86 MONTMOET. 



Then it is shewn in works on Finite Differences, that 



i(n = % + 'i^^Uo 4- -J — 2~ -^'^^^ + • • • 



, yi(?i--l) ...(??-m+l) .,„ 



i -» j 11 Uq . 



[m 



This formula Montmort gives, using A, B, C,... for Aw^j AV^, 



By the aid of this formula the summation of an assigned 



number of terms of the proposed series is reduced to depend on the 



,. ^ . ^ ,., n (n—1) ... (n — r+1) . 



summation of series of which — ^ — j — ^ ^ may be 



taken as the type of the general term ; and such summations have 

 been already effected by means of the Arithmetical Triangle and 

 its properties. 



152. Montmort naturally attaches great importance to this 

 general investigation, which is new in the second edition. He 

 says, page ^5^ 



Ce Problerae a, comme Ton voit, toute I'etendue et toute I'universa- 

 lite possible, et semble ne rien laisser a desirer sur cette matiere, qui n'a 

 encore et6 traitee par personne, que je s^ache : j'en avois obmis la de- 

 monstration dans le Journal des Sgavans du mois de Mars 1711. 



De Moivi'e in his Doctrine of Chances uses the rule which 

 Montmort here demonstrates. In the first edition of the Doctrine 

 of Chances, page 29, we are told that the "Demonstration may 

 be had from the Methodus Differentialis of Sir Isaac Xewton, 

 printed in his Analysis!' In the second edition of the Doctrine 

 of Chances, page 52, and in the third edition, page 59, the origin 

 of the rule is carried further back, namely, to the fifth Lemma of 

 the Princijna, Book iii. See also Miscellanea Analytica, page 152. 



De Moivre seems here hardly to do full justice to Montmort ; 

 for the latter is fairly entitled to the credit of the first explicit 

 enunciation of the rule, even though it may be implicitl}^ contained 

 in Newton's Princijna and Methodus Differentialis. 



153. Montmort's second part occupies pages 73 — 172 ; it re- 



