Thomas Jefferson Papers
Note: this document has content that may require expanded/print view for best results (icons above right)

# 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 6.72971 Log. f. 6.59432 0.135396 ÷ 10. 0.0135396 = 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. 6.89535 Log. f 6.59432 0.30103 which ÷ by Log. r 0.0135396 gives 22.23 y

(: 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.