A history of Greek mathematics - Wilbourhall.org

156 APOLLONIUS Ot PERGA
L, 1/ and M, M' respectively, then L'M, LM' are both parallel
to PQ (III. 44).
The first of these propositions asserts that, if the tangents at
three points P, Q, R of a parabola form a triangle pqr, then
Pr :rq = tQ: Qp = qp :pR.
From this property it is easy to deduce the Cartesian
equation of a parabola referred to two fixed tangents as
coordinate axes. Taking qR, qP as fixed coordinate axes, we
find the locus of Q thus. Let x, y be the coordinates of Q.
Then, if qp = x Y ,
qr = yv qR — h, qP — k, we have
s „ rQ == VvzV „ k -Vi = x i
x x
-x ~ Qp y 2/1
h-x x
'
From these equations we derive
x x
— hx, y* — ky ;
also, since — = ^x a we have f-
—
x
2/i-2/
x i 2/i
= 1.
By substituting for x 1} y x
the values V(hx), V(ky) we
obtain
©'+©'-•
The focal properties of central conies are proved in
III. 45-52 without any reference to the directrix ; there is
no mention of the focus of a parabola. The foci are called
'
the points arising out of the application ' (ra e/c rrjs irapafio\r)s
ytuo/xeua arj/ieTa), the meaning being that 8, S' are taken
on the axis AA' such that AS.SA' = AS'.S'A' = \pa .AA'
or CB 2 , that is, in the phraseology of application of areas,
a rectangle is applied to A A' as base equal to one-fourth
part of the ' figure ', and in the case of the hyperbola exceeding,
but in the case of the ellipse falling short, by a
square figure. The foci being thus found, it is proved that,
if the tangents At, A'r' at the extremities of the axis are met
by the tangent at any point P in r, v' respectively, rr' subtends
a right angle at S, S', and the angles rr'S, A'r'S' are equal, as
also are the angles rV/S", ArS (III. 45, 46). It is next shown
that, if be the intersection of r>S r/ , r'S, then OP is perpendicular
to the tangent at P (III. 47).
These propositions are