20 LECTURES AND ESSAYS [1869- 



splendid and fruitful in the history of human thought, 

 and well deserves the attention of metaphysicians. Only 

 let it be said that no criticism of Newton s time, which 

 starts from the arithmetical view of quantity, and urges 

 the old objections about infinite divisibility, and so forth, 

 is competent ; for the arithmetical theory is a product 

 of abstract reflection, and so stands on a lower platform 

 than the pure objective notion of Newton. 



There is no difficulty in comprehending the mathe 

 matical power which the conception of fluxions at once 

 puts in Newton s hands, if we remember that it is not in 

 any sense an extension of the theory of numbers that he is 

 seeking. It is true that the calculus has revolutionised 

 algebra as well as geometry ; but it has done so by trans 

 forming algebra from the abstract science of numbers to a 

 physical science the science of pure time. In Newton s 

 own mind, however, this conception was probably not 

 explicitly present. What he did see was, that all diffi 

 culties in geometry (and to Newton, as to the old geo 

 meters, geometrical magnitude is the type and exponent 

 of all magnitude whatsoever, when viewed with respect 

 to its generation) are reducible to the general form :- 

 &quot; Given the fluent as a function of time to determine the 

 fluxion and vice versa.&quot; 



The one class of problems that can be thoroughly 

 treated without explicit reference to a generation by flux, 

 is that which has for its geometrical type systems of 

 straight lines; and thus geometers were tempted to 

 introduce the fiction of indivisibles, in order to reduce 

 higher problems to this type. 



But these higher problems are not simply complicated 

 cases of the rectilineal type ; on the contrary, that type 

 is produced by one of the two essentially distinct elements 

 (generated and generating quantity), which usually 

 appear side by side, ceasing to be explicitly manifest. 



Take, for example, Newton s own instance at the 

 beginning of the &quot; De Quadratura.&quot; Suppose the abscissa 



