d 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. Then also the sum of all the
antecedents to all the consequents would be as 3 to 2. Now by
prolonging the arc DO until it meets AK at X, KX is the sum of the
antecedents. And by prolonging the arc KQ till it meets AD at Y, the
sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
Whence it would appear that the