“In this exhibition, 100 tables of the Encyclopédie di Diderot e d’Alembert are presented, which most directly recall the philosophy and geometry applied to nature, to other sciences and to man’s work: in architecture, astronomy, artistic techniques, music and tools, sciences and traditional techniques, military art and crafts of everyday life. It will also be possible to understand the life of the 18th century through the vision of all 2794 tables of Encyclopédie thanks to a specially equipped multimedia station”.

Prof. Renzo Baldoni, Museum director and exhibition curator

THE MUSEUM AND EXHIBITION

In the first nine years of life, the Museum of Computer Science and History of calculation in Pennabilli has carried out an intense cultural activity through publications, exhibitions, initiatives, meetings, which have contributed to spreading scientific culture among its visitors, mainly students and teachers. A living Museum should not only document and store instruments and archives, but must become a permanent education centre.

The Museum of Computer Science and History of calculation has so far realized and exhibited the following exhibitions:

  • the history of calculation
  • great mathematicians
  • a train of mathematicians
  • the stages and the men of the mathematics
  • the Pythagorean theorem
  • interesting numbers
  • the Genaille–Lucas rods
  • the history of computer science
  • computer applications
  • Exploring the Internet
  • virtual reality
  • the pendulum of Foucault
  • You, Moon
  • educational software
  • computer science in school
  • computer profession (in collaboration with French CNRS)
  • the Greek Pi:history and curiosity of a fascinating number
  • the Fractals: an autosomal world?

The exhibition La Mathematics in the Encyclopédie di Diderot e d’Alembert (1751-72)-100 tables is the first of a new series of initiatives aimed at “festing” the first 10 years of life of the Museum.

The Museum of Computer Science and History of calculation by Pennabilli from scientific museum wants to become more and more museum-laboratory, live museum, a significant local institution of permanent education. It is therefore intended to undertake an important renewal and strengthening of cultural activities based on the following three-year project:

  • Summer 2000: Encyclopédie di Diderot e D’Alembert-100table
  • autumn 2000: exhibition and scientific conference on “the theory of everything: a single law at the base of nature? ”
  • spring 2001: the famous curves
  • summer 2001: the polyhedra of Luca Pacioli
  • autumn 2001: the helmets of Escher
  • 1st semester 2002: the universal machine of Turing
  • 2nd semester 2002: the European roots of the computer

The celebrations of the tenth anniversary of the Museum hope can end with the creation of a Workshop of Intensity and with the publication of a volume, with attached cdrom, to be distributed to the visiting schools groups, libraries, universities and scientific research centers.

ENCYCLOPÉDIE

In 1751, with the title Encyclopédie, ou Dictionnaire raisonné des Sciences, des Arts et des Métiers, the first volume of the capital of French Enlightenment was published, destined to have enormous consequences on the cultural, political, social level. In the following two years, eight more volumes of text and two volumes of tables appeared. And overall it took thirty years before the most ambitious publishing company, for the contents and ends, of the whole 18th century had ended. The first edition of the monumental work was completed in 1781. Under the direction of Diderot, up to 1765, seventeen volumes of text and, until 1772, eleven volumes of tables (Recueil de Planches); without the collaboration of Diderot followed, between 1772 and 1777, four additional volumes of text and another volume of tables; finally in 1780-81, two volumes containing indexes (Tables analitiques) were published in Amsterdam. The first edition therefore included thirty-five tomes in folio: the twenty-one text volumes contained about 6000 items or articles (articles); each of these volumes was composed of about 1000 pages printed on two columns: in all therefore it was about 21,000 pages in folio. The twelve volumes of the tables contained 3132 engravings on copper accompanied by the relative captions and descriptions.

A total of 30,000 copies of the work were purchased in Europe. The sum of 1,158,000 livres was invested; but we know that the embroidery was 2.162,000 livres: the publishers, and in particular Le Breton, with the Encyclopédie enriched. What had been presented as one of the largest publishing companies in the century also became a big deal.

In 1747 Le Breton turned to Diderot proposing his work and Diderot immediately assicuously assimilated the collaboration of his friend D’Alembert especially for the editing of the part mathematics.

The 35-year-old Diderot was already known at that time as a man of letters rich in spirit and various interests, writer and philosopher, especially for his Pensées philosophiques (1746) and for the famous Lettre sur les Aveugles à l’usage de ceux qui voient (1749). Diderot soon sketched out the idea of developing a completely new and truly universal encyclopedia that should have been quite different from each other Previous, obtaining with the project a privilege of the king for the edition. So Diderot was able in 1750 to announce in the remarkable Prospectus the imminent publication of the first volume of Encyclopédie.

Fundamental importance was then the Discours préliminaire of D’Alembert placed in the opening of the first volume of the Encyclopédie which, proclaiming the philosophical program and the intellectual commitment of the whole work, is defined as the essential reason for the initial great success. The author was already at that time a scholar of European fame, an illustrious member of the academies of Paris, Berlin and London and friend of Frederick the Great. The Discours did nothing but complete its fame: it counts among the classical texts of Enlightenment (v. Appendix).

A work that presented itself with the vast goal of offering all human knowledge, carefully examined and exhibited critically, with the claim to be a real Summa of the scible “at the height of the century”, could obviously not be the result of the work of a few authors. The Encyclopédie is in fact a multi-handed work with the intervention of 178 authors. In the list of authors there are, together with many strangers, the most famous personalities of the century, from Diderot and D’Alembert to Voltaire, Rousseau, Montesquieu, Buffon, Quesnay, Turgot, Condorcet. The much heavier part of Encyclopédie’s writing and writing work was supported by Diderot, who devoted twenty-five years of his life to the work. D’Alembert retired in 1758 after the release of the seventh volume; Voltaire had preceded him; Rousseau followed him.

In the Encyclopédie, almost all the currents of pre-revolutionary French illuminator converge. The work had a dual purpose. As an encyclopedia, it had to illustrate the hierarchy and correlation of all human knowledge, to demonstrate the unity of knowledge in all possible coherent connections and thus to fulfil a particularly philosophical purpose. However, it must also be a Dictionnaire raisonné of sciences, arts and technology, and therefore contain the general principles on which these activities are based, as well as the essential features that characterize them as a whole and in the contents. The work was therefore to be a text of consultation, a “deposit of knowledge”, able to provide News on the most recent, for the time, stages of development of science and technology; it was to serve as a means of education, training, practical teaching for every type of reader in every field of knowledge and life, thus becoming also spokesperson for new religious, political and social ideas. The work, in the firm conviction of the authors, was conceived as a weapon of criticism against society and religion, as an arsenal of arguments and weapons of offence for the battle with which to realize, in the spirit of Enlightenment, a new order for the life of man, based on reason and freedom. Diderot's work can also be considered as a compendium of technical information, in the important position of technology. For the Encyclopédie, the technique is configured as the vehicle for the vulgarization of lighting ideas and, at the same time, gives shape to the tendency towards a refoundation of technical-industrial and manual activities by means of science, thus becoming scientific training and enlightenment important educational factors.

Overall, the work presents, in individual specialist articles and in the whole of the tables, a broad but static framework of the development stage reached by technology before the advent of the Industrial Revolution. The ideas supported and disseminated by Encyclopédie, relating to a secularization of knowledge and thought, to a refusal of the constituted authority, to the faith of the omnipotence of Reason and experience, but above all to the faith in science, progress and ability, for man, to complete every enterprise, corresponded perfectly to the thought already expressed in other texts of the Enlightenment literature.

The spiritual currents of Encyclopédie certainly contributed to paving the way for the bourgeois revolution of 1789; they lead to the ideas on science and education that the Revolution will make its own, in scientific conceptions that, considering the further developments in the history of technology, lead to the moment, of historical importance, in which the polytechnique école (1794), among which the truthbles foundurs or spiritual fathers can certainly be placed.

In it, the recognition of the unity of all technical activities, founded on mathematics and natural history, was expressed so valid that it continued to act throughout the industrial era.

Denis Diderot (1713-1784). French writer, born in Langres and dead in Paris. After completing his studies at the Jesuits, he left everything to study freely on his own, living ten years of lessons, translations and books. D’Alembert began the Encyclopédie; in 1749 he was locked up in Vincennes prison for his Letters on the Blind. As soon as he left prison he resumed the gigantic work of the Encyclopédie which ended in 1765. From 1773 to 1774 he stayed in St. Petersburg with Catherine II who had granted him a pension; he spent the last years in Paris working hard and died leaving many unpublished works. An interesting anecdote concerns the great mathematician Eulero and the atheist philosopher (or pantheist) Diderot. Invited by the great Catherine to visit her court, Diderot consecrated her ozi to convert courtiers to atheism; warned, the Empress charged Eulero to put the museruola to the frivolous philosopher. It was an easy mission, because talking about nutrition at Diderot, it was like talking to him Chinese. De Morgan tells us what happened: “Diderot was warned that an ingenuity mathematician possessed an algebraic demonstration of God’s existence and that he would expose it to the whole court if he wanted to listen to it; Diderot accepted with pleasure...Eulero left towards Diderot and said severely, with a tone of perfect conviction: “Lord, therefore God exists: answer!” This speech seemed to be sensible at Diderot's ears. Humiliated by the crazy laughter that welcomed his embarrassed silence, the poor philosopher asked Catherine for permission to return to France and she willingly agreed to it.

As a mathematician it was worth little; he published a mediocre collection of “Mémoires Mathématiques” (Paris, 1748), where in practice it deals only with the developer or evolving circle. In other fields, however, he collected well deserved laurels. His extensive production includes philosophical works, including:

Essay on merit and virtue, Philosophical Thoughts, Thoughts on the interpretation of nature; literary and artistic criticism, as: Speech on dramatic poetry, Richardson's Elogy, The Paradox on the Comedian, Salons; novels and stories, including: The religious, Rameau's nephew, James the fatalist, The two friends of Bourbonne; and dramas, such as: The natural son, the family father.

Jean Le Rond d’Alembert (1717-1783) is undoubtedly the most eminent personality that France produced during the first half of the 18th century. The most conspicuous part of d’Alembert’s scientific writings concerns mechanics (rational and celestial), not excluding hydrodynamics, physics mathematics (optics and acoustics) and mathematical pure analysis. At the age of 26 he published the Traité de Dynamique, with the general method to equate the problems of the dynamic that he carries and is destined to always bear the name of “the principle of d’Alembert”; he first explained the phenomenon of the precession of equinoxes, which has so much importance in astronomy, and could furnish to the hydraulic contributions of the highest value. His reflexions sur la causa générale des vents in which he was induced to congegnate the first reasoning to demonstrate the fundamental proposition of algebraic equation theory. Of considerable importance are his contributions to the infinitesimal mathematical analysis and suggested a full calculation application to the demonstration of Taylor’s formula that led Condorcet to designate that formula as d’Alembert’s theorem. The rectification of the regular asteroid led him to paradoxical consequences that were clarified in Italy (Mascheroni, Gratognini). The elliptical or reducible integrals led him to a clash (for priority rights) with V. Riccati. On differential equations, he demonstrated the existence of an integral factor for those of the first order and the integration of some of the upper order. He first dealt with differential equation systems.

In the theory of vibrating strings he gave a completely satisfactory order; on the other hand, the question of the meaning to be attributed to the logarithm of negative numbers, was, wrongly, opponent of Euler and supporter of G. Bernoulli. D’Alembert, despite the vastness and importance of his mathematical research, did not exert great influence on the masses, probably because he was a very little happy exhibitor and adopted the worst notations.

In the multitude of efforts to introduce rigor in infinitesimal calculation, few were on the right path. Among these the most remarkable were those of Alembert and, before that, of Wallis. In the article Différentiel of the Encyclopédie d’Alembert says: “Newton has never considered differential calculation as a calculation of infinitesimals, but as a method of the first and last reasons, that is, as a method to find the limit of these relationships.” D’Alembert defined the differential as “an infinitely small quantity or at least smaller than any assignable size”. He believed that Leibniz's calculation could be built on three rules for differentials, although he preferred to consider derivatives as a limit. In his search for the use of limits, like Euler, he says that 0/0 can be equal to any quantity you want. In another article, entitled Limite, d’Alembert says:” Limit theory is the true metaphysics of calculation...In differential calculation you never deal with infinitesimal quantities, but only with finite quantity limits. Therefore the metaphysics of infinite and infinitely small quantities, larger or smaller, is completely useless for differential calculation.”

The infinitesimals were simply a way of expressing itself that avoided the longest description in terms of limits. In fact, d’Alembert gave a good approximation of the correct definition of limit in terms of a variable quantity that approaches a fixed quantity less than any given quantity, although he also speaks of a variable that never reaches the limit. However, he did not give a formal display of the infinitesimal calculation that incorporated and used his ideas, which are fundamentally correct. Also d’Alembert remained in the vague on a number of points; he defined, for example, the tangent at a curve like the boundary of the secret when the two intersection points become one. This vagueness, especially in the ennunciation of the notion of limit, made many discuss whether a variable can reach its limit. For there was no explicit correct presentation of the infinitesimal calculation, d'Alembert warned those who were preparing to undertake the study with the phrase: "Go ahead, and faith will come to you."

D’Alembert distinguishes the convergent series from the divergent ones. In the article Série of the Encyclopédie says: “When progression or series approaches more and more to a finite quantity and, consequently, the terms of the series, or the quantities of which it is composed, are decreasing, it is said that the series is convergent and if you continue to infinity it will eventually become equal to this quantity. So, 1⁄2+1/4+1/8+1/16+... they form a series that always approaches 1 and that will become equal to it when you continue to infinity”.

In 1768 d’Alembert expressed doubts about the use of the series that are not convergent, saying in the Opuscules mathématiques: “As for me, I confess that all the reasonings based on the series that are not convergent... seem very suspicious to me, even when the results are in agreement with truths that have occurred in other ways.”

D’Alembert, in the article Fundamental contained in the volume VII (1757) of Encyclopédie, attacked Bernoulli. He did not believe that all odd periodic functions could be represented by a series such as, because the series is twice derived while this does not necessarily happen for an odd periodic function. However, even when the initial curve is sufficiently derived – and d’Alembert, in the work of 1746, required it to be twice – it is not necessarily representative in the form of Bernoulli. For the same reason, he criticised Euler's discontinuous curves. The dispute between d’Alembert, Euler and Bernoulli continued for a decade without an agreement. The essence of the problem was the extension of the class of functions that can be represented by the series of breasts or, more generally, by the Fourier series.

At the beginning of the century, most of the mathematicians believed that the different roots of the complex numbers introduced different types or orders of complex numbers and that there should be ideal roots whose nature were not able to specify but which could somehow be calculated. D’Alembert, on the other hand, in his memory, entitled Réflexions sur la causa générale des vents(1747), stated that each expression built from complex numbers through algebraic operations (including elevation to arbitrary power) is a complex number of form. The only difficulty he had in trying this statement was the case of. His proof of this had to be amended by Euler, Lagrange and others. In the Encyclopédie d’Alembert kept a discreet silence on complex numbers.

The 18th century was especially interested in science-related applications and since operational rules were intuitively safe, at least for real numbers, no one seriously concerned the foundations. Typical is the claim of d’Alembert in the article on the negative numbers of Encyclopédie. The article is not entirely clear and d’Alembert concludes by saying that “the algebraic rules of operations with negative numbers are generally allowed by everyone and recognized exact, whatever the idea that one can have on these quantities.” The various types of numbers, never properly brought to light, nevertheless acquired a stronger position in the Eighteenth century community. Also on the article Différence de l’Encyclopédie, about the letter of Newton to Collins of 1672 on the method of Sluse and Gregory for the tangents, d’Alembert writes: “This letter, which is sent to Leibniz, contains the method to find the tangents of the curves, but the application that it gives refers only to the curves whose equations do not present radicals. This letter therefore does not contain the differential calculation but only the calculation of Barrow for the tangents a little simplified. It is true that Newton mentions his method of finding tangents of all kinds of curves, geometrics and mechanics, there are or are not radicals in equations, but he merely declares it.”

Discussions about the foundations (or metaphysics, as was said at the time) of the calculation, inaugurated by Berkeley's criticism, were kept alive throughout the Eighteenth century and to it were the most authoritative mathematicians. D’Alembert clarified his positions in several articles of the Encyclopédie, in particular in the Limit entry, from which the following step was taken: “The limit theory is the basis of the true metaphysics of differential calculation.[...] Actually, the limit never coincides, or never becomes equal to the amount of which is the limit, but this approaches you more and can differ as little as you wish. The circle, for example, is the limit of circumscribed and inscribed polygons since it never confuses strictly with them although they can approach infinity. This notion can serve to clarify various mathematical propositions. For example, it is said that the sum of a decreasing geometric progression, whose first term is a and the second b, is; this value is not, properly, the sum of progression, is the limit of this sum, i.e. the quantity to which it can be approached as much as you want without ever getting there exactly. In fact, if and is the last term of progression, the exact value of the sum is aa-be/a-b, which is increasingly smaller than aa/a-b because in a geometric progression even if decreasing, the last term and is never =0: but, as this term is continually approaching zero, without ever coming, it is clear that zero is the limit and consequently the limit aa-be/a-b is.

For his defense of the Encyclopédie project, d’Alembert became famous as “the fox of the Encyclopedia” and, through the friendship of Voltaire and other “philosophes”, d’Alembert helped to open the way to the French Revolution. At the very young age of 24 he was elected to the Academy of Sciences, and in 1754 he became the perpetuel secrétaire, and in this capacity he was perhaps the most influential scientist in France.

THE CENTURY OF LIGHTS

In the 1600s, thanks to the brilliant scientific insights and the audacity of thinkers such as Newton, Galilei, Bacone, Locke and Cartesio many certainties rooted in the past began to crumble and soon would have been wiped out by the 18th century critical reason. With the disappearance of the Sun King in 1715, France finally liberates itself from the atmosphere of conservatism and intolerance that had characterized the last phase of the reign of Louis XIV. There were serious reasons for disappointment and discontent: the mercantile economy was questioned, agriculture was still backward, rural populations were stremned by hunger, epidemics and continuous wars; voices of blame were increasingly rising against luxury, parasitism and privileges of the noble class. The bourgeoisie was excluded from the privileges of aristocracy. In the course of the century, the demographic increase and the improvement of economic conditions helped to create a new faith in the reason and progress of humanity. Rationalism advocated a bitter battle against medieval concepts and customs. Thinkers such as Voltaire, d’Alembert and Maupertuis indicated in reason the necessary faculty to be able to stick to the facts, without losing themselves in non verifiable metaphysical theories. Locke’s empirism and the Newtonian need for a unitary explanation of natural phenomena were filmed in the most exquisitely philosophical field by E. Bonnot Abbot of Condillac (1714-1780). From Condillac's psychic premises, Helvétius (1715-1771) concluded that men, initially equal, were then shaped by the environment. Experiences, education, different engagements produce the conditions of inequality which is the basis of the distinction in social classes. To improve man must change his material conditions, first of all forms of government, such as monarchical absolutism, which are causes of miseries. In addition to the consequences of social policy and criticism, illuministic materialism opposed religious moralism.

The new French intellectuals, philosophes, formed a real opposition party that, despite the presence of theoretical differences rather strong within it, seemed however destined to animate a radical turning point in European history. The criticism of the religious, political and cultural institutions of the time, the restructuring of knowledge in new categories and systematics, together with the ambitious project to build a completely renewed, secular and guided society by the laws of nature constituted the central nodes of the illuminated philosophy that would lead to the revolution of 1789. The paths of the Illuminists fell against the fanaticism of Catholic culture, with its intolerance, its dogmas, its rigid hierarchical organization and against monarchical absolutism. The philosophy of the Lights involved every sphere of knowledge and led to a restructuring of conception, not only of politics and religion, but also of science, art, literature and economy.

“The Enlightenment is the exit of man from the state of minority that he must attribute to himself.[...] Knowing aude! Have the courage to use your own intelligence! This is the motto of Enlightenment.” Philosopher I. Kant wrote these pages in 1784, when the critical spirit that animated the battle of the Lights had been consolidated. The encyclopedia or reasoned dictionary of sciences, arts and crafts to which more or less well-known characters collaborated; the greatest merit goes to Diderot who wrote 1139 articles of history, art, philosophy and literature and ran much of the text of others in order to ensure a unity of thought. The voices of mathematics, physics and mechanics were written by d’Alembert. Other contributors include Montesquieu, Voltaire, Helvétius, Condillac, Rousseau (for music), Quesnay, Turgot (for the economy), Marmontel (for literary criticism) and Buffon (for the natural sciences).

In the technical field. scientific was evident the urgency that enliven the Encyclopedia to spread the knowledge of a renewed culture. The progress of mechanics and astronomy in the sixteenth and seventeenth century was consolidated in the next century on the basis of mathematical methods now established and found widespread diffusion thanks to the work not so much of scientists, as well as of literary. The same aims of the Encyclopedia enunciated by d’Alembert in the famous Preliminary Discourse, that is, the ordering and unification of knowledge, were in perfect harmony with the needs of scientists. According to the enlightened philosopher, through the use of three faculties, memory, reason and imagination, we come to the distinction of the three objects of knowledge, history, philosophy and fine arts, which represent the most general areas of the family tree of knowledge. In particular, philosophy coincides with the scientific knowledge of nature on a fundamental basis because it studies the properties of beings that we can know directly only through the senses. If the Newtonian system becomes part of the collective knowledge through the treatment of d’Alembert, in the scientific field those theories appear, since the publication of the Mathematic Principles (1687), the new paradigm of reference (and sometimes of confrontation) for the 18th century scientists, from Laplace to Gauss, from Eulero to Lagrange. Newton had operated a revolutionary synthesis that allowed to explain, with a single theory, the movements of celestial bodies and terrestrial phenomena.

In this direction scientific academies continued to study where applications and verifications followed: nature was observed, experimentally imitated and finally interpreted according to general laws formulated thanks to the apparatuses of mathematics, in particular the infinitesimal calculation of Leibniz and Newton.

In the field of physics, great importance covered the studies on electrical phenomena that led to the discovery by Coulomb (1736-1806) of the attraction and repulsion of the charges, to the idea of lightning by Franklin (1706-1790), to the demonstration of the existence of animal electricity through the experiments of Galvani (1737-1798) and to the invention of the first pile of Volta (1745-1827).

In the century of the Lumi the medicine took lessons from the facts interpreted with the quantitative criteria of mechanics and chemistry, making its own the need for interdisciplinary nature typical of the encyclopedic spirit. In natural sciences, the order of knowledge became a priority; the reports of scientific expeditions and the explorations of the colonies poured on the continent such a quantity of Information to impose the need for a rigorous classification of new plant and animal specimens. The most original conquest of 18th century science was the concept of evolution that, through Buffon and Lamarck, brought in the century Next to Darwin's theories.

The eighteenth century also marked a turning point for chemistry, which gained the dignity of autonomous science. The antecedent of modern chemistry can be indicated in the theory of flogisto, which allowed to convey attention to the phenomena of combustion and oxidation, although the proposed explanation presented many weak points from the beginning. Lavoisier (1743-1794) on the one hand demolished the theory of phlogist discovering the function of hydrogen in combustion processes, on the other hand introduced a quantitative survey of phenomena that allowed to elaborate a strict nomenclature of elements. Modern chemistry was thus founded, finally separated from physics. However, it should not be forgotten that the discoveries and inventions that characterized this fertile century were largely due to the considerable advances made by the technique. The 18th century science flourished in a social and economic context that posed an ever increasing number of practical questions, as evidenced by the enthusiasm with which the first achievements of the steam machine and the practical applications of studies on electricity were welcomed. The work of diffusion of the scientific knowledge implemented by Encyclopédie perfectly embodied the conception desired by Bacone a century before, according to which the social and public value of a fruitful science was realized in the union of theory and practice. The technical-artisanal knowledge, hidden in Renaissance workshops and handed down by the masters of the arts or specialized manuals, entered the relationship from the Seventeenth century onwards with scientific research and obtained an official recognition in the Encyclopédie. Diderot collected the necessary documentation to his intent by questioning the artisans in the workshops, sometimes getting the simplest machines to see how a work was born and to describe its production. The 11 volumes were dedicated to the tables of the arts and crafts; the techniques were so closer to the general public and were integrated into the cultural knowledge of hegemonics.

THE MATEMATICS OF THE FRANCE REVOLUTION

The age of revolutions did not affect only the sphere of politics. The French mathematicians who lived during the revolution not only gave numerous contributions to the whole of mathematical knowledge, but were largely the promoters of the main development lines of the explosive proliferation of mathematics in the century Next. Among the precursors of the French Revolution were Voltaire, Rousseau, d’Alembert and Diderot; in the field of mathematics, six men who would indicate the direction of future developments – Monge, Lagrange, Laplace, Legendre, Carnot and Condorcet – were also involved in revolutionary tumults.

In the 14th century Paris was one of the scientific centers of the world (the other was Oxford), but for a long time had lost this position. The University of Paris was behind the times. In France of the eighteenth century universities were not like today centres of mathematical studies, and it is difficult to mention even one mathematician of the eighteenth century who has carried out his activity, we say, at the University of Paris. Most French mathematicians of the time had relations with the church or with the army; others lived in the court of kings and princes or devoted themselves to private teaching. Among the many mathematical encyclopedias released in the last decades of the 18th century, the one that was most successful, judging by the numerous reissues, was the series of volumes of the Cours mathématique of Bézout. Lagrange had published his Mécanique analytique (1788), as well as several articles of algebra, mathematical analysis and geometry. Condorcet had published a De calcul intégral since 1765 and a Essai sur l’application de l’analyse à la probabilité des decisions rendues à la pluralité des voix in 1785. Monge had published numerous mathematical articles on the Acts of the Académie des Sciences. Through his many activities Monge had become, at the time of the revolution, one of the most famous French scientists. In fact, his reputation as a physicist and a chemist was perhaps superior to that of mathematician, as his geometry had not yet been recognized and evaluated to an appropriate extent to its importance. Géométrie descriptive, his masterpiece, had not been published because his superiors believed that he should be kept secret in the interest of national defence. Laplace and Legendre regularly collaborated with scientific periodicals, while Carnot in 1786 had already published a second edition of his Essai sur les machines en général, as well as poetic compositions and a work on fortifications.

At the beginning of the revolutionary period (1790), Talleyrand proposed the reform of weights and measures. The problem was demanded at the Académie des Sciences, which commissioned a committee. The committee included four mathematicians: Lagrange, Laplace, Legendre and Monge. The metric system represents, of course, one of the most tangible mathematical results of the French Revolution, but from the point of view of the theoretical development of mathematics its meaning is not minimally comparable to that of other contributions.

Condorcet belonged to the circle of intellectuals gathered around Voltaire and d’Alembert. He was a skilled mathematician and had published books on probability theory and integral computing. Confident in the perfectability of humankind and convinced that education would eliminate every vice, defended a system of public and free education, a wonderfully far-sighted idea, especially at those times. Condorcet perhaps owes its greatest reputation in the field of mathematics to the fact that it was a precursor of mathematics applied to social problems, especially through the application of the calculation of probability and statistics to such problems. With the advent of the Revolution, Condorcet's interests turned from mathematics to political and administrative problems. In 1792 he published his own scheme for a renewed educational system; but the proposal for a free education became the target of attacks and criticism. In 1794 he was arrested but was found dead in the cell, probably suicidal.

Most importantly, for the future of mathematics, were Monge's efforts to establish a school for the preparation of engineers. Thus the famous polytechnique ecole e Monge played an essential role in every stage of its creation. Monge held university courses on two mathematical disciplines both essentially new to traditional study programmes. The first was then known as stereotomy, corresponding to what is most commonly called descriptive geometry. In addition to the study of shadows, perspective and topography, Monge focused his attention on surface properties, including normal straights and tangent planes, and machine theory. Among the problems formulated by Monge v’era, for example, that of determining the intersection curve of two surfaces each of which is generated by a straight that in its motion intersects three skewers of space. Another problem was the determination of an equidistant space point from four dates. Problems like this are indicative of a change in education inherent in the French Revolution. We can say that while the 17th century was the century of the study of curves – the cycloid, the concoid, the catenary, the lemniscata, the equiangular spiral, the hyperboli, the parables and the spirals of Fermat, the pearls of Sluse, and many others – the 18th century was the century that really began the study of the surfaces. The birth of solid analytical geometry was partly due to the mathematical and revolutionary activities of Gaspard Monge. The lessons Monge held at the normal school in the academic year 1794-1795 were published under the title Géométrie descriptive. The concept behind the new descriptive geometry, i.e. the method of double orthogonal projection, is easy enough to understand. This simple process, today so common in the mechanical design, at the time of Monge provoked almost a revolution in the design technique used by military engineers. He also held a course on the “application of mathematical analysis to the geometry”. The abbreviated expression “geometry analytical” had not yet entered the current use; thus it was also for the expression “geometry differentile”, but the course held by Monge was essentially an introduction to this last discipline. Also in this field there were no school manuals, so Monge was forced to write and print his Feuilles d’analyse (1795) for the use of students. Here for the first time the analytical geometry in space actually acquired its own configuration and independence, this course formed the prototype of the current university programs of solid analytical geometry. In 1802 there appeared a large memory of Monge and Hachette entitled Application d’algèbre à la géométrie. The theorem with which this memory was opened is typical of a more elementary discussion of the topic. It is the generalization of the Pythagorean theorem much known in the eighteenth century: the sum of the squares of the projections of a flat figure on three planes between them perpendicular is equal to the square of the area of the figure. Monge and Hachette demonstrated this theorem exactly the same way that it is found in modern courses; in fact, the lightning-price admission volume could serve without difficulty as a text in a school of our century. The equations for the transformation of the coordinates, the conventional treatment of the straights and plans, the determination of the main plans of a quadrica, are arguments that are discussed extensively and completely. It is in the analytical geometry of Monge, more than in that of Clairaut and Eulero, which is for the first time a systematic study of the right in space.

Lagrange was so impressed by the work of Monge that, to what is said, he would exclaim: “With its application of mathematical analysis to geometry, this evil man will conquer immortality! ”.

Monge was a first floor figure of the French Revolution; however, the mathematician whose name was on the mouth of all the French during the Revolution was not Monge but Lazare Carnot.

Carnot was interested in the problems of education at every level, although it seems that he never taught. In 1797 Carnot wrote the Réflexions sur la métaphysique du calcul infinitésimal that enjoyed a wide popularity. In the second half of the eighteenth century none of the methods then commonly used in the infinitesimal calculation, neither the Newtonian one of the streams, nor the one of Leibniz based on the concept of differential, nor the one of d’Alembert who used the concept of limit, seemed satisfactory. The different methods adopted in the infinitesimal calculation, according to Carnot’s opinion, were nothing but simplifications of the old exhaustion method.

Today, however, the fame of Carnot is linked mainly to other works. In 1801 he published a work entitled De la corrélation des figures de géométrie, also characterized by a high degree of generality. Carnot tried to give the pure geometry a degree of universality comparable to that enjoyed by analytical geometry. In 1803 Carnot developed his theory of correlation between geometric figures in the Géométrie de position, a work that puts him next to Monge as the founder of the modern pure geometry. This passion for generalization that is found in Carnot’s work constituted the main driving force of modern mathematics, especially in our century. The topology in particular, interesting as it is to the properties of the figures that remain unchanged through a continuous deformation of the figures themselves, would fill Carnot with joy, if it could rise today, since he would recognize that it goes far beyond its correlation of geometric figures. The name Carnot is known among the mathematicians for a theorem that bears that name and that appeared in 1806 in a Essai sur la théorie des transversales. This theorem also represents a generalization of a result known since ancient times. A speculation led to financial ruin in 1809.

In 1794, the year of Terror, Legendre published his famous Eléments de géométrie, which had so much success in American schools. There were many fields where Legendre made significant progress, but they were almost all outside of the geometry: they were about the theory of differential equations, differential and integral calculation, function theory, number theory and applied psychology. He composed a three-volume treatise entitled Exercises du calcul intégral (1811-1819) which rivaled that of Euler for completeness and prestige; later he developed some parts of this treaty in three other volumes including the Traité des fonctions elliptiques et des intégrales eulériennes (1825-1832). Moreover, more importantly, he elaborated some fundamental analytical tools, which bear his name and proved very useful to physics mathematics. Legendre was also an important figure in the field of geodesia, where he developed the statistical method of minimum squares. The Institut's memories also contain one of Legendre's attempts to demonstrate the postulate of parallels, but among all his mathematical contributions Legendre preferred work on elliptical integrals and the theory of numbers. He published a Essai sur la théorie des nombres (1797-1798) in two volumes, the first treatise dedicated exclusively to this subject. His attention was attracted by Fermat's last theorem, and by 1825 he demonstrated his insolubility for n=5. Almost equally famous is a theorem on the congruous numbers published by Legendre always in the Essai of 1797-1798.

As soon as the Ecole Normale and Ecole Polytechnique were founded, Lagrange was invited to hold courses of mathematical analysis. The new system of studies required the preparation of notes or dispenses that summarised the content of the lessons: Lagrange drafted several series at various levels. For students of the normal school of 1795 he prepared and held lessons that would today be suitable for the first year of a higher algebra university course; the content of these lessons enjoyed a vast popularity that extended to America, where they were published in an English version called Lectures on Elementary Mathematics. For the upper-level courses of the ecole polytechnique, Lagrange held lectures of mathematics analysis and prepared a text that has always been considered a classic of mathematics: it is his Théorie des fonctions analytiques, which appeared in the same year that saw the publication of the Reflexions of Carnot: with both these works the year 1797 began the era of penalty in correspondence. Lagrange is generally considered as the most acute mathematician of the eighteenth century, comparable only to Eulero: his work presents, in fact, several aspects that cannot be easily illustrated in an elementary exhibition like ours. Among these should be noted the first, and perhaps larger, contribution of Lagrange: the calculation of variations. It was during the Berlin period that Lagrange published important memoirs about mechanical problems, such as that of the three bodies, for the first time fortified its new derivative calculation process and composed an important work on equation theory. In 1767 he published a memory on the method of calculating the approximate value of the roots of polynomial equations by means of continuous fractions; in another work of 1770 he considered the solvency of equations in terms of permutation carried out on their roots. It was this last work that had to lead to the theory of the groups, which later will have so great success, and to the demonstrations of Galois and Abel of non-resolvability, with ordinary methods, of the equations of grade greater than the fourth. The name of Lagrange is today linked to what is perhaps the most important theorem of group theory: whether or is the order of a subgroup g of an order G group O, then or is an O factor.

Lagrange found that the resolving equation of a fifth degree equation not only was not less than the fifth grade, as expected, but it was even a sixth grade equation. From this finding Lagrange drew the conjecture that polynomial equations higher than the fourth were not solved by the usual methods. He also elaborated the method of variation of parameters in the resolution of non-homogeneous linear differential equations.

Like many other modern first floor mathematicians, Lagrange also had a deep interest in the theory of numbers. In 1770 he published a demonstration of the theorem (of which Fermat claimed to have given a proof) according to which each number full-price admission positive is the sum of no more than four perfect squares; therefore this theorem is often known as the Lagrange theorem of the four squares.

He was a professor at normal school and polytechnique, but unlike Monge and Lagrange did not publish the texts of his lessons. His publications were mainly about celestial mechanics: in this field he affirmed himself as the most significant figure of the Next period in Newton and Napoleon, a great admirer of scientists, appointed him Minister of Public Education. The probability theory is debitrice in Laplace more than any other mathematician. From 1774 he wrote numerous memoirs on the subject, finally collecting the results gradually achieved in his classic Théorie analytique des probabilités of 1812. He drew this theory in all aspects and at every level, and in 1814 published his Essai philosophique des probabilités to offer an introductory exposure to the unskilled reader. Laplace wrote that “the theory of probability is only common sense expressed in numbers”; his analytique Théorie, however, reveals the hand of a great analyst who knows well the upper mathematical analysis. Among the many points on which Laplace drew attention in the newly cited work was the calculation of the lazy through the problem of the needle of Buffon, which had been almost forgotten for thirty-five years. This is often known as the Buffon-Laplace needle problem.

Laplace also drew Baynes' research into reverse probability from oblivion. In the Treaty of Laplace we find also exposed the theory of the minimum squares, which had been created by Legendre, with the addition of a formal demonstration that Legendre had failed to give. The analytique Théorie also contains the transformation of Laplace, of great utility in differential equations.

In a very technical writing of 1782 entitled Théorie des attractions des sphéroides et de la figure des planètes, also included in the Mécanique céleste, Laplace developed the concept of potential, which was very useful in the field of physics. The publication of the Laplace Heavenly Mechanics is usually considered as the culminating point of the Newtonian conception of gravitation.

Laplace and Lagrange, the two most important mathematicians of the Revolution, had in many respect opposite conceptions. For Laplace, nature constituted the essence, and mathematics represented only a bag of tools that he knew how to handle with extraordinary dexterity; for Lagrange the mathematics was a sublime art to himself. The magnitude of the heavenly Mechanics has often been described as difficult, but no one has ever said that it is elegant; the analytic Mechanics, on the contrary, has been called a “scientific poem” for the perfection and grandeur of its structure.

We close this small reflection with the year 1799, when Napoleon conquered the power and period of the Revolution can be considered concluded. This date, however, does not mark the end of the mathematical “our” activities: each of them continued to bring contributions to the mathematics. If we can draw a lesson from the successes two centuries ago it is that the things that really matter in the mathematics, and that have a lasting influence, are not those dictated by immediate practical needs.

Even in periods of great political and social upheavals are the things of “spirit”, in the French sense of the term, those that count more, and this spirit is perhaps best imparted by great masters. But perhaps even more important than this lesson is the moral illustrated by Carnot, that one should never lose heart, however disappointing is the political or intellectual situation.

100 TABLES IN EXHIBITION

The two volumes of the English Cyclopaedia or Universal Dictionary of Art and Sciences of Chambers, appeared in London in 1728 in two volumes, which constitutes the immediate Previous for the realization of the Encyclopédie, contained 30 tables; for the Encyclopédie 120 were planned. Diderot, already in the Prospectus, had elevated the number to 600, but altogether the engravings of the volumes of the Planches became 3000. Of these, about 2900 are dedicated to manufacturing or industrial processes. Diderot was firmly convinced that a look at the object or its faithful representation could be much more useful, on the information level, than a compilation of many pages. Diderot had perfectly understood how necessary technical design as an indispensable language for a rational dissemination of practical topics. Diderot sent designers to the factories to reproduce, with the utmost care, machines and tools and everything that could serve to increase the clarity of the description: a Titanic enterprise if you take into account the quantity of the objects and processes examined, even admitting that not all the engravings were drawn from original designs, but sometimes they had as models those of the Académie. As for the production processes, the number of illustrations would become infinity if all the steps were represented, for example, to transform an iron bar into a needle. The representation, in these cases, had to be limited to the really essential movements of the worker and therefore to the phases of the processing that were easier not only to draw but also to explain. “We have limited ourselves to representing the highlights,” says Diderot, “whose effective reproduction allows us to understand what were the other unseen phases of the sequence.” There is no wonder if, overall, this way of proceeding has however given rise to a very high number of tables. And it is remarkable that this challenging design and engraving work has been accomplished in a relatively short time.

The authors of the Encyclopédie were committed to giving a truly up-to-date view of the technologies in use. Excellent proposition that of course could not be completely realized. In fact, it was inevitable that a work of that magnitude and those claims should present unevenness, gaps and errors. As a matter of fact, it was only possible to give a rather static picture of the technological world “first”, although with a breadth, a clarity, a graphic and artistic quality up to that moment never seen, a static framework that seemed to call for change. It was here, as in the Descriptions, of the latest images of technical-scientific literature that, in contrast to what happened in technical drawings, mathematically and geometrically, presented in each process also man together with the tool or machine. These tables therefore also inform us about the environments and how hard they were then certain works and are configured, for a social history of work at that time, as a source of particular value which has not yet systematically drawn.

The need for illustrations found justification, according to the good principles of psychic philosophy, in the imperfection of technical languages in general, imperfection due to a long tradition of indifference to the “objects of life”: “The lack of custom both to write and to read writings on the arts makes it difficult to explain things in an intelligible way; it gives the need for illustrations. One could demonstrate with a thousand examples that a pure and simple linguistic vocabulary, well that it is done, cannot do without figures without incurring in obscure definitions and vague; the more so and more so this aid was necessary for us. A look at the object or its representation says more than a written page” (from “Prospectus” of Diderot).

In this exhibition are presented 100 tables, of 2794 of the work of Diderot and Alembert, which more directly recall the philosophy and geometry applied to nature, to other sciences and to the work of man: in architecture, astronomy, artistic techniques, music and tools, in the sciences and traditional techniques, in military art and in the crafts of everyday life. The “heart” of the exhibition is represented, of course, by the Tables of Mathematics, in the fifth tome of the “Raccolta”, performed under the direction of d’Alembert, which gave its explanation, except for Pascal’s arithmetic machine, presented by Diderot. What makes them especially interesting is the imbalance that distinguishes the series. The Algebra is not due to two tables, such as the Mathematical analysis. The hydraulic machines instead, with the explanations of Diderot, occupy twenty-five. How to say that pure nutrition affects publishers much less than its practical applications. The place in this series the manufacture of measuring instruments reflects the same concern. What Diderot had expressed since 1753 in the Interprétation de la nature: “The sphere of mathematics is a world of intellect, where what is assumed as absolute truth completely loses this prerogative when it comes back to the earth. It is deduced that it was up to the experimental philosophy to rectify the calculations of geometry, a conclusion accepted even by the mathematicians. But what is the purpose of correcting geometric calculation based on experience? Is it not easier to stick to the results of this? From that we see how the mathematics, especially the transcendental one, does not lead to anything precise without experience.” Following the thread of this reasoning the “mathematics” part of the Collection goes well beyond the series of tables of the beginning of the fifth tome, even though enriched by the Supplement: it in practice covers all the field of mechanical arts...

The 25 astronomy boards are worth much less than the corpus of articles dedicated to this science in the Encyclopedia. The tables of the fifth tome, prepared at the beginning of the enterprise, are almost all taken from Chambers. The best, and more numerous, concern instrumentation.

  • ARCHITECTURE AND CONSTRUCTION....................... 12 TABLES
  • ARTISTICAL TECHNIQUES............................................................................................ 6 TABLES
  • NTI............... 2 TABLES
  • MINERALOGIA: underground geometry....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
  • ASTRONOMY.........................................................................................................
  • GEOMETRY AND MATHEMATICS.................................................................................................................................................................................................................................................................................................................................... 19 TABLES
  • GNOMONICAL................................................................................ 9 TABLES
  • NAVIGATION............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
  • MECHANICAL................................
  • OTTI...................................................................................................
  • GEOGRAPHIC.................................
  • TOPOGRAPHIC AND CARTOGRAPHY................................................... 3 TABLES
  • LEGAL COURT.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
  • OROLOGER................................................................................ 1 TABLE
  • THE FUSION OF THE CAMPANE................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ 1 TABLE
  • MILITARY ART......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 4 TABLES
  • MARINA AND NAVAL ARCHITECTURE.....................

Note: A special note deserves the excellent CDROM “L’ENCYCLOPEDIE” of the DeAgostini Multimedia that presents life in the 1700s through the 2794 tables of the work of Diderot and D’Alembert.

  • Encyclopédie, ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers, par une Societé de Gens de Lettres. Mis en ordre et publié par M.Diderot, de l’Académie Royale des Sciences et des Belles Lettres de Prusse; et quant à la Partie Mathematique, par M. d’Alembert, de l’Académie Royale des Sciences de Paris, de cells de Prusse, et de la Societé Royale de Londres, B. Reissue in fac-simile 1966-67.
  • Idem, Italian translation by A. Calzolari,Ricci, Milan, 1970-78, 18 volumes.
  • Anthology of texts translated with the title Encyclopedia or Dictionary reasoned of Sciences, Arts and Crafts, Laterza, Bari, 1968.
  • Anthology, with the same title, edited by A.Pons, Feltrinelli, Milan,1966.
  • The Encyclopedia and the French Revolution of Orrei, The Editions of Work, Rome, 1946.
  • The Encyclopedia: History, Science, Ideology, Proust, Cappelli,Bologna,1978.

Curated by prof. Renzo Baldoni
Museum director