A history of Greek mathematics - Wilbourhall.org

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'
= 2PK. PK', since IPRK is half a right angle ;
therefore PK.PK' = \ CA 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 about 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
conies in existence, works which, as we learn from Pappus,
were considered worthy of a place, alongside the Conies of
Apollonius, in the Treasury of Analysis. Euclid flourished
about 300 B.C., or perhaps 10 or 20 years earlier; but his
Conies 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, crvve^rj)
with the conies \ Speaking of the relation of Euclid's Conies
in four books to this work, Pappus says (if the passage is
genuine) that Euclid gave credit to Aristaeus for his discoveries
in
conies and did not attempt to anticipate him or
wish to construct anew the same system. In particular,
Euclid, when dealing with what Apollonius calls the threeand
four-line locus, ' wrote so much about the locus as was
possible by means of the conies of Aristaeus, without claiming
completeness for his demonstrations \* We gather from these
remarks that Euclid's Conies was a compilation and rearrangement
of the geometry of the conies so far as known in his
1
Pappus, vii, p. 678. 4.