A history of Greek mathematics - Wilbourhall.org

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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