[Illustration]
These are the two cases in which Conic sections serve for refraction,
and are the same which are explained, in his _Dioptrique_, by Des
Cartes, who first found out the use of these lines in relation to
refraction, as also that of the Ovals the first of which we have
already set forth. The second oval is that which serves for rays that
tend to a given point; in which oval, if the apex of the surface which
receives the rays is D, it will happen that the other apex will be
situated between B and A, or beyond A, according as the ratio of AD to
DB is given of greater or lesser value. 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.
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