Exercitationum Mathematicarum libri quinque. . Quibus accedit Christiani Hugenii tractatus de ratiociniis in aleae ludo. [I: Propositionum Arithmeticarum et Geometricarum centuria; II: Constructio Problematum simplicium geometricorum; III: Apollonii Pergaei loca plana restituta IV: Organica conicarum Sectionum in plano descriptio; V: Sectiones miscellaneae triginta; Quibus accedit: Tractatus de ratiociniis in aleae ludo Christianus Hugenius]

The First Published Work on Probability Theory. First edition of Huygens' De ratiociniis in aleae ludo, the first printed treatise on probability. "Having heard in Paris about Pascal's work in probability problems, Huygens (1629-95) himself took up their study in 1656. This resulted in the Tractatus de ratiociniis ., a treatise that remained the only book on the subject until the eighteenth century. In his first theorems Huygens deduced that the 'value of a chance,' in the case where the probabilities for a and b are to each other as p:q, is equal to (pa + qb) / (p + q). He thus introduced as a fundamental concept: the expectation of a stochastic variable rather than the probability of a process (to put it in modem terms). Subsequent theorems concern the fair distribution of the stakes when a game is broken off prematurely. The treatise closes with five problems, the last of which concerns expected duration of play" (DSB). "If one looks on the first calculations of a probability by Cardan and by Galileo as unimportant and says that the real begetter of the calculus of probabilities is he who first put it on a sound footing, then we should pass over not only Cardan and Galileo but also Pascal and Fermat. For although Fermat, if provoked, could have done as much and more than his successors, yet the fact remains that his contribution was in effect the extension of the idea of the exhaustive enumeration of the fundamental probability set, which had already been given by Galileo. The scientist who first put forward in a systematic way the new propositions evoked by the problems set to Pascal and Fermat, who gave the rules and who first made definitive the idea of mathematical expectation was Christianus Huygens" (David, p. 110). "It was not until the tremendous researches of the 1690-1710 period which resulted in Essai d'Analyse sur les Jeux de Hasard (Montmort, 1708), Ars Conjectandi (James Bernoulli, 17l3), Calcul des Chances, à la statistique générale, àla statistique des décès et aux rentes viagères (Nicholas Struyck, 1713), and Doctrine of Chances (Abraham de Moivre, 1718), that Huygens' work was superseded, and then not entirely since James Bernoulli incorporated it in the Ars Conjectandi" (ibid., pp. 115-116). Huygens' treatise first appeared as an appendix to van Schooten's Exercitationum, which comprises five treatises, each with their own title-page, on various mathematical subjects. This was van Schooten's principal original work. "Book I contains elementary arithmetic and geometry problems . Book II is devoted to constructions using straight lines only and Book III to the reconstruction of Apollonius' Plane Loci on the basis of hints given by Pappus. Book IV is a revised version of van Schooten's treatment of the kinematic generation of conic sections, and Book V is a collection of individual problems. Worth noting, in particular, is the restatement of Hudde's method for the step-by-step building-up of equations for angular sections and the determination of the girth of the folium of Descartes: x3 + y3 = 3axy. Also noteworthy is the determination of Heronian triangles of equal perimeter and equal area (Roberval's problem) according to Descartes's method (1633)" (ibid.). RBH lists only one copy since Macclesfield, and that in a modern binding. Around 1564, Girolamo Cardano composed a work on probability entitled Liber de ludo aleae ('Book on Games of Chance'), but this was not published until 1663 as part of his Opera. Some time between 1613 and 1623, Galileo wrote about die-throwing problems but this remained unpublished until the 20th century. Italian writers of the fifteenth and sixteenth centuries, notably Pacioli, Tartaglia, and Cardano, had discussed the problem of the division of a stake between two players whose game was interrupted before its close (the so-called 'Problem of Points'). The problem was proposed to Pascal and Fermat, probably in 1654, by the gambler Chevalier de Méré. The ensuing correspondence be.…