The discovery of Hippocrates amounted to the discovery of the fact that from the relation
(1)it follows thatand if , [then , and]The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations
(2)[or equivalently...and the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).
Let AO, BO be straight lines placed so as to form a right angle at O, and of length respectively. Produce BO to and AO to .
The solution now consists in drawing a paraBOla, with vertex O and axis O, such that its parameter is equal to BO or , and a hyperBOla with O, O as asymptotes such that the rectangle under the distances of any point on the curve from O, O respectively is equal to the rectangle under AO, BO i.e. to . If P be the point of intersection of the paraBOla and hyperBOla, and PN, PM be drawn perpendicular to O, O, i.e. if PN, PM be denoted by , the coordinates of the point P, we shall havewhenceIn the solution of Menaechmus we are to draw the paraBOla described in the first solution and also the paraBOla whose vertex is O, axis O and parameter equal to . The point P where the two paraBOlas intersect is given bywhence, as before,
Thomas Little Heath
