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The plan uporì which this work was originally commenced, is continued in this second part of the course. As the single object is to provide for a class in college, such matter as is not embraced by this design is excluded. The mode of treating the subjects, for the reasons mentioned in the preface to Algebra, is, in a considerable degree, diffuse. It was thought better to err on this extreme, than on the other, especially in the early part of the course.
The section on right angled triangles will probably be considered as needlessly minute. The solutions might, in all cases, be effected by the theorems which are given for oblique angled triangles. But the applications of rectangular trigonometry are so numerous, in navigation, surveying, astronomy, &c., that it was deemed important, to render familiar the various methods of stating the relations of the sides and angles; and especially to bring distinctly into view the principle on which most trigonometrical calculations are founded, the proportion between the parts of the given triangle, and a similar one formed from the sines, tangents, &c., in the tables.
NATURE OF LOGARITHMS.*
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 power, be represented by a. Then taking a series of powers, both direct and reciprocal, as in Alg. 207 ;
* Maskelyne's Preface to Taylor's Logarithms. Introduction to Hutton's Tables. Keil on Logarithmis. Maseres Scriptores Logarithmici. Briggs' Logo arithms. Dodson's Anti-logarithmic Canon. Euler's Algebra. + See note A.
a', a, a, a, a, a, a, -, a-4, &c. The logarithm of a3 is 3, and the logarithm of a-is-1, of a' is 1,
of a is-2, of aois 0,
of a-3 is -3,&c. Universally, the logarithm of a' 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, 102, 10", 10°, 10-4, 10-2, 10--3, &c. Or 10000, 1000, 100, 10, 1, To To Ido, &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 a}, or 20.3333, is 0.3333,
of a3, or a"-6 6 6 6,
0.66 6 6, is 0.6666, of ať, or a..4285, is 0.4285,
of a?', or a3.6606, 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 cxponent 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=100.8 4 5 1
of 7 is 0.8451 20=101.30 10 therefore the of 20 is 1.3010 30=101.4771
logarithm of 30 is 1.4771 400=102.6 0 2
of 400 is 2.6020, &c. 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.
0 2 0
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 10 and 100, the log. (1 and 2, i.e. 1+the dec. part. 100 and 1000,
will be 2 and 3, 2+the dec. part. 1000 and 10000, between / 3 and 4, 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 u power. The characteristic, therefore, is the index of an index.