THE FUTURE OF MATHEMATICS. 41 



how we are to pass from one classification to the 

 other ; but if it were clear it would be already done, 

 and would no longer be the work of the future. 



The first example that comes to my mind is the 

 theory of congruents, in which we find a perfect 

 parallelism with that of algebraic equations. We 

 shall certainly succeed in completing this parallelism, 

 which must exist, for instance, between the theory of 

 algebraic curves and that of congruents with two 

 variables. When the problems relating to congruents 

 with several variables have been solved, we shall have 

 made the first step towards the solution of many ques- 

 tions of indeterminate analysis. 



Algebra. 



The theory of algebraic equations will long continue 

 to attract the attention of geometricians, the sides by 

 which it may be approached being so numerous and 

 so different. 



It must not be supposed that algebra is finished 

 because it furnishes rules for forming all possible 

 combinations ; it still remains to find interesting com- 

 binations, those that satisfy such and such conditions. 

 Thus there will be built up a kind of indeterminate 

 analysis, in which the unknown quantities will no 

 longer be whole numbers but polynomials. So this 

 time it is algebra that will model itself on arithmetic, 

 being guided by the analogy of the whole number, 

 either witii the whole pjolynomial with indefinite 

 coefficients, or witii tlu; whole polynomial with \\hole 

 coefiicients. 



