132. By this construction / az is a vanishing line, of which e is the centre, v e equal to its principal radial, and c e its auxiliary vanishing line (95) ; I and vs will obviously be the vanishing points of the diagonals of every square, lying in original planes having Lm for their vanishing line, the sides of that square being parallel and perpendicular to the intersecting line of its plane ; accordingly the quadrilateral fg h i is the image of such a square, lying in such a plane, and the line a $ being made equal to the given image of a diameter of the sphere, a 13 and a b are the images of equal original lines parallel to the picture and equally distant from it, or both lying in a plane parallel to that of the picture. If therefore an ellipse be described in f g hi, touching the sides in the points a/378, and having its transverse axis in c e, this ellipse will be the image of an original circle equal to a great one of the sphere, and having its plane parallel to that passing through the vertex and the centre of the sphere, or this original circle may be regarded as the oblique plan, on a plane parallel to it, of the section of the sphere by the vanishing plane, the projecting lines being parallel to the plane of the picture.
133. Draw v n perpendicular to v e, cutting e c in n, and through n draw a vanishing line perpendicular to e n, or having e n also for its auxiliary vanishing line; make n o, np, each equal to the auxiliary radial v n ; make e r, e s,1- in / each equal to the semi.conjugate axis of the ellipse last drawn, and complete the trapezium wxyz as the image of a square having op for its vanishing line, and its sides parallel and perpendicular to the intersecting line of its plane. An ellipse described in sox y r, having its transverse axis in e n, will be the outline of the sphere.
134. For n being the auxiliary vanishing point of the plane of the original of f J h i, o p is the vanishing line of all planes perpendicular to that original plane, and intersecting it in lines parallel to the plane of the picture. The original square of the quadrilateral tcxyz is there fore perpendicular to the plane of the original of fghi, or to the vanishing plane paasing through the vertex and centre of the sphere. Now it will be seen that the conjugate axis of the ellipse in fgh i is the oblique plan (59) of the chord of the tangents from the vertex to the section of the sphere by the vanishing plane, which chord of the tangents must be a (liameter of the small circle of the solid, consti tuting the original of its apparent outline ; this small circle being the base of the cone of rays tangential to its surface (62), and having its plane perpendicular to that of the vanishing plane passing through the vertex and centre of the sphere ; se xyz is consequently the image of the square circumscribing the circular base, and the inscribed ellipse that of the circle itself, or this ellipse is the outline of the sphere.
135. If the distance of the vertex (70) be supposed to be indefinitely great, compared to the magnitude of the object to be represented, the pyramid of rays may be conceived to become a prism, or the rays to be parallel. On this supposition the vanishing points of the lines of the original object would be indefinitely distant from the centre of the picture, and the images of parallel original lines would be parallels. The isometric projection of a parallelopiped (57) is obviously a limited case of this kind, the limitation being necessary from the object in view, which induces us to adopt that kind of projection. But there are occasions on which it is desirable to delineate rectilinear objects pictorially, which from their small relative size, and from other con siderations, do not require the application of perspective projection, and which would not be adequately represented by an isometric one. In such cases the draughtsman may readily accomplish his purpose by combining the principles of projection on co-ordinate planes with per spective, as in the following example.
136. Let a hexagonal figure, abcdefg, be drawn, with the condition that each pair of opposite aides shall be parallel, and consequently equal ; from the angles a, c, f draw lines parallel to the alternate sides, and meeting in a point d, and from the intermediate angles b, e,g draw lines parallel to the remaining sides respectively, and meeting in h. The figure thus formed will be the orthographic or orthogonal projection of a cube, under certain unknown conditions of inclination of the plane of projection to the projecting lines, and of these to the original planes of the solid.
137. The projections of the centres of each face of the cube, as q, may be found by drawing the diagonals, as ac, bd, and if lines be drawn through the centres of each pair of opposite faces, as pr, which lines will obviously be parallel to tho edges of the solid, and perpen dicular to the planes of the faces, they will pass through the vertices of right pyramids placed on each face. By making the altitude of these pyramids, as pq, equal to half the projection of the parallel edges bf, &e., of the solid, we obtain the remaining angles, 1, nt ,n, o,p,r, of the solid termed a rhomboidal dodecahedron, one diagonal of each face of which is one edge of the original cube.