Section CONTENTS. Section REFERENCES TO ALGEBRA. In the Trigonometry, Mensuration and Navigation, references are made to articles in the earlier editions of the Algebra. But in the revised edition, the numbers of the articles are nearly all different from the numbers of the corresponding articles in the former editions. In the following table, when any number which is referred to in the old editions is contained in a column on the left, the number of the same article in the revised edition will be found directly opposite, in the adjoining column. LOGARITHMS. SECTION I. 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 * 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; 2 αν, αν, αν, αν, α", α- ', α-, απ3, α-, &c. 1 The logarithm of a3 is 3, and the logarithm of a1 is —1, of a-2 is-2, of a-3 is-3, &c. of a1 is 1, of ao is 0, α 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, 4, 3, 102, 101, 10°, 10-1, 10-2, 10-3, &c. Whose logarithms are 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, 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 in this be increased or diminished, the power will be increased or diminished. |