178 
APOLLONIUS OF PERGA 
the value of A 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. MB or 
GB'. AD has its maximum value. 
The corresponding limiting value of A is determined by 
finding the corresponding position of D or M. 
We have B'G:MD = CM: AD, as before, 
= B'M.MA ; 
whence, since MD = CM, 
B'G: B'M = GM-.MA 
= B'M: B'A, 
so that B'M 2 = B'G. B'A. 
Thus M is found and therefore D also. 
According, therefore, as A is less or greater than the par 
ticular value of OG: AD thus determined, Apollonius finds no 
solution or two solutions. 
Further, we have 
AD = B'A + B'G- (B'D + B'G) 
= B'A+B'G-2B'M 
= B'A + B'G— 2 VB'A . B'G. 
If then we refer the various points to a system of co 
ordinates in which B'A, B'JSG are the axes of x and y, and if 
we denote 0 by (x, y) and the length B'A by h, 
A = OG / AD = y/{h + x — 2 Vhx). 
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 0 is on the parabola, while 
N'OM is the tangent at 0 ; for, as above, 
B'M: B'A = B'G: B'M = N'O : N'M, by parallels, 
so that B'A, N'M are divided at M, 0 respectively in the same 
proportion.