And in this latter case it is the same as that which Des Cartes calls his 3rd oval.
Now the finding and construction of this second oval is the same as that of the first, and the demonstration of its effect likewise.

But it is worthy of remark that in one case this oval becomes a perfect circle, namely when the ratio of AD to DB is the same as the ratio of the refractions, here as 3 to 2, as I observed a long time ago.

The 4th oval, serving only for impossible reflexions, there is no need to set it forth.
[Illustration] As for the manner in which Mr.Des Cartes discovered these lines, since he has given no explanation of it, nor any one else since that I know of, I will say here, in passing, what it seems to me it must have been.

Let it be proposed to find the surface generated by the revolution of the curve KDE, which, receiving the incident rays coming to it from the point A, shall deviate them toward the point B.Then considering this other curve as already known, and that its apex D is in the straight line AB, let us divide it up into an infinitude of small pieces by the points G, C, F; and having drawn from each of these points, straight lines towards A to represent the incident rays, and other straight lines towards B, let there also be described with centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at L, M, N, O; and from the points K, G, C, F, let there be described the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and let us suppose that the straight line HKZ cuts the curve at K at right-angles.
[Illustration] Then AK being an incident ray, and KB its refraction within the medium, it needs must be, according to the law of refraction which was known to Mr.Des Cartes, that the sine of the angle ZKA should be to the sine of the angle HKB as 3 to 2, supposing that this is the proportion of the refraction of glass; or rather, that the sine of the angle KGL should have this same ratio to the sine of the angle GKQ, considering KG, GL, KQ as straight lines because of their smallness.
But these sines are the lines KL and GQ, if GK is taken as the radius of the circle.

Then LK ought to be to GQ as 3 to 2; and in the same ratio MG to CR, NC to FS, OF to DT.