212 EULEn. 



435. A memoir by Euler, entitled Stir les Rentes Viageres, 

 immediately follows tlie preceding, occupying pages 1G5 — 175 of 

 the volume. 



Its principal point is a formula for facilitating tlie calculation 

 of a life annuity. 



Let A.,^^ denote the value of an annuity of one pound on the 

 life of a person aged 7n years, A,^_^^ the value of an annuity of 

 one pound on the life of a person aged m + 1 years. Then by 

 the preceding memoir, Art. 433, 



1 { {m-^l) 0)^ + 2) U+3) I 



. _ _1 {{m^ (m + 3) (»z + 4) ] 



"'^^ (m+1) 1 \ ^ X' "^ \' "^ ""J ' 



therefore (m) X A,,, = [m + 1) + {m + 1) J, 



Wi+l" 



Thus when A.^^ has been calculated, Ave can calculate A„^^^ 

 easily. 



Euler gives a table exhibiting the value of an annuity on 

 any age from to 94. But with respect to the ages 90, 91, 92, 

 93, 94, he says, 



Mais je ne voudrois pas consei'der a un entrepreneur de se mekr 

 avec de tels vieillards, a nioins que leur nombre ne fut assez consider- 

 able; ce qui est une regie generale pour tons les etablissemens fondes 

 sur les probabilites. 



Euler is of opinion that the temptations do not appear suf- 

 ficient to induce many persons to buy annuities on terms which 

 would be advantageous to the sellers. He suggests that defended 

 annuities might perhaps be more successful ; for it follows from 

 his calculations, that 350 crowns should purchase for a new born 

 infant an annuity of 100 crowns to commence at the age of 

 20 years, and continue for life. He adds, 



...et si I'on y vouloit employer la somme de 3500 ecus, ce seroit 

 toujours un bel etablissement, que de jouir cles I'age de 20 ans d'une 

 pension fixe de 1000 ecus. Ce^^endant il est encore douteux, s'il se 

 trouveroit plusieurs parens qui voiuh"oicnt bien faiie uu tel sacrifice 

 pour le lien de leurs enfans. 



