as the Value of the Ratio of Unity to any Number, is the Logarithm of the Ratio of Unity to that Number, so each Ratio is suppos'd to be measur'd by the Number of equal Ratiunculæ contain'd between the two Terms thereof; whence if in a continued Scale of mean Proportionals, infinite in Number, there be afsum'd an infinite Number of such Ratiunculæ, between any two Terms in the said Scale ; then that infinite Number of Ratiunculæ, is to another infinite Number of the like and equal Ratiunculæ between any other two Terms, as the Logarithm of the one Ratio, is to the Logarithm of the other.

But if instead of supposing the Logarithms compos'd of a Number of equal Ratiunculæ proportionable to each Ratio ; we shall take the Ratio of Unity to any Number to confitt always of the same infinite Number of Ratiunculæ, their Magnitudes in this Cafe will be as their Number in the former. Wherefore if between Unity and any two Numbers propos'd there be taken any Infinity of mean Proportionals, the infinitely little Augments or Decrements of the first of those Means in each from Unity will be Ratiunculæ, that is they will be the Fluxions of the Ratio of Unity to the said Numbers; and because the Number of Ratiunculæ in both are equal, their respective Sums or whole Ratios will be to each other as their Moments or Fluxions, that is the Logarithms of each Ratio will be as the Fluxion thereof. Consequently if the Root of any infinite Power be extracted out of any Number, the Difference of the said Root from Unity, shall be as the Logarithm of that Number. So that Logarithms thus produc'd may be of as many Forms as we please to afTume infinite Indices of the Power whose Root we seek. As if the Index be fuppos'd 100000, &c. we shall have the Logarithms invented by Neper ; but if the said Index bę 230258, &c. those of Mr. Briggs's will be produc'd.

Wherefore if it be any Number whatsoever and ñ infinite, then its Logarithm will be as itxu-I=

x5 +

&c. For the infinite 3

5 Root of 1*-* without ita Unciæ or prefixt Numbers




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is +x+xx*******XX, &c. and the celebrated binomial Theorem invented by Sir Ifaac Newton for determining them is I x*x^213, &c. or in this Cafe rather 1x**-**=*=***, &c for

being an infinitesimal is rejected; whence the infi nite Root of itxartet*-**+ **

3M 4n

g2 ÉC. and the Excels thereof above Unity, viz.



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&c. is the Augment of the first of the mean Proportionals between Unity and i-fix, which therefore will be as the Logarithm of the Ratio of 1 to 1+x, or 2


1000, &c.



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the Logarithm of 1+x. But as 17-**-i is a Ra. tiuncula it must be multiplied by 10000, &c. infinitely, which will reduce it to Terms fit for Practice, making the Logarithm of the Ratio of 1 to 1 **


tc. whence if the

3 4 İndex ni be taken 1000, &c. as in Naper's Form, the Logarithms will be fimply * f

+ Esc.

3 4 5' But as n may be taken at Pleasure the several Scales of Logarithms to such Indices will be as

> or reciproa cally as their Indices. Again, if the Logarithm of a decreasing Ratio bei

22/ fought, the infinite Root of 1-*=-* will be found by the like Method to be i-*

3n ****

, &c. which fubftract from Unity, and the Decrement of the first of the infinite Number of Pro. portionals will appear to be on x x x +**** B b 2



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fit, &c. which expresses the Logarithm of the Ratio of 1 to 1-* or the Logarithm of 1-* according to Neper's Form, if the Index n be put = 10000, &c. as before.

And to find the Logarithm of the Ratio of any two Terms, a the least and b the greater, it will be as a: b::

b 1:1**:*

'; or the Difference divided by the lesser Term when ’tis an increasing Ratio, and bra

when 'tis decreasing. b

Wherefore putting d=Difference between the two Terms a and by the Logarithms of the same Ratio may be doubly exprest, and accordingly is either ** d di d


d d' &c. or-xit

40 di

&c. both producing the same Thing. But if the Ratio of a to b be suppos'd to be divided into two parts, viz. into the Ratio of a to the arithmetical Mean between the two Terms, and the Ratio of the said arithmetical Mean to the other. Term b, then will the Sum of the Logarithms of those two Ratios be the Logarithm of the Ratio of a to b. Wherefore substitutings for at ab and it will be is:

& s-a d £::1:1-4, whence x=

And again as

be d 15:6::1:1-t-x, whence xa


ás d fubftituting“- for x, and we shall have for both Ratios

d2 2 a?





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d? +

&c. is the Logarithm of the 3 si

757 Ratio of a to b, whose Difference is d, and Sum s; which Series without the Index n, is by the bye, the


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std Fluent of the Fluxion of the Logarithm of suming d the flowing Quantity, for the Fluxion of the std 2 sd d


diť did Log. of is




d d3 ds. di +,&c. whose Fluent 2 x-t

&c. 3 so

std Neper's Logarithm of

and the same as above

Sed bating the Index n. This Series either way obtained converges twice as swift as the former, and consequently, is more proper for the Practice of making Logarithms ; thus, put a=! and b any Number at Pleasure, then d bel

ti = FI

which assume se, and then b

and because therefore we have for

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The Log. of b(=

2 šes to $e?,

+*&c. To illustrate this Theorem. Let it be required to find the Logarithm of 2 true to 7 Places.

Note, That the Index must be assum'd of a Figure or two more than the intended Logarithm is to have.


Here b=? = 2:eti=2-26:3=


whence = and ce=g.

Bb 3


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Whence Neper's Logarithm of 2 is

But ,69314714 multiplied by 3 will give, 207944142,
for the Logarithm of 8, inasmuch as 8 is the Cube or
third Power of 2, and the Logarithm of 8 plus the Log,
of 1 is equal to the Logarithm of 10, because 8
x1 = 10; wherefore to find the Logarithm of 15
we have beti

==, whence , and


The OPERATION ftands thus.


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Whence Neper's Logarithm of 1 4 is ,22314352
To which add the Logarithm of 8.

The Sum, viz. 2,30258494
is Neper's Logarithm of 10.

But if the Logarithm of
1o be made 1,000000, &c. as it is most conveniently
done in most of our Tables extant, then 2302585
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 sufficient, the Index
* would have been 2,30258, 50929, 94045, 68401,


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