332 BEGUELIN. 



thus he adopts one of D'Alembert's errors. He considers that if 

 the chances would have been equal according to the ordinary 

 theory, then when an event has happened t times in succession 

 it is ^ + 1 to 1 that it will fail at the next trial. 



(314. Beguelin applies his notions to the Petersburg Problem. 



n 

 Suppose there are to be n trials ; then instead of 3 which the 



common theory gives for the expectation Beguelin arrives at 

 112 2' 2' 2"-' 



2"^2"^2 + l'^2.3 + l^[4 + l ••• l^-l + l' 



The terms of this series rapidly diminish, and the sum to 

 infinity is about 2 J. 



615. Besides the above result Beguelin gives five other 

 solutions of the Petersburg Problem. His six results are not 

 coincident, but they all give a small finite value for the expecta- 

 tion instead of the large or infinite value of the common theory. 



616. The memoir does not appear of any value whatever; 

 Beguelin adds nothing to the objections urged by D'Alembert 

 against the common theory, and he is less clear and interesting. 

 It should be added that Montucla appears to have formed a 

 different estimate of the value of the memoir. He says, on his 

 page 403, speaking of the Petersburg Problem, 



Ce probleme a 6te aussi le siijet de savantes considerations metapliy- 

 siques pour Beguelin... ce metaphysicien et analyste examine au flam- 

 beau d'une metapliysique profonde plusieurs questions sur la nature du 

 calcul des probabilites... 



617. We have next to notice a memoir which has attracted 

 considerable attention. It is entitled An Inquiry into the pro- 

 bable Parallax, and Magnitude of the fixed Stars, from the Quantity 

 of Light which they afford us, and the particidar Circumstances of 

 their Situation, by the Rev. John Michell, B.D., F.RS. 



This memoir was published in the Philosophical Transactions, 

 Vol. LVII. Part I., which is the volume for 1767 : the memoir 

 occupies pages 234 — 264. 



