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II. Explanation of the Trigonometrical Tables
ART. 1. THE operations of Multiplication and Division, when they are to be often repeated, become so laborious, that it is an object of importance to substitute, in their stead, more simple methods of calculation, such as Addition and Subtraction. If these can be made to perform, in an expeditious manner, the office of multiplication and division, a great portion of the time and labor which the latter processes require, may be saved.
Now it has been shown, (Algebra, 233, 237,) that powers may be multiplied, by adding their exponents, and divided, by subtracting their exponents. In the same manner, roots may be multiplied and divided, by adding and subtracting their fractional exponents. (Alg. 280, 286.) When these exponents are arranged in tables, and applied to the general purposes of calculation, they are called Logarithms.
2. LOGARITHMS, THEN, ARE THE EXPONENTS OF A SERIES OF POWERS AND ROOTS.†
In forming a system of logarithms, some particular number is fixed upon, as the base, radix, or first power, whose logarithm is always 1. From this, a series of powers is raised, and the exponents of these are arranged in tables for use. To explain this, let the number which is chosen for the first
* Maskelyne's Preface to Taylor's Logarithms. Introduction to Hutton's Tables. Keil on Logarithms. Maseres Scriptores Logarithmici. Briggs' Logarithms. Dodson's Anti-logarithmic Canon. Euler's Algebra.
+ See note A.
power, be represented by a. Then taking a series of powers, both direct and reciprocal, as in Alg. 207;
αν, αν, απ, α', α", α-1, α-", απ3, απ', &c.
The logarithm of a3 is 3, and the logarithm of a1 is —1,
of a1 is 1,
of ao is 0,
of a 2 is-2,
Universally, the logarithm of aa is x.
3. In the system of logarithms in common use, called Briggs' logarithms, the number which is taken for the radix or base is 10. The above series then, by substituting 10 for a, becomes
104, 103, Or 10000, 1000,
102, 101, 10°, 101, 102, 10-3, &c.
Whose logarithms are
4, 3, 2, 1, 0, -1, -2, -3, &c.
4. The fractional exponents of roots, and of powers of roots, are converted into decimals, before they are inserted in the logarithmic tables. See Alg. 255.
The logarithm of a3,
or a0.3333, is 0.3333,
of a3, or ao.6666, is 0.6666,
of a, or a0.4285, is 0.4285,
of a's, or a 3.66 66, is 3.6666, &c.
These decimals are carried to a greater or less number of places, according to the degree of accuracy required.
5. In forming a system of logarithms, it is necessary to obtain the logarithm of each of the numbers in the natural series 1, 2, 3, 4, 5, &c.; so that the logarithm of any number may be found in the tables. For this purpose, the radix of the system must first be determined upon; and then every other number may be considered as some power or root of this. If the radix is 10, as in the common system, every other number is to be considered as some power of 10.
That a power or root of 10 may be found, which shall be equal to any other number whatever, or, at least, a very near approximation to it, is evident from this, that the exponent may be endlessly varied; and if this be increased or diminished, the power will be increased or diminished.
If the exponent is a fraction, and the numerator be increased, the power will be increased; but if the denominator be increased, the power will be diminished.
6. To obtain then the logarithm of any number, according to Briggs' system, we have to find a power or root of 10 which shall be equal to the proposed number. The exponent of that power or root is the logarithm required. Thus
7. A logarithm generally consists of two parts, an integer and a decimal. Thus, the logarithm 2.60206, or, as it is sometimes written, 2+.60206, consists of the integer 2, and the decimal .60206. The integral part is called the characteristic or index* of the logarithm; and is frequently omitted, in the common tables, because it can be easily supplied, whenever the logarithm is to be used in calculation.
By art. 3d, the logarithms of
10000, 1000, 100, 10, 1, .1, .01, .001, &c. are 4, 3, 2, 1, 0, -1, -2, -3, &c.
As the logarithms of 1 and of 10 are 0 and 1, it is evident, that, if any given number be between 1 and 10, its logarithm will be between 0 and 1, that is, it will be greater than 0, but less than 1. It will therefore have 0 for its index, with a decimal annexed.
Thus, the logarithm of 5 is 0.69897.
For the same reason, if the given number be between
1 and 2, i. e. 1+the dec. part.
10 and 100,
100 and 1000, 1000 and 10000,
2 and 3,
3 and 4,
2+the dec. part. 3+the dec. part.
We have, therefore, when the logarithm of an integer or mixed number is to be found, this general rule:
* The term index, as it is used here, may possibly lead to some confusion in the mind of the learner. For the logarithm itself is the index or exponent of a power. The characteristic, therefore, is the index of an index.
8. The index of the logarithm is always one less, than the number of integral figures, in the natural number whose logarithm is sought: or, the index shows how far the first figure of the natural number is removed from the place of units.
Thus, the logarithm of 37 is 1.56820.
Here, the number of figures being two, the index of the logarithm is 1.
The logarithm of 253 is 2.40312,
Here, the proposed number 253 consists of three figures, the first of which is in the second place from the unit figure, The index of the logarithm is therefore 2.
The logarithm of 62.8 is 1.79796.
Here it is evident that the mixed number 62,8 is between 10 and 100. The index of its logarithm must, therefore, be 1.
9. As the logarithm of 1 is 0, the logarithm of a number less than 1, that is, of any proper fraction, must be negative.
Thus, by art. 3d,
The logarithm of
10. If the proposed number is between its logarithm must be between
2 and 3. To obtain the logarithm, therefore, we must either subtract a certain fractional part from -2, or add a fractional part to -3; that is, we must either annex a negative decimal to
Thus, the logarithm
-2, or a posi
of .008 is either -2.09691, or 3+.90309.*
The latter is generally most convenient in practice, and is more commonly written 3.90309. The line over the index
That these two expressions are of the same value will be evident, if we subtract the same quantity, +.90309 from each. The remainders will be equal, and therefore the quantities from which the subtraction is made must be equal. See note B.