Vol.. 6, 1920 
STATISTICS: PEARL AND REED 
281 
be regarded as a hopeful line of approach to a true law of population 
growth. 
What we want obviously is a mathematical picture of the whole course 
of population in this country. It is not enough to be able to predict 
twenty or fifty years ahead as our logarithmic parabola is able to do 
satisfactorily, in one portion of the whole curve. How absurd equation 
(iii) would be over a really long time range is shown if we attempt to 
calculate from it the probable population in, say, 3000 A.D. It gives a 
value of 11,822,000,000. But this is manifestly ridiculous; it would mean 
a population density of 6.2 persons per acre or 3968 persons per square 
mile. 
It would be the height of presumption to attempt to predict accurately 
the population a thousand years hence. But any real law of population 
growth ought to give some general and approximate indication of the 
number of people who would be living at that time within the present 
area of the United States, provided no cataclysmic alteration of circum- 
stances has in the meantime intervened. 
It has seemed worth while to attempt to develop such a law, first by 
formulating a hypothesis which rigorously meets the logical requirements, 
and then by seeing whether in fact the hypothesis fits the known facts. 
The general biological hypothesis which we shall here test embodies as an 
essential feature the idea that the rate of population increase in a limited 
area at any instant of time is proportional (a) to the magnitude of the 
population existing at that instant (amount of increase already attained) 
and (6) to the still unutilized potentialities of population support existing 
in the limited area. 
The following conditions should be fulfilled by any equation which is to 
describe adequately the growth of population in an area of fixed limits. 
1. Asymptotic to a line y = k when = + 00 . 
2. Asymptotic to a line y = 0 when % = — ^ . 
3. A point of inflection at some point x = a and y = ^. 
4. Concave upwards to left oi x = a and concave downward to right 
oi X = a. 
5. No horizontal slope except 2it x = ^ co , 
6. Values of y varying continuously from 0 to as varies from— 00 to 
+ ^ ' 
In these expressions y denotes population, and x denotes time. 
An equation which fulfils these requirements is 
he""' 
when a, b and c have positive values. 
In this equation the following relations hold: 
= 4- 00 y = - (x) 
c 
