DEVELOPMENT OF MATHEMATICAL THOUGHT. 723 



tition of numbers had been studied, and many interesting 

 formulae had been found by induction, and subsequently 

 proved or not proved by a multitude of ingenious 

 devices. As in so many other directions of research 

 so also here, the genius of Gauss gave a great impetus to 59. 



Gauss's 



progress by the invention of a definite calculus and an * ory f 

 algorithm. This invention referred to the solution of ences - 

 what used to be known as indeterminate equations: to 

 find two or more numbers notably integers, which obey 

 a certain algebraical relation. For one large class of 

 these problems (which already occupied the ancient 

 geometers), viz., those of the divisibility of one number 

 by another (called the modulus) with or without residue, 

 Gauss invented the conception and notation of a con- 

 gruence. Two numbers are congruent if when divided 

 by a certain number they leave the same remainder. " It 

 will be seen," says Henry Smith, " that the definition of 

 a congruence involves only one of the most elementary 

 arithmetical conceptions that of the divisibility of one 

 number by another. But it expresses that conception 

 in a form so suggestive of analysis, so easily available in 

 calculation and so fertile in new results, that its introduc- 

 tion into arithmetic has proved a most important contri- 

 bution to the progress of the science." 1 Notably the 

 analogy with ordinary algebraic equations and the possi- 

 bility of transferring the properties and treatment of 

 these was at once evident. It became a subject of 



1 See Henry J. S. Smith in his 

 most valuable ' Report on the 

 Theory of Numbers' (Brit. Assoc., 

 1859-65, six parts. Reprinted in 

 ' Collected Math. Papers,' vol. i. 



pp. 38-364). It gives a very lucid 

 account of the history of this de- 

 partment of mathematical science 

 up to the year 1863. 



