PLANE TRIGONOMETRY. SCARCELY any department of Mathematics is more important, or more extensive in its applications, than Trigonometry. By it the mariner traces his path on the ocean; the geographer determines the latitude and longitude of places, the dimensions and positions of countries, the altitude of mountains, the courses of rivers, &c., and the astronomer calculates the distances and magnitudes of the heavenly bodies, predicts the eclipses of the sun and moon, and measures the progress of light from the stars. The section on right angled triangles in this treatise, may perhaps 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. As this treatise is intended to form a part of Day and Thomson's Course of Mathematics for the use of Schools and Academies, the references to Algebra are made to Thomson's Abridgment; and the references to Geometry, to Thomson's Legendre, as well as to Euclid's Elements. CONTENTS. Page 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, 189, 193,) 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., 232, 239.) 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 To explain this, let the number which is chosen for the use. |