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

THE CONICS, BOOKS VI, VII 169
The same is true if A A' is the minor axis of an ellipse and p
the corresponding parameter (VII. 2, 3).
If AA' be divided at H' as well as H (internally for the
hyperbola and externally for the ellipse) so that i^is adjacent
to A and H' to A', and if A'H: AH = AH' : A'H' = AA' :p,
the lines AH, A'H' (corresponding to p in the proportion) are
called by Apollonius homologues, and he makes considerable
170 APOLLONIUS OF PERGA
Secondly, Apollonius proves that, if PN be a principal
ordinate in a parabola, p the principal parameter, p' the
parameter of the ordinates to the diameter through P, then
p' = p + 4Al¥ (VII. 5); this is proved by means of the same
property as VII. 4, namely \p' - :PT' OP : PE.
Much use is made in the remainder of the Book of two
points Q and where AQ is drawn parallel to the conjugate
My
diameter CD to meet the curve in Q, and M is the foot of
the principal ordinate at Q
; since the diameter GP bisects
both A A! and QA, it follows that A'Q is parallel to GP.
Many ratios between functions of PP', DD' are expressed in
terms of AM, A'M, MH, MH', AH, A'H,&c. The first propositions
of the Book proper (VII. 6, 7) prove, for instance,
that PP' 2 : DD' = 2 ME': MH.
For PT 2 : CD 2 = NT:GN = AM: A'M, by similar triangles.
Also GP 2 :PT 2 = A'Q 2 :AQ 2 .
use of the auxiliary points H, H' in later propositions from
VII. 6 onwards. Meantime he proves two more propositions,
which, like VII. 1-3, are by way of lemmas. First, if CD be
the semi-diameter parallel to the tangent at P to a central
conic, and if the tangent meet the axis A A' in T, then
PT 2 : CD 2 = NT: CN. (VII. 4.)
Draw AE, TF at right angles to CA to meet CP, and let AE
meet PT in 0.
to CP, we have
or
Therefore PT 2 : CD
Then, if p' be the parameter of the ordinates
ip':PT=OP:PE (I. 49,50.)
= PT:PF,
*y .PF=PT\
2 = \p' . PF:£p'.
= PF: CP
= NT:GN.
CP
Therefore, ex aequali,
2 )
GP 2 : GD 2 = (AM : A'M) x (A'Q 2 : AQ
= (AM: A'M) x (A'Q 2 : A'M. MH')
x (A'M.MH': AM. MH) x (AM.MH : AQ 2 )
= (AM: A'M) x (AA': AH') x (A'M: AM)
x (MH':MH) x (A'H:AA'), by aid of VII. 2, 3.
Therefore PP' 2 : DD' 2 = MH' : MH.
Next (VII. 8, 9, 10, 11) the following relations are proved,
namely
(\)AA' 2 :(PP' + DD'f = A'H.MH':{MH'+V(MH.MH')} 2 ,
(2) AA' 2 : PP' .DD' = A'H : V(MH.MH'),
(3) A A' 2 : (PP' 2 + DD' 2 )
= A'H : MH± MH'.
The steps by which these results are obtained are as follows.
First, A A' 2 : PP' = A'H 2 : MH' (oc)
(This is proved thus
= A'H.MH':MH' 2 .
AA' 2 :PP' 2 =GA 2 :GP 2
= GJST.CT:CP 2
= A'M. A'A : A'Q
2 .