A MANUAL OF PRACTICAL SOLID GEOMETRY. ADAPTED TO THE REQUIREMENTS OF MILITARY STUDENTS AND DRAUGHTSMEN. COMPILED BY WILLIAM GORDON ROSS, MAJOR, ROYAL Engineers; PROFESSOR OF GEOMETRICAL DRAWING AND FORTIFICATION, ROYAL MILITARY CASSELL & COMPANY, LIMITED: LONDON, PARIS, NEW YORK & MELbourne. 1887. [ALL RIGHTS RESERVED.] EXE PREFACE. A FAMILIARITY with the principles and methods of Geometrical Drawing is essential for all those who would wish to represent, or record, on paper, the appearance, or proportions, of works designed by the architect and engineer. This little manual is compiled for the use of those who may wish not only to obtain a theoretical knowledge of that branch of Geometry which is usually termed Solid, or Descriptive, Geometry, but who may desire also to have the power of producing accurate and workmanlike drawings. The subject of Plane Geometry, also necessary to the draughtsman, has not been treated in this manual, as there already exist manuals of this branch of Geometrical Drawing which are within the reach of all. Up to the present, however, the treatises that have appeared on the subject of Solid Geometry have been either too elaborate, or too highpriced, to command any but a limited circulation. The object of the present work is to supply at a small cost the most essential and practical parts of what can be made a very complicated subject; in knowledge of which we in England compare unfavourably with our continental neighbours. Although of use to, and containing information for, all draughtsmen, this work is more especially intended to illustrate those parts of Solid Geometry which are of practical use to the military draughtsman. With this intention it follows in the main the lines of the course of Geometrical Drawing as studied at the Royal Military Academy, Woolwich. The convention, or modification of Orthographic Projection, known as the "System of Vertical Indices," being especially useful for the purposes of the military designer and draughtsman, has here received more attention than other methods; and the section relating to Defilade, although of no great extent, will be found in few other English works. Of this section of Solid Geometry the more practical and generally useful problems only are given. PRACTICAL SOLID GEOMETRY. THIS term comprehends the investigation and application of those geometrical principles and methods by which points, lines, and bodies of three dimensions may be correctly represented by points, lines; and figures, "projected," or delineated, on a single plane surface. The plane surface, on which such delineations are made by the draughtsman, is the sheet of paper on which he works. The general method by which these representations of the originals are produced is technically termed "Projection." Projection is of three kinds Radial or Perspective Projection, Orthographic or Perpendicular Projection, and Isometric Projection. For a general representation of an object a perspective view, corresponding with the image made by the object on the retina of the eye of the observer, is, in many cases, suitable enough. In perspective projection the lines, or visual rays, proceeding from all points of the original object, are supposed to converge in a single point, the "vertex," which is usually taken to be the eye of the observer. These visual rays, being intercepted by the interposition of a plane surface (commonly taken vertical) and the points obtained on the plane by the impact of the visual rays, being connected with each other in a manner and in an order similar to the manner and the order existing in the original object, a representation is produced of that original object which is called a perspective projection or picture. But this representation is much distorted when compared with the actual form of the original object; for not only are certain lines in the representation, corresponding |