NEWTON. 189 



and their respective partizans a bitter controversy as to priority of in- 

 vention, and charges of plagiarism were made and resented on either 

 side. Those who have impartially reviewed all the circumstances con- 

 nected with this dispute, have come to the conclusion that Newton 

 and Leibnitz each independently discovered the method which the one 

 called fluxions and the other the differential calculus. It so happened, 

 however, that the nomenclature adopted by the German philosopher 

 possessed special advantages, and in the hands of the Continental mathe- 

 matician the calculus made rapid progress. The methods which we 

 have seen were devised by Cavalieri and others for passing to the more 

 recondite problems, involved the principle of limits, while they avoided 

 the excessive tediousness of the methods employed by the ancient 

 geometers. We have also explained that some of these methods were 

 almost identical with cases of the fluxional or differential calculus. But 

 this last was essentially distinguished by presenting a general method 

 of investigating all the cases, and of offering the means of solving other 

 problems yet unattempted. The common principle of Newton's and 

 Leibnitz's methods is that of considering an algebraical quantity as in- 

 volving in a definite manner some other quantity which receives varying 

 values, and is hence termed the variable, of which the complex quantity 

 is called the function. Now, when the variable changes uniformly, the 

 function is usually such that its dependent changes proceed at a vary- 

 ing rate. Both Newton and Leibnitz demonstrated, each in his own 

 manner, that the rate of the variation of the function, compared with 

 the rate of the variable, could be expressed in terms of the variable. 

 Leibnitz proposed to compare the infinitely small elements of the 

 variable and its function generated simultaneously, and discovers the 

 finite ratio which, as he conceived, might exist between them. Thus, 

 for example, referring again to Fig. 83, we must suppose that when 

 the point p, which sweeps out the curve, has advanced an extremely 

 small distance, there will be formed the triangle like P a b, extremely 

 (infinitely) small, so that its hypotenuse may then be conceived as 

 coinciding with the curve ; then the two sides of the triangle formed 

 by the increase of the ordinate and that of the abscissa will give the 



same ratio as before. Leibnitz expresses this ratio by the symbol ^, 



which he terms the differential coefficient. 



Newton and Leibnitz give rules by which the differential coefficient 

 of any function may be found, the process being more or less long or 

 difficult according to the nature of the function. It has been indi- 

 cated by the explanations already given that \\\z fluxion or differential 

 coefficient is really the limit of the ratio towards which two quantities, 

 always connected by some definite relation, approach as the quantities 

 themselves are diminished. Neither of the inventors of the calculus, 

 however, expressly referred its principle to the notion of a limit. 

 Newton expressed the principle by ideas derivqd from mechanical 



