Popular Science Monthly/Volume 58/November 1900/The Population of the United States During the Next Ten Centuries

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IS it possible to predict with any degree of certainty the population of a country like the United States for a hundred years to come?

Doubtless the average intelligent person would say à priori that the growth of population is not a matter which can be made the subject of exact computation; that this growth is the result of many factors; and that those factors are subject to such great fluctuations that an estimate of the population a hundred years hence can be, in the nature of the case, only a rough guess.

It is true that the growth of population depends on a number of factors. It is also true that these factors vary in accordance with laws which are at present not known. Nevertheless it does not by any means follow that because the law of these variations is unknown we cannot represent the variations themselves by a mathematical equation. The problem of representing mathematically the law connecting a series of observations for which theory furnishes no physical explanation is one of the most common tasks to which the mathematician is called. And it does not in the least diminish the value of such a mathematical formula, for the purposes of prediction, that it is based upon no knowledge of the real causes of the phenomena which it connects together.

To illustrate: The black spots on the sun have been objects of the greatest interest to astronomers ever since Galileo pointed the first feeble telescope at his glowing disc. These spots, as observed from the earth, seem to pass across the disc from east to west as the sun rotates on its axis.

Among the problems with which the possessors of the first telescopes busied themselves were the observation of these spots for determining the period of the sun's rotation. The observation is a very simple one and consists merely in noting the time which elapses between successive returns of a spot to the central meridian of the disc. The earlier observers were astonished to find that the different spots gave different results for the rotation period, but it was only within the last thirty years that the researches of Carrington brought out the fact that these differences follow a regular law showing that at the solar equator the time of rotation is less than on either side of it.

The explanation which is generally accepted to account for this peculiar state of affairs is that the spots drift in the gaseous body of the sun and that this drift is most rapid near the equator and diminishes towards the poles. But this after all only pushes the explanation a little further back, and no satisfactory theory of this drifting of the spots has ever been reached. Doubtless the phenomenon is due to a large number of causes, acting together, whose resultant effect is shown in the motion of the spots as we see them.

However that may be, and although we are still unable to give any physical explanation of the phenomenon, a formula has been devised which fits the observations fairly well and which enables the astronomer to predict the motion of the spots with an accuracy comparable to the observations themselves. This formula is a complicated one, when written in its mathematical form, and involves a trigonometric function of the latitude of the spot raised to a fractional power.

Now no one pretends that this intricate formula expresses any real law of nature. But it does express the mathematical relation which connects together the observations, and by means of it the motions of the spots at different latitudes on the sun may be predicted with all desirable accuracy.

The problem of deriving an equation which shall represent the growth of the population of the United States during the past one hundred and ten years and which may be used to predict its growth through future decades, is exactly such a case as that of the sun's spots just mentioned. The observations in this case consist of eleven determinations of the population as given in the census returns from 1790 to 1890.

In studying these observations of population, taken at regular intervals of ten years, it occurred to me some years ago to examine them with some care in order to discover whether they were related to each other in any orderly way, and if so whether they could be represented by an equation of reasonable simplicity. It is evident that if an equation can be found which will fit the growth of population during the hundred years which intervened between 1790 and 1890 it would form the most probable basis for predicting the population of the future.

Somewhat to my surprise I discovered a comparatively simple equation which represented the census enumerations very closely and which, notwithstanding the fluctuations in the various factors which affect the growth of population, followed the general course of this growth with remarkable fidelity, as will be seen by the following table, which shows the population as given by the Census Bureau and as determined by the empirical formula. The discrepancies between the observed population and that computed from the formula are also given for the sake of an easy comparison. In each case the population is given to the nearest thousand, a figure far within the limit of error of the census count.

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 of this paper, viz., can the population of the United States an hundred years hence be predicted within reasonable limits of error?

During the past century the factors which govern the growth of population have fluctuated enormously; there have been wars and epidemics; there have been decades in which large numbers of emigrants landed upon our shores and there have been other decades in which emigrants were few; there have been years of plenty and years of want; booms and panics, good times and hard times have had their share in the century which has passed. Yet notwithstanding all these varying conditions, the growth of the population has been a regular and orderly one, so much so that it can be represented by a comparatively simple mathematical equation. Can this equation be trusted to predict the population in the decades which are to come?

How closely the formula will represent the population of the future will depend, of course, upon the continuance of the same general conditions which have held in the past. This does not mean that exactly the same factors are to operate, but that on the whole the change of one factor will be balanced by a change in another, so that in the main the character of the growth manifested during the past century will be continued. A decided change in the birth-rate or a widespread famine would bring out large discrepancies. But on the whole it may be expected that the experience of the last hundred years involves so many varying conditions that the general law of growth which satisfies that period will continue to approximate the development of the population for a considerable time to come.

This does not mean that any particular census enumeration of the future will be represented closely, but simply that in the main the computed values will follow the general growth of the population. The law of probabilities will lead one to expect at times considerable variations. The preliminary announcements from the Census Office, as given in the daily papers, indicate a result for 1900 of about 75,700,000 people, a value considerably below the computed one. This would mean that at this epoch the formula was not representing the actual growth, but does not at all indicate that it will cease to represent the general growth of the succeeding centuries. In any event this method furnishes the most trustworthy estimate which can be made for the future, since it gives the result which is mathematically most probable and which is based on all the data of the past. Carrying forward, therefore, the computation we obtain the following values for the most probable population of the future:

1900 77,472,000
1910 94,673,000
1920 114,416,000
1930 136,887,000
1940 162,268,000
1950 190,740,000
1960 222,067,000
1970 257,688,000
1980 296,814,000
1990 339,193,000
2000 385,860,000
2100 1,112,867,000
2500 11,856,302,000
2900 40,852,273,000

The law governing the increase of population, as generally stated, is, that when not disturbed by extraneous causes such as emigration, wars and famines, the increase of population goes on at a constantly diminishing rate. By this is meant that the percentage of increase from decade to decade diminishes. It will be noticed that the figures just given involve such a decrease in the percentage of growth. A simple differentiation of the formula gives as the percentage of increase of the population per decade 32 per cent, in 1790, 24 per cent, in 1880, 13 per cent, in 1990, while in one thousand years it will have sunk to a little less than three per cent. But according to the formula this percentage of increase will become zero, or the population become stationary, only after the lapse of an indefinite period.

The figures just quoted are, to say the least, suggestive. Forming, as they do, the most probable estimate we can make for the population of the future, they suggest possibilities of the highest social and economic interest. Within fifty years the population of the United States (exclusive of Alaska, of Indians on reservations and of the inhabitants of the recently acquired islands) will approximate 190 millions, and by the year 2000 this number will have swelled to 385 millions of people; while should the same law of growth continue for a thousand years the number will reach the enormous total of 41 billions.

How great a change in the conditions of living this growth of population would imply is, perhaps, impossible for us to realize. Great Britain, at present one of the most densely populated countries of the globe, contains about 300 inhabitants to the square mile. Should the present law of growth continue until 2900 the United States would contain over 11,000 persons to each square mile of surface.

With the growth of population our civilization is becoming more and more complex and the drafts upon the stored energy of the earth more enormous. As a consequence of all this, it would seem that life in the future must be subject to a constantly increasing stress, which will bring to the attention of individuals and of nations economic questions which at our time seem very remote.