94. It follows, therefore, that the line In which two original planes cut one another will have for its vanishing point that in which the two vanishing lines of the original planes cut each other, and that the intersecting point of the common intersection of two original planes will be that in which the intersecting lines of those planes cut each other.
95. Every vanishing plane le supposed to have an auxiliary one, per pendioular both to it and to the plane of the picture, and therefore passing through the distance of the picture : this auxiliary vanishing plane will cut the plane of the picture In an auxiliary vanishing line perpendicular to the principal one, and passing through its centre, and also perpendicular to the intersecting lines of the original planes. The line in which the auxiliary vanishing plane cuts the principal vanishing plane is termed the principal radial of the original plane, or planes, to which the vanishing plane pertains. This principal radial is obviously perpendicular to the principal vanishing line, and meets it in its centre, which will consequently be the vanishing point of all linen in the original planes perpendicular to their intersecting lines.
96. The principal radial will form, with the distance of the picture and with the auxiliary vanishing line, angles equal respectively to the complement of the angle, and to the angle itself, which the original planes make with the plane of the picture.
97. The auxiliary radial of any vanishing plane is one lying in the auxiliary vanishing plane, and perpendicular to the principal radial this auxiliary radial is that of all lines perpendicular to the original planes, the common vanishing point of which is the point in which the auxiliary radial meets the auxiliary vanishing line. This auxiliary vanishing point is the image, or projection of the points in which the auxiliary radial intersects all the original planes to which the principal vanishing plane is common.
98. The auxiliary vanishing plane, being perpendicular to the original planes, as well as to their vainishing plane, and to the plane of the picture, will Intersect those original planes in lines perpendicular to their intersecting lines, and parallel to their principal radial.
99. It follows from these definitions, that the vanishing lines of all planes perpendicular to one or more parallel original planes will pass through the auxiliary vanishing point of those planes.
100. If the original plane, or parallel planes, be perpendicular to the plane of the picture, their principal radial will coincide with the distance of the picture. Their auxiliary radial will be parallel to tho plane of the picture ; and the vanishing lines of all planes, perpen dicular to the original planes, will be parallel to each other, and perpendicular to the vanishing line of the original planes.
101. If a circle he supposed, described in an original plane on the point, as a centre, in which the auxiliary radial cute that plane, all lines touching that circle will be intersections with that original plane, of vanishing planes of other original planes, inclined to the first in a certain angle. The point in which any one of these tangents to the circle cuts the intersecting line of the first original plane will there fore be a point in the vanishing line, to be determined by each such vanishing plane respectively, while the vanishing Point of each such intersection in the original plane will be a second point in the vanishing line of the plane inclined to the former. Therefore this vanishing line will be determined, if these two points be found.
102. For if a right conical surface be imagined formed round the auxiliary radial as an axis, having the vertex for its apex, and its side forming with that axis the complement to the angle the secondary original planes form with the first ; the vanishing planes of these secondary original ones will touch that conical surface in a line, which will be the principal radial of each such vanishing plane; and there fore each vanishing plane will cut the first original plane in a line tangential to the circular section of the conical surface by that original plane. Again, the point in which these tangents to the circular section cut the intersecting line of the first original plane, will be common to the intersections with the plane of the picture of each vanishing plane respectively, or to the vanishing line as above stated.