484 MATHEMATICS 



Leaving the calculus, let us notice briefly the theory of functions 

 of real variables. The line of demarcation between these two sub- 

 jects is extremely arbitrary. We might properly place in the latter 

 all those finer and deeper questions relating to the number-system; 

 the study of our curve, surface, and other geometrical notions, the 

 peculiarities that functions present with reference to discontinuity, 

 oscillation, differentiation, and integration; as well as a very exten- 

 sive class of investigations whose object is the greatest possible 

 extension of the processes, concepts, and results of the calculus. 

 Among the many not yet mentioned who have made important 

 contributions to this subject we note: Fourier, Riemann, Stokes, 

 Dini, Tannery, Pringsheim, Arzela, Osgood, Broden, Ascoli, Borel, 

 Baire, Kopke, Holder, Volterra, and Lebesgue. 



Closely related with the differential calculus is the calculus of 

 variations; in the former the variables are given infinitesimal varia- 

 tions, in the latter the functions. Developed in a purely formal 

 manner by Jacobi, Hamilton, Clebsch, and others in the first part 

 of the century, a new epoch began with Weierstrass, who, having 

 subjected the labors of his predecessors to an annihilating criticism, 

 placed the theory on a new and secure foundation and so opened the 

 path for further research by Schwarz, A. Mayer, Scheffers, v. Esche- 

 rich, Kneser, Osgood, Bolza, Kobb, Zermelo, and others. At the 

 very close of the century Hilbert has given the theory a fresh im- 

 pulse by the introduction of new and powerful methods, which 

 enable us in certain cases to neglect the second variation and sim- 

 plifies the consideration of the first. As application he gives the 

 first direct and yet simple demonstration of Dirichlet's celebrated 

 Principle. 



Theory of Numbers Algebraic Bodies 



The theory of numbers as left by Fermat, Euler, and Legendre 

 was for the most part concerned with the solution of Diophantine 

 equations, that is, given an equation f(x, y, z, . . . ) =0 whose 

 coefficients are integers, find all rational, and especially all integral 

 solutions. In this problem Lagrange had shown the importance 

 of considering the theory of forms. A new era begins with the ap- 

 I>earance of Gauss's Disquisitiones arithmeticae in 1801. This great 

 work is remarkable for three things: (1) The notion of divisibility 

 in the form of congruences is shown to be an instrument of wonder- 

 ful power; (2) the Diophantine problem is thrown in the back- 

 ground and the theory of forms is given a dominant r61e; (3) the 

 introduction of algebraic numbers, namely, the roots of unity. 



The theory of forms has been further developed along the lines 

 of the Disquisitiones by Dirichlet, Eisenstein, Hermite, H. Smith, and 

 Minkowski. 



