VOL. XVII.] PHILOSOPHICAL TRANSACTIONS. ^\ 



It were needless to advertise, that the great trouble of working so many pro- 

 portions will be very much alleviated by using logarithms ; and that instead of 

 using Nni/ — Yj/u for the second term of the proportion, in finding the value 

 of three lives, it may suffice to use only Yyu, and then deducting the fourth 

 term so found out of the third, the remainder will be the present value sought ; 

 or all these fourth terms being added together, and deducted out of the value 

 of the certain annuity for so many years, will leave the value of the contingent 

 annuity on the chance of mortality of all those three lives. For example, let 

 there be three lives of 10, 30, and 40 years of age, proposed ; then the pro- 

 portions will be thus : 



As66l X 531 X 445, or 156190995, or Nnv, 



9, or 576, or Yyu for the first year, so 0,9434 to 0,00000348 

 for the second year, so 0,8900 to 0,00002462 

 for the third year, so 0,8896 to 0,00008 128 

 for the fourth year, so 0,792 1 to 0,000 1 6650 

 . for the fifth year, so 0,7473 to 0,00031071 

 for the sixth year, so 0,7050 to 0,0005 1051 



And so on, to the 6oth year, when we suppose the elder life of 40 certainly 

 to be expired ; from whence, till 70, we must compute for the first and second 

 only, and from thence to 90, for the single youngest life. Then the sum total 

 of all these fourth proportionals being taken out of the value of a certain annuity 

 for 90 years, being 1 6,58 years purchase, will leave the just value to be paid 

 for an annuity, during the whole term of the lives of three persons of the ages 

 proposed. And note, that it will not be necessary to compute for every year 

 singly, but that in most cases every 4th or 5th year may suffice, interpolating for 

 the intermediate years. 



It may be objected, that the different salubrity of places hinders this proposal 

 from being universal, nor can it be denied. But by the number that die, being 

 1174 per annum in 34000, it appears that about a 30th part die yearly, as 

 Sir William Petty has computed for London, and the number that die in 

 infancy, is a good argument that the air is but indifferently salubrious. So that 

 by what I can learn, there cannot perhaps be one better place proposed for a 

 standard. At least it is desired that in imitation hereof the curious in other 

 cities would attempt something of the same nature, than which nothing perhaps 

 can be more useful.* 



• This paper of Dr. Halley's lays a good foundation for that part of Political Arithmetic relating 

 to births, deaths, and the numbers of the people, as well as the calculation of annuities on lives. On 

 these principles several other ingenious men have since written, both in the latter volumes of the 

 Philos. Trans, and «laewbere, particularly Mr. Simpson and Dr. Price. Which volumes will here- 

 after afford an opportunity of comparing the several rules and deductions together. 



3 R 2 



