116 
CONIC SECTIONS 
at right angles) and A'A the axis of a rectangular hyperbola, 
P any point on the curve, PN the principal ordinate, draw 
PK, PK' perpendicular to the asymptotes respectively. Let 
PN produced meet the asymptotes in R, R'. 
Now, by the axial property, 
CA 2 = CN 2 -PN 2 
= RN 2 — PN 2 
= RP.PR' 
— 2 PK. PK', since ¿PRK is half a right angle ; 
therefore PK. PK' = \GA 2 . 
Works by Aristaeus and Euclid. 
If Menaechmus was really the discoverer of the three conic 
sections at a date which we must put at jibout 360 or 350 B.C., 
the subject must have been developed very rapidly, for by the 
end of the century there were two considerable works on 
conics in existence, works which, as we learn from Pappus, 
were considered worthy of a place, alongside the Conics of 
Apollonius, in the Treasury of Analysis. Euclid flourished 
about 300 B.C., or perhaps 10 or 20 years earlier; but his 
Conics in four books was preceded by a work of Aristaeus 
which was still extant in the time of Pappus, who describes it 
as £ five books of Solid Loci connected (or continuous, avvedi}) 
with the conics’. Speaking of the relation of Euclid’s Conics 
in four books to this work, Pappus says (if the passage is 
genuine) that Euclid gave credit to Aristaeus for his dis 
coveries in conics and did not attempt to anticipate him or 
wish to construct anew the s^me system. In particular, 
Euclid, when dealing with what Apollonius calls the three- 
and four-line locus, ‘ wrote so much about the locus as was 
possible by means of the conics of Aristaeus, without claiming 
completeness for his demonstrations ’. x We gather from these 
remarks that Euclid’s Conics was a compilation and rearrange 
ment of the geometry of the conics so far as known in his 
1 Pappus, vii, p. 678. 4.