100 Professor Wallace in Reply to the Remarks ofB, 



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 An- 

 alysis, for all are deduced from multiplication, which has 

 been known time immemorial. In fine Mr. B. seems to 

 adhere literally to the observation of Solomon, " nihil sub 

 sole novum." 



Mr. B. next observes that nearly the whole theory of 

 the functions, which I have named, is to be found in the 

 " Complement des Elemens d' Algebre" of La Groix, 

 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 

 case of the binomial theorem, viz. (l-l-z)"'=l-{-y2r-4" 



— i ^2;2-|-&c.=y(m) where mis supposed a whole 



positive number. La Croix says, " Parmi les difierentes 

 preuves qu' Euler a donnees de la generalite de la formule 

 du binome, la suivante tient le premier rang, parsa finesse 

 et sa brievete," (pa. 145. Ed. 3.) and then gives what Mr. 

 B. calls my fundamental theorem, deduced from a know- 

 ledge of the above expansion. Now/ (m) to be general, 

 even in this limited case, should include the expansions of 

 a '"^S log. (1 +z),Si7i. (1+2;), C6s. (l -i-z), he. Where 

 have 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 to represent. But in the results deduced from the 

 multiplication of Stainville's series, the expansions of the 

 binomial and multinomial follow, as is evident from p. 

 283 vol. V'll. no* 2 of the Journal, and in p. 284 the expan- 

 sions of e*'' , a "" , Log. (l+x), &c. 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 



