The seventeenth century witnessed the birth of modern science as we know it today. This science was something new, based on a direct confrontation of nature by experiment and observation. But there was another feature of the new science—a dependence on numbers, on real numbers of actual experience.
The pioneering practitioners of the new science knew that they were producing a new kind of knowledge and so they declared this newness in the titles of their books and articles. Thus we have Galileo’s Two New Sciences, Boyle’s New Experiments, Kepler’s New Astronomy, and Tartaglia’s New Science. When, in 1620, Ben Jonson presented a masque entitled “News from the New World,” his new world was not the newly found continent of North America, but the new world of science, the world revealed by the telescope of Galileo.
A numerical law is an exact statement that leads to prediction and test. For example, one of the laws discovered in the seventeenth century, in the age of the Scientific Revolution, was Hooke’s law, “Ut tensio, sic vis,” a statement concerning springs. It tells us that the tension in a spring is proportional to the stretching force. Let us suppose that we hang a spring from a hook and that the spring has a pan on which we can place various weights. Then Hooke’s law tells us that if the spring is stretched out by two inches by a weight of three pounds, then a weight of six pounds will stretch out the spring by four inches. At once the law can be tested. Such a test, implying the possibility of falsifying a theory, is a sign of modern science. According to Karl Popper, perhaps the foremost philosopher of science of the twentieth century, the property that a theory can be falsified by experiment is the very hallmark of science.
The ancients knew a few such numerical laws: the law of reflection of light, the law of the lever, and the law of buoyancy. But prior to the Scientific Revolution, the goal of science (or the study of nature) was not to seek laws of nature expressed in terms of numbers or number-relations. Those who created the new science of the seventeenth century not only found laws based on numbers but they were also willing to express these laws in terms of higher powers of numbers—squares and cubes.
Typical of the new science is the discovery by the astronomer Johannes Kepler (1571–1630) of what we know today as Kepler’s harmonic law. Kepler had been searching for a law that would explain why God, in creating the world according to the Copernican system, placed the planets where they are and caused them to move with the speeds that we can observe. Kepler sought in vain for many years to find some numerical law that would express the relationship between the celestial dimensions or distances and the speeds with which the planets move. Finally he found the answer. In his 1619 book The Harmony of the World he tells us that he discovered the harmonic law while delivering a lecture on astronomy to his students. Kepler found that for each planet, the cube of the average distance from the sun is proportional to the square of the period of revolution.
Kepler later found a similar law for the satellites of Jupiter. Today we know that such a harmonic law holds for any system of bodies that circulates around a central parent body. There are many applications of Kepler’s law; for instance, half a century later it gave Isaac Newton the clue to his discovery of the law of universal gravity.
GALILEO AND THE LAWS OF MOTION
According to Aristotle (384–322 B.C.E.), “To be ignorant of motion is to be ignorant of nature.” And indeed, studies of motion have been a fundamental part of thought throughout antiquity and during the Middle Ages. This motion, however, was not the kind of motion with which Galileo (1564–1642) was concerned and which we think of whenever we encounter the term “motion” today. The pre-Galilean thinkers were rather concerned with motion in the sense used by Aristotle. For them “motion” was any process in which there was a transition from any state or condition to another state. Thus the process of aging, the change in a person’s degree of wisdom, or the growth in weight of a boy could all be considered examples of motion. By contrast Galileo was concerned with physical motion, motion involving a change in place—the set of phenomena which we today commonly associate with the term. One of the major kinds of motion that Galileo studied was the motion of free fall, the falling motion of bodies that we can observe around us, right here on Earth.
In his founding treatise, the Dialogues Concerning Two New Sciences (1914; originally published 1638), Galileo boasted that he was setting forth “a very new science dealing with a very ancient subject.” He was aware that some superficial observations had been made, such as that a body in free fall is continuously accelerated, but nobody before him had discovered the laws of such acceleration, as he had done. No one before him, he declared, had discovered that “the distances traversed, during [successive] equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity [one].”1
A numerical example will help the reader to understand Galileo’s statement. Suppose that an object is let fall and that during the first second of time, it will fall through 16 feet. Then, according to Galileo, in the next second of time, it will fall through 3 × 16 feet; during the third second it will fall through 5 × 16 feet, and so on.
Galileo’s rule can be expressed differently, that the total distance fallen is proportional to the square of the total elapsed time. Thus, if a body falls freely through 16 feet in the first second, it will fall through 22 × 16 feet or 4 × 16 feet during the first two seconds, through 9 × 16 feet during the first three seconds, and so on.
The question then arises whether Galileo’s laws hold in nature or are merely the result of an abstract mathematical exercise. Galileo could not make an experimental test with freely falling bodies, in part because they move too swiftly. So he devised an experiment in which he “diluted” gravity, slowing down the motion of falling. For this purpose he used an inclined plane, a slanting flat board with a grooved track running down its length. He allowed a small metal ball to roll down the board at different inclinations and recorded the distance and times. In every case he found that the distances, starting from rest, were proportional to the square of the time.
In his own words:
We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former.2
Galileo presented the numerical values that he found in his experiments as proof that the laws of motion he had found were not abstract laws but actually occur in the physical world of nature. Thus he could proudly boast of an agreement to within “one-tenth of a pulse-beat.”
In the next generation, that of Newton’s Principia (1687), numerical data from experiment and observations were used in another way, to set requirements that revised or extended fundamental theory. An example is Newton’s development of perturbation theory in his attempts to deal with new and more precise observations. In the Principia, Newton showed how fundamental theory must be modified so as to account for the numbers given by experiment and observation. Two notable examples are the numbers representing resistance to motion and the motion of the Moon.
NUMBERS IN THE LIFE SCIENCES: DOES THE BLOOD CIRCULATE?
One of the tremendous advances in science made during the Scientific Revolution of the seventeenth century was William Harvey’s discovery of the circulation of the blood. Harvey (1578–1657) was a British physician who had been trained in Italy. In fact he was studying medicine at the University of Padua in the years when Galileo was using the newly invented telescope to revolutionize the science of astronomy.
In the early part of the seventeenth century, the age of Galileo and Kepler, the reigning concepts of human and animal physiology were still the ones set forth by Galen (c. 130–c. 200) in the second century. Galen believed that blood was manufactured continuously by the liver and then spread through the whole body with an ebb and flow motion like that of the tides in the ocean.
Harvey’s radical concept was that the heart acts like a pump. He showed by careful anatomical and physiological “exercises,” and by comparative data from various kinds of animals, that the heart pumps blood out into the aorta or primary artery; with each successive heartbeat the blood is pushed further and further out through the system of arteries; eventually this blood returns to the heart through the veins. In making this analysis, Harvey used the discovery of valves in the veins. These valves, as Harvey showed, permit the blood in the veins to move toward the heart but not away from the heart. In other words, the blood could not possibly ebb and flow in the system of veins.
In Harvey’s concept, basically the one in which we still believe today, the blood that returns to the heart through veins is not then directly pumped into the main system of arteries. Instead, this returning blood is pumped into the lungs, where it is aerated and then returns to the heart, where it is pumped once again into the aorta and the main system of arteries. One of Harvey’s telling refutations of the Galenic system, and the reason why Harvey appears in a book about numbers, is that he used an argument based on numerical evidence. Thus Harvey’s discovery shows us that the life sciences as well as the physical sciences made some use of numerical arguments.
Harvey’s quantitative argument against the Galenic system was based on two numerical quantities. The first was the capacity of the heart in human beings, in dogs, and in sheep. Then, multiplying this figure by the pulse rate, he computed how much blood is transferred from the heart—approximately 80 pounds of blood in each half hour for an average man. From these measures, Harvey wrote, it is manifest “that the beating of the heart is continuously driving through that organ more blood than the ingested food can supply, or than all the veins together at any given time can contain.”3
The critical reader will find two flaws in Harvey’s argument. The first is that he supposed the average human pulse rate to be 33 whereas the average or normal pulse is closer to 70 beats per minute. As a consequence, his calculations were off by a factor of 2. The second is that it is very difficult to measure the capacity of the heart. Harvey obtained a human heart from an anatomized cadaver and simply poured in water, measuring the amount of water that the heart could contain. But under these conditions the walls of the heart are much more flaccid than in the sturdy conditions of life. In fact, scientists did not make accurate measures of the heart’s capacity until well into the nineteenth century. Nonetheless, Harvey’s numerical argument proved the falsity of Galen’s basic postulates: There is simply no way in which the liver can continuously manufacture the quantity of blood being pumped out by the heart.
At more or less the same time that Harvey was studying the circulation, other scientists were applying numerical considerations to the life sciences. For example, Johannes van Helmont (1579–1644) of Belgium made quantitative studies of the nourishment of growing plants. In addition, Italian scientist Santorio Santorio, also known as Sanctorius (1561–1636), performed studies to determine the correlation of food intake and body weight. Sanctorius had a platform constructed which was supported by slings in such a way that it could be weighed while he sat on it (see figure 2.1). He then made careful measurements of his weight at regular intervals during a whole day, and thus was able to correlate the changes in his weight with the weight of solid food and liquids he ingested. He also weighed his solid and liquid excreta, and by numerical methods established the weight loss due to perspiration.
In assessing the novelty of a science based on numerical laws such as those of Galileo, Kepler, Harvey, and Sanctorius, we must keep in mind that although it was new to express laws of nature in terms of numbers, by the time of the Scientific Revolution, numbers had been used for centuries in various aspects of life. Surveyors and tax collectors had obviously been concerned with numbers. Jewelers and dealers in precious stones and metals had used numbers and measured their wares by use of a balance. Apothecaries had also used numerical weights in measuring medicines. Astronomers had measured stellar and planetary positions (altitude and azimuth or Right Ascension and declination) since at least the second century. Mapmakers had for centuries been locating positions of cities and natural features of the landscape in terms of two numerical quantities: longitude and latitude. And of course, astrologers had been making numerical calculations as part of their casting of horoscopes. But before the age of Kepler, Galileo, and Harvey, numbers were not used to express general laws of nature or to provide testable questions to test a scientific theory. This feature of the use of numbers in science set the new science of the Scientific Revolution apart from the traditional study of nature; in fact it defines the newness of the new science.
FIGURE 2.1 Sanctorius in his weighing chair, from his 1614 Medicina statica. Courtesy of the National Library of Medicine
The numerical character of the new science also appears in the invention of new instruments for measuring physical quantities. Among these are the pendulum clock for measuring time and the barometer for measuring air pressure. One of the great innovations of the seventeenth century was the invention of the telescope. Historians have explored how this new instrument changed our ideas about the Moon and the planets, and revealed the existence of new stars never before known. Less attention has been paid to the significant new level of precision attained when telescopes became equipped with micrometers. The telescope showed what planets are like and revealed details such as the mountains on the Moon, but it was only with the introduction of the micrometer that measurements could be made. This transformation of the telescope shifted the focus of astronomy from qualitative description to quantitative measurement.
A FIRST EXERCISE IN DEMOGRAPHY: HOW MANY PEOPLE CAN THE EARTH SUPPORT?
Many readers will be acquainted with the name of Antoni Van Leeuwenhoek (1632–1723). Leeuwenhoek is generally considered to be the father of microscopy because his pioneering discoveries established the microscope as a primary tool of biological science. He was by profession a draper who spent his life in the town of Delft in Holland. He was not educated in science, but he was a skilled artisan and instrument maker who constructed his own microscopes. (He left no records of his mode of producing the lenses that permitted high magnification.) Leeuwenhoek had an uncanny faculty of knowing what to look at, that is, what subjects would yield important information concerning different aspects of the life process. He also knew where to send his reports on his new discoveries—to the Royal Society in London, at that time the foremost scientific society in the world. They published his reports and honored him for his research by electing him a fellow of the Royal Society.
Leeuwenhoek was not particularly interested in demography, in problems of the world population. He came to this area because he was searching for a means of expressing in numbers the size of the objects he had been studying with his microscope. Leeuwenhoek saw that there are a number of ways of indicating quantity, as we may see by looking at cookbooks. A recipe might call for a “pinch” of salt, a tablespoon of this, or a teaspoon of that. Or, the recipe may call for half a pound of butter. Another way of indicating size would be by volume or area. Thus we might indicate the size of a Ping-Pong ball by the number of them that could be placed in a gallon pail.
Leeuwenhoek was concerned with the problem of size in relation to one of his most important discoveries: the spermatozoa. He wanted to give his readers some idea of the minuteness of these “animalcula.” On 25 April 1679, he wrote to the Royal Society concerning his discovery. His communication announced that the number of “little animals in the milt of a cod” (150 billion) was far greater than the total number of people that the earth could support.4
How did Leeuwenhoek compute the number of people the earth could support? In order to achieve this result, he did not resort to geometry, the mathematics of the university. Instead, he used arithmetic, a subject which he knew as a businessman, the arithmetic of shopkeepers.
Leeuwenhoek’s determination of the maximum number of people on the earth began with an estimate of the extent of the earth’s surface. He came up with 9,276,218 square miles. At that time the Dutch mile was reckoned at one-fifteenth of a degree or about 7.4 kilometers. He assumed that vast oceans occupied two-thirds of the total surface of the earth. He also estimated that two-thirds of the dry land was inhabited. Thus the inhabited part of the earth occupied 2,061,382 square miles. If we know how many people a square mile can support, then it is a simple job of multiplication to compute the maximum number of people that can live together on the earth.
To find out how many people could live in a square mile, Leeuwenhoek turned to the part of Holland he knew best. This was a region comprising North-and South-Holland and part of Brabant. This region is roughly in the shape of a rectangle whose area is 154 square miles. Holland did not have a national census until well into the next century, so Leeuwenhoek made use of the fact that a head tax or “capitation” had been levied in 1622. From this he could say that this area contained a population of about a million persons.
Leeuwenhoek was now in a position to determine the size of the maximum population of the earth. He said, let us “assume that the inhabited part of the earth is as densely populated as Holland.” He was of course aware that “it cannot well be so inhabited.” Since the inhabited part of the earth is “13,385 times larger than Holland,” the result is a maximum of “13,385,000,000 human beings on the earth.” In the present context, the way in which Leeuwenhoek arrived at his result is not of much interest. What is significant is that Leeuwenhoek’s desire to express in numbers the minuscule size of the animalcules he had discovered led him to produce the first recorded exercise in the numerical science of demography.
In the seventeenth century there arose a new and important industry using numbers—life insurance. A person would purchase an annuity by making payments to an insurer, and then after a certain date, the insured person would receive regular payments from the insurer for the rest of his life. The specific terms of purchase were agreed upon for each annuity. Both the insured and the insurer would want to get the most advantageous terms possible, and for this purpose they needed to know the average life expectancy of individuals in the insured’s age category. Of course, there was no way of predicting whether the individual insured would fit the average. Some people frowned on this kind of business because it was a form of gambling on a person’s life.5
The problem of supplying numerical tables for use in computing annuities was taken up by Edmund Halley (1656–1742), the astronomer after whom the comet is named, a skilled mathematician, and secretary of the Royal Society.
Halley began by ascertaining the number of burials in Paris and London in 1680: 24,441 in Paris and fewer than 20,000 in London. He found that the numbers of births and marriages were in the same proportion in each city.6 Such data, however, were subject to fluctuations, because of people moving between the country and the city. To supply reliable tables, Halley needed numbers for a stable population.
And so we can imagine the joy with which Halley would have received tables of the births, marriages, and deaths recorded in the town of Wroclaw (Breslau) in Silesia, now in Poland. These data had been compiled by Caspar Neumann (1648–1715), a pastor and an ecclesiastical judge in the reformed church. Believing that his data were important, Neumann sent them to Leibniz, the outstanding intellectual of the German-speaking world at the time. Leibniz, seeing no special value to these data, sent the figures on to Paris, where again their value was not recognized. A French correspondent of the Royal Society sent the compilation on to London, where it came into the hands of the secretary, Halley. Halley at once recognized that here were just the numbers he needed, based on careful records kept for a stable population. Few people moved to Wroclaw and few people moved away.
Halley used Neumann’s numbers as a basis for a memoir which is celebrated as having laid the foundations of a correct theory of the value of life annuities. The memoir was published in the Philosophical Transactions of the Royal Society for 1693. The memoir bore the title “An Estimate of the Degrees of the Mortality of Mankind, drawn from curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain the Price of Annuities upon Lives.” Table 2.1 is a partial summary of the data.
In recent decades, a group of scholars has been exploring how numerical considerations brought into being a science of statecraft and a mode of social analysis worthy of the name social science. Their endeavors have shown how quantitative considerations have entered the conduct of government and have given us a new level of understanding of social systems. Thanks to the research of these scholars, we are now aware of the stages in the creation of a new science—statistics—that has affected almost every aspect of our lives.7 For most readers, the word “statistics” will suggest a table of numbers, such as the data relating to the performance of members of a baseball team. This concept of statistics will seem quite proper since the singular “statistic” means a single numerical datum. The singular form, “statistic,” seems to have come into general usage only in fairly recent times, whereas the more general word “statistics” came into being in the seventeenth and eighteenth centuries. Furthermore, the noun “statistics” is singular rather than plural.
TABLE 2.1 SUMMARY OF DEMOGRAPHIC DATA FROM BRESLAU, 1687–1693, WITH ADDITIONAL DATA FROM NEUMANN FOR 1694
Number Surviving in the Year |
||||||||
Year of Birth |
Number Born |
1688 |
1689 |
1690 |
1691 |
1692 |
1693 |
1694 |
1687 |
1,186 |
940 |
884 |
792 |
738 |
708 |
680 |
662 |
1688 |
1,214 |
956 |
880 |
761 |
714 |
685 |
663 |
|
1689 |
1,191 |
954 |
849 |
743 |
690 |
655 |
||
1690 |
1,312 |
991 |
903 |
800 |
723 |
|||
1691 |
1,292 |
988 |
890 |
759 |
||||
1692 |
1,151 |
857 |
756 |
|||||
1693 |
912 |
|||||||
From E. Halley, “An Estimate of the Degrees of the Mortality of Mankind,”Philosophical Transactions of the Royal Society of London 17 (1693), pp. 596–610
A British historian, Keith Thomas, and his American counterpart, John Brewer, have carefully traced the increasing dependence of government on numbers for sixteenth-and seventeenth-century England. They find that during the seventeenth century the expanding economy of England and the problems of military statecraft created a pressing need for numerical information in different departments of government. Ministers of the Crown needed quantitative information on “all of the various resources of the different departments [of government] in order to exercise firm control over government policy.”8 The Parliament, “both as a policy maker and as the body dedicated to securing a responsible executive,” needed government statistics. Such statistics were also needed by various “occupational groups and special interests directly affected by state policies.” They were “eager to learn the grounds” on which decisions were being made. Additionally, according to Brewer, the general public had developed “a substantial appetite for the sorts of [numerical] information that only the very considerable resources of the state could provide.”
A striking example of the use of numbers in relation to the conduct of life is the regular tabulation and publication of Bills of Mortality. Such bills seem to have originated in outbreaks of the plague, when there was a great concern to know whether conditions were getting better or worse. The King, members of the Royal Court, and ordinary citizens wanted to learn whether it was safe to remain in town or whether to seek refuge in the country. Toward this end regular statements were published concerning the number of deaths. These were called Bills of Mortality. In London, following the plague epidemic of 1603, these Bills of Mortality came to be produced and published regularly, even during times of no epidemic. These broadsides listed the number of deaths in the week just past. According to John Graunt, who in the seventeenth century became the first person to analyze these Bills of Mortality, special attention was paid to “how the Burials increased, or decreased,” and “among the Casualties, what had happened rare, and extraordinary in the week current.” These bills were especially important in “the Plague-time,” as indicators of “how the Sickness increased, or decreased, so that the Rich might judge of the necessity of their removall” from London and so that “Trades-men might conjecture what doings they were like to have in their respective dealings.”9 These numerical tables were of special significance because they provided the database for what appears to have been the first truly statistical analysis in recorded history.
Historians generally agree that the first example of a statistical analysis, as we understand this term today, came in Graunt’s 90-page book entitled Natural and Political Observations Mentioned in a Following Index, and Made Upon the Bills of Mortality (London, 1662). Keep in mind that Graunt was not a university professor, nor even a college-educated amateur scientist, but rather a merchant engaged in a small business. He is said to have been a draper (a seller of cloth) or a haberdasher.
We learn about Graunt’s life through accounts of him by a contemporary, John Aubrey (1626–1697), who collected information concerning the lives and achievements of famous men he had known or heard about. From Aubrey we learn that as a young man Graunt customarily arose very early so as to spend several hours working up a good reading knowledge of Latin and French, but Aubrey does not tell us why Graunt wanted to learn these languages.
Graunt evidently became an important member of the London business community, active in civic affairs. He became an officer in an organization known as the “Trayned Band,” and Aubrey refers to him as “Captaine John Graunt” (afterward, major). His intellectual distinction was recognized in 1663 when he was elected a fellow of the Royal Society. According to Aubrey, Graunt was apparently a person of some influence; he obtained for his friend William Petty the post of professor of music in Gresham College and also a lucrative appointment to be “one of the Surveyors of Ireland.”10
Graunt’s business was wiped out by the Great Fire of London of 1666. Perhaps as a result of despair, Graunt then converted to Roman Catholicism. We may agree that at a time when Graunt “most needed the earthly help of those able to provide it, his spiritual beliefs gave them an excuse not to do so.”11
The path-breaking significance of Graunt’s contribution to knowledge was at once recognized by his contemporaries, and it was on the basis of his little book that he was elected a fellow of the Royal Society. Yet Graunt did not become a regular attendee at the weekly meetings of the society and he made no further contributions to knowledge; he even let his membership in the society lapse. He did, however, produce several expanded and revised editions of his book.
Graunt’s analyses were based, as the title of his book declares, on the London Bills of Mortality. He begins with a brief history of these bills. Until 1629, only summaries or totals were provided. Then the totals of christenings and burials were divided according to sex and the causes of death were given. In the bills for 1632, deaths were classified according to 63 different causes. For example, the 9,535 deaths listed for 1632 begin with 415 “Abortive and Stillborn” cases, one case of “Affrighted” death, and 628 “Aged.” There were 2,268 deaths of “Chrisomes” (newly baptized infants) and other infants, plus 267 deaths from “Dropsie and Swelling,” 1,108 from fever, and nine from “Scurvy, and Itch.”12 Graunt also makes the point that he was the first person to analyze these data and that the diagnosis of the cause of death was made by ignorant enumerators. As Graunt studied this collection of numerical data he was struck by the statistical regularities that appeared. Graunt found that “among the several Casualties some bear a constant proportion unto the whole number of Burials.” These included chronic diseases, accidents, and suicides. But he found that “Epidemical and Malignant Diseases,” such as the Plague, “do not keep that equality, so as in some Years, or Moneths [sic], there died ten times as many as in others.”13 One of his most interesting findings is that slightly more males than females are born each year.
One of the celebrated passages in Graunt’s book is his account of the way in which he used the numerical data in the Bills of Mortality to arrive at an estimate of the size of the population of London. In this exercise, he showed how to interpret these numbers, how to draw inferences from them. Graunt’s estimation of the size of the population of London is an outstanding example of statistical analysis. It shows the way in which tables of numbers can be made to yield statistical information. In other words, this is a pioneering attempt to interpret numerical data and cause them to yield a reasonable estimate of a significant social number.
Graunt’s computation of the London population draws the admiration of everyone who studies the history of statistics. Others before him had made guesses concerning the magnitude of such populations, but Graunt was the first person to base his figure on reasonable inferences from actual numerical data.
As his base figure, Graunt used the annual number of christenings as listed in the Bills of Mortality. This number, he pointed out, was subject to some error. It was arrived at from the entries in parish registers and so did not include the small number of Catholics, freethinkers, or other religious dissenters. Furthermore, there was no way of knowing the number of parents who, though nominally members of the established Church, had not had their children baptized.
Graunt’s first stage of computation was based on the conclusion that, on average, women of childbearing age would give birth to a child every other year. Or, as he put it, such “women, one with another, have scarce more than one Childe in two years.”14 Since there were on average some 12,000 children born in London each year, Graunt concluded that the total number of “Teeming women” (or women producing children) in London would be 2 × 12,000, or 24,000.
Next, he ascertained that the time span during which women gave birth extended over 24 years, from age 16 to age 40. He then took note that the number of women who were married ranged in ages from 16 to 76 years. Thus, the total number of women living in London would be about twice the number of women of childbearing age, or 2 × 24,000 = 48,000. Of course, Graunt’s 16 to 76 is a span of 60 years, which is somewhat more than 2 × 24. But Graunt was not attempting to produce an exact number so much as an order of magnitude.
The final number was based on another finding of Graunt’s. He found that the average family unit in London tended to consist of eight individuals: a mother and father, three children, and three servants or lodgers. Thus the total population of London would come out to be 8 × 48,000, or 384,000.
Is this a credible number? It’s a big number, but is it big enough? Graunt found two ways of checking the validity of his procedure. The first was to consider the actual number of inhabitants in certain parishes. In these, he found, there were three deaths annually per 11 families. He also knew that the total number of deaths in London was some 13,000 per annum. Hence, he concluded, the number of families in London must be (11/3) × 13,000 = 47,700, that is, about 48,000 families. This is the same number of families that he had arrived at by his earlier calculation and so it gave him confidence in his method and his concluding number.
Graunt’s second check on his procedure was based on the map of London. He assumed that a reasonable estimate of the total number of families living within the town walls was 12,000. He also found from the Bills of Mortality that the number of deaths outside the walls was three times the number of deaths within the walls. Hence, once again, he came out with 48,000 family units for London as a whole, a quarter of which lived within the town walls and three-quarters of which lived in the surrounding area.
A modern appraisal of Graunt’s results is that “they were perhaps not very far from the truth.” Graunt’s mortality rate of 3 out of 88 (corresponding to his eight members of the 11 families) would be expressed in our times as a rate of 34 per 1,000, which “is not improbable.”15 Of course, Graunt made some bold assumptions with which we can easily quarrel. But what is most important is not whether his work would pass muster today. Rather, what is historically significant is that he showed by example how numerical data could provide a basis for interpretation; he demonstrated how a statistical point of view could elicit conclusions of general interest from tables of specific numerical data.16
SIR WILLIAM PETTY AND POLITICAL ARITHMETIC
It would be difficult to find a pair as different in lifestyle, career, and influence as Graunt and Petty. Whereas Graunt was shy and retiring, dying in poverty and bankruptcy, Petty was a swashbuckling character, an adventurer, a confidant of the ruling monarch Charles II. He was a bold investor who amassed a fortune and died a wealthy man.
Sir William Petty (1623–1687) was in a real sense Graunt’s intellectual heir and successor. Whereas Graunt wrote a single work, Petty authored enough books to justify one written about his results by Sir Geoffrey Keynes, the brother of the economist.17 Petty’s multifaceted career has been summed up as follows. He appears, Keynes wrote, “as cabin boy on an English merchant ship, as peddler of sham jewelry in France, as naval cadet, as medical student in the Netherlands and in Paris, as reader in music in Gresham College, as Professor of anatomy at Oxford, as Fellow of the Royal Society, as educationist, inventor, Latin versifier, and ship builder, as physician to Cromwell’s army in Ireland, as surveyor and geographer of Ireland, and in the midst of all that as a man becoming ever more involved in administration, finance, and politics.”18
Petty was born in 1623 in Romsey, a rural community, where his father was a clothier who dyed his own cloth. Young Petty was fascinated by the work of artisans—smiths, watchmakers, carpenters, and so on. After an apparently sound schooling, Petty—at the age of 14—went to sea as a cabin boy. While cruising in the Mediterranean, Petty broke his leg and was put ashore at Caen, unable to get around and ignorant of the French language. How he managed it, we do not know for sure, but Petty healed and became fluent in French. He also improved his Latin. He evidently studied mathematics and navigation and when at the age of 20 he returned to England, he joined the Royal Navy.
Petty’s career in the Navy was short-lived, and in 1643 he returned to the Continent where he studied medicine and anatomy in the Netherlands and in Paris. He was so poor during these student years that, as he told John Aubrey, he had to live for a whole week on three pennies’ worth of walnuts. Back in England, Petty eventually (in 1649) became a Doctor of Medicine.
Somehow or other, Petty made the acquaintance of Graunt, who recommended him for a post in music at Gresham College, London. Later Petty became a professor of anatomy at Oxford and eventually was elected Master of Brasenose College in Oxford. Petty was interested in the new science coming into being during the Scientific Revolution and became one of the founders of the Royal Society of London.
At Oxford Petty became well known for his daring act of reviving a woman executed by hanging. She had been found guilty of murdering her illegitimate child. Her body was turned over to Petty to be used as a subject in the dissecting room. Petty was able to revive her, although she had been pronounced dead. Evidently she had been “inefficiently hanged.” Petty reported this medical phenomenon as “History of the Magdalen” and it was published in the popular press as News from the Dead in 1651.19
In 1652 Petty accompanied Cromwell’s troops into Ireland. He evidently didn’t serve in the simple capacity of doctor to the army, but rather as a surveyor, busy with evaluating the Irish estates that were being taken over by the British. Petty acquired land and leases for himself and so ended up with a minor fortune. Despite his association with Cromwell and his armies, Petty became a favorite of King Charles II after the Restoration. Perhaps Charles liked Petty’s wit, but it has also been suggested that he was grateful to Petty for having uncovered new sources of tax revenue to support the royal lifestyle. In any event, Charles thought well enough of Petty to grant him a knighthood in 1662.
From the point of view of the history of numbers, Petty is important for his early attempts to determine the population of England and of Ireland. Recognizing the significance of Graunt’s work, Petty advocated that “policy”—that is, the conduct of government—should be based on numerical data. Petty developed a theory of social and economic understanding and planning that he named “political arithmetic.” In fact, Petty wrote and published five works with the words “political arithmetic” in the title. In one of the most important of these, simply called Political Arithmetick (1690), he described this new form of statecraft as follows:
The Method I take to do this, is not yet very usual; for instead of using only comparative and superlative Words, and intellectual Arguments, I have taken the course (as a Specimen of the Political Arithmetick I have long aimed at) to express my self in Terms of Number, Weight, or Measure; to use only Arguments of Sense, and to consider only such Causes, as have visible Foundations in nature; leaving those that depend upon the mutable Minds, Opinions, Appetites, and Passions of particular Men, to the Consideration of others.20
It should be noted that Petty not only had a vision of a new statecraft based on numbers but also declared that the mathematics of numbers, that is, algebra, was the tool for making the analyses on which the new statecraft would depend.
“Political arithmetic,” as we shall see, became widely used during the eighteenth century to denote a statistically based statecraft—one based on numerical information concerning such entities as population or demography, natural resources, manufactures, exports and imports, and agriculture. The term fell into disuse in the nineteenth century. We may note that this subject is only imperfectly described as an arithmetic. The science of statistics, it has been observed, requires the mathematics of probability; arithmetic and even algebra do not suffice.21
Yet, whatever disciplines it encompassed, Sir William Petty’s “political arithmetic” was moving down a rational path that would not be followed by everyone.