•502 MR W. ROBERTSON SMITH ON HEGEL 



nascent principle of the fluent quantity — a notion, of course, made clear by the pre- 

 vious discussion of prime and ultimate ratios. The moments, in fact, are any quan- 

 tities 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 incre- 

 ments, while it is quite independent of the magnitude of the separate moments. 

 Now, says Newton, when A and B are diminished by half their moments, the 

 rectangle is AB — \ «B — \ bk + \ ab; and when A and B are increased by half 

 their moments, it is AB + ^ «B + \ bk -f £ ab ; and so to the increments a and b 

 in the sides corresponds an increment «B + bk in the rectangle. This demon- 

 stration is certainly very curt, and intended only for those who have mastered 

 Newton's fundamental notions, and may therefore be saved the tedium of a long 

 reductio ad absurdum. More at length, the proof would be of this kind. The 

 fluxion of the rectangle must, since the flow is continuous, be a definite quantity, 

 depending only on the magnitudes and fluxions of the sides at each moment. 

 Thus the fluxion of AB will be unchanged, if we suppose that from the values 

 A — ^ a, B — l b the sides flow with uniform velocity, equal to A and B, until 

 they attain the values A + I a, B + -*- b. In this case the increments a and b 

 will represent exactly upon the same scale the fluxions A and B. Meantime, the 

 rectangle has been flowing with a constantly increasing velocity, which at the 

 moment when the value iVB was reached, was the velocity Newton is seeking to 

 determine. The whole increment of the rectangle is aB + bk, which therefore 

 represents the', average velocity of the rectangle on the same scale as a,b repre- 

 sent the uniform velocities of the sides. Clearly the average velocity with which 

 the increment is described is greater than the velocity at the beginning of the 

 motion, and less than that at the end, and therefore, since the velocity is continuous, 

 is strictly the velocity at some intermediate point. But this point can be none 

 other than that at which the rectangle = AB, for were it any other point, we 

 could take a and b small enough to throw this point out, and there would still be 

 another point at which the fluxion of the rectangle must = aB + bk. But this is 

 contrary to the intuitive fact that the velocity is continuously increasing. To the 

 mathematician, however, this round-about process is unnecessary. He sees at 

 once that if the average velocity is independent of the duration of flow, and 

 depends solely on a certain point being included within the flow considered, the 

 velocity at that point must be strictly the average velocity, for in the limit the 

 two coincide. 



Now, Hegel, of course, did not see this, because he would not admit the 

 kinematical reality of fluxions. He, therefore, supposes that Newton wants to 

 find the differential of AB— a way of stating the problem which Newton would 



