But the first of the two next foregoing Series is manifeftly equal to the second xr.

ar

718

Scholium.

It is manifeft by viewing the foregoing Series in that y; for the Exponent of r in the leaft Term of the faid Series is o, in the leaft Term but one is 1, in the leaft Term but two is 2; and therefore Univerfally the Exponent of r in the nor greateft Term of the faid Series is n-1; wherefore ar 713 =).

And by Dividing each Part by r = s.

And by Dividing each Part of the 1ft Step by a, and then Extracting out of each the Root

Or by Multiplying each Part of the 1st Step by

yr

we have

a

a

*Now the Logarithm of " is (as fhall be fhew'd when we come to treat of Logarithms; but in the mean time we will suppose L to stand for the Word Logarithm) — n× L, r.

And the Logarithm of is L, y+L; r — L, a ;

a

where.

fore (fince the Logarithms of equal Quantities are equal) L, y†

L, r-L, anx L, r.

Consequently Li+L, r — L, a

L, T

—

is 2a + 4a+80+16a=30a = s-a

and satta+za+ax2=30a=8-16αx2. Horo 2 is the common Ratio.

Or

·sta-L, a
L, r

1. Therefore—>"+~ r =

pofition and Divifion.

And fince y = ara −1 and r = "—"; and fo arn-?

2. Therefore y x^~ = axsa
xs- 1H
-"-

= ax

ax s-a
3-y

VI. Given a, n, y; required r, s.

Solution.

=y

'=y=ar" (by Lemma 2, and its Scho

Since

kum.)

That is n

L, y — I., s — sr +i+l,

L, r

X. Given n, y, s; required r, a.

Solution.

Since, by Propofition VIII, sr"-37"~3 — yy." ~~Y';

172 I

2. And (by Propofition V.) a × 5-a" -2=9×5-"-.

Scholium.

If a befinite Number, great) Number; then (r being tion)y (by Propofition 1.)

and n an infinite (or infinitely
finite Number, and by Suppofi-
ar", must be an infinite Num-

ber; and confequently s (by Propofition I)

an infinite Number;

But if y be a finite Number, and n an infinite one, then (r being finite and 1.) a (by Propofition VII)

will be an infinitely little Number, and the only way we have to write fuch a Number, is by o.

r-I

n (because a is infinitely little in pofpe

Upon this Canon are founded fuch Queftions as are ufually propos'd in infinite Decreafing Geometrical Proportion, which are eafily folv'd by the faid Canon; to wit -S. As for In

r

yr

I

ftance;
If it were required to find the Sum of this Decreafing Geome-
A B A B

trical Proportion

AB

x bc bc

-I

bc

P 2

be finite Number, and be 1.

ofy