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Corresponding differentials are, as near as we please, proportionals to correspond- 

 ing and indefinitely small increments of variables, and to speak acctirately, they are 

 corresponding limits of such proportionals. 



The power and generality of this definition can only be understood 

 after a careful study of its consequences. It applies whatever the num- 

 ber of independent variables. It is the mathematical foundation of 

 Newton's conception of the state of change of variables, in which cor- 

 responding differentials are made to signify corresponding increments. 

 In other words, corresponding increments of a state of change of ixtriahles 

 are as near as we please, proportionals to corresponding and indefinitely 

 small increments of the varial)les. 



As an illustration of the method, consider z = xy, and as usual, let 

 A^. A?/i A 2^. denote any corresponding increments of .r, y, z. Then, 



As=.r Ay + .V A-i- + A-^' • Ay 



Let iV be a variable number which becomes indefinitely large in any 

 way whatever (as N^=\, 2, 3, 4, and so on indefinitely). Conceive A^» A2/> 

 to diminish as A' increases, so that their proportionals, N/\x, Nl\y, remain 

 finite and approach limits designated by dx, dy {l\x = dx/N-\-S'N^, 

 /\y^dy/ N-\- 5/N', for example). Then if dz denote the limit of the re- 

 maining proportional JV^ z, the equation from which it is to be determined 

 is N/\z = xN_^y -ir- yNr\x + N'/\x. l\y, which gives, by the theorems of 

 limit, dz = xdy + ydx. 



Here, the ratio dz/dx is absolutely indeterminate, since it depends upon 

 the values chosen for d.r, dy. 



Leibnitz rediscovered the calculus in 1676, and immediately published 

 his methods and spread them over Europe. His right to the title of inde- 

 pendent discoverer was disputed by the friends of Newton, because when 

 Leibnitz was just turning his attention to mathematics in 1673, he visited 

 London and consulted some manuscripts of Newton. Leibnitz's defense is 

 that he did not see the manuscript on the calculus, and his notes taken at 

 the time, and afterwards discovered, contain only references to Newton's 

 papers on optics. It is fortunate in respect to notation that we have 

 received the calculus from the hands of Leibnitz rather than Newton ; but 

 the history of the calculus, from Leibnitz on, revolves about objections to 

 his infinitesimal methods. In order to avoid those methods, Lagrange 

 recast the calculus into practically its present form. He regarded the 

 differentials of the independent variables as their small actual increments, 

 and the differential of a dependent variable as that part of its increment 

 which is of first degree when it is expanded in ascending powers of the 



