384 



TEE 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 formulas. It will be hardly neces- 

 sary to state that the growth of a country in population follows the lat- 



FlG. 2. 



ter rather than the former. In order to predict the future population, 

 it is necessary to determine what kind of a curve the previous num- 

 bers 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\ 



There are also certain terms of higher orders which may be omitted. 

 In this equation, P represents the population at any time, X the num- 

 ber of the decade and 8 } T and U are constants, which are to be de- 

 termined. 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. 



From these observations we form what are called " observation 

 equations " by substituting for P and X their proper values. 



