A history of Greek mathematics - Wilbourhall.org

'
'
Similarly
R'M
THE CONICS, BOOK III 153
2
: JM' = 2 AR'F'M' : A JFM',
2
whence R'J . JV' : R'M = P/P'P' : A R'F'M'.
But iTilf' 2 : OQ = 2 AR'F'M' : A
therefore, ea; aequali, RJ . Jr' : OQ
OQE
;
= 2 FJRF' : A
OQE.
It follows, since PJE'P' = JU'UR, and AOP2 7 = AOQE,
that iU . JV :
or JBJT . Jr : EV
OP = 2 E'J" . Jr' : OQ
.
JV
= OP 2 : OQ
2 ,
2 .
If we had taken chords Ri\, P'r/ parallel respectively to
OQ, OP and intersecting in I, an internal or external point,
we should have in like manner
RI .
Ir t
: RI
.
Ir( = OQ 2 : OP
2 .
As a particular case, if PP r
be a diameter, and Rr, Rr' be
chords parallel respectively to the tangent at P and the
diameter PP' and intersecting in /, then (as is separately
proved)
RI.Ir:RI.Ir' = p:PP'.
The corresponding results are proved in the cases where certain
of the points lie on the conjugate hyperbola.
The six following propositions about the segments of intersecting
chords (III. 24-9) refer to two chords in conjugate
hyperbolas or in an ellipse drawn parallel respectively to two
conjugate diameters PP', DD' ', and the results in modern form
are perhaps worth quoting. If Rr, Rr' be two chords so
drawn and intersecting in 0, then
(a)
in the conjugate hyperbolas
RO.Or RO . Or' _
CP ± l CD ~ 2 2 '
and (RO 2 + Or 2 (RO 2 )
+ Or' 2 : )
= CP 2 : CD
2 ;
(b)
in the ellipse
R0 2 + 0r' 2 OP + _ 2 ' ~CW~ " 4 *
R0 2 + Or 2