63. The general pyramid of rays will therefore be made up of a series of others, having one common vertex, and for their several bases the perimeter of a portion of continuous surface.
64. If we imagine these pyransida of rays cut by a plane, the common section will obviously be an outline of the object as it would present itself to an eye placed at the vertex, each line and point of the section coinciding with the corresponding lino and point of the original.
65. As long as the object and the spectator's eye retain the same relative position, it is immaterial in what direction or at what distance the plane cuts the pyramids of rays ; for the lines and points produced at each position of the plane must necessarily coincide with the originals when viewed from the vertex ; although the outlines on the plane would vary for each of its positions. But each of these different ontlinea would suggest to the mind the same original com bination of forms, provided it. be viewed from the true vertex, and cannot be a correct representation or imago of the object, if viewed from any other point.' 66. When wo revert, to the connection between this subject and drawing, in the common acceptation of the word, we shall point out the precautions that must be taken by the draughtsman, when applying the principles of projection to the pictorial delineation of objects, to prevent his drawing from appearing distorted when viewed indifferently from other than the correct point, which it must inevitably be on most occasions. lint at present, dismissing all considerations of light, vision, and art, we shall proceed to treat this branch of the subject of projections, termed perspective, in a purely geometrical manner.
67. Instead of the simple elements alone, which entered into the constructions for determining lines and points, referred to a co-ordinate plane, by parallel lines perpendicular to that plane, we have in per spective projection the additional elements of the convergence of the projecting lines, or rays, intersecting the plane at different angles, depending conjointly on the distance of their point of convergence from the original lines, and from the plane. This variation in the
conditions necessitates a different course of proceeding : in the former kind of projection the object of our constructions was to determine magnitudes ; in that we are about to consider, our object is to delineate apparent and not real form.
68. The following definitions are here given to avoid unnecessary repetitions. The plane, on which the projection is supposed to be formed, and which is represented by the drawing board or paper on which the constructions are made, will always be termed the plant of the picture.
69. The point of convergence of the rays, or projecting lines, or the vertex of the pyramids of rays, will be designated as the vertex.
70. The centre of the picture is the point in which a line, through the vertex, perpendicular to the plane of the picture, meets that plane; and the length of this perpendicular, from the vertex to the centre of the picture, is the distance of the picture or vertex : this term will also he applied to the line itself when we have occasion to refer to it.
71. The vertical plane is one passing through the vertex, parallel to the plane of the picture.
72. Let x Y Z ...and o x E s, in the figure, be the plane of the picture and the vertical plane; v in the latter being the vertex. Let BB" be any straight line taken as an elementary original object : the rays from every point in ti is" will lie in one plane, the intersection, b' b", of which with the plane of the picture will be the indefinite perspective pro jection or image of n B" : the projecting plane (9) passing through any original line le and the vertex will also intersect the vertical plane in a line v n, parallel to b' b". v n is the director of the original line.
73. If the original line n a" were parallel to the plane of the picture, and therefore also to the vertical plane, its indefinite image and director would be parallel to the original line. But if n B" be not parallel to the plane of the picture and vertical plane, it must intersect them both.