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The spirit of geometry. Quantification
and formalisation
Gilbert Faccarello∗
Entre esprits égaux et toutes choses pareilles, celui qui a de la géométrie
l’emporte et acquiert une vigueur toute nouvelle.1–Blaise Pascal
The huge economic, political and scientific transformations which took place in
France as in various countries in Europe during the seventeenth and eighteenth
centuries brought about their share of new problems, or at least the growing
awareness that old questions were assuming a new and fundamental character
that the new scientific spirit and the rise of knowledge could help solve. Some
∗Forthcoming in Gilbert Faccarello and Claire Silvant (eds), A History of Economic
Thought in France, volume 1, The Age of Enlightenment, London and New York: Routledge,
2023, pp. 172–238.
1“Between equal minds and all things being equal, he who has geometry prevails and
acquires a whole new vigour from it.”
1
The spirit of geometry. Quantification and formalisation 2
of these solutions were found in quantification, others in formalisation. This
evolution was the result, on the one hand, of the needs of expanding economic
activities and changing governmental practices and, on the other hand, of the
new attitude towards the world induced by the development of sciences. A
phrase drawn from the Bible (Wisdom of Solomon, XI, 20) – where, speaking
of justice, it is stated that God “arranged all things in right order by pro-
portion: by measure, by number, and by weight” – and transposed into the
new context became the scientific motto of the time and is to be found in
the writings of the period. Two expressions of it are celebrated examples of
this attitude: one is by William Petty (1623–1687) in his Political Arithmetick
– “instead of using only comparative and superlative words, and intellectual
arguments, I have taken the course . . . to express myself in terms of number,
weight, or measure” ([1676] 1690, Preface) – and the other by Isaac Newton
(1643–1727) in a notebook: “Numero pondere et mensura Deus omnia con-
didit”, that is, “God created everything by number, weight and measure”. The
need for quantification, and then attempts at formalisation, slowly extended
to “moral sciences” in general and to political economy in particular. It was
diversely felt, depending on both on the authors and the topics under exam-
ination. This need remained for a long time only programmatic, but partial
realisations were achieved. This chapter is an outline history of this evolution
in the French context.2
1Thinking 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 Négociant, ou Instruction
générale 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. This need in-
2In the secondary literature, there is sometimes a confusion between quantification,
using simple statistics, and the use of probability theory and/or algebra and analysis in
economic reasoning: they are sometimes all referred to as mathematisation. It is important
to differentiate them carefully.
The spirit of geometry. Quantification and formalisation 3
cluded the resources to be found in different places, the practicability and
safety of marine and overland routes, the states of the markets, or the knowl-
edge of the different kinds of money circulating within and between countries.
This led to the growth of technical knowledge in banking and financial oper-
ations – among them the significant emergence of life annuities and tontines
– and the establishment of the first stock exchanges. At the same time, the
need for some protection against risk was increasingly felt: hence the develop-
ment of insurance for different kinds of trade and personal property – marine
insurance in particular, for cargoes and ships, or fire insurance – and the slow
emergence of life insurance. All this implied, inter alia, collecting data on
trades, evaluating different kinds of risk and determining the lifespans of vari-
ous segments of the population and the life expectancy at different ages. The
rise of statistical thinking and the emergence of probability calculus, in part
due to these necessities, progressively provided solutions to these problems.
However, these developments did not always happen smoothly and they
encountered a number of social and political difficulties. Linked to the last-
ing 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. It is also possible to see in this reluctance the more
or less conscious transposition of a strict spiritual attitude into the secular
world.3In France, the strongest opposition was against life insurance. Among
the many contradictory verses from the Bible, a very few were retained by
theologians and interpreted as a condemnation of this new practice. Men’s
lives, the Church insisted, belong solely to God, and God is supposed to take
care of his people. Life insurance, which tries to calculate the duration of
human lives and so speculate on time, is in fact gambling with God and thus
immoral and impious. It implies a lack of faith, a mistrust in God. God is
3“The souls which belong to God should not have any assurance or foresight [ni assurance
ni prévoyance], they must act through faith, which has no clarity nor assurance in the
permanence of good works [dans la suite des bonnes œuvres]. They look at God, follow
him at all times and depend on the encounters that his providence creates” (J.-A. Duvergier
de Hauranne, abbé of Saint-Cyran, quoted by Dom Charles Clémencet, 1655–57, II, 32).
Saint-Cyran was one of the prominent Jansenist theologians – but this attitude is not specific
to Jansenist authors.
The spirit of geometry. Quantification and formalisation 4
the sole and real “life insurance”, protecting the poor and the needy4– God
insures those who pray, while insurance companies only insure those who pay!
This is the reason why, in France (as in Spain, Genoa or Holland, for example)
life insurance was prohibited. This prohibition was reasserted by the minis-
ter Jean-Baptiste Colbert in August 1681 in the celebrated Ordonnance de la
marine: “We forbid any insurance on the life of persons” ([1681] 1714, 257),
with this comment added by the editor of the 1714 republication ([1681] 1714,
257-8):
In some nations, these kinds of insurances are permitted5. . . 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 Émérigon (1716–1784), for example, in his 1783 Traité des
assurances et des contrats à 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 . . . can generate
crimes” (1783, I, 198). But times were changing and, a few years later, in
November 1787, the first life insurance company was finally authorised: the
Compagnie royale d’assurance sur la vie.6
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
4The question was still debated during the nineteenth and twentieth centuries (see for
example Tracy 1966), with, however, an outcome more and more favourable to life insurance.
On the Internet, the many results provided by the search for “God and life insurance”, for
example, are a good indicator of these debates.
5In England, the first life insurance company was established in 1706: the Amicable
Society for a Perpetual Assurance Office. One co-founder was William Talbot . . . Bishop
of Oxford.
6To inform the public and explain in detail the novelty of this kind of establishment, the
Compagnie published a 110-page prospectus (Compagnie royale d’assurances. Prospectus de
l’établissement des assurances sur la vie, 1788).
The spirit of geometry. Quantification and formalisation 5
of their countries, both for fiscal and for domestic and international politi-
cal 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”7– the word “pop-
ulation” only re-emerged in France in the mid-eighteenth century (Théré 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 policymak-
ing 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 cen-
tury and the beginning of the eighteenth, some prominent authors like Pierre
Le Pesant de Boisguilbert (1646–1714) and Sébastien Le Prestre de Vauban
(1633–1707) used more (Vauban) or less (Boisguilbert) serious empirical in-
vestigation to support their views and proposals.8After the death of Louis
XIV in 1715 and the hectic monetary and fiscal events during the Regency of
Philippe d’Orléans – John Law’s monetary “experiment” – the search for eco-
nomic and social information produced a rapidly expanding literature (Théré
1998). All fields were concerned – agriculture, taxation, money and bank-
ing, foreign trade, etc. Data collection and calculation often became part and
parcel of the development of political economy.
The quantitative aspect of the quest must not, however, conceal the
qualitative side: François Quesnay (1694–1774) and A. R. J. Turgot (1727–1781)
themselves paid due attention to it – the first with the Questions intéres-
santes sur la population, l’agriculture et le commerce proposées aux Académies
7For some aspects of these questions, see for example Esmonin (1964), Dupâquier and
Dupâquier (1985, Chapter 3) and Dupâquier (1988).
8Contrary to Boisguilbert in this field, Vauban was a real innovator and developed
a methodological reflection on data collecting and calculation – see for, example, Vilquin
(1975) and Meusnier (2003).
The spirit of geometry. Quantification and formalisation 6
et autres Sociétés savantes des Provinces, which he published in 1758 with
Étienne-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 Encyclopédie,
ou Dictionnaire raisonné des sciences, des arts et des métiers, 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, François Véron 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 col-
lecting (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 col-
lectors to obtain this information: people were reluctant to disclose it, most of
the time supposing that it was requested for fiscal or military purposes. Con-
sequently, the administration, dealing with local data, tried to generalise the
The spirit of geometry. Quantification and formalisation 7
results to more extended areas in a more or less questionable way. Towards the
end of the Ancien Régime 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 (1743–1794) in his
Résultats extraits d’un ouvrage intitulé: De la richesse territoriale du royaume
de France, 1791 (Perrot 1988, 2003). But all this was only a first and very pre-
liminary 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 Ingénieurs
des Ponts et Chaussées to the management of poverty relief and charity (Joël
1984; Etner 1987, Chapters 2 and 3) and even slavery (Oudin-Bastide and
Steiner, 2015). For these subjects, however, the task was a lot easier because
of the limited scope of the investigations.
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.9It was con-
sidered of the utmost importance, and some authors held it to be the basis
of political arithmetic itself. The chapter on political arithmetic that Jean-
Étienne Montucla (1725–1799) added to the 1778 reissue of the successful
Récréations mathématiques et physiques by Jacqes Ozanam (1640–1718) is en-
tirely 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
9It seems that the first occurrence of the word demography in French happened in 1855
in a book by Achille Guillard, Éléments de statistique humaine ou démographie comparée
(Paris: Guillaumin).
The spirit of geometry. Quantification and formalisation 8
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 Régime, 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 de-
creasing, especially if it was compared with the number of people who were
supposed to have lived during Greco-Roman antiquity, inferred from ancient
The spirit of geometry. Quantification and formalisation 9
texts. Yet the population was growing and the world had never been so pop-
ulous! 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 Encyclopédie. 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 con-
temporary 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, pref-
erences 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 ear-
lier (Ducreux 1977; Dupâquier and Dupâquier 1985, 108-9) by Iustus Lipsius10
(Joost Lips, 1547–1606), Giovanni Battista Riccioli11 (1598–1671) and Isaac
10 Admiranda, sive de Magnitudine Romana Libri Quattuor, 1597.
11 “De Verisimili Hominum Numero Superficiem Terræ Inhabitantium”, Appendix to Ric-
cioli’s Geographiae et hydrographiae reformatae libri duodecim, 1661, pp. 630–634.
The spirit of geometry. Quantification and formalisation 10
Vossius12 (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 mil-
lion 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. Étienne Noël
d’Amilaville (1723–1768), the author of the (long) entry “Population” in the
Encyclopédie, 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
géographique, historique et politique des Gaules et de la France, published be-
tween 1762 and 1770 (Expilly 1762–70), based his research on some partial
official inquiries and used the method of the multiplier: first a multiplier cal-
culated 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 multi-
12 Variorum Observationum Liber, 1685.
The spirit of geometry. Quantification and formalisation 11
plier of 25, he found that the population size was over 24 million. The depopu-
lation 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-François de La Michodière)
reached approximately the same conclusion. Moheau’s estimations, in his 1778
Recherches et considérations sur la population de la France (with the collabora-
tion of Antoine-Jean-Baptiste de Montyon), often considered as the first trea-
tise of demography, were similar.13 Some other writings are not to be neglected
either, like the “Mémoire sur la population de Paris et sur celle des provinces
de la France. Avec des recherches qui établissent l’accroissement de la popula-
tion de la capitale et du reste du royaume” by Jean-François Clément Morand
(1726–1784) – a physician and scientist who was in charge of the question
of population at the Académie royale des sciences – published in 1782 in the
Histoire de l’Académie 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
13 See, for example, Dupâquier and Dupâquier (1985, 175–87) and Bru (1988a). More
specifically: on Expilly, see Esmonin (1957); on Messance, Brian and Théré (1998); and on
Moheau, Rohrbasser (2003).
14 The survey part of this report was published in the Histoire de l’Académie for 1779,
but not the statistical part.
The spirit of geometry. Quantification and formalisation 12
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’Académie royale des sciences et belles-lettres of Berlin for
the year 1760: “Recherches générales sur la mortalité et la multiplication du
genre humain” and “Sur les rentes viagères”.15 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 Académie
15 Euler had already dealt with questions of population in his 1748 book, Introductio in
analysin infinitorum, as examples of applications of exponentials and logarithms (Euler 1748,
sections 110 and 111), with problems like: “If the number of men doubles every hundred
years, what is the annual [rate of] growth?” or “If the number of men increases every year
by its hundredth part, after how many years will their number be multiplied by ten?” (1748,
I, 80–81).
The spirit of geometry. Quantification and formalisation 13
royale des sciences, “Sur les naissances, les mariages et les morts” (1786),16 the
mathematician and scientist Pierre-Simon Laplace (1749–1827)17 – a younger
colleague of Condorcet at the Académie – 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.18
16 Laplace’s memoir is usually dated 1783 because it is included in the volume of the
Histoire de l’Académie royale des sciences for the year 1783, published in 1786. However, the
various volumes of the series Histoire de l’Académie royale des sciences sometimes included
memoirs not necessarily presented during the reference year. This is the case here with
Laplace’s memoir, which could not have been written in 1783 because the data used in it
include the number of births in Paris in 1784. Gillispie (1981, 393) notes that it was presented
to the Académie on 30 November 1785, with the title: “Mémoire sur la population de la
France” and then published the following year in the volume for 1783. Laplace summarised
his results on population in the tenth and final lecture on mathematics he gave at the
short-lived École normale (Laplace [1795] 1800).
17 On the impressive scientific work of Laplace, see Gillispie (1972, 1981).
18 As Condorcet wrote in 1781 in his assessment of another memoir by Laplace: “All
the questions pertaining to the probability calculus can be reduced to a single hypothesis,
namely that of a certain quantity of balls of different colours, mixed together, from which
one is supposed to pick randomly some balls in a certain order or proportion. If one supposes
The spirit of geometry. Quantification and formalisation 14
Suppose, Laplace writes, “an urn containing an infinity of white and black
balls in an unknown ratio and suppose that a first drawing yields pwhite
balls and qblack ones; suppose further that a second drawing yields q′black
balls but that the number p′of white balls pulled in this second drawing is
unknown” (1786, 696). Using the first ratio i=p/q, we can approximate p′
by pq′/q. The question is to determine the probability Pthat the (true) value
of p′lies between (pq′
q)(1 −ω)and (pq′
q)(1 + ω),ωbeing an arbitrary small
number: “or, what is equivalent”, that the error for pq′/q be no greater than
a=ωpq′/q (1786, 697). Vbeing an auxiliary variable defined by
V2=pqq′ω2
2(p+q)(q+q′)
Laplace (1786, 697–701) shows that
P= 1 −2
√π)Z∞
V
e−t2dt
and
p=2i2(i+ 1)q′2V2
a2−2i(i+ 1))q′V2
these equations being of course approximations obtained by deleting negligible
terms.19
Now let pand qbe, respectively, the population and the number of births to
be measured in some carefully chosen areas in order to determine the universal
multiplier i=p/q, and let q′be the total number of births in the country and
p′the true (but unknown) population of the realm. The question is: when
approximating p′by iq′=pq′/q, what is the necessary size of p to have a
the number of balls of each kind to be known, we have the ordinary probability calculus as
the geometricians of last century considered it, but if we suppose the number of balls of each
kind to be unknown and we want to estimate their proportion or their number . . . on the
basis of the number of balls of each kind we have picked, we have a new class of problems”
(Condorcet, “Sur les probabilités”, in Condorcet 1994a, 159).
19 For a modern presentation, see Dale (1999, 217–21) or Gorroochurn (2016, 79–83).
Laplace’s approach was criticised, but 150 years later (Pearson 1928).
The spirit of geometry. Quantification and formalisation 15
probability of 1,000 to 1 (P= 1,000/1,001) of not being mistaken by more
than a= 500,000 in the value of p′?
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 establish-
ment 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;
Dupâquier and Dupâquier, 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 Académie des
sciences in Paris, and by Gottfried Wilhelm Leibniz (1646–1716), 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)20 in his 1746 Essai sur les probabilités de la durée de
la vie humaine (Deparcieux 1746a; see also 1746b and 1760), with the same
distinction between median life and life expectancy, and significant method-
ological developments. Some decades later, a further step in the establishment
20 On Deparcieux, see, for example, Behar and Ducel (2003). Deparcieux became a
member of the Académie royale des sciences in 1746
The spirit of geometry. Quantification and formalisation 16
of better tables was made by another mathematician, a former collaborator of
Turgot and Condorcet and specialist in financial mathematics,21 Emmanuel-
Étienne 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 Académie 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 vérole à chaque âge et de celle
qu’un préservatif tel que la vaccine peut avoir sur la population et sa longévité
(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 based22 – an analysis refined in the 1806 book.
2Old and new political arithmetic
and “social mathematic”
As in the case of population – the field where the progress in techniques of
analysis was the most striking – calculations in the commercial and financial
fields also assumed a more theoretical flavour when they started to use the new
probability theory. The probabilistic approach also quickly permeated other
branches of the “moral sciences” and led to an ambitious programme: “social
mathematic”.
21 Duvillard published a book on life annuities (Duvillard de Durand, 1787) and was
involved in the management of the public debt. See Israel (1991, 1993, 1996) and Biondi
(2003). An important manuscript was discovered by Dell’Aglio and Israel and published in
2010 (Duvillard de Durand [1813] 2010): see Dell’Aglio and Israel (2010).
22 A positive review of the memoir, by three famous mathematicians (Joseph Louis La-
grange, Pierre-Simon Laplace and Adrien-Marie Legendre), is published as an appendix to
the 1806 book (Duvillard de Durand 1806, 205–10).
The spirit of geometry. Quantification and formalisation 17
Probability theory and socio-economic themes
Reasoning on risk and uncertain statements already had a long history,23 but
systematic reasoning was not developed until the second half of the
seventeenth century with Blaise Pascal (1623–1662), Christiaan Huygens and
Jakob Bernoulli24 (1654–1705) in particular. It was developed during the
eighteenth century by Abraham de Moivre (1667–1754) – a French protestant
who fled to England after Louis XIV’s revocation of the Edict of Nantes in
1685 – Thomas Bayes (1702–1761) and Richard Price in England,25 and Pierre
Rémond de Montmort (1678–1719), Condorcet and Laplace in France. These
authors first dealt with calculations of the probability of winning in “games
of chance” (“alea” being the Latin word for “game of dice”, or “chance”) or
questions linked to these games: what is a fair stake to participate in a
game, or how to divide the stakes in a fair way if the game is interrupted
before its end, etc. Currently used today to cover the entire field, the word
“probability” came in fact from the judicial field and referred, for example,
to already widely-discussed questions like the estimation of the veracity of a
testimony, the guilt or innocence of a person, or commercial and inheritance
problems. Jakob Bernoulli used the phrase “art of conjecturing” and proposed
to call it “stochastics”. The first books to be published were Jakob Bernoulli’s
Ars Conjectandi (in Latin, published posthumously in 1713) and Montmort’s
1708 Essay d’analyse sur les jeux de hazard (second edition: Montmort 1713).
But because of its recent origin and the increasingly technical aspects of the
subject, the diffusion of the theory was rather slow among men of letters and
philosophes, as this statement by Montesquieu shows:
The mathematician goes only from the true to the true or from the false
to the true by ab absurdo arguments. They do not know that middle
which is the probable, the more or less probable. In this respect, there is
not more or less in mathematics. ([1720–55] 1964, 957; [1720–55] 2012,
54)
23 There is now an outstanding recent literature on the history of probability theory,
from its beginnings to the end of the eighteenth century. See, for example, Coumet (1970),
Hacking (1975), Stigler (1986), Daston (1988) and Hald (2003).
24 Jakob Bernoulli and his brother Johann, both foremost mathematicians and friends of
Gottfired Wilhelm Leibniz (1646–1716), were associated members of the Académie royale
des sciences, and so, later, was their nephew Daniel Bernoulli.
25 “Roughly half of Bayes’s famous essay was written by Richard Price, including the
Appendix with all of the numerical examples” (Stigler 2018, 117).
The spirit of geometry. Quantification and formalisation 18
As in the case of insurance, the fact that the developments of probability
theory were initially linked to games of chance – an invention of the Devil
according to some theologians, and therefore considered with hostility by the
Church – did not at first favour its moral reputation. “Games of dice and
of cards . . . where the gain principally depends on chance, are not only
dangerous recreations”, François de Sales wrote in his influential Introduction
à la vie dévote, they are “bad and blameworthy, and . . . prohibited by
civil and ecclesiastical laws”. The gain made in this way is reprehensible:
“The gain, which should be the price of industry, is made the price of luck,
which does not deserve any price, because it does not depend on us” (Sales
[1608] 1730, 362–3). Moreover, these kinds of game are more than simple
“recreation”: they totally monopolise the attention of the players and are thus
“violent occupations”; and the winner’s joy is unfair because it implies the loss
and displeasure of somebody else ([1608] 1730, 363–4). It is significant that
in 1738, in the dedication26 of the second edition of The Doctrine of Chance,
Abraham de Moivre still had to rebut the presumption of amorality levelled
at probability theory.
There are many People in the World who are prepossessed with an
Opinion, that the Doctrine of Chances has a Tendency to promote Play,
but they soon will be undeceived, if they think fit to look into the
general Design of this Book . . . Your Lordship does easily perceive, that
this Doctrine is so far from encouraging Play, that it is rather a Guard
against it, by setting in a clear Light, the Advantages and Disadvantages
of those Games wherein Chance is concerned.
Later in the century, Condorcet felt obliged to stress that, if Pascal and
Pierre de Fermat (1607–1665) chose games of chance for the application of the
theory of combinations to contingent events, this is only because it was the
easiest way to proceed, and not because they were frivolous people
(Condorcet 1994a, 339). In his 1786 “Discours sur l’astronomie et le calcul
des probabilités”,27 he insisted that the only utility of this first application was
to prove how futile are all the expectations, of which those who give them-
selves over to these games are too often the dupes and the victims: per-
haps a mathematician, showing the ridiculousness of their speculations,
26 To Lord Carpenter (the dedication of the first edition, 1718, was to Newton).
27 “Discours” which he gave in December 1786 at the Lycée, a private institution of public
lectures for educated adults.
The spirit of geometry. Quantification and formalisation 19
would have a greater impact than a moralist stating their
disastrous consequences. (1994a, 602–3)
This kind of charge against probabilities, however, faded away as the
theory became more and more complex on the mathematical level and showed
its usefulness in practical life. Jakob Bernoulli’s intention – in the fourth and
unfinished part of Ars Conjectandi titled “Usum & applicationem
præcedentis doctrinæ in civilibus, moralibus & œconomicis” (“The use and
application of the preceding doctrine in civil, moral and economic matters”)
– was to apply the mathematical developments made for games of chance to
social and economic matters, and he considered it as the main part of his
work. In the same perspective, his nephew Nikolaus Bernoulli (1687–1759)
wrote his 1709 thesis (Dissertatio inauguralis mathematico-juridica de usu
artis conjectandi in jure) and published an abridged version of it
(“Specimina artis conjectandis, ad quaestiones juris applicatae”, 1711) to show
how probability theory could deal with legal and economic questions such as
the reliability of witnesses and of suspicions, marine insurance, the probability
of human life, life annuities, or the problem of the “absent” (after how many
years can an absent person be considered as dead?) – all fields where the
logic of the probable was already present. Daniel Bernoulli himself – nephew
of Jakob and cousin of Nikolaus – dealt with these matters in his celebrated
paper on the measure of risk, “Specimen Theoriae Novae de Mensura Sortis”
([1738] 1954).28 He applied his new method of evaluating risk to questions of
trade and insurance, for example to state the conditions of profitable insurance
on both sides, that is, both for the merchant who thinks about insuring his
trade and the insurer who insures the merchant; or to show that the merchant
can reduce his risk by dividing his merchandise and sending it on several boats
instead of one single ship, stating that this advice “will be equally service-
able for those who invest their fortunes in foreign bills of exchange and other
hazardous enterprises” ([1738] 1954, 31).
If some objections to the probabilistic approach still developed later in the
century, they had nothing to do with the former ones and were simply part
of the usual scientific debate. The most interesting were expressed by the
28 For a recent overview and some references, see Faccarello (2016a). The role of Buffon
in the discussion of the measure of risk and the so-called Saint Petersburg Paradox, recalled
in his “Essai d’arithmétique morale”, deserves to be mentioned (Buffon 1777, 75 et seq.). On
Buffon’s approach to political arithmetic, see, for example, Martin (1999a and 1999b).
The spirit of geometry. Quantification and formalisation 20
mathematician and philosopher Jean Le Rond d’Alembert (1717–1783),29 a
dissenting voice who induced his disciples Condorcet and Laplace to find new
solutions.
But could probability theory play a role in the development of political
economy proper, that is, go beyond calculations in specific fields of social and
economic life dealing with a great number of data (population, mortality tables,
insurances, quantification of risks, etc.)? Laplace was sceptical. In 1795, in
his last lesson at the short-lived École normale, devoted to probability, he
asked whether probability theory could also be applied to political economy
“and improve it”. His answer was rather negative as far as a priori reasoning
is concerned, and he subsequently remained vague and cautious about this
subject.30
The questions raised by this science are so complicated and depend on so
many unmeasurable or unknown elements that it is impossible to solve
them a priori. One can only have glimpses of them, and calculation, in
fields where it is possible, shows us how misleading they are. Let’s deal
with [political] economy as we did with physics, through experience and
analysis. On the one hand, consider the great number of truths about
nature that this method allowed us to uncover and, on the other hand,
the multitude of errors produced by the mania for systems; you will then
realise the necessity to always consult experience. It is a slow guide, but
always sound, and to drop it exposes us to the most dangerous errors.
(Laplace 1795 [1800], 73)
Condorcet, political arithmetic and “social mathematic”
With Condorcet, the question of the use of probability theory took on a new
dimension and received an ambitious answer. For him, the use of probabil-
ity theory in moral sciences is legitimate, as there is no difference in nature
between these sciences and those which had been successfully developed since
29 See Daston (1979) and Paty (1988). An important aspect of d’Alembert’s approach
was expressed in his controversy with Daniel Bernoulli on the debated question of smallpox
inoculation (D. Bernoulli 1760, 1766; d’Alembert 1761, 1767).
30 The allusion to political economy is still present in the 1812 republication of his lessons
in the second volume of the Journal de l’École Polytechnique and in the first edition, 1814,
of his Essai philosophique sur les probabilités – a development of his lecture on probabilities.
But “political economy” was subsequently deleted and replaced by “moral” or “moral and
political sciences”. On the Essai philosophique, see Bru (1986).
The spirit of geometry. Quantification and formalisation 21
the seventeenth century. Turgot, Condorcet wrote approvingly in the intro-
duction to his Essai sur l’application de l’analyse à la probabilité des décisions
rendues à la pluralité des voix, “was convinced that the truths of the moral
and political sciences are likely to have the same certainty as those which form
the system of the physical sciences” – astronomy being, for example, “close
to mathematical certainty” (1785, i). This conviction, however, needs to be
understood properly. While it implies that the nature of knowledge is basi-
cally the same in all fields of inquiry, it is such that nowhere it is possible to
find propositions that are absolutely certain. Insisting on the importance of
Turgot’s entry “Existence” in the Encyclopédie, Condorcet stressed that any
knowledge of the existence and properties of objects comes from our senses and
our ability to think about our sensations and combine them. The idea that
there exist constant laws for the various observable phenomena is only a hy-
pothesis: by nature, this knowledge can never produce any absolute certainty,
whatever the field of inquiry – mathematics included, because the hypothesis
also concerns human understanding, and not only external phenomena. This
approach only leads to a more or less strong confidence that these phenomena,
in the same circumstances, will happen again in the future.
This is the reason why, when he speaks of “certainty”, Condorcet does so only
metaphorically to express a great degree of “assurance” – the word “assurance”
was judged by him more suitable in this context (Condorcet 1785, xvi; 1994a,
523) and a better choice than the ambiguous phrase “certitude morale” (moral
certainty). “The knowledge that we call certain is . . . nothing other than
knowledge based on a very high probability” (1994a, 602) which it is, moreover,
meaningless to calculate in most cases (1785, xiv).31 Hence his statement that
all propositions belong to this part of the calculus of probability “where one
estimates the future order of events on the basis of their past order or, to be
31 For accurate developments on Condorcet’s related concept of “motif de croire” (grounds
for belief) and the distinction with the “sentiment de croire” (sentiment of belief) or
“penchant à croire”, see Baker (1975, 185–9). “It is of the utmost importance to
distinguish between the strength of the actual grounds for belief – the greater or lesser
frequency of the experiences involved in any case – and the force of the sentiment of
belief, which leads us to regard as constant any event that has often been repeated. This
natural and habitual sentiment . . . automatically increases in strength with the frequency
of the experiences involved . . . But it is also affected . . . by the intensity or force
of particular experiences (or impressions) upon the mind. As a result, since the intensity
of impressions has no relation to the probability of their recurrence, a disparity develops
between the strength of the sentiment of belief and the extent of the grounds for it” (1975,
187–88). See also Daston (1988, 213–18) and, on Condorcet’s general approach, Bru (1988b,
1994).
The spirit of geometry. Quantification and formalisation 22
more precise, the future order of unknown events on the basis of . . . known
events” (1994a, 291), and his reference to the classical urn model in probability
theory:
The reason to believe that, from ten million white balls and one black,
it is not the black one that I will pick up at the first go, is of the same
nature as the reason to believe that the sun will not fail to rise tomorrow,
and these two opinions only differ as to their lower or higher probability.
(Condorcet 1785, xi)
However, Condorcet did not follow the sceptical tradition (Rieucau 2003).
He believed in the progress and usefulness of knowledge, and he often de-
nounced “the absurdity of absolute scepticism” (1994a, 602). The systematic
collection of data and the organisation of accurate experiences permitted undis-
putable progress in the sciences, and what happened in physics or astronomy
would also happen in the new sciences of society. Over time, politics or po-
litical economy were likely to approach the same degree of assurance in the
truths they establish.
The question then is how to use probability theory and mathematics in a
legitimate way. Calculation should be handled cautiously because it can be
dangerous in the hands of “charlatans” (Condorcet 1994a, 337): in politics,
it is so easy to impress people with the use of numbers in order to influence
their opinions and choices. Some “ridiculous” applications of calculation to
political questions have also been made, but “how many applications, just as
ridiculous, have not been made in each part of physics?” (1785, clxxxix). All
this notwithstanding, the use of probability could no longer be dispensed with.
With its help, it is possible to reason in a more precise way and to avoid the
negative influence of vague impressions due to imperfect knowledge, prejudices,
interests or passions:
Almost everywhere [in the Essai] one will find results which comply with
what the simplest reason would dictate; but it is so easy to blur reason
with sophisms and vain subtleties that I would nevertheless feel happy if
all I have done is to support a single useful truth with the authority of a
mathematical proof. (1785, ii)
The spirit of geometry. Quantification and formalisation 23
But it is possible to go beyond “what reason alone can do”.32 In this per-
spective, he planned to develop “political arithmetic” into a systematic science.
The first attempts by Petty or Graunt were almost insignificant, he states, and
serious things only started with the works of Jan De Witt – “the illustrious and
unfortunate Jan De Witt sensed that political economy, as all science, must be
subjected to the principles of philosophy and to the precision of calculation”
([1794] 2004, 380) – and, above all, Jakob and Nikolaus Bernoulli. Hence the
wide and ambitious definition Condorcet gave of political arithmetic in the
supplement he wrote in 1784 to the Diderot entry “Arithmétique politique” on
the occasion of its republication in the first volume of the series Mathématiques
of the Encyclopédie méthodique: “Political arithmetic, in its wider sense, is the
application of calculation to political sciences” (1994a, 483). “The applica-
tion of the calculation of combinations and probabilities to these sciences [the
‘social art’] . . . is the only means to confer upon their results an almost
mathematical precision and to estimate their degree of certainty or likelihood”
([1794] 2004, 447).
It is this same science that Condorcet called “social mathematic” in his
1793 “Tableau général de la science qui a pour objet l’application du calcul
aux sciences politique et morales”:
I prefer the word mathematic, although now no longer used in the
singular, to arithmetic, geometry or analysis because these terms refer to
particular areas of mathematics . . . whereas we are concerned . . . with
the applications in which all these methods can be used . . . I prefer the
term social to moral or political because the sense of these words is less
broad and less precise. (1994b, 93–4)
Condorcet used probability theory in many fields, especially those where
decisions are to be taken in the face of uncertainty and imperfect information,
to estimate the outcomes of alternative choices. These applications, in line
with those of the Bernoullis, first regard the traditional fields of insurance, life
annuities, tontines or the problem of the “absent”, for example, (Crépel 1988,
1989) but also the activity of any entrepreneur who – as Richard Cantillon al-
ready insisted in his Essai sur la nature du commerce en général – always acts
32 During the French Revolution, Condorcet insisted that calculation would permit to
develop knowledge even if reason alone reached its limits. “We might even be nearing the
time when, in several branches of political sciences, everything that reason alone can do will
be at an end, and the application of calculus will be the sole means of progressing further”
(Condorcet [1792] 1847, 464–5).
The spirit of geometry. Quantification and formalisation 24
in an uncertain world. Generalising his analysis of the behaviour of both a mer-
chant and his insurer facing uncertainty and risk in maritime trade, Condorcet
conceived of any economic activity as an uncertain and risky undertaking –
“undertakings in which men expose themselves to losses in view of a profit”
(1994a: 396) – and used probability to describe the entrepreneur’s decisions
to invest (Rieucau 1998). A parallel is made with the traditional analysis
of “fair” games of chance, in which a fair stake is equal to the mathematical
expectation of gain: Condorcet explains that, as additional constraints and
calculations arise in economic activity, the analogy between a gambler and
an entrepreneur is somewhat misleading. As he wrote in his posthumously
published Éléments du calcul des probabilités:
When a merchant makes a conjecture [fait une spéculation] implying a
significant risk, it is not enough that his profit be such that the mean
value of his expectations be equal to his stake [sa mise] plus the interest
that a riskless trade would have brought him. In addition, he must have
. . . a very high probability that he would not suffer a loss in the
long run. To submit this kind of project to calculation, one should thus
determine, for the funds that each trader could successively employ in
such a risky trade, what is the excess of profit that he must obtain in
order either to have a sufficient probability not to lose his entire funds,
or to lose only part of them, or to just get them back, or to get them
back with a profit. (1805, 117)
But this statement concerning entrepreneurs in general mainly remained
programmatic. By contrast, a second field for “social mathematic”, related
to the collective level (public economics, social choice), is more developed.33
Of particular interest are Condorcet’s ideas about decision-making processes,
which originate in his discussions with Turgot, especially on juries and more
generally on any assembly susceptible to making decisions through voting and
some majority rule. At the Académie royale des sciences, the question of the
organisation of votes had already been discussed by Jean-Charles de Borda
(1733–1799) in a memoir presented on 16 June 1770 – of which we have no
record unless it is the one presented and published by the Académie in 1784
(Borda 1784) – on the occasion of a discussion of the rules to adopt for the
election of academicians. Borda criticised the usual way of organising a vote.
Suppose, he wrote, three candidates A,Band C, and 21 voters. Agets 8 votes,
33 For a general view, see, for example, Baker (1975) and McLean and Hewitt (1994). See
also Guilbaud (1952).
The spirit of geometry. Quantification and formalisation 25
B7 and C6. With a simple relative majority rule, Ais thus elected. But
suppose further that the 13 electors who voted in favour of Band Calso prefer,
respectively, Cand Bto A: in this case, the former result contradicts the
judgement based on these preferences (Borda 1784, 657–8). This is the reason
why a good form of election “must give the electors the means of pronouncing
on the merits of each candidate [“sujet”] and to compare them successively with
the merits of each of his competitors” (1784, 658–9).
This criticism is “very important and absolutely novel”, Condorcet wrote in
1784 in his presentation of Borda’s memoir in Histoire de l’Académie royale des
sciences (in Condorcet 1994a, 359; see also 1785, 119). But while accepting
the criticism , he found Borda’s proposal of a new method unsatisfactory. In a
nutshell, the Borda count consists in giving marks to candidates standing for
an election or running for an office. In the presence of ncandidates, each voter
gives a mark of n to the one arriving first in his or her preference, (n−1) to the
second, and so on, ending with a mark of 1 for the least preferred. The winner
is the candidate who gets the highest mark once all voters’ choices have been
added up.34 Condorcet’s opinion is that this method could sometimes suffer
from the same defect as the usual method it is supposed to correct (Condorcet
1785, clxxvii–clxxix; 1788, I, “Note première”).
“Social mathematic” and social choices
In dealing with how to take decisions in any kind of assembly, be it a
political assembly or a tribunal, Condorcet’s aim was to develop some ideas
presented by Jean-Jacques Rousseau (1712–1778) in Du contrat social (1762)
– a treatise Turgot himself had praised – and to clarify Rousseau’s concept of
“general will” (see, for example, Baker 1975, 229–31; Grofman and Feld 1988;
Estlund et al. 1989). It was not clear how this “general will” could be known,
especially when voters could not distance themselves from their own interests
and passions, from factions or lobbies. The “general will”, Rousseau stressed,
must be distinguished from the “will of all”:
34 This rule was adopted by law in 1796 for the election of new members of the Institut
national des sciences et des arts, an adoption reported in La décade philosophique, littériare
et politique on 7 August of the same year. This is what prompted the Spanish mathematician
José Isidoro Morales (1758–1818) to write his Memoria matemática sobre el cálculo de la
opinion en las elecciones (Morales 1797, “Prólogo”) and to send the book to the Institut
(McLean and Urken 1997, 155).
The spirit of geometry. Quantification and formalisation 26
[T]he general will is always right [droite] and always tends toward the
public utility. But it does not follow that the deliberations of the people
always have the same rectitude. . . . There is often a considerable
difference between the will of all and the general will. The latter considers
only the common interest, while the former considers private interest and
is merely a sum of particular wills. (Rousseau [1762] 2012, II, iii, 182)
Condorcet’s 1785 ambitious Essai sur l’application de l’analyse à la
probabilité des décisions rendues à la pluralité des voix deals with the
various ways to organise a vote, to fix the majority needed for the deci-
sion, and to estimate their relative advantages – building, as G.-G. Granger
(1956, Chapter 3) called it, a model of “homo suffragans”. The cases studied
are numerous, and here too, Condorcet’s project was only partly achieved:
starting with a set of strong simplifying hypotheses, the analysis becomes
mainly programmatic when some of these hypotheses are relaxed. In the first
part of the book (1785, xxi–xxii, 3), Condorcet supposes that voters (1) are
equally enlightened, (2) try honestly to answer the question asked (nobody
tries deliberately to influence others, there are no lobbies, no parties) and (3)
have only the public good in mind, ignoring their own private interests.
Rousseau had already set out these hypotheses in Contrat Social.
Condorcet’s approach is, however, more detailed and systematic, with some sig-
nificant differences: (1) the object of the vote can be any decision which needs
to be taken in the public or private sphere, and not necessarily a
“general object” (a law) as in Rousseau; (2) the outcome of the voting process
must not only be “right” and honest because emanating from the assembly of
virtuous citizens (nor must it deal with liberty and utility, as in the Ancient
peoples whose concern, in assemblies, was only to counterbalance the passions
and the interests of different groups in society): it must also comply with
“truth” – the voting process is a collective quest for “truth”, on the model of a
jury in a tribunal, what was called an epistemic approach to democracy; (3) in
the political sphere, Condorcet was in favour of a representative assembly: the
most important thing is the truth of the decision, and the size of the assembly
should be adapted according to the degree of enlightenment of its members;
(4) in this perspective, Condorcet introduced an additional and central vari-
able, the probability for each voter of making the “true” choice, and an addi-
The spirit of geometry. Quantification and formalisation 27
tional simplifying assumption: this probability is the same for all.35 Needless
to say, he was conscious that all these hypotheses were strong theoretical as-
sumptions and he progressively tried to relax some of them. On this basis,
nevertheless, two results are particularly striking (1785, 3–11): in the liter-
ature, they are known as the “jury theorem” and the “Condorcet effect” or
“paradox of voting”.
The jury theorem
Let v(vfor “vérité”, that is, truth) be the probability for each voter of
making the right choice, and e(efor “erreur” or error) the probability of
being mistaken: e= (1 −v). Suppose a dichotomous choice (for example, is
a person guilty or not guilty of a crime?) in which the number of voters is n
and qis the required majority in terms of a number of votes. Condorcet asked
two questions: (1) before the vote, what is the probability pof obtaining a
decision complying with the truth? (2) once the decision is made, what is,
for an external observer, the probability p∗that this decision complies with
the truth?36 In modern parlance (see, for example, Granger 1956, 105–106),
probability pis found using Bernoulli’s binomial distribution. It is the sum,
for all x,q≤x≤n, of the probability vx(1 −v)n−xthat a decision is true
when it obtains xvotes, multiplied by the possible number of occurrences of
this event: n
x!=n!
x!(n−x)!
that is,
p=
n
X
x=q
(n
x)vx(1 −v)n−x(1)
35 Condorcet also formulated the condition of independence of irrelevant alternatives
(Young 1988, McLean 1995).
36 “One should not confuse the probability of having a decision which is true, with the
probability that a decision which one supposes made complies with truth: the first is not
only contrary to the probability of having a wrong decision, but also to that of having no
decision; the second is only contrary to that of having a wrong decision” (Condorcet 1785,
xix–xx).
The spirit of geometry. Quantification and formalisation 28
Probability p* is found using the Bayes–Laplace theorem and is given by:
p∗=vq
vq+ (1 −v)q(2)
From Equation (1), p→1when n→ ∞ if v > 0.5;p→0if v < 0.5; and in
case v= 0.5,p= 0.5for all n. This is the “jury theorem”: in an assembly in
which the probability for each voter of making the right choice is greater than
0.5, the probability of the outcome being true increases with the number of
voters. Conversely, when v < 0.5, the probability of the outcome being true is
a decreasing function of this number (1785, xxiii–xxiv, 6–9). From Equation
(2) – in which the number of voters plays no role – and all other things being
equal, p∗is an increasing function of vand q.
The positive side of the story is the proposition that – under the very
restrictive conditions noted earlier – an assembly could collectively have a
degree of wisdom superior to that of its individual members, and, if v > 0.5,
this degree increases with the number of voters. This is the kind of statement
already made by Aristotle when, examining the different possible political
regimes, he declared that it is possible that many individuals, of whom no
one is “virtuous”, are collectively better than the best ones among them when
they are assembled (Politics, III, 11, 1281-a). Condorcet’s theorem could thus
be understood as a powerful argument in favour of democracy. However, if
v < 0.5, the opposite conclusion applies:
it could be dangerous to give a democratic constitution to an
unenlightened people: a pure democracy could even only suit a people
much more enlightened, much more free of prejudice than any of those
we know in history. (1785, xxiv)
A pure democracy would nevertheless be acceptable if decisions are
“limited to that which regards the maintaining of safety, liberty and
property, all objects on which a direct personal interest can enlighten
everybody” (1785, xxiv; see also 135) – these topics belonging precisely to
the “general” or “universal” objects in Rousseau’s approach. Alternatively, to
decide on an issue, the assembly could designate a committee composed of its
most enlightened members and then judge, not the decision itself, but whether
the decision is detrimental to justice or to certain fundamental human rights
(1785, 7). Condorcet, however, relativised the importance of the choice to be
The spirit of geometry. Quantification and formalisation 29
made between the different possible devices. For him, the key variable remains
the probability for each voter of being right or wrong: hence his tireless action
in favour of public instruction and science.
[T]he happiness of men depends less on the form of assemblies that decide
their fate than on the enlightenment of those who compose them, or, in
other words, . . . the progress of reason affects their happiness more
than the form of political constitutions does. (1785, 136; see also lxx)
The Condorcet effect, or paradox of voting
The second main point which attracted attention in the 1785 Essai is the
possible intransitivity of social choices resulting from the aggregation of
individual choices made by rational voters. Suppose, Condorcet wrote, that
voters have to rank three candidates or proposals A,Band C, through
pairwise comparisons (1785, 120–21). For each voter, there are a priori eight
possibilities (“XY ” meaning “Xis preferred to Y”): (1) AB,AC ,BC; (2)
AB,AC,CB; (3) AB,CA,BC; (4) AB,CA,CB; (5) BA,AC,BC; (6) BA,
AC,CB; (7) BA,CA,BC and (8) BA,CA,C B. Choices (3) and (6) are not
transitive and will not be chosen by a rational voter. But, at the aggregate
level, outcomes (3) and (6) are possible. Imagine that, among 31 voters, nine
vote for (1), two for (2), seven for (4), four for (5), six for (7) and three for (8).
Eighteen voters prefer AB against 13, 19 BC against 12, and 16 CA against
15, with the cyclic result ABCA.
Of course, this outcome has significant consequences for any social choice
theory based on the aggregation of individual choices, and Arrow’s so-called
impossibility theorem is well known, stating that there is no procedure for
the aggregation of individual choices guaranteeing a transitive social ranking,
while at the same time respecting some seemingly mild axioms expressing
“individualistic concerns” (social choice should reflect individual choices in
some minimal way). But this is not Condorcet’s approach: he did not think
that the paradox of voting was such an important problem, even when the
numbers of alternatives and voters grow – and it has been shown that the
probability of having a “Condorcet effect” quickly increases with them. He did
The spirit of geometry. Quantification and formalisation 30
not get locked in a logical dilemma, but proposed solutions to the impasse.37
In particular, in a three-alternative case, one simple solution (Condorcet 1785,
122) consists in referring to the total number of votes that each candidate or
proposal obtains against the two others. In the earlier example, AB and AC
obtain together 18 + 15 = 33 votes, BA and BC 13 + 19 = 32 votes and CA
and CB 16 + 12 = 28 votes. The winner is A.
In this respect, it is important to emphasise that Condorcet’s approach is
aimed at discovering “the truth”, even in decisions which do not deal with
justice but with choosing the right proposal or candidate. He was convinced
that on these occasions, thanks to reason and science, there exists a truth,
never imposed from above but which can be known provided those who decide
are enlightened enough and follow the right procedure. As Rousseau had
already insisted, a member of an assembly, when voting, must not express his
own preferences but decide whether the proposal under examination does or
does not comply with the common good. The “will of all” can differ from
the “general will” whenever individuals are unable to distance themselves from
their private or partisan interests. The same is true with Condorcet. He is
dealing with judgements. For him, contrary to Arrow later, the problem does
not consist in aggregating individual preferences and obtaining social choices
respecting the “particular wills” or “private interests” of voters because in this
case the result would be the “will of all”, not the “general will”. Two different
conceptions of democracy and the role of the State are in play here.
When he [a man] submits himself to a law which is contrary to his opinion,
he must say to himself: It is not here a question of myself alone, but of all;
I must therefore not behave according to what I believe to be reasonable,
but according to what all, distancing themselves, like me, from their
opinion, must consider as complying with reason and truth. (Condorcet
1785, cvii)
Condorcet’s ideas were subsequently discussed by two authors before the
theme sank into oblivion for decades: in 1794 by the Swiss mathematician
Simon Antoine Jean Lhuillier (1750–1840), and by Pierre Claude François
Daunou (1761–1840) who presented a memoir on this theme in 1800 at the
37 In modern terms, these solutions are the maximum likelihood estimation, Kemeny’s
rule or the search for a median in a metric space. See Black (1958, Chapter 18), Young
(1988, 1995) and Monjardet (2008).
The spirit of geometry. Quantification and formalisation 31
Classe des sciences morales et politiques of the Institut national des sciences
et des arts (Lhuillier 1794; Daunou 1803).38
3From calculation to calculus
The context
As has been shown, probability theory started to be applied to socio-economic
questions. But mathematics proper, and not probability, became fashionable
among the educated elite of the time. Mathematics and sciences, above all
astronomy, fascinated learned people in an almost exclusive way before ques-
tions of political economy and political philosophy became the centre of
attention in literary and philosophical circles: Voltaire, Émilie du Châtelet,
Georges-Louis Leclerc de Buffon or the young Diderot, for example, at some
point translated Newton or published in the field, and Rousseau himself was
proud of his mathematical knowledge. Nor must we forget that, at that time,
being involved in mathematics and science also had philosophical and even
theological implications. Moreover, mathematical operations were thought by
prominent scientists and philosophers to reflect the operations of the human
mind (Richards 2006). A version of this view is clearly stated in Condorcet.
Mathematics deals with identities, which are combined in different ways – but
they are not tautologies: “the various combinations of the same elements are
not the same thing” (Condorcet [1786] 1847–49, 470). The object of math-
ematical analysis “is nothing other than the various combinations of a sole
idea”:
In the most difficult equations . . . one can always arrive at two equal
terms, which . . . if one analyses them, will be the same combination
of this idea but stated in a different way. A science where all results are
only identical propositions, where all the terms are just arranged ideas of
a sole idea . . . must be exempt of any ambiguity, error or uncertainty.
A mind which is accustomed in this way to follow the truth on a sound
basis . . . forms the happy habit of sorting out and grasping the truth,
with whatever subject of science or conduct it is dealing. (in Condorcet
1994a, 238–9)
38 See, for example, McLean and Hewitt (1994) and McLean and Urken (1997).
The spirit of geometry. Quantification and formalisation 32
Mathematical analysis also has another advantage “independently of the
useful usages to which it could be applied”: that of “perfecting the mind”
(1994a, 239).
A revolution in mathematical thinking
The period saw a great change in mathematics itself. During the seven-
teenth and eighteenth centuries, it developed in a spectacular way in re-
lation to astronomy and physics, with a decisive shift in focus and meth-
ods: the old Euclidian geometry was downgraded and algebra and analysis
were put to the fore with the emergence of differential and integral calculus.
However, the terms “géométrie” and “géomètre”39 continued to be used to re-
fer to mathematics and mathematicians in general. Algebra was also called
“symbolic geometry, because of the symbols algebra uses in the solution of
problems” or “metaphysical geometry . . . because what is specific to meta-
physics is the generalisation of ideas, and not only does algebra express the
objects of geometry in general characters, but it can facilitate the application
of geometry to other objects” (D’Alembert, in Encyclopédie, vol. VII, 1757,
637).
The system of Newton largely replaced that of Descartes and, among
mathematicians, the creation of infinitesimal calculus, especially by Leibniz,
was developed by Jakob and Johann Bernoulli (the phrase “integral calculus”
was coined by the Bernoullis who convinced Leibniz to adopt it) and the main
mathematicians of the century like Euler and d’Alembert. Material advances
in probability theory were also based on infinitesimal calculus. These inno-
vations, however, were not accepted without difficulty. D’Alembert’s objec-
tions to probability theory have already been alluded to. Infinitesimal calcu-
lus was also challenged. A celebrated controversy was, for example, generated
by George Berkeley with his 1734 The Analyst, or A Discourse Addressed to
an Infidel Mathematician, where he criticised both Newton and Leibniz and
stated that infinitesimal magnitudes – let alone the system of higher-order in-
finitesimals – were incomprehensible and that some basic demonstrations were
flawed. The controversy was known on the continent and an echo of it was
39 A “géomètre” is “a person well-versed in geometry; but this name is also given in general
to any mathematician . . . Thus one says that Newton was a great geometrician, to say
that he was a great mathematician” (D’Alembert, in Encyclopédie, vol. VII, 1757, 627).
The spirit of geometry. Quantification and formalisation 33
still to be found in Alexandre Savérien’s 1766 Histoire des progrès de l’esprit
humain dans les sciences exactes, et dans les arts qui en dépendent (where
Berkeley is referred to as “a man of bad temper”).
Despite some lively debates, the temptation emerged to use mathematical
analysis in “moral sciences”. Was analysis not already an essential element in
astronomy or physics? This idea was at the very heart of Condorcet’s project
of “social mathematic”, even if he mainly dealt with probabilities. But it
also characterised some other programmatic discourses, such as that of Cesare
Beccaria (1738–1794)40 in Italy. The same year (1764) that he published Dei
delitti e delle pene – which brought him incredible fame in France and Europe
– he also published a short mathematical article on a fiscal aspect of smuggling,
“Tentativo analitico sui contrabbandi”, in Il Caffè, the journal of the Milanese
Enlightenment, beginning with these words:
Algebra being simply a precise and straightforward technique for
reasoning about quantities, it can therefore be employed not only in
geometry and in other mathematical sciences but also in the analysis
of anything that in some manner is capable of increasing or decreasing
. . . Therefore, political sciences can also make use of it, up to some
point. They deal with the debts and assets of a nation, taxes, etc., all
items which can be treated as quantities and subjected to calculation.
([1764] 1968, 149*)41
Beccaria, however, like Condorcet, thought that ridiculous misuses of
mathematics had to be cautiously avoided.
I said, “up to a point” because political principles are highly dependent
on many isolated wills and a wide variety of passions which cannot be
specified precisely: a policy composed of numbers and calculations would
be ridiculous and much more suitable to the inhabitants of the island of
Laputa than to present-day Europeans. ([1764] 1968, 149*)
Historical difficulties
Despite the novelty and difficulties of the task, various attempts to use
mathematics – or at least symbols – in economy were made in the second
40 On Beccaria, see especially Bianchini ([1982] 2002).
41 An asterisked page number means that the translation has been modified.
The spirit of geometry. Quantification and formalisation 34
half of the century.42 But in France as in other countries like England and
above all Italy, a great difference with what happened in probability theory
must be noted. While probabilities were used in their most recent and complex
developments (for example, the Bayes–Laplace theorem on inverse probabil-
ity), attempts at some formalisation of economic theory used at best simple
algebra and not the most recent achievements in mathematical analysis. More-
over, the authors who gave impetus to the application of probability theory
to socio-economic subjects were the same who developed this theory. But this
was not the case with those who used mathematical analysis to deal with po-
litical economy: even when they were engineers or mathematicians, they were
not leading scientists and contented themselves with using simple tools, and
this could explain some shortcomings.
This last point is best illustrated by the new mathematical concept of
function. With the development of infinitesimal calculus, mathematicians
dealt with increasingly complex functions: but, during most of the century,
these functions were always specified, and not written in a general form like
f(x). Still in 1758 in his Histoire des mathématiques, Montucla reported the
definition of the new concept – for the first time stated by Johann Bernoulli
in 1718 – where the examples43 were carefully specified:
With the present-day mathematicians, we call here function any
expression made in some way of constant and variable magnitudes; for
example, p(aa ±xx),aa +xx,mx +mm −1,xx, etc., are functions of
x. (1758, II, 301n)
Yet the use of one letter to express a general functional form had already
been proposed by Johann Bernoulli and made more precise by Alexis Claude
Clairaut (1713–1765) and Euler in memoirs presented in 1734 – respectively, at
the Académie royale des sciences in Paris and the Imperial Academy in Saint
Petersburg (Clairaut 1736; Euler 1740). Clairaut used “different signs like
Πx,Φx, . . . to express different functions in general” (1736, 197) and Euler
introduced the notation f(x)– in the present case f(x
a+c)– which is still used
today (1740, 186–7). But these notations were only to impose themselves some
decades later – they were not used (to our knowledge) by their authors in their
42 Two classical surveys are still useful: Robertson (1949) and Theocharis (1961).
43 The notation aa meaning a2, etc.
The spirit of geometry. Quantification and formalisation 35
didactic books like Clairaut’s Éléments d’algèbre (1746) or Euler’s Éléments
d’algèbre (first published in Russian in 1770, then in German, and translated
into French in 1774 by Johann III Bernoulli44 ). Moreover, a clear distinction
prevailed between pure or abstract mathematics and “mathématiques mixtes”
or applied calculus: in applications (at that time almost exclusively to physics),
functions had to be cautiously specified in order to find solutions to specific
problems and calculate the value(s) of the relevant unknown(s). This practice
had important consequences on attempts to formalise some propositions in
political economy, as the use of mathematics in this field fell within the domain
of mixed mathematics.
Implicit formalisation
Before coming to technical aspects linked to the use of mathematics in
political economy, it is worth noting that, even in the absence of any
symbol or equation, the language of mathematics could also permeate social
and economic discourse. During our period, this language was sometimes used
without explicit formalisation and was not necessarily metaphorical: a simple
literary explanation was thought sufficient to explain a theoretical point. Two
outstanding examples of this practice are to be found in Turgot and Condorcet
(see Chapter 6, this volume). The first is the development Turgot gave to the
theory of value and price, the equilibrium exchange ratio between two com-
modities (the “appreciative value” and, by extension, the interest rate) in a
bilateral monopoly being determined by the “average esteem value”, that is, a
situation in which the difference in the esteem values of the quantity of the
received good over that of the quantity of the good given in exchange is equal
for both parties. The second example is the determination by Condorcet of
the equilibrium amount of public expenditure and taxation: referring to the
idea of a decreasing marginal utility of income (or wealth), he stated that the
equilibrium condition is that the utility brought by the additional amount of
public expense be equal to the “pain” or disutility generated by the additional
amount of taxes necessary to finance it.
There are some other instances in which the use of mathematical language
is more striking but somewhat metaphorical. The first and perhaps most
intriguing is to be found in Rousseau’s Contrat social (1762), when the author
44 Grandson of Johann and nephew of Daniel.
The spirit of geometry. Quantification and formalisation 36
is dealing with the concept of “general will” as distinct from the “will of all”.
It has been suggested above that this concept needed to be more accurately
explained, and that this was precisely Condorcet’s aim in his 1785 Essai. To
state his views, Rousseau (1762, Book 2, Chapter 3: “Whether the General
Will can Err”) had recourse to the mathematical image of the “plusses and
minuses” of particular wills, which cancel each other out to give the general
will.
There is often a considerable difference between the will of all and the
general will. The latter considers only the common interest, while the
former considers private interest and is merely a sum of particular wills.
But take away from these same wills the pluses and minuses, which
mutually cancel each other out, and the remaining sum of the differences
is the general will. ([1762] 2012, 182)
Commentators were puzzled by these sentences, until it was noticed that
they refer to integral calculus: Rousseau’s thought is explained on this
basis (Philonenko 1984, 1986; see Dobrescu 2010 for some developments). But
another, still more intriguing passage is to be found in Book III (Chapter 1,
“On Government in General”), when Rousseau explains the relations existing
between the government, the sovereign and the people. The relation between
the sovereign and the people
can be expressed as the relationship between the extreme terms of a
continuous proportion whose proportional mean is the government. The
government receives from the sovereign the commands which it then gives
to the people, and in order for the state [the people] to be in proper
equilibrium it is necessary, taking everything else into account, for the
product or power of the government taken by itself to be equal to the
product or power of the citizens, who are sovereigns from one perspective
and subjects from another. ([1762] 2012, 206)
Among other things, Rousseau was trying to show that many forms of
government can exist, depending on the number of the citizens, but once the
number is fixed, only one form suits. This passage was not understood until
Marcel Françon (1949) showed that Rousseau’s sentences and language were
perfectly correct according to the mathematical vocabulary used in his time to
deal with proportions, and were designed to illustrate an important theoretical
point.
The spirit of geometry. Quantification and formalisation 37
Allusive mathematics was also used by a central figure in the lively debate
about free trade in grain: Ferdinando Galiani (1728–1787). He too refers to
infinitesimal calculus (Faccarello 1998) and more especially to what was called
at that time, after Leibniz, the calculus of “maximis et minimis”, that is, the
extrema on a curve. The method of “maximis et minimis” was defined by
d’Alembert, in volume X of the Encyclopédie, 1765, as the method to discover
the point, the place or the moment at which a variable quantity becomes the
greatest or the smallest possible, given its law of variation (its function). In his
Dialogues sur le commerce des bleds, and in his polemical stance against the
physiocrats, Galiani states that “all problems in political economy boil down
to doing some good for the people” but also that “there is no good which is not
mixed with some evil which often weakens or balances it” (1770, 228). How to
solve “the equation of the problem”?
When in a problem there are several unknowns, the equation becomes
indeterminate or belongs to the class of problems we call maximis and
minimis, and such in fact are all political problems. It is a question of
doing the greatest possible good with the least possible evil . . . There
is a point, a limit before which the good is greater than the evil; if you
exceed it, the evil becomes greater than the good. (1770, 229–30)
Later on, Galiani returned several times to this question. In De’ doveri de’
principi neutrali verso i principe guerregianti, he generalised the approach to
morals.
Every moral question . . . is a composite problem always amounting
to how, in a given case, one can obtain the greatest good for oneself at
the cost of the smallest possible injury to others, or again how one can
obtain the greatest good for other men at the cost of the least trouble to
oneself. (1782, pp. 16-17)
This kind of problem, expressed by an “undefined” equation, cannot be solved
in general but only when particular data are specified. It is thus impossible to
determine whether, in general, total freedom of the external grain trade is to
be granted. But this is possible if the place and time are known. Galiani thus
describes what an economist today is tempted to see as a first expression of
the reduced form of a model, with its constraints and parameters.
It is not because a problem is indeterminate that it cannot be resolved.
It is resolved by means of a general equation, itself also indeterminate,
The spirit of geometry. Quantification and formalisation 38
composed of several unknowns and comprising all cases. When the
unknowns . . . are specified in the particular circumstances of cases,
this equation adapts itself to each circumstance and resolves the cases.
(1782, 35n)
The language of “maximis et minimis” is also to be found later in other
authors45 and finally forms another version of the maximising attitude already
stressed in the literature, or an additional argument in favour of
Boisguilbert’s theological basis or Turgot’s sensationist approach. It is also
linked to Leibniz’s theological views and rehabilitation of the old and
disparaged idea of the existence of final causes in nature, an approach
developed by Pierre Louis Moreau de Maupertuis (1698–1759) with his (at
that time) much-debated “principle of least action”. This principle stated that,
in physics, for any change happening in nature, the quantity of “action” needed
by this change is always as small as possible, that is, a minimum: in the pro-
duction of its effects, nature does so always by the simplest means. It could be
easily adapted to the moral world, where actions were taken by men precisely
with a view (the final cause) of an efficient result. Hence the many statements
that can be found in certain authors: Achilles-Nicolas Isnard (1748–1803), for
example, (more on him below) who, in his Traité des richesses, stated that the
utilisation of productive resources must be such that “the costs are as small as
possible relative to production” (1781, I, xv; 2006, 72)46 in all sectors, includ-
ing agriculture (1781, I, 47; 2006, 130); that as man cannot enjoy anything
without labour, “he has to aim to enjoy as much as possible with the least
labour and at the lowest possible cost. This consequence is engraved in all
the tables of the laws of human industry” (1781, I, 9; 2006, 90); and that, in
exchanges, man is thus inclined to “give as little as possible in order to receive
as much as possible” (1781, I, 19; 2006, 100).
45 A reference to this method was again made by Malthus in 1814: “Many of the questions
both in morals and politics seem to be of the nature of the problems de maximis et minimis
in Fluxions; in which there is always a point where a certain effect is the greatest, while on
either side of this point it gradually diminishes” (1814, 30).
46 The reference “2006” refers to van den Berg (2006), where substantial excerpts of
Isnard’s writings are translated.
The spirit of geometry. Quantification and formalisation 39
Early attempts
Attempts to use algebra and somewhat to formalise economic reasoning in
France did not happen in a vacuum but in a European environment that it is
important to mention, in order to put these developments into an appropriate
historical perspective. The first attempt, in 1753–1754, made by Forbonnais
was rather timid and mainly characterised by the use of simple symbols in
monetary matters. Moreover, it was not original: the use of symbols in “moral
sciences”, while unusual, was not new. It had already been implemented in
Italy, in the field of monetary theory, in the wake of the scientific impulse
given by Galileo Galilei, by Giovanni Ceva (1647–1734), an administrator and
mathematician, and Trojano Spinelli (1712–1777), a man of letters (Ceva 1711,
Spinelli 1750).47 Ceva, for example, in his De re numaria (1711), used letters –
G,H,I,K,Lfor different periods of time, aand dfor amounts of “population”,
band efor quantities of money and c,hand ffor external values of money (that
is, in terms of commodities) – and simple algebra to calculate, for example,
c:fwhich shows the evolution of the value of money as a function of the other
variables and is supposed to give some indications for the economic policy of
the Prince.48 His reasoning, moreover, is presented in terms of definitions,
axioms, scholia, theorems and corollaries.
There is no reason to believe that Forbonnais knew the books by Ceva or
Spinelli in adopting a similar approach in the monetary field. His purpose
was, however, less ambitious and limited to an analysis of the exchange rates
between currencies, at par or unbalanced. He first did so in his 1753 entry
“Change” in the third volume of the Encyclopédie – republished as Chapter 8
of his 1754 Éléments du commerce – and in a new text (Chapter 9 of Éléments),
“De la circulation de l’argent”. Suppose, he wrote, the following bilateral ex-
change rates between three currencies: a=b,b=cand c=a. In this example,
47 For a thorough analysis, see Bianchini ([1982] 2002, Chapters 1 and 2). Ceva’s text is
written in Latin. Moreover, the meaning of some terms he uses – for example “population”
– is peculiar and can generate misinterpretations.
48 A very simple use of symbols, but in a far less interesting way – the calculation of
simple statistical means – was also made by the Venetian Giammaria Ortes in his 1757
Calcolo sopra il valore delle opinioni umane. Ortes wanted to express the “opinion” that a
society has of some groups, of which it is composed, that is, the importance of each group
in the social order and access to riches. This attempt simply boils down to expressing this
“opinion” by the per-capita income of a group (for example, the “noble families”, or the
“men of letters”, etc.), the global income of a group and the number of its members being
expressed by letters, and then replaced by figures ([1757] 1804, 270–78).
The spirit of geometry. Quantification and formalisation 40
exchanges are at par and there is no incentive to exchange currencies with each
other. But if, for example, c=a+d(dhere being an increment of cand not
a new currency) all other things being equal, arbitrages will be made until a
new equilibrium at par will be reached (1754, II, 61–2). The analysis is then
adapted to deal with the metallic contents of coins and their variations (II,
76–9). A much more interesting use of algebra in the field of monetary theory,
and more in the spirit of Ceva, was made later by Henry Humphry Evans Lloyd
(1718–1783) – a Welsh military officer who served several continental powers
in Europe and was closely connected to the Milanese Enlightenment – in An
Essay on the Theory of Money (1771).
Mathematics had also been used in moral philosophy since the beginning of
the eighteenth century by the Scottish philosopher Francis Hutcheson (1694–
1746) in An Inquiry into the Original of Our Ideas of Beauty and Virtue
. . . With an attempt to introduce a Mathematical Calculation in Subjects of
Morality (Hutcheson 1725) and to a lesser extent in An Essay on the Nature
and Conduct of the Passions and Affections (1728).49 Hutcheson was looking
for “a Universal Canon to compute the Morality of any actions, with all their
Circumstances” (1725, 168) when judging some action. In this perspective, he
stated several “axioms” like the following:
The moral Importance of any Character, or the Quantity of publick Good
produc’d by him, is in a compound Ratio of his Benevolence or Ability:
or (by substituting the initial Letters for the Words, as M=Moment
of Good, and µ=Moment of Evil ), M=BA . . . The Virtue then
of Agents, or their Benevolence, is always directly as the Moment of
Good produc’d in like Circumstances, and inversly as their Abilitys: or
B=M/A. (1725, 168–9)
This kind of approach is important because it was to be adopted some decades
later, first in Italy and then in France, to deal with the theory of value and
prices. It is first characterised by an absence of reflection on the way symbols
are taken for concepts and then concepts changed into variables and measured:
49 See Brooks and Aalto (1981). Mathematics is used in the three first editions of the
Inquiry, in “Treatise II. Concerning Moral Good and Evil”, Subsections XI and XII of Section
III (1725, 168–74; 1726, 182–90; and 1729, 186–93) and at the end of the book (1725,
267; 1726 and 1729, 290) – some similar formalisation also being used in Hutcheson (1728,
304–305). It disappears in the fourth edition, 1738, but the reasoning remains the same
and literary sentences express the same mathematical relations. A French translation was
published in 1749, but of the fourth edition.
The spirit of geometry. Quantification and formalisation 41
in the above quotation, for example, benevolence is simply noted by Band
ability by A, as if they were measurable magnitudes or statistical data. It is
also characterised by the use of the simplest functional relationships between
these variables: the “moment of good” is defined as Bmultiplied by A– in
other words, an increasing function is represented by a multiplication, and
a decreasing function by a division. For the reasons noted earlier, authors
were still unable to deal with general functional forms: they specified the
functions every time, and in these applications, they always chose the simplest
form. Even Daniel Bernoulli partially followed this approach in his 1738 Saint
Petersburg paper on risk: to express the hypothesis that the variation in the
utility, dy, of a variation in income (or wealth) dx is a decreasing function
of this income (or wealth) x, he wrote that dy is inversely proportional to x,
multiplie