28o 
STATISTICS: PEARL AND REED 
Proc. N. a. S. 
Predicted population in 1910 = 89,128,000. 
Deviation of prediction from actual, 1910 = — 2,844,000. 
Percentage error = 3%. 
Data for 1790 to 1890, inclusive: 
y = 9,013,800-6,242,170^ + 839,782:^2 + 19,744,300 log % (vii) 
Predicted population in 1910 = 91,573,000. 
Deviation of prediction from actual, 1910 = — 399,000. 
Percentage error = 0.4 per cent. 
Data for 1790 to 1900 inclusive: 
y = 8,748,000 -5,880,890a; + 821,00^^ 4. 18,232,100 log ^ (viii) 
Predicted population in 1910 = 91,148,000. 
Deviation of prediction from actual, 1910 = —824,000. 
Percentage error = 0.9%. 
Beginning with 1860 (equation (iv)) and coming down to 1900, our 
hypothetical statistician would have been only once in error as much as 
1% in his prediction of the 1910 population by this logarithmic parabola. 
The one larger error is for the 1880 curve, where apparently the aberrant 
counts of 1860 and 1870 exert an undue influence. 
Altogether it seems justifiable to conclude that : 
1. A logarithmic parabola of the type of equation (ii) describes the 
changes which have occurred in the population of the United States in 
respect of its gross magnitude, with a higher degree of accuracy than any 
empirical formula hitherto applied to the purpose. 
2. The accuracy of the graduation given by this logarithmic parabola is 
entirely sufficient for all practical statistical purposes. 
II 
Satisfactory as the empirical equation above considered is from a 
practical point of view, it remains the fact that it is an empirical expression 
solely, and states no general law of population growth. Insofar it is 
obviously an undesirable point at which to leave the problem of the 
mathematical expression of the change of population in magnitude. 
It is quite clear on a priori grounds, as was first pointed out by Malthus 
in non-mathematical terms, that in any restricted area, such as the United 
States, a time must eventually come when population will press so closely 
upon subsistence that its rate of increase per unit of time must be re- 
duced to the vanishing point. In other words, a population curve may 
start, as does that shown in figure 1, with a convex face to the base, but 
presently it must develop a point of inflection, and from that point on 
present a concave face to the x axis, and finally become asymptotic, the 
asymptote representing the maximum number of people which can be 
supported on the given fixed area.^ Now, while an equation like (ii) 
can, and will in due time, develop a point of inflection and become con- 
cave to the base it never can become asymptotic. It, therefore, cannot 
