Professor Wallace in Reply to the Remarks oj B, 99 



leads to results in the higher analysis, which have not been 

 rigorously and logically established, either by Newton, 

 Leibnitz, or any of their followers, down to Lagrange. 

 The whole of their methods, notwithstanding the applica- 

 tion of the principle of exhaustions,of indivisibles, of the the- 

 ory of limits, of prime and ultimate ratios, the expansion of 

 binomials, multinomials, &c. in point of perspicuity and log- 

 ical precision, are still liable to the objections of Berkeley, 

 their reasoning being more or less infected with thefallacia 

 suppositionis, or as he calls it shifting of the hypothesis. 

 Even Lagrange in expanding the form y (x+«) whch he 

 makes the principal fonndation of his theory of analytical 

 functions, is liable to similar objections. " It appears in 

 short to me (says Woodhouse in his Preface to his Principles 

 of Analytical Calculation) that M. Lagrange has generalized 

 too hastily, and inhis general form and demonstration has 

 virtually included properties which he makes the consequen- 

 ces of that form and demonstration."" The results deduced 

 from the simple multiplication of Stainville's series are not 

 liable to the objection of the fallacia suppositionis. The 

 laws of the expansion of binomial or multinomial func- 

 tions are not assumed, as in most cases in Lagrange's meth- 

 od. They follow as consequences from the results ob- 

 tained, and these results are applicable to almost every 

 department of analytical functions. These were my mo- 

 tives in calling the attention of Mathematicians to some of 

 the properties, deduced from the multiplication of these 

 series, in your Journal. • 



But Mr. B. observes that " these series can hardly be 

 called new." Admitting this, is the application of them 

 the less important ? The reason however which Mr. B. 

 gives why they are not new, since they may be produced 

 by the expansion of a binomial, has certainly the merit of 

 novelty in it, if no other. Will Mr. B. then pretend to as- 

 sert, that no series can be new which results from the ex- 

 pansion of a binomial ? The expansion of the binomial 

 itself results from common multiplication in algebra, and 

 even the whole body of the modern analysis may be dedu- 

 ced from it. Woodhouse in the Preface to the work al- 

 ready cited, p. 25, remarks, that " between the differen- 

 tial Calculus and the rule for multiplication, the interval is 

 not immense. It is that compendious method of addition 



