" Itepee it follows, that if the proposed right line be per pendicular to the plane of projection, its projection will he reduced to a single point, which will be that in N1 hick it falls upon the ],lane.
Figure 2.—The projections a b, a' 11, of the same indefi nite right line, A B, being given upon two planes, not parallel, L M N L m B Q, this right line is determined ; for, if from one of the projections, a 1), we imagine a plane perpendicular to L tsi N the position of this plane being known, it would necessarily pass through the right line A a. The position of this right line, which is found at once upon both the known planes, consequently at their mutual intersection, is therefoi•e absolutely determined.
What we have just stated is to be understood as inde pendent of the planes of projection, and takes place, whatever may be the angle formed In the two planes. But it' the angle horned by the two planes of projection be very obtuse, that formed by their perpendiculars will be very acute : very trifling mistakes in this respect, will, in practice, lead to very grave errors in determining the position of the right line. lo obviate this cause of inaccuracy, at least in the absence of better means, it is usual to have the planes of projection perpendicular to each other : besides which, as most artists, who use projections, are finniliar with the position of a hori zontal plane and the direction of a plumbline, they gene rally represent one plane of projection as horizontal, and the other vertical.
" The necessity of representing, in draw ings, the two pro jections upon the same sheet, and in larger operations upon the same area, has f;irther determined artists to represent the vertical plane as turning and folding down, as upon a hinge, at its intersection with the horizontal plane, so as that the two may form but one plane, upon which they construct their projections.
"The vertical projection is thus, in filet, traced upon a horizontal plane, and it must ever be kept ill mind, that it must be corrected and put in its place, by turning it one fourth of a revolution round its intersection with the hori zontal plane : to do w hich accurately, care must be taken to trace this intersection very plainly upon the design.
" Thus, in Figure 2, the projection a' b' of the right line A B. could not be drawn upon it real vertical plane ; but if we colic:eke the plane to be turned about the right line L m, so as to bring it in contact at L 1,1 we shall readily execute the vertical projection 0' //.
Besides the I:le:Hides of execution presented by this dis position, it possesses the additional advantage of abridging the labour of projections. Thus, suppose the points a, a', to he the horizontal and vertical projections of the point A, the plane indicated by the right lines A a, A a', will be perpen dicular to the two planes of projection at the same time, because it passes along the right lines perpendicular to them : consequently, it will be perpendicular to their common inter section a m ; and the right lines a c, a' c, according to which they cut these two planes, will be themselves perpendicular to L B.
" Now, if the vertical plane be turned about r N qs upon a hinge, the right line a' c, still continues perpendicular to ; and the case is still the same, when the vortical plane, laid down, assumes the position c a". Thus the two right lines a c, c a". passing both by the point c, and being both perpendicular to L m, are in prolongation to each other ; the ease is similar with the right lines h a, D as to every other point, as B. Hence it follows, that when we hai, e obtained the horizontal projection of a point, the projection of this same point upon the vertical plane, supposed to be laid down, will be in the right line drawn along by the horizontal pro jection perpendicularly to the intersection, 1. m, of the two planes of projection, and this reciprocally.
"This result very frequently Occurs in practice.
"'We have hitherto considered the right line, A a, Figure 2, as indefinite ; in which case we should only have to do with its direction ; hut we must now consider it as terminated by the points A a, which will bring us to take its extent into our calculation. We shall, therefore, examine how this may be deduced from a knowledge of its two [injections.