| Observed | Computed | ||
| Year. | Population. | Population. | Discrepancy. |
| 1790 | 3,929,000 | 4,012,000 | 83,000 |
| 1800 | 5,308,000 | 5,267,000 | 41,000 |
| 1810 | 7,240,000 | 7,059,000 | 181,000 |
| 1820 | 9,634,000 | 9,569,000 | 65,000 |
| 1830 | 12,866,000 | 12,985,000 | 119,000 |
| 1840 | 17,069,000 | 17,484,000 | 415,000 |
| 1850 | 23,192,000 | 23,250,000 | 58,000 |
| 1860 | 31,443,000 | 30,468,000 | 975,000 |
| 1870 | 38,558,000 | 39,312,000 | 754,000 |
| 1880 | 50,156,000 | 49,975,000 | 181,000 |
| 1890 | 62,622,000 | 62,634,000 | 12,000 |
The smallness of the discrepancies and the consequent close agreement of the formula with the observations show that the growth of the population has been a regular and orderly one. There are, however, two residuals which have abnormally large values. The census of 1860 shows a population of 975,000 larger than the computed value, while that of 1870 falls 754,000 below that of the computed value.
The explanation of these discrepancies is not far to seek. The devastating effect of the war would show itself in the census of 1870 and succeeding years. The effect would be to give 1870 a smaller observed value than would be expected. This is precisely what we find to be the case, the census of that year falling 754,000 short of the computed value. An abnormally small value in 1870 would, of course, have its effect on the population of succeeding decades and would also give an apparent difference of opposite sign to the observed population in 1860.
There is, however, good reason to believe that the population in 1870 as determined by the census was much smaller than the actual population at that time. Mr. Robert Porter, in Census Bulletin No. 12, October 30, 1890, makes the statement that the census of 1870 was grossly deficient in the Southern States and that a correct and honest enumeration would have shown at that time a much larger population than that actually returned by the Census Bureau. There are, of course, no means of ascertaining exactly the extent of these omissions, but there is no question that the population as computed by the formula for 1870 is far nearer the truth than the value given by the census for that year.
However this may be, it is evident that the formula represents the general law of growth which held between 1790 and 1890 with an accuracy almost comparable with that of the census determinations themselves. The question of immediate interest, however, is not whether the growth of population during the last century can be represented by a mathematical formula, but it is that which stands at the beginning
