come practised in dealing with written questionsa point not to be lost sight of by those who are preparing for examinations. The Second Edition has been enlarged by the addition of chapters on the straight line and plane with explanatory diagrams and exercises, on tangentplanes, and on the cases of the spherical triangle. It is hoped that the work, thus rendered more complete, may prove still more useful as a classbook and means of self-instruction to the various and constantly increasing classes of students for whom it is designed. It was originally intended as an aid in teaching the Mechanical Drawing Class at the Royal School of Mines from Professor Bradley's Elements of Practical Geometry. The authors of this work were associated with him in his duties at King's College, London, and the Royal Military Academy, and learnt practically the value of his treatise; but the cost of that work has rendered it inaccessible to many for whom the present book may be available. A greater number of diagrams have not been added, in order that students may be thrown upon their own resources, and encouraged to consider the principles upon which their work proceeds, more than they would probably do if there were figures always at hand for reference. October, 1871. ELEMENTARY SOLID OR DESCRIPTIVE GEOMETRY. ELEMENTARY EXPLANATIONS, DEFINITIONS AND THEOREMS. "The object of Descriptive Geometry is the invention of methods by which we may represent upon a plane having only two dimensions, namely length and breadth, the form and position of a body which possesses three dimensions, namely length, breadth, and height.” HALL'S Elements of Descriptive Geometry. IN Descriptive Geometry Solids are represented by their drawings or projections on two planes conceived at right angles to one another, intersecting in a line called the ground line xy, and named from their usual positions the horizontal and vertical planes of projection. See Fig. 1. These planes form four dihedral angles. Drawings or projections on the horizontal plane are called plans; on the vertical plane elevations. The plan or horizontal projection of any point A in space is the foot of the perpendicular let fall from point A on the horizontal plane, and is marked a; the elevation of A is marked a' and is the point of intersection of the perpendicular from A on the vertical plane. See Fig. 1. E. G. I The perpendicular Aa from the given point A, upon the given plane is termed the projector or projecting-line of that point, and the plane its plane of projection. Plane (H) Fig. 1. Fig. 1. (H) (v) The plane ABab containing the given straight line AB, and perpendicular to the plane of projection, is called the projecting plane of that line; and the intersection ab of these two planes the projection of the line, i. e, its plan, or in the case of AB and a'b' its elevation. The projection of a line may also be defined to be the sum of the projections of its points. If the given line be a curve, not lying in a single plane, the Superficies containing all its projectors for a given plane of projection, is termed the projecting surface of the curve: |