92 LECTURES AND ESSAYS [1869- 



fluxions matter of mere definition, and thereafter to seek 

 for them a geometrical or other practical meaning. To 

 this end, Lagrange proves (but in a way not convincing) 

 that it is always possible to expand f(x + i) in ascending 

 powers of i, and defines the coefficients of the expansion 

 as derived functions (fluxions) of / x. Hegel, to simplify 

 matters, expands, not f(x + i), but/(# + 1), i.e. takes unity 

 as the increment of x. In view of this change, his process 

 becomes what I have formerly called &quot; an excessively 

 clumsy adaptation of the method of Lagrange.&quot; Yet 

 Dr. Stirling without comparing, without even professing 

 to compare, Hegel with Lagrange, &quot; sees no reason for 

 assuming Hegel not to have correctly described what he 

 had simply before him&quot; (viz. Lagrange s method), and 

 finds that in using the word adaptation I have &quot; mis 

 apprehended what it is all about &quot; and &quot; fooled myself 

 into a formal wrangling against a mathematical ratiocina 

 tion that nowhere exists.&quot; That is, Hegel shall make the 

 mathematical change from i, which is sense, to 1, which is 

 nonsense, and the mathematician who checks him shall 

 be &quot; fooling himself.&quot; x 



The next step is to find the use of the fluxions after we 

 have derived them, and this Hegel does, still in dependence 

 on Lagrange, for the special problem of the tangents of 

 curves. As Lagrange in this problem uses from the very 

 first x + i, and not Hegel s useless x + 1, there is something 

 touching in the naiveti with which Hegel here throws him 

 self into the arms of his friend. But soon his natural 

 independence of spirit revives, and, having signalised 

 himself by proposing to throw away a constant in one of 

 Lagrange s equations, by which the meaning of the equa 

 tion is utterly changed, he is emboldened to tamper again 

 with the i t to which he seemed for a moment to have 



1 In re-reading my paper, I observe with regret that on p. 507, 1. 10, 

 i has been printed instead of 1. This erratum obscures the magni 

 tude of Hegel s divergence from Lagrange, but should have been at once 

 observed by any one who had his eye on the quite explicit statements 

 of Hegel (iii. pp. 326, 327) to which I refer. 



