The Doctrine of Permutations and Combinations; Being an Essential and Fundamental Part of the Doctrine of Chances as It Is Delivered by Mr. James Bernoulli, in His Excellent Treatise on the Doctrine of Chances, Intitled, Ars Conjectandi, and by the

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1795 Excerpt: ...and the number of terms in the said new series is m + a, instead of m + 1, which is the number of terms in each of the said foregoing seriefes, DEMONDEMONSTRATION. 32. This will appear by setting down the said series that has been just described; which may be done as/ollows: m m m--l. m m--l m--z m m--l m--l m--X.. 1+-+-X +-X x +-X X X 3 + kc. 1 1 2 I 2 312 3 4 m m m--l m m--l m--2 I +-+-X +--X X 4 ic. I 12 1-3 In these two rows of terms it is evident, in the first place, that the terms in the upper row, after the two first terms i and consist of two, three, and four, and more, factors, everj new term having one more factor than the term next before it j and, 2dly, that the terms in the lower row that stand immediately under the third, fourth, fifth, and other following terms in the upper row, consist of one factor less than the corresponding terms, or terms immediately over them in the upper row; and, 3dly, that the terms in the lower row consist of the very fame factors as the corresponding terms in the upper row, excepting that they want the last factors of the said terms in the upper row. And hence it follows, that, in order to reduce the terms in the lower row to the fame denomination as the terms in the upper row, we must multiply them by factors that (hall have the fame denominators as the last, or additional factors in the tipper row, and which must have their numerators equal to their denominators, so as to make each of them equal to i, to the end that the magnitudes of the said lower terms may not be altered by the multiplication of them by the said, new factors. Thus, for example, the second term of the low Qt lower row, to wit, y, must be multiplied into the factor--t in order to bring it to the same denomination as the third term in the up...