42 HALLEY. 



to the table which he deduced from them. Halley's table is in the 

 following form: 



1 



2 

 3 

 4 



1000 

 855 



798 

 760 



The left-hand number indicates ages and the right-hand num- 

 ber the corresponding number of persons alive. We do not feel 

 confident of the meaning of the table. Montucla, page 408, under- 

 stood that out of 1000 persons born, 855 attain to the age of one 

 year, then 798 out of these attain to the age of two years, and 

 so on. 



Daniel Bernoulli understood that the number of infants born 

 is not named, but that 1000 are supposed to reach one year, then 

 855 out of these reach two years, and so on. Hist de VAcad. ... 

 Paris, 1760. 



^D. Halley proceeds to shew the use of his table in the calcu- 

 lation of annuities. To find the value of an annuity on the life of 

 a given person we must take from the table the chance that he 

 will be alive after the lapse of n years, and multiply this chance 

 by the present value of the annual payment due at the end of 

 n years ; we must then sum the results thus obtained for all values 

 of n from 1 to the extreme possible age for the life of the given 

 person. Halley says that " This will without doubt appear to 

 be a most laborious Calculation." He gives a table of the value 

 of an annuity for every fifth year of age up to the seventieth. 



^Q. He considers also the case of annuities on joint lives, or 

 on one of two or more lives. Suppose that we have two persons, 

 an elder and a younger, and we wish to know the probability 

 of one or both being alive at the end of a given number of years. 

 Let N be the number in the table opposite to the present age of 

 the younger person, and R the number opposite to that age in- 

 creased by the given number of years ; and let N=R-\- Y, so that 

 Y represents the number who have died out of N in the given 

 number of years. Let n, r, y denote similar quantities for the 

 elder age. Then the chance that both will be dead at the end 



