sun and that this drift is most rapid near the equator and diminishes towards the poles. But this after all only pushes the explanation a little further back, and no satisfactory theory of this drifting of the spots has ever been reached. Doubtless the phenomenon is due to a large number of causes, acting together, whose resultant effect is shown in the motion of the spots as we see them.
However that may be, and although we are still unable to give any physical explanation of the phenomenon, a formula has been devised which fits the observations fairly well and which enables the astronomer to predict the motion of the spots with an accuracy comparable to the observations themselves. This formula is a complicated one, when written in its mathematical form, and involves a trigonometric function of the latitude of the spot raised to a fractional power.
Now no one pretends that this intricate formula expresses any real law of nature. But it does express the mathematical relation which connects together the observations, and by means of it the motions of the spots at different latitudes on the sun may be predicted with all desirable accuracy.
The problem of deriving an equation which shall represent the growth of the population of the United States during the past one hundred and ten years and which may be used to predict its growth through future decades, is exactly such a case as that of the sun's spots just mentioned. The observations in this case consist of eleven determinations of the population as given in the census returns from 1790 to 1890.
In studying these observations of population, taken at regular intervals of ten years, it occurred to me some years ago to examine them with some care in order to discover whether they were related to each other in any orderly way, and if so whether they could be represented by an equation of reasonable simplicity. It is evident that if an equation can be found which will fit the growth of population during the hundred years which intervened between 1790 and 1890 it would form the most probable basis for predicting the population of the future.
Somewhat to my surprise I discovered a comparatively simple equation which represented the census enumerations very closely and which, notwithstanding the fluctuations in the various factors which affect the growth of population, followed the general course of this growth with remarkable fidelity, as will be seen by the following table, which shows the population as given by the Census Bureau and as determined by the empirical formula. The discrepancies between the observed population and that computed from the formula are also given for the sake of an easy comparison. In each case the population is given to the nearest thousand, a figure far within the limit of error of the census count.
