724 SCIENTIFIC THOUGHT. 



interest to determine the residues of the powers of 

 numbers. A number is said to be a quadratic, cubic, 

 or biquadratic residue of another (prime) numljer (the 

 modulus) if it is possible to find a square, cube, or bi- 

 quadratic number which is congruent with the first 

 number. The theory of congruences was a new calculus : 

 as such it was, like the theory of determinants or of in- 

 variants or the general theory of forms, a tactical device 

 for bringing order and simplicity into a vast region of 

 very complicated relations. Gauss himself wrote about it 

 late in life to Schumacher.^ " In general the position as 

 regards all such new calculi is this — that one cannot 

 attain by them anything that could not be done without 

 them : the advantage, however, is, that if such a cal- 

 culus corresponds to the innermost nature of frequent 

 wants, every one who assimilates it thoroughly is able — 

 without the unconscious inspiration of genius which no 

 one can command — to solve the respective problems, 

 yes, even to solve them mechanically in complicated 

 cases where genius itself becomes impotent. So it is 

 with the invention of algebra generally, so with the 

 differential calculus, so also — though in more restricted 

 regions — with Lagrange's calculus of variations, with 

 my calculus of congruences, and with Mobius's calculus. 

 Through such conceptions countless problems which 

 otherwise would remain isolated and require every time 

 (larger or smaller) efforts of inventive genius, are, as it 

 were, united into an organic whole." But a new calculus 

 frequently does more than this. In the course of its 



^ See ' Briefwechsel,' &c., vol. iv. p. 147 ; also Gauss's ' Werke,' vol. viii. 

 p. 298. 



