4 NUMBERS IN THE AGE OF REASON
The latter part of the eighteenth century is known as the age of the American and French revolutions. This was also the time of a growing concern with numbers, especially national numbers. In Britain the century’s end witnessed the completion of Sir John Sinclair’s 20-volume census of Scotland, a monument of statistical information and analysis. In the United States, the Federal Constitution required that a census be taken every 10 years.1 This was the first time in history that a nation established a regular census. And during the decade from 1789 to 1800, the French government became increasingly aware that numerical information was crucial for statecraft. To carry on the wars that engaged the new French republic, a count of men by age categories was needed to ensure a supply of military personnel. Also, the cost of warfare demanded exact and reliable numerical data on actual and prospective tax revenues to support the war effort.
Aside from these practical applications, to which we will return in our discussion of Thomas Jefferson and Benjamin Franklin, the eighteenth century witnessed important studies on plant and animal physiology and writings on human happiness. These express an ideal of quantification, even though they do not end up in tables of numerical data.
One important thinker of the early eighteenth century was a Scottish philosopher, Francis Hutcheson (1694–1747). Although Hutcheson’s works were widely read and frequently reprinted in the early eighteenth century, modern readers are more familiar with his contemporaries of the Scottish Enlightenment, such as the philosopher David Hume and the economist Adam Smith.
Hutcheson’s first book, An Inquiry into the Original of our Ideas of Beauty and Virtue (1725), has been described by Garry Wills as “a sturdy piece of prose, as well as a strict logical exercise—it set the standard for expository writing that David Hume and Adam Smith, Thomas Reid and Adam Ferguson would maintain.”2 On the basis of this work, Hutcheson won a professorship of moral philosophy at the University of Glasgow, where he had an illustrious and influential career.
Here we are not particularly concerned with Hutcheson’s place in the history of philosophy, but rather with his attempt to analyze the moral sense by setting up algebraic expressions. For example, in his discussion “to find a universal Canon to compute the Morality of any Actions, with all their Circumstances, when we judge of the Actions done by our selves … [according to] Propositions, or Axioms,”3 he proposes the following algebraic expression:
In this equation, B is “Benevolence, or Virtue in any Agent,” M is the agent’s “Moment of publick Good,” meaning “the Quantity of publick Good produc’d by him,” A is “his natural Ability,” and I stands for “private Interest.”
Hutcheson used this algebraic relationship to translate several commonsense notions about morality into mathematical language. The first is that if two people have the same natural ability to do good (A), the one who produces more public good (M) is more benevolent (B). Conversely, if two people produce the same amount of public good, the one with more ability is less benevolent (since it was in that person’s ability to do more). The plus/minus sign in the equation allowed Hutcheson to factor in self-interest.
Suppose, for example, that an act is good for the public and also directly benefits the person performing that act. Compare this with an act that is good for the public yet directly bothersome, or even harmful, to the person involved. According to Hutcheson’s formula, the first person’s benevolence is less than the second person’s benevolence. In the first case, the person’s self-interest offsets the public good (M – I); in the second case, it boosts it (M + I). Hutcheson did not assign actual numbers to these entities, but assumed that they could be quantified or reduced to numbers.
Another significant reason to include Hutcheson in a history of numbers is that he concluded from his algebra that “in equal Numbers, the Virtue is as the Quantity of the Happiness, or natural Good.”4 That is, he taught that “Virtue is in a compound Ratio of the Quantity of Good, and Number of Enjoyers.” This led him to the important conclusion that “that Action is best, which accomplishes the greatest Happiness for the greatest Numbers.” Here is a precursor, by more than 50 years, to Jeremy Bentham’s (1748–1832) utilitarian philosophy of “the greatest happiness for the greatest number.”
In this new Hutchesonian human science, happiness is an important aspect of human behavior and existence. Recall the phrase in Jefferson’s Declaration of Independence concerning the right to “life, liberty and the pursuit of happiness.” The Hutcheson formula with its suggestion of calculability gave expression to beginnings of a numerical human science.
For the thinkers of the Enlightenment, this concept of the calculability of human happiness had specific implications, some of which are not obvious to us today. Happiness was a measurable quantity in two dimensions: both the amount of happiness within an individual and the sum of happiness within a group could be quantified. It was a concept central to the development of a new human science rooted in numbers.
HALES’S NUMERICAL PLANT AND ANIMAL SCIENCE
In the early eighteenth century, approximately a hundred years after William Harvey’s pioneering work on the circulation of blood, Stephen Hales (1677–1761) offered a spectacular example of the application of numbers to the life sciences. A clergyman and amateur scientist, Hales is often called the founder of plant physiology. Hales believed that the phrase from the Book of Revelation that God had made the world by weight and measure implied that the way to understand the world of nature must be to weigh and to measure; that is, to base the science of nature on numerical data.
One quantity that Hales measured was the pressure of the sap in vines. In the early 1700s many people believed there was a circulation or an ebb and flow of sap in plants, a kind of analogue of the circulation of the blood in animals. Of course, there is no such circulation, because plants have no organ corresponding to the animal heart. However, the sap does rise.
Indeed, Hales made a major discovery about the movement of sap, as the result of a happy accident. An ardent gardener, he relates that he had pruned some vine stems in the springtime and was alarmed that the cut vines were bleeding or oozing sap. To remedy this situation, he fastened little caps or covers made of pig’s bladder around the cut tip of the vines. To his great surprise he observed that these caps or covers were expanding. Some kind of pressure was driving the sap upward in the springtime.5
FIGURE 4.1 Stephen Hales’s measurement of sap pressure (note that the vines are connected with three open manometers), from Hales’s Vegetable Staticks. Courtesy of the Botanical Library, Harvard University
Hales tells us that he at once changed conditions so as to convert his qualitative observations into numerical data. To this end he replaced the bladder caps by manometers (pressure gauges) (see figure 4.1), and thus became the first person in history to measure the phenomenon we know as root pressure, the pressure that drives the sap upward and outward to nourish the plant. All of us know of this flow of sap in the spring, since the sap of maple trees is collected from sugar-maple trees and then boiled down to produce maple syrup. Hales made other contributions to plant physiology in his efforts to create quantitative science. His concern with numbers led him to be the first person to measure blood pressure in animals. Unfortunately, his method of determining blood pressure involved exsanguinating the animal. This procedure obviously could not be used for medical diagnosis in human beings. The poet Alexander Pope, Hales’s neighbor, simply could not understand how so nice a man as Hales could perform experiments in which animals, as subjects for measurement, were allowed to bleed to death.
THOMAS JEFFERSON: A LIFE REGULATED BY NUMBERS
In the eighteenth century, the growing interest in using numbers to regulate and quantify ordinary experience began to find expression at the highest levels of political and social thought. Thomas Jefferson’s intellectual world and his daily life were regulated by numbers to a degree that seems astonishing to a reader in the twenty-first century. Skilled in mathematics, he was, for instance, a master of the Newtonian calculus of fluxions and delighted in numbers and in calculation. Almost every aspect of his life was reduced to numerical observations and calculations. It is one great merit of Garry Wills’s book on Jefferson and the Declaration of Independence to have described in detail Jefferson’s commitment to numerical rules and quantitative observations. Wills traces Jefferson’s concern with numbers to their possible sources, among them the writings of Sir William Petty, the prophet of a numerically based polity.
Jefferson kept detailed numerical records of his farming and gardening activities, of meteorological conditions, and of any other facet of daily life that could possibly be quantified. He exhibited his love of precision in a letter to Abigail Adams of 22 August 1813. He wrote that he had “ten and one-half grandchildren, and two and three-fourths great-grandchildren,” adding “these fractions will ere long become units.”6
He wanted to have almost every aspect of life reduced to numbers, and even his political thought was pervaded by this approach.
As a man of science of the post-Newtonian era, Jefferson knew that numerical evidence modifies abstract theory—a feature of exact science from the seventeenth century to the present. Jefferson was concerned for numbers not simply for their own sake, but as numerical evidence in the post-Newtonian manner. This idea appears again and again in Jefferson’s writings. For example, he wrote how the “degrees”—that is, the actual numbers—“fix the laws of the animal and vegetable races, which may exist with us.”7
At Monticello, Jefferson meticulously made and recorded daily observations (“with a good degree of exactness”) of the meteorological conditions, the measurements of temperature, winds, rainfall, and barometric pressure. His data for a seven-year period from 1 January 1810 to 31 December 1816, he wrote, were so exact and continuous that they enabled him “to deduce the general results.” There were in all “three thousand nine hundred and five” numerical observations.
“During the same seven years,” he wrote, “there fell six hundred and twenty two rains, which gives eighty nine rains every year, or one for every four days.” Furthermore, “the average of the water falling in the year being 47-1/2 inches, gives fifty three cents [i.e., hundredths] of an inch for each rain, or ninety three cents for a week.”
One of the questions that interested Jefferson was whether a wheelbarrow with two wheels was more efficient than the customary one with a single wheel. Performing actual tests was the only way to find out. Jefferson’s account of his measurements shows how he understood the importance of time and motion studies, two centuries before Frederick Taylor (1856–1915) did his pioneering research in the same field.8
Jefferson’s extreme fascination with numbers also extended to politics, and could lead him to proposals that seem absurd to a reader in the twenty-first century. It is well known that Jefferson believed in periodic rebellions as a force for political renewal. As he wrote to James Madison in 1787, “I hold it, that a little rebellion, now and then, is a good thing, and as necessary in the political world as storms in the physical.”9 It is not as well known that he did not think we have a right to impose our ideas and regulations, and even our laws and contracts, on future generations. This is the sense of his many expressions of the need for future generations to free themselves from the chains of the past. The future, he held, should not be bound by what binds us in the present.
Such statements imply a need for future revolutions. Jefferson extended this proposed limitation to all pacts, constitutions, and agreements, believing that “the earth belongs in usufruct to the living.”10 (The American Heritage Dictionary defines “usufruct” as “the right to enjoy the profits and advantages of something belonging to another as long as the property is not damaged or altered in any way.”) Being Jefferson, he quantified these notions and concluded that “every constitution then, and every law, naturally expires at the end of 19 years.” In practice, the actual time would be only approximately 19 years. Jefferson rounded out this number to 20. On this subject Jefferson drew on the work of the French naturalist, Comte de Buffon (1707–1788).
In the spirit of charity, let us not explore the chaotic consequences that would follow had Jefferson’s 19-year rule been put into practice. But this numerical fantasy is of interest in showing how Jefferson’s thinking tended to be conditioned by his deep concern with numbers. Here is the seed of Jefferson’s dictum that a society needs a revolution every 20 years.
Consider Jefferson’s judgment on Shays’ Rebellion, an uprising of economically depressed farmers in western Massachusetts that took place in 1786–87. “The late rebellion in Massachusetts,” he wrote, “has given more alarm than I think it should have done.”11 He based this conclusion on simple numerical calculations. “Calculate that one rebellion in 13 states in the course of 11 years” is not very great, he wrote. This numerical ratio is the same as “one for each state in a century and a half.” In fact, he wrote, “No country should be so long without one,” adding that there is no “degree of power in the hands of government” that can “prevent insurrections.” Writing from Paris, he took note that in France, despite “all it’s [sic] despotism” and its “two or three hundred thousand men always in arms,” he had been witness to “three insurrections in the three years” he had been there, “in every one of which greater numbers were engaged than in Massachusets [sic] and a great deal more blood was spilt.”
Jefferson’s concern with numbers and his arithmetical skill were of real importance for political action during his service as Secretary of State. One of the numerical problems addressed by Jefferson was an apparently simple one: how to assign to each state a fair number of representatives in Congress. This numerical exercise had to meet the provisions of the Constitution, which requires a review of apportionment every decade on the basis of the decennial census of the population. The Constitution, however, provides no details concerning the way an apportionment is to be made. A few rules were set forth. Here is the relevant text of the Constitution:
Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative; and until such enumeration shall be made, the state of New Hampshire shall be entitled to chuse [sic] three, Massachusetts eight, Rhode Island and Providence Plantations one, Connecticut five, New York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three.12
There are certain conditions imposed by Article I. One restriction is that the representation should not exceed one for every 30,000 in the population. Women and children are to be counted even though they had no voting rights. Furthermore, it is provided that Negro slaves are to be included in the count but only at the rate of three-fifths of a white person. A further condition is that each state must have at least one representative. The same reckoning is to be applied to the obligation of the states to support the expenses of the federal government.
The average reader of these conditions will assume that apportionment is a matter of simple arithmetic. Most Americans assume that the composition of the House of Representatives is determined by a just and fair apportionment of representatives according to the populations of the individual states. In fact, however, very few people understand how that apportionment is determined in actual practice. Determining the number of representatives for each state poses a mathematical as a well as a political problem, one that has been the cause of intense argument, debate, and analysis during most of our country’s history.
The problem of apportionment eventually has enlisted the creative efforts of some of the country’s most able mathematicians. Apparently simple and obvious solutions have proved to involve major political issues that have raised great passions of debate. The problem of dividing responsibility for the expenses of running the federal government does not present the same issues that beset apportionment, because a money obligation can be divided down to the last cent, but a representative cannot be divided.
Suppose that a given state has one-thirteenth of the total national population. Its share of the federal expenses can be easily and fairly reckoned at one-thirteenth of the whole.
In apportionment, however, there is no such easy division. Suppose that there are 120 seats in the House and that a certain state has one-thirteenth of the national population. Simple arithmetic gives this state one-thirteenth of the 120 representatives. Dividing 120 by 13 yields 9 3/13 or 9.231, a little more than nine representatives. Similar calculations for each state will yield an integral number of representatives plus a fractional or decimal part. Obviously, a state cannot have a “fractional” representative.
The simplest solution might seem to be to stick to the integral numbers and discard the fractional parts. This solution, however, would be unfair to a state entitled to, say, 1.987 representatives, reducing that state’s representation from almost two to just one, a loss of nearly half of its just share. A loss of the fractional or decimal part would be less significant for a state entitled to 12.121 representatives, since its loss would be less than about one-tenth of its just share.
There are other possible solutions: one would be to give an extra seat to the state with the highest fraction, then to give the next extra seat to the state with the next highest fraction, continuing until the total number of representatives adds up to 120. An alternative would be to use a similar procedure, but start by giving an extra seat to the most populous state, rather than to the state with the highest fraction.
There are still other possibilities. Jefferson favored trying various divisors other than 120, discarding all fractions. Eventually, by trial and error, a divisor can be found that will yield a total number of 120 representatives.
Objections can be (and have been) raised to each of these methods, perhaps for favoring either the small or the large states or favoring the South versus the North. Clearly, there is no simple and obvious way to satisfy the requirements of the Constitution.
Congress first attempted to devise a system of apportionment in 1792 in legislation entitled “An Act for the apportionment of Representatives among the several States according to the first enumeration.”13 Thomas Jefferson analyzed this proposal for President George Washington and found such serious flaws that Washington vetoed the proposal. This was the first presidential veto in the history of our country, one of only two times that Washington exercised his veto power, showing how seriously Washington regarded the problems raised by Jefferson.
This incident provides a striking example of how blueprints for government may introduce technical problems that are beyond the scope of knowledge and forethought of the founders. It also displays the mathematical acumen of Thomas Jefferson, whose analysis of the proposed method provided both the grounds for Washington’s veto and the basis for the system that was finally adopted.
Among the reasons that Jefferson objected to the first bill proposing an apportionment was that it contravened what he considered a basic principle of good government. The bill said nothing whatsoever about the actual method used in the proposed assignment. Only because Jefferson was skilled in arithmetic and the manipulation of numbers was he able to figure out how the apportioning had been done. (His analysis was confirmed when the method used by Congress was revealed by Alexander Hamilton to Washington in support of the bill.) Hence, as Jefferson quite correctly argued, when the next apportionment was needed, following the census of 1800, Congress would not know how it had been done the first time. Since the method had not been specified, the way was open to change the procedure at will the next time around. As Jefferson wrote, the bill “seems to have avoided establishing [the procedure] … into a rule, lest it might not suit on another occasion.” Perhaps, he observed, “it may be found the next time more convenient to distribute them [the residuary representatives] among the smaller states; at another time among the larger states; at other times according to any other crotchet which ingenuity may invent, and the [political] combinations of the day give strength to carry.” Jefferson wanted any law to contain an explicit method which “reduces the apportionment always to an arithmetical operation, about which no two men can ever possibly differ.” This requirement of having a sound and unambiguous method was a fundamental characteristic of all good Enlightenment science, expressed simply and beautifully by Linnaeus, “Method [is] the soul of science.”
Jefferson’s application of arithmetic to a problem of statecraft is a splendid example of what the eighteenth century knew by William Petty’s term “political arithmetic.” This was a subject of great significance for Jefferson and many of his contemporaries in America and Europe, not the least of whom was Benjamin Franklin.
Benjamin Franklin, like Jefferson, was fascinated by numbers, although he did not suffer from Jefferson’s mania for numerations. He tells us in his autobiography that when he was about 16 years of age, he was “asham’d” of his “Ignorance in Figures,” that is, arithmetic or reckoning, a subject which he “had twice fail’d in learning when at School.” To remedy this situation, he obtained a book on arithmetic, which he went through by himself “with great ease.” Practical applications were important. As a young tradesman in Philadelphia, he studied accounting and in his autobiography, Franklin advocated that young women be taught accounting. He had in mind that a woman might need to take over her husband’s business in case he became ill or were to die.
His great success as a shopkeeper shows that he had mastered the subject. But sometimes Franklin used his skill with numbers simply to ease his boredom. As a young man, serving as clerk to the Pennsylvania Legislature, he found himself wearied by “sitting there to hear Debates in which as Clerk” he “could take no part” and which were often “unentertaining.” So, to pass the time, he amused himself by performing numerical calculations for fun, producing numbers to fill out what are called magic squares and circles.
A magic square is a square array of numbers, in which the sum of the numbers in each column and in each row is the same and is equal to the sum of the numbers in each of the diagonals. A famous magic square appears in Albrecht Dürer’s sixteenth-century engraving of Melencolia I. Here (see figure 4.2) the sum of the numbers in each row (e.g., 16 + 3 + 2 + 13) is 34. The sum of the numbers in each column (e.g., 16 + 5 + 9 + 4) is also 34, as is the sum of the numbers in each of the two diagonals (16 + 10 + 7 + 1 or 13 + 11 + 6 + 4). (See figure 4.3.)
Later on, Franklin wrote of how he had “acquired such a knack at it,” that he could “fill the cells of any magic square, of reasonable size, with a series of numbers as fast as I could write them.”14 Finding the traditional magic squares to be too “common and easy,” he devised more complex forms of magic squares “with a variety of properties.” He proudly described one of these as “the most magically magical of any magic square ever made by any magician.” Indeed, not content with his own form of magic square, Franklin went on to invent a magic circle. So proud was he of his magic squares and circles that Franklin included one of each in the revised and expanded 1769 edition of his book on electricity.15
FIGURE 4.2 Albrecht Dürer, Melencolia I, 1514. Courtesy of the Fogg Art Museum, Harvard University Art Museums, Gift of William Gray from the collection of Francis Calley Gray
FIGURE 4.3 The magic square from Dürer’s Melencolia I.
Of course Franklin did not invent the concept of a magic square, nor do we know exactly how he learned about these arithmetic curiosities. However, some years later he described his encounters with two books that mentioned magic squares. In a letter to his London patron, Peter Collinson, Franklin explained how he had encountered a book on the subject in the library of James Logan, a Philadelphian who owned the most extensive collection of scientific books in Colonial America. Logan had shown him a “remarkable” book on magic squares. It was, Franklin recalled, “a folio French book, filled with magic squares, wrote, if I forget not, by one M. Frenicle.”16 Logan also showed Franklin “an old arithmetical book, in quarto, wrote, I think, by one Stifelius, which contained a square of 16.” Franklin recalled how “that evening,” at home, he produced a magic square of 16 with unusual properties, which he later reproduced in his book on electricity. This is the same Stifel (or Stifelius) whom we have seen proving Pope Leo X to be the anti-Christ because the letters in his name could add up to 666, the “number of the Beast.”
Franklin’s appreciation of the role of numbers in statecraft was manifested early in his career. In 1729 (when he was 25 years old) he wrote a pamphlet on the need for paper currency. Franklin based his ideas on currency on Petty’s “political arithmetic,” and later, stimulated by Petty’s ideas, he developed the concept of a labor theory of value.
The labor theory of value, in Franklin’s words, holds that “Trade in general being nothing else but the exchange of Labour for Labour, the value of all things is … most justly valued by Labour.”17 That is, the value of any commodity depends on the labor needed to produce it. This theory eliminates such factors as capital investment and market scarcity.
The full title of Franklin’s tract is A Modest Enquiry into the Nature and Necessity of a Paper-Currency. In it, Franklin aimed to show that value need not be measured only in terms of the precious metals, gold and silver. These have long been admired, he wrote, for their “Fineness, Beauty, and Scarcity.” But even though, commonly, all things are valued by silver equivalent, silver itself is “of no certain permanent Value, being worth more or less according to its Scarcity or Plenty.” Therefore, “It seems requisite to fix upon something else,” something “more proper to be made a Measure of Values, and this I take to be Labour.”18
To make this concept clear, Franklin gave an example, actually a “close paraphrase” of two passages in Petty’s Treatise of Taxes (1662).19 In this example, it is supposed that two men are at work, one raising corn, the other “digging and refining” silver; at the end of a year (or other suitable interval of time), one man will have grown 20 bushels of corn while the other will have produced 20 ounces of silver. Therefore an ounce of silver is worth the labor of growing a bushel of corn. The actual worth of silver may decline, as happened with the discovery of the mineral riches of the New World, but this does not directly affect the rate of labor expended in producing a bushel of corn.
Franklin’s concern with numbers as the exact expression of exact information is evident in the pages of the Pennsylvania Gazette, the weekly newspaper that Franklin edited and published in his late twenties and early thirties. On this topic I draw on information assembled by the dean of Franklin scholars, J. A. Leo LeMay, for his forthcoming magisterial biography of Franklin. LeMay found that the pages of the Gazette regularly displayed numerical data. Franklin, as a man of his times, read books with quantitative data, the eighteenth-century works that advanced the subject of political arithmetic. Franklin also drew on compilations of data concerning trade and shipping.
Thus he informed his readers about the work of Joshua Gee, who had had a certain success in London for his books containing numerical data on trade and population. On 5 January 1731, Franklin published in the Gazette what LeMay describes as “information concerning all ships that had entered or cleared out of the major colonial ports” except for Southern ports for which there was apparently no information readily available. This shipping news was not a space filler, but a subject of real importance, and on the front page Franklin called his readers’ attention to these data. “In this paper,” he announced, “we exhibit an Account for one Year, of all the Vessels entered and cleared, from and to what places, in the Ports of Philadelphia, Amboy, New-York, Rhode-Island, Boston, Salem and New Hampshire.” These data, he explained, would enable the “ingenious Reader” to “Make some Judgment of the different Share each Colony possesses of the several Branches of Trade.” To provide a complete picture of the situation, Franklin printed an extract from Gee’s Trade and Navigation of Great Britain Considered, dealing with the trade between Britain and the American colonies. A supplement appeared on 12 April 1731 with trade data for Barbados. Franklin’s regular publication of such data concerning trade was a popular feature of the Pennsylvania Gazette.
According to LeMay, Franklin’s interest in the numerical aspects of trade “paralleled his interest in demography.” Thus, on 20 November 1729, he began to publish data on the number of burials in Philadelphia, first week by week, and then totaled for the year. So that his readers might grasp the significance of these numerical data, he published comparative numbers for Boston, Berlin, Amsterdam, and London.
On 6 August 1731, Franklin published data on the inhabitants of Breslau, noting that one twenty-ninth of the population died each year. The data for Boston, he found, showed that “not above a 40th Part of the People of that Place die yearly, as a medium.” In evaluating the use of such numerical information, we should keep in mind the warning of James Cassedy, a historian of early American medical statistics. Cassedy found that mortality data were regularly skewed to an optimistic view of the healthy conditions of life in Colonial American cities.20
In the present context, these signs of Franklin’s early concern with demographic data are worth noting because, as we shall see shortly, Franklin’s continuing interest in demography led to his devising a law of population increase and eventually provided a basis for a policy that he advocated for Britain and the Colonies.
Franklin’s interest in numbers appeared in his contributions to the new science of demography, the numerical study of populations. One major figure in this area was Thomas Malthus (1766–1834), an English pioneer who set forth the basic rule of this new science at the end of the eighteenth century. Malthus’s rule states that if there were no barriers to the growth of human populations, there would be a geometric increase in the size of any population. Another way of saying this is that an unchecked population will double in size every so many years.
This rate of growth, also called exponential, produces an increase at an unbelievably large rate. A story is told of a native of India who had performed an important service for his king. The king offered him any reward he wished. He replied that he was a humble man and would ask only for a grain of rice to be placed on the first square of a chessboard, then two grains on the second square, then four on the third, and so on until the last square. The king was sorry that his loyal subject had asked for such an apparently small gift. But, in fact, this program—if actually carried out—would have required more rice than was available in all of India. This example shows the enormous rate of increase in a geometric expansion.
Malthus also is credited with a second and related law—that the maximum possible increase in the food supply is in a simple arithmetic ratio. That is, in two years you can double the food supply, in three years triple it, and so on. This is a vastly slower rate of increase than the geometric or exponential growth rate mentioned above.
Darwin extended these laws of Malthus from human populations to all animal and vegetable populations. Then, in a flash of insight, Darwin recognized that all of the individuals of such geometrically increasing populations couldn’t survive. And so Darwin was led to the doctrine of “natural selection” and the theory of evolution by “natural selection.” Clearly, the law of population growth has been of enormous importance for science.
In the second edition of his book on population, published in 1803, Malthus declared that the rule of population doubling had been stated a half-century or so earlier by Benjamin Franklin in his writings on demography. It is difficult to be sure who first stated this important law, but many thinkers of the late eighteenth century associated this law with Benjamin Franklin.
Franklin wrote two works on the new science of demography. The first was a pamphlet entitled “Observations Concerning the Increase of Mankind.”21 Written in 1751, this essay was first published in Boston in 1755 and reprinted soon after in London. Later, with various revisions, it appeared in a dozen or so different publications, becoming one of Franklin’s most often reprinted essays.
The “Observations” was written in response to the British Iron Act of 1750, which restricted the manufacture of iron in the American colonies. Franklin’s argument employed political arithmetic, based on the idea that policy issues should be determined by a statistical analysis of numerical data.
Franklin’s important contribution to demography centered on his clearly stated rule that, under the American conditions of relatively unchecked growth, the population would double every 20 or 25 years. The major part of this pamphlet is devoted to numerical demographic data concerning population—births, deaths, and marriages—from all over the world. Franklin’s law of population growth is thus firmly rooted in numbers and it leads to the conclusion that land is needed for this expanding population. Additionally, Franklin’s law leads to an important principle of policy: the recognition that there will be a time when “the greatest Number of Englishmen will be on this Side of the Water.”
Accordingly, Franklin concluded that an expansionist policy in North America was a necessity and that British America was destined to become the most populous and important part of the British empire. After 1751, the “increase of mankind” became the very core of Franklin’s faith in the inescapable growth of American power, either within the framework of the British empire or without and even against it. Clearly Franklin’s theoretical or scientific work on demography led to practical political considerations of national policy.
A second publication in which Franklin expounded his ideas on population is the so-called Canada pamphlet, actually entitled “The Interest of Great Britain Considered, With Regard to her Colonies, And the Acquisitions of Canada and Guadaloupe.”22 This work was composed and published in 1760, while Franklin was colonial agent in London. The occasion was the impending favorable conclusion of the Seven Years’ War, known by Americans as the French and Indian War. As victors, the British would be able to annex either Canada or Guadeloupe.
In his booklet, Franklin included the earlier “Observations,” providing a very strong argument for the acquisition of Canada. The increase in population, of which he had written almost 10 years earlier, would require new regions into which the population could spread, thus peopling a greater part of North America with British colonials.
Franklin argued that with protection from French foes and their Indian allies, and with the availability of cheap land, there would be a natural increase of population. The consequence would be an ever-expanding market for British manufactured goods.
Suppose now, Franklin argued, that Britain chose Guadeloupe and that there was no room for natural expansion in British North America. In this case, the colonists would be “confined within the mountains.” Under these conditions, the natural increase of population would cause the population density to increase until it became as great as that of Britain. The cost of land would rise and wages would fall. The industries—extractive, plus agriculture and hunting—would, under these circumstances, no longer be as profitable as before and the colonists would be forced to turn to manufacturing.
Under these conditions, Americans would become producers rather than consumers. Hence Americans would depend less and less on the mother country. It was clearly to the advantage of Britain to annex Canada. Historians generally agree that Franklin’s numerically based argument was a major influence on Britain’s decision to acquire Canada rather than Guadeloupe.
FRANKLIN ON NUMBERS AND SMALLPOX
A spectacular example of Franklin’s use of numbers is provided by his public advocacy of inoculation. Franklin was particularly sensitive about this controversial practice because, when his son Francis died of smallpox, rumors circulated that the boy had contracted the disease through inoculation. To counter such rumors, Franklin published a notice in the Pennsylvania Gazette, announcing that his son had contracted smallpox in the normal or natural way.23
Smallpox has been essentially eliminated from the world disease picture, but in Franklin’s day and earlier, smallpox epidemics would sweep through Europe and America and there was real fear of this cause of death. At that time inoculation was the only known method of prevention.
Inoculation differs in an important way from the later practice of vaccination. The term vaccination has its root in the word “vaccinia,” meaning cowpox, a rather mild disease related to smallpox. In vaccination, the patient is exposed to the viral material of cowpox lesions, causing him or her to become immune to cowpox; this immunity also confers on the patient an immunity to the related and more deadly smallpox. Edward Jenner (1749–1823), a British physician, discovered this method of preventing smallpox in the late eighteenth century.
In inoculation, on the other hand, the patient is exposed to the viral material of smallpox itself, taken from the lesions of a smallpox patient. This method usually gives the patient a relatively mild case of smallpox, which does not kill the patient but leaves him or her immune to the disease.
Inoculation had its own dangers, and some patients who had been inoculated died of the infection. A well-known case was that of Jonathan Edwards, the celebrated colonial intellectual figure and preacher in Northampton, Massachusetts. Appointed president of the college now known as Princeton, where there was a raging epidemic of smallpox, Edwards had himself and his family inoculated before moving to New Jersey, only to die from the inoculation before taking office. But inoculation was not generally fatal. The Dutch physician Jan Ingenhousz, a friend of Franklin’s, was employed for many decades as physician to the royal family of Austria. His chief job was to perform inoculations, and his record was perfect: not a single death from inoculation!
In America feelings ran high on whether to inoculate one’s family. One way for a person to decide whether to have his children inoculated was to appeal to numbers. But what were the actual numbers? What was the probability of getting smallpox in an epidemic, and how did these numbers compare with his or her chances of death? And how did these numbers compare with the probability of death resulting from inoculation? These topics were central to Franklin’s short essay on smallpox and inoculation, published in his newspaper in 1736.
Franklin began by referring to a “current Report, that my Son Francis, who died lately of the Small Pox, had it by Inoculation.”24 Franklin was concerned that some people might be deterred “from having that Operation perform’d on their children” on the basis of “that Report (join’d with others of the like kind, and perhaps equally groundless).” Accordingly, Franklin did “hereby sincerely declare,” that his son had not been inoculated, but had “receiv’d the Distemper in the common Way of Infection.” Franklin added that he supposed the report of his son’s death must have arisen from “its being my known Opinion, that Inoculation was a safe and beneficial Practice; and from my having said among my Acquaintance, that I intended to have my Child inoculated, as soon as he should have recovered sufficient Strength from a Flux with which he had been long afflicted.”
In Part Three of his autobiography he reiterated these sentiments:
In 1736 I lost one of my Sons a fine Boy of 4 Years old, by the Small Pox taken in the common way. I long regretted bitterly & still regret that I had not given it to him by Inoculation; This I mention for the Sake of Parents, who omit that Operation on the Supposition that they should never forgive themselves if a Child died under it; my Example showing that the Regret may be the same either way, and that therefore the safer should be chosen.
Several decades later, when Franklin was in London as agent for several American colonies, he was active in promoting inoculation. He joined forces in this endeavor with William Heberden, a prominent London physician, who—like Franklin—was a fellow of the Royal Society. The two of them produced a pamphlet, for distribution in the American colonies, which set forth the way to perform inoculations. Franklin wrote an accompanying essay on the history of the practice and gave numerical evidence to show that this practice was relatively safe. Franklin was very proud of this essay and had it reprinted in 1759 as a separate work. The pamphlet was entitled “Some Account of the Success of Inoculation for the Smallpox in England and America.”25 Franklin’s essay begins with a history of the practice of inoculation in New England. At the outset, Franklin showed his deep understanding of the use of numbers in any controversy. He explains that the data on this topic tended to be unreliable. The “practice of Inoculation always divided people into parties,” he wrote, because some people would be “contending warmly for it” with “others as strongly against it.” Those opposed to inoculation would assert that “the advantages pretended were imaginary,” that the “Surgeons, from views of interest, conceal’d or diminish’d the true number of deaths occasion’d by Inoculation, and magnify’d the number of those who died of the Small-pox in the common way.” Accordingly, the job of reporting on the number of deaths by inoculation and ordinary smallpox was turned over to town constables who had to submit their numbers under oath.
Our interest in Franklin’s pamphlet centers on the use of numbers in a policy debate and on Franklin’s recognition that in evaluating any medical practice, the test must be based on numbers. Those of us old enough to remember the introduction of the Salk vaccine for polio will recall that the preliminary tests were numerical, based on a massive nationwide statistical analysis.
Franklin’s goal was simple and straightforward. He wanted to give anxious parents evidence that it was safe to have their children inoculated. The data he assembled were most impressive. For example, Franklin presented data he obtained from “Dr. Archer, physician to the Small-pox hospital here.” During the period from its founding to 31 December 1758 there had been given a total of 1,601 inoculations. From these, only six recipients had died. During these same years the number of “Patients who had the Small-pox in the common way” were 3,856, of whom 1,002 had died. In other words, the risk of death from the mild case of smallpox produced by inoculation was minuscule (about 3 out of 800), whereas the chance of death from smallpox taken “in the common way” was rather high (about one out of four).
Data from the Foundling Hospital were even more impressive, Franklin noted. In this hospital, the practice was that “all the children admitted, that have not had the Small-pox, are inoculated at the age of five years.” There had been 338 children inoculated since this practice had been put into effect. Of this number, Franklin reported, only two had died. One of this pair had, in fact, not died from the inoculation, but had been a victim of a “worm fever.”
“On the whole,” Franklin concluded, “if the chance were only as two to one in favour of the practice among children, would it not be sufficient to induce a tender parent to lay hold of the advantage?” But, “when it is so much greater, as it appears to be by these accounts (in some even as thirty to one),” then no parent would any longer “refuse to accept and thankfully use a discovery God in his mercy has been pleased to bless mankind with.”
Franklin was not always in dead earnest when writing about numbers. In 1755, he penned a letter about marriage to Catherine Ray, a young woman to whom he was sending some “fatherly Advice.”26 This letter shows his delight in making puns, comparing the ingredients of a happy marriage to the four basic operations of arithmetic.
“You must practise Addition to your Husband’s Estate,” Franklin wrote, “by Industry and Frugality” and “Subtraction of all unnecessary Expences.” As to “Multiplication,” he wrote, “I would gladly have taught you that myself,” but “you thought it was time enough, and wou’dn’t learn”; now it would be the husband who “will soon make you a Mistress of it.” As to “Division,” however, “I say with Brother Paul, Let there be no Divisions among ye.”
Franklin and Jefferson brought their fascination with numbers into the creation of the new republic, which emerged as a political entity just as their importance as instruments of policy was becoming clear even to people who were not geniuses.