192 BE MOIVRE. 



he might have obtained immediately by putting the proposed ex- 



I 2)n 

 pression in the equivalent form — - . 



De Moivre then gives the general theorem for the approximate 

 summation of the series 



1 1 , 1 , 1 , 



We have already noticed his use of a particular case of this 

 summation in Art. 276. 



De Moivre does not demonstrate the theorem ; it is of course 



included in the wellknown result to which Euler's name is usually 



attached, 



^ _ r , 1 11 du^ 1 1 d^u^ 



See Novi Coram.... Petrop. Vol. xiv. part 1, page 137 ; 1770. 

 The theorem however is also to be found in Maclaurin's 

 Treatise of Fluxions, 1742, page 673. 



335. We return to the Doctrine of Chances, to notice what is 

 given in its pages 243 — 254 ; see Art. 324. 



In these pages De Moivre begins by adverting to the theorem 



obtained by Stirling and himself He deduces from this the 



following result : suppose 7i to be a very large number, then the 



/I 1\" 

 logarithm of the ratio which a term of I ^ + ^ ) , distant from 



the middle term by the interval I, bears to the middle term, 



2P 

 is approximately . 



This enables him to obtain an approximate value of the sum of 

 the I terms which immediately precede or follow the middle term. 

 Hence he can estimate the numerical values of certain chances. 

 For example, let n = 3600 : then, supposing that it is an even 

 chance for the happening or failing of an event in a single trial, 

 De Moivre finds that the chance is '682688 that in 3600 trials, 

 the number of times in which the event happens, will lie between 

 1800 + 30 and 1800-30. 



