Spinoza: Letter 40 and Letter 39

Below the pertinent, optically related passages from letter 40, for reference sake. Spinoza in this letter attempts to answer a follow-up question posed to him by his friend Jarid Jelles, regarding an answer he had already supplied in Letter 39. Both letters from Jelles are lost. The ultimate subject I take to be the seventh discourse of Descartes La Dioptrique (that passage here). The translation is Shirley’s:

Spinoza’s 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. But they are considered to be so because the object is so far from us that the aperature of the telescope, in comparison with its distance, can be considered as no more than a point. 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. 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. For from any point on an object a line can be drawn passing through the center of a circle, although for that purpose the aperature 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.

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 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.

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

Voorburg, 25 March 1667

 

And here is an English Translation of Letter 39, to which the above letter has followed:

Letter 39

To the most humane and sagacious Jarig Jelles, from B.d.S.

[The Original, written in Dutch, is lost. It may be the text reproduced in the Dutch edition of the O.P. The Latin is a translation.]

Most humane Sir,

Various obstacles have hindered me from replying any sooner to your letter. 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.

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. 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. 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. If you desire to see the proof of this I am ready to send it to you whenever you wish.

Voorburg, 3 March 1667

[trans. Samuel Shirley altered]

And the Latin Text of both Letters 39 and 40 here