36 LECTURES AND ESSAYS [1869- 



the continuous quantity of the higher analysis and of 

 actual nature, and the discrete quantity of arithmetical 

 abstraction. 1 He thus fell into the delusion, that a writer 

 like Lagrange who, from the extreme nominalistic stand 

 point of the eighteenth century, seeks to make analysis 

 a merely formal instrument, in no way expressing the 

 essence of things, and who, for example, boasts that in his 

 Mecanique Analytique one will find no such unnecessary 

 incumbrances as figures Hegel, I say, imagined that such 

 a writer had really reached a higher generality than 

 Newton, when he had only reached an untenable extremity 

 of one-sided abstraction, and hence, without a moment s 

 hesitation, resolved that by simply treating the successive 

 differential coefficients as the successive derived functions 

 obtained by expanding y in terms of x, we shall be quit 

 &quot; of the formal categories of the infinite, and of infinite 

 approximation, and of the equally empty category of 

 continuous magnitude &quot; (iii. 320). 



The differential calculus, then, is a special branch of 

 mathematics which has to deal (by purely arithmetical 

 methods) with qualitative forms of quantity, i.e., says 

 Hegel, with relations of powers. A power, it should be 

 said, means with Hegel a quantity raised to a higher 

 power than the first, and the link between the clauses of 

 the foregoing sentence is as follows : &quot;In the equation 



- =; a the relation of y to % is an ordinary quantity, and - 



a common fraction, just like T, so that the function is 

 only formally one of variable magnitudes. On the 



y2 y 



contrary, if =p, - has no determinate quotient, and, in 



x x 



fact, x has no ratio to y, but only to y 2 . Now the relation 



1 Hegel absolutely identifies analysis with arithmetical process 

 &quot; Auf analytische d. i. ganz arithmetische Weise &quot; (iii. 328). Had 

 Hegel ever studied the treatment of incommensurables in ordinary 

 algebra ? If algebra is &quot; ganz arithmetisch,&quot; the whole doctrine of 

 indices is false. 



