102 Professor Wallace in Reply to the Remarks of B. 



Theorie analytique des probabilites, 1814. These theories,as 

 is well known, are applied with singular success in the 

 most abstruse physical researches, whilst in the works of 

 English Mathematicians, scarcely a word, relative to them, 

 has been given. Lately, however, in the Phil. Trans, and 

 some few other works of less note, several have distinguish- 

 ed themselves in these investigations. In calling the at- 

 tention of the American reader to those important enqui- 

 ries, it is to these works, and also to La Croix's work on 

 the Differential and Integral Calculus, 2d. Ed. particularly 

 the 3d. vol. published in 1 8 1 9, that I would refer for general 

 information on these subjects, and not to the Complement 

 des Elemens d* Algebre, however useful as a school book, — 

 To those who take a pleasure in proceeding from the most 

 elementary principles to the most remote conclusions, it 

 must be highly interesting to discover the different ways 

 in which the same truths may be established, and to pur- 

 sue those methods, which from the most simple principles 

 lead to to the most general results. 



As to originality in these investigations, it is extremely 

 difficult to do impartial justice to the merits of authors. 

 Whoever was the first author of the expansion of a binomi- 

 al, or the law on which such series depend, I should con- 

 sider as having the greatest claim' to originality and inven- 

 tion ; as all the improvements in the modern analysis 

 principally depend on the developement and application 

 of series. Hence Lagrange gives Fermat the honour of first 

 exhibiting the germ of the new Calculus (Calcul des fonc- 

 tions p. 325.) whilst Laplace gives almost the whole merit 

 to Wallis. In his Theorie analytique des probabilites (2d. 

 Ed. 1814,) one of the most profound and elegant analyti- 

 cal performances in existence, speaking, in the preface to 

 this work, of the Arithmetic of Infinities of Wallis, he says, 

 that it is " V un des ouvrages qui ont le plus contribue 

 au progres de 1' analyse, et on V on trouve le Germe de la 

 theorie des integrales definies, V une des bases de ce nou- 

 veau calcul des probabilites.'' Wallis published his Arith" 

 metica Infinitorum in 1657. (He was born in 1616, ninety- 

 one years before Euler existed.) Laplace gives his re- 

 markable theorem in p. 465. of the additions to his Theory 

 of Probabilities, where he shews how nearly it is connect- 

 ed with the modern calculus, particularly the theory of 



