LAPLACE. 557 



Laplace arrived at the result (1) ; but the assumption is used in 

 such a way as to diminish very decidedly the apprehension of any 

 erroneous consequences : the assumption, so to speak, is made to 

 extend over an indefinitely small interval instead of over a finite 

 interval. 



Poisson had however previously considered the question in his 

 Memoire sur la proportion des naissances des deux sexes; this 

 memoir is published in the Memoires...de VInstitut, Yol. ix, 1830 ; 

 there he uses Bayes's theorem, and proceeds as we have done in 

 establishing (2), but he carries the approximation further: he 

 arrives at the result (3). See page 271 of the memoir. 



Thus the result (3) is demonstrable in two ways, namely, by 

 the assumed inversion of James Bernoulli's theorem, and by 

 Bayes's theorem. As Poisson in his latest discussion of the ques- 

 tion adopted the inversion of James Bernoulli's theorem, we may 

 perhaps infer that he considered the amount of assumption thus 

 involved to be no greater than that which is required in the use of 

 Bayes's theorem. See Art. 552. 



In a memoir published in the Cambridge Philosophical Trans- 

 actions, Vol. VI. 1837, Professor De Morgan drew attention to the 

 circumstance that Laplace and Poisson had arrived at the result (1) 

 by assuming what we have called an inversion of James Bernoulli's 

 theorem ; and he gave the investigation Avhich, as we have said, 

 depends on Ba3^es's theorem. Professor De Morgan however over- 

 looked the fact that Laplace had also implicitly given the result 

 (2), and that Poisson had arrived at the result (3) by both 

 methods. It will be found on examining page 428 of the volume 

 which contains Professor De Morgan's memoir, that his final 

 result amounts to changing v'"^ into v in the second term of the 

 value of V in Poisson's result (3). Poisson, however, is correct ; 

 the disagreement between the two mathematicians arises from the 

 fact that the approximations to the values of fi and v which Pro- 

 fessor De Morgan gives towards the top of the page under con- 

 sideration are not carried far enough for the object he has in 

 view. 



In the Treatise on Probability by Galloway, which is con- 

 tained in the Encyclopcedia Britannica, reference is expressly made 

 to Professor De Morgan's memoir, without any qualifying remark ; 



