THE HISTORY OF MATHEMATICS IN THE NINETEENTH 



CENTURY 



BY PROFESSOR JAMES P. PIERPONT OF YALE UNIVERSITY 



THE extraordinary development of mathematics in the last century 

 is quite unparalleled in the long history of this most ancient of 

 sciences. Not only have those branches of mathematics which were 

 taken over from the eighteenth century steadily grown, but entirely 

 new ones have sprung up in almost bewildering profusion, and 

 many of these have promptly assumed proportions of vast extent. 



As it is obviously impossible to trace in the short time allotted to 

 me the history of mathematics in the nineteenth century even in 

 merest outline, I shall restrict myself to the consideration of some 

 of its leading theories. 



Theory of Functions of a Complex Variable 



Without doubt one of the most characteristic features of mathe- 

 matics in the last century is the systematic and universal use of the 

 complex variable. Most of its great theories received invaluable aid 

 from it, and many owe their very existence to it. What would the 

 theory of differential equations or elliptic functions be to-day without 

 it, and is it probable that Poncelet, Steiner, Chasles, and von Staudt 

 would have developed synthetic geometry with such elegance and 

 perfection without its powerful stimulus? 



The necessities of elementary algebra kept complex numbers 

 persistently before the eyes of every mathematician. In the eight- 

 eenth century the more daring, as Euler and Lagrange, used them 

 sparingly; in general one avoided them when possible. Three events, 

 however, early in the nineteenth century changed the attitude of 

 mathematicians toward this mysterious guest. In 1813 Argand 

 published his geometric interpretation of complex numbers. In 

 1824 came the discovery by Abel of the imaginary period of the 

 elliptic function. Finally Gauss in his second memoir on biquadratic 

 residues (1832) proclaims them a legitimate and necessary element 

 of analysis. 



The theory of function of a complex variable may be said to have 

 had its birth when Cauchy discovered his integral theorem 



ff(x)dx=Q 



published in 1825. In a long series of publications beginning with 

 the Cows d' Analyse (1821), Cauchy gradually developed his theory 

 of functions and applied it to problems of the most diverse nature; 



