AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 507 



the equation x = y"; 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 incommensurables.] The simple and yet comprehensive way of representing 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 " wholly functions of the potentiation and 

 the power." 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 "Potenzenbestimmung" got by con- 

 sidering the second term of this series as the first derived potence-function of?/. 

 In short, the true mathematical commencement in this part of analysis is no more 

 than the discovery of the functions determined by the expansion of a power. 



We see at once that this is simply an excessively clumsy adaptation of the 

 method of Lagrange, which is based on the proposition thatf(x + i) can always 

 be expanded in a series of ascending integral powers of i, and then defines the 

 successive fluxions [or derived functions] of fx with reference to the series. 

 Hegel adds to Lagrange nothing but confusion, and a degree of vagueness which 

 is quite pitiable; and, of course, his method has the same fundamental fallacy as 

 that of Lagrange, in so much &sf(x + i) cannot always be expanded as Lagrange 

 proposes, or what comes to the same thing, the details of the calculus cannot be 

 deduced by processes purely arithmetical from the definition (for it is no more) 



dx n n—l 



-t— — n x . I do not, therefore, think it needful to go into details on this part 



of Hegel's method. The really important point is the use to be made of these 

 magical " Potenzenbestimmungen," which, according to Hegel, depends on the 

 discovery of concrete relations which can be referred to these abstract analytical 

 forms. Hegel proceeds as follows: — 



There is always a fall of one dimension in passing to the first derived function. 

 Hence the calculus is useful in cases where we have a similar fall in the powers. 

 We are also to remember that, by differentiating an equation, we get not an 

 equation but a relation. Whenever, then, we wish to investigate relations con- 

 nected with any equation, but of a lower dimension, we have room for the cal- 

 culus. A case in point is the investigation of the relations between the tangent, 

 subtangent, and ordinate, for example, in a curve of the second degree. These 

 relations are linear, while the equation contains squares. They depend, there- 

 fore, on the first derived function (pp. 341, 342, 344). 



That such a statement is mere guess work is clear, if we observe that 



by a linear relation Hegel means indifferently the ratio of two straight lines, 



or a ratio involving only first powers of x and y. Or, again, since the value 



of the radius of curvature is also on Hegel's principles linear, why does 



vol. xxv. PART II. 6 p 



