*THE POPULAR SCIENCE MONTHLY*

definite one, and may be correctly expressed by saying that the space passed over varies as the square of the time.

These illustrations will be sufficient to make clear the distinction between mathematical and empirical formulæ. It will be hardly necessary to state that the growth of a country in population follows the latter

rather than the former. In order to predict the future population, it is necessary to determine what kind of a curve the previous numbers representing the population will plot. As has been stated above, this curve comes out a parabola and the equation for this parabola is

*P = S + TX + UX ^{2}*.

There are also certain terms of higher orders which may be omitted. In this equation, *P* represents the population at any time, *X* the number of the decade and *S, T* and *U* are constants, which are to be determined. The determination of these constants involves a somewhat technical process, which may be briefly stated as follows: We first write down the population of the United States for each decade since the census began to be taken.

Year | Population | Year | Population | ||||

1790 | 3.9 | millions | 1850 | 23.2 | millions | ||

1800 | 5.3 | 1860 | 31.4 | ||||

1810 | 7.2 | 1870 | 38.6 | ||||

1820 | 9.6 | 1880 | 50.2 | ||||

1830 | 12.9 | 1890 | 62.6 | ||||

1840 | 17.1 | 1900 | 76.3 |

From these observations we form what are called "observation equations" by substituting for *P* and *X* their proper values.