48'8 PHILOSOPHICAL THANSACTIONS. [anNO 1 092-3. 



chances that one of the two persons is living, Yy the chances that they are 

 both dead ; Ry the chances that the elder person is dead and the younger 

 living; and rY the chances that the elder is living and the younger dead. 

 Thus, two persons of 18 and 35 are proposed, and after 8 years these chances 

 are required. The numbers for 18 and 35 are 6lO and 490, and there are 50 

 of the first age dead in 8 years, and 73 of the elder age. There are in all 610 

 X 490 or 298900 chances ; of these there are 50 X 73 or 3650 that they are 

 both dead. And as 298900 to 2989OO — 3630, or 295250 ; so is the present 

 value of a sum of money to be paid after 8 years, to the present value of a 

 sum to be paid if either of the two live ; and as 560 X 73, so are the chances 

 that the elder is dead, leaving the younger ; and as 4 1 7 X 50, so are the chances 

 that the younger is dead, leaving the elder. Wherefore as 6 10 X 490 to 560 

 X 73, so is the present value of a sum to be paid at 8 years end, to the sum 

 to be paid for the chance of the younger's survivance; and as 610 X 490 to 

 417 X 50, so is the same present value to the sum to be paid for the chance of 

 the elder's survivance. 



This possibly may be yet better explained by expounding these products by 

 rectangular parallelograms, as in fig. 2, pi. 12, where A B or CD represents the 

 number of persons of the younger age, and DE or BH those remaining alive 

 after a certain term of years; whence CE will answer the number of those dead 

 in that time ; so AC, BD may represent the number of the elder age ; AF, BI 

 the survivors after the same term ; and CF, DI those of that age that are dead 

 at that time. Then shall the whole parallelogram ABCD be N n, or the pro- 

 duct of the two numbers of persons, representing such a number of persons 

 of the two ages given ; and by what was said before, after the term proposed 

 the rectangle HD will be as the number of persons of the younger age that 

 survive, and the rectangle AE as the number of those that die. So likewise 

 the rectangles A I, FD will be as the numbers, living and dead, of the other 

 age. Hence the rectangle HI will be as an equal number of both ages surviving. 

 The rectangle FE being the product of the deceased, or Yy, an equal number 

 of both dead. The rectangle GD or %, a number living of the younger age, 

 and dead of the elder. And the rectangle AG or rY a number living of the 

 elder age, but dead of the younger. This being understood, it is obvious, 

 that as the whole rectangle AD or N w is to the gnomon FABDEG or N n — 

 Yy, so is the whole number of persons or chances, to the number of chances 

 that one of the two persons is living; and as AD or N « is to FE or Yy, so 

 are all the chances to the chances that both are dead ; whence may be computed 

 the value of the reversion after both lives. And as AD to GD or Ry, so the 

 whole number of chances to the chances that the younger is living and the 



