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 dehnitions and of unambiguous axioms ; pro- 

 ceeding from these by rigorous reasoning to the theorems 

 of the new science.^ 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 — formuhe into 

 geometrical conceptions — the geometrical construction of 

 algebraical expressions. It is the inverse operation of 

 the former. Tu tliis inversion of any given operation operations, 

 lies the soul and principle of all mathematical progress, 

 both in theory and in application." The invention of 



10. 

 Inverse 



' lleferring sixjcially to the 

 defiDition of a " function " or 

 mathematical dependence, a con- 

 ception introduced l»y Kuler, but 

 not rigorou.sly defined by him, 

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

 cr)mmenceraent du sifecle, I'idde 

 de fonction etait une notion Ji 

 la fois trop restreinte et trop 

 vague. . . . Cette definition, il 

 fallait la donner : car I'analyse ne 

 jjouvait qu'ii ce prix actjuerir la 

 parfaite rigueur. " In its generality 

 tliid task wa« performed in the 

 last third of the century by 

 Weierstrass, but tlie necessity of 

 this criticism of the formulic in- 

 vented by modern mathematics 

 dates from the api)earance of 

 Cauchy's ' Momoire sur la thdorie 

 dcs intcgiales ddfinies' of 1814, 

 which Legcndre reported on in this 

 sense, but which wxh not published 

 till 182f). 



- The operations referred to are 

 generally of two kinds : first, there 

 is the operation of translating 

 geometrical relations, intuitively 

 given, int<5 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 ca.se the 

 new relations arrived at reijuire 

 to be interpreted, and thi.s inter- 

 pretation leads nearly always to 

 an extension of knowledge or to 

 novel conceptions. A simjde ex- 

 ample of the first kind presents 

 itself in the geometrical construc- 

 tion of the higher powers of 

 quantity. Having agreed to ilefine 

 by a tiie length of a line, by a^ 

 an area, what is the meaning of 

 n^ n* . . . a" ? Can any geomet- 

 rical meaning be attaoiied to these 

 symbols ? An example of the 



