A history of Greek mathematics - Wilbourhall.org

162 APOLLONIUS OF PERGA
Similarly, if CM meets P'N' in K,
P'G 2 = N'G 2 + P'N' 2
= 2 AH'F'G + 2{AMKN')
= 2(AMHG) + 2AHH'K.
Therefore, by subtraction,
P'G 2 -PG 2
= 2AHH'K
= HI.(H'I±IK)
= HI. (HI± IK)
= HP
CA+AM
CA
- AW 2 AA '±P -
~
'
which proves the proposition.
AA'
If be any point on PG, OP is the minimum straight line
from to the curve, and 0P / increases as P' moves away from
P on either side; this is proved in V. 12. (Since P f G > PG,
Z GPP' > Z GP P f ;
therefore, a fortiori, Z OPP' > Z OP'P,
and OP' > OP.)
Apollonius next proves the corresponding propositions with
reference to points on the minor axis of an ellipse.
'
If p' be
the parameter of the ordinates to the minor axis, p'=AA' 2 /BB' ',
or i/= OA 2
/GB. If now E' be so taken that BE'=ip',
then BE' is the maximum straight line from E' to the curve
and, if P be any other point on it, E'P diminishes as P moves
farther from B on either side, and E'B' is the minimum
straight line from E' to the curve. It is, in fact, proved that
~ 6
f/
E'B 2 - E'P 2 = Bn 2 .
P
> where Bn is the abscissa of P
(V. 16-18). If be any point on the minor axis such that
BO > BE', then OB is the maximum straight line from to
the curve, &c. (V. 19).
If g be a point on the minor axis such that Bg > BG, but
Bg < f p\ and if Gn be measured towards B so that
Cn : ng = BB' :
p',
then n is the foot of the ordinates of two points P such that
Pg is the maximum straight line from g to the curve. Also,