MENAECHMUS’S PROCEDURE 
115 
For, let the right-angled cone HOK (Fig. 3) he cut by a 
plane through A'AN parallel 
to the axis OM and cutting the 
sides of the axial triangle HOK 
in A', A, N respectively. Let 
P be the point on the curve 
for which PN is the principal 
ordinate. Draw OC parallel 
to HK. We have at once 
\ 
PN 2 = HN. NK 
= CN 2 — CA 2 , since MK = OM, and MN = 00 = CA. 
This is the property of the rectangular hyperbola having A'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 axés. 
R 
A' 
c 
p 
R 
N 
Fig. 4. 
If (Fig. 4) OR, OR' be the asymptotes (which are therefore 
i 2