[ 70 ] 
Thofe who are well verfed in the nature of loga- 
rithms, I mean thofe that can deduce them from the 
do&rine of fluxions and infinite feries, will eafily 
apprehend, that the quantity here called a, is that which 
fome call the hyperbolic logarithm ; others, the natural 
logarithm : it is what Mr. Cotes calls, the logarithm 
whofe modulus is i : laftly, it is by fome called 
Nepers logarithm. And, to fave the reader fome 
trouble in the pra&ice of this laft theorem, the 
mod neceffary natural logarithms, to be made ufe of 
in the prefent difquifttion about lives, are the fol- 
lowing: 
If r = i. 04, then will cc—O. 0392207. 
r — 1.05, " " - a = O. O4879OI. 
r— i.o( 5 , - - - ct = o. 05 825 89. 
It is to be obferved, that the theorem here found, 
makes the values of lives a little bigger, than what 
the theorem found in the firft problem of my book 
of annuities on lives, does 5 for, in the prefent cafe, 
there is one payment more to be made, than in the 
other 5 however, the difference is very inconfiderablc. 
But, altho’ it be indifferent which of them is ufed, 
on the fuppofition of an equal decrement of life to 
the extremity of old-age 5 yet, if it ever happens, 
that we Ihould have tables of obfervations, concern- 
ing the mortality of mankind, intirely to be depended 
upon, then it would be convenient to divide the 
whole interval of life into fuch fmaller intervals, 
as, during which, the decrements of life have been 
obferved to be uniform, notwithftanding the decre- 
ments in fome of thofe intervals fhould be quicker, 
or flower, than others ; for then the theorem here 
1 found 
