A history of Greek mathematics - Wilbourhall.org

178 APOLLONIUS OF PERGA
the value of X in order that the solution may be possible.
Apollonius begins by stating the limiting case, saying that we
obtain a solution in a special manner in the case where M is
the middle point of CD, so that the rectangle CM .
AID or
CB' . AD has its maximum value.
The corresponding limiting value of X is determined by
finding the corresponding position of D or M.
We have
whence, since
B'C :MD = CM: AD, as before,
= B'M:MA;
MD = CM,
B'C:B'M = CM:MA
= B'M:B'A,
so that B'M 2 = B'C.B'A.
Thus M is found and therefore D also.
According, therefore, as X is less or greater than the particular
value of OC: AD thus determined, Apollonius finds no
solution or two solutions.
Further, we have
AD = B'A + B'C- (B'D + B'C)
= B'A + B'C-2B'M
= B'A + B'C- 2 VB'A .
B'C.
If then we refer the various points to a system of coordinates
in which B'A, B'N' are the axes of x and y, and if
we denote by (x, y) and the length B'A by h,
X = 00/AD = y/(h + x-2Vhx).
If we suppose Apollonius to have used these results for the
parabola, he cannot have failed to observe that the limiting
case described is that in which is on the parabola, while
N'OM is the tangent at ;
for, as above,
B'M : B'A
= B'C:B'M = N'O :
N'M,
by parallels,
so that B'A, N'M are divided at M,
proportion.
respectively in the same