8. Evolution by Logarithms.-In evolution we proceed by the following

RULE X.

1o. Find the logarithm of the number whose root is to be found. 2°. Divide this logarithm by the index of the given root; the quotient will be the logarithm of the required root.

3. Find from the tables the corresponding number.

In cases like the preceding, in which the characteristic is negative, and not divisible by the given index, it is necessary to increase the negative characteristic by as many units as will render it divisible, and then carry so much to the decimal. Thus, in the preceding example we consider

4 = 6 + 2; and therefore

3) ☎ + 2.51268

2 +0.83756

9. Tables of Logarithmic Sines. In order to apply logarithmic calculations to trigonometrical quantities, it is necessary to construct tables of the logarithms of the natural sines, cosines, &c. As all the sines and cosines, all the tangents from o° to 45°, and all the cotangents from 45° to 90°, are less than unity, the logarithms of these quantities have negative characteristics. In order to avoid the necessity of entering negative numbers, 10 is added to every logarithm before it is registered in the tables of logarithmic sines. Thus, sin2 45° = 0.5

Therefore

logarithm of sin 45° = 1.84949

If to this we add 10, we find the tabular log sin 45°

To find the log sine, log cosine, log tangent, or log cotangent, of an angle.

RULE XI.

1o. Find from the tables the log sine, log cosine, &c., which corresponds to the degrees and minutes.

2. Multiply the tabular difference by the seconds, and divide by 60.

3°. If the required quantity be a log sine or log tangent, add the result to the last figures obtained in 1o; if it be a log cosine or log cotangent, subtract.

The result will be the required log sine, log cosine, &c.

The log sine, log cosine, log tangent, or log cotangent, being given, to find the angle.

RULE XII.

1. Find in the tables the next lower log sine, log cosine, &c., and note the corresponding degrees and minutes.

2. Subtract this from the given log sine, log cosine, &c., multiply the difference by 60, divide by the tabular difference, and consider the result as seconds.

3°. If the given value be that of a log sine or log tangent, add these seconds to the degree and minutes; if it be that of a log cosine or log cotangent, subtract.

The result will be the required angle.

EXAMPLES.

1. Given log sine = 9.35624; find the angle.

2. Given log cotang = 10.11234; find the angle.

As the tabular log is equal to the true logarithm, plus 10, it is evident that before using log sines, log cosines, &c., in numerical computations, it is necessary to subtract 10 from the values entered in the tables.

10. Gauss's Logarithms.—By means of these tables the logarithms of the sum and difference of two numbers may be immediately derived from the logarithms of the numbers themselves. The table consists of three columns, A, B, and C. The column A extends from o to 2, proceeding by thousands; from 2 to 3.4, proceeding by hun

dreds; and from 3.4 to 5, proceeding by tenths. Here it terminates, since the numbers in the column B vanish, and those in column C become equal to those in A for 5 and all other higher numbers. The construction of the table is explained as follows:-If we suppose any number A, in the first column = log m, then for this and the corresponding numbers B and C in the second and third columns, we have

Being given the logarithms of two numbers a and b, of which the first is greater than the second, to find the logarithm of their sum.

RULE XIII,

1o. Subtract the less from the greater logarithm.

2°. With the difference enter column A, and find the corresponding number in column B.

3o. Add the number so found to the greater logarithm.

The result is the logarithm of the sum.

The rule may be proved as follows:-Let a =

mb, then

Taking the logarithms of both sides, and substituting for log 1 +

(

I

m

its equal B,

log (a+b)= log a + B

In a similar manner it may be shown that

log (a + b) = log b + C

As the tabular differences corresponding to the columns B and C are complementary, that is to say, their sum is always equal to 100, or 1000, or 10000, the method of proportional parts may be applied as follows, having regard to one of these differences only.

RULE XV.

1o. From the difference of the logarithms take the next lower log which can be found in either of the columns B or C.

2°. To the remainder annex two, three, or four ciphers, as the case requires, and divide by the tabular difference.

3°. From the quotient take the remainder.

The result is always to be subtracted from the corresponding logarithm in the other column.

EXAMPLE.

1. The logarithms of two numbers 0.25042 and 0.19033; find the logarithm of their difference.

2. The logarithms of two numbers are 3.44134 and 1.21352; find

the logarithm of their difference.