A history of Greek mathematics - Wilbourhall.org

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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 .
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' =GA 2 2 :GP 2
= GJST.CT:CP 2
= A'M. A'A :
A'Q
2 .