1873] STIRLING AND HEGEL 75 



product, of an exponential magnitude, and so on, flows 

 mechanically therefrom.&quot; That is, the method of the 

 calculus is settled (absolvirt) when the fluxion of % H is 

 known. And Dr. Stirling must see that if I know the 

 fluxion of xy, I have only to put y = % to get the fluxion of 

 x 2 , from which, by an ordinary mathematical induction, 

 I can again rise to # 3 , # 4 , and finally to x n , so putting myself 

 in possession of the whole method. Thus surely the vital 

 nature of the deduction of the fluxion of the product xy 

 is abundantly manifest, and if Newton deceived himself 

 in this matter the whole structure of his calculus is built 

 on delusion. For Newton certainly imagined that his 

 method rested on a system of strict mathematical deduc 

 tion, in which &quot; the very smallest errors are not to be 

 neglected &quot; (Introd. ad quadraturam curvarum), and the 

 keystone of this system of deduction is just the fluxion 

 of a product. 



Will it then be believed that under these circumstances 

 Dr. Stirling flatly denies that Hegel in this matter accuses 

 Newton of any error whatever ? The only errors of 

 Newton to which Hegel has made any allusion are, we 

 are told, two quite different mistakes, made known to the 

 philosopher by Lagrange and Schubert respectively. For 

 in the matter of the fluxion of the product, Hegel admits 

 the result to be right, and objects only &quot; to the formal expedient 

 in justification of it,&quot; that is, to the mathematical proof of 

 the result (p. 109). Is not Dr. Stirling aware that the 

 gravest charge of error that can be brought against a 

 mathematician is by universal consent just this, that he 

 has brought out a true result by expedients formally 

 incorrect ? Were Dr. Stirling a man of less moral earnest 

 ness, one would be tempted to see in such an argument 

 a mere play of conscious sophistry. But manifestly in 

 this case the apparent jest is put forth in bitter earnest. 

 It is even made matter of grievous accusation against me 

 that I have &quot; enormously misrepresented &quot; Hegel, &quot; con 

 founding logical unsatisfactoriness with mathematical 



