DEVELOPMENT OF MATHEMATICAL THOUGHT. 695 



inverse of differentiation, led early to investigations of 

 the kind just mentioned. The experience that ordinary 

 fractions might be expressed by decimal fractions i.e., 

 by finite or infinite series led to the inverse problem 

 of finding the sum of such series and many other 

 answerable and apparently unanswerable problems. The 

 older method of research consisted in treating these 

 problems when and as they arose : new chapters were 

 accordingly added to the existing chapters of the 

 text-books, dealing with special functions or mathe- 

 matical expressions. It was only towards the end of 

 the eighteenth century, and at the beginning of the 

 nineteenth, that Lagrange, Gauss, and Cauchy felt and 

 proclaimed the necessity of attacking the question gener- 

 ally and systematically ; the labours of Euler having 

 accumulated an enormous mass of analytical knowledge, 

 a great array of useful formulae, and amongst them not 

 a few paradoxes which demanded special attention. I 

 have already had occasion to refer to the problem of 

 the general solution of equations as an instance where, 

 in the hands of Abel, the tentative and highly ingenious 

 attempts of earlier analysts were replaced by a method- 

 ical and general treatment of the whole question. 

 Another chapter of higher mathematics, the investiga- 

 tion of expressions which presented themselves in the 

 problems of finding the length of the arc of an ellipse, 

 and which opened the view into the large province of the 

 so-called higher transcendents, gave Abel further occasion 

 of laying new foundations and of creating a general 

 theory of equations or of forms. 



But yet another interest operated powerfully in the 



