# Calculation of Population Increase, [October 1801]

# Calculation of Population Increase

[October 1801]

the | Census of | 1791. | was | 3,929,326. | wanting | 70,474 | of 4. millions |

that | of | 1801. | is | 5,366,786. | includg. | 10,000 | for Maryld & |

100,000. Tennissee |

<calling the 1st. four millions & the last 5,000,000 in 10. years it is in the geometrical ratio of 2¼ per annum and would take1 31. years to double>

in a series of Geometrical progressionals

given | the 1st. term | f. | = | 3,929,326 |

the no. of terms | t | = | 10 | |

the last term | l | = | 5,366,786 | |

required the ratio | r |

from the nature of geometrical progression we have this equation

f × rt = l. then rt = | l | and Log. r × t = Log. l.–Log. f |

f |

Log. r = | Log. l.–Log. f |

t |

Log. l | 6.7297142 | |

Log. f. | 6.5943180 | |

.1353962 | ||

÷ 10. | .0135396 | = Log. r = Log. of 1.031667 = r |

given | f | = | 3,929,326 | |||

l | = | 3,929,326 | × 2 | = | 7,858,652 | |

r | = | 1.031667 | ||||

required | t |

f × rt = l rt = | l | Log. r × t = Log. l.–Log. f |

f |

t = | Log. l.–Log. f |

Log. r |

Log. l. | 6.8953480 | ||

Log. f | 6.5943180 | ||

.3010300 | |||

which ÷ by | Log. r | .0135396 | gives 22.23 y |

MS (DLC: TJ Papers, 232:41570); undated; entirely in TJ’s hand; endorsed by TJ: “Census.”

TJ apparently made these calculations sometime after 17 Sep. 1801, the day he received W. C. C. Claiborne’s letter of 4 Aug., which included Claiborne’s estimate of 100,000 for Tennessee’s population. In December, when Madison conveyed the second census to TJ for transmittal to Congress, there was still no return for Tennessee. The return for part of Baltimore County, Maryland, was lacking until 19 Nov., which may account for TJ’s addition of 10,000 people for Maryland along with the 100,000 for Tennessee in his calculations (Madison to TJ, 8 Dec.). TJ had not completed the computations printed above by 3 Oct., since in his letter to William Short on that day he forecast a “duplication” of the population “in 23. or 24. years.” To Wilson Cary Nicholas on 25 Oct., TJ reported the doubling time of the population as 22 years and three months. That was the exact result of his figuring in the document printed above, which means he completed his computation by 25 Oct. 1801.

Geometrical Progression: in this document, TJ reckoned the growth of population as a geometric progression—the ratio of increase is considered to be constant, but the population grows by a greater number of individuals each successive year because the base population against which the rate of growth is multiplied grows larger. TJ used logarithms to perform those computations. The document has three sections, separated by horizontal rules at the left margin. In the first section, TJ rounded the total for the 1790 census up to 4,000,000 and rounded his estimate for the 1800 census total down to 5,000,000. Without showing his arithmetic, he determined the annual rate of increase over ten years, using the rounded figures, to be 2.25%, which would result in a doubling of the population in 31 years. He subsequently voided that estimate, striking through it with three diagonal strokes. The middle and final sections of the document contain his calculations using specific figures for the two censuses, rather than the rounded totals. From the increase to 5,366,786 from 3,929,326 over ten years—ten terms in his figuring above—he computed an annual rate of growth of 3.1667% (each year the population was 1.031667 times what it had been the year before). In the final section, he used that rate of increase to calculate how long it would take the population to reach 7,858,652, or twice the total of the 1790 census. The result, by his analysis, was 22.23 years.

The idea that human populations increase by geometric progression gained widespread notice from Thomas Robert Malthus’s Essay on the Principle of Population, first published in 1798. However, well before Malthus wrote that treatise, Leonhard Euler and others applied geometric progression, which had been used to figure compound interest and for other applications in commerce and finance, to the problem of calculating population growth. By the 1770s and 1780s the principle of geometrical increase of population had wide acceptance among people who wrote about and discussed demography. The arithmetical skills needed to calculate progressions, though, remained “a matter beyond most demographers and nearly all less specialized commentators on population” (James C. Riley, Population Thought in the Age of the Demographic Revolution [Durham, N.C., 1985], 131–5). In the 1750s, Benjamin Franklin stated in his Observations Concerning the Increase of Mankind, Peopling of Countries, &c. that the American population doubled every 25 years. European and British writers, including Malthus, fastened upon Franklin’s period of doubling for North America, which was considerably shorter than the doubling period calculated for some regions of Europe. Franklin, however, based his estimate on a combination of suppositions rather than the computation of the actual rate of growth. Madison paid some notice to doubling rates in unsigned essays on emigration and population he wrote for Philip Freneau’s National Gazette in 1791, but apparently got his figures from Franklin and other sources. That same year, William Barton wrote a long paper on population and longevity in the United States. Barton treated factors that affect population growth, such as birth and death rates, and like Franklin and Madison he did not venture into the mathematics of geometric increase. TJ apparently got the mathematical wherewithal to calculate geometric progressions from William Small, his professor at the College of William and Mary. Logarithms were among the practical skills of arithmetic that TJ, in 1799, deemed worthwhile to master (he classified mathematical functions beyond logarithms and quadratic equations as a “delicious,” but mostly superfluous, “luxury”). Earlier, writing under Query VIII of his Notes on the State of Virginia, he did not refer specifically to the geometric progression of population, but took it for granted. Using data from the period 1654 to 1772—and employing the word “term,” as in the document above—he computed that the population of Virginia was doubling every 27.25 years. Although he did not show his calculations, he must have worked out that geometric progression by a process similar to what he followed in the document above. In 1792, TJ wrote that the enslaved black population of the U.S. doubled “in about 25. years.” He did not provide clues to indicate if he computed that rate himself or got the information by other means (Riley, Population Thought, 132–6; Thomas Robert Malthus, An Essay on the Principle of Population, ed. Patricia James, 2 vols. [Cambridge, 1989], 1:11–12, 295; Thomas Robert Malthus, An Essay on the Principle of Population, ed. Philip Appleman, 2d ed. [New York, 2004], 21; Patricia James, Population Malthus: His Life and Times [London, 1979], 103, 106–7, 380; Patricia Cline Cohen, A Calculating People: The Spread of Numeracy in Early America [Chicago, 1982], 112–15; , 14:110–12, 114, 117–18; , Transactions, 3:25–62, 134–8; William Barton, Observations on the Progress of Population, And the Probabilities of the Duration of Human Life, in the United States of America [Philadelphia, 1791]; , No. 667; William Petersen, Malthus [Cambridge, Mass., 1979], 61, 147–8; , 1:54; Autobiography in , 1:4; , 82–7; Vol. 19:584; Vol. 22:473; Vol. 24:98; Vol. 31:126–7).

Malthus’s Essay, with its central assertion that the growth of population, increasing at a geometric rate, far outpaces the increase of resources needed to sustain life, appeared anonymously in 1798. TJ saw a review of the work and discussed its argument with Thomas Cooper, but he did not read that version of the Essay. Only after a revised and expanded second edition appeared in 1803, bearing its author’s name, did TJ read Malthus. Later he acquired the book for his library ( ; Drew R. McCoy, “Jefferson and Madison on Malthus: Population Growth in Jeffersonian Political Economy,” , 88 [1980], 259–76; , No. 2938; Thomas Cooper to TJ, 16 Feb. 1804; TJ to Cooper, 24 Feb. 1804).

1. TJ here canceled “between.”