example, B A) take any convenient point, A, from which let fall A D perpendicular to X X, and on B D as a diameter describe a circle. From D let fall perpendiculars, D e, D F, on the two given

traces. From the point e, thus found on the opposite trace to that on which the point A was assumed, let fall e E perpendicular to X X; join E A, cutting DF in G. From G draw G H perpendicular to XX, cutting the circle in H; DBH will be the required angle.

26. Given (in fig. 7),

the Traces of a Plane, BA, BC, to Draw the

Traces of another Plane which shall be Parallel to the given Plane, and at a given Perpendicular Distance from it in either

Direction. Complete the construction described in Article 25. Join D H

(this represents the perpendicular distance of the point D in the axis from the given plane); then from H, along HD (or along D H produced, according to the direction in which the new plane is to lie), lay off the given perpendicular distance between the planes, H K. From K draw K M parallel to H B, cutting X X in M. From M draw M N parallel to B C, and ML parallel to B A; these will be the traces of the plane required.

Or otherwise:-Complete the construction described in Article 24 (see fig. 8). Af is the rabatment of the intersection of the given plane with a plane, A D e, perpendicular to the vertical trace B C. Through A draw A M perpendicular to Af, and make A M equal to the given distance between the planes; draw M N parallel to Af, cutting X X in N In De produced take D O equal to D N. O is a point in the trace of the plane required. Through O draw O P parallel to B C, cutting X X in P; and through P draw P Q parallel to B A. OPQ is the plane required.

27. Given (in fig. 9), the Traces of Two Planes, CA d and C Bd, to Draw the Projections of their Line of Intersection.-The traces of

the required line are C and d, where the traces of the given planes intersect. From those points respectively let fall C c and

d D perpendicular to X X; join C D, cd; these will be the projections required.

28. To Find the Projections of the Point where a Straight Line Intersects a Plane (the traces of the line and of the plane being given), it is only necessary to draw the traces of two planes traversing the given line in convenient directions, and find the projections of the lines in which those two planes cut the given plane; the intersections of those projections will be the projections of the point required.

Fig. 9.

B_X

D

29. Given (in fig. 10), the Traces of Two Planes, CA d, C B d,

to Find the Angle between them. From either of the intersections of the traces (say d) let fall d D perpendicular to X X; draw DC, joining D with the other intersection of the traces. Through any convenient point, I, in D C, draw GIH perpendicular to DC,

cutting A C in G and BC in H. Along X X lay off D E = DC, and Di DI; join dE (this will be the length of the line of intersection of the planes). From i let fall ik perpendicular to d E; in IC take I Kik; join K G, KH; GK H will be the angle required.

When the traces of the two given planes are inconveniently placed for the completion of the figure, we may substitute for either pair of traces another pair of traces parallel to them, and more conveniently placed.

30. Given (in fig. 10), the Traces, A d and A C, of a Plane; also the Traces, d and C, of a Straight Line in that Plane; to Draw the Traces of a Plane which shall Cut the given Plane in that Line at a given Angle.-From either of the traces of the straight line, as d, let fall d D perpendicular to X X; draw the straight line D C, joining D with the other trace, C, of the straight line. Through any convenient point, I, in D C, draw I G perpendicular to D C, cutting C A in G. In X X lay off DE DC and Di = DI; join d E, and on it let fall the perpendicular i k. In IC take I Kik; join K G. Then draw K H, making G K H = the given angle, and cutting G I, produced if necessary, in H. Draw CH, cutting X X in B, and join B d; these will be the traces of the plane required.

31. Given (in fig. 11), the Traces of a Plane, A B C, and the Projections of a Point, G, g, to Draw the Traces of a Plane Traversing the given Point, and Parallel to the given Planc.-Through either of the projections of the given point (say G) draw GH parallel to the corresponding trace of the given plane, and cutting X X in H. (This will be one of the projections of a line through the given point, parallel to the trace A B of the given plane.)

Through H draw HD perpendicular to X X; and through g draw g D parallel to X X, cutting HD in D (g D will be the projection and D one of the traces of the line before mentioned). Through D draw DE parallel to C B, cutting X X in E; and through E draw E F parallel to B A; D E F will be the traces of the required plane.

32. Given, the Traces of a Plane, E F, ED (in fig. 11), and One Projection of a Point in that Plaue, to Find the other Projection of that Point. Suppose g, the vertical projection of the point, to be given. Draw g D parallel to X X, cutting E D in D. From D let fall D H perpendicular to X X. From g draw g G perpendicular to X X, and from H draw H G parallel to E F; the intersection of those lines, G, will be the required horizontal projection of the given point.

33. Given (in fig. 12), the Traces, A B C, of a Plane, and the Projections, D, d, of a Point, to Draw the Projections of a Perpendicular let Fall from the Point on the Plane. From one of the projections of the given point (say D) draw D E F perpendicular to the corresponding trace, B A, of the given plane, and cutting BA in E, and X X in F. From E let fall Ee perpendicular to

X X; from F draw F

perpendicular to X X, cutting the trace BC in f; join fe; from d draw d g perpendicular to B C, cutting

X

A

fe in g; and from g draw g G perpendicular to X X, cutting DF in G. D G and dg will be the projections of the perpendicular required.

34. Given (in fig. 13), the Pro jections of a Point, D, d, and those of a Straight Line, A B, Bab, to Draw the Traces of a Plane which shail Traverse the Point, and be Perpendicular to the Line.-Through one of the projections of the given point (say D) draw D G perpendicular to A B (the corresponding projection of the given line), cutting X X in G. Through G draw Gg perpendicular to X X; through d, the other projection X X, cutting Gg in g; through g cutting X X in C; and through C draw CF perpendicular ECF will be the traces of the required plane.

Fig. 12. of the point, draw dg parallel to draw E C perpendicular to a b,

C

and finally, by Article 27, find the intersection of those two planes.

to A B.

35. Given, the Projections of a Point and of a Straight Line, to Draw the Projections of a Perpendicular let Fall from the Point upon the Straight Line.Find by the preceding rule the traces of a plane traversing the given point, and perpendicular to the given line; then, by Article 23, find the traces of a plane traversing the given point and line; projection of the line of

36. Given, the Projections of Two Straight Lines that are neither