Mortality data, risk probability, and the psychology of assent in the enlightenment smallpox debate

David Spadafora
Pinehurst, North Carolina


Portrait of Voltaire who wrote on inoculation and data
Nicolas de Largillière, Portrait of Voltaire, ca. 1724. Source.

The present health crisis is hardly the first to provoke significant controversy about preventing and treating widespread disease. Debate over epidemic-related data, its reliability, and its uses has a long history. So does concern about the psychological elements involved in securing assent from physicians and an endangered population for the use of specific prevention and care regimens. Connected episodes relevant to both of these important considerations, and to today’s debates, come from the Enlightenment era’s confrontation with smallpox. 


Voltaire and the philosophes: champions of inoculation       

Voltaire (1694-1778), a lifelong hypochondriac, had become seriously ill with smallpox in 1723. During his roughly two years in England as an exile later that decade, he paid special attention to attempts there at preventing smallpox outbreaks by means of inoculation. His Philosophical Letters or Letters concerning the English Nation (1733-34), a key document of the Enlightenment, famously devoted a chapter to inoculation (also known as variolation, or l’insertion de la petite vérole), seeking to persuade the French to take up this practice. The chapter is thought to have been drafted as early as 1727.

Here he argued that if inoculation had been used in France as in England and Turkey, thousands of French lives would already have been saved. To buttress this claim, he offered data. At least 60 out of 100 people worldwide are infected with smallpox (presumably, he meant at some point during their lifetimes). Of these 60, at least 20 die (according to the death rate of the most favorable years, he stipulated), and 20 others (who presumably do not die from the disease) are permanently disfigured. Thus, he concluded (in mathematically incomprehensible terms), “there you have one-fifth of humankind killed or disfigured by this disease, most probably.” By contrast, he contended, of those inoculated in England or Turkey (whence the English practice derived), no one dies unless infirm or already dying of another disease, no one has enduring pox marks, and—if the inoculation is done correctly—the inoculated do not get smallpox a second time.1

These are strong claims, and they direct Voltaire’s readers toward a clear-cut decision: get an inoculation yourself, and support inoculation for everyone. With this commentary, Voltaire provided strong momentum for what turned out to be a decades-long effort by him and some of his intellectual compatriots, the philosophes, to argue vigorously for the use of inoculation in France. It is not too strong a word to describe this effort as “propaganda,” and not just because of Voltaire’s flawed arithmetic. The philosophes were eager to persuade individual French physicians, the French medical establishment, and the French government of the efficacy and safety of inoculation. Especially in the 1750s, 1760s, and early 1770s, with leadership from Voltaire and the astronomer and geographer Charles-Marie de la Condamine (called the “Don Quixote of inoculation” who “never stops campaigning” for it), they bent every effort toward the achievement of their goal.2

The proponents of inoculation believed that they were being modern and scientific—“following the science,” they might have said and nearly did say—and that their country was sadly behindhand in this and other matters. “Is it to France that we owe the telescope, the fire-engine, [knowledge about] gravitation, knowledge about light, inoculation, the grain-sowing machine, condoms?” Voltaire asked rhetorically.3 Those opposing inoculation, sometimes called anti inoculistes, including the Parlement of Paris and the Sorbonne’s faculty of theology, were represented as being especially benighted—guilty of “imbecility” or incapacitated judgment.4 The Marquis de Chastellux described the opponents of inoculation as a party that “sees with regret the progress” that has been made by inoculation, because in France “we embrace new opinions with difficulty.” As Cosimo Alessandro Collini wrote to Voltaire, owing to “prejudices,” people have “closed their ears to your lessons and your advice. The calculations show how one can prove an arithmetic rule, have proved that you were right. . . .” “How can the evidence of a calculation not suffice to carry conviction in the hearts of men . . . ?” “We read with horror,” said the Encyclopédie of Diderot and d’Alembert in its article on inoculation, “of the centuries of darkness, and we call barbarous the superstition of the druids . . . ; and in this century so polite, so full of light that we call ourselves the century of Philosophy, we don’t notice that our ignorance, our prejudices, our indifference for the well-being of humanity stupidly consign to death each year in France alone 25,000 subjects. . . .”5

Among the opponents of inoculation, the clergy were the philosophes’ true bêtes noires, in this matter as in so much else. Voltaire depicted them as standing astride the path of medical progress, insisting on acceptance of the dictates of providence, which had willed the disease. The Theology Faculty of the Sorbonne had indeed at first issued a condemnation of inoculation,6 and the argument emphasizing providence did have its place in anti-inoculation argumentation. The anonymous author of a tract on inoculation published in 1757 stated the providential case forthrightly: the procedure “tempts God, seeks to weaken the order established by Providence, [and] prevent its judgments,” and would encroach on “the rights of the God” to bring this disease while undertaking to shield those who “by order of providence” are “naturally destined to have it.”7 In fact, it is easier to find English clerical proponents of this argument than French, partly because predestination was much more a Protestant than a Catholic doctrine. A sermon preached by Edmund Massey in London in July 1722, for example, was frequently referenced by French commentators on both sides of the issue. It contended that inoculation was a diabolical operation that sought to banish providence out of the world, so as to “sinfully endeavour to alter the Course of Nature by . . . presumptuous Interposition” while promoting vice and immorality.8

Philosophe proponents of inoculation also attacked the Parlement of Paris, the chief French law court, for its decision in June 1763 to restrain the practice temporarily, in the midst of an outbreak that some suspected was caused by inoculation. The Parlement sought the judgment first of the Sorbonne’s Faculty of Medicine and then of its Faculty of Theology as to whether inoculation was medically useful and religiously appropriate. The debate inside of the medical faculty, led by Joseph de l’Epine for the antis and Antoine Petit for the pros, was long, intense, and bitter. Meanwhile, the philosophes grumbled about hidebound doctors and guffawed over the presumed right of theologians to judge a scientific matter.9

In this matter as in others, the philosophes used every arrow in their quiver—private correspondence, publications, and speeches given in the Paris Académie des sciences—to hit the target at which they were aiming. The Faculty of Medicine finally decided in 1768 in favor of quasi-toleration for the practice of inoculation. Eventually, royal examples helped bring them even closer to their goal: Empress Catherine of Russia in October 1768,10 after urging by Voltaire, and Louis XVI and his brother, after his predecessor, King Louis XV, had died of smallpox in 1774.


English data collection: Jurin, Nettleton, Scheuchzer, and Massey

The source of the figures cited by Voltaire remains uncertain, but it seems very likely that at some point he became aware of data collected and analyzed by London physician James Jurin, M.D. These data appeared initially in 1722, first as a letter printed in the Philosophical Transactions of the Royal Society and then in pamphlet form, and thereafter through annually-issued pamphlets titled An Account of the Success of Inoculating the Small Pox for the Year. While he was in England, Voltaire made connections with leading English intellectuals such as Bolingbroke and Pope who could have led him to Jurin’s work. At some point he became personally acquainted with Jurin, because the two later corresponded familiarly about physics, not long before Voltaire was made a Fellow of the Royal Society.11

Voltaire could have learned about English inoculation activity and Jurin’s data before he ever set foot in England, however. In 1723, a decade prior to the appearance of Voltaire’s chapter on smallpox-death prevention, the French doctor Jean de la Coste had published a small book in favor of inoculation. It received favorable reviews in the Mercure late that year and in the Jesuit-run Journal de Trévoux the next year—in the latter case with a mention of Jurin and some data from him.12 In addition, accounts of English inoculation practices and results appeared as early as 1724 in French summaries of the activities of the republic of letters.13 Voltaire kept up with these literary journals and might well have read any or all of the relevant material prior to his time in England.14

In any case, we can say with confidence about Voltaire’s figures that they look very much like those collected and published by Jurin, who served for most of the 1720s as Secretary of the Royal Society. This lofty official position, second only to that of Isaac Newton, allowed Jurin both to reach a wide audience of readers and to motivate those readers to respond to him, in order to develop a basic understanding of the comparative risk of inoculation. The dispute over inoculation being “of the utmost Importance to Mankind,” he hoped that “without Passion, Prejudice, or Private Views, it may be fairly and maturely examin’d,” and he believed that his investigative and reportorial approach provided for that objective and temperate examination. Specifically, his work would “form an Estimate of the Hazard, which all Mankind, one with another, are under of dying of the natural Small Pox, [so] that, by comparing this with the Hazard of Inoculation, the Publick may be enabled to form a Judgment, whether or no the Practice of Inoculation tends to the Preservation of Mankind, by lessening the Danger to which they are otherwise liable.”15

In his original letter to the Royal Society about this topic, Jurin noted that one way to develop the requisite data would be an annual house-to-house census of cases and outcomes in several large towns and parishes, conducted annually over several years by “a careful Person.”16 He chose instead to rely upon a network of observers he considered “credible Persons”—medical practitioners such as apothecaries, physicians, and surgeons, along with gentry and clergy—with himself as the central node receiving their observations. His call for communication on this topic asked “All Persons concern’d in the Practice of inoculating the Small-Pox . . . to keep a Register of the Name, Age and Habitation of every Person inoculated, the Manner of the Operation, the Days of Sickening and of the Eruption, the Sort of Small-Pox is produc’d, and the Event.” He also requested details about the eruptions and about anyone who died after inoculation. Each observer’s report was to be sent to him in late winter for publication in the spring.17 This appeal was soon printed in France as well.18

His correspondents gave him data that he analyzed and assembled, in anonymized form, into tabulated categories. This procedure allowed him to say that at least one-fourteenth (7 percent) of all deaths are attributable to smallpox, that about two-elevenths (18 percent) of all smallpox cases are fatal, and that between one-sixtieth (1.8 percent) and one ninety-first (1.1 percent) of those inoculated will die.19 After half a decade of work, these statistical conclusions remained basically unaltered: 16.3 percent or roughly one-sixth of the people known to have contracted smallpox had died of the disease (a running tally of 2,957 out of 18,089); of the 100 people known to have had “perfect” inoculation in 1726, only 1 had died, which brought to 14 the running tally of deaths among the 724 people known to have been inoculated (about 2 percent). Thus, “there still remains a great Advantage on the Side of Inoculation, above the Hazard of taking the Disease in the Natural Way. . . .”20

This was, in fact, a novel way of assessing a treatment, and it constitutes “arguably the first use of numerical evidence to evaluate a medical practice” in Britain.21 To be sure, Jurin had many predecessors as commentators on the usefulness of inoculation, and some on smallpox data. John Arbuthnot had recently published figures from the Bills of Mortality, showing that 8-9 percent of London’s deaths were owing to smallpox, and that the mortality rate for natural smallpox was about 10 percent. He had gone on to assert (without providing data) that the mortality rate for those inoculated was only 1 percent.22 Dr. Thomas Nettleton came close to offering comparative mortality data. In his December 1722 communication with the Royal Society, he provided figures from various Midlands and southern Yorkshire towns showing that in “this last year” about one-fifth of the 3,405 people known to him to have had smallpox had died, whereas “out of sixty one who have been inoculated hereabouts, not one has died.”23 In fact, it appears to have been Nettleton’s urging of more extensive data collection that prompted Jurin’s project. But that project does seem to have been truly innovative in terms of its emphasis on statistical probability.

After Newton died in 1727 and was succeeded by Dr. Hans Sloane as President of the Royal Society, Sloane’s librarian, the young Swiss botanist and physician Johann Gasper Scheuchzer, succeeded Jurin as the leader of the smallpox data-collection effort, inheriting Jurin’s archive of materials. On reviewing them, he concluded publicly that, “both from Reason and Experience,” the case for inoculation is “highly probable.” The data he presented for the years 1727 and 1728 showed the mortality rate for natural smallpox remaining constant, with a small increase in mortality among those inoculated during the past two years (3 of 121, or 2.5 percent). Scheuchzer discussed each of the three deaths in detail, concluding that the “Odds remain as much in favour of this Practice” as before. His pamphlet contended that at least 113 lives had been saved by inoculation between 1721 and 1728, and that more than 2,000 deaths had occurred owing to the failure to inoculate.24 Whether this additional projection was Scheuchzer’s is unclear: he died suddenly in April 1729, and Sloane took responsibility for publishing his report, the last in the series begun by Jurin.

Despite both participation by and also support from the many correspondents of Jurin and Scheuchzer, the aggregated data did not remain unchallenged. The toughest-minded criticism came from Isaac Massey, Apothecary to Christ’s Hospital School in London, with its fine program in mathematical education. He had already attacked inoculation on medical grounds, in a 1722 pamphlet where he also proposed a research project for the Royal College of Physicians that he surely thought would support his opposition. His idea was that every person inoculated should be obliged “to have his Name, and Place of Abode, entered in a Register for that Purpose; wherein the Time and Successes, good or bad, should be also Registred [sic], and if afterwards any should live to have the Small Pox, some Care should be taken effectually, that the College be acquainted therewith, and a Memorandum be made of it. This would clear up all Doubts. . . .”25 Perhaps this concept influenced Jurin in the initial development of his project.

In 1723, having by then read Jurin’s report to the Royal Society, Massey wrote another pamphlet pointing out what he thought were serious errors in the project. He contended that Jurin’s data taken from the Bills of Mortality was “misapply’d,” because the Bills made no distinction between the deaths of well-cared-for patients and those of indigent or already unhealthy people with little or no access to care. Further, the impoverished sick were unlikely to be able to afford inoculation, and they were “rejected by the inoculators, as improper” for the procedure. In short, “to form a just Comparison, and calculate right in this Case, the Circumstances of the Patients, must and ought to be as near as may be on a Par.” If appropriate adjustments in the raw data of the Bills of Mortality were made to take such key issues into account, thus making the two groups more comparable, the mortality rate for natural smallpox would decline to about 2.6 percent (1 death out of 42 natural smallpox victims). Having proposed a much narrower difference in probability of death between the two groups, Massey went on to point out that the reliability of the reported data could be generally challenged, on the grounds of the easy confusing of smallpox symptoms and those of other diseases such as scarlet fever, swinepox, and measles.26

In his review of the final report by Jurin, he made the same basic argument about the problem with Jurin’s superficially “great Odds,” namely, the non-comparability of healthy, well-cared for patients versus the sick and indigent. He also adduced data from the children at Christ’s Hospital and in his private practice, concluding that the mortality rate in people ages 5–18 is about 1 in 40.27

By the later 1720s, therefore, key figures in the English medical establishment were making strong claims, based on the harvesting of data from hundreds of cases, about the probability of inoculation’s success. Those claims were being contested on the same grounds that would eventually lead, centuries later, to normalization processes for the review of patient data. Data-driven analysis of disease was accepted by both sides, however, and, as we have seen, it would soon be promulgated by a leading philosophe to advance the cause of inoculation.28


David Hartley who wrote on inoculation and data
David Hartley, Engraving by William Blake in the 1791 edition of Hartley’s Observations on Man. Source.

David Hartley enters the debate

Nevertheless, inoculation became decreasingly popular in the immediately succeeding years, whether on its home turf in England or in France.29 In 1733, the same year as the initial publication of Voltaire’s Lettres philosophiques, a young English physician carrying on the Jurin statistical tradition took the case for inoculation farther as he faced a smallpox outbreak in his town.

David Hartley (1705–57), remembered principally for his major contribution to the early development of physiological psychology, enjoyed a long and prosperous career as a physician, despite not having the M.D. degree. Although it seems to have included attendance at many lectures in chemistry, mathematics, and medicine in other colleges, his study as an undergraduate student at Jesus College, Cambridge culminated in a bachelor of arts degree rather than bachelor of medicine in January 1726. The next year he became a Fellow of his college, and in January 1728 he received the M.A. degree. At some point during his Cambridge career, he studied mathematics and natural philosophy under the famous blind Lucasian Professor Nicholas Saunderson,30 who worked on the topic later known as Bayes’ Theorem, about the probability of events in circumstantial context. He was practicing medicine in Newark-on-Trent in Nottinghamshire by later 1728 or early 1729, supplementing his income with a mastership at the Newark Grammar School. Within a year, he moved to the resort town of Bury St. Edmunds in Suffolk, there establishing a successful medical practice. It likely included two major families, the Cornwallises and the Townshends.31 In the fall of 1735, he relocated his practice to what is now the Leicester Square neighborhood of London, and in 1742 he moved again, for the last time, to the rapidly rising spa-town, Bath. While practicing in London and Bath, he came to know many members of the aristocracy, including the Duke and Duchess of Newcastle, whom he treated.32

In both Newark-on-Trent and Bury St. Edmunds, Hartley encountered local smallpox epidemics, and he exchanged letters with Nettleton, Sloane, and Jurin about the disease as well as its prevention through inoculation.33 It is not known if he first actually inoculated patients himself, but his good understanding of the practice and his convictions about its efficacy did lead to his first publication, a January 1733 pamphlet in favor of inoculation. Here he adduced what he saw as its chief virtue: “it will probably remain in the Town [Bury] a very long Time still, to the vast Detriment of all Trade and Business, and perhaps the Loss of the Assizes, Sessions, Fair and other publick Meetings: Whereas if Inoculation was generally practiced, it might be got through the Town in a very short Time.” This concern about the wide social benefit of particular courses of action by individuals marked Hartley’s later thinking, too, as he became a notable Enlightenment figure—although not, as a rather traditional Christian, a philosophe per se. In this small pamphlet, he observed that with inoculation “We have then a most happy Opportunity of serving Mankind at the same Time that we serve our selves,” because “it is very probable that if Inoculation was generally practiced, we should arrive at such Perfection in it [i.e., in the procedure], that few or none would die.”34

Indeed, probability lay at the center of Hartley’s argument. The data that had been collected to this point revealed, he averred, that the utmost hazard of dying of inoculated smallpox was 1 chance in 50, and probably as little as 1 in 60 or 80 or 100. By contrast, he said, the odds of dying of natural smallpox had been shown to be 1 in 6, and in the case of an outbreak in Uxbridge, 1 in 3. Using these differential odds, and assuming that Bury St. Edmunds had a population of 3,000 people who had not already contracted smallpox, upon inoculation of everyone perhaps 60 people would die from the procedure, whereas without any inoculations 500 would die from natural smallpox—and thus 440 people “may be clearly saved by Inoculation.”35

One by one he disputed the received arguments against inoculation. No one who has had it is known to have gotten the disease again. Inoculation does not lead to more harmful long-term side effects (“bad Consequences, as Consumptions, Boils, and Blotches, weak Eyes, &c.”) than does natural smallpox. Other distempers are not known to have been transmitted by the procedure, especially in light of the fact that they are most likely transmitted by particulate matter in the air rather than by physical contact.36 Far from inoculation being sinful by virtue of endangering human lives voluntarily, it would be a greater sin not to be treated, inasmuch as not having inoculation leads to greater endangerment by a factor of four (1 in 50 chances of dying versus 4 in 50). Can it be truly sinful to utilize a “Practice which would restore Trade and Business, and the Means of living to many, who now want it, which would save great numbers of Lives, and which might be of singular Service to Mankind”? To trust in providence is not to do nothing but rather “to do all we can for our selves, and then to believe we shall be taken care of.”37

In establishing the case for inoculation, Hartley also looked beyond the loss of life itself to relevant psychological issues.

If besides the Loss of so many Lives we consider the Fears and Uneasinesses of People for themselves, of Parents, Children, Relations and Friends for them, their own Uneasiness at not being able to visit their Friends under this Distemper, the Unfitness for Business which it lays many under, &c. one has still greater Reason to wish that this was the general Practice in Children, from the Age of 2 or 3, to that of 7 or thereabouts.

He recognized that every innovation finds resistance, in medicine as in religion and government. Such “Opposition has generally been the Fate of every such Thing,” he contended, including the relatively new use of quinine to treat malaria and of mercury for syphilis, “either out of Interest and Dishonesty, or Ignorance, Folly, and Superstition.” His own times should reflect on “what Censures Posterity may justly pass upon us in the present Point.” But he remained hopeful that his fellow townspeople would be persuaded by his argument that the appropriateness of inoculation “is made out to be probable” in his pamphlet.38


Martin Warren’s challenge to Hartley   

Statistical probability and psychological ease: these were the fundamental positive grounds on which Hartley rested his case. That case did not persuade the citizens of Bury St. Edmunds, however, and inoculation remained out of favor there, and in England generally, for another decade. Hartley’s failure owed more than a little to Martin Warren, M.D. (1696-1735), another Bury doctor who replied to Hartley’s piece with his own scathing pamphlet. Warren, too, was a Cambridge man, graduating B.A. from Emmanuel College seven years before Hartley. They surely knew each other, inasmuch as Warren, too, became a fellow at his college and went on to get both the M.A. and the M.D. degrees while Hartley was at Cambridge.39 The two would likely have been in some science- and medicine-oriented lectures together.

Warren’s vituperative commentary seems likely to reflect the hostility of an older, degreed doctor toward a younger, recently arrived competitor who did not hold the M.D. degree yet had an obvious “Fondness of being thought Somebody.” The pamphlet made a series of other belittling personal slights, in rejecting Hartley’s argument about economic well-being as the “specious Pretext of a tender Concern for the Detriment of all Trade,” referring to his “cowardly” suppression of a catalog of maladies consequent to inoculation, and mentioning sarcastically the “little Acquaintance he has with Men of our Profession.” He also observed that in Hartley’s citation of the Uxbridge situation he had committed a “very illogical, and unscholarlike” error by drawing a “universal Inference from a particular Instance.”40

Warren completely rejected inoculation, which he described as a practice “justly exploded and condemned by all rational Men,” one “found to be unsuccessful, insecure, and not to answer the Intention” of its practitioners and advocates. In today’s terminology, Warren might be called an “inoculation denier.” His pamphlet deployed the standard arguments against the practice, including religious considerations, contradicting Hartley at every turn. In the matter of the statistical dimension of Hartley’s argument, Warren took two approaches. First, he denigrated it altogether. Inoculation, in Warren’s acerbic commentary, had

daily sunk into disuse and contempt, and of late has scarce been mentioned amongst Us, till our Author on a reliance of his mathematical Skill, and a thorough Acquaintance with the Doctrine of Chances, undertook to strike a new Light, to open our Eyes, and by plain and easy Calculations, to evince the Reasonableness and Security of it, even to a Demonstration.

Hartley himself is sarcastically referred to as “our exact Numerist” and “a Man of . . . great Figures” who seeks to have “over-powered” his readers “by a Demonstration of Numbers.” These labels take on especially negative, anti-mathematical meaning in light of a quotation from the Italian physician and anatomist Giorgio Baglivi (1668-1707) that Warren cited toward the end of his pamphlet: “‘Mathematicks, Rhetorick, Arithmetic, &c. are as serviceable to an accurate History of Diseases, as the Art of Painting is to a Musician.’”41

Having belittled the data and their presenter, Warren next sought to show the actual shortcomings of Hartley’s and Jurin’s data and the virtues of his own. Even if Hartley’s computations from the data he used were “True and Just,” he had “partially represented” or entirely “suppressed” many facts and material circumstances that would change those data. Worse still, Jurin’s data, so important to Hartley’s case, could not be relied upon because his correspondents practiced a “disingenuous Industry to deceive him.” Then, notwithstanding his condemnation of Hartley’s logical posture in citing the Uxbridge outbreak data, Warren went on to cite his local data so as to undermine Hartley’s overall mortality ratio for natural smallpox. In Bury, Warren contended (without giving a source), a total of 1,683 people had been known to have contracted smallpox to date, of whom only 124 had died, for a mortality ratio not of 1:6 (as in Jurin’s data and Hartley’s pamphlet) but of 1:13. From these deaths, Warren contended, 79 ought to be subtracted for various reasons, particularly the poverty or advanced age of some victims, which then yielded an even lower mortality ratio of 1:27. Furthermore, violating once more his announced logical principle of not generalizing from single instances, he noted immodestly that in his own practice the mortality ratio was lower than 1:70, likely because of the excellent care and guidance his patients had received before they contracted the disease. In short, according to Warren, even if the mortality ratio among the inoculated was accepted to be 1:50, that ratio would be higher than the one his patients had experienced with proper preventive care against natural smallpox.42


Hartley: probability, case studies, and psychology        

Prima facie, this was a devastating critique of Hartley’s publication. Little wonder that Hartley’s friend William Warburton, later Bishop of Gloucester (and an eventual public supporter of inoculation), wrote privately that the Hartley pamphlet “quite spoiled his character here [i.e., Bury] as a writer.”43 As far as is known, Hartley did not again publish on inoculation, although he did send the pamphlet—“a little hasty incorrect paper,” as he then called it44—to Dr. Sloane. The two corresponded about the topic, revealing that Hartley eventually inoculated four people at Bury, who survived the treatment. His letter on the subject was read at a Royal Society meeting in December 1734.45 By then Hartley was turning his attention to other topics, including the treatment of rabies (then called “hydrophobia”) and eventually kidney stone, from which he himself suffered throughout his adult years. It was on his investigation and treatment of the stone that he made his name as a physician.

There is reason to think, however, that the publication for which he is most famous, Observations on Man (1749), was stimulated in a notable way by his encounter with inoculation-related statistics and his failed effort to promote the practice. This big book, which builds on both Lockean and Newtonian concepts and is a key document of the English Enlightenment, marks the earliest stages of a science of physiological psychology. The word “psychology” may be used here for the first time in its modern sense, when Hartley writes of “Medicine and Psychology, or the Theory of the human Mind, with that of the intellectual Principles of Brute Animals” as being parts of natural philosophy, which seeks to “decipher the Laws by which the external World is governed, and thereby to predict or produce such Phænomena, as we are interested in.”46 The Observations’ development of both the theory of association of ideas and also its connection with physical nerve impulses or “vibrations” had profound influence on a wide array of important thinkers during the next 75 years, including Joseph Priestley, Benjamin Rush, Samuel Taylor Coleridge, and James Mill.

Notwithstanding Hartley’s professional concentration on medicine and his research interest in the physiology of the nervous system, he had been deeply attracted from childhood to the study of moral philosophy and religion—as reflected in the very conception of Observations on Man. The book also exhibited to careful readers Hartley’s enduring interest in mathematics, which he cared about far above humanistic studies such as rhetoric and poetry. In the mid-1730s, he said during a conversation with James Byrom that all who “taste mathematics quit poetry and such nonsense for mathematics,” and that in the future “mathematics and natural philosophy will overbear oratory; man will never quit mathematics, so necessary to human life.”47 This orientation to mathematics likely owed much to Hartley’s experience with Saunderson, whose textbook on algebra he later was instrumental in publishing.48

Hartley’s mathematical interests included not only geometry, algebra, and calculus but also probability, the “doctrine of chances,” with which, as we have seen, he was familiar as early as the pro-inoculation pamphlet. The contents of Observations on Man make clear that he became even more knowledgeable about the topic in succeeding years. Saunderson himself may have spurred Hartley to explore probability in the first place: he is known to have brought to Cambridge in 1711 a copy of the initial, Latin edition of Abraham De Moivre’s Doctrine of Chances. Certainly, Hartley had read De Moivre (1718 English edition) for himself by the time he wrote the Observations on Man during the later 1730s, and there is evidence of his citing “a fair Calculation according to the Doctrine of Chances” in 1736.49 The sophistication of the relevant pages in Observations on Man makes it clear that he had a solid understanding of basic probability, and even that he was in touch with whoever it was that discovered Bayes’s Theorem.50

What propelled his deepening understanding of probability, and his desire to introduce it into a large book about the psychology of association, its physiology, and the moral and religious situation of man? The answer could well be inoculation and his experience with making a case for it in 1733. To be sure, the topic of inoculation comes up in Observations on Man only once, and then incidentally.51 But contextual issues in his discussion of medicine and probability suggest strongly that his personal experience with attempting to persuade people to make a medical-treatment decision underlay his longish discussions of the doctrine of chance.

For Hartley, the general problem faced by medicine (or “physic”) was “Having the Symptoms given, to find the Remedy.” In some instances, this problem could be solved “empirically and directly by the Histories of Distempers, and of their Cures.” In others, however, all the learning and experience of even the ablest physicians “either cannot find Histories sufficiently similar, or none where the Event [i.e., outcome] was successful.” Here the issue would need resolution “rationally and indirectly,” a method that he also called “explicit and scientifical.” When lacking “the most probable Method of explaining . . . the Symptoms of Distempers,” doctors must “fansy something in the place of these”; that is, “where Practice is silent, Physicians must and will have recourse to some Theory, good or bad,” and deductive reasoning from presumed explanation to physical effects will then take the place of pure inductive reasoning from those effects to their underlying causes.52 In most cases, “rational assent” to any proposition (including a proposed medical explanation of symptoms or cure) developed by either induction or deduction comes before “practical assent.” But in some cases the reverse occurs, whereby rational assent is “generated and cemented most firmly by the Prevalence of the Practical.”53 This reverse process “is particularly observable in the Regards paid to Medicines, i.e. in the rational and practical Assent to the Propositions concerning their Virtues”—a comment that surely reflected Hartley’s experience with both inoculation and treatments for calculi.

Hartley believed that the rules for ascertaining truth and advancing knowledge, and thereby reaching assent to or dissent from propositions, could be mathematized. The “great Business in all Branches of Knowlege [sic]” being to “reduce, unite, and simplify our Evidences,” it appeared to him “not impossible, that future Generations should put all Kinds of Evidences and Inquiries into mathematical Forms,” which would lead to the coalescence or integration of the various disciplines. For now, however, the science of probability or “doctrine of chances” offered an excellent beginning in that direction. It suggested that when the evidences for any proposition or purported fact were somehow dependent on one another, the probability of that proposition or fact was lower than in instances where the evidences were independent of each other. That is, wherever these evidences

be not necessary to support each other, but concur, and can, each of them, when established upon its own proper Evidences, be applied directly to establish the Proposition, Fact, &c. in Question, the Deficiency in the Probability of each must be very great, in order to render the Proposition perceptibly doubtful; and this holds so much the more, as the Evidences are more numerous.

If evidence-dependency makes probability weak, evidence-independency makes it strong. Hence, “a Report passing from one original Author through a Variety of successive Hands loses much of its Credibility, and one attested by a Variety of original Witnesses gains, in both Cases, according to the Number of successive Reporters, and original Witnesses.”54

Recognizing this distinction, and looking back at his pamphlet and Jurin’s work, Hartley might well have concluded that the apparent probability of their proposition—the advisability of inoculation—may have appeared comparatively weak to readers because its evidence, however imaginatively assembled through Jurin’s correspondence project, had passed from hand to hand to hand.  A few specific cases, mostly anonymized, were cited by Jurin and Hartley, of course, but the great bulk of cases was aggregated into a set of data. For all his logical shortcomings, Warren removed a step in this kind of process when he discussed the local cases with which he was personally familiar right in Bury, perhaps fostering thereby the believability of his dissent from Hartley’s pro-inoculation proposition.

At this early stage of data-use in medicine, a relative absence of case histories would have loomed large in the eyes of those judging the Hartley-Warren debate. Case histories had already enjoyed a long and significant impact on the evolution of medicine, and in the early modern era, as the medieval exemplum or casus was transformed into the historia, the careful recording of these particularized relations by doctors and their reading by other doctors became an expectation of the profession.55 Hartley surely recognized this fact, as the sections of his Observations on Man that discussed natural and civil history made clear. It is, he wrote, “a practical Error of great Importance” to assume that well-attested “historical Evidences are inferior to mathematical ones. They are equal, as far as we have any thing to do with them; i.e. can judge of them, or be influenced by them.”56

It was doubtless for this reason that Hartley shifted his approach when in 1739 he came to write his longest piece about the stone, promoting Mrs. Stephens’s treatment: it featured no data tables but rather 131 case histories (including his own), mostly written by the patients themselves or recorded by him from patients’ oral accounts, along with some already printed or “communicated to me by good hands.” Quite differently from Jurin’s procedure, all but two of these cases provided patient names and places of residence.57 As Hartley put it in a letter to one of the scholarly journals of the day, each case “singly taken, favours the dissolving Power of the [Stephens] Medicines; Three or Four taken together become a strong Presumption; but the united Force of the whole cannot be thought less than a full Proof, by those who know them accurately.”58 The comparative outcomes of his two different approaches to seeking assent would likely have confirmed his views: although he could not persuade Bury St. Edmunds to inoculate, his advocacy for Mrs. Stephens’s treatment for calculi helped instrumentally to yield (in 1740) a £5,000 bounty to her from donors and Parliament, by means of which she agreed to make public the formulation of her lithontriptic.59

Hence, despite his attraction to mathematics in general and probability in particular, Hartley was ready to countenance alternative forms of evidence as long as their individual elements did not depend heavily upon one another. For whether as data tables or as case history narratives, evidence that persuaded its beholder either to assent or to dissent always involved the association of ideas, that basic psychological process central to Hartley’s Observations on Man. Rational assent to a proposition, he wrote, proceeds “from a close Association of the Ideas suggested by the Proposition, with the Idea, or internal Feeling, belonging to the Word Truth; or of the Terms of the Proposition with the Word Truth.” When mathematics is involved, “it is mere Association again, which appropriates the Word Truth to the Coincidence of the Words, or Symbols, that denote the Numbers.” Indeed,

a mathematical Proposition, with the rational Assent or Dissent arising in the Mind, as soon as it is presented to it, is nothing more than a Group of Ideas, united by Association, i.e. than a very complex Idea. . . .  And this Idea is not merely the Sum of Ideas belonging to the Terms of the Proposition, but also includes the Ideas, or internal Feelings, whatever they be, which belong to Equality, Coincidence, Truth, and in some Cases, those of Utility, Importance, &c.60

The development of Hartley’s associationist psychology, therefore, was closely interlaced with his interest both in mathematical probability and also in issues of achieving assent—issues that arose when he attempted to persuade first his Bury fellow-townspeople and then potential donors and the English Parliament of the efficacy, respectively, of inoculation and of Mrs. Stephens’s lithontriptic. Here, it seems reasonable to conclude, the problem of interpreting inoculation data eventually helped lead to a view of assent that conformed to the psychology of association.


Portrait of Daniel Bernoulli who wrote on inoculation and data
Joh. Niklaus Grooth (1723-1797), Portrait of Daniel Bernoulli, 1790. Source.

D’Alembert versus Bernoulli: probability, psychology, and assent

A different but parallel development connected to smallpox, inoculation, data, and psychology took place in France some years later, this one involving another great and influential Enlightenment figure, Jean le Rond d’Alembert (1717-83). The illegitimate child of two aristocrats, his mother left him as a newborn at a Paris church, which presented him to a foundling orphanage, after which he was brought up by a nurse with whom his father placed him.  A first-rate education paid for by his father culminated in the receipt of a bachelor of arts degree in 1739 from the Jansenist Collège Mazarin, after which he studied law for two years. He then pursued his real passion, mathematics and the physical sciences, developing deep knowledge especially in mechanics and dynamics and publishing major works in these fields. In recognition of his intellectual accomplishments, he was elected to the Académie des sciences (1741) and eventually to the Académie française (1754). The well-roundedness of his knowledge, which included expertise in music theory, led to his participation in the famous salon of Mme. du Deffand. It brought him into touch with the philosophes, and later made possible his appointment as co-editor with Diderot of the great Encyclopédie. D’Alembert oversaw the mathematics and science component of the encyclopedic project and wrote hundreds of articles, some provocative, on an array of topics. His “Preliminary Discourse” to the Encyclopédie is considered to be among the most characteristic and revealing statements of the French Enlightenment.

In the Académie des sciences, d’Alembert became engaged in debates about probability, particularly with Daniel Bernoulli (1700-82), member of the famous Swiss mathematical family. Bernoulli had argued as early as 1738, in what has remained a renowned paper, that the then-prevailing rule for establishing risk in aleatory matters such as gambling worked only if the values calculated in it pertained not to price (which, presumably, is the same for everyone) but to the utility that the outcome in question yields. “Mathematicians evaluate money in proportion to its quantity, but in practice people with common sense evaluate money in proportion to the utility they can obtain from it,” he asserted. Indeed, in his view it was not realistic to measure the value of a risk without considering its utility, utility being related to the pleasure received—and therefore “dependent on the particular circumstances of the person making the estimate.” Accordingly, precise generalizations about categories of risks are implausible, “since the utility of an item may change with circumstances.” But it did seem plausible to Bernoulli to think of differential risk in terms of “imperceptibly small growth” in an individual’s wealth, which “proceeds continuously by infinitesimal increments,” thus allowing for the construction of a curve that could be used to analyze risk and reward. The changing area under the curve would reflect willingness to take risk, and also allow for measurement involving people who are more or less affected by a potential gain or loss, and whose anxiety about such loss is inevitably different.61

The introduction of individual circumstances into the mathematical analysis of risk was innovative and proved influential. By the time d’Alembert published in 1754 an Encyclopédie article on the game of chance Croix ou pile (Heads or Tails), he had given consideration to the issue raised by Bernoulli. He found himself in agreement that it would be desirable to introduce into the assessment of risk-taking “moral considerations,” such as players’ different wealth circumstances. For it is “certain . . . that of two men of unequal wealth who are playing a fair game following the ordinary rules, he who is the less rich risks more than the other.” Unfortunately, he believed, such considerations were “nearly impossible to submit to calculation because of the diversity of the circumstances,” as a result of which one must “employ abstraction and resolve the problems mathematically, while supposing moreover the moral circumstances to be perfectly equal. . . .”62

D’Alembert developed this idea within the context of inoculation in a 1761 paper on risk calculation. Here he observed that the two risks that were always compared actually had different temporal scope: “the risk of dying from inoculation, little as it is, is a present danger, and the risk of dying from natural smallpox (although greater) is a distant danger, which spreads itself over the whole period of life. . . .” This distinction holds obvious psychological importance to someone trying to make a decision about receiving inoculation. D’Alembert added to the resulting complexity of decision-making by contending as well that no one possesses sufficiently reliable knowledge to ascertain, even approximately, what risk he or she will run at each subsequent age of dying from natural smallpox in the course of life. Even if one could achieve the precision needed to answer such a question, how would one then incorporate into a total risk calculation (tallying every month’s perceived risk) the particular risks of times in the distant future? “A problem,” he admitted, “that appears to me to be insoluble, and whose solution moreover, were it possible, would be differently probable for each individual, considering the circumstances in which each finds himself.”63 Here we encounter in a different and even more complicated form the individuation of risk to which he and Bernoulli had already been pointing when they were thinking about games of chance.

As the debate between these two great mathematicians continued in the 1760s, it focused exclusively on smallpox and inoculation. A paper by Bernoulli, written in 1760 (without its existing introduction) but formally published only in 1766 by the Académie des sciences, explored in great detail the data for the risk of death by natural smallpox, compared with the risk of inoculation. He began with a contrast: if smallpox could be prevented, of a whole generation consisting of 13,000 children about 1,000 would be saved from death—but life expectancy in this group would be increased by an average of only two years. The first of these facts, he said, would interest many more people than would the second, because the former’s impact seemed more concentratedly immediate and more personal, whereas in the case of the latter the effect was distributed and impersonal. He could understand that ordinary people, “the vulgar,” would not be struck by the change in life expectancy, but he found it remarkable that elite people do wonder if it is worth the trouble of undergoing a procedure such as inoculation in order to prolong their lives, on average, by two years.64

In taking this tack, Bernoulli directly introduced the psychology of individual risk-taking into the consideration of a social problem that he considered truly important for the well-being of humanity, demanding all the knowledge that could possibly be brought to bear on it. He asserted that his paper did not aim at defending or recommending inoculation, but merely intended to “shed new light” on the issues—although he also said that what he knew about inoculation data “placed him on the side of those who thought the procedure to be strongly useful.”65 His paper did in fact augment the statistical examination of inoculation in important ways, partly by opening up consideration of the procedure’s net increases to life expectancy at particular ages (again examining the area under a curve, here constructed by data points for those ages, while using the assumption of a constant rate of mortality from smallpox). Partly, too, his paper’s contribution lay in showing that the existing mortality data differed among European cities, and that the French population would be substantially higher without smallpox deaths, yielding great advantages to the state and the French economy. The paper also argued vigorously for the future collection of data about what was for Bernoulli a key statistical desideratum: the ages at death of those who succumbed to smallpox. Even so, the sophisticated tables of data he presented went far beyond the simpler ones of Jurin and Scheuchzer in the 1720s.66

D’Alembert read Bernoulli’s paper in manuscript form, without the introduction that was added later. On November 12, 1760, he presented to the Académie des sciences his own paper on the analysis of inoculation and smallpox data. In a letter to Voltaire the next spring, he summarized the presentation as follows.

I have been forgetting to ask you if you have received a piece that I have written on inoculation, . . . in which I believe to have proved, not that inoculation is bad, but that its partisans have pretty badly reasoned to this point, and don’t suspect it. This piece, very clear as to what I believe, and very impartial, was read six months ago at a public assembly of the Académie des sciences, and appeared to me to have made a positive impression on the audience.67

He went on to note that “a doctor in Clermont en Auvergne, having inoculated his son, the son died from the inoculation, and the father died from remorse. This fact, if true, would be very detrimental to inoculation, although at bottom it would not be decisive.” Obviously, d’Alembert treaded carefully across this relation of an anecdote to his old friend and fellow-philosophic-warrior, inasmuch as Voltaire had been heartily pro-inoculation for decades. But the private comment underscored what Voltaire had known about him for some time, and what his remarks to the Académie and subsequent publications made plain: namely, that d’Alembert was (in today’s terms) an “inoculation skeptic.” His skepticism followed from his conviction that “those who have tried to calculate the advantages of inoculation, little skilled in analysis, are mistaken” in their “point of view on the question,” and this error, he believed, applied even to Bernoulli himself.68

He also faulted Bernoulli for taking a strong position on inoculation when the data remained inadequate, as Bernoulli acknowledged and regretted.69 The issue of reliable data bothered him more than it did his Swiss colleague. As d’Alembert put it in a later paper, “one has to date only very imperfect knowledge, a lack of sufficient facts and observations.” The key example he cited, as Bernoulli had done, was not knowing the annual smallpox mortality rates at particular ages. It was, he insisted, “absolutely contrary to unanimously recognized experience” to say that the mortality rate from smallpox was the same for every age cohort,” yet this is precisely what Bernoulli had done—along with everyone else to date.70

The issue of method dogged d’Alembert’s interest in probability analysis as much as did the issue of inadequate data. He averred that there were many relevant questions to be answered over which calculations might not in the end even “have any hold,” because one might “not be able ever to determine moral issues in a manner that is precise and rigorous.” Serious “doubts about the calculus of probability” being able to lead to conviction or assent in particular cases worried him, but he expressed publicly the hope that his reservations would lead other mathematicians to more sophisticated work.71

In any case, he pointed again and again to the psychological complexity of inoculation-related risk, and therefore to “considerations relative to the situation of individuals,” as in this long and revealing comment.

The difficulty of appreciating the disadvantage of succumbing to smallpox at a time more or less distant becomes all the greater if one considers that this consideration will be and ought to be quite different for each particular age, relative to one’s current age, to his situation, to his manner of thinking & feeling, to the needs his family, his friends, his fellow citizens could have for him. Imagine, for example, that it is announced to someone that if he doesn’t get inoculated, he will die from smallpox at the end of 20 years; it is certain that these 20 years of life of which he is assured could be for him, or appear to him, more or less advantageous relative to the circumstances in which he finds himself. . . . [T]here will not be perhaps two individuals who value this advantage the same way.72

He also wrote with some feeling about the psychology of inoculation decision-making by fathers and mothers. Such a decision, he wrote, “will depend not only on the degree to which [a father] loves his son, but on the manner in which he loves him, if as his son or as his heir; if by tenderness or only by duty; if for his well-being, or the well-being of the State: the decision will depend again on the circumstances in which this father finds himself as to his son. . . .”73 By this point, he could not have disagreed more with the position of a pro-inoculation commentator who wrote, “It is not at all here a moral question, it is a matter of calculation, let us not make a case of conscience out of an arithmetical problem.”74

Regarding individuals, d’Alembert also made the related political observation that others had, in his opinion, too much ignored. “The interest that the State in general has about Inoculation has been too much confounded with that of individuals; yet these two interests can be very different.” For example, what if the state were willing to consider sacrificing through inoculation one in five citizens in order to keep the other four healthy and vigorous down to the age of 100? The state inherently considers its citizens “indifferently,” d’Alembert contended, and therefore it could make this sacrifice without hesitation, whereas “for each individual, the interest of his self-preservation is the first of all.”75

D’Alembert continued to think and write about these topics in afteryears, publishing his reflections in his Mélanges de littérature. There he spoke of the debate over inoculation as having become “an affair of party,” with almost violent overtones, as contestation divided “enlightened men.” The “Adversaries of Inoculation call its partisans Murderers, and these treat their antagonists as bad Citizens.” D’Alembert wanted to show, as a skeptic, that inoculation had been “badly defended in certain respects, and more badly attacked in others.” Novice mathematicians (presumably including his old friend, Voltaire), he wrote, “are perhaps not sufficiently struck as they ought to be by the difficulty of this problem” of calculating comparative risk. They “believe that they are able to evaluate, at least approximately, the sum of the risks involved, by calculations founded on vague and gratuitous assumptions.”76 But the matter, he knew and said, was vastly more complicated when people in real life with real but differing psychological contexts had to assent to or dissent from a proposition involving risk—either that inoculation was comparatively safe or that it was comparatively dangerous.


Conclusion: “Uncertainties and Perplexities”        

Voltaire had thought that the case for inoculation was open and shut, a matter of science and numbers leading in one, and only one, enlightened direction. As the century wore on, the debate over this practice became increasingly complicated, as d’Alembert and Hartley illustrate. The pro-inoculation forces eventually won the day. But the deliberations and arguments of Enlightenment intellectuals like Voltaire, Hartley, Bernoulli, and d’Alembert remind us today that probability based on data is nearly always debatable, that reliable data and the tools for analyzing them are forever developing, and that data do not vitiate the importance of psychology in decision-making when a health threat confronts millions of people. As Hartley put it, “the application of mathematical Knowlege [sic] is just as much exposed to the several Kinds and Degrees of Uncertainty, as that of any other. . . .  [I]n the Theories of Chemistry, of manual Arts and Trades, of Medicine, and, in general, of the Powers and mutual Actions of the small Parts of Matter, the Uncertainties and Perplexities are as great, as in any Part of Science.”77 They remain so even now.



  1. Voltaire, Lettres philosophiques, ed. Gustave Lanson, 2nd edn., 2 vols. (Paris, 1915), 1:134-36. In his commentary on this chapter of Voltaire’s book, Lanson appropriately contends that the author must have meant one-fifth of the 60 percent who contract smallpox, not one-fifth of the entire population of the globe:  Lettres philosophiques, 1:147 n. 27. Thus, the mortality rate from smallpox would be about 12 percent, in Voltaire’s intended terms.
  2. Arthur H. Rowbotham, “The ‘Philosophes’ and the Propaganda for Inoculation of Smallpox in Eighteenth-Century France,” University of California Publications in Modern Philology, 18:4 (1935), 265-90; Theodore Besterman, Voltaire, 3rd edn. (Chicago, 1976), 542. The descriptions of Condamine come from Louis Petit de Bachaumont, Mémoires secrets de Bachaumont (Paris, 1874), 139, 127 (entries for March 24, 1765 and October 19, 1764). To be sure, not every philosophe was a proponent of inoculation: the physician and advanced thinker Julien Offray de la Mettrie was opposed to it: John McManners, Death and the Enlightenment: Changing Attitudes to Death among Christians and Unbelievers in Eighteenth-Century France (Oxford, 1985), 46.
  3. Voltaire to Jean le Rond d’Alembert, May 4, 1759, Digital correspondence of Voltaire, letter D8286.
  4. Voltaire to d’Alembert, Mar. 15, 1769, and d’Alembert to Voltaire, Aug. 7, 1763, Digital correspondence of Voltaire, letters D15516 and D11345.
  5. [François-Jean de Chastellux], Réponse à une des principales objections qu’on oppose maintenant aux Partisans de l’Inoculation de la petite Vérole, 2-3, printed in Angelo Gatti, Réflexions sur les préjugés qui s’opposent aux progrès et à la perfection de l’inoculation (Brussels, 1764); Collini to Voltaire, March 27, 1760, Digital correspondence of Voltaire, letter D8822; Théodore Tronchin, “Inoculation,” in Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des métiers, 17 vols. (Paris, 1751-57, and Neuchâtel, 1765), 8:767. Ironically, early opponents of inoculation in England noted that unlike the English, the French (and Italians) had “never swallowed the Bait” of knowledge of inoculation in Turkey: William Wagstaffe, A Letter to a Freind [sic]; Shewing The Danger and Uncertainty of Inoculating the Small Pox, 2nd edn. (1722; London, 1722), 37-38.
  6. John McManners, Church and Society in Eighteenth-Century France, 2 vols. (Oxford, 1998), 1:306; idem, Death and the Enlightenment, 46.
  7. Anon., L’Inoculation de la Petite Vérole Déferée à l’Église et aux Magistrats (Paris, 1756), 60-72.
  8. Edmund Massey, A Sermon against the Dangerous and Sinful Practice of Inoculation (London, 1722), 15, 27, 29. There certainly were English clergy who supported inoculation, including David Some as early as the mid-1720s: The Case Of Receiving Small-Pox by Inoculation, ed. Philip Doddridge (written 1725; London, 1750).
  9. Rowbotham, “The Philosophes and the Propaganda for Inoculation,” 271-77; Elise Lipkowitz, “The Physicians’ Dilemma in the 18th-Century Smallpox Debate,” Journal of the American Medical Association, 230:17 (November 5, 2003), 2329:1-2330:2; Charles-Marie de la Condamine, “Suite de l’histoire de l’inoculation de la petite vérole, depuis 1758 jusqu’en 1765,” Histoire de l’Académie royale des sciences (Paris, 1765), 505-32. The Société Royale de Médicine, a less ancient and more progressive group than the Faculté, was favorably disposed to inoculation, on the whole. It tended to have more youthful members and was in competition with the Faculté, which has been described as “an intellectually narrow and moribund scientific institution.” Only a few philosophes, such as the physician Jean Darcet, were members of both groups: Alan Charles Kors, D’Holbach’s Coterie: An Enlightenment in Paris (Princeton, 1976), 247, 220.
  10. Catherine II to Voltaire, December 28, 1768, Digital correspondence of Voltaire, letter D15396.
  11. Jurin to Voltaire, Feb. 1, 1740 and June 23, 1741, Digital correspondence of Voltaire, letters D2157 and D2504. Voltaire’s membership in the Royal Society came in 1743.
  12. Review of Jean de la Coste, Lettre sur l’inoculation, Mercure (November 1723), 943-49; Mémoires pour l’Histoire des Sciences et des beaux Arts [Journal de Trévoux], 24:6 (Juin 1724), 1073-90 (article lii; for Jurin, 1084-85).
  13. Bibliothèque Angloise, 11:1 (Jan. 1724), 275-77; Michel de la Roche, Mémoires Littéraires de la Grande Bretagne, 11 (Jan.-June 1724), 271-92, 293-94 (for statistics, 294), and 12 (July-Dec. 1724), 487-505 (for statistics 494).
  14. Lanson speculates on Voltaire’s relevant reading in Lettres philosophiques, 138-39.
  15. James Jurin, “A Letter to Dr. Caleb Cotes-Worth . . . Containing a Comparison between the Danger of the Natural Small Pox, and of That Given By Inoculation,” Philosophical Transactions of the Royal Society, 32:374 (1722-23), 214-16. The fullest account of Jurin’s project is given in Andrea A. Rusnock, “The Weight of Evidence and the Burden of Authority: Case Histories, Medical Statistics and Smallpox Inoculation,” in Roy Porter, ed., Medicine in the Enlightenment (Amsterdam, 1995), 289-315.
  16. Jurin, “Letter to Dr. Caleb Cotes-Worth,” 222.
  17. James Jurin, An Account of the Success of Inoculating the Small-Pox in Great-Britain, for the Year 1726. With A Comparison between the Miscarriages in that Practice, and the Mortality of the Natural Small-Pox (London, 1727), Advertisement (unpaginated, at end of pamphlet).
  18. Bibliothèque Angloise, 11:1 (Jan. 1724) 276-77.
  19. Jurin, “Letter to Dr. Caleb Cotes-Worth,” 214-15, 217-19, 223-24.
  20. Idem, Account of Success of Inoculating 1726, 16-21, 24.
  21. Rusnock, “Weight of Authority,” 289. On smallpox and inoculation data collection and analysis in general for the eighteenth century, see Edward Huth, “Quantitative evidence for judgments on the efficacy of inoculation for the prevention of smallpox: England and New England in the 1700s,” Journal of the Royal Society of Medicine, 99 (May 2006), 262:-66:2.
  22. [John Arbuthnot], Mr. Maitland’s Account of Inoculating the Small Pox, 2nd edn. (London, 1722), 18-21.
  23. “Part of a Letter from Dr. Nettleton, Physician at Halifax, to Dr. Jurin, R.S. Secr[.,] concerning the Inoculation of the Small Pox, and the Mortality of that Distemper in the natural Way” (Dec. 16, 1722), Philosophical Transactions, 32:374 (1722-23), 211-12. On Nettleton, see Arthur Boylston, “Thomas Nettleton and the dawn of quantitative assessments of the effects of medical interventions,” Journal of the Royal Society of Medicine, 103:8 (August 1, 2010), 335-39.
  24. John Gasper Scheuchzer, An Account of the Success of Inoculating the Small-Pox in Great Britain, the Years 1727 and 1728 (London, 1729), 11, 13-25, 27, 37.
  25. Isaac Massey, A Short and Plain Account of Inoculation, 2nd edn. (1722; London, 1723), 21. Edmund Massey was the nephew of Isaac (Boylston, “Nettleton and the dawn of quantitative assessments,” 337), and he went to Cambridge from Christ’s Hospital in 1708. Edmund explicitly mentioned Isaac and the Short and Plain Account of Inoculation, in his A Letter to Dr. Maitland, in Vindication of the Sermon against Inoculation (London, 1722), 22 n. The Christ’s Hospital connection also involved James Jurin, who was educated there before going on to Cambridge in 1702, and who later served as a tutor to a Christ’s Hospital scholar.
  26. Idem, A Letter to the Learned James Jurin, D.D. R.S. (London, 1723), 1-3, 5, 9, 14, 16.
  27. Idem, Remarks on Dr. Jurin’s Last Yearly Account of the Success of Inoculation (London, 1727), 3, 6-7.
  28. Argument over data by the public as well as the mathematically sophisticated continued into the 1770s:  M.A., “Interesting Observations on the Increase of the Small Pox” (December 29, 1774), Gentleman’s Magazine, 44 (1774, Supplement), 612:1-2; Richard Price to William Adams, Feb. 14, 1776, in the Correspondence of Richard Price, ed. Bernard Peach and D.O. Thomas, 3 vols. (Durham, N.C., 1983-94), 1:242 and n.6.
  29. Martha Ellen Webb, “The Early Medical Studies and Practice of Dr. David Hartley,” Bulletin of the History of Medicine, 63:4 (Winter 1989), 628.
  30. Curiously, Saunderson himself is said to have been taught mathematics at an early age by two men, one of whom was Thomas Nettleton: Boylston, “Nettleton and the dawn of quantitative assessments,” 338; John Crabtree, A Precise History of the Parish and Vicarage of Halifax (Halifax, 1836), 278.
  31. Mary Hartley to William Gilpin, 1796, in Rebecca Warner, ed., Original Letters (Bath, 1817), 106.
  32. By far the best account of his medical background and the beginning of his practice is Webb, “Early Medical Studies and Practice of Hartley,” 618-36, but see also Richard C. Allen, David Hartley on Human Nature (Albany, N.Y.), 53, 72, 76, 412-13 n. 26; [David Hartley, Jr.], “A Sketch of the Life and Character of Dr. Hartley,” in Herman Andrew Pistorius, Notes and Additions to Dr. Hartley’s Observations on Man, 3rd edn. (London, 1801), 1:5-19; Mary Hartley to William Gilpin, 1796, in Warner, ed., Original Letters, 108-09; Richard Parkinson, ed., The Private Journal and Literary Remains of John Byrom [pub. as vols. 32, 34, 40, and 44 of The Chetham Society, Remains Historical and Literary], 2 vols. in 4 parts (Manchester, 1854-57), 2:1:104 (entry for April 4, 1737).
  33. Webb, “Early Medical Studies and Practice of Hartley,” 631-32; Allen, David Hartley, 411 n. 21; David Hartley, Reasons Why the Practice of Inoculation Ought to be introduced into the Town of Bury at Present (Bury St. Edmunds, 1733), 5, 13.
  34. Hartley, Reasons Why Inoculation, 1, 6.
  35. Ibid., 3-4, 6.
  36. Ibid., 7-15.
  37. Ibid., 16-18.
  38. Ibid., 7, 20.
  39. ACAD: A Cambridge Alumni Database, s.v. “Martin Warren,” https.//
  40. Martin Warren, An Answer to a Pamphlet, Entituled, Some Reasons Why the Practice of Inoculation Ought to be introduced into the Town of Bury at Present (Bury St. Edmonds, 1733), 5, 4, 20, 8, 23, 15.
  41. Ibid., 3, 16, 4, 7, 25.
  42. Ibid., 7, 9-11.
  43. Warburton to Dr. Robert Taylor, May 27, 1737, as quoted in Webb, “Early Medical Studies and Practice of Hartley,” 633 n. 132.
  44. Hartley to Sloane, January 27, 1733, as quoted in Webb, “Early Medical Studies and Practice of Hartley,” 633 and n. 136.
  45. Webb, “Early Medical Studies and Practice of Hartley,” 635.
  46. David Hartley, Observations on Man, His Frame, His Duty, and His Expectations, 2 vols. (London, 1749), 1:354 (Prop. 88).
  47. Byrom, Private Journal and Literary Remains, 2:1:109-10 (entry for April 15, 1735).
  48. Hartley to John Lister, June 5, 1738 and April 20, 1741, in “The Correspondence of Dr. David Hartley and Rev. John Lister,” Transactions of the Halifax Antiquarian Society, [vol. 32] (Oct. 4, 1938), 238-39, 260.
  49. Stephen Stigler, “Who Discovered Bayes’s Theorem?,” American Statistician, 37:4:1 (Nov. 1983), 294:1; Hartley, Observations on Man, 1:338 (Prop. 87); Hartley to Lister, May 15, 1736, Calderdale Archives, Calderdale Central Library, Halifax, as quoted in Allen, David Hartley, 43.
  50. Hartley, Observations on Man, 1:335-39 (Prop. 87).
  51. Ibid., 1:136 (Prop. 29)
  52. Ibid., 1:264, 266 (Prop. 78).
  53. Ibid., 1:324-24, 330 (Prop. 86).
  54. Ibid., 1:344, 351, 335-36 (Prop. 87).
  55. Gianna Pomata and Nancy G. Siraisi, “Introduction,” Historia: Empiricism and Erudition in Early Modern Europe (Cambridge, Mass., 2005), 14, 22-23; Pomata, “Praxis Historialis: The Uses of Historia in Early Modern Medicine,” in Historia, 112-13, 122-23; Chiara Crisciani, “Histories, Exempla, and Anecdotes: Michele Savonarola from Latin to Vernacular,” in Historia, 310.
  56. Hartley, Observations on Man, 1:363 (Prop. 88).
  57. David Hartley, A View of the Present Evidence For and Against Mrs. Stephens’s Medicines, as a Solvent for the Stone. Containing A Hundred and Fifty-five Cases. With Some Experiments and Observations (London, 1739), esp. vii and “Advertisement”.
  58. Idem, “An Account of the Contribution for making Mrs. Stephens’s Medicines public; with some Reasons for it, and Answers to the most remarkable Objects made against it,” History of the Works of the Learned, 1 (June 1738), 448.
  59. Allen, David Hartley, 54-64; John Byrom to Mrs. John Byrom, June 14, 1739, in idem, Private Journal, 2:1:244. This venture of Hartley’s saw him propose what we would call clinical trials of Mrs. Stephens’s formula, which did occur in 1739. See Hartley, “Account of the Contribution for making Mrs. Stephens’s Medicines public,” 444. Hartley contended that, in order to benefit the suffering as expeditiously as possible, “if these medicines should be effectual, their Publication ought not to be deferred a Moment longer than necessary”:  ibid., 451.
  60. Hartley, Observations on Man, 1:324, 326, 328 (Prop. 86).
  61. Daniel Bernoulli, “Exposition of a New Theory on the Measurement of Risk,” trans. Louise Sommer, Econometrica 22:1 (Jan. 1954), 24, 33, 25-26 (secs. 3-5, 19, 5-6).
  62. Jean le Rond d’Alembert, “Croix ou pile,” Encyclopédie, 4:513.
  63. Jean le Rond d’Alembert, “Sur l’application du Calcul des Probabilités à l’inoculation de la petite Vérole” (“Onzième mèmoire”), in Opuscules mathématiques, ou Mémoires sur différens sujets de géometrie, de méchanique, d’optique, d’astronomie, 8 vols. (Paris, 1761-80), 2:30, 56. D’Alembert returned to the issue of the calculation of comparative risk going forward from a particular age, in this case 30, in a later publication, once again insisting how hard it is to make such a calculation: Jean le Rond d’Alembert, “Doutes & questions sur le calcul de probabilités,” in Mélanges de Littérature, d’Histoire, et de Philosophie, 5 vols. (Berlin and Amsterdam, 1753-67), 5:322.
  64. Daniel Bernoulli, “Essai d’une nouvelle analyse de la mortalité causée par la petite Vérole, & des avantages de l’Inoculation pour la prévenir,” Mémoire de Mathématique et de Physique, in Histoire et Mémoires de l’Académie des sciences (Paris, 1766), Part 2, 2.
  65. Ibid., 43, 8.
  66. Ibid., 8, 14, 22, 26, 4, 44-45. On Bernoulli’s views, see Lorraine Daston, Classical Probability in the Enlightenment (Princeton, 1988), 83-85. My thinking about the issues in the debate between Bernoulli and d’Alembert is framed largely on the basis of Daston’s valuable analysis.
  67. D’Alembert to Voltaire, April 9, 1761, Digital correspondence of Voltaire, letter D9731.
  68. D’Alembert, “Avertissement,” in Opuscules mathématiques, 4:ix.
  69. For his comments to a December 1762 session of the Académie, see Daston, Classical Probability, 84-85.
  70. D’Alembert, “Doutes & questions sur le calcul des probabilités,” 5:322, 336; idem, “Vingt-troisième Mémoire,” eighth letter, Opuscules mathématiques, 4:102.
  71. D’Alembert, “Vingt-septième Mémoire,” sec. I, and “Vingt-troisième Mémoire,” sixth letter, Opuscules mathématiques, 4:307-08, 91. See also Keith Michael Baker, Condorcet: From Natural Philosophy to Social Mathematics (Chicago, 1974), 174-76.
  72. Jean le Rond d’Alembert, “Doutes & questions sur le calcul de probabilités,” 5:325-26.
  73. Ibid., 5:361-62.
  74. Anon., L’Inoculation de la Petite Vérole Déferée, 113-14.
  75. D’Alembert, “Doutes & questions sur le calcul de probabilités,” 5:345-46.
  76. Ibid., 5:307-09.
  77. Hartley, Observations on Man, 1: 363-64 (Prop. 88).



DAVID SPADAFORA is a historian of modern European thought and a retired professor and academic administrator. Most recently, he was President of the Newberry Library in Chicago, Illinois.


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