d'alembert. 265 



478. D'Alembert thus sums up his results, on his page 24 : 



Concluons de toutes ces reflexions j 1". que si la regie que j'ai donnee 

 dans Y Encyclopedie (faute d'en connoitre une meilleure) pour deter- 

 miner le rapport des probabilites au jeu de croix et pile, n'est point 

 exacte a la rigueur, la regie ordinaire pour determiner ce rapport, Test 

 encore moins; 2". que pour parvenh a une tlieorie satisfaisante du cal- 

 cul des probabilites, il faudroit resoudre plusieurs Problemes qui sont 

 peut-etre insolubles; savoir, d'assigner le vrai rapport des probabilites 

 dans les cas qui ne sont pas egalement possibles, ou qui peuvent 

 n'etre pas regardes conime tels; de determiner quand la j)robabilite 

 doit etre regardee comme nulle ; de fixer enfin comment on doit estimer 

 I'esperance ou I'enjeu, selon que la probabilite est 2)lus ou moins grande. 



479. The next memoir by D'Alembert which we have to 

 notice is entitled Sur^ Vwpplication du Calcid des Prohahilites a 

 Vinocidation de la petite Verole ; it is published in the second 

 volume of the Opuscules. The memoir and the accompanying 

 notes occupy pages 26 — 95 of the volume. 



480. "We have seen that Daniel Bernoulli had written a 

 memoir in which he had declared himself very strongly in favour 

 of Inoculation ; see Art. 398. The present memoir is to a certain 

 extent a criticism on that of Daniel Bernoulli. D'Alembert does 

 not deny the advantages of Inoculation ; on the contrary, he is 

 rather in favour of it : but he thinks that the advantages and 

 disadvantages had not been properly compared by Daniel Ber- 

 noulli, and that in consequence the former had been overestimated. 

 The subject is happily no longer of the practical importance it 

 was a century ago, so that we need not give a very full account 

 of D'Alembert's memoir ; we shall be content with stating some 

 of its chief points. 



481. Daniel Bernoulli had considered the subject as it related 

 to the state, and had shewn that Inoculation was to be recom- 

 mended, because it augmented the mean duration of life for 

 the citizens. D'Alembert considers the subject as it relates to 

 a private individual : suppose a person who has not yet been 

 attacked by small-pox ; the question for him is, whether he will 

 be inoculated, and thus run the risk, small though it may be, 

 of dying in the course of a few days, or whether he will take his 



