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By N. Keyfitz, Hal Caswell

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000022. In a population closed to migration (which the earth as a whole must be) the rate of increase must equal the birth rate (say b) less the death rate d: r = b − d, or b = r + d. 022 per thousand. Over the million years birth and death rates must on the average have been this close. 00047. 47 per thousand on the average; again simple arithmetic gives an idea of how close, on the whole, birth and death rates have been over historical time. To show further how exceptional is the situation in which death rates in many places have dropped to 20 per thousand while births stay at 40, let us see how short a time it would have taken to arrive at the world’s present 4 billion population with birth rates that are 20 per thousand higher than death rates.

000022. In a population closed to migration (which the earth as a whole must be) the rate of increase must equal the birth rate (say b) less the death rate d: r = b − d, or b = r + d. 022 per thousand. Over the million years birth and death rates must on the average have been this close. 00047. 47 per thousand on the average; again simple arithmetic gives an idea of how close, on the whole, birth and death rates have been over historical time. To show further how exceptional is the situation in which death rates in many places have dropped to 20 per thousand while births stay at 40, let us see how short a time it would have taken to arrive at the world’s present 4 billion population with birth rates that are 20 per thousand higher than death rates.

It tells us that a population increasing at 1 percent doubles in 70 years, and so forth. 2 = 22 years. 032)22 = 10,216,000. 032)44 = 20,429,000. The formula n = 70/100x is plainly good enough for such hypothetical calculations. n= The Period of Compounding. All this is based on a definition of x by which the ratio of the population at the end of the year to that at the beginning of the year is 1 + x. Interest calculations are said to be compounded annually on such a definition. 5). Suppose that x is compounded j times per year; then at the end of 1 year the population will have grown in the ratio (1 + x/j)j .

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