A history of Greek mathematics - Wilbourhall.org

A
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
ordinate. Draw 00 parallel
to HK. We have at once H,
M N
PN = HN.NK ^
2
—- mn. MK —MN 2 m±y
2
Fjg g
Q
P v
\
= 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.
R
n
A
Fig. 4.
If (Fig. 4) CR, CB! be the asymptotes (which are therefore
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