I^APLACE. 591 



the probability that in fjb trials the number of times in which the 

 event happens will lie between 



jpiJb — T sl'lfii? (1 —p) and p>iJi + T V2/x^j> (1-2^) 



2 f"^ 

 is approximately -j- e'^^ dt 



Hence to make the limits as close as possible we must have 

 p [^-p) as small as possible, and thus p = -^. This, we say, ap- 

 pears to have been Laplace's process. It is however wrong ; for 

 p (1 — ^;) is a maximum and not a minimum when p=-^. More- 



over we have not to make r V2yC6p (1 - p) as small as possible, 

 but the ratio of this expression to p\x. Hence we have to make 



sj p n _ 27) \ 



—^—^ ^ as small as possible ; that is we must make 1 as 



small as possible : therefore p must be as great as possible. In 



the present case ^ = — ; we must therefore make this as great 



as j)ossible : now in the solution of the problem 2r is assumed 

 to be not greater than a, and therefore we take 2r = a as the 

 most favourable length of the rod. 



Laplace's error is pointed out by Professor De Morgan in 



Art. 172 of the Theory of Prohahilities in the Encyclopcedia 



Metro])olitana. The most curious point however has I believe 



hitherto been unnoticed, namely, that Laplace had the correct 



result in his first edition, where he says ...et il est facile de voir 



2r . 

 que le rapport — qui, pour un nombre donne de projections, 



rend I'erreur a craindre la plus petite, est I'unite . . . The original 

 leaf was cancelled, and a new leaf inserted in the second and third 

 editions, thus causing a change from truth to error. See Art. 932. 



Laplace solves the second part of Buffon's problem correctly, 

 in which Buffon himself had failed; Lai^lace's solution is much 

 less simple than that which we have given in Art. 650. 



