i8 73 ] DISCUSSION ii 



somewhat similar) and say, in effect, that it is essentially 

 erroneous. The fact, however, is that, as in far greater 

 matters, Newton here shows his profound knowledge of 

 the question in hand ; and adopts, without any parade, 

 a method which gives the result true to the second order of 

 small quantities. The metaphysicians cannot see this, 

 and Dr. Stirling speaks with enthusiastic admiration of 

 the clear-sightedness and profundity of Hegel in detecting 

 this blunder, and for it &quot; harpooning &quot; Newton ! 



What Newton seeks is the rate of increase of a quantity 

 at a particular instant. Instead of measuring it by the 

 rate of increase after that instant (as the metaphysicians 

 would require) he measures it by observing, as it were, 

 for equal intervals of time before and after the instant in 

 question. 



Any one who is not a metaphysician can see at once 

 the superior accuracy of Newton s method, by applying 

 both methods to the case of a rapidly varying velocity ; 

 such as that of a falling stone, or of a railway train near 

 a station. 



In reference to what Professor Tait had said, Mr. 

 Sang remarked that the line of argument attributed to 

 Newton had been used by John Napier before Newton s 

 birth. Napier s definition of a logarithm runs thus 

 (Descriptio, lib. i. cap. i. def. 6), (Constructio, 23, 25) 

 that if two points move synchronously along two lines, 

 the one with a uniform velocity (arithmetice), the other 

 (geometrice) with a velocity proportional to its distance 

 from a fixed point, the distance passed over by the first 

 point is the logarithm of the distance of the second from 

 the fixed point. In order to compare this variable 

 velocity at any instant with the constant velocity, he 

 takes a small interval of time preceding, and another 

 succeeding the given instant, shows that the true velocity 

 is included between the two velocities thus obtained, and 

 (28, 31) takes the arithmetical mean as better than either, 

 and as true (inter terminos). 



