[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 curve KDE was of such a nature that
having drawn from some point which had been assumed, such as K, the
straight lines KA, KB, the excess by which AK surpasses AD should be
to the excess of DB over KB, as 3 to 2. For it can similarly be
demonstrated, by taking any other point in the curve, such as G, that
the excess of AG over AD, namely VG, is to the excess of BD over DG,
namely DP, in this same ratio of 3 to 2. And following this principle
Mr. Des Cartes constructed these curves in his _Geometric_; and he
easily recognized that in the case of parallel rays, these curves
became Hyperbolas and Ellipses.
Let us now return to our method and let us see how it leads without
difficulty to the finding of the curves which one side of the glass
requires when the other side is of a given figure; a figure not only
plane or spherical, or made by one of the conic sections (which is the
restriction with which Des Cartes proposed this problem, leaving the
solution to those who should come after him) but generally any figure
whatever: that is to say, one made by the revolution of any given
curved line to which one must merely know how to draw straight lines
as tangents.
Let the given figure be that made by the revolution of some curve such
as AK about the axis AV, and that this side of the glass receives rays
coming from the point L. Furthermore, let the thickness AB of the
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