International Symposium
Bruno de Finetti Centenary Conference

Rome, November 15-17, 2006

November 15, 2006  (from 12 a.m. to 6 p.m.)

Library of Department of Mathematics "G. Castelnuovo"
Università di Roma La Sapienza
piazzale Aldo Moro 5 - Roma

Exhibition of Historical books in Probability                        


 Probability from Cardano to de Finetti

Gerolamo Cardano-Scratch by himself


Ma sopra tutte le invenzioni stupende, qual eminenza di mente fu quella
di colui
che s'immaginò di trovar modo di comunicare
i suoi piú reconditi pensieri a qualsivoglia altra persona,

benché distante per lunghissimo intervallo di luogo e di tempo?
parlare con quelli che son nell'Indie,
parlare a quelli che non sono ancora nati
né saranno se non di qua a mille e dieci mila anni? e con qual facilità?

con i vari accozzamenti di venti caratteruzzi sopra una carta.
 galileo   Galileo Galilei Linceo


The early period of Probability

This section of the exhibition contains some of the most important contributions to probability from the very beginning to the 19th century.  The main source on the development of probability in this period is Todhunter's fundamental book A History of the Mathematical Theory of Probability , 1865.

1. The very beginning - XVII century

Girolamo Cardano
(1501-1576),  Liber de Ludo Aleae, in Opera Omnia, 1663

Galileo Galilei (1564-1642),  Discorso sopra il gioco dei dadi,  in Opere, 1744

Blaise Pascal (1623-1662),  Lettres à M. Fermat, in Oeuvres, 1779

Christiaan Huygens  (1629-1695),  De Ratiociniis in Ludo Aleae, in Opera varia, 1724


The exhibition starts with  Liber de Ludo Aleae  (The book on Games of chance)  by Girolamo Cardano (1501-1576).  The book was published in 1663,  long after the author's death, in the first volume of the Opera Omnia (Opera moralia, pages  262-276).

“Cardano's treatise occupies fifteen folio pages, each containing two columns. It may be considered as a gambler's manual. It contains much miscellaneous matter connected with gambling, such as description of games and an account of the precautions necessary to be employed in order to guard against adversaries disposed to cheat: the discussion relating to chances forms but a small portion of the treatise.  In Chapter XIII Cardano shows the numbers of cases which are favourable for each throw that can be made with two dice (the quantity of interest is the sum) . He also treat correctly the case of three dice. Furthermore Cardano treats the "Ludo fritilli, (a not completely clear variant of the previous case) giving a corresponding list of the number of favourable cases.”

(taken freely from Todhunter's fundamental book A History of … Probability ,1865)


The second author presented is Galileo Galilei   (1564-1642) with Considerazioni sopra il Giuoco dei Dadi   (Considerations on the game of  dice). The date of this piece is unknown; it first appeared in the edition of Galileo’s work published at Florence in 1718.  

“Galileo wrote it answering the following question at the request of a friend: with three dice the number 9 and 10 can each be produced  by six different combinations, and yet experience shows that the number 10 is oftener thrown than the number 9. Galileo makes a careful analysis of all the cases which can occur, and he shows that out of the 216 possible cases 27 are favourable to the appearance of the number 10, and 25 are favourable to the appearance of number 9.” 

(from Todhunter's fundamental book A History of…Probability ,1865)


            The next author is Blaise Pascal (1623-1662). The letter to Fermat presented in the exhibition was written on July 29, 1654. In the same year, 1654, the Traité du triangle arithmetique ("Properties of the Figurate Numbers"), had been printed, but it was not published until 1665, after the author’s death [a table of the arithmetical triangle, from the Ouvres, 1779, is shown in the exhibition].  The year 1654 is usually considered as the birth’s year of the Theory of Probability. 

“It appears that the Chevalier de Méré proposed certain questions to Pascal; and Pascal corresponded with Fermat on the subject of these questions. […] Three letters of Pascal to Fermat on this subject, which were all written in 1654, were published in the Varia Opera D. Petri de Fermat, 1679. […] Pascal’s first letter indicates that some previous correspondence had occurred which we do not posses; the letter is dated July 29, 1654. In this letter Pascal discusses  the problem to which […] he attached the greatest importance. It is called in English the Problem of Points, and it is thus enunciated: two players want each a given number of points in order to win; if they separate without playing out the game, how should the stakes be divided between them?”


The problem proposed by de Méré to Pascal was not new: it had been posed by Luca Pacioli (1445-1515), but his solution was not satisfactory. After giving the now famous solution to this problem, Pascal treats briefly another problem proposed by de Méré, and which is now known after his name. In modern terms we can rephrase the problem as follows:  is the probability of getting at least one six by throwing four times a die equal to the probability of getting at least a double six by throwing twenty-four times a couple of dice?      


The solutions given by Pascal to these problems are based on combinatorial calculus, the subject of his famous Traité du triangle arithmétique. 


Montmort (Essay d'analysis sur les jeux de hazard, 1708) was the first to attach Pascal's name to the triangle. Behind the Traité were many related investigations spanning many centuries and many countries and the triangle has had several names, e.g. in Italy it is called after Nicolò Tartaglia (1499-1547) .

(from Earliest Known Uses of Some of the Words of Mathematics)

« Elle doit la naissance à deux Géomètres français du dix-septième siècle, si féconde en grands hommes et en grandes découvertes, et peut-être de tous les siècles celui qui fait le plus d’honneur à l’esprit humain. Pascal et Fermat se proposèrent et résolurent quelques problèmes sur les probabilités. »  

Laplace, page 3 of  Théorie Analytique des Probabilités (1812)

« Une problème relatif aux jeux de hasard, propose a un austère janséniste par un homme de monde (Chevalier de Méré a été l’origine du calcul des probabilités. »

 Poisson, Recherches.. page 1



            Pascal did not published anything on Probability during his life. The first to publish a treatise (in 12 pages) on this subject, in 1657, was Christiaan Huygens (1629-1695).


Huygens wrote the treatise De Ratiocinis ... in his mother language, and his instructor in Mathematics, van Schooten, translated it into Latin.


The term Expectatio appears in the Latin translation by van Schooten and expectation appears in the English translation of the Latin, while Huygens' Dutch text has no term that translates as "expectation;" it speaks rather of the "value" of a game.


(from Earliest Known Uses of Some of the Words of Mathematics)



Huygens’ treatise contains fourteen propositions and ends with five problems [a copy of the problems taken from the Opera varia, 1724 is shown in the exhibition] . The first proposition asserts "If I expect a or b, and have an equal chance of gaining either of them, my Expectation is worth (a + b)/2." “The third proposition asserts that if a player has p chances of gaining a and q chances of gaining a, his expectation is (pa+qb)/(p+q).  […] Propositions IV—VII discuss the Problem of Points. His method is similar to Pascal’s. […] The fourteen proposition consists in the following problem. A and B play with two dice on the condition that A is to have the stake if he throws six before B throws seven, and that B is to have the stake if he throws seven before A throws six; A is to begin, and they are to throw alternately; compare the chances of A and B.” The solution given by Huygens is based on the third proposition, and therefore Huygens has not to show that the game will eventually end.



The treatise ends with “the following five problems:


(1) A and B play with two dice on this condition, that A gains if he throws six, and B gains if he throws seven. A first has one throw, then B has two throws, then A has two throws, and so on until one or the other gains. Show that A’s chance is to B as 10355 to 12276.


(2) Three players A, B, C take twelve balls, eight of which are black and four white. They play on the following condition; they have to draw blindfold, and the first who draws a white ball wins. A is to have the first turn, B the next, C the next, then A again, and so on. Determine the chance of the players.


(3) There are forty cards forming four sets each of ten cards; A plays with B and undertakes in drawing four cards to obtain one of each set. Show that A’s chance is to B’s as 1000 to 8139.


(4) Twelve balls are taken, eight of which are black and four are white. A plays with B and undertakes in drawing seven balls blindfold to obtain three white balls. Compare the chance of A and B.


(5) A and B take each twelve counters and play with three dice on this condition, that if eleven is thrown A gives a counter to B; and if fourteen is thrown B gives a counter to A; and he wins the game who first obtains all the counters. Show that A’s chance is to B’s as 244140625 is to 282429536481.”


(from Todhunter's fundamental book A History of … Probability ,1865)


2. The first books and the first asymptotic results - XVIII century


Jakob Bernoulli  (1654-1705),  Ars Conjectandi,  published posthumously, 1713

Pierre Rémond de Montmort  (1678-1719), Essay d'analyse sur les jeux de hazard,  II edition, 1713

Abraham de Moivre  (1667-1754) , The Doctrine of Chance, 1718

Gottfried Wilhelm Leibniz  (1646-1716),  Lettres a M. Montmort, in  Opera omnia,  1768

Johann Bernoulli (1667-1748), a letter to Leibniz on De Moivre and Montmort,


in Virorum celeberr. Got. Gul. Leibnitii et Johan. Bernoullii Commercium philosophicum et mathematicum, 1745



“The treatise by Huygens continued to form the best account of the subject until it was superseded by the more elaborated works of  J. Bernoulli, Montmort, and de Moivre.”


 (from Todhunter's fundamental book A History of … Probability ,1865)


The works of  Jakob Bernoulli, Montmort, and de Moivre are to be considered the first books on Probability, and all of them appeared at the beginning of the 18th century, a little more than half a century after Huygens’ treatise. All the five problems proposed by Huygens were considered by the above authors.


   It may be interesting to note that “Jakob Bernouilli solves the second problem of Huygens’ text [(2) Three players A, B, C take twelve balls, eight of which are black and four white. They play on the following condition; they have to draw blindfold, and the first who draws a white ball wins. A is to have the first turn, B the next, C the next, then A again, and so on. Determine the chance of the players ] on three suppositions as to the meaning; first he suppose that each ball is replaced after it is drawn; secondly he suppose that there is only one set of twelve balls, and that the balls are not replaced after being drawn; thirdly he supposes that each player has his own set of twelve balls, and that the balls are not replaced after being drawn.


    In solving the fourth problem of Huygens’ text  [ (4) Twelve balls are taken, eight of which are black and four are white. A plays with B and undertakes in drawing seven balls blindfold to obtain three white balls. Compare the chance of A and B. ] de Moivre took the meaning to be that A is to draw three white balls at least. Montmort had taken the meaning to be that A  is to draw exactly three white balls. Johann Bernouilli in his letter to Montmort (published in the second edition of Montmort’s book) took the meaning to be that A is to draw three white balls at least. Jakob Bernouilli had considered both the meanings.”   

(from Todhunter's fundamental book A History of … Probability ,1865)


The most important book of the exhibition is the Ars Conjectandi,  by Jakob Bernoulli  (1654-1705).  Jakob Bernoulli died in 1705 before finishing his opera. The fourth part of the opera proposed to apply the Theory of Probability to questions of interest in morals and economic sciences, but was left unfinished.  The publisher had asked to finish it first to the Jakob Bernoulli's brother, Johann Bernoulli, and then to Nicolas Bernoulli, a nephew of Jakob and Johann. Nicolas did not consider himself adequate to the task, and the opera was published as the author had left if, eight years after the author's death, in 1713.

“The Ars Conjectandi is divided into four parts. The first part consists of a reprint of the treatise of Huygens' work, completed with some annotations by Jakob Bernoulli (the first page of this part is shown in this exhibition). The second part is devoted to the theory of permutations and combinations. The third part consists of the solution of various problems relating to the games of chance. The fourth part, as already recalled, was unfinished; the most remarkable subject contained in it is the enunciation and investigation of we now call Bernoulli’s Theorem. It is introduced in terms which show a high opinion of its importance:”


« Hoc igitur est illud Problema, quod evulgandum hoc loco proposui, postquam jam per vicennium pressi, et cujus tum novitas, tum summa utilitas cum pari conjucta difficultate omnibus reliquis hujus doctrinae capitibus pondus et pretium superaddere potest. »

Bernoulli Ars conjectandi, page 227    


 (taken from Todhunter's fundamental book A History of …Probability ,1865)


Abraham de Moivre (1667-1754) published The Doctrine of Chance: A method of calculating the probabilities of events in play in 1718, although a shorter version in Latin, De mensura sortis, had been presented to the Royal Society and published in the Philosophical Transactions in 1711. Francis Robartes, who later became the Earl of Radnor, suggested to de Moivre that he present a broader picture of the principles of probability theory than those which had been presented by Montmort in Essay d'analyse sur les jeux de hazard (1708). Clearly this work by Montmort and that by Huygens which de Moivre had read, contained the problems which de Moivre attacked in his work and this led Montmort to enter into a dispute with de Moivre concerning originality and priority (the argument with Montmort appears to have been settled amicably).

The Doctrine of Chance appeared in expanded editions in 1718, 1738 and 1756. The first edition was dedicated to Newton (this is the edition shown in the exhibition). The definition of statistical independence appears in this book together with many problems with dice and other games:

« Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other. Two events are dependent, when they are so connected together as that the Probability of either's happening is alter'd by the happening of the other. »     The Doctrine of Chance (1738 edition)


Among other problems, the "gamblers' ruin" problem appears as Problem LXV in the 1756 edition. In A history of the mathematical theory of probability (1865), Todhunter says that probability:

“... owes more to [de Moivre] than any other mathematician, with the single exception of  Laplace.

The 1756 edition of The Doctrine of Chance contained what is probably de Moivre's most significant contribution to this area, namely the approximation to the binomial distribution by the normal distribution in the case of a large number of trials. De Moivre first published this result in a Latin pamphlet dated 13 November 1733 aiming to improve on Jakob Bernoulli's law of large numbers. The work contains « ... the first occurrence of the normal probability integral. He even appears to have perceived, although he did not name, the parameter now called the standard deviation ... » (Hacking, Biography in Dictionary of Scientific Biography , New York 1970-1990)

In Miscellanea Analytica (1730) appears Stirling's formula [ n! » c nn+½ e-n ] (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre wrote:

« I desisted in proceeding farther till my worthy and learned friend Mr James Stirling, who had applied after me to that inquiry, [discovered the value of the c=(2π)½]. »

      (taken from the  biography of de Moivre -The Mac Tutor History of Mathematics archive)



Pierre Rémond de Montmort (1678-1719) published the Essay d'analyse sur les jeux de hazard  for the first time in 1708, three years after Bernoulli’s death, and about half a century after Huygens’ De Ratiociniis … . “Montmort divides his work into four parts: I. Theory of combinations; II. Games of chance depending on cards; III. Games of chance depending on dice; IV. Solutions of various problems in chance, including the five problems proposed by Huygens.”  The second edition, shown in this exhibition, was published in 1713, the same publication year of the Ars Conjectandi. The second edition contains a fifth part containing one letter from Johann Bernoulli to Montmort, some letters from Nicolas Bernoulli to Montmort, and from Montmort to the young Bernoulli.


A peculiarity of this book is the following: “The name of Montmort does not appear in the title page or in the work, except once on page 338, where it is used with respect to a place.”


Montmort explains in the preface that “his studies has arisen from the request of some friends that he would determine the advantage of the banker at the game of Pharaon;  he had been led on to compose a work which might compensate for the loss of Jakob Bernoulli’s work”, due to his premature death. “Montmort refers briefly to his predecessors, Huygens, Pascal, and Fermat. He says that his work is intended principally for mathematicians, and that he fully explains the various games which he discusses because, «pour l’ordinaire les Sçavans ne sont pas Joueurs ».”


After the preface follows an Advertissement, where he says that two small treatises on the subject had appeared since his first edition; namely a thesis by Nicolas Bernoulli De arte Conjectandi in Jure, and a memoir by de Moivre, De mensura sortis .” The latter had been published in 1711, and “Montmort seems to have been displeased with the terms in which  reference was made to him by de Moivre”.



«M. de Montmort fut vivement piqué de cet ouvrage, qui lui parut avoir été entiérement fait sur le sien, et d’apres le sien. »

(Fontenelle, Eloge de M. de Montmort, 1719) 


Montmort died of small pox at Paris in 1719, one year after the publication of the Doctrine of Chance, by de Moivre. He was 31 years old.


(taken freely from Todhunter's fundamental book A History of …Probability ,1865)






Gottfried Wilhelm Leibniz (1646-1716) did not contribute much to probability, nevertheless he was very interested in the subject. Among others, Leibniz corresponded with Montmort, with Jakob Bernoulli and his brother Johann Bernoulli. In the exhibition one can find a letter from Leibniz to Montmort, in which he records a very favourable opinion of the Essay d'analyse . He says however 



« … J’aurois souhaité les loix des Jeux un peu mieux decrites, et les termes expliqués en faveur des étranges et de la postérité. »

Leibniz, 1708    





In the exhibition one can also find a letter from Johann Bernoulli (1667-1748) to Leibniz:



« … Dominus Moyavraeus, insignis certe Geometra, qui haud dubie adhuc haeret Londini, luctans, ut audio, cum fame et miseri, quas ut depellant, victum quotidianum ex informationibus adolescentum petere cogitur. O duram sortem hominis! Et parum aptam ad excitanda ingenia nobilia; quis non tandem succumberet sub tam iniquae fortunae vexationibus?  vel quodnam ingenium etiam fervidissimum non algeat tandem?  Miror certe Moyvraeum tantis angustistiis pressum ea tamen adhuc praestare, quae praestat. »

Johann Bernoulli, 26 April 1710






We end this part of this brief note by recalling that in the preface Montmort explains why he did not devote “a part to the application of his subject to political, economical, and moral questions, in conformity with the known design of Jakob Bernoulli; […] and he thus states the conditions under which we may attempt the applications with advantage:” 



« 1° borner la question que l’on se propose une petit nombre de suppositions, établies sur des faits certains ; 2° faire abstraction de toutes les circonstances ausquelles la liberté de l’homme, cet écueil perpétuel de nos connaissances, pourroit avoir quelque part. »

Montmort, Essai d’Analyse …   



The unfinished design of Jakob Bernoulli to apply probability to political, economical, and moral questions started to be implemented mainly after 1763, when Richard Price published posthumously the single paper by Thomas Bayes (1702-1761): An Essay towards solving a Problem in the Doctrine of Chances (Philosophical Transactions of the Royal Society of London 53 (1763), 370-418).  This work may be considered as a starting point for the application of probability to this kind of questions.   


The Essay considers the problem: "Given the number of times in which an unknown event has happened and failed: Required the chance of the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named." Or, in modern terms: given the outcomes of a number of Bernoulli trials, find the posterior distribution of the probability of a success. For the prior Bayes took a uniform distribution on the unit interval. […]

Inverse Probability was the term used in the 19th and early 20th centuries for the probability found when reasoning from effects to causes, direct probabilities being used when reasoning from causes to effects. […] Today the terms Bayes's Formula, Rule and Theorem are associated with a basic theorem on conditional probability. "La règle de Bayes" appears with this meaning in 1843 in A. A. Cournot's Exposition de la Théorie des Chances et des Probabilités (pp. 158-9). Cournot says the rule is "attributed to Bayes." It is not in the Essay but comes from Laplace: it is his VI principle for the case where the causes are unequally probable: see Théorie Analytique des Probabilités (1814, pp. xiv-xv in the edition on Gallica.) Laplace's terminology, of "cause" or "hypothesis" for the event whose conditional probability is to be found, survived well into the twentieth century. Thus in J. L. Coolidge's An Introduction to Mathematical Probability (1925) "Bayes' Principle" sits in the chapter on "the probability of causes." However "der Bayessschen Satz" in A. N. Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung (1933, p. 46) is just a theorem about events.


 (from Earliest Known Uses of Some of the Words of Mathematics)





In the exhibition, in a small separate section devoted to a few modern translations of antique books, one can find a German translation of Bayes contribution published in the interesting Ostwald’s collection Klassiker der exakten Wissenschaften (in an edition of 1908).



3. Probability as a degree of belief and applications - XVIII and XIX century

Gottfried Wilhelm Leibniz
  (1646-1716),  Nouveaux Essaies sur l'entendement humain, in Oeuvres philosophiques latines et françoise, 1765

Marie Jean Antoine de Condorcet
 (1743-1794), Essai sur l'Application de l'Analyse à la Probabilité des Décisions rendues à la pluralité des voix., 1785

Pierre Simon de Laplace  (1749–1827), Théorie Analiytique des Probabilités, III edition, 1820, containing the Essai philosophique sur les probabilités as an Introduction.

Siméon Denis Poisson  (1781-1840), Recherches sur la probabilité des jugements en matière criminelle et en matière civile, 1837




As already recalled, G. W. Leibniz was very interested in the subject, and especially in its applications.  This is clearly explained in many of his contributions. In the exhibition it is shown the Nouveaux Essais sur l'entendement humain ("New Essays on Human Understanding")


Nouveaux Essais sur l'entendement humain was a chapter-by-chapter rebuttal by Gottfried Leibniz of John Locke's major work, An Essay Concerning Human Understanding. It is one of only two full-length works by Leibniz (the other being the Theodicy). Like many philosophical works of the time, it is written in dialogue form. It was originally written in 1704, but Leibniz refrained from publishing it at that time in deference to Locke's death that same year, and it did not see print until 1764, at which time much of Leibniz's philosophy was out of favour.

(from Wikipedia Nouveaux Essais sur l'entendement humain)    




In the exhibition it is shown the beginning of Chapter XVI-Livre IV. The external margins contain the following indications



« De differens degrés de probabilité en Jurisprudence, en Medicine , dans les Mathematiques » [sic] 


Leibniz , Nouveaux Essais sur l'entendement humain, 1704 (1765)    



It may be interesting to observe that in Nouveaux Essais ..., the title of Livre IV is De la Connaissance, and that (in this Livre) the titles of Chapters XIV, XV, and XVI are Du Jugement , De la Probabilité, and De degrés d’Assentiment, respectively.  

            The book  Essai sur l'Application de l'Analyse à la Probabilité des Décisions ..., by Marie Jean Antoine de Condorcet  (1743-1794) was published in 1785, a few years before the French Revolution, and less than ten years afterwards Condorcet perished a victim of the Revolution.  The book is divided into five parts; the first and second part are devoted to examine the probability that a “correct” decision is taken under different conditions (Hypotheses). The conditions concern the number of voters, the majority in order to take a decision, and the probability to take the correct decision (vérite)  or the wrong one (erreur). The third part is devoted to the inverse problem. As an example we quote the first of the thirteen preliminary problems in this part:


Soient deux évènements seul possibles A et N, dont on ignore la probabilité, et qu’on sache seulement que A est arrivé m fois, et N, n fois. On suppose que l’un des deux évènements arrivés, et on demande la probabilité que c’est l’évènement A, ou que c’est l’évènement N, dans l’hypothèse que la probabilité de chacun de deux évènements est constamment la même.  

Condorcet, Essai sur l'Application de l'Analyse à la Probabilité des Décisions..,1785    




With respect to these preliminary problems Condorcet makes the following historical remark



L’idée de chercher la probabilité des évènements futurs d’après des évènements passés, paroît s’être présentée a Jaques Bernoulli e à Moivre, mais ils n’ont donné dans leurs ouvrage aucune méthode pour y parvenir. Bayes et Price en ont donné une dans les Transaction philosophiques, années 1764 et 1765, et M. de la Place est le premier qui ait traité cette question d’une manière analytique.


Condorcet, Essai sur l'Application de l'Analyse à la Probabilité des Décisions…,1785

page LXXXIII    





The fourth part is devoted to examine the modifications which the results of the preceding part of his book require, before they can be applied to practice. For instance we cannot in practice assume that all voters are of equally skill and honesty.  […]  Condorcet asserts that the fifth part is devoted to some examples. But it would be rather more correct to describe this part as furnishing some additions to the preceding investigations than as giving examples of them.

(from Todhunter's fundamental book A History of …Probability ,1865)


Pierre Simon de Laplace  (1749–1827) published the  Théorie Analiytique des Probabilités in 1812. The third edition, 1820, (shown in the exhibition) contains the Essai philosophique sur les probabilités as an Introduction. His main contributions concern generating functions, the normal approximation, and the development of the now called Bayes’ rule.


The first edition of Laplace's Théorie Analytique des Probabilités was published in 1812. This first edition was dedicated to Napoleon-le-Grand but, for obvious reason, the dedication was removed in later editions. The work consisted of two books […] The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, [the now called] Bayes's rule, and remarks on moral and mathematical expectation.

Later editions of the Théorie Analytique des Probabilités also contains supplements which consider applications of probability to: errors in observations; the determination of the masses of Jupiter, Saturn and Uranus; triangulation methods in surveying; and problems of geodesy in particular the determination of the meridian of France. Much of this work was done by Laplace between 1817 and 1819 and appears in the 1820 edition of the Théorie Analytique.

 (from Laplace’s biography -The Mac Tutor History of Mathematics archive)



In the 1730s De Moivre had found the normal approximation to the binomial. Then in the early 19th century Laplace, Poisson, Cauchy and others worked on normal approximations in connection with the theory of least squares. Laplace's "On the probability of errors of the mean results of a great number of observations and of the most advantageous mean results" Théorie Analytique des Probabilités livre II, chapitre IV, p. 309 was the most influential treatment from that era.

 (from Earliest Known Uses of Some of the Words of Mathematics)



In Recherches sur la probabilité des jugements en matière criminelle et en matière civile, 1837, Siméon Denis Poisson  (1781-1840) generalized Bernoulli’s Theorem (to the case of trials with unequal probabilities) and introduced  the expression "law of large numbers" .


In Recherches sur la probabilité des jugements en matière criminelle et matière civile, an important work on probability published in 1837, the Poisson distribution first appears. The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times. He also introduced the expression "law of large numbers". Although we now rate this work as of great importance, it found little favour at the time, the exception being in Russia where Chebyshev developed his ideas.

 (from Poisson’s biography -The Mac Tutor History of Mathematics archive)

A new era began with Pafnuty Lvovich Chebyshev  (1821-1894). The collection of his works Oeuvres , 1899-1907, [shown in the exhibition] was edited by his two students M. A. Markov and N. Sonin.  The collection contains some papers devoted to probability and in particular the two papers (1867, 1887)  where Chebyshev gave the proof of  the inequality which has taken its name.


The inequality in the discrete case had been already proved by L. J. Bienaymé in  Considérations à l'appui de la découverte de Laplace ... Comptes Rendus de l'Académie des Sciences, 37, (1853), 309-324.   In the later literature Bienaymé's contribution was often overlooked, thus A. A. Markov refers to "die Ungleichheit von Tschebyscheff." in Wahrscheinlichkeitsrechnung (1912, p. 56).

 (from Earliest Known Uses of Some of the Words of Mathematics)


In the second paper (1887) Chebyshev extends the result to the continuos case


… and gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace. Of this Kolmogorov wrote

The principal meaning of Chebyshev's work is that through it he always aspired to estimate exactly in the form of inequalities absolutely valid under any number of tests the possible deviations from limit regularities. Further, Chebyshev was the first to estimate clearly and make use of such notions as "random quantity" and its "expectation (mean) value".

 (from Chebyshev’s biography -The Mac Tutor History of Mathematics archive)



The birth of modern Probability

This section is divided into two parts.

4a.  The first part starts with some fundamental books on probability, published between the end of the 19th century and the beginning of the 20th century, such as

Calcul de Probabilités (1888), by J. Bertrand 

Calcul de Probabilités (II edition 1912), by H. Poincaré  [the I edition appeared in 1896]

 Wahrscheinlichkeitsrechnung und ihre Anwendung ...   ( II edition in two volumes 1908-10), by E. Czuber (1851-1925)  [the I edition in one volume appeared in 1902]


Wahrscheinlichkeits-Rechnung (1912, German translation from Russian), by  A. Markov

The books by J. Bertrand, H. Poincaré, E. Czuber, and A. Markov are listed in


Calcolo delle Probabilità, (I edition 1919 in one volume, II edition 1925 in two volumes) by G. Castelnuovo (1865-1952),

as the most important books the author, Guido Castelnuovo, consulted while writing his own book on probability.  This list appears just after the preface of the II edition. In the revised reprint of the II edition of 1933, Castelnuovo mentions some other books. Most of the books in Castelnuovo’s list are presented in this Section.


For many years Castelnuovo's book was the (almost) unique book on probability, written in Italian. The book had a fundamental role for Bruno de Finetti, who started to study probability by reading this book and the book by Czuber.

[see the text of his last lesson, Scientia, 1976] 

[see also D.Cifarelli and E. Regazzini  de Finetti's Contribution to Probability and Statistics (in JSTOR)]


Castelnuovo quotes also the book by L. Bachelier, Calcul des probabilités (1912), but he states that the book by Bachelier has been written using an approach different from that of his own book: "Notevole per elevatezza di mezzi matematici impiegati, ma redatto secondo un indirizzo diverso da quello del mio Trattato". In the list of the most important books consulted by Calstelnuovo there is also the book  Eléments de la théorie des probabilités, by Émile Borel.  This book does not appear in the exibition, while the exibition will show some of the volumes of the collection

Traité du Calcul des Probabilités et de ses Applications (1925), edited E. Borel

and in particular

Volume1 I. Principes et formules classiques du calcul des probabilités, by E.Borel (1871-1956), Leçons professées à la Faculté des Sciences de Paris. Redigées par R.Lagrange. 159 p.

Volume3   I. II. Les principes de la Théorie des probabilités, by Maurice Fréchet (1878-1973).

In the revised reprint of 1933, Castelnuovo also quotes the following books, published after the second edition of 1925

 Calcul de Probabilités(1925),  by Paul Lévy (1886-1971)

 Wahrscheinlichkeitsrechnung und ihre Anwendung ...(1931), by Richard Von Mises (1883-1953)


At that time the collection Traité du Calcul des Probabilités et de ses Applications, edited E. Borel, was not yet complete.  Furthermore the second list contained also the books by A.Fischer, The mathematical Theory of Probabilities (1925) and by J.L.Coolidge, An Introduction to mathematical Probability (1925), not presented in the exhibition.




 4.b  The second part of this section contains some works, which played a fundamental role in the birth of modern probability. In particular we present here two papers of the authors that are considered as the fathers of what is called Brownian (or Wiener) process, L. Bachelier (1870-1946) and A. Einstein (1879-1955) [the paper by Wiener “Differential spaces”(1923) is not presented here]. Furthermore there is a paper by Percy John Daniell (1889-1946), which is a fundamendal link between the Theory of Probability and Integration.


We quote here only the first one to be published: Théorie de la spéculation (1900), by  L.Bachelier [Annales Scientifiques de l'École Normale Supérieure Sér. 3, 17 (1900), p. 21-86]


      The rest of this part concerns de Finetti and Kolmogorov, and presents the papers on stochastic processes with independent increments [now known as Lévy’s processes], that these authors published in The Rendiconti of the Accademia delle Scienze in the years 1929, 1930, 1931, and 1932. [These papers were presented to the Accademia by Guido Castelnuovo]



We quote here only the paper by Bruno de Finetti  (1906-1985)

Le funzioni caratteristiche di legge istantanea.

Atti della Reale Accademia Nazionale dei Lincei. Rendiconti Classe di Scienze Fisiche, Matematiche e Naturali, (6), 12 (1930), pp. 278-282

and the paper by Andrei Nikolaevich Kolmogorov   (1903-1987)

Sulla forma generale di un processo stocastico omogeneo (Un problema di Bruno de Finetti).

Atti della Reale Accademia Nazionale dei Lincei. Rendiconti Classe di Scienze Fisiche, Matematiche e Naturali, (6), 15 (1932), pp. 805-808.


This part ends with

Grundbegriffe der Wahrscheinlichkeitsrechnung (1933),  by A.N. Kolmogorov

and with

Bruno de Finetti’s "La prévision: ses lois logiques, ses sources subjectives," Annales de l'Institute Henri Poincaré, 7, (1937) 1-68.

These two works are two cornerstones in the modern theory of probability: the one by Kolmogorov in the axiomatization of probability, and the one by de Finetti in the subjective interpretation of probability.


Interest in EXCHANGEABLE random variables dates from the 1920s with Bruno de Finetti (1906 - 1985) the most influential contributor. However "exchangeable" emerged only slowly as the standard term. De Finetti used "équivalent" in his most widely read work, "La prévision: ses lois logiques, ses sources subjectives," Annales de l'Institute Henri Poincaré, 7, (1937) 1-68. L. J. Savage The Foundations of Statistics (1954) used the term "symmetric." Pólya suggested the term "échangeable" and it appeared in a 1943 book by Fréchét. De Finetti took up the term and "exchangeable" was adopted in the 1964 English translation of his 1937 work. David (2001) cites M. Loève's Probability Theory (1955) for the first occurrence of "exchangeable" in the English literature.

[This entry was contributed by John Aldrich, based on J. von Plato Creating Modern Probability (1994).]

COHERENT in subjective probability theory. The term is derived from the "cohérence" of B. de Finetti's "La prévision: ses lois logiques, ses sources subjectives," Annales de l'Institute Henri Poincaré, 7, (1937) 1-68. The English term is found in the mid 1950s, most conspicuously in Abner Shimony "Coherence and the Axioms of Confirmation," Journal of Symbolic Logic, 20, (1955), 1-28.

The term "consistency" was used in F. P. Ramsey's treatment of subjective probability, "Truth and Probability" (1926) (published in The Foundations of Mathematics and other Logical Essays (1931)): the calculus of probabilities can be "interpreted as a consistent calculus of partial belief."

(Based on a note to the translation of de Finetti (1937) in H. E. Kyburg Jr. & H. E. Smokler (eds) Studies in Subjective Probability (1964))  

 (from Earliest Known Uses of Some of the Words of Mathematics)

de Finetti and Probability


The first work in this section is the copy of Introduzione matematica alla statistica metodologica, the manuscript of the Lessons that Bruno de Finetti gave at the Istituto Centrale di Statistica, when he was 24 years old. A dedication to G. Castelnuovo can be found in the front page.  

Then one can find some works on the subjective interpretation of probability, in chronological order, such as Probabilismo: Saggio critico sulla teoria delle probabilità e sul valore della scienza (1931) and Sul significato soggettivo della probabilità (1931). This part ends with

     Compte rendu critique du colloque de Genève sur la théorie des probabilités (1939)

Which is presented together with the collection of the

Colloquie consacré a la théorie des probabilités / présidé par Maurice Fréchet. (1938)
containing contributions by
M. Fréchet, W. Heisenberg, G. Pólya, P. Cantelli, I.F. Steffensen, W. Feller, P. Lévy, R.Von Mises, A. Wald, H. Cramér, O. Onicescu,  E. Hopf , E. Slutsky,  S. Bernstein, H. Steinhaus, B. de Finetti, J. Neyman, V.Glivenko, E. Dodd, G. Jordan, N. Obrechkoff.


The collection is devoted to a conference which was held in Genève in the week 10-15 October 1937, thanks to an anonymous sponsor. Some of the invited Speakers could not participate to the conference, nevertheless they contributed with a paper in this collection.

Bruno de Finetti booklet is a precious complement to the conference: it contains also the discussions of some of the participants to the conference, and in particular some in which Doeblin was involved. This booklet is the translation of a paper by Bruno de Finetti, and the Italian version had appeared in the Giornale dell'Istituto Italiano degli Attuari. The French edition is completed with the addition of the abstracts of some invited speakers which could not participate. Doeblin contributed to this French edition by  writing some of these abstracts.



The exhibition continues with some papers on insurance problems, such as

Il problema dei “pieni”.

Giornale dell'Istituto Italiano degli Attuari, (11) 1940, pp. l-88


Impostazione individuale e impostazione collettiva, del problema della riassicurazione.

Giornale dell'Istituto Italiano degli Attuari, 13 (1942), pp. 28-53

which have been now discovered in the international scientific community.



In the part dedicated to insurance we find also a small contribution that de Finetti gave to the Enciclopedia Treccani. In the Appendix of Enciclopedia Treccani (1949/1960) he wrote the voice Attuariale, matematica. He never mentioned it: this small contribution was not even quoted in the list of his works prepared by himself, and his daughter Fulvia did not know anything about it. It has been added to the list of his work only very recently [w.r.t. November 15, 2006]. For sure this fact means that he did not give much importance to this small work, but also shows his attitude towards them.  

It was not easy to find out this contribution in the Enciclopedia, because in the general index was quoted as Matematica, but at this voice is due to Tricomi. Maria Teresa Liuzza, of the Istituto Treccani, helped Giovanna Nappo to find out it. She also kindly provided the following list concerning quotations of Bruno de Finetti in the Enciclopedia Treccani.


volume XXIV pag. 284,

voce NASCITA (tabella relativa ai saggi di fecondità matrimoniale - formula di Tait), autore Luigi Galvani;

volume XXVII pag. 928,

voce POPOLAZIONE ( formula descrittiva dell'evoluzione di una popolazione considerata come funzione del tempo)autore Ugo Giusti;

Appendice II vol. 1 pag. 742,

voce CURVE STATISTICHE (Annali di statistica ) - autore  Rodolfo Benini;

Appendice III vol. 2 - pagg. 485-486 –

voce PROBABILITA' (CALCOLO DELLE)- autore Giuseppe Pompilj;

Appendice IV vol. 1 pag. 189,

voce ATTUARIALE, MATEMATICA – autore Giuseppe Ottaviani;

Appendice IV vol. 3 pag. 53, 

voce PROBABILITA' (CALCOLO DELLE), autore Giorgio dell'Aglio;

Appendice V  vol. 2 pag. 330,

voce FRATTALI (Autosomiglianza), autore Luigi Accardi;

Appendice V  vol. 2 pag. 691,

voce INFERENZA STATISTICA  (Metodi Bayesiani , probabilità soggettiva), autore Ludovico Picconato;

Appendice V vol. 4  pag. 272, 

voce PROBABILITA' (CALCOLO DELLE), autore Luigi Accardi;

Appendice V vol. 5 pagg. 211-212, 

voce STATISTICA, autore Ester Capuzzo.



The exhibition contains also the famous textbook Matematica logico intuitiva (III edition, 1959), but especially the book Teoria delle probabilità. Besides the edition of 1970, the Library has also two copies of a preliminary version (in two chapters), containing some corrections of the author. A few years after the 1970 edition (in Italian) the book was translated into English by A. Smith and A. Machì.  The exhibition ends with some  books dedicated to de Finetti (or named after de Finetti’s Representation Theorem), and a few collections of his works. The last one (Opere scelte in 2 volumes, 2006) has been published by U.M.I. this year (one century after his birth) and is presented in the Bruno de Finetti Centenary Conference, just before the opening of the exhibition.










In the exhibition, in a small separate section devoted to a few modern translations of antique books, one can find


an English translation of Cardano’s Liber de Ludo Aleae (in an edition of 1961),


a German translation of Bayes-Price contribution, published in the interesting Ostwald’s collection Klassiker der exakten Wissenschaften (in an edition of 1908),


a German translation of Laplace’s book “Theorie Analitique…”, published in 1932, again in Ostwald’s collection Klassiker der exakten Wissenschaften,


and finally the English version of Kolmogorov book “Grundbegriffe…”




definetti ride

Probability from Cardano to de Finetti



A special thank to the staff of the Library: Angelo Bardelloni, Carolina Del Bufalo, Alessandra Foti, Ninfa Inguì, Chiara Tullio, and especially to Adele Piccolomo, which coordinated their (and my) work, besides working hard on the project; a thank also to Lucilla Vespucci which was fundamental in the starting steps of the organization.

Their work and their suggestions were precious (and essential) for the realization of the exhibition.
Giovanna Nappo

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LINKS to web-sites on the Hystory of  Probability and Statistics

·        Figures from the History of Probability and Statistics

·        A short history of Probability

·        The Beginnings of Modern Probability Theory  (section 4 of A Handbook for the Early History of Modern Science by Ian C. Johnston)

·        Studies in the History of Statistics and Probability I and II   (from  Materials for the History of Statistics )

·        Life and Work of Statisticians    ( from  Materials for the History of Statistics )

LINKS to web-sites containing digital copies of Historical Books on Mathematics

Cornell University Library. Historical Math Monographs

University of Michigan Historical Math Collection

Livres Numérisés Mathématiques (LiNuM)


LINKS to papers on Bruno de Finetti (available only from the web-site



LINKS to papers of  Bruno de Finetti (some of them are available only from the web-site