MENAECHMUS AND CONICS
253
but to discover, the existence of curves having the properties
corresponding to the Cartesian equations. He discovered
them in plane sections of right circular cones, and it would
doubtless be the properties of the principal ordinates in
relation to the abscissae on the axes which he would arrive
at first. Though only the parabola and the hyperbola are
wanted for the particular problem, he would certainly not
fail to find the ellipse and its property as well. But in the
case of the hyperbola he needed the property of the curve
with reference to the asymptotes, represented by the equation
xy — ab; he must therefore have discovered the existence of
the asymptotes, and must have proved the property, at all
events for the rectangular hyperbola. The original method
of discovery of the conics will occupy us later. In the mean
time it is obvious that the use of any two of the curves
x 2 = ay, y 2 = hx, xy = ah gives the solution of our problem,
and it was in fact the intersection of the second and third
which Menaechmus used in his first solution, while for his,
second solution he used the first two. Eutocius gives the
analysis and synthesis of each solution in full. I shall repro
duce them as shortly as possible, only suppressing the use of
four separate lines representing the two given straight lines
and the two required means in the figure of the first solution.
First solution.
Suppose that AO, OB are two given straight lines of which
AO > OB, and let them form a right angle at 0.
Suppose the problem solved, and let the two mean propor
tionals be OM measured along BO produced and ON measured
along AO produced. Complete the rectangle OMPN.
Then, since AO : OM = OM: ON = ON: OB,
we have (1) OB. OM = ON 2 = PM 2 ,
so that P lies on a parabola which has 0 for vertex, OM for
axis, and OB for latus rectum;
and (2) AO .OB = OM.ON= PN. PM,
so that P lies on a hyperbola with 0 as centre and OM, ON as
asymptotes.