24 JOURNAL OF THE 



at any point is the chord produced through this point and its " consecutive 

 point," so that dy -^- dx gives the slope of the tangent at the point. This 

 is of course wrong, but it exactly balances the other error above, for from 

 the last equation we find the slope, so determined for the curve j = ax"^ -\- b, 

 to be (2 rt-.i"), which we know to be correct by the strict method of limits. 



The great French author, Lagrange, says in this connection, "In regard- 

 ing a curve as a polygon of an infinite number of sides, each infinitely 

 small, and of which the prolongation is the tangent of the curve, it is 

 clear that we make an erroneous supposition; but this error finds itself 

 corrected in the calculus by the omission which is made of infinitely small 

 quantities. This can be easily shown in examples, but it would be, per- 

 haps, difficult to give a general demonstration of it." 



Bishop Berkele}-, long before Lagrange, showed this secret compensa- 

 tion of errors in a particular example, his endeavor being particularly to 

 ''show hoiu error may bring forth truth, though it ca7tnot bring forth 

 scietice. ' ' 



Leibnitz, in attempting a defense of his theor}-, stated that "he treated 

 infinitely small quantities as i^icojnparables, and that he neglected them 

 in comparison with finite quantities 'like grains of sand in comparison 

 with the sea '; a v.ew which would have completeh- changed the nature of 

 his analysis by reducing it to a mere approximative calculus." See 

 Comte's Philosophy of Mathematics, Gillespie, p. 99. 



The demonstration given above in the case ofy"(.r) = ax^ -{- b can be 

 made general, as follows : 



From the exact equation (9) above, following the notation of Leibnitz 

 where dy and dx are taken as identical with AjF and A-^', we have exactly, 



dy 



- ==/! (.r) 4- zt>. 

 dx 

 The followers of Leibnitz, in differentiating, throw away the term w as 

 nothing and pretend to write exactly, 



dy 



- =f' i-r); 

 dx 



but as they make another error by calling the ratio of the increments dy 

 and dx the slope of the curve, we thus find the latter to equal/ ^ (j), which 



Ay 

 was assumed above to equal lim. — or the slope of the curve; so that the 



AX 

 two errors, for au}- function, balance each other and w^e reach a correct 

 result. 



As the truth of any result, as given by the Leibnitz method, can only 

 be tested, in a similar manner to the above, by comparing with a result 

 known to be correct by use of the method of limits, it would seem to be 

 inexcusable not to found the calculus upon this latter method. After- 



