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Pascal's manuscript on conic sections was founded on the work of Desargues and is now lost, but it was seen by Descartes and Leibniz. Descartes could not believe that the work was written by the boy and felt that it must have been written by the father instead. Here occurred Pascal's famous "mystic hexagram'' theorem of projective geometry: If a hexagon be inscribed in a conic, then the points of intersection of the three paira of opposite sides are collinear, and conversely (see Figure 59). He probably established the theorem, in Desargues' fashion, by first proving it true for circle and then passing by projection to any conic section. Although the theorem is one of the richest in the whole of projective geometry (see Problem Study 9.11), we probably should take lightly the often told tale that Pascal himself deduced over 400 corollaries from it. The manuscript was never published, and probably never completed, but in 1640 Pascal did print a one-page broadside, entitled Essay pour les coniques, announcing some of his findings, only two copies of this famous leaflet are known to be still in existence, one at Hanover among the papers of Leibniz, and the other in the Bibliotheque Nationale at Paris. Pascal's "mystic hexagram" theorem is involved in the third lemma of the leaflet. Pascal's Traité du triangle arithmetique was written in 1653 but was not printed until 1665. He constructed his "aritbmetical triangle" as indicated in the adjoined figure. Any element (in the second or a following row) is obtained as the sum of all those elements of the preceding row lying just above or to the left of the desired element. Thus, in the fourth row, 35 = 15+10+6+3+1 The triangle, which may be of any order, is obtained by drawing a diagonal as indicated in the figure. The student of college algebra will recognize that the numbers along such a diagonal are the successive coefficients in a binomial expansion. For example, the numbers along the fifth diagonal, namely 1, 4, 6, 4, 1, are the successive coefficients in the expansion of 〖(a+b)〗^4. The finding of binomial coefficients was one of the uses to which Pascal put his triangle. He also used it, particularly in his discussions on probability, for finding the number of combinations of n things taken r at a time (see Problem Study 9.13 (g)), which he correctly stated to be n!/(r!(n-r)!)   , where n! is our presen-day notation *for the product n(n-1)(n-2)…(3)(2)(1).
There are many relations involving the numbers of the arithmetic triangle, several of which were developed by Pascal (see Problem Study 9.13). Pascal was not the originator of the arithmetic triangle, for such an array had appeared in a number of prior works, the oldest known reference being a work of 1303 by the Chinese algebraist Chu Shr-kié. It is because of Pascal's development of many of the triangle's properties and because of the applica- tions which he made of these properties that the array has become known as Pascal's triangle. In Pascal's treatise on the triangle appears one of the earliest acceptable statements of the method of mathematical induction.
Although the Greek philosophers of antiquity discussed necessity and contingency at length, it is perhaps correct to say that there was no mathematical treatment of probability until the latter part of the fifteenth century and the early part of the sixteenth century, when some of the Italian mathematicians attempted to evaluate the chances in certain gambling games like that of dice. Cardano, as was noted in Section 8-8, wrote a brief gambler's guidebook in which some of the aspects of mathematical probability are involved. But it is generally agreed that the one problem to which can be credited the origin of the science of probability is the so-called problem of the points. This problem requires the determination of the division of the stakes of an interrupted game of chance between two supposedly equally skilled players, knowing the scores of the players at the time of interruption and the number of points needed to win the game. Pacioli, in his Suma of 1494, was one of the first writers to introduce the problem of the points into a work on mathematics. The problem was also discussed by Cardano and Tartaglia. But a real advance was not made until the problem was proposed, in 1654, to Pascal, by the Chevalier de Méré, an able and experienced gambler whose theoretical reasoning on the problem did not agree with his observations. Pascal became interested in the problem and communicated it to Fermat. There ensued a remarkable correspondence between the two men, in which the problem was correctly but differently solved by each. Pascal solved the general case, obtaining many results through a use of the arithmetical triangle. Thus it was that in their correspondence Pascal and Fermat laid the foundations of the science of probability.
Pascal's last mathematical work was that on the cycloid, the curve traced by a point on the circumference of a circle as the eircle rolls along a straight line (see Figure 60). This curve, which is very rich in mathematical and physical properties, played an important role in the early development.
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