100 Professor Wallace im Reply to the Remarks of B. 
which is the low basis of the most towering speculations, 
the humble Origin of the Sublime Geometry.” Now this be- 
ing the case, according to Mr. B’s. mode of reasoning, 
there is nothing new in the modern improvements in dn- 
alysis, for all are deduced from multiplication, which 
been known time immemorial. In fine Mr. B. seems to 
adhere literally to the observation of Selena: us mihi sub 
um.” 
Mr. B. next observes that nearly the whole theory of 
the functions, which I have named, is to be found in the 
“Complement des Eléméns a Algebre”? of La Croix, 
where it is stated that the method was first given by Euler. 
Now any person who reads La Croix will find, that Euler 
pre-supposes the knowledge of the expansion of a binomi- 
al function, and the results which he has given do not in- 
clude a single case of a transcendent function, and were 
only given as examples of the applications of the simplest 
the binomial theorem, viz. (1+2z)"=1+7z2+ 
m (n=) +&c.=f ) where m is ones a whole 
= 
B. « m Ae 
ledge of ny: abor ve expansion. Now (m) to be general, 
even in this limited case, shone joeleas the ae of 
a '**, log. (1+2z), Sin. (1+2), Cos. (142), & Where 
hve! these binomials been expanded by Euler, by any of 
the ordinary operations of algebra, such as multiplication ? 
Before Euler deduced his fundamental theorem, these ex- 
pansions should have been given, whatever m might be 
made torepresent. But in the results deduced from the 
multiplication of Stainville’s series, the expansions of the 
binomial and multinomial follow, as is evident from 
283 vol. VII. no. 2 of the Journal, and in p. 284 the expan- 
sions of e**,a”*, Log. (1+2), &e. are given, e being the 
base of Napier’s System of Logarithms, and e* =a where 
A==log. a to Napier’s System. The expansions of circu- 
lar functions into series might be deduced in a similar 
