Page:Popular Science Monthly Volume 76.djvu/389

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These observation equations follow:

PSM V76 D389 Formulas defining the first ten us census between 1800 and 1890.png

Normal equations for each constant are formed from these observation equations by multiplying each equation by the coefficient of the constant concerned in the equation and adding. This gives us three equations containing three unknown quantities. These unknown quantities are determined by any method and substituted in the general formula for and respectively. For example, in 1900, before the census returns for that year were available, the process above outlined yielded the following equations:

PSM V76 D389 Formula for estimating the results of the 1900 us census.png

When these equations are solved, it is found that

If we substitute these in the formula, we get

which is the forecast for 1900.

(It should be observed that in this work the year 1790 was considered 1, and 1800 was taken as the origin.)

This estimate proved somewhat low, as the census returns reported 76.3 millions for 1900. This indicates that the population of the country is growing a little more rapidly than would be indicated from its past history.

While the government authorities are at work on the census for 1910, it will be interesting to try this method of forecasting, and to see how well our results will compare with those to be announced later on. I have made a number of equations which are supposed to represent empirically the growth of the population of our country. These have been made in various ways, but all depend upon the parabolic formula, and the method outlined above.

PSM V76 D389 Formulas for estimating the results of the 1910 census.png

The equations yield the following values for the census of 1910: