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. 1777 Excerpt: ...will be x5--+ 5X--1 = Here a--o, b----5, c =z o, = + 5, m + 1, and mi--1, and the formula gives-Ii 1 =-----+ r, 5+5--15.--5 which is one root of the equation, but manifestly cannot be. that answering the case proposed; agreeable to what was observed in Rem. 2. Therefore assuming m$ 2, the sirst operation gives n& '209, ...
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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. 1777 Excerpt: ...will be x5--+ 5X--1 = Here a--o, b----5, c =z o, = + 5, m + 1, and mi--1, and the formula gives-Ii 1 =-----+ r, 5+5--15.--5 which is one root of the equation, but manifestly cannot be. that answering the case proposed; agreeable to what was observed in Rem. 2. Therefore assuming m$ 2, the sirst operation gives n& '209, and the second m$ = '209056926, for the required value of At; or the length of the chord of; 2 degrees, when the radius is unity, which is true to the, last sigure. 13. As it may be of some use to the young Algebraist, I shall now endeavour to illustrate, by a sew Examples, the Newtonian method of obtaining the roots of literal equations., Example 1. Given, y--axy--ay--x = o. Requited the value of y, in a series composed of the powers of a and x, with their coessicients. The The first thing to be done is to tabulate the equation, which is easily effected as follows: Make a right angle DAB (sig. 10.), and from A towards B write down all the powers of y, and from A towards D all the powers of x, as high as in the given equation, beginning from unity. Divide the whole into small squares or parallelograms, as in the scheme, and insert the terms of the equation in their corresponding squares or parallelograms, that is, in those which have the fame powers of x and y; then circumscribe the signisicant parallelograms with the polygon FBCDE, and the equation is tabulated or prepared for extraction. Now, in order to determine the sirst term of the series, make any two terms which are placed in two adjacent angles f the polygon equal to o; that is, . '. y--a?y--o, --ay--Xs = O, y-3 = O, The two sirst equations give the sirst terms of a series for y when At is supposed very little in respect to a; and the last, when a is little in respect to te. We sha...
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All Editions of The Rationale of Circulating Numbers: With the Investigations of All the Rules and Peculiar Processes Used in That Part of Decimal Arithmetic (1777)