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PASCAL, Blaise (1623-1662). Trait du triangle arithmetique, avec quelques autres petits traitez sur la mesme matiere. Paris: Guillaume Desprez, 1665.
4o (227 x 172 mm). Folding woodcut diagram. (Some very light even browning, one lower margin chipped, possibly washed.) Later vellum; full morocco folding case.
FIRST EDITION OF THE FIRST COMPREHENSIVE STUDY OF THE ARITHMETICAL TRIANGLE, and its applications to combinatorial analysis and the calculus of probability. Pascal's correspondence with Fermat between July and October 1654 marks the beginning of the calculus of probability. The focus of their discussion was two main problems in gaming: the first concerned the probability that a player will obtain a certain face of the die in a given number of throws; the second was to determine for any game involving several players the portion of the stakes to be returned to each player if the game is interrupted. Although Fermat solved these problems, his method consisted of only combinatorial analysis. Pascal, however, discovered the advantages of the systematic application of reasoning by recursion. His recourse to mathematical induction appears in the final section of the Trait
Although the principle of the arithmetical triangle was already known at the time, Pascal was the first to make a comprehensive study of it, deriving from it the greatest number of applications. It has been pointed out that Pascal's concern went beyond the purely mathematical aspect of the problems of probability, and was to link decisions and uncertain events. Composed partly in French and partly in Latin, the Trait has a complex structure; it was printed in 1654 but not distributed until 1665, with the addition of the preliminaries and table. VERY RARE. Tchemerzine V, p. 61; Norman 1651.
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