A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE CONICS, BOOKS IV-V 159
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 show-
160 APOLLONIUS OF PERGA
most remarkable of the extant Books. It deals with normals
Next Apollonius takes points G on the axis at a distance
to conies regarded as maximum and minimum straight lines
from A greater than ^p, an(^ ne proves that the minimum
drawn from particular points to the curve. Included in it are
straight line from G to the curve (i.e. the normal) is GP,
a series of propositions which, though worked out by the
where P is such a point that
purest geometrical methods, actually lead immediately to the
(1) in the case of the parabola NG =
determination of the evolute of each of the three conies ; that
\p ;
is to say, the Cartesian equations to the evolutes can be easily
(2) in the case of the central conic NG : GN = p. A A' ;
deduced from the results obtained by Apollonius. There can
be no doubt that the Book is almost wholly original, and it is
a veritable geometrical tour de force.
ing that
Apollonius in this Book considers various points and classes
for the parabola
of points with reference to the maximum or minimum straight
( 1
P'G = 2 PG 2 + NN' 2 ;
lines which it is possible to draw from them to the conies,
(2) for the central conic P'G = 2 PG 2 + NN
AA
/2 V '
.
.
j;
.A XL
generally and for certain particular cases that, if in an ellipse
or a hyperbola AM be drawn at right angles to AA' and equal
to J p, and if CM meet the ordinate PN of any point P of the
curve in H, then PN = 2 2 (quadrilateral MANH)
;
this is a
L^R
A on either side ; he proves in fact that, if PN be the ordinate
J P
from P,
(1) in the case of the parabola PE 9 - = AE 2 + AN 2 ,
(2) in the case of the hyperbola or ellipse
PE 2 = A&
AA
+ AN* P
•
A r, ,
AA
where of course p = BB' 2 /AA\ and therefore (AA / ±p) / A A'
As these propositions contain the fundamental properties of
is equivalent to what we call e 2 , the square of the eccentricity.
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
AG > %p, measure GN towards A equal to \p. Let PN be
OP increases as P moves farther from A (V. 7).
i. e. as the feet of normals to the curve. He begins naturally
with points on the axis, and he takes first the point E where
AE measured along the axis from the vertex A is \p, p being
the principal parameter. The first three propositions prove
lemma used in the proofs of later propositions, V. 5, 6, &c.
Next, in V. 4, 5, 6, he proves that, if AE = \p, then AE is the
minimum straight line from E to the curve, and if P be any
other point on it, PE increases as P moves farther away from
It is also proved that EA' is the maximum, straight line from
E to the curve. It is next proved that, if be any point on
the axis between A and E, OA is the minimum straight line
from to the curve and, if P is any other point on the curve,
the ordinate through N, P / any other point on the curve.
Then shall PG be the minimum ^line from G to the curve, &c.