30 LECTURES AND ESSAYS [1869- 



increments, 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 - JflB - |6A + %ab ; and when A and B are increased 

 by half their moments, it is AB + J#B + JfrA + \db ; and 

 so to the increments a and b in the sides corresponds an 

 increment 0B + bA in the rectangle. This demonstration 

 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 absur- 

 dum. More at length, the proof would be of this kind. 

 The fluxion of the rectangle must, since the flow is con 

 tinuous, 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 - %b the sides flow with 

 uniform velocity, equal to A and 6, until they attain the 

 values A + %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 AB was reached, was the velocity 

 Newton is seeking to determine. The whole increment of 

 the rectangle is aB + bA, which therefore represents the 

 average velocity of the rectangle on the same scale as a, b 

 represent 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 = aE + bA. But this is 

 contrary to the intuitive fact that the velocity is con 

 tinuously increasing. To the mathematician, however, 



