•506 MR W. ROBERTSON SMITH ON HEGEL 



quality. And further, there was in Hegel a rigid determination not to see the 

 real qualitative difference between the continuous quantity of the higher analysis 

 and of actual nature, and the discrete quantity of arithmetical abstraction.* 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 explanding y in terms of x, we 

 shall be quit " of the formal categories of the infinite, and of infinite approxima- 

 tion, and of the equally empty category of continuous magnitude" (hi. 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 : — " In the equation 



- = a the relation of y to x is an ordinary quantity, and - a common fraction, 

 just like j , so that the function is only formally one of variable magnitudes. On 



2 



the contrary, if ^- =p, - has no determinate quotient, and, in fact, x has no ratio 



to y, but only to y 1 . Now the relation of a magnitude to a power is not a quantum, 

 but qualitative." It is needless to say that the man who could make "no con- 

 stant ratio" identical with " no ratio," and who did not see that Jpx has a 

 definite value for each value of x, or who did not see that p is a quantum, though 

 not of the same dimensions as y 2 (which probably was what confused Hegel), is 

 hardly fit to construct a new theory of the calculus. But let us pass on. 



The subject matter of the calculus is then, we are to believe, equations in 

 which one variable appears as a function of a second, one of these at least occur- 

 ring in a power higher than the first. In such a case the variation of the variables 

 is qualitatively determined, and therefore continuous. It would be vain to ask 

 why; but since we are told that in the equation s — ct there is no scope for differ- 

 entiation, - not being qualitative, we may at least conclude that Hegel does not 



regard uniform motion as continuous ! 



So far as the principle goes it is quite sufficient, continues Hegel, to consider 



* Hegel absolutely identifies analysis with arithmetical process — " Auf analytische d. i. ganz 

 arithmetische Weise" (iii. 328). Had Hegel ever studied the treatment of incoramensurables in 

 ordinary algebra 1 If algebra is " ganz arithinetisch," the whole doctrine of indices is false. 



