553 



995. With respect to the history of the result obtained in 

 Art. 994, we have to remark that James Bernoulli began the 

 investigation ; then Stirling and De Moivre carried it on by the aid 

 of the theorem known by Stirling's name ; and lastly, the theorem 

 known by Euler's name gave the mode of expressing the finite 

 summation by means of an integral. See Arts. 123, 334, 335, 423. 

 But it will be seen that practically we use only the first term 

 of the series given in Euler's theorem, in fact no more than 

 amounts to evaluating an integral by a rough approximate quadra- 

 ture. Thus the result given by Laplace was within the power of 

 mathematicians as soon as Stirling's Theorem had been published. 



Laplace, in his introduction, page XLII, speaking of James 

 Bernoulli's theorem says, 



Ce theoreme indique par le bon sens, 6tait difficile a demontrer par 

 I'Analyse. Anssi I'illustre geometre Jacques Bernoulli qui s'en est 

 occup6 le premier, attachait-il une grande importance a la demonstra- 

 tion qu'il en a donnee. Le calcul des fonctions generatrices, applique 

 a cet objet, non-seulement d^montre avec facilite ce theoreme ; mais de 

 phis il donne la probabilite que le rapport des evenemens observes, ne 

 s'ecarte que dans certaines limites, du vrai rapport de leurs possibilites 

 respectives. 



Laplace's words ascribe to the theory of generating functions 

 the merit which should be shared between the theorems known 

 by the names of Stirling and Euler. 



We may remark that in one of his memoirs Laplace had used 

 a certain process of summation not connected with Euler's 

 theorem : see Art. 897. 



996. Laplace gives the following example of the result ob- 

 tained in Art. 993. 



