From 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.

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.

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 seventeenth and eighteenth centuries, it developed in a spectacular way in relation to astronomy and physics, with a decisive shift in focus and methods: 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 “geometrie” and “geometre”^[151] continued to be used to refer 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 metaphysics 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 Encyclopedie, 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 innovations, however, were not accepted without difficulty. D’Alembert’s objections to probability theory have already been alluded to. Infinitesimal calculus 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 infinitesimals - were incomprehensible and that some basic demonstrations were flawed.

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)^[152] 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 con- trabbandi”, in Il Caffe, 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 sci­

ences 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*)^[153]

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 ofthe task, various attempts to use mathematics - or at least symbols - in economy were made in the second half of the century.^[154] 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 probability), attempts at some formalisation of eco­nomic theory used at best simple algebra and not the most recent achievements in mathematical analysis. Moreover, 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 political 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 mathematiques, Montucla reported the definition of the new concept - for the first time stated by Johann Bernoulli in 1718 - where the examples^[155] were carefully specified:

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 Aca- demie royale des sciences in Paris and the Imperial Academy in Saint Peters­burg (Clairaut 1736; Euler 1740).

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 (Chapter 6, this volume). The first is the development Turgot gave to the theory of value and price, the equilibrium exchange ratio between two commodities (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.

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 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 Dobresco 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 Franςon (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.

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 Encyclopedie, 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, 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, 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 authors^[157] 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 production 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 Traite 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)^[158] in all sectors, including 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).

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 1753-54, 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 (1712­1777), a man of letters (Ceva 1711, Spinelli 1750).^[159] Ceva, for example, in his De re numaria (1711), used letters - G, H, I, K, L for different periods of time, a and d for amounts of “population”, b and e for quantities of money and c, h and f for external values of money (that is, in terms of commodities) - and simple algebra to calculate, for example, c:f which 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.^[160] 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 Encyclopedie - republished as Chapter 8 of his 1754 Ele­ments du commerce - and in a new text (Chapter 9 of Elements'), “De la circulation de l’argent”. Suppose, he wrote, the following bilateral exchange rates between three currencies: a = b, b = c and c = a. In this example, exchanges are at par and there is no incentive to exchange currencies with each other. But if, for example, c = a + d (d here being an increment of c and 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).^[161] 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 = BxA.... 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: in the ear­lier quotation, for example, benevolence is simply noted by B and 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 B multiplied 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, multiplied by a constant b:

Condorcet criticised this kind of formalisation, for example, in his “Notes sur la these de Nicolas Bernoulli”, where he referred to Daniel’s paper, while at the same time accepting the idea of a decreasing additional utility of income or wealth (Condorcet 1994a, 582-4). But his reaction was more explicit in an exchange of letters with Pietro Verri (1728-1797) on the occasion of the publication of Verri’s Meditazioni sulla economia politica in 1771. In the fourth section of his book, Verri, probably inspired by some formulation found in Richard Cantillon’s Essai sur la nature du commerce en general (1755),5⁰ adopted a mathematical language to describe how the prices of commodities are fixed. They depend, he wrote, on the confrontation of supply and demand in markets and, while admitting that his state­ment is approximative, he symbolised the supply of a commodity by the number of sellers, and the demand for it by the number of buyers, and stated that the price varies like the ratio of the number of sellers to the number of buyers.

Allow me to use the language of the science which deals with quantities.

The number of sellers being equal, prices are proportional to the number of buyers: the number of buyers being equal, prices rise in proportion of the diminution of the number of sellers; combining the two proportions and sup­posing that the numbers of sellers and buyers are not equal... the prices of things are directly proportional to the number of buyers and inversely pro­portional to the number of sellers.

(Verri 1772, 33-4)

In a letter dated 7 November 1771, Condorcet disagrees.

You say that the price is inversely proportional to the number of sellers and proportional to that of buyers. I know that the price increases when the num­ber of buyers increases, and that it decreases when the number of sellers rises; but is it in the same ratio? I do not think so. Thus it seems to me that, in this case... the language of mathematics, far from leading to more precise ideas, is misleading.

(in Condorcet 1994a, 70-71)

Several months later, Verri sent him the (already) sixth edition of his successful book, containing an important novelty: the (anonymous) intervention of the math­ematician Paolo Frisi - a corresponding member of the Academie royale des sci­ences - who added mathematical notes in various places, indicated by “N.D.E.” (that is, editor’s notes), and a 17-page appendix, “Estratto del libro intitolato An Essay on the Theory of Money”, in which he summed up the principal propositions of Henry Lloyd’s book, compared them to Verri’s and tried to show that they pro­ceeded from the same spirit and were complementary. The variation of a price is formalised by comparing two situations: the first, where the price of a commodity is P, the number of sellers V(Vfor “venditori”) and the number of buyers C (C for “compratori”); the second where the same variables are p, v and c. The variation in price is written as (Frisi 1772, 37):

50 “It is clear that the quantity of Produce or of Merchandise offered for sale, in proportion to the demand or number of Buyers, is the basis on which is fixed or always supposed to be fixed the actual Market price” (Cantillon [1755] 1931, 119).

where A, B, M, m, n are constants, M, m and n being positive. But he states that imposing realistic constraints on the variables - such as P = 0 when C = 0 - boils down to using the simplest formulation above. When Verri introduces money - the “universal commodity”, the value of which is U - Frisi applies his formula to the variation in its value (1772, 84), and he does the same with the interest rate I which is supposed to depend on the supply O (“offerta”) and demand R (“ricerca”, that is, search) of money (1772, 92). With the same conventions as before:

It is also interesting to note that, in the context of Verri’s analysis of the beneficial effects of an increasing circulation of money, which is supposed to have real effects on the production of wealth, Frisi alludes to Maupertuis’s principle of least action in order to state, in a very loose analogy, a formula for the maximum wealth in a country.^[162]

In a second letter to Verri, Condorcet’s reaction is still negative. The variations in the (relative) numbers of buyers and sellers are not the sole elements influencing prices, and even if this were true, “how could we prove that the price is equal to the ratio of these two numbers multiplied by an invariable quantity?” (in 1994a, 72). As for Frisi’s formalisation, “the conditions with which he determines his func­tion are not sufficient: an infinity of other functions could satisfy them” (1994a, 73).^[163] In the realm of mixed mathematics, things are not so simple and a good deal of empirical data is required, as was the case with astronomy. They are, however, extremely difficult to obtain when the number of variables is large or cannot be measured. The use of the language of geometry

can lead to deal in an abstract way with questions which must only be dealt with on the basis of experience... and the most brilliant and truthful ideas... cease being so when you subject them to this analytical rigour and you seek to consider them as abstract numbers. The quantity of universal commodity, the quantity of a specific commodity, can be referred to as numbers; but the urge to buy and the urge to sell are not amenable to any calculation, and yet the variations in prices depend on this moral quantity which in turn depends on opinion and passions.... Thus, in all economic problems where quantity is involved, we should be very happy when we know that one is increasing and the other decreasing... one is positive and the other negative, large or small, and not attempt to measure them.

(1994a, 73-4)

As we know, Condorcet was not hostile to the application of mathematics to the moral sciences. He rather used probability theory, and also calculus, but the subject was carefully circumscribed and presented well-defined technical aspects, as in the case of life annuities, for example. This was also the case for the theory of taxa­tion: in a long note to his 1786 Vie de M. Turgot, he developed mathematically the consequences of a tax reform which would replace all indirect taxes with a direct tax on the net product of land (Condorcet 1786, 129-37; 1994a, 629-34). On the eve of the nineteenth century, however, as we will see below, the same approach he had criticised in the 1770s was still prevailing. But before coming to this point, some physiocratic ambiguities must be addressed.

Physiocratic ambiguities

Physiocrats are well known for their calculations, starting from the entries “Fer- miers” and “Grains” Quesnay wrote for the Encyclopedie, with his tables show­ing his strategic distinction between the “prix commun du vendeur” and the “prix commun de l’acheteur”, and the celebrated “Tableau economique” which, for the first time, gave a visual representation of the working of an economy (Chapter 5, this volume). But while they are sometimes (erroneously) associated with the use of mathematics and the formalisation of the economic discourse, nothing is further from the truth than the famous assertion by Georges Weul- ersse, the historian of physiocracy, and some commentators after him, that the physiocrats were the subjects of a mathematical daemon. It was only after a long period of neglect by economists that several authors drew inspiration from them, from the Tableau or its variant, the “Formule arithmetique du Tableau economique”, to present formalised relations between sectors - including Karl Marx with his schemes of reproduction of social capital - or a mathematical translation of the Quesnaysian schemes.^[164] [165]

However, with the Tableau, the meaning of the word “calculation” had some­what changed. It was not only a question of collecting and calculating data, but of establishing theoretical relations between them and obtaining a global theoretical view of the economy. It is in this sense that Mirabeau’s eulogy of calculation in Philosophie rurale is to be understood:

The Tableau Economique is the first rule of arithmetic invented to reduce to an exact and precise calculation... the perpetual implementation of this commandment of the Lord: In the sweat of thy face shalt thou eat bread.... For the economic science, calculations are just like bones for the human body. Nerves, blood vessels, muscles invigorate it and give it motion.... Economic science is developed in depth through observation and reasoning; but, without calculations, it would always be an uncertain science, confused and everywhere subject to mistake and prejudice.

(Mirabeau 1763, xix-xx)

This approach is further developed by Guillaume Grivel (1735-1810) in one of the two entries “Arithmetique politique” of the Encyclopedie methodique,⁵ stress­ing the political and philosophical role played by calculation: political arithmetic, Grivel writes, consists in expressing “the general interest of humanity” through principles “subjected to calculation, and confirmed by the results of calculation” (Grivel 1784, 241). Ultimately, however, all this refers to the computation and meaning of statistics and not yet the idea of using (simple) mathematics in political economy. As regards this use, three opinions present interesting variations: they are those of Quesnay himself, Pierre Samuel Dupont de Nemours (1739-1817) and Charles Richard de Butre (1725-1805).

Quesnay against the apostasy of "evidence”

Quesnay was hostile to the use of mathematics. He preferred simple arithmetic and a visual representation of the physiocratic theory. It is true that he thought he could convince a greater number of people with tables, pictures and figures. But his attitude is more interestingly the result of his epistemological position as regards the nature of geometry and the recent developments in mathematics. As a matter of fact, he was exclusively in favour of the old Euclidian geometry, the “geometrie demonstrative”, and against the new “geometrie metaphysique”, that is, algebra and infinitesimal calculus. His opinion was clearly stated in his last work,^[166] Recherches philosophiques sur l'evidence des verites geometriques, avec un projet de nouveaux elemens de geometrie (1773), in which he endeavoured to develop traditional geometry: “I wholly adopt Euclid’s demonstrative geometry and that of Mr Clairaut as long as it is demonstrative” (1773, ix). “One speaks... of my geometry, as if I had another demonstrative geometry than that of Euclid. I declare that I do not have any other one” (1773, xi).

The statements contained in this usually neglected book are not so surprising in Quesnay’s works: they are best understood when linked to the entry “Evidence” Quesnay published in 1756 in volume VI of the Encyclopedie, which lays the foun­dations of his epistemology - a narrow Lockean interpretation of sensationist phi­losophy. For him, “evidence”^[167] is one of the two bases of certainty - the other being faith - and the foundation of knowledge. It is defined as “a certitude which is so clear and manifest in itself that the mind cannot deny it” (1756, 146; 2005, 61), and is based on sensations:

“Evidence” necessarily results from a personal observation of our own sensa­tions I thus mean with “Evidence” a certainty to which it is as impossible

not to be subjected as it is impossible for us to ignore our present sensations.

(1756, 146; 2005, 61-2)

But our sensations generate two kinds of truth: “real truths” and “ideal or purely speculative truths”. The first are “those which consist in exact and self-evident relations that the real objects have with the sensations they provide”, that is, which are generated by real objects. The latter, on the contrary, refer to “the relations that sensations have between them”, independently of real objects: such as, for exam­ple, “the metaphysical, geometrical, logical or conjectural truths inferred from fac­titious or general abstract ideas. Dreams, deliriums and insanity also produce ideal truths” (1756, 151; 2005, 74). Now the certainty of our knowledge only originates in real truths: ideal or speculative truths are deceiving and uncertain.

It is in fact self-evident that factitious ideas do not have any relation to the objects, such as we perceived them through the use of senses: consequently, the truths they present cannot teach us anything about the reality and prop­erties of objects, nor about the properties and functions of the sensitive being - contrary to what we can only get when we grasp the real and exact relations between these objects and our sensations, and between our sensa­tions and our sensitive being. The certainty of our natural knowledge thus only consists in the “evidence” of real truths.

(1756, 151; 2005, 74)

It is on this basis that Quesnay opposes algebra and “the geometry of the imper- ceptibles”, that is, infinitesimal calculus, which are simply ideal or speculative truths. In a nutshell, metaphysical geometry loses touch with reality and can­not help us to understand it, when it does not simply create confusion. It only deals with abstractions which do not correspond to any real magnitude that our senses can grasp and measure because “these abstractions rule out any measur­able object” (1773, 32).^[168] Exceeding the limits set by nature, it cannot produce any certain knowledge, adds errors to our ignorance (1773, ii) and is just “un badinage d’esprit” (“banter of the mind”) (1773, 26). Simple calculations are sometimes useful, but calculations in themselves “are neither causes nor effects”, as Quesnay stressed in a note to his first “Probleme economique”: “In all sci­ences, the certainty consists in the ‘evidence’ of objects. If we do not reach this ‘evidence’ which supplies calculation with facts and data likely to be counted and measured, calculation will not correct our errors” (in Quesnay 2005, I, 614). The unavoidable and essential role of the senses in connection with real objects is again stressed: “If we go beyond the testimony of the senses, we leave the sphere of ‘evidence’” (1773, vi) and thus of knowledge.

There is for us no other “evidence” than that which is decided by the senses.... Senses are allegedly deceiving: yes; but senses themselves disabuse us, and only they can disabuse us: only they can assure us of the accuracy of geo­metrical demonstrations or uncover the errors which lead us to reject them.

(1773, v-vi)

The true geometry is thus “the science of visible magnitudes” (1773, vii) and must only deal with them: “demontrer c’est montrer”, Quesnay forcefully stresses, play­ing with the words “montrer” (to show) and “demontrer” (to demonstrate) - to demonstrate is to show (1773, v). There is no other geometrical “evidence” than that which compares given limited magnitudes to other given limited magnitudes (1773, xxx). In dealing with pure abstractions, metaphysical geometricians “attack the certainty of all physical knowledge on which men base their conduct and actions for their safety and preservation” and are like those sophists who “through abstract reasoning denied the existence of [external] bodies, of motion, etc.” (1773, 24-5). This leads to “the apostasy of the very ‘evidence’ of realities, which is the last excess of delusion of the human intellect” (1773, 64).

Quesnay’s position was probably not isolated, but it was nevertheless never taken seriously, all the more so since the results of his investigations in geometry (such as the problems of trisecting an angle or squaring the circle) were judged puzzling, if not ridiculous. In his “Eloge de M. Quesnay” published in Mercure de France, d’Alembert was ironic as regards Quesnay’s ‘“lucubrations geometriques”, and rather perfidious. A “too scathing mathematician”, he wrote,

said that the leader of a sect should not get involved in writing on geometry while not knowing it; because this damned science is the measure of the accuracy of the mind; and who talks nonsense in mathematics - where a good mind never errs - is more than suspect of not perfectly reasoning about other fields, where it is easier to err. It would have been too harsh and unjust to severely apply this apopththegm to an illustrious old man, consumed by his work. So nobody spoke in this way.

(D’Alembert 1778, 155-6)

Condorcet’s (1786) “Discours sur les sciences mathematiques” can well be seen as an answer to Quesnay and those who rejected the “new geometry”. Abstraction is for him a necessary step for achieving exact analysis. Mathematics removes all indi­vidual properties from the object it considers, which becomes “an abstract relation of numbers or magnitudes” and is referred to “by a letter or a line”. The real object is forgotten, “it ceases to exist for the mathematician. These signs, which are appar­ently arbitrary, are the sole object of his meditations” and it is only after completing his operations that he can apply his result to the real object under examination.

The indisputable truths found through this method seem at first sight to be only intellectual and abstract truths.... One could be tempted to believe that they do not belong to physical nature. But this would be a mistake, since they are real truths when the phenomenon to which you applied them exists in the universe just as you supposed it; and if your hypothesis is not rigor­ously accurate, the same methods will let you know to what extent the result of your calculations could diverge from nature, within which limits the real truth is included, and what is the degree of probability that it does or does not lie within narrower limits.

([1786] 1847-49, 469-70)

Despite Quesnay’s position, however, some appeals to mathematics and formalisa­tion, while extremely rare, were not completely absent from the physiocratic literature.

Dupont on political curves

Dupont de Nemours tried to go a step further and admitted that mathematics could have a useful role to play. Since the 1760s, his ambition was to provide a theory of prices - he was impressed by Quesnay’s distinction between the “prix commun du vendeur” and the “prix commun de l’acheteur” - but he had never been able to develop it. However, he expressed some views on the evolution of agricultural prices in a memoir titled “Des courbes politiques” (Dupont de Nemours [1774] 1892)^[169] which remained unpublished: the only manuscript we know can be found in a letter Dupont sent in 1774 to a son of Carl Friedrich von Baden. In this text, Dupont exam­ines the impact of the establishment of an excise tax on the price of some commodi­ties or services. According to the physiocratic doctrine, this tax falls in last resort on the “produit net”, diminishes the gross returns of the farmers and affects agriculture in a negative way. Suppose an excise tax on corn, finally paid by the producer. The before-tax price of corn, its “prix naturel” (say, £1,500), is the maximum price for the farmer, and the after-tax price (£1,000) the minimum - note that what Dupont calls “price” here is rather the total revenue of the farmer. If the tax is removed, the price received by the farmer returns to its former level. As for consumers, they always pay in the end the same price for the final commodity (say, £1,600) - this “price” being in fact their total expense for the farmer’s production. This price is a bit higher than the natural price for the farmer, depending on the number of necessary craftsmen, mer­chants or retailers who intervene between the first producer and the final consumer.

However, Dupont stresses, things are complicated to analyse in detail. (1) While the lowest and highest prices for the farmer are known, it is of interest to analyse step by step the progressive impact that the establishment or removal of the excise tax has on the price of corn, that is, the succession of market prices which pre­vail after the initial (negative or positive) shock took place, until the final price is reached. (2) The final consumer also has to face a fluctuating price because it is initially impacted by the disequilibrating fiscal shock, but this price fluctuates around the usual natural final price, £1,600. (3) The establishment or removal of the excise tax does not impact one sector only, but also progressively all other branches of production, through the use of the commodity as a means of produc­tion or of consumption for the workers - thus temporarily impacting their pro­duction costs, disturbing their prices and incomes and interacting with each other. (4) These movements of prices and incomes depend on the way the tax or its removal is passed on to buyers each time, which in turn depends on the many pos­sible commercial strategies of the agents.

On this basis, reasoning on the removal of the excise tax and with the hypothesis of a single specific and general commercial strategy in all sectors, Dupont draws a curve (on the left, Figure 8.1) showing how the price for the farmer (his total rev­enue) progressively rises from its minimum (£1,000) to its maximum (£1,500), the successive periods of time being indicated along the r-axis. A symmetrical curve (on the right on the same graph) roughly symbolises the related increasing pur­chasing power of the farmers and landowners - it is just the duplication of the first curve and does not add anything to the analysis - and finally a “serpentine” (above on the right) expresses the fluctuations in the price for the final consumer, around its normal level (£1,600).

Even disregarding the ambiguities of the text, Dupont’s result is puzzling and purely illustrative: the material difficulty he had to face is that of dealing with a

Figure 8.1 Stylised representation of Dupont’s political curves

representation of a partial equilibrium, all other things not being equal, that is to say, taking into account the general interdependence of markets at the same time. He was convinced, however, that such a curve could be rigorously obtained and that it should be possible to find its equation - with a view to a future “general theory of prices”. Hence his plea in favour of mathematics: “To state that higher geometry [mathematics] is not applicable to politics would be as absurd as... to claim that one could not apply it to mechanics or hydraulics” when these sci­ences were in their infancy (Dupont de Nemours [1774] 1892, 290). But being unable to achieve this task by himself, he wrote that he spoke to “the great Daniel Bernoulli”, trying to involve him in this adventure. Bernoulli probably only gave a polite answer.

Dupont again presented his memoir on 1 May 1796 at the Classe des sciences morales et politiques of the Institut national des sciences et des arts. An echo of this session is to be found in the report of the discussion, kept in the archives of the institution. The positive role of mathematics was again stressed.

The citoyen Dupont (de Nemours)... in a memoir titled Des courbes poli­tiques... intended to show the utility of calculation - even that of the most advanced geometry [mathematics] - applied to problems of political economy for which, with the simple use of ordinary logic, we only obtain a moral and vague result. He endeavours to prove that the greatest part of these problems, like those regarding the motion of the [celestial] bod­ies, gather a mass of data which influence each other, the effects of which can be submitted to geometric analysis, the results only being liable to be expressed by curves.

(Report quoted by Israel 1991)

However, this insistence on the role of “higher geometry” must be treated with serious reservations. As a matter of fact, Dupont did not think that the use of math­ematics was likely to bring new results: the draft of the curves, he stated in 1774, shows that “I am close to the solution. I am speaking of the geometrical [math­ematical] solution, because we have known the political one for a long time and the geometricians will not teach us anything in this respect” ([1774] 1892, 299). Formalisation is only useful as a tool to convince people of the accuracy of the theory: if they find some pleasure and interest in finding the equation of the curves, mathematicians will provide physiocratic theory with the support of the great num­ber of people who care about their opinion ([1774] 1892, 300).

Butre S manuscripts

Another physiocrat, Butre, felt the need to use mathematics in questions of political economy, but this time in a more interesting way: in order to develop theoretical insights on the basis of a reformulation of the views Quesnay had stated in “Ana­lyse de la Formule arithmetique du Tableau economique” and the first “Probleme economique”. Butre was a former collaborator of Quesnay and Mirabeau and made statistical and economic calculations for them. He was a military man by profes­sion, trained in usual mathematics, became a specialist in agriculture and agricul­tural accounting and published articles on this subject in Ephemerides du citoyen. He used algebra in 1766-67 in two unpublished works: his contribution to the Turgot prize competition about the effects of indirect taxes and a short treatise, Elemens d’economie politique, the manuscripts of which were recently discov­ered by Loi'c Charles and Christine There (Charles and There 2016a, 2016b). His approach thus broke with that of Quesnay, and it is symptomatic that Butre drew inspiration from Clairaut’s Elemens d’algebre (Charles and There 2016b, 314-15 and Appendix)^[170] while Quesnay, as noted earlier, instead praised Clairaut’s Ele- mens de geometrie (“as long as it is demonstrative”). This is not, however, the only difference between them: with the view of more rigorously developing Quesnay’s system, Butre made changes to its theoretical structure, modifying the number of classes from three to four, specifying their intrasectoral expenditures and introduc­ing international trade (Charles and There 2016a, 141-5).

The way in which he applied mathematics to uncover new theoretical results - and not just to confirm what was already known, as Dupont was ultimately to propose later - was novel in France but presents obvious similarities to Beccaria’s 1764 approach in his essay on smuggling.^[171] Like mathematicians - Euler, as noted

Figure 8.2 Dupont’s political curves (reproduced with permission of Landesarchiv Baden- Wurttemberg, Karlsruhe, Germany)

earlier, applied this method to questions of population in 1760 - Butre lists ques­tions and provides answers (both being sometimes rather unclear). His “Probleme 1er fondamental” is a good illustration thereof. Given “the total value of annual productions of a nation, the costs and gains of the food trade... and those of industry, and the... ratio between the spending on food and on manufactured goods”, the problem is to find “what the ratio between the [value of] raw subsist­ence goods and the [value] of raw materials should be to have an equal balance in food, primary goods and industry” (in Charles and There 2016b, 323), that is, an equilibrium between supply and demand in each sector. Butre’s development is not really an answer to this question but shows how simple algebra is used.

Let a be the value of the production of the productive sector (agriculture). This production, according to Butre, is not yet ready for consumption and must be trans­formed by the sterile sectors (food industry, craftsmen and manufactures) into consumption goods. If x is the value of agricultural products used by craftsmen and manufacturers, a - x is consequently the value of them used in the food indus­try. Let g be the “cost and gains” of the food sector (the “cost” is the additional cost over the used raw agricultural products) and f those of manufacturing. Hence, (a - x + g) is the value of the production of food and (x + f the value of industrial goods produced and consumed by the nation. Finally, n is “the ratio of expenses in food to expenses in manufactured goods”. Hence,

The age of ambition

During the last quarter of the eighteenth century, and parallel to Condorcet’s devel­opments in probability theory and “social mathematic”, three ambitious attempts

Conclusion: the higher the level of the duty t, the lower is x and the more smuggling is encouraged (the point of indifference is easier to reach).

There were intellectual relationships between the French and the Milanese Enlightenment, especially through the brothers Alessandro and Pietro Verri and Paolo Frisi. The periodical of the Milanese intel­lectuals, Il Caffe, was known, and Beccaria himself send a copy of a reprint of it to Morellet in early 1766. Beccaria’s masterwork, Dei delitti e dellepene, was widely discussed in France in 1765-66.

were made to introduce symbols and algebra into political economy. They deal with the core of the theory and mark the real beginning of formalised economic reasoning. Two different approaches were implemented. The first was adopted by Achilles-Nicolas Isnard, an engineer, who investigated the structural relationships in a multi-sectoral economic system and the relative price system: his develop­ments are to be found in his 1781 Traite des richesses.

The second route was taken by Charles-Franςois de Bicquilley (1738-1814), a soldier, and Nicolas-Franςois Canard (1754-1833), a teacher, in order to determine, inter alia, the equilibrium prices of commodities. This was done in Canard’s Princi- pes d'economiepolitique (1801) and Bicquilley’s Theorie elementaire du commerce (1804). Bicquilley’s Theorie was published three years after Canard’s Principes, but it was written at approximately the same time. The manuscripts of the first versions of these two books were in fact submitted the same year, 1799, to the Institut national des sciences et des arts, but not to the same Classe. Bicquilley submitted his manu­script to the first Classe of the Institut (mathematics and physics: the new Academy of Sciences) on 28 August, and Canard to the second Classe, that of moral and politi­cal sciences (a new academy which included political economy: see Chapter 9, this volume), after the Classe launched a competition on 4 October about the impact of taxation. While Canard’s work was occasionally discussed during the nineteenth and twentieth centuries, Bicquilley’s Theorie elementaire du commerce went unnoticed and was only rescued from oblivion by Pierre Crepel during the 1990s - it seems that the book was printed but never put on sale (Crepel 1998c).^[172]

Isnard and the intersectoral structure of the economy

Achilles-Nicolas Isnard was a Ponts et Chaussees civil engineer trained in alge­bra and analysis - a competence he also used in the more classical field of the management of the public debt (Isnard 1801) when he was a member of the Tribunat - but he was also interested in moral and political philosophy: his Traite des richesses was in fact followed by Catechisme social (1784), a sensationist- based pamphlet, and Observations sur le principe qui a produit les revolutions de France, de Geneve et d'Amerique dans le dix-huitieme siecle (1789), the purpose of which was to refute Rousseau’s “fatal principle” that “the law is the act or expres­sion of the general will” (1789, 5). In his Traite, he intended to fight the physi­ocrats, their theory of the exclusive productivity of agriculture and their related view of the economy: in his opinion, they were confusing production with crea- tion^[173] and, in the Tableau economique, they wrongly represented the landowners

“as being seated on a throne and distributing on both sides salaries to the other classes” (1781, I, 41n; 2006, 124n*).^{[174] [175]}

To this end, he represented the economy as a physical and interconnected system of branches. Each branch uses several commodities as means of production in order to produce a given amount of a particular commodity. This amount should at least replace the total quantity of this good used as means of production in the various branches. There is usually a “surplus” over this quantity, which forms the “richesse disponible” (“available” or “disposable wealth”) that can be used for final consump­tion and to increase the level of production. As an example, Isnard supposed a system formed of two branches and two commodities M and M’ and reasoned in terms of “mesures” (“measures”) of M or M’, that is, in terms of (physical) units.

Let 40M and 60M’ be two kinds of production: one supposes that to produce 40M, a consumption of 10M + 10M’ is necessary, and to produce 60M’ one must consume 5M +10M’; to produce the sum of these two productions, a consumption of 15M + 20M’ is thus required.

(1781, I, 36; 2006, 118*)

In modern terms,

The surplus produced in the economy is thus 25M + 40M’. It is important to note that, despite some clumsy expressions like the sign + used here or the phrase “value of the disposable wealth” to refer to 25M + 40M’, Isnard is perfectly aware that M and M’ are heterogeneous commodities that cannot be added or compared: “to compare heterogeneous things, one has to find between them some relation of homogeneity, that is to say, to find some homogeneous qualities” (1781, I, 16; 2006, 96). This task is performed by the introduction of money or, at least, with the choice of a numeraire. The system of relative prices allows the comparisons: and, as a result, it will simultaneously determine the distribution of the global surplus in the economy, that is, the share of the value of the surplus that the producers receive in each branch. Let M” be “a common measure” and suppose that M = M” and M’ = 2M”, so that M = 0.5M’.

The producers of 40M would have to spend, for the costs, the value of 30M or 30M” and could freely dispose of 10M or 10M”. The producers of 60M’ would have to spend, for the costs, the value of 12.5M’ or 25M” and could freely dispose of 47.5M’ or 95M”.

(1781, I, 36; 2006, 118*)

Now suppose that the relative prices change because M’ = 3M” instead of 2M”. As a result, the distribution of the surplus is modified: “the disposable revenue of the own­ers or producers of 40M will be equal to zero, and that of the producers of 60M’ will be equal to the total mass of disposable wealth” (1781, I, 36; 2006, 118*). As a con­sequence, it is clear that the physiocratic exclusive productivity of agriculture is only a special case of the distribution of the surplus, depending on given relative prices.

The total sum of the disposable products depends absolutely upon the needs of nature. She requires a certain portion of the general mass of wealth; she leaves the rest to the enjoyment and the needs of man. The part each producer draws from that portion which is destined for enjoyment, is relative to the particular values of the products. (1781, I, 37; 2006, 118)

But how are prices determined? Isnard alludes to the competition of capitals and the resulting uniformity of their equilibrium rates of remuneration - a condition that, if considered as a constraint for the determination of prices, would have led to the Ricardian and Marxian system of “prices of production”. But Isnard focuses instead on the equality of supply and demand in markets. In a barter economy, the supply of any commodity must be equal to the demand for it, expressed by the quantities of other commodities offered in exchange.

However, since the offers are composed of several heterogeneous commodi­ties, it is not possible to deduce from the equality... the relation between two particular commodities. To find the relation between commodities taken two by two, one would have to formulate as many equations as there are commodities. The first member of those equations would contain the quan­tity of commodities, and the second the sum of offers.

(1781, I, 19; 2006, 100)

Suppose three commodities M, M’ and M”, of which the respective quantities a, b and c are offered for sale - noted by Isnard aM (that is, a units of M), bM’ and cM”. The owners of M offer the quantities maM and naM, respectively, for some units of M’ and M”; the owners of M’ offerpbM’ and qbM’, respectively, for some units of M and M”; and the owners of M” offer rcM” and scM”, respectively, for some units of M and M’ (m, n,p, q, r, s being proportions, with m + n = p + q = r + s = 1). Equilibrium in markets entails that

Hence the equilibrium relative prices (1781, I, 20, 2006, 100):

Now, to see how relative prices change, “one only has to assume various numbers in place of the algebraic quantities we have just used, assuming always, as we have done up to now, that the quantities of commodities remain constant” (I, 21; 2006, 102). As for Boisguilbert or Turgot, only relative prices matter: the word “value”, Isnard writes, when applied to wealth, must not be taken in an absolute sense and just refers to the exchange ratios between units of commodities. If we want to speak of absolute value, we should speak of utility: “The word which properly expresses the absolute meaning that can be given to it is utility” (1781, I, 17n; 2006, 98n).

Isnard further complexified the analysis, detailing the various components of the costs of production, analysing the labour market, introducing taxation, etc. His developments are not always free from ambiguity, but his basic achievement remains remarkable. No wonder that William Stanley Jevons and Leon Walras regarded him as one of their “precursors”, and that he could also be listed later as a “forerunner” of input-output analysis.

Canard and the "latitude”

Nicolas-Franςois Canard taught various disciplines (rhetoric, philosophy, grammar and his favourite field: mathematics^[176]) before being appointed in 1795 professor of mathematics in the newly founded Ecole centrale, then Lycee, of Moulins in the Departement of Allier. He was not one of those prominent figures who illuminated French intellectual life from the end of the reign of Louis XVI to the Restoration. Nevertheless, he was in contact with some of them - Joseph Lakanal (1762-1845) in particular, a “founding father” of the Institut national des sciences et des arts - who were influential in political circles and, after 1795, at the newly founded Insti­tut. A letter by Lakanal, written in the very first days of January 1801, testifies that Canard was his friend and mentions their common intellectual interests. They taught philosophy and mathematics together before the Revolution and subscribed to the new philosophical ideas:

Perhaps it would be well to note that, at this time, we were often running the risk of losing our position for having substituted the doctrine of Condillac for the mess of words without ideas with which, in our schools, everything except reason was satisfied.

(Lakanal, quoted in Israel 1996)

As regards formalisation, Canard published two books which are the result of two competitions launched by the Classe des sciences morales et politiques of the Insti­tut. The first competition, in October 1799, asked the following question: “Is it true that, in an agricultural country, any kind of tax falls in last resort on the landowners and, if this is the case, do indirect taxes fall on these same landowners with an addi­tional burden?”. The second, in 1801, asked “What are the means to improve the institution of the jury in France?”. In both competitions, Canard was distinguished. In the first, he won the prize in January 1801 with a manuscript, Essai sur la cir­culation de l’impot, which, substantially enlarged, was published the same year as Principes d’economiepolitique. In the second, in April 1802, he had to share the prize with a magistrate; his manuscript was published some months later with the title Moyens deperfectionner le jury. In both books, he made use of some formali­sation - the second to a much lesser extent: it is moreover on criminal justice and deals with some (very simple) probabilities.

The Principes were a criticism of some central aspects of the then moribund physiocratic doctrines, especially the exclusive productivity of agriculture, and were intended to prove that the impact of taxation affects not only the landown­ers. In order to advance his argument on a sound basis, Canard first developed his views on the origin of wealth, income distribution, the working of markets and the determination of prices along a sensationist line akin to Turgot’s and Condillac’s. In a nutshell - we only focus here on Canard’s use of formalisation - in a regime of free trade, prices depend on supply and demand, and more specifically on the strength of the respective needs to sell or buy, on the degree of competition within sellers and within buyers and on a general maximising attitude of agents: “all indi­viduals are inclined to get the greatest possible number of enjoyments and conse­quently to obtain the greatest quantity... of wealth” (Canard 1801, 27). Each seller tries to fix the highest possible price for his commodity, and each buyer, instead, endeavours to obtain the commodity he wants at the lowest possible price.

Suppose a market with a certain number of sellers and of buyers: “there will necessarily be a difference between the price asked by the former and the price offered by the latter. This difference between the highest and lowest prices forms a latitude” (1801, 28). Sellers and buyers are thus led to bargain and fight to get the greatest share of this “latitude”. It is in this context that Canard uses symbols and simple algebra in the same vein as Hutcheson and Frisi did before, with the same shortcomings - no reflection is made about measurement, increasing and decreas­ing functions are represented by multiplications and divisions, and when there is more than one variable, they are simply multiplied together.

Let L be the latitude, x the share of it that the sellers try to add to the lowest price (that is, their share of the latitude), (L - x) the share that the buyers try to subtract from the highest price, B the need of the buyers and b that of the sellers, N the number of buyers (“their competition”) and n that of the sellers. Focusing on the buyers, Canard states that the share x they have to pay to the sellers increases with their need B and their competition N, while the share (L - x) they would like to subtract from the sellers’ highest price increases with the sellers’ need to sell b and their competition n. A similar reasoning could be made from the sellers’ side. He then writes (1801, 29):

How can these equations, which are usually misinterpreted, be understood? The first, which Canard does not explain, could intuitively represent the balance of opposite forces. But what he says of the second is more interesting and precise: he states that this equation^[177] “expresses the equality of the moments of two opposite forces which equilibrate”. And he adds, “The entire theory of political economy refers to the principle of the equilibrium of these two forces, just as the entire stat­ics refers to the principle of equilibrium of the lever” (1801, 30-31). Reference is thus made to physics and more specifically to the moment of a force, and this helps us understand the equilibrium condition.^[178] In a nutshell, the moment of a force measures the capacity of a force to generate some movement around an axis. It is the product of the force and the “moment arm” (in French “bras de levier”, “levier” meaning lever) which is the distance between the moment centre and the perpen­dicular of this centre to the line of action of the force. Two forces equilibrate when they have the same moment around the same axis, with opposite algebraic signs.

Canard simply applies the same definition to political economy. The buyers’ situ­ation is symbolised by x / BN: the share of the latitude they have to give up is a decreasing function of BN (the product BN representing “the need and competition of the buyers”), x being the “distance” between the lowest price and the point of agreement, that is, the moment arm of the buyers. The sellers’ situation is represented by ( L - x) / bn (the product bn represents “the need and the competition of the sell­ers”): the share they have to surrender to the buyers is a decreasing function of bn, (L - x) being the “distance” between their highest price and the same point of agree­ment, that is, the moment arm of the sellers. Moreover, Canard calls bn “the force of the buyers” in the bargaining process and BN “the force of the sellers” (1801, 29). Hence the equilibrium condition, where each force is multiplied by its moment arm: bnx = BN (L - x). If it is not satisfied, there is still an incentive to move and the bargaining continues until a balance is reached. The equilibrium value of x then is

The extreme values of x are x = L if bn = 0, and x = 0 if BN = 0. In the first case, the need to sell is “as small as possible” and/or competition between sellers is nil: this is the case of “the sellers’ monopoly”. In the second, “the competition or the need of buy­ers is as small as possible”: this is “the opposite monopoly of the buyers” (1801, 31).

Now what are the lowest and highest possible prices for a commodity? The low­est is simply the “necessary” or “natural” wage S of the producer, that is, a subsist­ence wage. The highest depends on whether the commodity is of absolute necessity to the buyers: if it is not, any price increase will diminish the number of purchases until an equilibrium is reached where the gain generated by the increase is equal to the loss due to the reduction of demand; if it is a necessary good, the price will be limited by the necessary wage, or otherwise wages will have to rise or riots will hap­pen. The general formula for the price P is thus

Canard then develops his analysis, introducing in particular the structure of the production system, that is, the many branches necessary to produce a commod­ity, and thus the markets for intermediate products and means of production. But, like Dupont and contrary to Isnard, he does not succeed in taking the intersectoral structure of the economy into account and ends up with complicated and rather meaningless formulas. As for the question asked by the Institut for the competition, it is answered in Chapter VIII of the book (1801, 153-202), where Canard shows that the burden of a tax is shared by all kinds of income (apart from the subsistence wage) according to the respective “forces” of the agents.

Canard’s memoir triggered a discussion in the second Classe of the Institut, probably because of its use of mathematics and some questionable developments. In his report, Pierre-Louis Rrnderer (1754-1835) - an economist and politician who played an important role in intellectual and political life during the revolutionary and Napoleonic periods - was rather reluctant and expressed the hope that the memoir would provoke new publications which could “confirm what is right in it, correct what is inaccurate and compensate for what is incomplete” (15 January 1801, in Rα'dere∣' 1857, 601). As for the long analysis of the memoir published in Memoires de l'Institut national des sciences et arts, it is just an uncomfortable paraphrase of some aspects of Canard’s ideas, its author declaring that a proper analysis is very difficult to do (Institut national des sciences et des arts 1802, 16-25). After the dis­cussion, Lakanal sent a letter to the Classe to protest against Rmderer’s criticisms and against the reservations expressed in the statement of the award: he stressed instead the novelty of Canard’s analysis based on the fundamental equation of the equilibrium of opposing forces (Lakanal, in Israel 1996). Finally, Joachim Lebreton (1760-1819), another member of the Classe, published a long review of the Princi- pes in La decadephilosophique, litteraire etpolitique, the journal of the Ideologues (Lebreton 1802): it forms a laudatory summary of Canard’s propositions, but with­out any reference to their mathematical expression.^[179]

Despite reticence, Canard’s Principes circulated and quickly benefited from sev­eral translations,^[180] which was exceptional at that time. Francis Horner (1778-1817) published a detailed and critical review of the French edition in the second issue of the Edinburgh Review (Horner 1803). He was especially critical of Canard’s use of algebra, which formed in his view an “injudicious and unskilful pedantry... which diverts an instrument from its proper use, and attempts to remove those landmarks by which the sciences are bounded from each other”. Canard, he states, “has only trans­lated, into a language less readily understood, truths, of which the ordinary enuncia­tion is intelligible and familiar to all” (1803, 439). Horner is not hostile to the use of algebra in political economy but, he stresses, a subject may also possess a mathemat­ical precision “without requiring, or even admitting, the symbolic representations of algebra” (1803, 440). Analogies borrowed from mathematics can be used, but only as illustrations or ornaments. In this respect, “the frugal and classic taste, with which Beccaria has interspersed allusions of this nature, forms a contrast to the pedantry and profusion with which M. Canard has overloaded his composition” (1803, 440).

Bicquilley, mathematics and probability theory

One can easily guess what Horner would have said, had he known Charles-Franςois de Bicquilley’s Theorie elementaire du commerce, which is certainly the most ambitious attempt to formalise the economic discourse during the Enlightenment. His Theorie elementaire is full of symbols and calculations and looks a bit like a mathematics book with his definitions, theorems and corollaries - a vocabulary already present in Ceva’s 1711 De re numaria. More importantly, he uses both algebra and probability theory, but with the same shortcomings as those already noted in various writings from Hutcheson’s Inquiry onwards.

Bicquilley was not unknown in scientific circles (Crepel 1998a). In 1783, he published a successful didactic book on probability theory, Du calcul des proba­bilities (Toul: Joseph Carez), not devoid of insights, which was translated into Ger­man five years later and reprinted in 1805 (Paris: Courcier). Condorcet presented the book at a meeting of the Academie royale des sciences in April 1783 and cer­tainly drew some inspiration from it for his own research (Condorcet 1994a, 203­209). In 1786, with a memoir titled Theorie des assurances relatives au commerce de mer, which remained unpublished, Bicquilley shared with the mathematician Sylvestre-Franςois Lacroix (1765-1843) a prize in the competition launched by the Academie about the theory of marine insurance. Theorie elementaire du commerce was written and submitted to the first Classe of the Institut in 1799. Its purpose was clear: “we still do not have an elementary theory of commerce treated as a mathematical science and in the generality of its principles. Yet no subject seems more liable to be dealt with in this way... and worthy of being a part of public education” ([1804] 1995, 62). After a positive report written by three celebrated mathematicians - Charles Bossut, Jean-Baptiste Delambre and Joseph-Louis Lagrange - its author was encouraged to publish his work, which he did after some revisions. The report, inserted into the publication, does not express any reserva­tions, contrary to what happened to Canard in the second Classe.

The subjects pertaining to commerce have been dealt with until now with simple reasonings drawn from their relationships with morals, politics, the respective interests of the contracting parties, and supported by arithmeti­cal calculations.... The citoyen Bicquilley intended to apply in general the analytical science to this subject.... With this method, the language of commerce is freed from a good many loose, obscure and uncertain phrases; it acquires the precision and accuracy of mathematics.

([1804] 1995, 64-5)

The Theorie is dense and rather short (128 pages). The greatest part of the 239 sections, spread out over seven chapters, is devoted to the central themes of value, price, money and the working of markets (including the concept of “quantite dou- teuse”, or uncertain quantity), the last two chapters being devoted to basic consid­erations on insurance and the “direction of trade” (that is, an analysis of the causes of an easy or difficult circulation of such or such commodity, including monopoly or what we now call monopsony and the multifaceted interventions of the State). The fundamental part of the book, however, typical of Bicquilley’s use of for­malisation, is the determination of the prices of commodities, on which most of the developments are based:^[181] “the analysis of commerce, either general or particular, of one or several traders or of an entire people, must be based on the same princi­ples as that of a single market” ([1804] 1995, 73).

Let us thus take the simplest case and suppose a bilateral exchange. The buyer’s “search” for a commodity, noted R (“recherche”), expresses “the desire he has to possess it”; and the “abundance” Q he faces in the market is the supply of it (§ LXXIII). For an individual, the value of a commodity, noted U, is “the quantity of esteem” that is attached to it (§ XXI). To signify that this value is an increasing function of the search and a decreasing function of the supply, and following a device already met in other authors, the value of the commodity is defined as “the ratio of its search to its abundance” (Theorem 1, § LXXIV), that is, U = R / Q. Money is analysed in the same way, but more as an analogy: for an individual, S being his search for money and N the abundance of it, the value of a unit of money is defined as S / N (§ LXXVIII).

Now let D be the quantity this individual wants to buy and P the total price of this quantity. The value for him of the quantity D is D ( R / Q). But, in exchange, he must give the sum P of money, the value of which is for him, in a similar way, P( S / N). He will be willing to buy if there is a positive difference between the esteem value of what he gets and the esteem value of what he gives: this difference is what Bicquilley calls “convenance” (convenience) (§ XX). The “convenance” of the buyer is thus D ( R / Q) - P ( S / N). Now let r, q, s, n be the analogous variables for the seller who offers his commodity for sale and is thus in search of money. The “convenance” of the seller is P(5 / n) - D ( r I q) (§ LXXXIII, Theorem 3).

The exchange takes place if the two conveniences are positive. What will be the unit price of the commodity, that is P/D, called “cherte” (dearness)? We must have simultaneously: hence

The value of P / D must thus lie within the following interval:

Bicquilley sees perfectly well that in this case, the equilibrium price is still not totally defined, and he discusses some special cases. He then introduces two other key variables: (1) Hand h, respectively, the “habilete” (ability or skill) of the buyer and the seller in the bargaining process; and (2) dropping the hypothesis of a bilat­eral monopoly, A and V, the number of buyers (“acheteurs”) and of sellers (“ven- deurs”), which express, as for Verri and Frisi, the degree of competition within these groups. “The relation of the convenience of the buyer and that of the seller in the markets is directly related to their skill, and inversely to their competition” (§ XCV, Theorem 5). With the introduction of these additional variables, the equi­librium price is now defined as (§ XCVI, Theorem 6)

The demonstration is of course basically flawed for the reasons already stated, but it nevertheless remains fascinating. And what is particularly remarkable is that, in fact, Bicquilley is formalising Turgot’s theory of value and price. What is called here “convenance” is exactly what Turgot dealt with in “Valeurs et monnaies”, and the condition for an exchange is the same in both authors. However, Bicquilley’s final solution is different, because of the introduction of the additional variable “habilete” into a bilateral exchange and because of the kind of formalisation. It is impossible to know whether he had knowledge of Turgot’s essay, which was not yet officially published but which, like some of his other writings, circulated in certain circles.

4.