Thinking in terms of number, weight and measure

The needs of trade

Producers and merchants were the first concerned with the need for reliable data, quan­titative but also qualitative: Le parfait Negotiant, ou Instruction generale pour ce qui regarde le commerce de toute sorte de marchandises, tant de France que des pays estrangers, for example, by Jacques Savary (1622-1690), first published in 1675, was re-edited many times until 1800.

However, these developments did not always happen smoothly and they encoun­tered a number of social and political difficulties. Linked to the lasting controver­sies about usury, for example, some contracts - especially life annuities - were suspected by the Church of concealing some form of usury. But one of the most striking facts was the reluctance of the clergy to accept insurance: gambling was condemned by the Church and insurance was equated with a game of chance.

In some nations, these kinds of insurances are permitted^[117]... but among us all these agreements are unlawful, condemned and against morality, from which an infinity of abuses and deception would ensue; for who can vouch for other people’s lives? There is no promise nor remedy for fate and death.

Echoes of the prohibition were still to be found decades later. The lawyer Balthazard- Marie Emerigon (1716-1784), for example, in his 1783 Traite des assurances et des contrats a la grosse, stressed that life insurance is not genuine insurance: it is only a bet on people’s lives. “Man is priceless.... The life of a man is not an object of trade; and it is odious that his death becomes the matter of a mercantile speculation

And this kind of bet...

The needs of politics

The Government, too, felt a strong need for reliable data. To various degrees in history, princes have been eager to obtain precise knowledge of the resources of their countries, both for fiscal and for domestic and international political reasons. That need, together with the constant need to raise loans and manage a growing public debt, proved fundamental during the formation and expansion of modern States. The “number of inhabitants”^[119] - the word “population” only re-emerged in France in the mid-eighteenth century (There and Rohrbasser 2011) - their wealth and income, the level of economic activity in different sectors and the techniques employed therein, the state of the balance of trade, and all other such data were essential for good economic policymaking but also, on the international stage, for expressing the political strength of the monarchy. This is the reason why the state of the public finances and the population size were supposed to be confidential data, in order to avoid disclosing the real forces of the country to foreign States, whether allies or, above all, enemies.

With the development of economic activities and the problems posed by eco­nomic policymaking, the prohibition on public discussion of these matters was, however, less and less respected. Authors of the growing literature on the sub­ject had to produce their own data. At the end of the seventeenth century and the beginning of the eighteenth, some prominent authors like Pierre Le Pesant de Boisguilbert (1646-1714) and Sebastien Le Prestre de Vauban (1633-1707) used more (Vauban) or less (Boisguilbert) serious empirical investigation to support their views and proposals.^[120] After the death of Louis XIV in 1715 and the hectic monetary and fiscal events during the Regency of Philippe d’Orleans - John Law’s mon­etary “experiment” - the search for economic and social information produced a rapidly expanding literature (There 1998).

The quantitative aspect of the quest must not, however, conceal the qualitative side: Franςois Quesnay (1694-1774) and A. R. J. Turgot (1727-1781) themselves paid due attention to it - the first with the Questions interessantes sur la popu­lation, l'agriculture et le commerce proposees aux Academies et autres Societes savantes des Provinces, which he published in 1758 with Etienne-Claude Marivetz (in Quesnay 2005, 334-87); and the second with the long series of detailed ques­tions he asked two young Chinese visitors in 1766, in order to learn more about the state of the Chinese economy and society (in Turgot 1913-23, II, 523-33). But quantification predominated. For some, it was a kind of new Grail, initiated abroad by authors like John Graunt (1620-1674) and William Petty and called “Political Anatomy” and “Political Arithmetick” by the latter. Denis Diderot (1713-1784), in the Encyclopedie, ou Dictionnaire raisonne des sciences, des arts et des metiers, adapting the entry “Political arithmetic” of Ephraim Chambers’s Cyclopaedia, stressed the importance of this approach.

Political arithmetic... aims at investigations useful to the art of governing peoples, such as those about the number of men who inhabit a country, the quantity of food they need, the work they can do, their lifespan, the fertility of land, the frequency of shipwrecks, etc. One easily understands that from these and many other similar discoveries, acquired by calculations based on some well ascertained experiences, a skilled minister would draw many consequences for the perfection of agriculture, for domestic as well as for

foreign trade, for the colonies, for the circulation and use of money, etc....

Sir Petty, English, was the first to publish essays under this title.

(Diderot 1751, 678)

Some years later, Franςois Veron de Forbonnais (1722-1800), in a polemical stance against the physiocrats, insisted that the knowledge of such data is essential to avoid erroneous theoretical speculations and their damaging consequences for economic policy: “the false philosophy generalises everything; the observation of facts is neglected” and the public good, which is the aim, “is definitively lost” (1767, I, iii).

But obtaining quantitative information was certainly no easy task. From the end of the seventeenth century to the First Empire, the history of data collecting (Gille 1980, Chapter 1) is a long enumeration of attempts implemented at various levels (local, provincial and, in the end, national), without really reliable results due to the great diversity of the provinces, the fragmentary character of the data and the bad or good will of those who were supposed to obtain and convey the requested information. Nor was it easy for the local collectors to obtain this information: peo­ple were reluctant to disclose it, most of the time supposing that it was requested for fiscal or military purposes. Consequently, the administration, dealing with local data, tried to generalise the results to more extended areas in a more or less ques­tionable way. Towards the end of the Ancien Regime and during the Revolution, however, more serious attempts were made, for example, to estimate the size of the population or the national income of the country. For the latter, the most relevant was made by the chemist and social scientist Antoine Laurent Lavoisier (1743­1794) in his Resultats extraits d'un ouvrage intitule: De la richesse territoriale du royaume de France, 1791 (Perrot 1988, 2003). But all this was only a first and very preliminary step in building and publishing official statistics. In the meantime, one of the main subjects of the time, public finances, had been under scrutiny and Jacques Necker (1732-1804), a Swiss banker, had caused a sensation when, as the Finance Minister of Louis XVI, he published his Compte-rendu au Roi on the state of the budget - one of the causes of his disgrace and which led him to publish in 1784 De l'administration des finances de la France, a detailed three-volume book on public finances, economic policy and plans for reform.

Finally, a kind of cost-benefit calculation was used by many authors in different fields, from the construction of roads and canals by the Ingenieurs des Ponts et Chaus- sees to the management of poverty relief and charity (Joel 1984; Etner 1987, Chap­ters 2 and 3) and even slavery (Oudin-Bastide and Steiner 2015).

A strategic variable: population

Perhaps the field where calculations proved to be relatively less difficult - but widely discussed - was what was later to be called demography.^[121] It was considered of the utmost importance, and some authors held it to be the basis of political arithmetic itself. The chapter on political arithmetic that Jean-Etienne Montucla (1725-1799) added to the 1778 reissue of the successful Recreations mathema- tiques et physiques by Jacqes Ozanam (1640-1718) is entirely dedicated to ques­tions of population, with some brief questions about life annuities - all in the form of problems to be solved by the reader.

Since politics has been enlightened as to what constitutes the real strength of states, much research has been done on the number of men in each country, in order to determine their populations. Moreover, as nearly all governments have been obliged to borrow large sums of money, mainly in the form of life annuities, it has been natural to examine the progression of mortality in the human race, in order to proportion the interest on these loans to the prob­ability of the annuity’s extinction. It is to these calculations that we give the name Political arithmetic.

(Montucla 1778, 245)

While a census of the population was conceivable, it was in fact unworkable, except at a local level. Fortunately, some data existed in the form of registers of births (baptised people), marriages and deaths (burials) managed by the clergy in the parishes, a copy of them being supposedly transmitted since 1736 to the royal administration. Supposing that they were reliable - this was not always the case - they could be used to calculate the total number of inhabitants of a town, a province or the whole country through the indirect method of a “universal multiplier” based, for example, on the number of “feux” (dwellings) but usually on the number of births. Once a census had been made in a carefully chosen sample of parishes or towns, the division of the number of inhabitants by the number of births in these places during the same period gave the value of this multiplier. It was then easy to approximate the size of the population of a province or of the realm by applying this multiplier to the number of births in these areas. The method was, however, far from perfect because of the inaccuracy of some data, the possible variations in the average lifespan in different places, the different local behaviours of people towards marriage and the size of the family or the local flows of emigration or immigration - the multiplier could thus have different values in different places. It is true that, at that time, there was a widespread belief in the uniformity of the laws of fecundity and mortality, and in any case, the universal multiplier method was the only practicable one. In this context, in August 1772, the Comptroller General Joseph Marie Terray launched the first national survey, the aim of which was to know, every year, the total number of baptisms, marriages and burials in the realm.

The case of population provides us with a celebrated example of erroneous beliefs generated by a lack of reliable data. During the Ancien Regime, and in the eighteenth century in particular, a great number of authors were convinced that the population of France - but also of the world - was dramatically decreasing, espe­cially if it was compared with the number of people who were supposed to have lived during Greco-Roman antiquity, inferred from ancient texts. Yet the popula­tion was growing and the world had never been so populous! To various degrees, this depopulation thesis was shared by prominent authors like Quesnay and Victor Riqueti de Mirabeau (1715-1789), and it was also expressed in the Encyclopedie. There were, however, a few dissenting views, including that of Voltaire, for exam­ple. The most striking statement of the thesis is certainly due to Charles-Louis de Secondat de Montesquieu (1689-1755) in his most celebrated books: Lettres persanes (1721, Letters 112 to 122) and De l'esprit des lois (1748, Book XXIII).

Thou hast perhaps not considered a thing which is a continual subject of wonder to me. How comes the world to be so thinly peopled, in comparison to what it was formerly? How hath nature lost the prodigious fruitfulness of the first ages?... According to a calculation, as exact as can be made in mat­ters of this nature, I find there is hardly upon the earth the tenth part of the people that there was in ancient times. And what is very astonishing, is, that it becomes every day less populous: and, if this continues, in ten ages it will be no other than a desert. This is... the most terrible catastrophe that ever happened in the world.

(1721 [1964], 121)

How could this supposed depopulation be explained? In France, some contemporary factors, advanced by Quesnay, were the incessant wars, and some major political events such as the 1685 Revocation of the Edict of Nantes (Quesnay [1757-58] 2005, 259-60). Another factor lay in the way of life, preferences and tastes of the people, especially the wealthy classes, which played a part in their attitude towards family (Montesquieu [1748] 1964, 687). But, Montesquieu asserted, among the most pow­erful factors were the nature of the political regime and the religion prevailing in it (Faccarello 2020, 89-90). In this perspective, the conviction shared by many authors was that the decline of the population was the obvious sign of a bad government and of the necessity for reforms. It seems, however, that the origin of the depopulation the­sis - in addition to the political conviction that it should necessarily be so because of the nature of the government - lies in books published some decades earlier (Ducreux 1977; Dupaquier and Dupaquier 1985, 108-9) by Iustus Lipsius^[122] (Joost Lips, 1547­1606), Giovanni Battista Riccioli^[123] (1598-1671) and Isaac Vossius^[124] (1618-1689), in which, based on a misinterpretation of some texts from Roman antiquity, the popula­tion of the Roman Empire and the city of Rome was tremendously overestimated (the population of Rome was 9.37 million according to Riccioli and 14 million accord­ing to Vossius), and that of France sometimes hugely underestimated (5 million for Vossius, Riccioli being closer to the reality with 20 million).

A century later, authors were still dealing with these sorts of numbers and the many sources drawn from antiquity and accounts of travels. Etienne Noel d’Amilaville (1723-1768), the author of the (long) entry “Population” in the Encyclopedie, is a good example. Examining many of the debates about the size of the population, he concluded that they are blind alleys: it was high time to drop calculations - “based on too fanciful assumptions” - and “to speak philosophi­cally” (1765, 89). With the conclusion that, for him, the population of the world had always been... constant.

From these principles it follows that the population in general must have been constant, and that it will be so to the end; that the sum of all men taken together is equal today to that of all the periods that we may choose in antiq­uity... and to what it will be in the centuries to come; that finally, with the exception of those terrible events in which plagues have sometimes devas­tated nations, if there have been times when more or less scarcity has been noted in the human race... it is not because its totality has diminished, but because the population has changed place, which makes the decreases local.

(d’Amilaville 1765, 91)

Advances in calculation

It was thus obvious that some serious investigations had to be undertaken. Many attempts were made, the most accurate and successful being those of Jean-Joseph Expilly (1719-1793), Louis Messance (1734-1796) and Jean-Baptiste Moheau (1745-1794). Expilly, in the six volumes of his Dictionnaire geographique, his- torique etpolitique des Gaules et de la France, published between 1762 and 1770 (Expilly 1762-70), based his research on some partial official inquiries and used the method of the multiplier: first a multiplier calculated on the supposed number of “feux” in the country, then on the number of baptisms. With this latter and more accurate method, and with a birth multiplier of 25, he found that the population size was over 24 million. The depopulation thesis was thus radically contested - although without immediate effect, except that Expilly, criticised, lost the financial support of the government and could not publish the 7th and final volume of his Dictionnaire (see, however, Expilly 1780). With a more refined methodology, the works by Messance (1766, 1788) (with the collaboration of Jean-Baptiste-Franςois de La Michodiere) reached approximately the same conclusion. Moheau’s estima­tions, in his 1778 Recherches et considerations sur la population de la France (with the collaboration of Antoine-Jean-Baptiste de Montyon), often considered as the first treatise of demography, were similar.^[125] Some other writings are not to be neglected either, like the “Memoire sur la population de Paris et sur celle des provinces de la France. Avec des recherches qui etablissent l’accroissement de la population de la capitale et du reste du royaume” by Jean-Franςois Clement Morand (1726-1784) - a physician and scientist who was in charge of the question of population at the Academie royale des sciences - published in 1782 in the Histoire de l’Academie for 1779 (Morand 1782).¹4

Montesquieu and Wallace stated that the human species had decreased since the times that we call ancient. Hume and Voltaire argued the other side; and, against or in favour of the augmentation of the population in Europe since the beginning of this century, we find an almost equal number of authorities... M. Morand... believes he can conclude that the population of France has considerably increased over the past forty years...; after reading his work, it is difficult not to be of the same opinion.

(Condorcet, report on Morand, in Condorcet 1994a, 143)

But these developments in data collection and calculations still left several pend­ing problems linked to the determination of the universal multiplier. In a polemical exchange with Moheau, published in 1778 in Mercure de France, M.J.A.N. Caritat de Condorcet (in Condorcet 1994, 131-4) pointed out two of them: (1) even if we believe that the ratio between the number of births and the population is a general law of nature, the way in which the partial census - on which the multiplier is based - is carried out deserves much more attention as to the number and quality of places used than it was possible to implement at that time; (2) in these circum­stances, in all probability, the size of the population determined in this way deviates from reality: with the implicit suggestion to calculate this probability.

On the first point, the prominent Swiss mathematician Leonhard Euler already had a radical attitude. In 1767, he published in French two short texts in the Histoire de l’Academie royale des sciences et belles-lettres of Berlin for the year 1760: “Recherches generales sur la mortalite et la multiplication du genre humain” and “Sur les rentes viageres”.^{[126] [127] Euler did not believe in universal laws of fecundity and mortality. Noting that, in different places, the registers of births and deaths at each age usually give different information, he proposed to follow a general approach independent of data, that is, to determine some mathematical formulas which could then be applied to local data to obtain answers to questions about population:}

these registers differ widely according to the diversity of the towns, villages and provinces... and for this same reason the solutions to all these questions differ widely according to the registers on which they are based. This is the reason why I intend to deal here in a general way with most of these ques­tions without limiting myself to the results that the register of a certain place provides: it will then be easy to apply [the formulas found in the general investigation] to any place that we want.

(Euler 1767a, 144)

The analysis then proceeds in terms of simple algebra, and formulas are found to answer such questions as: “Given a number of men, all of the same age, find how many will still probably be alive after a number of years”, “find the probability that a man of a certain age will be still alive after a number of years” or “for given hypotheses of mortality and fecundity, together with the total number of people alive, find how many of them will be of each age.” The approach is further devel­oped in the memoir on life annuities (1767b).

But another notable development took place one decade after Terray launched his large-scale inquiry. In a memoir presented at the Academie royale des sciences, “Sur les naissances, les mariages et les morts” (1786),^[128] the mathematician and scientist Pierre-Simon Laplace (1749-1827)^[129] - a younger colleague of Condorcet at the Academie - tried to estimate the accuracy of the value of the universal births multiplier, the determination of which “is the most delicate point” in the calcula­tion of the population size (Laplace 1786, 694). Obtaining a good approximation of “the true ratio of the population to the number of annual births” (1786, 694), and consequently of the population of the country, depends on the size of the census used to calculate it: the larger (smaller) the number of observations, the more the result will express the action of permanent (variable) causes.

The population in France, deduced from the number of annual births, is thus only a probable result, and consequently potentially flawed. It is up to the analysis of chance to determine the probability of these errors and the size of the census so as to make it very likely that these errors be maintained within narrow limits.

(1786, 695)

The question was thus to determine, for a given value of the multiplier and with a high probability, the required number of observations in order that the error in the resulting size of the population be inferior to a given number - in other words, Laplace imagines what would be called later the confidence interval. In this perspective, he used his former results in the theory of probability - especially on inverse probability - and in the approximation of mathematical formulas which are functions of large numbers. The model he used is the urn model, at that time almost universally accepted in probability theory.^[130]

Laplace (1786, 697-701) shows that

and

these equations being of course approximations obtained by deleting negligible terms.^[131]

For a value of the birth multiplier equal to 26 and given the average number of annual births for the years 1781 and 1782 (q = 973, 054.5), the number of inhabit­ants would be 25,299,417.

Now, my analysis shows that, to have a probability of 1,000 to 1 of not being mistaken by more than half a million in this estimation... the census which provided the basis for the determination of this factor 26 should have been of 771,469 inhabitants. If 26.5 is taken as the ratio of population to births, the num­ber of inhabitants in France would be 25,785,944 and, to have the same prob­ability of not being mistaken by more than half a million, the factor 26.5 should be determined after a census of 817,219 inhabitants. It follows that, if we want to have on this subject the accuracy that its importance requires, the census should be based on one million or one million and two hundred thousand inhabitants.

(Laplace 1786, 696)

Finally, during this period, decisive progress was made during the repeated attempts to establish tables of mortality. This was an important task because of the con­stant need for public finances to raise money, often in the form of tontines or life annuities - and, at the end of the period, for the establishment of life insurance contracts. In the seventeenth century, several authors in Britain, the Netherlands, Germany, etc. tried to build such tables. The pioneering attempt by John Graunt was followed by many others (Behar 1976; Dupaquier and Dupaquier 1985, Chapter 6). Significant advances were made by the Huygens brothers - Lodewijk (1631-1699) and Christiaan (1629-1695) - in their mutual correspondence when Christiaan was at the Academie des sciences in Paris, and by Gottfried Wilhelm Leibniz (1646­1716), with the distinction between the concepts of median life and life expectancy: but they remained unfortunately unknown for a long time (Leibniz’s manuscripts were published in 1866 and the correspondence between the Huygens in 1920). In this context, the main public progress was made by the mathematician Antoine Deparcieux (1703-1768)^[132] in his 1746 Essai sur les probabilities de la duree de la vie humaine (Deparcieux 1746a; see also 1746b and 1760), with the same distinction between median life and life expectancy, and significant methodological develop­ments. Some decades later, a further step in the establishment of better tables was made by another mathematician, a former collaborator of Turgot and Condorcet and specialist in financial mathematics,^[133] Emmanuel-Etienne Duvillard de Durand (1755-1832). His calculations - first made during the Revolution in a (lost) memoir on the establishment of a National savings bank presented in 1796 at the first Class of the Institut national des sciences et des arts (the Republican successor to the Aca- demie royale des sciences: see Chapter 9, this volume) - found their outcome in the appendix to his book, Analyse et tableaux de l’influence de la petite verole a chaque age et de celle qu’un preservatif tel que la vaccine peut avoir sur la population et sa longevite (1806, 159-98) where Duvillard estimates the extension of life expectancy at each age due to the vaccination against smallpox. Moreover, in the same 1796 memoir, he proposed a mathematical expression of a law of mortality on which his tables are based^[134] - an analysis refined in the 1806 book.

2.