31. These theorems on projections would be useless to the practical geometrician so long as the co-ordinate planes are supposed to retain their relative situation in space; to enable him to make the requisite constructions on the projections, and to determine the unknown mag nitudes entering Into the original solids by means of the projections of the known ones, he supposes the one co-ordinate plane turned round on the common intersection v z till the two planes coincide in one and the same plane by this supposition the relations to Y 2 of the lines, points, and traces, on the plane which is supposed to be turned round, remain unaltered; while the principles on which the projections are made the correct interpretation of the new relations which the projections of original points and lines on one plane assume with regard to the projections of the same points and lines on the other plane, when these two co-ordinate planes are supposed to coincide In one.
32. The same method of bringing two planes into one may be applied, or rather conceived to be applied, to the projecting plane of any original point or line, this projecting plane being supposed turned round on the projection of the line till the plane coincide with the co ordinate plane, that is to say, a construction can be made with the projection of a line founded on this supposition, by which a line may be found representing the original line as brought into the co-ordinate plane; and by an analogous construction, an original plane may be constructed as if turned round on its trace till It coincide with the co ordinate plaue.
33. This principle may be carried still farther : thus it construction can be made, founded on the supposition that an original plane has been turned round on its intersection with another euch plane till they coincide, and this compounded plane, If we may use such an expression, has been again turned round on its trace till it has been brought into the common plane of projection.
34. It must hence be understood that the practical application of the theory of projections is entirely synthetical, that is, the draughts man, first drawing a line to represent v z, proceeds from this simple assumption to draw the projections of oertain points and lines of a solid, on which he proposes to operate, from their known, assumed, or given relations to each other, and from their conventional relations to the supposed co-ordinate planes, which may in every case be conceived to be so situated as to facilitate these constructions. Having thus got the projections of known or given lines, he proceeds from these data to ascertain the absolute magnitudes of lines and angles depending on these given ones, by making the constructions alluded to, founded on the supposition of projecting lines and planes being turned round on the projections determined by them, till they coincide with the co ordinate planes.
35. If a plane be turned round on its intersection with another, a line in the former will make the same angle with that intersection, when the two planes are brought into one, that the line made with that intersection when the planes were in situ. The two lines which are the intersections of the projecting plane of a point (5) with the co-ordinate planes in situ, which lines have been 'shown to be equal respectively to the projecting lines (4) of that point, will be both perpendicular to v z, and therefore will coincide in one line perpen dicular to that line is, when the two co-ordinate planes coincide in one.
38. The two co-ordinate planes in situ form four dihedral angles, and an original point may he situated in any one of these ; that is to say, of a system of related original points referred to those planes, some one or more may be in different dihedral angles : it is essential that the learner should know how to assign the relative situation of the original points in space from the relative of their plans and elevations to T Z.
37. Let us distinguish the four dihedral angles time :— therefore will be a point in one trace of the plane sought : and since this trace must be perpendicular to the elevation of the line, Mn drawn through q perpendicular to a p will be that trace. The same con struction, applied, mutatis' mutandis, to the other projection of the point, will furnish a point in the horizontal trace of the plane sought, which trace must be drawn through R perpendicular toe P. The two traces thus found will intersect each other in a point of v z (22).
Given a plane a m, m n, and a point a,a : to draw a line through the point perpendicular to the plane, and to determine the point in which this line cuts the given plane.
40. Through a,'a draw lines perpendicular to the given traces 1,313 for the indefinite projections of the perpendicular sought (29) : from the point x, in which AN cuts Ts, draw N n perpendicular to Y Z for the intersection with the other co-ordinate plane of the plan-projecting plane of the perpendicular (9) ; and from a, in which A P cuts ML, draw a 1 perpendicular to v z : the point 1 is the elevation of the point in which the plan.projecting plane of the perpendicular cuts the trace sat, m / ; and is is that in which the same plane cuts the trace m n. Consequently is / is the elevation of the intersection of the same plan projecting plane with the original plane. Now it is obvious that the point sought must lie in this intersection ; consequently the point p, in which ap outs n 1, must be the elevation of the point in which the perpendicular intersects the given plane.