ough the centres of the same ellipses,
and if through these centres there are drawn also the diameters LD,
PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
to the aforesaid BK, ON, AR. And because the three ellipses are
similar and similarly disposed, and have their diameters LD, PP, QQ
parallel, it is certain that their conjugate diameters BK, ON, AR,
will also be parallel. And the centres K, N, R being, as has been
stated, in one and the same diameter of the spheroid, these parallels
BK, ON, AR will necessarily be in one and the same plane, which passes
through this diameter of the spheroid, and, in consequence, the points
R, O, A are in one and the same ellipse made by the intersection of
this plane. Which was to be proved. And it is manifest that the
demonstration would be the same if, besides the points O, A, there had
been others in which the spheroid had been touched by planes parallel
to the straight line BM.

CHAPTER VI

ON THE FIGURES OF THE TRANSPARENT BODIES

Which serve for Refraction and for Reflexion

After having explained how the properties of reflexion and refraction
follow from what we have supposed concerning the nature of light, and
of opaque bodies, and of transparent media, I will here set forth a
very easy and natural way of deducing, from the same principles, the
true figures which serve, either by reflexion or by refraction, to
collect or disperse the rays of light, as may be desired. For though I
do not see yet that there are means of making use of these figures, so
far as relates to Refraction, not only because of the difficulty of
shaping the glasses of Telescopes with the requisite exactitude
according to these figures, but also because there exists in
refraction itself a property which hinders the perfect concurrence of
the rays, as Mr. Newton has very well proved by experiment, I will yet
not desist from relating the invention, since it offers itself, so to
speak, of itself, and because it further confirms our Theory of
refraction, by the agreement which here is found between the refracted
ray and the reflected ray. Besides, it may occur that some one in the
future will discover in it utilities which at present are not seen.

[Illustration]

To proceed then to these figures, let us suppose first that it is
desired to find a surface CDE which shall reassemble at a point B rays
coming from another point A; and that the summit of the surface shall
be the g