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it is evi. efpect to

in re, f them es and

number of terms being indefinitely small in com-
parison of n : it is also evident, that they ’may
be rejected in all the rest of the terms of the
feries; because these terms (by reason of the in-
definite smallness of e) bear no asignable propor-
tion to the preceding ones. Hence we have 1 +

nes n4e4
ne +
+

&c. = N; but ne is.

2.3.4 (=L) the hyperbolic logarithm of itel" (or N) by what has been already specified: therefore 1 + L L L4

LS

&c.' – N. 2.3 2.3.4 2.3.4.5 2. E.I.

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2

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PROP. II.
To determine the byperbolic logarithm (L) of any given

number (N).
It appears from the preceding Prop. that itt
L L}

&c. is =N: therefore, if x + i be. "2.3

L' L L+ put = N, we shall have L

+

2.3 2.3.4 &c: =*; and consequently, by reverting the fe

qo
ries, L =*
+

&c. 4

6

+

2

+

. w 18

+

2

X: E. 1.

OTHERWISE.

Because 1 + el" = N (by the definition of low garithms) we shall have i te=

Ni = 1+x)", by putting 1 + x=N, and m=s. Therefore,

+ x1 being = .1 + mx + m.

n

I

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I

11?

2

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2

+

m

43 &c. where, 112

3 being rejected in the factors m - 1,7-2, m-— 32 &c. as indefinitely small in comparison of 1, 2, 3, &c. the equation will become e = 1118 –

172x? mx? 712x4

C&c. whence-(=ne = L) = *

& 3

404 +

+

&c, the very fame as be3 4 fore,

2

e

4

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But this series, tho' indeed the most eafy and natural, is of little use in determining the logarithms of large numbers ; since, in all such cases, it diverges, instead 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 series that will always

! converge.

First, then, let the number whose logarithm you would find be denoted by ; where it is manifest (however great that number may bę) * will be always less than unity • moreover, let 1 + " (as before) be the term of the logaritismic progression agreeing with the proposed number, or, which is the fame, let 17" =

-: then, (by taking the root on both sides) we shall have item

PET"

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I

(by

(by making m =

mm of m.

1

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2

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mx3

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+

I

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&c. where m being re

3 jected in the factors on - 1, m - 2, &c. (as before) our equation will become i te=1-mx &c. whence x +

&c. 2 3

2 3 4 =ne = the hyperbolic logarithm of Which series, it is manifest, will always converge, let the value of

be ever so great; because * will be always less than unity.

But it is further obfervable that this series has exactly the same form (except in its signs) with that above for the logarithm of 1 + x; and that, if both of them be added together, the series 2x+ 283 285

2x7 +

&c, thence arising, 'will be more 3 5 ng simple than either of them; since one half of the terms will be intirely destroyed thereby. Therefore, because the sum of the logarithms of any two numbers is equal to the logarithm of the product of those numbers, (see Article 1.) it is mani

273 felt that 2x + + &c. will truly express

3 5.

1+% the logarithm of

x 1+%. Which

*] feries converges, still, faster than xt

+

&c.

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

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But now, to determine what this value must be, make

1 +* – N, and then * will be found = N

but if the quantity proposed be a fraction N+1

P 1+ instead of a whole number, make

Q -x

P-Q: either of which and you will have x =

P+Q values, substituted in the foregoing series 2x +

2X5

&c. will give the hyperbolic logarithm 3 5 of the respective number.

Example. Let it be proposed to find the hyperbolic logarithm of the number 2. Here x being =

, and x == ; we

2x3

+

2

I

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Which values being respectively divided by the numbers, 1, 3, 5, 7, 9, &c. and the several quotients added together, (see the general series) we shall have ,346573590 &c. whose double, being ,69314718@ &c. is the hyperbolical logarithm of the number 2,

After

45 After the very fame manner the hyperbolic logarithm of any other number may be determined; '/ but, as the series converges, nower and flower, the higher we go, it is usual, 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 simple, that has occurred to me, especially, when a great degree of accuracy is re, quired.

It is thus. Let a, b and c denote any three numbers in arithmetical progreffion, whose common difference is unity, then, a being =b-I and cab+1, we shall have ac=b-I, and con

ac +I sequently

+

Whence, by the nature of logarithms, we "likewise have 2 log. blog

ac +I - log, c = log. : but the logarithm of ac +", by putting

I

SX, will be

2ac+I
2x5 2x7
4 +

&c. (by what has been already 3 5 7 thewn): which being denoted by S, we shall

ac

ac

ac

2%

+ 1

ac

log. b = } log. a+ log. c + IS: 7 have {log. a = 2 log. b log. <

(log. c. = 2 log. b- log. a -- S.)

S.

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

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

this

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