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 + ^AN (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 M, where AQ is drawn parallel to the conjugate 
diameter CD to meet the curve in Q, and M is the foot of 
the principal ordinate at Q ; since the diameter CP bisects 
both AA! and QA, it follows that A'Q is parallel to CP. 
Many ratios between functions of PP', DD' are expressed in 
terms of AM, A'M, MH, MB', AH, A'H,kc. The first pro 
positions of the Book ? proper (YIL 6, 7) prove, for instance, 
that PP' 2 : DD' 2, = MH':MH. 
For PT 2 : CD 2 == NT : GN = AM : A'M, by similar triangles. 
Also CP 2 : PT 2 = A'Q 2 : A Q 2 . 
Therefore, ex aequali, 
CP 2 : CD 2 = {AM : A'M) x (A'Q 2 : AQ 2 ) 
= (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 YIL 2, 3. 
Therefore PP' 2 ; DD' 2 = MH' : MH. 
Next (VII. 8, 9, 10, 11) the following relations are proved, 
namely 
(1) AA' 2 :(PP' + DD') 2 =A'H.MH':{MH'+ V(MH.MH')} 2 , 
(2) AA' 2 : PP'. DD' = A'H : V(MH. MH')~ 
(3) AA' 2 : (PP' 2 + DD' 2 ) = A'H : MH+ MH'. 
The steps by which these results are obtained are as follows. 
First, AA' 2 : PP' 2 = A'H : MH' (a) 
= A'H.MH':MH' 2 . 
(This is proved thus : 
AA' 2 : PP' 2 = GA 2 :GP 2