216 DANIEL BERNOULLI. 



here we may suppose that there are x^ quantities, each equal to 

 a + iCj, and x^ quantities each equal to a — iCg. The arithmetical 

 mean is 



'^2 (^ + ^l) + ^1 (<^ - ^2) 



— J 



^1 + ^2 



that is a. The geometrical mean is the quantity which we had 

 to shew to be less than a. 



884. Daniel Bernoulli proposes to determine what a man 

 should stake at a wager, in order that the wager may not be 



disadvantageous to him. He takes the case in which 'p^—]p^ — -^ . 

 Then we require that 



(a + icj^ {a — x^^ — a. 



This leads to x^ — — . 



Thus x^ is less than x^ and less than a. 



885. Daniel Bernoulli now makes an application to in- 

 surances. But this application will be more readily understood if 

 we give first a proposition from Laplace which is not in Daniel 

 Bernoulli's memoir. Suppose that a merchant has a fortune 

 physique equal to a, and that he expects the sum x to arrive 

 by a ship. Also let p be the chance that the ship will arrive 

 safely, and lei q = l —p. 



Suppose that he insures his ship on the ordinary terms of 

 mathematical equity ; then he pays qx to the insurance company, 

 so that he has on the whole a + x — qx, that is a -\-px. 



Suppose however that he does not insure ; then his fortune 

 physique is (a + xfa'^. We shall shew that a-\-px is greater 

 than {a + xYa^. 



Laplace establishes this by the aid of the Integral Calculus, 

 with which however we may dispense. We have to shew that 



(a + xYa^ is less than a +px, 



that is that (1 + - ) is less than 1 + -^ . 



\ a/ a 



