bers, is 1 +*+**+xxx+xxxx, &c. and the celebrated binomial Theorem invented by Sir Isaac Newton for

x

2n

-+

n

2n

-3 in this Cafe rather I X

X

&c.for

3 4 5 being an Infinitesimal, is rejected; whence the infinite

x2 x3 Root of 1 X*" I +

+

&c. and the 31 4n

x2 x3 Excess thereof above Unity, viz..

3n 4n &c. is the Argument of the first of the mean Proportionals between Unity and. 1 **, which therefore will be as the Logarithm of the Ratio of 1 to 1 +*, or as the Logarithm of 1 +*. But as 1+*- 1 is a Ratiuncula, it must be multiplied by 10000, &c. infinitely, which will reduce it to Terms fit for Practice, makeing the Logarithm of the Ratio of 1 to it * =

+

&c. whence if the

3 4 Index n be taken 1000, &c. as a Neper's Form, the

x4 Logarithms will be fimply * -- +

+

3 4 5 &c. But as n may be taken at Pleasure, the several Scales of Logarithms to such Indices will be as 1000, &c. reciprocally as their Indices.

Again, if the Logarithm of a decreasing Ratio be

1000, &c.

x2

I

2

2

or

n

n

2n

sought, the infinite Root of Iow* = 1-* will be found by the like Method to be i

3 **, &c. which fubtract from Unity, and the Decre4n ment of the first of the infinite Number of Proportionals will appear to be

&c. 3 4

which

1

&c. or

262

which expresses the Logarithm of the Ratio of 1 to 1-X, or the Logarithm of 1-4, according to Neper's Form, if the Index n be put =10000, &c. as before.

Aid to find the Logarithm of the Ratio of any two Terms, a the leffer, and b the greater, it will be as a : 1

b

(b-a ::1:1+ *; whence 1+x= -; and x = the Difference divided by the leffer Term when 'tis an

b. increasing Ratio, and when 'tis decreasing.

b Wherefore, putting d=Difference between the two Terms. a and b, the Logarithms of the fame Ratio may be doubly exprefied, and accordingly is either d

d3 04 Х

+ *

d d th d -X + +

&c. both producing

3 + b the fame Thing.

But if the Ratio of a to b be supposed 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 subfticuting is for Latib, and it will be, {s:

d @::1:1-*; whence * =

es and again, ass:b::1:1 + x ; x =

음 26

d : Therefore substituting for Thall have the Logarithms of those Ratios; viz. d

c. and + + +

262

363 3

464

I
2

5,24 =)

S

(20= =)

*, we

d3

2x

+

ss-dd

13

the Logarithm of the Ratio of a to b, whose Difference is d, and Sums s; which Series, without the Index n, is, by-the-bye, the Fluent of the Fluxion of the

std Logarithm of assuming d, to be the Aowing

sed Quantity, for the Fluxion of the Logarithm of

std 2sd

d dad d41 død is

+ +

s?

+, &c. whore d a3

d? Fluent 2 x -t

+
+

&c. is Neper's

757' and the same as above, abating

sod 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=1, and b

any

Number at Pleasure ; then d which assume

and then b = Ite. 6+1

d and because =l, therefore have we for

353

555

Logarithm of std,

S

THE OREM I.

2

n

te The Log.ofb=

Xet es + šos

+ . To illustrate this Theorem: Let it be required to find the Logarithm of 2 true to 7 Places. Note, That the Index must be assumed of a Fi

gure or two more than the intended Logarithm is to have.

EXAMPLE.

Mere (b='+e=2; therefore i te=(1x2 =) 2--21; and 3e= (2-1=) 1; whence e = 1, and le=

The

Whence Neper's Logarithm of 2 is ,6931471
But ,69314718, multiplied by 3, will give 2,07944154
for the Logarithm of 8, inasmuch as 8 is the Cube or
third Power of 2; and the Logarithm of 8 + Log. of
his equal to the Logarithm of io, because 8 x1 = 10;
wherefore to find the Logarithm of 15 we have b=
ite

==; whence e = 1, and ee =

The OPERATION stands thus :

e

IIIIIIIIرد

= =,I1IITTI

137 174
es

1693

21

45725
339

3
,11157178

Whence Neper's Logarithm of 1 is ,22314356
To which add the Logarithm of 8, 2,07944154
The Sum, viz.

2,30258510
Neper's Logarithm of 10. But if the Logarithm of
to be made 1,000000, &c. as it is for Conveniency

2302585
done in most of the Tables extant, then

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
n would have been 2,30258, 50929, 94045, 68401,

1

Here b ==

79914, &c. and its Reciprocal, yz. = 0,43429, 44819, 03251, 82765, 11289, &c. which, by the Way, is the Subtangent of the Curve expressing Briggs's Logarithms; from the Double of which the faid Logarithms may be had directly.

For, because I = 0,4342944, &c. *** =,868588 9638, &c. which put=m, and then the Logarithm of b

& C. 3 5 7 9

EXAMPLE. Let it be required to find Briggs's Logarithm of 2. = 2:::e=

and ee = 5.

2

ņ

n

Ite

3 me

mes

me?

mel

me +

+

+

Let it be required to find Briggs's Logarithm of 3. Now because the Logarithm of 3 is equal to the Logarithm of 2 plus the Logarithm of 1 (for 2 X= 3), therefore find the Logarithm'of 1, 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, inasmuch as it will not converge so fast as the Logarithm of 1; for the smaller the Fraction represented by e, which is

deduced