1873] HEGEL AND THE CALCULUS 29 



half an hour would suffice to learn the calculus. Certainly 

 he might have employed a good many hours in unlearning 

 his false conceptions of it. 



Hegel has next something to say about the way in which 

 mathematicians have developed the details of the calculus. 

 Since none of them had a clear notion of the matter in 

 hand, their proofs, we are told, are very weak. They 

 always fall back into methods merely approximate, 

 subjecting infinitely small quantities to the laws of finite 

 quanta, and yet rejecting them as relatively unimportant, 

 in despite of these laws. Of course, adds Hegel, we need 

 not look for the rigour of demonstration of the old geo 

 metry, for the analysis of the infinite is of a nature 

 essentially higher than that geometry. However, mathe 

 maticians have sought this rigour, and they have all 

 failed. Of course, it would be easy for any one to point 

 out numerous mathematicians who have failed ; but let 

 us simply ask whether Newton has done so. Hegel 

 unhesitatingly affirms that he has, and Dr. Stirling is 

 jubilant at the discovery. 



The error is supposed to lie in the deduction in Prin. ii. 

 Lem. 2, of the fluxion of a product. The statement of 

 Newton is as follows : If A B be two quantities increas 

 ing continuously, and their moments or rates of change a 

 and b, the moment or change of the rectangle A.B is 

 aB + bA. By moment Newton does not mean the incre 

 ment actually received in any time, however short, but the 

 nascent principle of the fluent quantity a notion, of 

 course, made clear by the previous discussion of prime 

 and ultimate ratios. The moments, in fact, are any 

 quantities proportional to the rates at which A and B 

 are flowing the products of the fluxions of A and B by 

 an arbitrary increment of time. If moments, then, are 

 called increments, the meaning is increments which would 

 be received if the rate of flow remained constant, and the 

 ratio of two moments is simply the ratio of the fluxions, 

 and therefore equal to the limit of the ratio of the actual 



