DEVELOPMENT OF MATHEMATICAL THOUGHT. 639 



this second and more hidden line of research the logical 

 side of the new development. It corresponds to the work 

 which Euclid performed in ancient geometry, the framing 

 of clear definitions and of unambiguous axioms ; pro- 

 ceeding from these by rigorous reasoning to the theorems 

 of the new science. 1 But the translation of geometrical 

 and mechanical conceptions into those of generalised 

 arithmetic or algebra brought with it a logical problem 

 of quite a novel kind which has given to modern mathe- 

 matics quite a new aspect. This new problem is the re- 

 translation of algebraical i.e., of general formulae into 

 geometrical conceptions the geometrical construction of 

 algebraical expressions. It is the inverse operation of 

 the former. In this inversion of any given operation 

 lies the soul and principle of all mathematical progress, 

 both in theory and in application. 2 The invention of 



10. 



Inverse 

 operations. 



1 Referring specially to the 

 definition of a " function " or 

 mathematical dependence, a con- 

 ception introduced by Euler, but 

 not rigorously defined by him, 

 M. Poincare" says, loc. cit. : " Au 

 commencement du siecle, 1'idee 

 de fonction e'tait une notion a 

 la fois trop restreinte et trop 

 vague. . . . Cette definition, il 

 fallait la douner : car 1'analyse ne 

 pouvait qu'a ce prix acquerir la 

 parfaite rigueur." In its generality 

 this task was performed in the 

 last third of the century by 

 Weierstrass, but the necessity of 

 this criticism of the formulae in- 

 vented by modern mathematics 

 dates from the appearance of 

 Cauchy's ' Mdmoire sur la theorie 

 des integrates definies ' of 1814, 

 which Legendre reported on in this 

 sense, but which was not published 

 till 1825. 



2 The operations referred to are 

 generally of two kinds : first, there 

 is the operation of translating 

 geometrical relations, intuitively 

 given, into algebraical relations ; 

 and, secondly, the operation of 

 extending algebraical relations by 

 going forward or backward in the 

 order of numbers, usually given 

 by indices. In each case the 

 new relations arrived at require 

 to be interpreted, and this inter- 

 pretation leads nearly always to 

 an extension of knowledge or to 

 novel conceptions. A simple ex- 

 ample of the first kind presents 

 itself in the geometrical construc- 

 tion of the higher powers of 

 quantity. Having agreed to define 

 by a the length of a line, by a 2 

 an area, what is the meaning of 

 a 3 a 4 ... a" ? Can any geomet- 

 rical meaning be attached to these 

 symbols ? An example of the 



