c 97)

this Problem to solve j vi%. The Chord of an Arc bring givcn, te find theCImdes an other Arc, that Jhall be to the first as n to i. I.ct y be theChord given,% the Chord required ; now the Arc belonging to the Chord y i»,

%y ’ y y '

'-- &c. And the Arc belonging to the Chord

l lid® b °

f > /

- 4 --— 8 cc. The first of these Ares is to the

AO d 4 Iizd 6

second as 1 to n ; therefore multiplying the Extreams and Means together, welhall have this Equation

5 K'

-- &C. — n y

II id®

11 id®

Compare these two Series with the two Series of the Theorem»

and you will find 4 = 1, b — o, c — — -, d =0, e =

J 6 d d 40 d 4

n 3 n

&C. v n , h — o , i — , k. — O, / = -, m = O, &c.

6 6 d d 40 d+

n — n 3 I — n n

hence z. will be = n y -+ — - 3 \ &c. or n y -+■ - y y A, Lcc.

6 d d i X g d d

Supposing A|to denote the whole preceding Term, which will be the lame Se-ries as Mr. Newton has first found.

By the fime Method, this genera! Problem may be solved ; The Abfiijsecorreffonding to a certain Area in any Curve bring given , to find the Abfiijse ,vohose corresfonding Area fhall be to the first in a given Patio.

The Logarithmic Series might also be found without borrowing any otherIdea, than that Logarithms are the Indices of Powers : LettheNumber,whoseLogarithm we enquire, be i - 4 - y, suppole its Log. tob eaz.-hbz.z.-bc z. 3 , &c. Let there be another Number 1 - 4 - y ; therefore its Logarithm

will beay-Pbyy-+cyl, &c. Nowif I -+ z. = 1 - 4 - y\ n , ^ follow*

that a z. ~+ b z. z. - 4 - c y ? , &C. : ay ■+ b y y -+• c y \ &c. : : n : 1. that is,5CC. =nay-+nbyy-Pncyl, &c. thereforewe may find a Value of L exprest by the Powers ofjr j again, since 1 - 4 - r.

— 1 - 4 - y ”, therefore z. r= 1 H- ys — 1, that is t = ny ~b " X ”-- y y

1 i

n n — in — i . .

- 4 - — x - x - y ?, &c. Therefore z. is doubly exprest by the

Powers of y. Compare these two Values together, and the Coefficients a, b, c>&c. will be determin’d, except the first a which may be taken at Pleasure*and gives accordingly, ali the different Species of Logarithms,

O

XXIII. De