f TC
together with CB is equal to DB, as was stated in the last
construction: then the equation will be between _(2/3)y + sqrt(xx + aa
- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
seen that DC is an ellipse, of which the axis DO is to the parameter
as 9 to 5; and therefore the square on DO is to the square of the
distance between the foci as 9 to 9 - 5, that is to say 4; and finally
the line DO will be to this distance as 3 to 2.

[Illustration]

Again, if one supposes the point B to be infinitely distant, in lieu
of our first oval we shall find that CDE is a true Hyperbola; which
will make those rays become parallel which come from the point A. And
in consequence also those which are parallel within the transparent
body will be collected outside at the point A. Now it must be remarked
that CX and KS become straight lines perpendicular to BA, because they
represent arcs of circles the centre of which is infinitely distant.
And the intersection of the perpendicular CX with the arc FC will give
the point C, one of those through which the curve ought to pass. And
this operates so that all the parts of the wave of light DN, coming to
meet the surface KDE, will advance thence along parallels to KS and
will arrive at this straight line at the same time; of which the proof
is again the same as that which served for the first oval. Besides one
finds by a calculation as easy as the preceding one, that CDE is here
a hyperbola of which the axis DO is 4/5 of AD, and the parameter
equal to AD. Whence it is easily proved that DO is to the distance
between the foci as 3 to 2.

[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 an