694 



SCIENTIFIC THOUGHT. 



distinguish two which are very prominent, and are 

 roughly represented by the two schools just referred 

 to. In the first place, a function can be formally 

 defined as an assemblage of mathematical symbols, 

 each of which denotes a definite operation on one 

 or more quantities. These operations are partly direct, 

 like addition, multiplication, &c. ; partly indirect or 

 inverse, like subtraction, division, &c. Now, so far 

 as the latter are concerned, they are not generally and 

 necessarily practicable, and the question arises, When 

 are they practicable, and if they are not, what mean- 

 ing can we connect with the mathematical symbol ? 

 In this way we arrive at definitions for mathematical 

 functions which cannot immediately be reduced to the 

 primary operations of arithmetic, but which form special 

 expressions that become objects of research as to their 

 properties and as to the relation they bear to those 

 fundamental operations upon which all our methods of 

 calculation depend. The inverse operations, represented 

 by negative, irrational, and imaginary quantities ; further, 

 the operations of integration in its definition as the 



a certain finality when Fourier 

 introduced his well-known series 

 and integrals, by which any kind 

 of functionality or mathematical 

 dependence, such as physical pro- 

 cesses seem to indicate, could be 

 expressed. The work of Fourier, 

 which thus gave, as it were, a sort 

 of preliminary specification under 

 which a large number of problems 

 in physical mathematics could be 

 attacked and practically solved, 

 together with the stricter defini- 

 tions introduced by Lejeune Dir- 

 ichlet, settled for a time and for 

 practical purposes the lengthy dis- 

 cussions which had begun with 



Euler, Daniel Bernoulli, d'Alem- 

 bert, and Lagrange. The above- 

 named chapter, written by Prof. 

 Pringsheim, gives an introduction 

 to the subject showing the historical 

 genesis of the conception of function 

 and the various changes it was sub- 

 jected to, and then proceeds to 

 expositions and definitions mostly 

 taken from the lectures of Weier- 

 strass (see p. 8), whereas Cayley's 

 article introduces us to the elements 

 of the general theory of functions 

 as they were first laid down by 

 Riemann in the manner now com- 

 monly accepted. 



