APPENDIX. 615 



them each takes his own stake. Then according to the ordinary- 

 principles we estimate the expectation of A at 



m {a + i) +2^^ 

 m + n +p ' 



so far as it depends upon the game which is to be played. Or if 

 we wish to take account of the fact that ^ has already paid down 

 the sum a, w^e may take for the expectation 



mia +h) ■\- pa ,. , . mh — na 

 — a, that is, . 



Eizzetti however prefers another definition ; he says that A has 



m chances out of 7n -{-n + p of gaining the sum h ; so that his 



7726 



expectation is . Rizzetti tries to she^v that the ordinary 



m + 71 -\-p 



definition employed by Montmort and Daniel Bernoulli leads to 

 confusion and error ; but these consequences do not really follow 

 from the ordinary definition but from the mistakes and unskil- 

 fulness of Rizzetti himself 



The memoir does not give evidence of any power in the sub- 

 ject. Rizzetti considers that he demonstrates James Bernoulli's 

 famous theorem by some general reasoning which mainly rests 

 on the axiom, Effectus constans et immutabilis pendet a causa 

 constante, et immutabili. On his page 224 he gives wdiat he con- 

 siders a short investigation of a problem discussed by Huygens 

 and James Bernoulli ; see Arts. 33, 103 : but the investigation is 

 unsatisfactory, and shews that Rizzetti did not clearly understand 

 the j)roblem. 



1056. I am indebted for a reference to the memoir noticed 

 in the preceding Article to Professor De Morgan who derived it 

 from Kahle, Bihliothecce Philosopliice Struviance...GoiimgQn, 1740. 

 2 Vols. 8vo. Vol. I. p. 295. Professor De Morgan supplied me 

 from the same place with references to the following works which 

 I have not been so fortunate as to obtain. 



Andrew Budiger, De sensu falsi et veri, lib. I. cap. xii. et 

 lib. III. : no further description given. 



Kahle himself Elementa logicce probabilium, methodo mathe- 

 matica...'RRlse MagdeburgiccT, 1735, 8vo. 



