144. If a spectator stand opposite two or more long horizontal parallel lines, as those of the façade of a long building, or of a garden wall, for example, he very palpably perceives the apparent convergence of these parallels in both directions, as they recede from him to the right and left ; on reflection, he Is therefore convinced that the apparent form of the really parallel straight lines are curves, produced by the varying angles under which the equal ordinates between the parallels are seen, as they become more and more distant from the eye.
145. The parallel projections of such long horizontal lines, which would result from the plane of the picture being assumed parallel to the originals, would reassume their natural apparent curvature, If viewed from the correct vertex; but if not, their parallelism would offend the eye as being at variance with daily experience, and still more would any attempt to draw on a plane the apparent curvature of the lines in question be reprehended as being contrary to the verdict of the judgment, which decides that the originals, being straight lines, ought not to be represented by curves.
146. The draughtsman, consequently, must never assume his plane of the picture parallel to the longest Bide of a building, fic., however much he may be tempted to do so from the facility of making his con structions under this condition, when the projections of such a side would subtend at the vertex an angle of more than 15* or 20°.
147. Keeping these conditions in view, the draughtsman may assume the distance of his picture, or its distance from the vertex, entirely according to his own convenience, since it is only the absolute magni tude of the image or projection which is altered by the different distances of the picture, the figure of the image being similar on all parallel planes, as long as the vertex and object remain the same. For the sake of facility of construction, be will generally assume his plane of the picture as coinciding with some principal vertical line of the object or model.
148. The shadow of any object is obviously the projection of it on a surface, by converging on parallel lines or rays, according as the lumi nary is supposed to be at a finite or at an infinite distance, as the sun may be considered to be as regards terrestrial objects. When, there fore, we have obtained the projection of an object by the principles just explained, they will also enable us to obtain the projection of its shadow on one or more planes or surfaces, as supposed to be cast by an artificial light or by the sun ; the problem being simply to deter mine the projection of the intersection of a pyramid or prism of rays passing from a given or assumed point through the points of a projected object.
149. If the object be perspectirdy projected, and the luminary be the sun, the vanishing point of the parallel rays, whose direction must be given or assumed, will represent the sun, since that vanishing point is the image of a point infinitely distant.
150. Although our power of forming correct conceptions of the true form of an object, as derived from a projection or pictorial representa tion of it, is much increased by the addition of light and shade, and of shadows of the object correctly projected by rules identical with those by which its outline was obtained, yet as soon as we thus approach the domain of a higher art, that of painting, the mathematical precision of the shadows we should obtain by our rules must yield to more important considerations connected with the art alluded to. Hence it is that the applies the geometrical principles for finding the true shadows of the engine, building, or analogous object, the outline of which he has delineated ; for at an early stage of his practice in drawing he ought to have acquired sufficient knowledge of art to be able to add to his outline the effect of light and shade without any gross violation of truth of nature, and with a better pictorial effect than he could ensure by geometrical rules. We shall consequently only give two simple examples relating to the projection of shadows, rather as affording additional illustrations of the prin ciples of projections, than for any practical utility as regards the specific subject of shadows.
151. Let the line o s, es, passing through the centre c, c, of a sphere, be given as the direction of the solar rays; it is proposed to determine the shadow of that sphere on the given plane a at n. It is obvious that the problem is to determine the section of the right cylindrical surface, formed by the system of parallel rays, which are tangential to the spherical surface, by the plane a se a ; and that the great circle of the sphere passing through the points in which these rays touch it will be the base of the cylinder, and the boundary between the illuminated hemisphere and that in shadow.