46 SAUVEUR. 



5 5 



at his first throw is ^ , and ^ x 86 = 5 ; thus taking 5 from 86 we 



may consider 81 as left for B. Now B's chance of success in a single 



throw is ^ ; thus — x 81, that is 5 J, may be considered the value 

 oO oO 



of his first throw. 



Thus Caramuel assigns 5 to J. and 5 J to B, as the value of 



their first throws respectively ; then the remaining 25f he proposes 



to divide equally between A and B. This is wrong : he ought to 



have continued his process, and have assigned to A for his second 



5 6 



throw ^ of the 25f , and then to B for his second throw -^ of the 



remainder ; and so on. Thus he Avould have had for the shares of 

 each player an infinite geometrical progression, and the result 

 would have been correct. 



It is strange that Caramuel went wrong when he had the 

 treatise of Huygens to guide him ; it seems clear that he followed 

 this oruidance in the discussion of the Problem of Points for Uvo 

 players, and then deserted it. 



74. In the Journal des Scavans for Feb. 1679, Sauveur gave 

 some formulae without demonstration relating to the advantage of 

 the Banker at the game of Bassette. Demonstrations of the for- 

 mulae will be found in the Ars Conjectandi of James Bernoulli, 

 pages 191 — 199. I have examined Sauveur's formulae as given 

 in the Amsterdam edition of the Journal. There are six series 

 of formulae ; in the first five, which alone involve any difficulty, 

 Sauveur and Bernoulli agree : the last series is obtained by simply 

 subtracting the second from the fifth, and in this case by mistake 

 or misprint Sauveur is wrong. Bernoulli seems to exaggerate the 

 discrepancy when he says, Qu5d si quis D.ni Salvatoris Tabellas 

 cum hisce nostris contulerit, deprehendet illas in quibusdam locis, 

 praesertim ultimis, nonnihil emendationis indigere. Montucla, 

 page 390, and Gouraud, page 17, seem also to think Sauveur more 

 inaccurate than he really is. 



An eloge of Sauveur by Fontenelle is given in the volume 

 for 1716 of the Hist, de F Acad.... Paris. Fontenelle says that 

 Bassette was more beneficial to Sauveur than to most of those who 



