DE MOIVRE. 189 



ters consist substantially of translations of portions of the DoctHne 

 of Chances, and so do not call for any remark. The sixth Chapter 

 is entitled De Kumero PiDictorum in Tesseris; it relates entirely 

 to the formula of which we have given the history in Art. 149. 



331. The seventh Chapter of the Responsio... is entitled Solu- 

 tiones variorum Prohlematum ad Sortem spectantium. This Chapter 

 gives the solutions of nine problems in Chances. The first eight 

 of these are in the Doctrine of Chances ; nothing of importance is 

 added in the Miscellanea Analytica, except in two cases. The first 

 of these additions is of some historical interest. Suppose we take 

 an example of the Binomial Theorem, as {p + qY', one term w^ill 

 be 28p^q^: then De Moivre says, page 218, 



...at fortasse nesciveram hujus termini coefficientem, nimiruni 28, 

 designaturam numerum permutationuni quas literse ^9, p, p, p, p, p, q, q, 

 productum /)^ q' constituentes subire possint ; immb vero, hoc jam din 

 mihi erat exploratum, etenim ego fortasse primus omnium detexi co- 

 efficientes annexas productis Binomii, vel Multinomii cujuscunque, id 

 denotare quotenis variationibus literse producti positiones suas inter se 

 permuteut: sed utrum illud facile fuerit ad inveniendum, j)ostquani 



lex coefficientium ex productis continuis :r- x — ^r— x —= — x (tc. 



1 z o 4 



jam perspecta esset, aut quisquam ante me hoc ipsum detexerit, ad rem 



prassentem non magni interest, cum id monere suffecerit banc proprie- 



tatem Coefficientium a me assertam fuisse et demonstratam in Actis FJd- 



losophicis Anno 1697 impressis. 



The second addition relates to Problem XLix. of the Doctrine 

 of Chances; some easy details relating to a maximum value are 

 not given there which may be found in the Miscellanea Analytica, 

 pages 223, 224. 



332. The ninth problem in the seventh Chapter of the Re- 

 sponsio ... is to find the ratio of the sum of the largest p terms 

 in the expansion of (1 + 1)" to the sum of all the terms ; p being 

 an odd number and 7i an even number. De Moivre expresses 

 this ratio in terms of the chances of certain events, for which 

 chances he had already obtained formulae. This mode of ex- 

 pressing the ratio is not given in the Doctrine of Chances, being 

 rendered unnecessary by the application of Stirling's Theorem ; 



