section with the given plane will be the line whose vertical

Through a given point, to draw a plane parallel to a given plane.

22. Let a, a' be the projections of the given point; ßb Bb', the traces of the given plane. Then, since the intersections of two parallel planes with a third plane are also parallel, therefore the traces of the required plane must be parallel to those of the given plane.

Through the given point a, a', and in the required plane, suppose a line to be drawn parallel to the horizontal trace of the plane; it will be parallel to ẞb; its horizontal projection ad must pass through a, and be parallel to ßb;

its vertical projection must pass through a', and be parallel

to xy.

Let d' be the vertical trace of this line, it is .. a point in the vertical trace of the required plane; draw :. d'c' parallel to ẞb', then d'c' is the vertical trace of the required plane: produce c'd' to meet xy in y, draw yc parallel to Bb, it will be the horizontal trace of the required line.

To verify this construction, suppose through the given point a line to be drawn parallel to Bb', of which the projections are ae and a'e'; the point e, where it meets the horizontal plane, must be on the horizontal trace of the required plane.

PROBLEM 9.

To draw a plane through three given points.

23. Since the points are given, their projections are also given; let these be a, a, b, b', c and c'. Join therefore the points, taken two and two, by the lines ab, a'b', bc, b'c',

ac, a'c'; which, having each of them two points in the required plane, will be wholly contained in that plane. Find, therefore, the three points of intersection of these lines with the planes of projection, and we shall have the

two traces of the required plane; the three points of each trace must be in the same straight line, and the two traces must intersect the ground line in the same point.

Thus let the line ab, a'b' meet the planes of projection in h, and ~; the line ac, a'c', in h, and,; the line bc, b'c', in hand, consequently the horizontal trace of the required plane will pass through the points h, h1, h3, and the vertical through v, v,, v,, and these two traces shall cut xy in the same point t.

COR. If the line joining two of the points be parallel to one of the planes of projection, as, for instance, the vertical plane, the preceding construction could not be executed. But, in that case, the vertical trace of the required plane is parallel to that line; or, what is the same thing, to its vertical projection.

PROBLEM 10.

To draw a plane through a given point and through a given

line.

24. Let a and a' be the projections of the given point; be and b'c' those of the given line; ad and a'd' those of a

line through the point and parallel to the given line, and therefore situated in the required plane.

Let cand e be the horizontal, and b' and d' the vertical, traces of these lines; .. the lines ce, b'd', are the traces of the required plane, and produced must meet xy in the same point.

PROBLEM 11.

Todraw a plane through a given line parallel to a given line.

25. In the first line ab, a'b', take any point, f, f", and through it draw a line parallel to the second line cd, c'd'.

The required plane must contain this line, and the line ab, a'b'; and the traces of these lines will determine the traces of the required plane. (Prob. 9, 10.)

PROBLEM 12.

Through a given point to draw a plane parallel to two given lines.

26. Through the given point draw lines parallel to the given lines; the intersections of these lines with the planes of

projection, will give two points of each trace of the required plane.

PROBLEM 13.

To draw a line through a given point which shall meet two

given lines.

27. First, find the traces of a plane passing through the point, and through one of the lines; then the traces of a second plane, also passing through the point and the other line; the line required, which ought to be in both planes at the same time, is therefore their common intersection. Thus, let a, a' be the projections of the given point

bc, b'c', and de, d'e', those of the lines; let fa, ag', be the traces of the first plane; fß, Bg, those of the second plane : then gf, g'f', will be the projections of their intersection, or of the line required.

To verify this construction, we must have, 1°. That this intersection should pass through the given point. 2°. It should meet the first given line. 3°. It ought also to meet the second line.