5 o POPULAR SCIENCE MONTHLY. 



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 de- 

 terminations of the population as given in the census returns from 

 1790 to 1890. 



In studying these observations of population, taken at regular in- 

 tervals 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 equa- 

 tion 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 equa- 

 tion 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 popu- 

 lation 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. 



