i87s] HEGEL AND THE CALCULUS 37 



of a magnitude to a power is not a quantum, but quali 

 tative.&quot; It is needless to say that the man who could 

 make &quot; no constant ratio &quot; identical with &quot; no ratio/ 

 and who did not see that J$x 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 occurring 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 



differentiation, - not being qualitative, we may at least 

 t 



conclude that Hegel does not regard uniform motion as 

 continuous ! 



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

 Hegel, to consider the equation y = x n ; the advance to 

 more complicated functions is quite mechanical. Now 

 both y and x are really numbers, and so may be expressed 

 as sums. (This, of course, is a very bold assumption, as 

 Hegel says nothing of the possible case of incommensur- 

 ables.) The simple and yet comprehensive way of repre 

 senting x as a sum is to write it as binomial. Now expand 

 x n as a binomial function, and we have a series of terms 

 which are &quot; wholly functions of the potentiation and 

 the power.&quot; The differential calculus seeks the relation 

 between these terms and the original components of x. 

 As we are not concerned with the sum, but merely with 

 the relation of the terms of the expansion, it would be best 

 simply to expand (x + i) n , and to define the particular 

 &quot; Potenzenbestimmung &quot; got by considering the second 

 term of this series as the first derived potence-function 

 of y. In short, the true mathematical commencement in 



