A history of Greek mathematics - Wilbourhall.org

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A
174 APOLLONIUS OF PERGA
2AGPT: (GL) = V(CN. NT) :
GN
r a
= PN.^:GN (1.37,39)
= PN.GT:GT.GN .^ CA
= 2AGPT.CA.GB;
therefore (GL) = GA . GB.
The remaining propositions of the Book trace the variations
of different functions of the conjugate diameters, distinguishing
the maximum values, &c. The functions treated are the
following
p', the parameter of the ordinates to PP' in the hyperbola,
f
according as A is (1) not less than^j, the parameter corresponding
to A A', (2) less than p but not less than -|p, (3) less
than \p
(VII. 33-5).
PP f^p\ as compared with AA' ^p> in the hyperbola (VII. 36)
or the ellipse (VII. 37).
PP'+p' „ „ AA f +p in the hyperbola (VII.
38-40) or the ellipse (VII. 41).
PP'.p'* „ „ AA'.p in the hyperbola (VII. 42)
or the ellipse (VII. 43).
PP' 2 +p' 2 „ „ AA' 2 +p 2 in the hyperbola, according
as (1)
A A' is not less than
p, or (2) ^4J/ or < p f
(VII. 51).