104 SUPPLEMENT TO CHAP. VIT. [CHAP. TJ. 



crements, since the increments are then limiting increments, and can be no 

 smaller without disappearing. 

 We have thus 



dy 



~ = 5 x 2 x, 

 d x 



which is an exact relation between the increments upon this supposition. 

 From this we have d y = 5x2 x d x. 



If now we sum up all the increments d y, then by virtue of the supposi- 

 tion we have made, / d y must equal y. We thus suppose y to flow, as it 



were, unbrokenly along by the consecutive increments d y, just as the side 

 of a triangle moving always parallel to itself, and limited always by the 

 sides, describes the area of that triangle, while the change d y of its length 

 is the difference between two immediately contiguous positions. Upon 



d y 

 this supposition, we repeat, -= is the limit of the ratio of the increments, 



a '*c 



which limit is, as we see from (4), equal exactly to 5 x 2 x. We do not 

 reject or throw away d x from the right of that equation " because of its 

 small size with reference to 2 *," nor, thus rejecting it upon one side of the 

 equation, do we retain it upon the other " in order to retain a trace of 

 the letter *" !; but simply pass to the limit, and then, according to our 

 fundamental principle above, equate those limits themselves. But if 

 / d y = y, then the integral of 5 x 2xdx, or/5 x 2xdx = y = 5x*. 



By "differentiating," as we say, equation (1) we get (5), and by "inte- 

 grating" (5) we obtain (1). 



Hence we see the appropriateness of the term "fluent" given by Newton 

 to the quantity d y or 2 x d x. So also we see the appropriateness of the 



dy 

 term " ultimate ratio " * for -7 itself. 



d x 



* Liebnitz undoubtedly discovered the calculus independently of Newton, 

 but he considered (fa; as a quanity so " infinitely" small that in comparison 

 with a finite quantity it could be disregarded " as a grain of sand in compari- 

 son with the sea." We see, indeed, from eq. (4) that if d x upon one side be 



d x 



zero, we get the same value for - as before. But if d x is zero on one side, 



ay 



it should be zero on the other side also. No matter how small we suppose d x 

 to be, we have no right to get rid of it by disregarding it. That Liebnitz rec- 

 ognized this cannot be doubted, and he was therefore inclined to consider his 

 method as approximate only. But to his surprise he found his results exact, 

 differing from the true by not even so much as a " grain of sand." There was 

 to him ever in his method this mystery, nor could he conceive what these 

 quantities could be which, though disregarded, gave true results. Bishop 

 Berkeley challenged the logic of the method, and adduced it as an evidence of 

 "how error may bring forth truth, though it cannot bring forth science." 

 Strange to say, even the disciples of Newton were unable to answer Berkeley 

 without taking refuge in the undoubted truth of their results. And yet New- 

 Ion in his Principia lays it down as the corner-stone of his method, that 

 " quantities which during any finite time constantly approach each other, and 



