By Todhunter, I. (Isaac)
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Extra resources for A History of the mathematical theory of probability from the time of Pascal to that of Laplace
Had not yet received an express name, and he proposes to call them triangulo- triangulairea. b order is n(1&+I) ... nd that immediately to the left of it. Thus . 10=4+6, 35=20+15, 126=70+56, ... The properties of the numbers are developed by Paseal witt,. great skill and distinctness. h order: the sum is equal to the number of the combinations of n + r -1 things taken r at a time. and Pascal establishes this by an inductive proot 18 PASCAL AND FEro~T. 23. Pascal applies his Arithmetical Triangle to various subjects; among these we have the Problem of Points, the Theory of Combinations, and the Powers of Binomial Quantities.
3) Suppose that A 'Wants '11. - 2 points and B wants n points. An interesting relation holds between the second and third examples which we will exhibit. ' PASCAL AND FERMAT. 19 Let M denote the number of cases which are favourable to A and N the number of cases which are favourable to B. Le~ 'I'=2n-2. ~ =~ say. A is entitled to 28 2'"+~ • 8 F . -2- , that IS to 2'" (2'" + ~); so that he may be considered to have recovered his own stake and to have won the fraction ~ of his adversary's stake. In the third example we have M +N= 2'"-1, M-N= 2l:=.!
CHAPTER VI. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1670 AND 1700. 69. THE present chapter will contain notices of various contributions to our subject, which were made between the publication of the. treatise by Huygens and of the more elaborate works by James Bernoulli, Montmorl, and De Moivre. 70. A Jesuit named John Caramuel published in 1670, under the title of Mathesis Biceps, two folio volumes of a course of Mathematics; it appears from the list of the author's works at the beginning of the first volume that the entire course was to have comprised four volumes.