DANIEL BERNOULLI. 227 



Halley's table begins with 1000 at the end of the first year, 

 and does not say to what number of births this corresponds. 

 Daniel Bernoulli gives reasons for assuming this to be 1300, 

 which accordingly he takes ; see Art. 64?. 



404. On page 21 of the memoir, Daniel Bernoulli says that 

 the following question had been asked: Of all persons alive 

 at a given epoch what fractional part had not been attacked 

 by the small-pox ? The inquirer himself, who was D'Alembert, 

 estimated the number at one-fourth at most. Daniel Bernoulli 

 himself makes it about two-thirteenths. He intimates that it 

 would be desirable to test this by observation. He adds, 



Voici un autre theoreme qui pourroit servir h la verification de 

 nos principes. Si de tous les vivans on ne prend que Tenfance et la 

 jeunesse, jusqu'a I'age de seize ans et demi, on trouvera le nombre 

 de ceux qui auront eu la petite verole a pea-pres egal au nombre de 

 ceux qui ne I'auront pas eue. 



405. Daniel Bernoulli gives another interesting investigation. 

 Bequired to find the number of survivors at a given age from 

 a given number of births, supposing the small-pox altogether 

 extinguished. Retain the notation of Article 402 ; and let z be 

 the number who would have been alive at the age x if there had 

 been no small-pox, the original number of births being supposed 

 the same. 



The whole mortality during the element dx of time being 



9 fix 

 — d^, and the mortality caused by the small-pox being , we 



II tit 



sd'Oc 

 have for the mortality in the absence of small-pox — d^ . 



But this mortality arises from a population f ; and we must mul- 

 tiply it by g to obtain the mortality which would arise from a 

 population z. Hence, finally, 



mn,> 



dz d^ s dx 



therefore — = -p -\r -z. — • 



z ^ g tnn 



15—2 



