TRACT 21.

LOGARITHMS.

341

rithms (or artificials, as he always calls them in his tract onthe construction), and four in the naturals. A specimen ofthe table is as here follows:

Radical Table.

Terms'

1st Column.

2d Column.

69th Column.

Naturals.

Artificals

Naturals.

Artific.

Naturals.

Artificial:"

X

10000000.0000

0

9900000.0000

100503.3

504885S.8900

6834225.8

2

9995000.0000

5001.2

9895050.0000

105504.6

5046333.4605

6839227.1

3

9990002.5000

10002.5

9890102.4750

110505.8

5043811.2932

6844228.3

4

9985007.4987

15003.7

9885157.4237

115507.1

5041289.3879

6849229.6

5

9980014.9950

20005.0

9880214.8451

120508.3

5038768.7435

6854230.8

&c

&c till

&c

& c

&c

&.C

& c

-21

9900473.5780

100025.0

9801468.8423

’200528.2

4998609.4034

6934250.8

Having thus, in the most easy manner, completed the radi-cal table, by little more than mere addition and subtraction,both for the natural numbers and logarithms; the logarithmicsines were easily deduced from it by means of the 2d theorem,namely, taking the sum and difference of each tabular sineand the nearest number in the radical table, annexing 1 ci-phers to the difference, dividing the result by the sum, thenhalf the quotient gives the difference between the logarithmsof the said numbers, namely, between the tabular sine andradical number; consequently, adding or subtracting thisdifference, to or from the given logarithm of the radical num-ber, there is obtained the logarithmic sine required. And thusthe logarithms of all the sines, from radius to the half of it,or from 90° to 30°, were perfected.

Next, for determining the sines of the remaining 30 de-grees, he delivers two methods. In the first of these he pro-ceeds in this manner: Observing that the logarithm of theratio of 2 to 1, or of half the radius, is 6931469’22, of 4 to 1is the double of this, of 8 to 1 is triple of it, &c; that of 10to 1 is 23025842.34, of 20 to 1 is the sum of the logarithmsof 2 and 10; and so on, by composition for the logarithms ofthe ratios between 1 and 40, 80, 100, 200, &c, to 10000000;he multiplies any given sine, for an arc less than 30 degrees,