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

THE COFICS, BOOK V 161
We have P'N' 2 = p
. AN' = 2 NG . AN'
;
and NG 2 = NN' 2 + NG 2 ±2NG. NN',
according to the position of N'.
Therefore P'G 2 = 2NG.AN+NG 2 + NN' 2
and the proposition is proved.
= PN 2 + NG 2 + NN' 2
= PG 2 + NN' 2 ;
(2) In the case of the central conic, take G on the axis such
that AG > \p, and measure GN towards A such that
NG:GN = p:AA / .
Draw the ordinate PN through N, and also the ordinate P'N'
from any other point P'.
We have first to prove the lemma (V. 1, 2, 3) that, if AM be
drawn perpendicular to A A' and equal to \p, and if CM,
produced if necessary, meet PN in H, then
PN = 2
2 (quadrilateral MANE).
This is easy, for, if AL(= 2AM) be the parameter, and A'L
meet PN in R, then, by the property of the curve,
PN = AN.NR
2
= AN(NH + AM)
= 2 (quadrilateral MANH).
Let GH, produced if necessary, meet P'N' in H'. From H
draw HI perpendicular to P'H'
Now, since, by hypothesis, NG :
NH = NG, whence also
H'N' = N'G.
GN
— p:AA'
= AM:AC
= HN:NC\
Therefore NG 2 = 2AHNG, N'G 2 = 2 A H'N' G.
And PN 2 = 2(MANH);
therefore PG = NG 2 2 + PN 2 = 2 (AMHG).
1523.2 M
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)
CA+AM
= HP
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,