BY WILLIAM SMYTH, A. M., PROFESSOR OF MATHEMATICS IN BOWDOIN COLLEGE. 1. Elementary Algebra. This treatise is designed for beginners. The first principles of the science are familiarly explained, and applied to the investigation of the rules and solution of the questions of Arithmetic. 2. Treatise on Algebra. This is an extended treatise on the subject embracing the theory of the higher equations. With the following parts of the course it is designed for the use of Colleges and Schools. 3. Plane Trigonometry, Surveying, and Navigation. The Trigonometry is devoloped in a manner to give a clear idea of the nature and use of the Trigonometrical Tables. The Surveying is designed to give a comprehensive view of the subject, embracing surveys of every extent, from the simple field to an extended territory, including also those required for Canal routes, Railroads, &c. Tables for this work are published in a separate volume. 4. Analytical Geometry. The design of this treatise is to explain the origin and true nature of the Analytical Geometry; to show its adaptation to the investigation of the truths of Geometry, and thus to prepare the way for the introduction of the Calculus. 5. Elements of the Differential and Integral Calculus. In this treatise the student is led by a very natural and easy process to a general knowledge of the principles of the Calculus, to see in what it consists, and to discern its power as the highest instrument of Mathematical investigation. All the parts of this Series of Elementary Works are arranged with reference to each other, and are adapted, it is believed, to conduct the student with facility and advantage over the wide extent of mathematical topics which it embraces. BOSTON: SANBORN, CARTER AND BAZIN, (Successors to B. B. Mussey, & Co.) SIG. 1 ELEMENTS OF PLANE TRIGONOMETRY, SURVEYING AND NAVIGATION: BY WILLIAM SMYTH, A. M., PROFESSOR OF MATHEMATICS IN BOWDOIN COLLEGE. BOSTON, SANBORN, CARTER AND BAZIN. PORTLAND,- -SANBORN AND CARTER. 1855. KE11335 HARVARD UNIVERSITY Entered according to act of Congress, in the year 1855, by WILLIAM SMYTH, A. M. in the Clerk's Office of the District Court of the District of Maine. PRESS OF J. GRIFFIN, PREFACE. In developing the subject of Plane Trigonometry, the object is to determine all the parts of a plane triangle from certain of these parts which are given. We commence with the inquiry, how many of the parts must be known in order that the rest may be found. This being answered, the solution which first presents itself is the determination of the required parts by a construction of the triangle. In order to this we must have measures, both for the sides and angles. These being obtained, we proceed next to the construction, and solve all the different cases which may occur. In the performance of this work the student cannot fail to see the imperfection of the method, and the necessity, in order to accuracy, of a solution by arithmetical calculation. But here a difficulty presents itself, arising from the impracticability of combining, in the same numerical calculation, angles or the arcs by which they are measured, and straight lines; quantities, which having no common measure, cannot be compared with each other. The problem not admitting of direct solution, some indirect method must be employed. On reflecting upon the means for accomplishing this object, it will occur that if a series of triangles can be found having angles of every possible magnitude, and the sides of which, or the ratio they bear to each other, are given, then any proposed triangle may be calculated from some one of these by means of a simple proportion. The inquiry now is, how can such a series of triangles be formed. Commencing with the rightangled triangle, as the most simple, we see that such a series may with facility be constructed in the quadrant of a circle; and with the advantage that the radius of the quadrant will be the common hypothenuse in all the triangles. But can the remaining sides of these triangles, or their ratio to |
