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But we must now observe, that there are various forms or species of logarithms; because it is evident that what has been hitherto said, in respect to the properties of indices, holds equally true in reJation to any equimultiples, or like parts, of them; which have, manifestly, the same properties and proportions, with regard to each other, as the indices themselves. But the most simple kind of all, is Neiper's, otherwise called the byperbolical.

The byperbolical logarithm of any number is the index, of ihat term of the logarithmic progreffion agreeing with the proposed number, multiplied by the excess of the common ratio above unity,

Thus, if e be an indefinite small quantity, the hyperbolic logarithm of the natural number agreeing with any term 1 tel" of the logarithmic progression 1, i te, itel', i tel, itel*, &c. will be expressed by ne,

PROPOSITION I.

The hyperbolic logarithm (L) of a number being gi

ven, to find the number itself, answering thereto.

I

n I

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2

Let 17 el" be that term of the logarithmic progression 1, it el, I tel", itel', 1 + 14, &c, which is equal to the required number (N). Then, because it el is, universally, = 1 + ne + n. az -.e tn.

e3 &c. we shall, also,

3 have it ne tn. 1. e*.+ n. &c. = N. But, because n (from the nature of logarithms) is here supposed indefinitely great, it is evident, first, that the numbers connected to it by the fign--, may be rejected, as far as any assigned

number

nI

n

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2

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3

ntet

2

number of terms being indefinitely small in comparison of n: it is also evident, that they may be rejected in all the rest of the terms of the series; because these terms (by reason of the indefinite smallness of e) bear no aflignable proportion to the preceding ones. Hence we have it

ne? m3 e3 ne +

to

&c. -N: but ne is 2.3

2.3.4 (EL) the hyperbolic logarithm of 1 +el" (or N) by what has been already specified: therefore it L L3

LS
L+ + + + &c. N.

2.3
2.3.4

2.3.4.5 2. E. I.

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

number (N).

2

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It

appears from the preceding Prop. that 1 +L L’ L3

&c. is =N: therefore, if x to be 2.3

L2 L3 L' put = N, we shall have Lt + +

2.3.4 &c. = x; and consequently, by reverting the sex2

** ries, L = * +

+

&c. 3 4 5

6

2

2.3

2. E. I.

OTHERWISE. Because I tel" N (by the definition of logarithms) we shall have ite=N=1+x)"; by putting 1 + x = N, and m =-. Therefore, 1 + x]" being = i + mx + m. X2 + m .

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n

m

2

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m

2

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I

m

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m

2

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2

2

X3 &c. we have e = mx + m
3
x2 4 m
.

203 &c. where,

3 an being rejected in the factors m-1, m-2, m-3 &c. as indefinitely small in comparison of 1, 2, 3 &c. the equation will become e = mx — + mx3

&c. whence C(=ne = L)

mx2

2

mx4

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m

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&c. the very same as be5

3

2 fore.

But this series, tho' indeed the most easy 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.

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First, then, let the number whose logarithm you would find be denoted by ; where it is manifest (however great that number may be) x will be always less than unity : moreover, let it-el" (as before) be the term of the logarithmic progression agreeing with the proposed number, or, which is the same, let it el"

: then, by taking the root on both sides) we

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n

2

M

m

- m.

2

&c.

4

e

1

I

I

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(by making m= = I - mx + m

- I x2

*? &c. where m being re

3 jected in the factors m – 1, m 2, &c. (as before) our equation will become 1+1=1 — mx

mp3
&c. whence * +

+ +
3

3 =ne = the hyperbolic logarithm of Which series, it is manifeft, will always converge, let the value of be ever so great ; because x will be always less than unity.

But it is further obfervable that this series has exactly the same form (except in its figns) with that above for the logarithm of 1 + x; and that, if both of them be added together, the series 2x + 2x3

2x7 +

&c. thence arifing, will be more 3 5 7 simple than either of them; lince 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 chose numbers, (see Article 1.) it is mani

275 felt that 2x +

+ &c. will truly express

3 5 the logarithm of 1 + x

*I+x. Which feries converges, fill, faster than x + ** + &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.

2x5

+

or

1

2

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But now, to determine what this value muft be,
make
1 + x

= N, and then * will be found =
I
N-I
NtI
+
; but if the quantity proposed be a fraction

P 1 +*
instead of a whole number, make

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

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

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

Example. Let it be proposed to find the hyper-
bolic logarithm of the number 2.
Here x being = =, and *; we

2+1
shall have

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2

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:+)

-. =>333333333 &c.
** (**) = 9037037037 &c.

***) 5004115226 &c.
Ś*) = ,000457247 &c.'
x?)

,0000 50805 &c.
X"(=;*') ,000005645 &c.

X.?) ,000000627 &c. mas (=**) = ,000000069 &c. &c.

&c.

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Which values being respectively divided by the
numbers, 1, 3, 5, 7, 9, &c. and the several quo-
tients added together, (see the general series) we
shall have ,346573590 &c. whole double, being
,693147180 &c. is the hyperbolical logarithm of
the number 2.

After

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