DEVELOPMENT OF MATHEMATICAL THOUGHT. 693 



other. The question arises, What are we to under- 

 stand under this term ? What is a mathematical 

 function or dependence ? The question was approached 

 by the great analysts of the second half of the 

 eighteenth century. A preliminary answer which served 

 the requirements of a very wide field of practical 

 application was given by Fourier at the beginning of 

 the nineteenth century. Since that time the question 

 has been independently treated by two schools of 

 Continental mathematicians. Of these the first was 

 founded by Cauchy in France, and is mainly represented 

 by Bernhard Riemann and his numerous pupils in 

 Germany ; the other centres in the Berlin school, 

 headed by Weierstrass, and goes back to the work 

 of Lagrange. 



The interests which have led to this modern branch 46. 



Theory of 



of mathematical research are various, but we can Functions. 



1 The literature suitable for intro- 

 ducing the student of mathematics 

 to the modern theory of functions 

 which plays in analysis, i.e., the 

 doctrine of variable quantity, a 

 part of similar importance to that 

 which the theory of forms plays in 

 algebra is so enormous, the sub- 

 ject being approached from so 

 many sides by different writers, 

 that it seems worth while to refer 

 to two expositions which may be 

 read with profit, and which do not 

 require extensive mathematical 

 knowledge. First and foremost I 

 would recommend Cay ley's article 

 on " Functions " in vol. ix. of the 

 'Ency. Brit.' Then there is the 

 chapter on " Foundations of the 

 General Theory of Functions," con- 

 tained in the 2nd volume of the 

 German ' Mathematical Encyclo- 

 pedia,' written by Prof. Pringa- 



heim. Cayley's article intro- 

 duces the general theory after 

 giving a short summary of the 

 more important " known " func- 

 tions, including those which pre- 

 sented themselves in the first half 

 of the nineteenth century, and 

 which I referred to in dealing with 

 the work of Abel and Gauss (see 

 note, p. 648). The treatment of 

 these latter functions, which had 

 been brought to a certain degree of 

 perfection by Jacobi, had made it 

 evident that more general aspects 

 had to be gained and broader 

 foundations laid. But ever since 

 the middle of the eighteenth 

 century another development of 

 mathematical ideas had been going 

 on which started from the solution 

 of a problem in mathematical 

 physics namely, that of vibrating 

 strings, which led in the sequel to 



