xxviii.] ANALOGY. 63' 



Analogy in the Mathematical Sciences. 



Whoever wishes to acquire a deep acquaintance with 

 Nature must observe that there are analogies which con- 

 nect whole branches of science in a parallel manner, 

 and enable us to infer of one class of phenomena what 

 we know of another. It has thus happened on several 

 occasions that the discovery of an unsuspected analogy 

 between two branches of knowledge has been the starting- 

 point for a rapid course of discovery. The truths readily 

 observed in the one may be of a different character from 

 those which present themselves in the other. The analogy, 

 once pointed out, leads us to discover regions of one 

 science yet undeveloped, to which the key is furnished by 

 the corresponding truths in the other science. An in- 

 terchange of aid most wonderful in its results may thus 

 take place, and at the same time the mind rises to a higher 

 generalisation, and a more comprehensive view of nature. 



No two sciences might seem at first sight more different 

 in their subject matter than geometry and algebra. The 

 first deals with circles, squares, parallelograms, and other 

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

 Prior to the time of Descartes, the sciences were developed 

 slowly and painfully in almost entire independence of each 

 other. The Greek philosophers indeed could not avoid 

 noticing occasional analogies, as when Plato in the Thsee- 

 tetus describes a square number as equally equal, and a 

 number produced by multiplying two unequal factors 

 as oblong. Euclid, in the 7th and 8th books of his Ele- 

 ments, continually uses expressions displaying a conscious- 

 ness of the same analogies, as when he calls a number 

 of two factors a plane number, e-TUTreSo? dptOfjos, and 

 distinguishes a square number of which the two factors are 

 equal as an equal-sided and plane number, iaoirXsvpos 

 Kal e7ri7re8o? opiOfji6<;. He also calls the root of a cubic 

 number its side, ir\evpd. 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 



