A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

To prove (1)
R'l 2 : IW-H'Q 2 : QH
THE CONICS, BOOK III 155
we have
= 2 AH'F'Q :
AHFQ
= H'TU'R' :
Also R'T 2 : TR = 2 R'U' 2 : UR = AR'U'W 2 :
and jRT a : Ti? 2 = TW 2 : TW = ATH'W 2 :
HTUR
(III. 2, 3, &c).
A220TT,
A TWIT,
so that R'T 2 :TR 2 = ATH'W' - AE'CHF: ATi^TF- A-RETTF
= H'TU'R':HTUR
= R'l 2 : i7? 2 , from above.
To prove (2) we have
RV 2 : 7iT 2 = RU 2 : R'U' 2 = ARUW: AR'U'W,
and also
= HQ 2 : QH' = AHFQ 2 :
AH'F'Q
= HTUR * :
H'TU'R',
so that
RV 2 : VR' 2 = HTUR + ARUWiH'TU'R' + AR'U'W
= ATHWiATHW
= TF 2 : TIP
= RO 2 : OR'
2
2 .
Props. III. 30-6 deal separately with the particular cases
in which (a) the transversal is parallel to an asymptote of the
hyperbola or (6) the chord of contact is parallel to an asymptote,
i.e. where one of the tangents is an asymptote, which is
the tangent at infinity.
Next we have propositions about intercepts made by two
tangents on a third : If the tangents at three points of a
parabola form a triangle, all three tangents will be cut by the
points of contact in the same proportion (III. 41) ;
if the tangents
at the extremities of a diameter PP' of a central conic
are cut in r, r' by any other tangent, Pr . P'r' = CD 2 (III. 42)
if
the tangents at P, Q to a hyperbola meet the asymptotes in
* Where a quadrilateral, as HTUR in the figure, is a cross-quadrilateral,
the area is of course the difference between the two triangles
which it forms, as HTW ^ RUW.
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