A history of Greek mathematics - Wilbourhall.org

146 APOLLONIUS OF PERGA
(2) In the case of the central conic, we have
AR'UW = ACF'W - AGPE.
(Apollonius here assumes what he does not proye till III. 1,
namely that AGPE = ACQT. This is proved thus.
We have GV: GT = GV 2 : CP
therefore
so that
AGQV: ACQT = ACQV: AGPE,
ACQT = AGPE.)
Therefore AR'UW' = ACF'W' * ACQT,
and it is easy to prove that in all cases
Therefore R'M .
AR'MF'=QTUM.
MF' = QM(QT + MU).
2
;
(I. 37, 39.)
Let QL be drawn at right angles to CQ and equal to p'.
Join Q'L and draw MK parallel to QL to meet Q'L in K.
Draw CH parallel to Q'L to meet QL in H and MK in N.
Now
But QT :
so that
RfM: MF' = OQ:QE
— QL : 2 QT, by hypothesis,
MU
= QH:QT.
= CQ : CM = QH: MN,
(QH + MN) :{QT + MU) = QH:QT
= R'MiMF', from above.
where i)a
is the parameter of the principal ordinates and p the parameter
of the ordinates to the diameter
PV.
If the tangent at the vertex A meets
VP produced in E, and PT, the tangent
at P, in 0, the proposition of Apollonius
proves that
But
therefore
Thus
0P:PE = p:2PT.
OP _ i
PT 2
=p.PE
= p.AN.
QV 2 : QD' 1 = PT 2 : PN
= p
.
2 , by similar triangles,
AN:p u
. AN