ELISHA MITCHELL SCIENTIFIC SOCIETY. 23 



Comparing the last equation with 

 dy =f^ (.r) dx, 

 obtained from an equation above, we see that when dx = A.r, 

 dy == Ai' — Zi'dx. 

 Thus we have proved analytically, when dx = A.r that 

 dy is never equal to aj (except when f{x) represents a 

 straight line) and the difference is exactly represented on 

 the figure by the distance IS. 



Mauy of the older writers, following the lead of Leibnitz, assumed that 

 dy and aJ'. as well as ds and {^s, were identical when dx = A-f, and the 

 error is perpetuated to this day by possibly the majorit}' of the most recent 

 writers; thus Williamson, in his Differential Calculus (6th Ed., 1887), page 

 3, says, "When the increment or difference is supposed infinitely small 

 it is called 2i differential.''' Similarly in a recent American treatise on 

 the calculus by Bowser, the same definition is given. 



Professor Bowser defines " consecutive values of a function or variable 

 as values which differ from each other by less than any assignable quan- 

 tity."' He then adds. "A differential has been defined as an infinitely 

 small increment or an infinitesimal; it may also be defined as the differ- 

 ence between two consecutive values of a variable or function." 



As there are an infinite number of values lying between o and "any 

 assignable quantity," however small, it follows that such differentials are 

 simply quite small finite quantities. 



The differentiation of a function, as y = ax- -\- b, would then pro- 

 ceed after the method of Leibnitz, as follows : 



y := ax'^ + b. 



Give to .r and y the simultaneous infinitesimal increments dx and dy, 

 y -^ dy = a (x -\- dx)^ -\- b. 

 Subtracting the first equation from the last, we have, 

 dy =a (2 xdx -f- dx'^). 



Now from the nature of infinitesimals, it is regarded by the followers of 

 Leibnitz as evident that dx^ can be neglected in comparison with 2 xdx, 

 because the square of the infinitesimal dx is infinitely small in compari- 

 son with the variable itself, whence, 



dy ^ 2 ax dx. 



It is scarce!}' necessary to remark to the reader that, for an exact result, 

 we cannot make dx = o in part of an equation without making it zero 

 throughout; so that the equation is fundamentally wrong. 



When we go to the applications to curves, however, another error is 

 made, of an opposite character to the first, so that by this secret compen- 

 sation of errors the result is finally correct. Thus a curve is regarded as 

 a polygon whose sides connect "consecutive points " and a tangent line 



