240 EULER. 



chance that at least one card is in its right place becomes equal 

 to 1 — e~\ where e is the base of the Napierian logarithms ; this is 

 noticed by Euler : see also Art. 287. 



431. The next memoir by Euler is entitled Recherches g^ne- 

 rales sur la mortalite et la multiplication du genre humain. This 

 memoir is published in the volume for 1760 of the Histoire de 

 V Acad. ... Berlin ; the date of publication is 1767: the memoir 

 occupies pages 144 — 164. 



432. The memoir contains some simple theorems concerning 

 the mortality and the increase of mankind. Suppose N infants 

 born at the same time ; then Euler denotes by (1) N the number 

 of them alive at the end of one year, by (2) N the number of 

 them alive at the end of two years, and so on. 



Then he considers some ordinary questions. For example, 

 a certain number of men are alive, all aged m years, how many 

 of them will probably be alive at the end of n years ? 



According to Euler's notation, (m) N represents the number 

 alive aged m years out of an original number N\ and {m + n) N 

 represents the number of those who are alive at the end of n 



more years ; so that — , . is the fraction of the number 



aged m years who will probably be alive at the end of n years. 

 Thus, if we have a number M at present aged m years, there will 



probably be — -. — —- M of them alive at the end of n years. 



433. Then Euler gives formulae for annuities on a life. Sup- 

 pose M persons, at present each aged m years, and that each 

 of them pays down the sum a, for which he is to receive x 



1 



annually as long as he lives. Let - be the present worth of the 



unit of money due at the end of one year. 



(m -f 1) 

 Then at the end of a year there will be M ' . ^ of the 



(in) 



persons alive, each of whom is to receive x : therefore the present 



worth of the whole sum to be received is - M —- -r — . 



X {m) 



