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number of terms being indefinitely fmall in comparifon of n it is alfo evident, that they may be rejected in all the rest of the terms of the feries; because thefe terms (by reafon of the indefinite fmallnefs of e) bear no affignable proportion to the preceding ones. Hence we have 1 + &c. N: but ne is

ne +

n2e2

+

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n4e4

2 2.3 2.3.4

I

(=L) the hyperbolic logarithm of 1+el" (or N) by what has been already fpecified: therefore 1+

L+ + +

L2 L3

L+

L5

+

&c. N.

2

2.3 2.3.4

2.3.4.5

Q. E. I.

PROP. II.

To determine the hyperbolic logarithm (L) of any given

number (N).

It appears from the preceding Prop. that 1+L
L2 L'

--

+ &c. is N therefore, if x + 1 be.

2.3

putN, we shall have L

L' L3 L+ + + +

2

2.3 2.3.4

&c. x; and confequently, by reverting the fe

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Because garithms) we shall have 1 + e = N by putting 1+=N, and m =

+ being = 1 + mx + m.

n

m

x2 + 1.

N (by the definition of lo

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being rejected in the factors m-1, m — 2, m — 32 &c. as indefinitely fall in comparison of 1, 2, 3, &c. the equation will become e mx - +

e

mx2

&c. whence ne = L) = x

m

mx3 mx+

(=

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2

&c, the very fame as be

But this feries, tho' indeed the most eafy and natural, is of little ufe in determining the logarithms of large numbers; fince, in all fuch cafes, it diverges, inftead of converging. It will be proper, therefore, to give, here, the invention of other methods, which authors have had recourse to, in order to obtain a feries that will always converge.

you would find be denoted by

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First, then, let the number whofe logarithm ; where it is manifeft (however great that number may be) * will be always lefs than unity; moreover, let 1 (as before) be the term of the logarithmic progreffion agreeing with the propofed number, or, which is the fame, let 1+e" =

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n

then, (by taking the root on both fides) we

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m

=

mx + m.

2

(by making m

M-I M-2

m.

2

3

* &c. where m being re

jected in the factors m-1, m2, &c. (as before) our equation will become 1 + ei mx

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mx3

2

3

m

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&c. whence x + + + &c. 2 3 4

ne the hyperbolic logarithm of

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Which feries, it is manifeft, will always converge,

let the value of be ever fo great; because x

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But it is further obfervable that this feries has exactly the fame form (except in its figns) with that above for the logarithm of 1 + x; and that, if both of them be added together, the feries 2x+ 2x5 2x7

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3

+ +&c. thence arifing, will be more

5 7

fimple than either of them, fince one half of the terms will be intirely deftroyed thereby. Therefore, because the fum of the logarithms of any two numbers is equal to the logarithm of the product of thofe numbers, (fee Article 1.) it is mani&c. will truly express

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not only because the even powers are here deftroyed, but because x, in finding the logarithm of any given number (N), will have a lefs value.

But

But now, to determine what this value must be,

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P+Q

values, substituted in the foregoing feries 2x + 2x3 2x5 +:

3 5

&c. will give the hyperbolic logarithm

of the refpective number.

Example. Let it be propofed to find the hyperbolic logarithm of the number 2.

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Which values being refpectively divided by the numbers, 1, 3, 5, 7, 9, &c. and the feveral quotients added together, (fee the general feries) we fhall have 346573590 &c. whofe double, being ,693147180 &c. is the hyperbolical logarithm of the number 2,

After

After the very fame manner the hyperbolic logarithm of any other number may be determined; but, as the feries converges, flower and flower, the higher we go, it is ufual, in computing of tables, to derive the logarithms we would find, by help of others already known; for which there are various methods; but the following is the most commodious and fimple, that has occurred to me, especially, when a great degree of accuracy is required.

It is thus. Let a, b and c denote any three` numbers in arithmetical progreffion, whofe common difference is unity; then, a being b-1 and cb+1, we fhall have ac-b-1, and conb2 ac + 1

sequently

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Whence, by the nature

of logarithms, we likewife have 2 log. b-log.

a-log,

log, clog.

ac + I

: but the logarithm

of ac+, by putting

2x3

ас

*2x5

2x7

ac

I

2ac + I

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+ + &c. (by what has been already

3 5

7

fhewn): which being denoted by S, we fhall

log.blog. a+ log. c + S.
2 log. blog. c — S.
2 log, blog. a-
S.

have log. a

log.

As an example hereof, let it be proposed to find the hyperbolic logarithm of 3.

Then, the hyperbolic logarithm of 2 being already found,693147180 &c. that of 4, which is the double thereof, will also be known. Therefore, taking a 2, b = 3, and 4, we fhall, in

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