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.