DE MO IV RE. 191 



cui Tabulae subesset error perpetuus in quinta quaque figura decimali 

 sum mar um : quae cum pro bumauitate sua monuisset, bis subjunxit 

 seriem celerrime convergentem, cujus ope summse logaritbmorum tot 

 numerorum naturalium quot quis sumere voluerit obtineri possent ; 

 res autem. sic exposita fuerat. 



Then follows a Theorem which is not quite coincident in 

 form with what we now usually call Stirling's Theorem, but is 

 practically equivalent to it. De Moivre gives his own investiga- 

 tion of the subject, and arrives at the following result : 



log 2 + log 3 + log 4 + ... + log (m — 1) 



= (m-2)logm-m + ^^-3g^3+j260;^.-l^ 

 , 1111 



12 ' 360 1260 ' 1680 



With respect to the series in the last line, De Moivre says 

 on page 9, of the Supplement to the Miscellanea A nalytica . . . quae 

 satis commode convergit in principle, post terminos quinque pri- 

 mes convergentiam amittit, quam tamen postea recuperat... The 

 last four words involve an error, for the series is divergent, 

 as we know from the nature of Bernoulli's Numbers. But De 

 Moivre by using a result which Stirling had already obtained, 



111 



arrived at the. conclusion that the series 1 — :r^ + -rr^ — =-^— r + ... 



12 360 1260 



is equal to - log 27r ; and thus the theorem is deduced which 



we now call Stirling's Theorem. See Miscellanea Analytica, 

 page 170, Supplement, page 10. 



334. De Moivre proceeds in the Siqyplement to the Miscellanea 

 Analytica to obtain an approximate value of the middle coefficient 

 of a Binomial expansion, that is of the expression 



(W2+1) (m+2)... 2m 

 m [m — 1) ... 1 



He expends nearly two pages in arriving at the result, which 



