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

THE CONICS, BOOK V 163
if P' be any other point on it, P'g diminishes as P / moves
farther from P on either side to B or B\ and
/"££' tM 2 -(7i? 2
2 J* '2 , 2
p g
2__p
2
g
= nn J____ 2 Qr ^2 __
If be any point on P# produced beyond the minor axis, PO
is the maximum straight line from to the same part of the
ellipse for which Pg is a maximum, i.e.
the semi-ellipse BPB\
&c. (V. 20-2).
In V. 23 it is proved that, if g is on the minor axis, and gP
a maximum straight line to the curve, and if Pg meets A A'
in G, then GP is the minimum straight line from G to the
curve ; this is proved by similar triangles. Only one normal
can be drawn from any one point on a conic (V. 24-6). The
normal at any point P of a conic, whether regarded as a
minimum straight line from G on the major axis or (in the
case of the ellipse) as a maximum straight line from g on the
minor axis, is perpendicular to the tangent at P (V. 27-30);
in general (1) if be any point within a conic, and OP be
a maximum or a minimum straight line from to the conic,
the straight line through P perpendicular to PO touches the
conic, and (2) if
0' be any point on OP produced outside the
conic, O'P is the minimum straight line from 0' to the conic,
&c. (V. 31-4).
Number of normals from a point.
We now come to propositions about two or more normals
meeting at a point. If the normal at P meet the axis of
a parabola or the axis AA' of a hyperbola or ellipse in G, the
angle PGA increases as P or G moves farther away from A,
but in the case of the hyperbola the angle will always be less
than the complement of half the angle between the asymptotes.
Two normals at points on the same side of A A' will meet on
the opposite side of that axis ;
and two normals at points on
the same quadrant of an ellipse as i5 will meet at a point
within the angle AGB f (V. 35-40). In a parabola or an
ellipse any normal PG will meet the curve again; in the
hyperbola, (1) if A A' be not greater than p, no normal can
meet the curve at a second point on the same branch, but
M 2
164 APOLLONIUS OF PERGA
(2) if A A' > p, some normals will meet the same branch again
and others not (V. 41-3).
If P 1
G v P 2
G 2
be normals at points on one side of the axis of
a conic meeting in 0, and if be joined to any other point P
on the conic (it being further supposed in the case of the
ellipse that all three lines 0P 1}
0P 2
, OP cut the same half of
the axis), then
(1) OP cannot be a normal to the curve;
(2)
is
if OP meet the axis in K, and PG be the normal at P, AG
less or greater than AK according as P does or does not lie
between P x
and P 2
.
From this proposition it is proved that (1) three normals at
points on one quadrant of an ellipse cannot meet at one point,
and (2) four normals at points on one semi-ellipse bounded by
the major axis cannot meet at one point (V. 44-8).
In any conic, if M be any point on the axis such that AM
is not greater than J^>,
and if be any point on the double
ordinate through M, then no straight line drawn to any point
on the curve on the other side of the axis from and meeting
the axis between A and M can be a normal (V. 49, 50).
Propositions leading immediately to the determination
of the evolute of a conic.
These great propositions are V. 51, 52, to the following
effect
If AM measured along the axis be greater than \p (but in
the case of the ellipse less than AG), and if MO be drawn perpendicular
to the axis, then a certain length (y, say) can be
assigned such that
(a) if OM > y, no normal can be drawn through which cuts
the axis ; but, if OP be any straight line drawn to the curve
cutting the axis in K, NK < NG, where PN is the ordinate
and PG the normal at P
;
(b) if OM = y, only one normal can be so drawn through 0,
and, if OP be any other straight line drawn to the curve and
cutt ing the axis in K, NK < NG, as before ;
(c) if 0M