AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 501 



mathematicians try to get over this by saying that p is not really = q , but is 



only a definite value, to which g comes as near as you please. Of course, 

 if this is so, it is as evident as anything can be that the difference between p and 

 ~ is not a quantitative one. But, adds the philosopher, naively enough, that 



doesn't help one over J£ = q . Suppose now that we were to say j x really = p (a 



definite quantity), as, in fact, mathematicians do say, then it is obvious that fix 



couldn't have been = 0. Or if, finally, it is conceded that y = ° (which Hegel 



seems to think most likely, since by and dx vanish together), then what can 

 p be? 



Now, can any one say that the man who devised this argument knew what 

 he was doing? When did any mathematician suppose that after evaluation 



q is indeterminate ? Or had Hegel never read Newton's first lemma, with its 

 " fiunt ultimo sequales"? Or, again, if Hegel allows that there is no quantita- 

 tive difference between p and ^, why does he assume a qualitative one ? Or, above 



all, why try to explain Newton's doctrine without ever deigning more than a 

 contemptuous glance at the one central point of the whole ? Hegel boasts that 

 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 unim- 

 portant, in despite of these laws. Of course, adds Hegel, we need not look for 

 the rigour of demonstration of the old geometry, for the analysis of the infinite 

 is of a nature essentially higher than that geometry. However, mathema- 

 ticians 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 increasing continuously, and their moments or rates of change a and b, 

 the moment or change of the rectangle AB is «B + bA. By moment Newton 

 does not mean the increment actually received in any time, however short, but the 



