448 PHILOSOPHICAL TRANSACTIONS. [aNNO 1754. 



ing of the publication of the antiquities found at Herculaneum, he says, Spero 

 che il primo tomo non tarder^ molto tempo ad uscire ; and then mentions some 

 particular things that had been lately discovered among the ruins ; a little brass 

 bust of some unknown philosopher, of the manner excellent, and is perfectly 

 well preserved : a statue of an orator, in marble ; and another brass bust, on a 

 term, of a youth, with very beautiful hair, and the whole excellent. The artist 

 has put his name to the latter, AnOAAilNIOI APXIOT AQHNAIOi; EnoiHIE. 

 He says, that the workmen were then just entering on some nobleman's house, 

 as appeared by the rich Mosaic pavements, &c. and that they were in hopes it 

 would prove a very good new mine. 



LJCII. On the Value ofanAnnuilyforLife, and the Probability of Survivor- 

 ship. By Mr. James Dodson. p. 487. 



• The writers on the subject of annuities on lives have justly distinguished them 

 into 2 kinds : in the first, the annuitant is entitled to receive a payment, if he 

 be alive on the day on which it becomes due ; but if he dies on the preceding 

 day, or sooner, his heirs have no claim to any part of the payment, so to have 

 become due ; but in the second, if the annuitant dies at any intermediate time 

 between the days of payment, his heirs are to receive a part of the annuity, pro- 

 portional to the time elapsed between the preceding day of payment and the an- 

 nuitant's decease. This latter kind of annuities have been distinguished from the 

 former, by the words, secured by a grant of lands ; because, where lands are 

 leased for lives, the conditions are generally such as are above described. 



The values of the first kind of annuities have been investigated on principles 

 purely arithmetical ; but in order to perform the latter, fluxions have been used, 

 Mr. D. conceives, without any necessity : but as the investigation of the former 

 may be usefully made a part of the latter, he first recites the method of perform- 

 ing that, and then proceeds to attempt the other on the same principles. 



If, with De Moivre, we suppose the decrements of life to be equal (viz. 

 that out of a number of persons, alive at a given age, equal to the number of 

 years that a person of that age has a possibility of living, there will die one in each 

 year, till they are all extinct) ; then, out of a number of chances equal to that 

 number of persons, which may, for instance, be 36, all but one are favourable, 

 in the first year, to any individual ; and consequently it is 35 to 1 that he receives 

 one payment of the annuity, by living till it becomes due ; that is, the probability 

 of his receiving it, is -§4, and of not receiving it, ^V- 



Again ; since, by supposition, there dies but 1 person in the first year, and 1 

 in the lA ; there are but 1 chances, in the 36, against his receiving the 2d pay- 

 ment, by living till it becomes due ; and consequently f-i will be the probability 

 of his receiving that also ; the probability of his dying in that year being -j?^-, as 



