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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. 1798 Excerpt: ...required. L. 148 13 y BS?, / Proposition XXIX. 127. If we assume any quantity at pleafure, and therefrom derive a series of terms, by equal multipliers or divisors-, the terms of the series are faid to be in Continued Geometrical Progression (99): And the sum of all the terms of the series, except the last term, will have the fame ratio to the sum of all the terms, except the sirst term; as the common multiplier or its reciprocal to Unity. Demonstration. Let a, v, represent the two extreme terms in a continued geometrical progression, whereof r or--is the common multiplier, n the number, and s the sum of all the terms: Then, according as we take a or v for the leading quantity, the series will be expressed by the following terms, viz. A. a, ar, ar, ar3, ar, a r l (=v) C v v v v v B 2 iv 'r rl' r r L Qor(8)vr-1, r', v r-t, vr, v'r -' (=) And from these series, it appears by inspection, that s--a = j--v x r (A) or s--a =j-dxi (B); and therefore (16) r the ratio s--a: s--v--r: i (A), or the ratio s--a: s--v rr 1: 1 (B). r E. D. 128. Corol. 1. Since a rn 1 or v r n 1 =(1(127) therefore, the ratio 1: r nI 3= a: v, or the ratio 1: r n1--v: a (16). Thus: in any continued geometrical progression, there is the fame ratio between unity and that power of the common mon multiplier or divisor, whereof the exponent is the excess of the number of terms above unity, as there is between the two extreme terms. 129. Corol. 2. Hence, in a continued geometrical progrelsion, if any three of the sive quantities a, v, r, n, and s, are supposed to be Given, the remaining two may be determined, as in the following table, where the expressions belong to an increasing progression, and may be applied also to a decreasing progression, by making the quantities a and v c...

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