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, and this with so much difficulty that we can always augment it more easily by one of the others, as you shall now discover. So let us see how nature has neglected it; for causing rays such as VKR and YKT to curve inward toward S on the surfaces BCD and 123… [for 123 see figure below] 

Lens P

…she [Nature] has made the image RST a bit smaller than if she had caused them to curve outward, as they do toward 5 on the surface 456, or if she had let them all be straight. We also need not consider the first of these three things, when the objects are not at all accessible: but, when they are, it is obvious that, the closer we look at them, the larger are the images that form at the back of our eyes; so that, inasmuch as Nature has not given us the means of looking at them closer than about half a foot or a foot away, in order to add to this all that we can through art, we need only interpose a lens such as that which is marked P, mentioned above, which causes all the rays coming from the closest possible point to enter into the eye as if they came from another point more distant. Now by this means, the most we can bring about is that there be only a twelfth or a fifteenth as much space between the eye and the object as would be there otherwise; so that the rays will come from the different points of this object, intersecting twelve or fifteen times closer to it – or even a little more, for they will no longer begin to cross on the surface of the eye, but rather on that of the lens, to which the object will be a bit closer – will form an image whose diameter will be twelve or fifteen times greater than it would be if we were not making use of this lens”

Perhaps here we see the issues at stake in Spinoza’s conception of the appropriateness of the spherical lens. Descartes invokes the exercise of bringing an object closer to the eye, and thereby increasing the size of the image-object on the retina, but does not mention that such a closer approximation would increase the size of the angle at the surface of the lens [is this Spinoza’s point?]. When Descartes then describes how the use of lens P accomplishes much the same type of action, here too is not mentioned that the size of the angle is increased during magnification.

Descartes continues:

“and as a result its surface will be approximately two hundred times greater, which will make the object appear approximately two hundred times more distinctly; by this means it will also appear much larger, not exactly two hundred times as large, but more so or less in proportion to how far away we judge it to be. For example, if in looking at the object X through the lens P, we dispose our eye C in the same way that it should be disposed in order to see another object twenty or thirty paces away from it, and if, having besides no knowledge of the location of this object X, we judge it to be truly at thirty paces, it will seem more than a million times greater than it is; thus it will be possible to turn a flea into an elephant, for it is certain that the image formed by a flea on the back of the eye, when the flea is so close it it, is no less great than that which is formed there by an elephant, when it is thirty paces away. And on this alone is founded the entire invention of these small flea glasses made of a single lens, whose use is now quite common everywhere…”

It is here in Descartes explication of magnification that he stumbles upon the single lens microscope, a likely device that Spinoza may have in mind, one that demands a spherical lens if only due to the extremely small glass pieces involved, and the glass-thread bead technique in making their objectives. Spinoza does not mention microscopes in his objection to Descartes, but one cannot help but think that he has in mind the entirely spherical balls of glass (sometimes ground into convex-convex shapes), which are used in their construction. If indeed this is the text that Spinoza – and Jelles – are investigating, the slide from telescope to microscope no doubt also occurred in Spinoza’s mind as well, as he seeks to argue for the superiority of the spherical lens. If tiny balls of glass [even drops of water] can work effectively as microscopic lenses, so much more does their ubiquitous efficacy seem to be enforced. It is these micro-spheres that are able to take rays from a wide range of angles, as the specimen is pressed right up against them, which despite distortions are able to produce dramatic magnification. And Descartes brings up the point of their shape right away:

“…although we have not yet discovered the true shape that they should have; and because we usually know that the object is quite near when we use them to look at it, it cannot appear as large as it would if we were to imagine it farther away.”

After a description of lensed magnification that would be capable in a filled-with-water tube, Descartes continues with a plan of filling the tube with a solid transparent body…

“Now, inasmuch as it would be very inconvenient to join water against the eye in the manner in which I have just explained, and even though since we cannot know precisely what the shape of the membrane BCD which covers it [see first diagram posted above], we would not know how to determine exactly that of the lens GHI, to substitute it instead; still, it will be better to make use of another invention, and by means of one or of many lenses or other transparent bodies – also enclosed in a tube but not joined to the eye so exactly that a little air does not remain between the two – to bring it about that from the entrance of this tube the rays coming from a single point of the object bend, or curve, in the way that is required in order to make them reconverge in another point, near the place where the middle of the back of the eye will be when this tube is placed in front of it. Then, again, these same rays, in coming out of this tube, bend and straighten themselves in a such a manner that they can enter into the eye the same as if they had not been bent at all, as if they merely came from some nearer place. And then, those which will come from various points, having intersected at the entrance of this tube, do not come apart at the exit, but go toward the eye in the same way as if they came from a larger or closer object. Thus, if the tube HF [see same diagram] is filled with a completely solid lens, whose surface GHI is of such a shape that it causes all the rays coming from point X, once in the lens, to tend towards S; and if it causes its other surface KM to bend them again in such a way that they tend from there towards the eye in the same way as if they came from the point x, which I assume to be so located that the lines xC and CS have between them the same proportion as XH and HS; then those which come from point V will necessarily intersect the rays from point x on the surface GHI, in such a way that, since they are already distant from them when they are at the other end of the tube, the surface KM will not be able to bring them together, especially if it is concave, as I suppose it to be; instead it will reflect them toward the eye, in nearly the same way as if they came from point Y. By means of this they will make the image RST as much larger  as the tube is longer, and there will be no need, in order to determine the shapes of the transparent bodies we will wish to use for this effect, to know exactly the shape of the surface BCD”

This is perhaps the contact point for Spinoza’s agreement with Jelles at the end of letter 39:

“So the case is as you describe; that is, if no account is taken of anything except the length 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.”

Lastly, Descartes ends up with the standard figure of the telescope:

“But, because there would again be some inconvenience  in finding lenses or other such bodies sufficiently thick to fill the entire tube HF, and sufficiently clear and transparent so that they would not impede the passage of light because of this, 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 KLM should cause. And on this alone is founded the entire invention of the telescopes composed of two lenses placed in the two ends of a tube, which gave me occasion to write this Treatise.”