120. If A, n be two points in A 11, such that each is in the chord of the tangents from the other point produced, then, from the properties of the circle, A E, B 0 Will be equal respectively to the tangents A N, 13 L, drawn from those points; and the square on A B is equal to the Bum of the squares on A N, 13 L, or on a E, n E. E therefore lies in the circumference of a circle described on A B as it diameter. Since the angle A v n, made by the directors of A L, n x is a right angle by con struction ; the images of A L, n a' will be perpendicular to each other, and parallel, respectively, to those of the tangents A a, A as; B L, n K having the same station points with the chords K L, at N. Again, since A t is harmonically divided in K and e, and n in at 'and c, the image of K L will be bisected by that of o, and the image of at N will be also bisected by the image of e (87) : hence those images being diameters to the ellipse, mutually bisecting each other, and parallel reciprocally to the tangents which arc the images (S6) of A N, A as, n K, Br., the images of K L, at a must be conjugate diameters, and since those diameters are perpendicular to each other, they must be the axes.
121. If v', the foot of the director v v', coincided with n, or if v v' were in the auxiliary vanishing plane, the perpendicular to v E would be parallel to A n, and r q, s tt would be the originals of the axes, which accordingly would be parallel and perpendicular to the inter secting line. But in every other position of v r', with reference to the circle, these axes most be oblique to that intersecting line, while the angles they form with it will vary according to the distance of v' from D, and according to the length of the director v v'.
122. The points o and E will not be common to two or more con centric circles, the originals of the axes of the elliptic projections of concentric circles will not be in the same straight lines, nor will they have the same station points, except in the case of v' and n coinciding, when the originals of the axes will bo parallel and perpendicular to A n.
123. If A 13 touched or cut the original circle, the originals of the axes, ke., of the parabolic or hyperbolic projections might be found on the same principles : but as these curves do not often occur in practical perspective drawing, we shall not dwell on the subject.
124. The only solids with curved surfaces that need be considered are, the cylinder, the cone, and the sphere.
125. If a line be conceived to page through the vertex, parallel to the axle of a cylinder, whether right or oblique, two planes pa sh% through this parallel will touch the cylinder in two lines of its surface, also parallel to its axis, which will be the originals of the straight out line of the perspective projection, or image, of that cylinder.
126. These two tangential planes will cut the piano of the base of the solid, or that of any section of it whatsover, in two lines, which will be tangents to the curve of that section. And the parallel to the axis through the vertex is obviously the radial of that axis, which, by its intersection with the plane of the picture, will determine the vanishing point of that axis ; and this vanishing point, it must be remembered, is the image of the point, in any original plane, cutting the cylinder in which the two tangents to the curve of the section in that plane meet, which have been shown to be the originals of the outline of the solid.
127. If therefore the image of the base or of any section of the cylinder by a plane be obtained, lines drawn tangents to this image through the vanishing point of the axis will give the straight parts of the outline of the solid; these outlines must also be tangents to every other curve which is the image of any section of the original cylinder.
128. If a line pass through the vertex and the apex of a cone, and meet the plane of its base, or any other plane cutting the cone, two lines drawn through the point of intersection tangents to the curve of the section will be the intersections with that plane of two others passing through the vertex and tangential to the surface of the solid, and these two planes will touch the cone in lines which will be the originals of the outline of its image.
129. The ray just mentioned passing through the apex of a cone is analogous to the radial of a cylinder passing through the vertex, the cylinder being considered as a cone, with its apex infinitely distant.
130. If the line through the vertex and the apex of a cone, or the ray of that apex, be parallel to the plane of its base, or of any section, the tangents to the base lying in its plane, or in that of such section, must be drawn parallel to that ray, and the image of the apex will be the vanishing point of these parallel tangents.
131. Let C be the centre of the picture ; a b, bisected in e, being given as the image of a diameter, parallel to the plane of the picture, of a sphere c, therefore being the image of its centre (SS). Draw an indefinite line through c and e, and c v perpendicular to it, equal to the assumed distance of the picture ; take any point e at pleasure in e c, but as far from e as convenient ; draw a $ through f perpendicular to c e, making e a, f a, equal to e a, e b. Join v e and set off its length each way from e to 1 and at along a line perpendicular to c e.