214 DANIEL BERNOULLI. 



relative value of money ; of these Daniel Bernoulli's has attracted 

 most notice. 



Suppose a person to possess a sum of money x, then if it re- 

 ceive an increment dx, Daniel Bernoulli estimates the relative 

 value of the increment as proportional to dx directly and x in- 



ndx 



versely ; that is, he takes it equal to where Jc is some con- 



stant. Put this equal to Jt/ ; so that 



■J ruCtX 



dy = ; 



therefore y — T^ log ^ + constant 



= Ti log - say. 



Laplace calls x the fortune physique and y the fortune morale. 

 "We must suppose a some positive quantity, for as Daniel Bernoulli 

 remarks, no man is absolutely destitute unless he is dying of 

 hunger. 



Daniel Bernoulli calls y the emolumentum, a he calls summa 

 honorum, and x — a he calls lucrum. 



880. Suppose then that a person, starting with a for his fortune 

 physique, has the chance p^^ of gaining a?^, the chance p^ of gaining 

 x^, the chance p^ of gaining x^, and so on ; and suppose the sum 

 of these chances to be unity. Let 



Y= hp^ log {a + x^ + hp^ log {a-\-x,^ -\- hp^ log (a + i^Cg) + . . . — ^ log a. 



Then Bernoulli calls Y the emolumentum medium, and Laplace 

 still calls Y the fortune morale. Let X denote the fortune 

 physique which corresponds to this fortune morale ; then 



Y=h log X—h log a. 



Thus X = (a + cc/^ (« + xf^ {a + x^""' . . . 



And X—a will be according to Laplace V accroissement de la 

 fortune physique qui procurerait a Tindividu le menie avantage 

 moral qui r4sidte pour lui, de son expectative. Daniel Bernoulli 

 calls X—a the liicru7n legitime expectandum seu so7^s quwsita. 



