A history of Greek mathematics - Wilbourhall.org

180 APOLLONIUS OF PERGA
treatment of this problem in all its particular cases with their
Swpio-fxoi 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 contact,
and the length of the parallel semi-diameter are given
(see Conies, 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 (SLoopicr/ievr] rofirj), two Books.
The general problem here is, Given four points A, B, G, 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 x 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.GP=\.BP.DP,
where A, B, C, 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 determination
of the limits of possibility and the number of possible
solutions for different positions of the point-pairs A, C 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-
1
Pappus, vii, pp. 642-4.