DANIEL BERNOULLI. 215 



381. Daniel Bernoulli in his memoir illustrates his hy- 

 pothesis by drawing a curve. He does not confine himself to the 



case in which ^ = 7c log - , but supposes generally 2/ = </> (x). 



Thus the ordinary theory of mathematical expectation amounts to 

 supposing that the curve becomes a straight line, or (p (x) a 

 linear function of x. 



382. After obtaining the value of X which we have given 

 in Art. 380, the remainder of Daniel Bernoulli's memoir consists 

 of inferences drawn from this value. 



383. The first inference is that even a fair game of chance 

 is disadvantageous. Suppose a man to start with a as his fortune 

 physique, and have the chance p^ of gaining x^, and the chance 

 p^ of losing x^. Then by Art. 380, the fortune physique which he 

 may expect is 



{a + a?/' (a - x^^^ ; 



we have to shew that this is less than a, supposing the game to be 

 mathematically fair, so that 



Daniel Bernoulli is content with giving an arithmetical ex- 

 ample, supposing i>i =/>2 = 2 • I^aplace establishes the proposition 



generally by the aid of the Integral Calculus. It may be proved 

 more simply. We have 



x^ x^ 



■^^ ~ iCj + a?/ ^^"aj^ + iCg' 



and we have to shew that 



[[a-^x^'^'ia-x^"''^^' is less than a. 



Now we may regard x^ and x^ as integers. Thus the result 

 we require is true by virtue of the general theorem in inequalities 

 that the geometrical mean is less than tlie arithmetical mean. For 



