[ 93 ]' 
there (hall be a furvivor at the end of the ifl, 2d, 
3d, &c. moments^ from this time to the end of the 
poflible exigence of furvivorfliip. This coincidence 
every one converfant in thefe.fubjeds muff fee, upon 
refledfing, that both thefe fenfes give the true prefent 
value of a life-annuity fecured by land, without 
intereft of money 
^ The fum of the probabilities that any given lives will attain 
to the end of the ift, 2J, 3d, &c. years from the prefent time to 
the utmoft extremity of life (for initance, -p -f- +.1, &c. 
to — 22i for lives of 40, by the hypothefis) may be called 
their expeSiation^ or the number of payments due to them, as 
yearly annuitants. The fum of the probabilities that they will at- 
tain to the end of the ift, 2d, 3d, &c. half years (or, in the par- 
ticular cafe fpecified, + -i|. -P 1 -®, Sic. = half 
years, or 22^ years\ is their expectation as heilf^ yearly annuitants. 
And the fums juft mentioned of the probabilities of their attain- 
ing to the end of the ift, 2d, 3d, moments (equal in the fame 
particular cafe to 23 years) is properly their expe^ation of life^ 
or x}ne.\x expert ation as annuitants fecured by land. 
Mr. De Moivre has concealed the demonftrations of the 
rules he has given for finding thefe expeSlations of life, and only 
intimated, in general, that he difeovered them by a calculation 
deduced from the method of fluxions, p. 66, of his Treatife on 
Annuities. It will, perhaps, be agreeable to fome to fee how 
eafily they are deduced in this method upon the hypothefis of an 
equal decrement of life. 
Let X ftand for a moment of time and n the complenient of any 
affioned life. Then Sic. will bethepre/ent 
<= n n n 
probabilities of its continuing to the end of the ift, 2d, 3d, &c, 
moments ; and the probability of its continuing to the end 
n 
of X time. will therefore be the fiuxion of the fum of 
n 
the probabilities, or of an area reprefenting this fum, whofe 
This 
