or these two
constructions, as may be easily known, come back to the first one
which was shown before. And it is manifest by the last method that
this curve is the same that Mr. Des Cartes has given in his Geometry,
and which he calls the first of his Ovals.

It is only a part of this oval which serves for the refraction,
namely, the part DK, ending at K, if AK is the tangent. As to the,
other part, Des Cartes has remarked that it could serve for
reflexions, if there were some material of a mirror of such a nature
that by its means the force of the rays (or, as we should say, the
velocity of the light, which he could not say, since he held that the
movement of light was instantaneous) could be augmented in the
proportion of 3 to 2. But we have shown that in our way of explaining
reflexion, such a thing could not arise from the matter of the mirror,
and it is entirely impossible.

[Illustration]

[Illustration]

From what has been demonstrated about this oval, it will be easy to
find the figure which serves to collect to a point incident parallel
rays. For by supposing just the same construction, but the point A
infinitely distant, giving parallel rays, our oval becomes a true
Ellipse, the construction of which differs in no way from that of the
oval, except that FC, which previously was an arc of a circle, is here
a straight line, perpendicular to DB. For the wave of light DN, being
likewise represented by a straight line, it will be seen that all the
points of this wave, travelling as far as the surface KD along lines
parallel to DB, will advance subsequently towards the point B, and
will arrive there at the same time. As for the Ellipse which served
for reflexion, it is evident that it will here become a parabola,
since its focus A may be regarded as infinitely distant from the
other, B, which is here the focus of the parabola, towards which all
the reflexions of rays parallel to AB tend. And the demonstration of
these effects is just the same as the preceding.

But that this curved line CDE which serves for refraction is an
Ellipse, and is such that its major diameter is to the distance
between its foci as 3 to 2, which is the proportion of the refraction,
can be easily found by the calculus of Algebra. For DB, which is
given, being called _a_; its undetermined perpendicular DT being
called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
-2ay + yy_). But the nature of the curve is such that 2/3 o