OF all branches of statistics, those which relate to the population of a country or city are of most general interest. The interest felt in the question of our national population culminates every ten years when the census is taken. To be sure, no great importance is to be attached to mere numbers; yet we can not help feeling a little pride if we belong to the biggest religious denomination, the biggest university, or the biggest country.
The plots in Fig. 1 represent the growth in population of various countries as indicated by the census of 1900. The curves were made by Mr. W. R. Wilcox and were printed in the census report for that year.
An examination of these curves shows that for the most part the growth of a country is constant; for example, the lines representing France, Spain, Sweden and Norway, Turkey and Italy, are nearly straight. This indicates that while the population of those countries is increasing slightly, there is no great gain from year to year. The population of the United States is represented by a curve which is well known in mathematics. In the chart below I have redrawn this curve, and with it one which is a true parabola.
It will be seen that these two are strikingly similar. Now if the population of the United States increased in such a manner as always to follow this parabolic form, the census enumeration would be unnecessary, as one could predict the future population from the past. Unfortunately, however, this is not the case; and it is only by a somewhat tedious method that we are able to predict the future population with any degree of certainty. There are two kinds of formula?—rational and empirical. A rational formula is one which is mathematically true under all conditions. The fact that the volume of a cylinder equals π multiplied by the square of the radius, multiplied by the length, is a fact that does not depend upon any external circumstances whatever. On the other hand, the value of the acceleration due to gravitation is not a constant quantity, but differs with the latitude and altitude of the observer. This latter is one of the most important physical constants in nature and a great deal of time arid money have been expended in determining its value. While there is no mathematical formula that expresses this value, an empirical formula has been devised in which, if one substitutes the latitude and altitude of the place of observation, a