138. By previously constructing the projection of a cube in the manner just described, the sides of which will furnish a scale of the ratio of the projections of any lines parallel to the edges of that cube, the projection of any parallelopiped may be obtained, and from this again the image of any symmetrical solid deduced. In this manner the forms of crystals can be drawn with the most perfect accuracy, and a most distinct conception obtained of them and of the relative position of their planes. And by analogous constructions diagrams of the theorems of solid geometry may be drawn, which would greatly facili tate the study of analytical geometry.
139. It has been stated that perspective projection Is principally employed to furnish a pictorial outline of a building, machine, &o., or to convey an Idea of en object of that description to the spectator, but to do this the perspective outline must excite in his mind the ideas of the real forms of that object in their relative situations, such as would be excited by the object itself, when viewed from a given point. But there are limitations to the apparent forms of objects, arising from the structure of the eye and the laws of vision, which the draughtsman must never lase sight of, when he practically applies the purely geo metrical principles we have deduced, or otherwise he may produce an accurate projection of an object which would be perfectly unintelligible to an ordinary spectator ; as the outline of the sphere, deduced In the preceding example, would be to an uninitiated eye.
140. Since the eye can only embrace at one time a very limited field of view, in order to see the whole of an object without changing the place of the eye, the spectator must not be nearer to it than a certain distance, for otherwise he would have to turn his head to see the successive parts, and at each such change of position the apparent forms of those parts just escaping from his view would undergo a con siderable modification, arising from the structure of the eye itself. Few persons are aware of these modifications, owing to the effects of habit and the result of the judgment, which induce us unconsciously to assign the real and constant forma we know the parte of the object to possess to the apparent forms under which those parts are seen. Indeed It requires a considerable degree of abstraction and education of the eye to make the mind cognisant of the fact, that it is never the real form of an object that presents Itself, a truth familiar to' artists, who know that persons when first attempting to draw an object before them by eye, invariably draw it as they know it to be, and not as they really see It.
141. We have stated that the perspective projection of an object is rarely viewed from the precise point from which alone it ought to be viewed, so that the forms in the projection may suggest the ideas of the original forms whence they were deduced ; consequently the out line should not in any part deviate greatly from what we may call the average form under which the true one would present itself to the eye.
To effect this accordance the draughtsman must assume his point of view, or vertex, at such a proportional distance from the object itself, or from the imaginary model of it, that the rays from the points of it farthest apart, may not contain an angle greater than 60° at most, and, if circumstances allow of it, of not more than In short the pyramid of rays from an object to the vertex should be included within a cone the angle at the apex of which is not greater than that above named.
142. The distance of the vertex from the object being determined from these considerations, and its position with respect to the various parts of the original object decided on, by the conditions of the kind of view of that object it is proposed to delineate, the position of the plane of the picture should, generally speaking, be perpendicular to the axis of the cone or pyramid of rays before alluded to ; but the fol lowing principles must determine more accurately its situation.
143. From the frequency of their occurrence under circumstances favourable for the observation, the eye is accustomed to the apparent convergence of long horizontal parallel lines, as in streets, aisles of cathedrals, long avenues of trees, or walls, 8:e., but perpendicular parallel lines are rarely if ever long enough to cause this optical effect. Now we have proved that the projections of parallel lines never can be parallel unless the originals are parallel to the plane of the picture; if therefore the draughtsman were to assume that plane not parallel to the vertical lines of a building, &c., the convergeuce of the projections of these lines would offend tho eye of a person looking at his drawing, as being at variance not only with his judgment of the real parallelism of the lines in question, but even with his daily uncultured observa tion. But there is another optical phenomenon regarding the appear ance of long parallel lines, which we must briefly allude to, because it throws considerable light on the distinction between the apparent forms of objects as seen by the eye, which forma are functions of the angles solely under which the original forms are seen, and the figures on a plane, resulting from the section by that plane of the pyramids of rays from those original forms, which sectional forms are functions of the arcs subtending those angles.