514 Mr. Gompertz on the nature of the function 
ceeding by small intervals of time, v^hatever the law pf 
mortality may be, provided the intervals be not greater than 
certain limits : I now call the reader's attention to a law 
observable in the tables of mortality, for equal intervals of 
long periods ; and adopting the notation of my former paper, 
considering L to express the number of living at the age x, 
and using x for the characteristic of the common logarithm ; 
that is, denoting by x(L) the common logarithm of the 
number of persons living at the age of x, whatever x may 
be, I observe that if x fL) — X fL , h ^ (L . ] — x fL , ] 
IV+sJ’ be all the same; that is to say, 
if the differences of the logarithms of the living at the ages 
n, n + m ; n m, n + ; n + 2w, n + sw; &c. be con- 
stant, then will the numbers of living corresponding to those 
ages form a geometrical progression ; this being the funda- 
mental principle of logarithms. 
Art. 2. This law of geometrical progression pervades, in 
an approximate degree, large portions of different tables of 
mortality ; during which portions the number of persons 
living at a series of ages in arithmetical progression, will be 
nearly in geometrical progression ; thus, if we refer to the 
mortality of Deparcieux, in Mr. Baily's life annuities, we 
shall have the logarithm of the living at the ages 15, 25, 35, 
45, and 55 respectively, 2,9285; 2,88874; 2,84136; 2,79379; 
2.72099, for X ; x ^LJ ; x ^Lj ; &c. and we find 
X ^Lj — X ^LJ = , 04738 A (LJ — X = , 04757, and con- 
sequently these being nearly equal ( and considering that for 
small portions of time the geometrical progression takes place 
very nearly ) we observe that in those tables the numbers of 
