A history of Greek mathematics - Wilbourhall.org

MENAECHMUS'S PROCEDURE 113
particular hyperbola which we call rectangular or equilateral,
and also to obtain its property with reference to its asymptotes,
a considerable advance on what was necessary in the
case of the parabola. Two methods of obtaining the particular
hyperbola were possible, namely (1) to obtain the hyperbola
arising from the section of any obtuse-angled cone by a plane
at right angles to a generator, and then to show how a
rectangular hyperbola can be obtained as a particular case
by finding the vertical angle which the cone must have to
give a rectangular hyperbola when cut in the particular way,
or (2) to obtain the rectangular hyperbola direct by cutting
another kind of cone by a section not necessarily perpendicular
to
a generator.
(1) Taking the first method, we draw (Fig. 2) a cone with its
vertical angle BOG obtuse. Imagine a section perpendicular
to the plane of the paper and passing through AG which is
perpendicular to OB. Let GA produced meet GO produced in
A\ and complete the same construction as in the case of
the
parabola.
Fig. 2.
In this case we have
1523.2
PN 2
= BN.NG = AN.NG.
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