ADVERTISEMENT. THE following Treatise of Plane Trigonometry and Mensuration is the third of a series which is to constitute a General Course of Mathematics for the use of the gentlemen cadets and the officers in the senior department of this Institution. The Course, when completed, will comprehend the subjects whose titles are subjoined: I. Arithmetic and Algebra.* II. Geometry.* III. Plane Trigonometry with Mensuration.* IV. Analytical Geometry with the Differential and Integral Calculus, and the Properties of Conic Sections. V. Practical Astronomy and Geodesy, including Spherical Trigonometry.* VI. The Principles of Mechanics, and VII. Physical Astronomy. Royal Military College, 1845. * Published. PREFACE. THE direct object of Plane Trigonometry is the discovery of the relations which subsist among the sides and angles of rectilineal triangles. Whether the investigation of these relations is conducted geometrically or analytically, the values of the results immediately deduced from them are numerical. The processes also of deducing the expressions of the results, and of combining them to obtain formulæ for the solutions in complex propositions, are algebraical or arithmetical. The applications of Trigonometry in the higher branches of mathematical and physical science are purely analytical. For these reasons, and to preserve uniformity of method, it has become the usual practice to treat the whole subject of Plane Trigonometry analytically. In Mensuration, the general problem is, to determine the relation which any proposed geometrical magnitude bears to an arbitrary unit of the same kind. This relation is numerical. If general, it is expressed by algebraic symbols: if particular, by a number. The processes of combining the quantities and deducing the values of the results, as in plane trigonometry, are algebraical or arithmetical. Hence the analytical seems likewise the most convenient method of investigating those relations of magnitude which are contemplated in Mensuration. It is, therefore, adopted in the sections relating to this subject, as well as in those which relate to trigonometry, in the following treatise. The whole work is divided into nine Sections. Of these, the first and second are devoted to the consideration of the manner of expressing geometrical magnitudes by numbers. The investigations of a considerable number of important trigonometrical formulæ and series are given in the third. The fourth contains rules for the computation of the trigonometrical tables. |