 5:34 pm

    Thank you for your precisions about Gabbey’s comments.
    I wouls like to give you an alternative answer to the question of a possible source of Spinoza’s conception of “cones of rays” and also anti cartesian point of view aout spherical lenses.
    Just see Thomas Hobbes Elementa philosophiae pars II de Homine 1658, chap. 8. and 9.

    • Kevin von Duuglas-Ittu August 14, 2012 at 6:08 pm

      Thank you for reading closely josé. I am somewhat removed from this study of mine by time, but still am interested. I don’t have the Hobbes source at hand. My two questions would be:
      1. Could Hobbes have had Kepler as a source as well?
      2. Does Hobbes have reference to the imperfect construction of the human eye as Kepler does?

      • josé Médina August 19, 2012 at 8:27 am

        As Descartes says that in optics, Kepler has been his master, i think that all “modern” thinkers in the early XVIIth have approved the keplerian idea that vision is made on the very retina and not on the pupill?. The cone of ray is an ancient idea from Euclid. The real novelty is the retinal role in vision. Then Two possibilities: either the cartesian thesis : the soul sees. either the hobbesian the body sees. that is in this context thatHobbes tried to explain the phenomenon of vision without idealistic dualism.
        What I wanted to suggest is that when Ferrier recognised his failure in performing the anaclastic lens, the problem was : is optics a pure mathematical science or a physical one? and Hobbes suggest that spherical lenses are anyway better than an impossible hyperbolial one because the focus is a physical point not a mathematical one.
        By the way, Hobbes has read Kepler and discussed his thesis in an optical latin manuscript (Harl 6796) an another english manuscript harl 3360)recently visible on the British Library digitalzed manuscript.

      • Kevin von Duuglas-Ittu August 19, 2012 at 8:37 am

        Very interesting José. Then both Spinoza and Hudde – a Spinoza collaborator I discuss who uses the term “mechanical point” for the point of focus (is this a Hobbesian term?) are following Hobbes. Though it still seems that Spinoza has in mind Kepler when speaking about how imperfect the eye is, unless this is Hobbes notion as well.

      • josé Médina August 20, 2012 at 3:17 am

        It is precisely because Kepler corrects the form of the eye and thinks ti an hyperbola that Hobbes in his manuscript says: as there is some discussions about the form of the eye and nothing is too sure without adequete means of observation, I shall continue supposing the eye circular but I know that many people have different configutations. So Evidently, Kepler remains the Reference read by bith Spinoza and Hobbes.
        By the way, note tha it is the same Jarig Jelles who asks Spinoza the difference between Hobbes and Spinoza. ( letter 50) we are on a political matter but the idea to compare Hobbes with Spinoza shows that ther is a connexion in the Jelles/ Spinoza discussion.
        Best regards

      • Kevin von Duuglas-Ittu August 20, 2012 at 3:31 am

        Great references José. I do think it adds a lot to bring Hobbes into the Spinoza optical discussion, thank you for that. As to the braiding of the political and the optical I have seen that Leibniz himself seemed to make an rather curious “optical” reply to Spinoza’s TPT: . It appears that all authors were struggling with the power of the spherical trope. There was a sense in the era by which clarity and confusion were reigning analogies. That is one reason why I find Spinoza’s resistance to optical metaphors in non-optical topics so notable. Perhaps this was do to the fact that he made lenses and instruments and did not just theorize about them. Thank you also for returning me to these earlier thoughts I had.

      • josé Médina August 20, 2012 at 6:17 am

        we are now returning to the importance in Spinoza’s thought of the imagination./ vision;
        i dont know whether you understand french, but this link might interest you.
        let me know,
        kind regards

    • Kevin von Duuglas-Ittu August 14, 2012 at 6:26 pm

      p.s. not sure if you found it, but this was my full treatment of these two letters, with some reference to Kepler:

  2. Kevin von Duuglas-Ittu August 20, 2012 at 6:26 am

    Alas José, I do not read French (!), but it sounds like an essay topic i would enjoy. I have always thought that Spinoza’s critique (as well as much less acknowledged “praise”) of the imagination is one of the most neglected parts of his Philosophy. After all he called mathematics (“number”) merely an “aid to the imagination”. He was able to both embrace imagination but also also critique it, and I wonder if his work in optics gave him a unique vantage point (to use an optical metaphor) to do so.

    • josé Médina August 20, 2012 at 6:48 am

      Cone of ray as i told you is a common place of classical greek optics. the problem is how ti interpret it. Medieval optics neglect totally oblique rays as the only one non refracted is the axe perpendicular. Kepler returns on the argument and notes that all the points send a cone of ray in order to make an image corresponding point by point on the bottom of the retina. Te revolution is double: to see is to see a picture inverted on the retina as in acamera obscura, and what we see is an image imago that Kepler leaves to physicians because to complicated. he then study geometrically the pictures at the bottom on the retina.
      Hobbes is a bit apart. and singular;
      Returning on the old medieval classical optics, he recuperates the idea of the prevalence of the perpendicular ray. on the other way, he accepts the keplerian invention of the retina picture. He then make a mixture of his own and sustains that we see by a rebound from the retina towards the object through a visusal line passing through the center of the eye, ( whence the importance that it souhl be a circle). In doing so, Hobbes succeed in non accepting the idealistic dualism of Descartes who says that it is the soul that sees and not our body. hobbes ‘s materialism gives a different explaination: it is our imaginations that sees. but the optical illusions are mathematically determined. nothing then arbitrary or free.
      Evidently hobbesian thesis is too odd and nobody retained it. Spinoza has a keplerian and cartesian standard abour the cone refracted in the eye tii the bottom of the retina.
      if you give me a e-mail adress ‘ll send you images.

      • Kevin von Duuglas-Ittu August 20, 2012 at 6:56 am

        sure. My email is kevin.vonduuglasittu AT

        Mostly I was concerned with appreciating just what Spinoza was describing his his two optical letters which either have been completely ignored or when considered deeply criticized: . This lead me to an investigation of the actual lenses and methods of grinding (and even instruments) Spinoza may have used, as it struck me that his techniques – given that they made up much of his daily activity – indeed may have influenced or informed his philosophy, for better or worse.

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