DANIEL BERNOULLI. 2l7 



Let » = where m and n are integers. 



Then we know that [{l + ^)" l'^ 1^^ is less than 



m +n 



by the theorem respecting the geometrical mean and the arith- 



metrical mean which we quoted in Art. 383 ; and this is what we 



had to establish. 



It follows that the merchant can afford without disadvantage 



to increase his payment to the insurance company beyond the 



sum qx. If we suppose f to represent the extreme additional 



sum, we have 



f = a +jyx — (a + ic) V. 



886. We now return to Daniel Bernoulli. We have seen 

 that a merchant can afford to pay more than the sum qx for 

 insuring ; but it may happen that the insurance company demand 

 more than the merchant can afford to pay. Daniel Bernoulli 

 proposes this question : for a given charge by the insurance com- 

 pany required to find the merchant's fortune, so that it may 

 be indifferent to him whether he insures or not. 



Retaining the notation of the last Article, let e be the charge 

 of the insurance company ; then we have to find a from the 



equation 



a-\-x — e = {a + xYa^ 



19 



Daniel Bernoulli takes for an example a?= 10000, e=800,^= ^ ; 



whence by approximation a— 5043. Hence he infers that if the 

 merchant's fortune is less than 5043 he ought to insure, if greater 

 than 5043 he ought not to insure. This amounts to assuming 

 that the equation from which a is to be found has only one 

 positive root. It may be interesting to demonstrate this. We 

 have to compare 



a-\-x — e with {a + ic)^a^ 



where a is the variable, and x is greater than e. 



