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But,69314714 multiplied by 3 will give, 207944142
for the Logarithm of 8, inafmuch as 8 is the Cube or
third Power of 2, and the Logarithm of 8 plus the Log.
ofis equal to the Logarithm of 10, because 8
X110; wherefore to find the Logarithm of 1
we have b

= 1 = {, whence e ➡, and

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2

Whence Neper's Logarithm of 4 is,22314352
To which add the Logarithm of 8.

2,07944142

The Sum, viz.. 2,30258494
is Neper's Logarithm of 10. But if the Logarithm of
10 be made 1,000000, &c. as it is most conveniently
done in most of our Tables extant, then 2302585

n

1,000, &c. Whence n 2302585, &c. is the Index
for Briggs's Scale of Logarithms; and if the above
Work had been carried on to Places fufficient, the Index
would have been 2,30258, 50929, 94045, 68401,

I

79914, &c. and its Reciprocal, viz. 1= 0,43429,

n

44819, 03251, 82765, 11289, &c. which by the Way, is the Subtangent of the Curve expreffing Briggs's Logarithms; from the double of which the faid Loga rithms may be had directly.

I

For because =0,4342944,,

n

n

,868588

9638, &c. which put=m, and then the Logarithm

of b

me3 mes me' me9

me + + +

5.

EXAMPLE.

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7

9.

Let it be required to find Briggs's Logarithm of z.

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Let it be requir'd to find Briggs's Logarithm of 3 ; now because the Logarithm of 3 is equal to the Logarithm of 2 plus the Logarithm of 12 (for 2 x 12 = 3) therefore find the Logarithm of and add it to the Logarithm of 2 already found, the Sum will be the Logarithm of 3, which is better than finding the Logarithm of 3 by the Theorem directly, inafmuch as it will not converge so fast as the Logarithm of 1; for the fmaller the Fraction reprefented by e, which is deduc'd

B b 4

deduc'd from the No. whofe Logarithm is fought, the fwifter does the Series converge.

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Here b

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Briggs's Logarithm of 1

49

I

176091256

To which add the Logarithm of 2,301029990
The Sum is the Logarithm of 3 = 0,477121246

Again, to find the Logarithm of 4, because 2 x 2 =4,
therefore the Logarithm of 2 added to itself, or multi-
plied by 2 the Product 0,60205998 is the Logarithm of 4.
To find the Logarithm of 5, becaufe
therefore from the Logarithm of 19
Subtract the Logarithm of z

5,

1,000000000

"

301029990

There remains the Logarithm of 5 =

,698970010

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The Sum will be the Logarithm of 6,778151236 Which being known, theLogarithm of 7, the next Prime Number, may be eafily found by the Theorem; for because 6 x 7, therefore to the Logarithm of 6 add the Logarithm of and the Sum will be the Logarithm of 7.

Here b

EXAMPLE.

I

Aandee 10!

m=

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,954242492

The Sum is the Logarithm of 9

And the Logarithm of 10 having been determined to be 1, 0000000, we have therefore obtained the Logarithms of the first ten Numbers.

After the fame manner the whole Table may be conftructed, and as the prime Numbers increase, fo fewer Terms of the Theorem are required to form their Logarithms; for in the common Tables which extend but to feven Places, the firft Term is fufficient to produce the Logarithm of 101, which is compos'd of the Sum of the Logarithms of 100 and 100, because 100 X10 101, in which cafe b

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2019

I-e

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whence in making of Logarithms according to the preceding Method, it may be obferv'd that the Sum and Difference of the Numerator and Denominator of the Fraction whofe Logarithm is fought, is ever equal to the Numérator and Denominator of the Fraction reprefented by e; that is, the Sum is the Denominator, and the Difference which is always Unity, is the Numerator; confequently the Logarithm of any Prime Number may be readily had by the Theorem, having the Logarithm either next above or below given.

Tho', if the Logarithms next above and below that Prime are both given, then its Logarithm will be ob→ tained fomething cafier, For half the Difference of the

4

Ratios

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Ratios which conftitute the 1st Theorem, viz. (n =)

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255 454

656

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8533

&c. is the Logarithm of

the Ratio of the arithmetical Mean to the geometrical Mean, which being added to the half Sum of the Logarithms next above and below the Prime fought, will give the Logarithm of that Prime Number, which for Diftinction's Sake may be call'd Theorem the fecond, and is of good Dispatch, as will appear hereafter by an Example.

But the best for this Purpofe is the following one, which is likewise deriv'd from the fame Ratios as Theorem the firft. For the Difference of the Terms be tween a b and 4 ss or‡aa+žab+4 b b, is a o

-2

— 1 ab + 1 b b = 1⁄2 a—1 b = 4, dd, 1, and the Sum of the Terms a b and ss being put =y, therefore (fince y in this Cafes and d=1) it follows that

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Ratio of a b to xz, whence X +

7y

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n y 3y3 55

&c. is the Logarithm of the Ratio of 1⁄2 s to ✔ab

which converges exceeding quick and is of excellent Ufe for finding the Logarithm of Prime Numbers, having the Logarithms of the Numbers next above and below given, as in Theorem the Second.

EXAMPLE.

Let it be required to find the Logarithms of the Prime Number 101, then a 100 and b 101, whence y=20401, put=m=4342944819, &c. then

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