OH. XVI] THREE GEOMETRICAL PROBLEMS 339 



double that of the old one (from which it followed that the 

 volume was increased eight-fold) or by placing a similar cube 

 altar next to the old one. Whereupon, according to the legend, 

 the indignant god made the pestilence worse than before, and 

 informed a fresh deputation that it was useless to trifle with 

 him, as his new altar must be a cube and have a volume exactly 

 double that of his old one. Suspecting a mystery the Athenians 

 applied to Plato, who referred them to the geometricians. The 

 insertion of Plato's name is an obvious anachronism. Eratos- 

 thenes* relates a somewhat similar story, but with Minos as 

 the propounder of the problem. 



In an Arab work, the Greek legend was distorted into the 

 following extraordinarily impossible piece of history, which I 

 cite as a curiosity of its kind. "Now in the days of Plato," 

 says the writer, " a plague broke out among the children of 

 Israel. Then came a voice from heaven to one of their prophets, 

 saying, ' Let the size of the cubic altar be doubled, and the 

 plague will cease ' ; so the people made another altar like unto 

 the former, and laid the same by its side. Nevertheless the 

 pestilence continued to increase. And again the voice spake 

 unto the prophet, saying, ' They have made a second altar like 

 unto the former, and laid it by its side, but that does not pro- 

 duce the duplication of the cube.' Then applied they to Plato, 

 the Grecian sage, who spake to them, saying, ' Ye have been 

 neglectful of the science of geometry, and therefore hath God 

 chastised you, since geometry is the most sublime of all the 

 sciences.' Now, the duplication of a cube depends on a rare 

 problem in geometry, namely...." And then follows the solu- 

 tion of Apollonius, which is given later. 



If a is the length of the side of the given cube and x that 

 of the required cube, we have a? =* 2a s , that is, x : a = V2 : 1. 

 It is probable that the Greeks were aware that the latter ratio 

 is incommensurable, in other words, that no two integers can 

 be found whose ratio is the same as that of $2 : 1, but it did 

 not therefore follow that they could not find the ratio by 



* Archimedii Opera cum Eutocii Commentariis, ed. Torelli, Oxford, 1792, 

 p. 144 j ed. Heiberg, Leipzig, 1880-1, vol. m, pp. 104—107. 



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