ANALOGY. 289 



No two "sciences might seem at first sight more 

 entirely discrete and divergent in their subject matter 

 than geometry and arithmetic, or algebra. The first deals 

 with circles, squares, parallelograms, and various other 

 forms in space ; the latter with mere symbols of number, 

 the symbols having form indeed, but bearing a meaning 

 independent of shape or size. Prior to the time of Des- 

 cartes, too, the sciences actually were developed in a slow 

 and painful manner in almost entire independence of each 

 other. The Greek philosophers indeed could not avoid 

 noticing occasional analogies, as when Plato in the 

 Thaeetetus describes a square number as equally equal, 

 and a number produced by multiplying two unequal 

 factors as oblong. Euclid, in the jih and 8th books of 

 his Elements, continually uses expressions displaying a 

 consciousness of the same analogies, as when he calls a 

 number of two factors a plane number, eV/Tre^o? apti^s, 

 and distinguishes a square number of which the two 

 factors are equal as an equal-sided or plane number, 

 la-6-7r\evpo<s KOLL eTTiTredos apiO/ji6s. He also calls the root 

 of a cubic number its' side, Tr\evpa. In the Diophantine 

 algebra many problems of a geometrical character were 

 solved by algebraic or numerical processes ; but there 

 was no general system, so that the solutions were of an 

 isolated character. In general the ancients were far more 

 advanced in geometric than symbolic methods ; thus 

 Euclid in his 4th book gives us the means of dividing 

 a circle by purely geometric or mechanical means into 2, 

 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30 parts, but he- was 

 totally unacquainted with the theory of the roots of unity 

 exactly corresponding to this division of the circle. 



During the middle ages, on the other hand, algebra ad- 

 vanced beyond geometry, and modes of solving equations 

 were painfully discovered by those who had no notion 

 that at every step they were implicitly solving important 



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