CRITICISMS BY BRITISH WRITERS 265 



called tedious and prolix. Of the commentators on 

 the method of fluxions, Maclaurin is to be esteemed 

 most acute and judicious, but his Introduction 

 exhibits rather the exertions of a great genius 

 struggling with difficulties, than a clear and distinct 

 account of the subject he was discussing." To 

 remove this prolixity, it was proposed, conformably 

 to the notions of Newton, to cali in the doctrine of 

 prime and ultimate ratios or of limits. Euler and 

 D'Alembert, on the other hand, rejected motion, but 

 retained limiting ratios, failing, however, in supply- 

 ing a satisfactory explanation therefor. Wood- 

 house is the earliest English mathematician who 

 speaks in respectful and appreciative terms of 

 the services to mathematics rendered by Bishop 

 Berkeley. In fact, Woodhouse admits as valid 

 some of Berkeley's objections which had been 

 declared invalid. The methods of treating the 

 calculus **all are equally Hable to the objection of 

 Berkeley, concerning the fallacia suppositionis, or 

 the shifting of the hypotJiesis.'' Thus, in fluxions 

 and the method of limits, x is increased by /, and, 

 in the case of x'"^ the increment of the function, 



divided by /, is inx"'~^-\ — ^^ ^;ir'"~VH-, etc. ; then, 



z 



putting, /=o, there results mx"'~^. But since the 

 expansion of {x-^-i)'" was effected '*on the express 

 supposition, that i is some quantity, if you take 

 /=o, the hypothesis is, as Berkeley says, shifted, 

 and there is a manifest sophism in the process " 

 (p. xii). 



