180 
APOLLONIUS OF PERGA 
1 Pappus, vii, pp. 642-4. 
treatment of this problem in all its particular cases with their 
8iopL<y¡xoL could present no difficulty to Apollonius. 
If the two straight lines are parallel, the solution of the 
problem gives a means of drawing any number of tangents 
to an ellipse when two parallel tangents, their points of con 
tact, and the length of the parallel semi-diameter are given 
(see Conics, III. 42). In the case of the hyperbola (III. 43) 
the intercepts made by any tangent on the asymptotes contain 
a constant rectangle. Accordingly the drawing of tangents 
depends upon the particular case of our problem in which both 
fixed points are the intersection of the two fixed lines. 
(y) On determinate section (SuopLcrpirr] ro/xri), two Books. 
The general problem here is, Given four points A, B, C, D on 
a straight line, to determine another point P on the same 
straight line such that the ratio AP. CP : BP. DP has a 
given value. It is clear from Pappus’s account 1 of the contents 
of this work, and from his extensive collection of lemmas to 
the different propositions in it, that the question was very 
exhaustively discussed. To determine P by means of the 
equation 
AP. CP — A. BP. DP, 
where A, B, G, D, A are given, is in itself an easy matter since 
the problem can at once be put into the form of a quadratic 
equation, and the Greeks would have no difficulty in reducing 
it to the usual application of areas. If, however (as we may 
fairly suppose), it • was intended for application in further 
investigations, the complete discussion of it would naturally 
include not only the finding of a solution, but also the deter 
mination of the limits of possibility and the number of possible 
solutions for different positions of the point-pairs A, G and 
B, D, for the cases in which the points in either pair coincide, 
or in which one of the points is infinitely distant, and so on. 
This agrees with what we find in Pappus, who makes it clear 
that, though we do not meet with any express mention of 
series of point-pairs determined by the equation for different 
values of A, yet the treatise contained what amounts to a com-