342 THREE GEOMETRICAL PROBLEMS [CH. XVI 



having for its asymptotes the tangent at the vertex of the 

 parabola and the axis of the parabola. Then the ordinate and 

 the abscissa of the point of intersection of these curves are the 

 mean proportionals between I and 21. This is at once obvious 

 by analysis. The curves are a? = ly and xy = 2l 2 . These cut in 

 a point determined by x* = 21" and y" = 4/*. Hence 

 I : x = x : y = y : 21. 



The solution of Apollonius*, which was given about 220 B.C., 

 was as follows. The problem is to find two mean proportionals 

 between two given lines. Construct a rectangle OADB, of 

 which the adjacent sides OA and OB are respectively equal to 

 the two given lines. Bisect AB in G. With G as centre describe 

 a circle cutting OA produced in a and cutting OB produced in 

 b, so that aDb shall be a straight line. If this circle can be so 

 described, it will follow that OA : Bb = Bb : Aa = Aa : OB, that 

 is, Bb and Aa are the two mean proportionals between OA 

 and OB. It is impossible to construct the circle by Euclidean 

 geometry, but Apollonius gave a mechanical way of describing it. 



The only other construction of antiquity to which I will 

 refer is that given by Diodes and Sporusf. It is as follows. 

 Take two sides of a rectangle OA, OB, equal to the two lines 

 between which the means are sought. Suppose OA to be the 

 greater. With centre and radius OA describe a circle. Let 

 OB produced cut the circumference in G and let A produced 

 cut it in D. Find a point E on BG so that if DE cuts AB 

 produced in F and cuts the circumference in G, then FE = EG. 

 If E can be found, then OE is the first of the means between 

 OA and OB. Diodes invented the cissoid in order to determine 

 E, but it can be found equally conveniently by the aid of 

 conies. 



In more modern times several other solutions have been 

 suggested. I may allude in passing to three given by HuygensJ, 



* Archimedis Opera, ed. Torelli, p. 137 ; ed. Heiberg, vol. m, pp. 76—79. 

 The solution is given in my History of Mathematics, London, 1901, p. 84. 



t Ibid., ed. Torelli, pp. 138, 139, 141 ; ed. Heiberg, vol. m, pp. 78—84, 

 90-93. 



t Opera Varia, Leyden, 1724, pp. 393—396. 



