j6 'Treatise of Algebra^

a b —a

and consequently — —- - by equal Division: And

b c — b

in the very same Manner it will appear that — — -—— y

e d—c

c d — c abc

and — =- p &c. Therefore , — &c. being

d t — a bed 6

all equal, by Hypothesis, it is evident that -—

c—b d—e

d—c

e \ _ (esc. must be likewise equal.

THEOREM IX.

In any contlnutd geometrical Proportion (as r, 3,9,27,81.)the Product of the two Extremes, and that of every othertwo Terms , equally distant from them , are equal.

For the Ratio of the first Term to the second, beingthe same as that of the last but one to the last by Hy-pothesis, these four Terms are in Proportion ; and there-fore, by Theorem I, the Rectangle of the Extremes isequal to that of their two adjacent Terms: And, afterthe very fame Manner, it will appear, that the Rectangleof the third and last but two, is equal to that of theirtwo adjacent Terms, the second and last but one, and soof the rest ; whence the Truth of the Proposition is ma-nifest.

THEOREM X.

The Sum of any Number of Quantities, in continued geome-trical Proportion, is equal to the Difference of the Rect-angle of the second and lajl Terms and the Square of thefirst, divided by the Difference of the first andsecond Terms.

For, let the first Term of the Proportion be denotedby a, the common Ratio by r, the Number of Termsby n, and the Sum of the whole Progression by x : Thenit is manifest that the second Term will be expresied byetxr, or ar; the third by ar x r, or ar* ; the fourth

by ar 2 x r, or arl, and the » t!l or last by ar~~ l ; and

therefore