38 LECTURES AND ESSAYS [1869- 



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 that f(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 funda 

 mental fallacy as that of Lagrange, in so much as f(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 



dx n 

 definition (for it is no more) -y- = nx n ~ l . 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 &quot; Potenzenbestimmungen,&quot; 

 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 connected with any equation, 

 but of a lower dimension, we have room for the calculus. 

 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, 

 therefore, on the first derived function (pp. 341, 342, 



344). 



That such a statement is mere guesswork is clear, if 

 we observe that by a linear relation Hegel means indiffer- 



