DANIEL BEKNOULLI. 221 



Let X represent the sum which a person with the capital a 

 might give without disadvantage for the expectation of A \ then x is 

 to be found from 



(a + 1 — a?)^ (a + 2 — a?)^ (a + 4 — x)^ (a + 8 — x)^ . .. = a. 



Put a — X — a \ thus 



{a + 1)^ [a + 2)^ (a' + 4)^ {a + 8)tV ... - a' = a;. 



Then if a is to have any large value, from what we have 

 already seen, x is small compared with a, so that we may put a for 

 a \ and we have approximately 



a; = (a + 1)^ (a + 2)^ (a + 4)^ (a + 8)^... -a. 



Laplace reproduces this part of Daniel Bernoulli's memoir with 

 developments in pages 439 — 442 of the Theorie...des Proh. 



391. Daniel Bernoulli's memoir contains a letter addressed to 

 Nicolas Bernoulli by Cramer, in which two methods are suggested 

 of explaining the paradox of the Petersburg Problem. 



(1) Cramer considers that the value of a sum of money is not 

 to be taken uniformly proportional to the sum ; he proposes to 

 consider all sums greater than 2^"^ as practically equal. Thus he 

 obtains for the expectation of B 



1 2 4 2^^ 



2*^2^ 



"■" 02 "■" 03 "•"••• • '^"•5 



924 924 924 



' 926 "^ 927 "' 928 "1" •••• 



The first twenty-five terms give 12 J; the remainder constitute 



a geometrical progression of which the sum is ^ . Thus the total 

 is 13. 



(2) Cramer suggests that the pleasure derivable from a sum 

 of money may be taken to vary as the square root of the sum. 

 Thus he makes the moral expectation to be 



2 a/I + J V2 + g v/4 + ^ V8 +■ . . . , 



that is j^ . This moral expectation corresponds to the sum 



