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 

 meaning 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 in- 



a certain finality when Fourier Euler, Daniel Bernoulli, d'Alem- 



introduced his well-known series bert, and Lagrange. The above- 



and integrals, by which any kind named chapter, written by Prof, 



of functionality or mathematical Pringsheim, gives an introduction 



dependence, such as physical pro- I to the subject showing the historical 



cesses seem to indicate, could be genesis of the conception of function 



expressed. The work of Fourier, \ and the various changes it was sub- 



which thus gave, as it were, a sort i jected to, and then proceeds to 



of preliminary specification under expositions and definitions mostly 



which a large number of problems taken from the lectures of Weier- 



in physical mathematics could be strass (see p. 8), whereas Cay ley's 



attacked and practically solved, , article introduces us to the elements 



together with the stricter defini- ' of the general theory of functions 



tions introduced by Lejeune Dir- j as they were first laid down by Rie- 



ichlet, settled for a time and for i mann in the manner now commonly 



practical purposes the lengthy dis- ! accepted, 

 cussions which had begun with 



