# ALEAE GEOMETRIA, the Geometry of Chance by Blaise Pascal

The "calculation of probabilities" began four years after the death of René Descartes [4][5][6][7][8][9][10], in 1654, in a correspondence between Blaise Pascal and Pierre Fermat [1][11][13][14][16][21][22][32]. They exchanged letters on elementary problems of gambling, in this case a problem of dice and a problem of "parties".  Pascal and Fermat were particularly interested by this problem and succeeded in "Party rule" by two different methods.  One understands the legitimate pride of Pascal in his address of the same year at the Académie Parisienne created by Mersenne, to which he presented, among "the ripe fruit of our Geometry" (“les fruits mûrs de notre Géométrie” in french) an entirely new treaty, of an absolutely unexplored matter, the distribution of chance in the games. In the same way, Pascal in his introduction to “Les Pensées” wrote that "under the influence of Méré, given to the game, he throws the bases of the calculation of probabilities and composes the Treatise of the Arithmetical Triangle. If Pascal appears at first sight as the initiator of the calculation of probabilities, watching a little closer, its role in the emergence of this theory is more complex. However, there is no trace of the word probabilities in Pascal's work. To designate what might resemble what we now call calculation of probabilities, one doesn’t even find the word in such a context. The only occurrences of probability are found in “Les Provinciales” where he referred to the doctrine of the Jesuits, or in “Les Pensées”. We do not find in Pascal's writings, the words of “Doctrine des chances”, or “Calcul des chances”, but only “Géométrie du hasard” (geometry of chance). In 1654, Blaise Pascal submitted a short paper to "Celeberrimae matheseos Academiae Parisiensi" (ancestor of the French Royal Academy of Sciences founded in 1666), with the title "Aleae Geometria” (Geometry of Chance) or “De compositione aleae in ludis ipsi subjectis", that was the seminal paper founding Probability as a new discipline in Science. In this paper, Pascal said “… et sic matheseos demonstrationes cum aleae incertitudine jugendo, et quae contraria videntur conciliando, ab utraque nominationem suam accipiens, stupendum hunc titulum jure sibi arrogat: Aleae Geometria” that we can translate as “By the union thus realized between the demonstrations of mathematics and the uncertainty of chance, and by the conciliation of apparent contradictions, it can derive its name from both sides and arrogate to itself this astonishing title: Geometry of Chance” (« … par l’union ainsi réalisée entre les démonstrations des mathématiques et l’incertitude du hasard, et par la conciliation entre les contraires apparents, elle peut tirer son nom de part et d’autre et s’arroger à bon droit ce titre étonnant: Géométrie du Hasard ». We can observe that Blaise Pascal attached a geometrical sense to probabilities in this seminal paper.  As Jacques Bernoulli, we can also give references to another Blaise Pascal document entitled “Art de penser” (the “Logique” of Port-Royal), published the year of his death (1662), with last chapters containing elements on the calculus of probabilities applied to history, to medicine, to miracles, to literary criticism, to events of life, etc.
In “De l'esprit géométrique », the use of reason for knowledge is thought on a geometric model. In geometry, the first principles are given by the natural lights common to all men, and there is no need to define them. Other principles are clearly defined by definitions of names such that it is always possible to mentally substitute the definition for the defined [23][24][25]. These definitions of names are completely free, the only condition to be respected is univocity and invariability.  Judging his solution as one of his most important contributions to science, Pascal envisioned the drafting of a small treatise entitled “Géométrie du Hasard” (Geometry of Chance). He will never write it. Inspired by this, Christian Huygens wrote the first treatise on the calculation of chances, the “De ratiociniis in ludo aleae” ("On calculation in games of chance", 1657). We can conclude this preamble by observing that seminal work of Blaise Pascal on Probability was inspired by Geometry. The objective of GSI conference is to come back to this initial idea that we can geometrize statistics in a rigorous way.
We can also make reference to Blaise Pascal for this GSI conference on computing geometrical statistics, because he was the inventor of computer with his “Pascaline” machine. The introduction of Pascaline marks the beginning of the development of mechanical calculus in Europe. This development, which will pass from the calculating machines to the electrical and electronic calculators of the following centuries, will culminate with the invention of the microprocessor. But it was also Charles Babbage who conceived his analytical machine from 1834 to 1837, a programmable calculating machine which was the ancestor of the computers of the 1940s, combining the inventions of Blaise Pascal and Jacquard’s machine, with instructions written on perforated cards, one of the descendants of the Pascaline, the first machine which supplied the intelligence of man.

References:
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[2] BARBARESCO, F., “Les densités de probabilité « distinguées » et l’équation d’Alexis Clairaut:                                                regards croisés de Maurice Fréchet et de Jean-Louis Koszul », Colloque GRETSI’17, Juan-Les-Pins-September 2017
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