Cambridge: PRINTED BY C. J. CLAY, MA. AT THE UNIVERSITY PRESS. GEO. MANN RICHARDSON, ...... C LIBRARY Leland Stanford, Jr. UNIVERSITY PREFACE. I HAVE endeavoured, by putting the subject in a simple, concise, and systematic form, to give to this treatise the elementary character which is required in a book intended for beginners, and at the same time to make it sufficiently comprehensive to meet the wants of a more advanced class of students. The difficulties which hinder beginners I have found to be chiefly of two kinds. One of these arises from the want of sufficient knowledge of solid geometry; the study of projections, as a practical subject, being begun too commonly before the student has made himself acquainted with the geometrical principles on which the solutions of the problems depend. To begin in that way is, I think, to make a mistake; for, without a knowledge of first principles, it is impossible to get such a grasp of the subject as will make it the useful and effective instrument which it ought to be. I have, therefore, considered it best to devote the first chapter to some theorems on the straight line and plane, and to introduce occasional theorems in the other parts of the work; my object being to establish the principles before giving their applications... The other difficulty to which I have alluded lies in the inability of the learner to realise from their projections 67043 the positions in space of points and lines. It is one which requires considerable time and thought to overcome. I have tried, however, to reduce it as much as possible by giving two figures with each problem of Chapter II.; one of these figures being a perspective, representing the points, lines, and planes in their true positions, and the other their projections, and the ordinary solution of the problem. I trust that by carefully comparing these figures the student may be led by easy steps to connect the two things and obtain a clear idea of the methods employed in Descriptive Geometry. I have little doubt that any one who masters the first two chapters will find his after-course both interesting and comparatively easy. I may add that I have never lost sight of the practical nature of the subject, and have introduced only so much theory as seemed to me necessary to place the practice on a proper footing. J. B. M. MANCHESTER, April, 1878. CONTENTS. Theorem I. Two straight lines which cut one another are in one plane Theorem II. If two planes cut one another, their common section is a Theorem III. If a straight line be perpendicular to each of two straight lines at their point of intersection, it shall also be per- PAGE Theorem IV. Every plane which contains the normal to another plane is perpendicular to that plane Theorem V. If two planes be perpendicular to one another, every line drawn in one of them perpendicular to their common section Theorem VI. If two planes which cut one another be both perpendi- cular to a third plane, their common section shall be perpendicular Theorem VIII. If two straight lines be parallel, and one of them be perpendicular to a plane, the other shall be perpendicular to it Theorem XI. If a straight line be parallel to a plane, it shall be parallel to the line in which any plane containing it cuts the first Theorem XII. If two parallel planes be cut by another plane, their Theorem XIII. Planes to which the same straight line is perpendicular Theorem XIV. If two straight lines which meet one another be parallel respectively to two other lines which meet, but are not in the same plane with the first two, the first two and the other two shall Theorem XVII. If two straight lines be at right angles to one another, their projections on a plane parallel to one of them shall also be Theorem XVIII. If the projections of two straight lines be at right angles to one another, and one of the lines parallel to the plane of projection, the two lines shall be at right angles to one another INTRODUCTORY Problem I. Given the traces of a straight line, to find its projections Problem II. Given the projections of a straight line, to find its traces Problem III. To determine the projections of a straight line which passes through a given point, and is parallel to a given straight Problem IV. Given the projections of two points, to determine the Problem V. Given the projections of a point, and a line through the point, to lay off a given distance from the point along the line |