188 DE MOIVRE. 



tingiilshed men ; he was also admitted as a member of the Koyal 

 Society. De Moivre sent to Montmort a copy of the Doctrine of 

 Chances when it was pubKshed, and about two years afterwards 

 Montmort died. 



De Moivre quotes the words of Fontenelle which we have 

 already given in Art. 136, and intimates that these words 

 induced him to undertake a comparison between his own labours 

 and those of Montmort, in order to vindicate his own claims. As 

 the Doctrine of Chances was written in English it was not readily 

 accessible to all who would take an interest in the dispute; and 

 this led De Moivre to devote a section to the subject in his Mis- 

 cellanea Analytica. 



329. The second Chapter of the Responsio... is entitled De 

 Methodo Diferentiarum, in qua exhihetur Solutio Stirlingiana de 

 media Coefficiente Binomii. The general object is to shew that 

 in the summation of series De Moivre had no need for any of 

 Montmort's investigations. De Moivre begins by referring to a 

 certain theorem which we have noticed in Art. 152; he gives some 

 examples of the use of this theorem. He also adverts to other 

 methods of summation. 



Montmort had arrived at a very general result in the summa- 

 tion of series. Suppose u\'^ to denote the n^^ term of a series, 

 where u^ is such that A'^w^ is zero, ni being any positive integer ; 

 then Montmort had succeeded in summing any assigned number 

 of terms of the series. De Moivre shews that the result can be 

 easily obtained by the method of Differences, that is by the method 

 which we have explained in Art. 151. 



The investigations by Montmort on the summation of series to 

 which De Moivre refers were published in Vol. xxx. of the Philo- 

 sophical Transactions, 171 7. 



This Chapter of the Responsio,.. gives some interesting details 

 respecting Stirling's Theorem including a letter from Stirling 

 himself. 



330. The third Chapter of the Responsio... is entitled De Me- 

 thodo Comhinatio7ium ; the fourth De Permutationibus ; the fifth 

 Combinationes et Permutationes idterius consideratce : these Chap- 



