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

MENAECHMUS'S PROCEDURE 115
For, let the right-angled cone HOK (Fig. 3) be cut by a
plane
through A'AN parallel
to the axis OM and cutting the
sides of the axial triangle HOK
in A f , A, JV" respectively. Let
P be the point on the curve
for which PN is the principal
A
ordinate. Draw 00 parallel
P v
to HK. We have at once \
H,
M N
PN = HN.NK ^
2
—- mn. MK —MN 2 m±y
2
Fjg g
= CN 2 -CA 2 ,
since MK = OM, and MN = 0(7= 0^.
This is the property of the rectangular hyperbola having A'
as axis. To obtain a particular rectangular hyperbola with
axis of given length we have only to choose the cutting plane
so that the intercept A 'A may have the given length.
But Menaechmus had to prove the asymptote-property of
his rectangular hyperbola. As he can hardly be supposed to
have got as far as Apollonius in investigating the relations of
the hyperbola to its asymptotes, it is probably safe to assume
that he obtained the particular property in the simplest way,
i. e. directly from the property of the curve in relation to
its axes.
Fig. 4.
If (Fig. 4) CR, CB! be the asymptotes (which are therefore
12
R
Q
n
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.