340 THREE GEOMETRICAL PROBLEMS [CH. XVI 



geometry: in fact, the side and diagonal of a square are 

 instances of lines whose numerical measures are incommen- 

 surable. 



I proceed now to give some of the geometrical constructions 

 which have been proposed for the duplication of the cube*. 

 With one exception, I confine myself to those which can be 

 effected by the aid of the conic sections. 



Hippocratesf (circ. 420 B.C.) was perhaps the earliest mathe- 

 matician who made any progress towards solving the problem. 

 He did not give a geometrical construction, but he reduced the 

 question to that of finding two means between one straight 

 line (a), and another twice as long (2a). If these means are x 

 and y, we have a : x = x : y = y : 2a, from which it follows that 

 x 3 = 2a s . It is in this form that the problem is always presented 

 now. Formerly any process of solution by finding these means 

 was called a mesolabum. 



One of the first solutions of the problem was that given by 

 ArchytasJ in or about the year 400 B.C. His construction is 

 equivalent to the following. On the diameter OA of the base 

 of a right circular cylinder describe a semicircle whose plane is 

 perpendicular to the base of the cylinder. Let the plane con- 

 taining this semicircle rotate round the generator through 0, 

 then the surface traced out by the semicircle will cut the 

 cylinder in a tortuous curve. This curve will itself be cut by 

 a right cone, whose axis is OA and semi-vertical angle is (say) 

 60°, in a point P, such that the projection of OP on the base of 

 the cylinder will be to the radius of the cylinder in the ratio of 

 the side of the required cube to that of the given cube. Of 

 course the proof given by Archytas is geometrical ; and it is 

 interesting to note that in it he shows himself familiar with the 

 results of the propositions Euc. Ill, 18, III, 35, and xi, 19. To 



* On the application to this problem of the traditional Greek methods of 

 analysis by Hero and Philo (leading to the solution by the use of Apollonius's 

 circle), by NicomedLa (leading to the solution by the use of the conolioid), and 

 by Pappus (leading to the solution by the use of the cissoid), see Geometrical 

 Analysis by J. Leslie, Edinburgh, second edition, 1811, pp. 247 — 250, 453. 



t Proclus, ed. Friedlein, pp. 212—213. 



% Archimedis Opera, ed. Torelli, p. 143 j ed. Heiberg, vol. m, pp. 98 — 103. 



