294 Remarks on Professor Wallace^s reply to B. 



iremely short and simple manner by the comnnon operations 

 of multipHcation. 



Now it is a fact, nolwiths landing the positive declaration of 

 Professor Wallace to the contrary, that Euler's demonstration 

 is not restricted to this very simple and easy to be demon- 

 strated case ; but is general for all values of the exponent, 

 whether integer, fractional, positive, negative, or surd, and 

 it is characterized by La Croix as being elegant and rigor- 

 ous.* Moreover, every thing that can be obtained from the 

 multiplication of these new series can be easily deduced 

 from Euler's method by a perfectly rigorous process. 



Professor Wallace complains that a wrong title was given 

 to his paper. This could not have been known to B. ; and if it 

 had been known, it would in nowise have affected the justice 

 of his remarks. The fact would still have been that Pro- 

 fessor Wallace had republished as a late discovery what had 

 been known by mathematicians for half a century. The 

 only difference, as it now appears, is, that Mr. Stainville 

 gave it as new for ihe first time in 1818, and Professor Wal- 

 lace for the second time in 1824 — Euler's having been pub- 

 lished in 1775. When B. read the first communication of 

 Professor Wallace, the thought never occurred that the 

 method was not claimed as a new process, particularly as the 

 words "Aezu Algebraic Series'*^ occur at the top of every page,* 

 and the very guarded acknowledgment of his obligation to Mr. 

 Stainville in vol. vii. p. 28.5, is so connected with the account 

 of numerical faculties, integrations, derivations, and the no- 

 tices of other series besides those denoted ^y fa, that it did 

 not attract particular attention as referring exclusively to 

 this last series. The whole notice of Mr. Stainville is compri- 

 sed in this brief sentence, "M. de Stainville of the Polytechnic 

 School has given the series in this communication in Volume 

 IX. of Gergonne's Annals; and from the extensive ap- 

 plication of which they are susceptible, the subject is de- 

 serving of farther investigation." This is not very definite 

 because there are several series in that paper. Admitting 

 it, however, to refer particularly to the series denoted by /a, 

 the question will then occur — who discovered all its properties 

 and made all the deductions from it? was it Mr. Stainville, 



* La demonstration prec^dente ne laisse rien a disirer du c6t6 de la ri- 

 gueur et de l'616gance. — Comp. des. Eliment. p. 163. 



f In justice to Professor Wallace it should be remarked, that no part 

 of this title can be attributed to him, as there was none accompanying 

 his ipanuscript. The title mentioned was inserted at the time of publi- 

 cation.— En. 



