A history of Greek mathematics - Wilbourhall.org

120 CONIC SECTIONS
Proof from Pappus.
The proofi in the case where the given ratio is different from
unity is shortly as follows.
Let S be the fixed point, SX the perpendicular from S on
the fixed line. Let P be any point on the locus and PN
+—
KAN SK,'
A' XV K'S
perpendicular to SX, so that SP is to NX in the given
ratio (e);
thus e 2 = (PN 2 + SN NX 2 ) :
Take K on SX such that
2 .
e 2 = SN 2 :NK 2 ;
then, if K f be another point on SN, produced if necessary,
such that
NK = NK',
e 2 : 1 = (PN 2 + SN 2 ) NX 2 = SN 2 : : NK
= PN 2 :(NX -NK 2 2 )
= PN XK 2 : . XK'.
2
The positions of N, K, K' change with the position of P.
If A, A' be the points on which N falls
with X respectively, we have
when K, K' coincide
SA.AX = SN: NK = e:l= SN:NK'= SA': A'X.
Therefore SX :
SA = SK :SN = (l+e):e,
whence (1 +e) :e = (SX-SK) :
= XK:AN
(SA
-SN)