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Excerpt from 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 Celebrated Dr. John Wallis, of Oxford, in a Tract Intitled From the Subject The next Traél: in this Colleé'tion relates to the Rational Numbers that will exprefs the Sides of Right-angled Tri angles, and contains two methods of finding as many fets of numbers as we pleafe that fhail have this propeny. The firft of thefe methods begins in page-417, and ends in page 431, and the fecond reaches from page 431 to page 448 after which I have inferred a Table of the Squares of the feveral natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 81c, as far as 100, together with two additional columns adjoining to the column of the faid fquares, in the former of which I have fet down the differences of the faid fquares, and in the latter the differences of thofe differences, or the fecond differences of the fquares'themfelves; which fecond differences are all equal to each Other, and to the number 2. This Table begins in page 449, and is aecom panied with fome remarks which extend 19 page 4. 57. This Trad has a confiderable refemblance to fome parts of the foregoing Difcourfe of Dr. Wallis, and may afford fome amufement to fuch readers as are fond of contemplating the properties of numbers. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1795 edition. Excerpt: ...it must be that of 29, (or else 37 must be laid aside;) But this brings in 13 once, for which we may have a second at the Square of 3, but then we cannot have a third without a fourth, at the Square of 191. Therefore (waving that at the Square of 3) we must take both (if at all) at the Square of 191. Now this brings in 7 a fourth time, which calls for a fifth and sixth: One of these we might have at the Square of 2; but then we cannot have a sixth without a seventh. Therefore (waving that at 2) we must (if at all) take both at the Square of 67. But here, beside a second 31 (for which we may have a third at the Square of 4, or of 5, ) we have 3 a fourth time (which will make up, not a Cube, but the Triple of a Cube, ) which is not to be ad mitted, because we cannot have a fifth and sixth. And consequently, the Square aside, (as not to be joined either with that that of 439 or 29;) but (waving that) we must have recourse to the other two (at 29 and 439) for Tripling of 67. Now here we have 13 once; and therefore must have it twice more; not from the Square of 3, (because, as before, if we take a second here, we cannot have a third without a fourth;) but from that of 191. Which doth not only supply 13 twice; but also 7 and 31 which were also wanting: So that we have now a second Cube, such as was desired j whose Components are, 3, 7, 13, 19, 31, 67, thrice taken. And the Square whence it arifeth, is that of 7 x 11 X 29 X 163 X 191 x 439. And if, from the remaining Square of 2, 4, 3, 5, 37, 67, we could form a third; this, Compounded with the last foregoing (as Prime to it) would form a fourth. But this cannot be, because no Prime doth here thrice occur but only 7 and 31: And neither of these can be thrice taken, without being incumbered with.