A history of Greek mathematics - Wilbourhall.org

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160 APOLLONIUS OF PERGA
Next Apollonius takes points G on the axis at a distance
from A greater than ^p, an(^ ne proves that the minimum
straight line from G to the curve (i.e. the normal) is GP,
where P is such a point that
(1) in the case of the parabola NG = \p ;
(2) in the case of the central conic NG :
GN = p. A A' ;
and, if P' is any other point on the conic, P'G increases as P f
moves away from P on either side ; this is proved by showing
that
( 1
for the parabola P'G 2 = PG 2 + NN' 2 ;
(2) for the central conic P'G 2 = PG 2 + NN /2 .
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As these propositions contain the fundamental properties of
the subnormals, it is worth while to reproduce Apollonius's
proofs.
(1) In the parabola, if G be any point on the axis such that
Let PN be
AG > %p, measure GN towards A equal to \p.
the ordinate through N, P / any other point on the curve.
Then shall PG be the minimum ^line from G to the curve, &c.