A history of Greek mathematics - Wilbourhall.org

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
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