Professor Wallace in Reply to the Remarks of B. 101 



manner, as Kramp has done by means of his Factoriels, (a 

 name by which he designates series similar to Stainville'a) 

 and extended their application to definite integrals, &c. 



When a writer generalises too hastily it is difficult to 

 form an idea of the truth or legitimacy of his conclusions, 

 and that Eulerhad, in no small degree, this failing, will be 

 evident to any one who reads his works. Lagrange (Le- 

 mons surle Calcul des fonctions, 1806, p. 409) speaking of 

 the expansion of some functions by Euler, shews, that he 

 takes as granted what he ought to prove. 'Car (says he) 

 la Demonstration qu' on trouve dans le tome XV des M)vi 

 Commentarii de Petersbourg, est si compliquee, qu' il est 

 difficile de juger de sa justesse et de sa generalite." Does 

 not this observation hold, in its full force, against Mr. B. 



also, in asserting that, /a x/6=/(«+0 as deduced from 

 particular and limited premises. 



If Euler has not been completely successful in the above 

 and similar instances, still it would be an injustice to the 

 memory of so great a map, not to acknowledge that he has 

 contributed as much, if not more than any other individual 

 of his time, to the modern improvements in almost every 

 department of the new Calculus. In vols. XV. and XVI. 

 ofthe.M)i>. Comm. Petrop. he has published his Integral 

 Calculus, in tom. XVI. of which S. 28, the integral Jdx 

 {log. ^)~2—-\/n; is given, and long before in tom. V. des 

 anciens memoires de Petersbourg p. 44. The integral 

 /e~*"c?a;=|V'«^isof the same form, and is integrated be- 

 tween the limits a;=o and a;= cd ; which shews that he had 

 been in possession ofthe^erm of the theory of definite in- 

 tregals, one of the most useful, though one of the most 

 difficult, in the modern analysis, and which has since been 

 so far extended by Laplace in the Memoirs of the 

 Institute, by Legendre in his " Exercises de Calcul 

 Integral,'' by Brisson, Poisson, &c. in the Journal of 

 the Polytechnic School, by Herschel in the Phil. Trans. 

 1814, and others who have pursued and still pursue this in- 

 teresting investigation. Similar researches have been 

 carried on by different authors, and under various denom- 

 inations, and Laplace has further generalized these theo- 

 ries in his Calcul des fonctions generailces, exhibited in vol. 

 8. of the Journal of the Polytechnic School, and in his 



