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

THE COmCS, BOOK I 145
or CQ in the case of the central conic, bisects all chords as
RR' parallel to the tangent at Q. Consequently EQ and CQ
are diameters of the respective conies.
In order to refer the conic to the new diameter and the
corresponding ordinates, we have only to determine the parameter
of these ordinates and to show that the property of the
conic with reference to the new parameter and diameter is in
the same form as that originally found.
The propositions I. 49, 50 do this, and show that the new
parameter is in all the cases p', where (if is the point of
intersection of the tangents at P and Q)
(l)
whence
0Q:QE = p':2QT.
In the case of the parabola, we have TP = PV = EQ,
AEOQ = APOT.
Add to each the figure POQF'W ';
therefore QTW'F' = CJEW ' = AR'UW,
whence, subtracting MUW'F' from both, we have
Therefore R'M .
But R'M : MF'
therefore R'M 2 : R'M
AR'MF' = ejQU.
M F' = 2QT. QM.
= 0Q:QE = p':2 QT, by hypothesis
.
And R'M. MF' = 2QT .
MF'
= p' . QM
:2QT. QM.
QM, from above ;
therefore R'M 2 = p' . QM,
which is the desired property. 1
1
The proposition that, in the case of the parabola, if p be the parameter
of the ordinates to the diameter through Q, then (see the first figure
on p. 142)
0Q:QE = p:2QT
has an interesting application ;
for it enables us to prove the proposition,
assumed without proof by Archimedes (but not easy to prove otherwise),
that, if in a parabola the diameter through P bisects the chord QQ' in V,
and QD is drawn perpendicular to PV, then
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,
= QH:QT.
MU = 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
1523.2 L
QV*:QD* = p:pa ,