CH. XVI] THREE GEOMETRICAL PROBLEMS 341 



snow analytically that the construction is correct, take OA as 

 the axis of x, and the generator of the cylinder drawn through 

 as axis of z, then with the usual notation, in polar co-ordinates, 

 if a is the radius of the cylinder, we have for the equation 

 of the surface described by the semicircle r = 2a sin 8 ; for that 

 of the cylinder r sin 8 = 2a cos (f> ; and for that of the cone 

 sin 8 cos <f> = ^. These three surfaces cut in a point such that 

 sin 8 8 = ^, and therefore (r sin 8)" = 2a 3 . Hence the volume of 

 the cube whose side is r sin 8 is twice that of the cube whose 

 side is a. 



The construction attributed to Plato* (circ. 360 B.C.) de- 

 pends on the theorem that, if CAB and DAB are two right- 

 angled triangles, having one side, AB, common, their other 

 sides, AD and BG, parallel, and their hypothenuses, AG and 

 BD, at right angles, then if these hypothenuses cut in P, we 

 have PG : PB = PB : PA = PA : PD. Hence, if such a figure 

 can be constructed having PD = 2PC, the problem will be 

 solved. It is easy to make an instrument by which the figure 

 can be drawn. 



The next writer whose name is connected with the problem 

 is Menaechmusf, who in or about 340 B.C. gave two solutions 

 of it. 



In the first of these he pointed out that two parabolas 

 having a common vertex, axes at right angles, and such that 

 the latus rectum of the one is double that of the other, will 

 intersect in another point whose abscissa (or ordinate) will 

 give a solution. If we use analysis this is obvious; for, if 

 the equations of the parabolas are y* = 2ax and a? = ay, they 

 intersect in a point whose abscissa is given by x* = 2a 3 . It is 

 probable that this method was suggested by the form in which 

 Hippocrates had cast the problem : namely, to find x and y so 

 that a : x = x : y = y : 2a, whence we have a?=ay and y* = 2ax. 



The second solution given by Menaechmus was as follows. 

 Describe a parabola of latus rectum I. Next describe a rect- 

 angular hyperbola, the length of whose real axis is 42, and 



* Archimedis Opera, ed. Torelli, p. 135 ; ed. Heiberg, vol. in, pp. 66—71. 

 t Ibid., ed. Torelli, pp. 141—143; ed. Heiberg, vol. ra, pp. 92—99. 



