41. The plan P of the point may be obtained by drawing pp perpendicular to y z (35), to cut A N, the indefinite plan of the line ; or by applying the foregoing construction, mutatis mutandis, to the other projections.
To draw a line through a glren point P,p, to make any proposed angle with a given line A B, a b.
42. If the proposed line is to be parallel to the given one, lines drawn through the projections of the given point, parallel respectively to those of the given line, will be the projections of the line sought (23) ; but if the lines are not to be parallel, join P,p with any two points A, a, 8,6. taken at pleasure in the given line. AB P will therefore be the plan, and a by the elevation of the triangle thus formed. Find the traces, ma, an, of the plane of this triangle, by finding the points in which any two of its sides intersect the co-ordinate planes (19), since these points must lie in the traces required.* 33. Our limits will only admit of two or three examples of ele mentary constructions to illustrate the subject of projections, referring to the theorem on which each step of the construction is founded.
Given a point Pp in a given line A r, ap, to draw a plane through P,p perpendicular to the given linut 39. Draw a line r Q through r, perpendicular to the plan of tho line, for that of a line parallel to the co-ordinate plane, and lying in the plane sought ; then (20) p q, parallel to v z, will be the elevation of parallel. The linelp ct,pq meets the co-ordinate plane in Q,q (13) g 43. Draw m n perpendicular to Y Z, to cut the traces anywhere at pleasure in points at, n ; the line m m,'n is consequently the traces of a plane assumed at pleasure as perpendicular to both co-ordinate planes (22), and cutting the given plane as L is in a lino, the projections of which, of course, coincide with the traces of the plane. The length of this line, or the real distance between the points as n when in situ, is obtained by making m m' in Y Z equal to m at ; then the hypothenuse in' it is the intersection of the given plane with the assumed plane, brought into the plane of projection by the rotation on ra n of this assumed plane.
44. From M and a, as centres, with m' n, L n for radii respectively, describe area cutting in Join X n', a le' ' • the triangle it re euuwe quently (32) repreaenta that portion of the given plane st L II inter cepted between the co-ordinate planes and the line x in, In is, brought into the co-ordinate plane by being turned round on the trace X L, and by this rotation, bringing the original of the triangle, r • n, pa b, along with it. To draw this triangle as thus brought into the co-ordinate planes, produce rn, pb, to cut the two traces in n and e respectively, make a e' equal to Le, join De. In the same manner the lines a' p' a' are obtained, constituting the original triangle as brought into the co-ordinate plane in the manner described.
45. The points • a, B b, Pp, lying in the original plane, will describe arcs of circles during the rotation of that plane on its trace: the planes of these circles must obviously be perpendicular to the original plane and to the co-ordinate plane, and consequently cut the co-ordinate plane in lines or tracer perpendicular to u L, that of the original plane. llence if lines be drawn through A, u, and r, perpendicular to x L, they will pass through a', 6', and p , since the traces of these planes will be the projections of all lines lying in them, and therefore of the circular arcs alluded to (21), iu which the points a', b', and p' lie. By this means the points a', b', and p' may be found, or verified if previously obtained on any other principle.
48. Draw p' a' to make the proposed angle with a' b'• ; then the plan A, and elevation of the point a, in which the proposed line meets the given one in the given angle, may be determined from a' by the converse proceeding to that by which a', L', and p' were obtained. And lastly, e A, pa will be the projections of the line as required.
The foregoing construction might have been made with the trace an instead of L x; but the triangle, when brought into the co-ordinate plane ou the supposition of the rotation of the plane of that triangle on a, would not coincide with a' b' p'.