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 of 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.
83