i8 73 ] HEGEL AND THE CALCULUS 31 



this roundabout 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 in 

 cluded 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, there 

 fore, supposes that Newton wants to find the differen 

 tial of AB a way of stating the problem which Newton 

 would have rejected as misleading. The differential 

 can be nothing else than (A + dA)(B + dB) - AB. But 

 Newton writes instead of this (A + $dA) (B + JdB) - 

 (B- |^A)(B - JdB), thereby making an error in so 

 elementary a process as the multiplication of two bi 

 nomials ! But where is Hegel s justification for saying 

 that what Newton is seeking is (A + dA)(B + dB) - AB ? 

 Newton says nothing about differentials at all ; his a is, 

 as we have seen, not the infinitely small increment of A, 

 but an arbitrary multiple of the fluxion of A, which need 



not be infinitely small. Newton s - - is, if you please, 



_ T , (A + dA) (B + dB) - AB 

 ~IK~ &quot; ; 



but even this, which is very different from what Hegel 

 writes, is simply a different, by no means a more funda 

 mental, view of the problem than Newton s. 



Dr. Stirling tells us that Hegel s expression is what 

 Newton s says his is, &quot; the excess of the increase by a whole 

 dA and dB. But what Newton says is only that when 

 the sides are increased from A - \a and B - J&, through 

 increments a and b the rectangle increases by aB + bA. 

 That this is true surely cannot be denied. In fact 

 (A + a) (B + b) - AB would have represented not the 

 velocity at value AB, but the average velocity of the 

 rectangle during the interval between values AB and 



