410 CALCULATIONS OF POPULATION IN JUNE, 1900 



1860, and the second making allowance for extraordinary losses 

 (marked I in the table). All differences, ratios, etc., in which 

 the census population for 1870 enters are put in parentheses. 



In forming the four columns of calculated population the 

 formula is applied to the figure found for the beginning of the 

 decade to obtain the natural increase. To this is added the ascer- 

 tained immigration for the decade. The result is the total in- 

 crease, and hence is derived the calculated figure for the end of 

 the decade. 



The second column of the table shows the census population 

 in millions and decimals of a million for the dates in the first. 

 The third column gives successive differences of the second. The 

 fourth shows the immigration, and the fifth the difference be- 

 tween the two preceding, here called " natural increase." The 

 percentage formed by that increase, compared with the popula- 

 tion at the beginning, next appears. This shows a general falling 



off in value, as noted above, while its reciprocal, denoted -A— in 



Ap 



the seventh column, is the number represented by e + fp + gp 2 

 in our formula. The remaining columns give the four calcula- 

 tions of population, as already explained. Each calculated value 

 is followed by the correction reducing it to the value actually 

 found. 



■Another word as to the meaning of the coefficient e, the num- 

 ber of people that increased by a unit (as 2.862 to 3.862, 2.279 to 

 3.279, and so on) in ten years, when the population was very 

 scanty. The length of time in which the population would double 

 itself under those circumstances, by natural increase, is found 



by ascertaining what power 2 is of 1 -f- — . The four calcula- 

 tions give 23, 19, 21, and 26 I years respectively. The time of 

 doubling lengthens without limit as the population increases, 

 and the effect of coefficients other than e appears. 



Comparison of Four Calculations. — All calculations agree in in- 

 dicating a large deficiency for the census population in 1870, 

 which was 898,000 by the last and nearly double as much by 

 the first. The 1890 census was also short by a less amount, 

 while that of 1880 gave an excess according to all four. The 

 census figures from 1820 to 1860 are much more easily recon- 

 cilable. None of these is more than a hundred thousand in 

 error by more than one of the four calculations, and only one 

 census, that for 1830, errs in the same direction according to all 



