TRIGONOMETRY.

BOOK I.

LOGARITHMS.

1. THE LOGARITHM of a number is the exponent of the power to which a given fixed number must be raised in order to produce the first number.

2. The BASE of the system is the fixed number.

3. The base, in the common system of logarithms, is 10.

It thus appears that, in the common system, the logarithm of every number between 1 and 10 is some number between 0 and 1; that is, a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2; that is, 1 plus a fraction. The logarithm of every number between 100 and 1,000 is some number between 2 and 3; that is, 2 plus a fraction; and so on.

4. By means of negative exponents the application of logarithms may be extended, in the common system, to numbers less than 1. Thus, since

From this it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1; that is, -1 plus a fraction. The logarithm of every number between 0.1 and 0.01 is some number between 1 and -2; that is, -2 plus a fraction. The logarithm of every number between 0.01 and 0.001 is some number between -2 and -3; that is, -3 plus a fraction; and so on.

5. In the common system, as the logarithms of all numbers which are not exact powers of 10 are incommensurable with those numbers, their values can only be obtained approximately, and are expressed by decimals.

6. The integral part of any logarithm is called the CHARACTERISTIC, and the decimal part is sometimes called the MAN

TISSA.

7. The characteristic of the logarithm of ANY NUMBER GREATER THAN UNITY, is one less than the number of integral figures in the given number.

For it has been shown (Art. 3) that the logarithm of 1 is 0, of 10 is 1, of 100 is 2, of 1000 is 3, and so on.

8. The characteristic of the logarithm of ANY DECIMAL FRACTION is a negative number, and is one more than the number of ciphers between the decimal point and the first significant figure.

For it has been shown (Art. 4) that the logarithm of 0.1 is -1, of 0.01 is -2, of 0.001 is -3, and so on.

NOTE. In general, whether the given number be integral, fractional, or mixed, the characteristic of the logarithm of any number expressed decimally is the distance of the first, or left-hand, significant figure from the units' place, being positive when that figure is on the left of the units' place, and negative when on the right.

GENERAL PROPERTIES OF LOGARITHMS.

9. The logarithm of a PRODUCT is equal to the sum of the logarithms of its factors.

For let M and N be any two numbers, x and y

their respective

logarithms, and a the base of the system. Then, by definition (Art. 1), we have

Multiplying equations, member by member, we have

Therefore,

MN at a" = ax + ".

log (MXN)=x+y= log M+log N.

10. The logarithm of a QUOTIENT is equal to the logarithm of the dividend diminished by that of the divisor.

For, by Art. 9, we have

M= = a*, N = a".

Dividing the first equation by the second, member by member, we have

11. The logarithm of any POWER of a number is equal to the product of the logarithm of the number by the exponent of the power.

For let m be any number, and take the equation (Art. 9)

M=a*,

then, raising both sides to the mth power, we have

12. The logarithm of the ROOT of any number is equal to the

logarithm of the number divided by the index of the root.

For, let n be any number, and take the equation (Art. 9)

M=a*,

then, extracting the nth root of both sides, we have

NM=Nw=a.

13. Hence, by means of logarithms, we can perform multiplication by addition, and division by subtraction; also, we can raise a number to any power by a single multiplication, and extract any root of a number by a single division.

14. All numbers, integral, fractional, or mixed, having the same succession of significant figures, have logarithms with the same decimal part.

For since the logarithm of 10 is 1, the product of any number by 10 will have a logarithm increased by 1; and, likewise, the quotient of any number divided by 10 will have a logarithm diminished by 1; and, 1 being an integer, the logarithms will differ only in their characteristics.

15. The negative sign placed over the characteristic indicates that it alone is negative, the decimal part being always positive.

TABLE OF LOGARITHMS.

16. A Table of Logarithms usually contains all the whole numbers between 1 and a given number, with their logarithms. The accompanying table contains the logarithms of all numbers from 1 up to 10,000, calculated to six places of decimals.

17. In the table, the characteristics of the logarithms of the first 100 numbers are inserted; but for all other numbers the decimal part only of the logarithms is given, while the characteristic is left to be supplied by inspection, according to the principles already furnished (Art. 7, 8).

18. The numbers are in the column headed N, and their logarithms, or the decimal parts of their logarithms, are opposite

on the same line. When the first two figures of the decimal are the same for several successive logarithms, they are not repeated for each, but, being used once, are then left to be supplied. 19. In the column headed D are the mean or average ences of the ten logarithms against which they are placed.

TO FIND THE LOGARITHM OF ANY NUMBER.

differ

20. When the given number is any integer of ONE or TWO figures.

Look on the first page of the table, and opposite the given number will be found the logarithm with its characteristic. Thus,

the logarithm of 63 is 1.799341;

21. When the given number is any integer of THREE FIG

URES.

Look in the table for the given number, and opposite the same, in the column headed 0, will be found the decimal part of the logarithm, to which must be prefixed 2 as the characteristic (Art. 7). Thus,

the logarithm of 110 is 2.041393;

22. When the given number is any integer of FOUR figures, either with or without ciphers annexed.

Find the first three figures of the given number in the column headed N, and, opposite to them, in the column headed by the fourth figure, will be found the decimal part of the logarithm; to which the characteristic, as determined by Art. 7, must be prefixed. Thus,

the logarithm of 4901 is 3.690285;

23. When the given number is any integer of FIVE or MORE figures.

Find the logarithm of the first four figures as in Art. 22,