If, now, we add angles B and C, and take the sum from 180′, the remainder will be the angle BAC.

Hence, BAC = 180° . (56° 26'+20°)

= 103° 34'.

5. Two sides, the one 18 and the other 21, and the angle opposite the side 24 equal to 76°, are given, to find the remaining side and the other two angles.

Let x denote the angle opposite the side 18. Then, sin. 76° : sin. x, (Prop. 4, Trig.).

24 18

whence the angle opposite the side 18 is 46° 41′ 45′′.

Adding this to the given angle, and taking the sum from 180°,

we get 57° 18′ 15′′ for the third angle.

To find the remaining side, denoted by y, we have

6. The three sides of a triangle are 18, 24, and 20.815; required the angles.

This problem may be solved by Prop. 6, or by Prop. 8, Trigo nometry.

First. By Prop. 6.

In the triangle ABC, make CB =

24, AC 20.815, and AB 18.

Then,

:

=

24 38 815 :: 2.815: CD

=

- BD.

CD-BD =

109.264225
24

= 4.5527.

B

It will be observed that Examples 5 and 6 refer to the same triungle, and that in Example 5 the angle B was 57° 18′ 15′′. This slight discrepancy in the results should be expected, on account of the small number of decimal places used in the computations.

Hence, cos. A = .78800, and 4 (Table II, page 59) = 38° very nearly; the angle A is therefore equal to 76°, which agrees with Example 5.

7. Given, the three sides, 1425, 1338, and 493, of a triangle; required, the angle opposite the greater side, using the formula for the sine of one half an angle.

Make a = 1425, b = 1338, and c = 493; then the L▲ is opposite the side a, and the formula is

Then we have s = 1628, s―

in which s denotes the half sum of the three sides.

b = 290, s— C= 1135, (s—b)

Therefore, A = 44° 56′ 28.5", and A = 89° 52′ 57′′; little less than a right angle.

In these seven examples we have shown that it is possible to solve any plane triangle, in which three parts, one at least being a side, are given, without the aid of logarithms. But, when great accuracy is required, and the number of decimal places employed is large, the necessary multiplications and divisions, the raising to powers, and the extraction of roots, become very tedious. All of these operations may be performed without impairing the correctness of results, and with a great saving of labor, by means of logarithms; but, before using them, the student should be made acquainted with their nature and properties.

LOGARITHMS.

Logarithms are the exponents of the powers to which a fixed number, called the base, must be raised, to produce other numbers.

The exponent of a number is also a number expressing how many times the first number is taken as a factor.

Thus, let a denote any number; then a indicates that a has been used three times as a factor, a' that it has been asel four times as a factor, and a" that it has been thus 130d n times.

Now, instead of calling these numbers 3, 4, exponents, we call them the logarithms of the powers a3,

To multiply a2 by a3, we have simply to write a, giving it an exponent equal to 2 + 5; thus, a2 x a = a*.

5

Hence, the sum of the logarithms of any number of factors is equal to the logarithm of the product.

12

To divide a1 by ao, we have only to write a, giving it an exponent equal to 12-9; thus, a 12 a° = a3; and, generally, the quotient arising from the division of am by 2", is equal to a

m_n.

Hence, the logarithm of a quotient is the logarithm of the dividend diminished by the logarithm of the divisor.

If it is required to raise a number denoted by a3, to the fifth power, we write a, giving it an exponent equal to 3×5; thus, (a3)=a5, and, generally, (a") manm ̧

Hence, the logarithm of the power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

To extract the 5th root of the giving it an exponent equal to

number a3, we write a, ; thus, Va3=a3, and,

5

generally, to extract any root of a number, we divide the exponent of the number by the index of the root, and the quotient will be the exponent of the required root.

Hence, the logarithm of a root of a number is equal to the quotient obtained by dividing the logarithm of the number by the index of the root.

Now, understanding that by means of a table of logarithms we may find the numbers answering to given logarithms, with as much facility as we can find the logarithms of given numbers, we see from what precedes that multiplications, divisions, raising to powers, and the extraction of roots, may be performed by logarithms; and the utility of logarithms, in trigonometrical computations, mainly consists in the simplification and abridgment of these operations by their use.

The common logarithms are those of which 10 is the base; that is, they are the exponents of 10.

it follows that in this, as in all other systems, the logarithm of 10.

From what precedes, it is evident that the logarithm of any number between 10 and 100 must be found between 1 and 2; that is, its logarithm is 1 plus a number less than 1; and any number between 100 and 1000, will have for its logarithm 2 plus some number less than 1, and so on. The fractional part of the logarithm of a number is expressed decimally.

The entire number belonging to a logarithm is called its index. The index is never put in the tables, (except from 1 to 100), and need not be put there, because we always know what it is. It is always one less than the number of digits in the integer. Thus, the number 8754 has 3 for the index to its logarithm, because the number consists of 4 digits; that is, the logarithm is 3 and some decimal.

The number 347.921 has 2 for the index of its logarithm, because the number is between 347 and 348, and 2 is the index for the logarithms of all numbers over 100, and less than 1000.

All numbers consisting of the same figures, whether integral, fractional, or mixed, have logarithms consisting of the same decimal part. The logarithms differ only in their indices.

24*