Benjamin Franklin and Exponential Growth
Benjamin Franklin’s formal education ended at age 10 and he frequently lamented his lack of mathematical training. He wrote in his Autobiography that he failed in arithmetic during his one year at George Brownell’s school and that his friend Collins far “outstript” him in mathematical learning. Through self-study, however, Franklin mastered and developed mathematical and statistical ideas and he became one of the early pioneers of statistics in America.
One of the mathematical concepts Franklin used often — but that confused many of his contemporaries — was exponential growth. Classic examples of exponential growth are:
The wheat (or rice or pennies) on the chessboard story
Spread of an epidemic in an unvaccinated, previously unexposed population
Population growth with unconstrained resources
Compound interest
Franklin wrote about the last two cases, and is famous for his legacy involving compound interest. Before investigating what Franklin did, however, let’s briefly review what exponential growth is.
Exponential Growth and the Chessboard Story
Exponential growth is the key point of the famous chessboard story. A mathematical consultant solves a major problem for the king, who offers payment of $20,000 for the service. The consultant, however, has an alternative proposition.
“Sire,” says the consultant, “my wants are humble and are satisfied with mere pennies. You see that chessboard. Simply put one penny on the first square, two pennies on the second square, four pennies on the third, and so on, with each square having double the number of pennies as the square before.”
The king thinks about the few pennies that would be put on the first chessboard row and readily agrees. But the king had slept during his math classes and did not realize how quickly doubled amounts grow. The board contains 255 pennies in the first eight squares, 65,535 pennies in the first two rows, and 16,777,215 pennies in the first three rows. These would accumulate to 18,446,744,073,709,551,615 pennies on the entire chessboard, or more than 30,000 times the combined wealth of first 100 individuals on Forbes’s Real-Time Billionaires List.
In the family-friendly version of the story, the king realizes by about the third row that this humble request is going to cost him more than he had thought. He and the consultant have a good laugh, the consultant agrees to accept the $20,000 originally offered (or, alternatively, the hand of the prince or princess in marriage), and the king rejoices that he has such a clever adviser who has taught him something new.
In another version of the story the king, furious that he has been tricked and humiliated in front of his courtiers, orders the consultant to be exiled (or in some variants, executed). Perhaps a penalty of exile is a bit extreme, but the consultant did violate the ethical codes of the American Mathematical Society and the American Statistical Association, which emphasize the importance of honesty and integrity in their respective professions. The consultant should have set a fair price for services rendered, and not attempted to take unfair advantage of knowledge the consultant possessed but the king did not.*
Linear growth occurs if a quantity grows at a constant rate — for example, if each square had two pennies more than the previous square. The key feature of exponential growth is that a quantity grows at a rate that is proportional to its current size. In mathematical terms, let xt denote the value of a quantity at time t. The quantity has exponential growth over time if xt = x0 bkt, where x0 is the starting value at time 0, b is a fixed value greater than 1, and k is a fixed positive constant. It’s called exponential growth because the time t is in the exponent of the expression, unlike linear growth where xt = x0 + at. and the quantity increases by a constant amount a each year.
The chessboard problem exhibits exponential growth where the number of pennies in square t is 2t-1 — each square has twice as many pennies as the previous square. The total number of pennies on the board after square t is 2t - 1. For compound interest, where a 5% interest rate is compounded annually, the value of an initial investment x0 in year t is xt = x0 1.05t — the current value of the investment is multiplied by 1.05 each year.
Franklin on Exponential Population Growth
The British Iron Act of 1750 was intended to protect Britain’s iron industry by restricting the manufacture of iron products in the colonies. The American colonies were to export pig iron to Britain duty-free, where it would be made into implements that would be shipped back at a profit. This act was viewed in America as an attempt to curb colonial growth, and Van Doren (1938) argues that it spurred Franklin to write Observations Concerning the Increase of Mankind, Peopling of Countries, &c in 1751.
Franklin mused that population growth in Europe was limited by the available resources: “Europe is generally full settled with Husbandmen, Manufacturers, &c. and therefore cannot now much increase in People.” He viewed America, however, as having almost boundless resources, which would allow people to marry earlier and to produce more children. In paragraph 22, Franklin posited a doubling of the English population (he did not include Native American and Black persons in his reckoning) every 25 years:
There is in short, no Bound to the prolific Nature of Plants or Animals, but what is made by their crowding and interfering with each others Means of Subsistence. Was the Face of the Earth vacant of other Plants, it might be gradually sowed and overspread with one Kind only; as, for Instance, with Fennel; and were it empty of other Inhabitants, it might in a few Ages be replenish’d from one Nation only; as, for Instance, with Englishmen. Thus there are suppos’d to be now upwards of One Million English Souls in North-America, (tho’ ’tis thought scarce 80,000 have been brought over Sea) and yet perhaps there is not one the fewer in Britain, but rather many more, on Account of the Employment the Colonies afford to Manufacturers at Home. This Million doubling, suppose but once in 25 Years, will in another Century be more than the People of England, and the greatest Number of Englishmen will be on this Side the Water.
Franklin realized that without external constraints, population would grow at an exponential rate. He was not quite sure about the value of the constant k, however, and conjectured in paragraph 4 that the population might double every 20 years:
Hence Marriages in America are more general, and more generally early, than in Europe. And if it is reckoned there, that there is but one Marriage per Annum among 100 Persons, perhaps we may here reckon two; and if in Europe they have but 4 Births to a Marriage (many of their Marriages being late) we may here reckon 8, of which if one half grow up, and our Marriages are made, reckoning one with another at 20 Years of Age, our People must at least be doubled every 20 Years.
In paragraph 4, Franklin posits an exponential growth model where the population t years after 1750, xt, can be calculated as xt = x0 2t/20. The initial population doubles after 20 years, doubles again in another 20, and so on. He assumes that everyone marries young and each couple has four children that survive to adulthood. If the parents are in their teens or early twenties when the children are born, the population of each family will be doubled approximately every 20 years.
Franklin’s data sources. Of course, Franklin did not have U.S. census data on which to base his calculations (statistics from the first census were published in 1793), but he had been interested in population statistics for a long time. At age 16, Franklin had cited William Petty’s writings about London mortality statistics in his “Silence Dogood” column. Franklin also studied the writings of Edmond Halley, who, when not discovering comets, made critical observations about the population statistics of Petty and John Graunt.
Franklin studied and published data about population growth in the colonies. In 1749, his Pennsylvania Gazette published data on the number of houses in Philadelphia. That same year he printed data from New Jersey censuses in Poor Richard (reproduced in Lemay, 2007, p. 244). At the bottom of the table he wrote: “Total of Souls in 1737, 47369; Ditto in 1745, 61403; Increase 14034. Query, At this Rate of Increase, in what Number of Years will that Province double its Inhabitants?”
Franklin unfortunately did not give the reader enough information to be able to answer his query. You need at least three time points of data to be able to estimate growth over time, and Franklin gave only two. The curve linking two points could have any shape: straight line, exponential growth with b=2, exponential growth with b=3, or any other value, and with only two years, you cannot distinguish among the possible rates of growth.
If you assume linear growth, the population is estimated to increase by 1754.25 persons each year, and it would take 35 years to double the population of 61403. That is clearly not the growth rate Franklin had in mind. He had access to demographic data from other cities that would have supported an exponential growth model with b = 2. With b = 2, k is estimated to be log2(61403/47369)/8 = 0.047, where log2 is the base 2 logarithm.** This corresponds to the population doubling about every 1/0.047 » 21 years, and this calculation on New Jersey and other population data is likely behind Franklin’s conjectures about growth in the “Observations” pamphlet.
Was Franklin right about exponential growth of the US population? Let’s look at the data. Figure 1(a) shows the estimated U.S. population from 1650 to 2020. It certainly exhibits the sharp J-shaped curve associated with exponential growth, but is the population doubling every 20 to 25 years?
Figure 1. Estimated US population from 1650 to 2020 in (a) original scale (b) log base 2 scale.
If the population is doubling at regular intervals, we would expect a plot of the logarithm (base 2) of population vs. year to exhibit a straight line. And, indeed, we see a straight line in Figure 1(b) for the period from 1700 to about 1900. We can estimate the time-to-double by fitting a linear regression predicting log2(population) from year for 1700 to 1900, obtaining the red line in Figure 1(b). The fitted equation log2(population) = – 53.55 + 0.042t corresponds to the exponential growth model
With this model, the population between 1700 and 1900 doubles every 1/0.042 = 23.7 years. Franklin’s conjectures about the growth rate were right on target!
Legacy of Franklin’s work. Half a century later, Thomas Malthus cited Franklin’s observations in his 1798 An Essay on the Principle of Population. Von Valtier (2011) writes that through Malthus, Franklin influenced Charles Darwin’s development of the theory of natural selection. The statistical genetics research of Ronald A. Fisher, who developed analysis of variance and maximum likelihood methods in statistics, was also influenced by Malthus’s work and thus, indirectly, by Franklin.
Benjamin Franklin and Compound Interest
Franklin’s legacy also includes a concrete demonstration of exponential growth. In a codicil of his will, he left 1,000 pounds each to the cities of Philadelphia and Boston. But he specified that the money was not to be spent right away — it was to be invested at 5 percent interest and allowed to grow. Franklin wrote: “If this plan is executed and succeeds as projected without interruption for one hundred Years, the Sum will then be one hundred and thirty-one thousand Pounds” (Vierling, 1923). Franklin’s calculation multiplied the original 1,000 pounds by 1.05 raised to the 100th power (= 131.5).
Franklin directed that the loans should be made to “young married artificers under age of twenty-five years,” recognizing that young men who had recently learned a craft would need capital to grow their businesses. After 100 years, each city could spend 100,000 pounds on public works, reinvesting the remainder in small loans to young married artisans for another 100 years.
When the trusts were dissolved in 1990, the Philadelphia funds had grown to $2 million and the Boston funds to $4.4 million (Butterfield, 1990). If, however, the cities had followed Franklin’s instructions and grown the money at 5% per year, the value of each trust in 1990 should have been the equivalent of (31,500) x (131.5) = 4.1 million pounds sterling (approximately $18.6 million, using the approximation from https://measuringworth.com that one pound in 1791 was the equivalent of $4.55).
Yenawine (1995) compares Franklin’s calculation with the actual value of each trust for each year from 1791 to 1991 and describes why the funds missed their targets. Both cities started out as Franklin had specified, but over time the demand for loans from artisans was less than Franklin anticipated. Controversies and lawsuits arose over the appropriate uses of the funds, and their annual growth rate was less than 5%.
Franklin had anticipated in his will, however, that assumptions are not always met: “Considering the accidents to which all human Affairs and Projects are subject in such a length of Time, I have perhaps too much flattered myself with a vain Fancy, that these dispositions, if carried into execution, will be continued without interruption, and have the Effects proposed.”
Copyright © 2025 Sharon L. Lohr
Footnotes and References
*Although he respected mathematics, Franklin had a less favorable opinion of some of the mathematicians he encountered. Chapter 7 of Franklin’s Autobiography describes a member of his club for mutual improvement, the Junto: “Thomas Godfrey, a self-taught mathematician, great in his way, and afterward inventor of what is now called Hadley's Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was forever denying or distinguishing upon trifles, to the disturbance of all conversation. He soon left us.”
**Von Valtier (2011, p. 177) writes: “While there is no documentary evidence to prove Franklin competent with logarithms, it is highly likely that he was. Logarithmic tables had been introduced more than a century earlier, and Franklin's personal library contains, still extant, a dissertation on logarithms.”
Butterfield, F. (1990). From Ben Franklin, a gift that’s worth two fights. The New York Times, April 21, pp. 1, 9.
Feldman, R.W. (1959). Benjamin Franklin and mathematics. The Mathematics Teacher, 52, 125-127.
Fox, J. (2024). After failing math twice, a young Benjamin Franklin turned to this popular 17th century textbook. Smithsonian magazine, online, December 19.
Franklin, B. (1916). Autobiography of Benjamin Franklin. With illustrations by E. B. Smith and edited by F. W. Pine. New York: Henry Holt and Company. First published as Mémoires de la vie privée de Benjamin Franklin, translated by Jacques Gibelin. Paris: F. Buisson Libraire, 1791.
Franklin, B. (1755). Observations Concerning the Increase of Mankind, Peopling of Countries, &c. Boston: S. Kneeland.
Isaacson, W. (2003). Benjamin Franklin: An American Life. New York: Simon & Schuster.
Lemay, J.A.L. (2009). The Life of Benjamin Franklin, Volume 3: Soldier, Scientist, and Politician 1748-1757. Philadelphia: University of Pennsylvania Press.
Munson, R. (2025). Ingenious : A Biography of Benjamin Franklin, Scientist. New York: Norton.
Pasles, P.C. (2008). Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey. Princeton, NJ: Princeton University Press.
Van Doren, C. (1938). Benjamin Franklin. New York: Viking Press.
Vierling, F. (1923). The Franklin trust. Journal of Accountancy, 36, 15-26.
Von Valtier, W.F. (2011). ‘An extravagant assumption’: The demographic numbers behind Benjamin Franklin’s twenty-five year doubling period. Proceedings of the American Philosophical Society, 155, 158-188.
Yenawine, B.H. (1995). Benjamin Franklin’s Legacy of Virtue: The Franklin Trusts of Boston and Philadelphia. Ph.D. dissertation, Syracuse University.