a ie i ala a 
a 
Professor Wallace in Reply to the Remarks of 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 enema: 
The whole of their methods, notwithstanding the applica- 
tion of the principle of exhaustions s,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 the fallacia 
suppositionis, or as he calls it shifting of the hypothesis. 
Even Lagrange in expanding the form f (x+2) whch he 
makes the principal fonndation of his theory of analytical 
functions, is liable to similar objections. “It appears in 
short to me ea W oodhouse i in bee Preface to his Principles 
of Analytical C 
too hastily, and inhis os aa ‘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 
partment 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 
— in your Journal. — 
r. B. observes’ 
called new.” Admittin 
at ‘these series can hardly be 
is the application of them 
on however which aie 
merelty i in it, if no other. will B. then Boris to as- 
that no series can be new neh results from the ex- 
Fo A of a binomial ? ; expan of the binomial 
itself results from common m tipli 
ready cited, p. 25, r that ‘‘ between the differen- 
tial Calculus and oy ae ‘for Puitptication, the interval is 
notimmense. Itis that compendious method of addition 
