of the United States and Territories of North America. 77 
of the last columns of the preceding table, and as the fact is 
worthy of being particularly remembered, it may be more expli- 
citly stated as follows : 
In 1790, 1 
In 1800, ( 
In 1810, C 
In 1820, J 
the relation of the 
Slaves to the Free 
Persons, were as 
1 
10 to 40 
10 to 49 
10 to 51 
10 to 53 
Many other interesting relations might possibly be deduced 
from the tables relating to the slave population ; but it is time 
to hasten to the consideration of the facts which this survey of 
the American population has afforded, relative to the numbers 
devoted to agriculture, commerce, and manufactures *. 
* In Counsellor Cooper’s Letters on the Slave Trade, it is remarked, “ that the 
proportion of deaths among slaves has been determined, from a series of observa- 
tions, to be about 1 in 20.” Adopting this, therefore, as the most probable datum 
to which we can at present refer, we may determine from it what proportion of 
births is necessary, in order to produce the slave population, at the different pe- 
riods referred to. For this purpose, let A denote the amount of the slave popula- 
tion, at any given period, A' its amount at any succeeding time, and n the interval 
in years. Let also — represent the rate of mortality, and — the annual ratio 
of births ; then, from the formula of population, 
A' 
= a(i + ^Y, 
\ mx J 
we may deduce, by the application of logarithms, 
✓ m — x\ log A^ — log A 
1 1 i — I — -) 
\ mx / n 
the latter number of which, being a known function, may be denoted by log CL 
hence, the preceding equation will become, 
,og ( 1 + ^r) = log0 
and, by passing from logarithms to numbers, there will arise, 
1 + ■ 
and which, by reduction, produces 
= O : 
m (O — 1) + 1 ’ 
a general formula for the annual ratio of births. 
From the actual enumerations of the slaves, we deduce the following results t 
(1790 to 1800,) (1.0254 
For the period from -j 1800 to 1810, v the value of O is ■< 1,0291 
( 1810 to 1820, j ( 1.0259, 
and which values of O, being substituted in the preceding formula, and also the va- 
lue of m (20), there will arise the following values of x ; viz. 
From 
