CHAMBERS'S EDUCATIONAL COURSE,-EDITED BY W. AND R. CHAMBERS. SOLID AND SPHERICAL GEOMETRY, AND CONIC SECTIONS. BEING A TREATISE ON THE HIGHER BRANCHES OF SPHERICAL GEOMETRY OF PLAYFAIR; SECTIONS OF WEST; AND VARIOUS IMPROVEMENTS AND ADDITIONS. BY A. BELL FORMERLY MATHEMATICAL MASTER IN DOLLAR INSTITUTION. t 1 EDINBURGH: AND SOLD BY ALL BOOKSELLERS, 1837. PREFACE. This volume contains the Higher Branches of Synthetical Geometry. It consists of Treatises on Solid Geometry, Spherical Geometry, Spherical Trigonometry, the Projections of the Sphere, Perpendicular Projection, Linear Perspective, and Conic Sections. The first three Treatises are those contained in Playfair's Edition of Euclid's Elements, with some alterations. Several useful Definitions, Scholia, Corollaries, and Propositions, have been added. Instead of the first four propositions on Spherical Geometry, other four, from the excellent System of Mathematics by West, have been substituted, as they contain a more full exposition of principles. From the same work another important proposition, the eighteenth, has been added. The Spherical Trigonometry has been improved by inserting the twelfth and thirteenth, for other propositions which are somewhat intricate. These two are also taken from West's system, but the expressions in the demonstrations are considerably altered. The propositions which these have displaced, rather injure the symmetry of the system, as they require the aid of Analytical Trigonometry. A second demonstration has been given of the fifth proposition of the second book of Solid Geometry, depending on the principle established in the twenty-seventh proposition of the additional fifth book of the former volume. The two Treatises on the Projections of the Sphere are also adopted from West's system, except the problems on the Stereographic Projection, which, with the Treatises on Perpendicular Projection and Perspective, have been composed expressly for this work. The Treatise on Conic Sections is taken from the same work; and as the Corollaries are numerous, and most of them important, but undemonstrated in the above work, except a few, demonstrations have been added to those that required them, in order to remove unnecessary obstacles to the progress of the student, who will find a sufficient field for exercise in the undemonstrated theorems and problems annexed for this purpose to this Treatise. The Treatises on Projections have been added on account of their utility in some branches of practical science and of art; the Projections of the Sphere being necessary in Spherical Trigonometry, and in Nautical and Practical Astronomy; and Perpendicular Projection and Perspective being indispensable in constructing the diagrams in Geometry of three dimensions, and figures of objects in various branches of the arts and of philosophy |