plane. Now make this third plane turn round aa till it coincide with the horizontal plane; the point a will not be changed, a a' will coincide with xy, and the point a will coincide with a" if aa"-aa', consequently, the line joining a and a' in space, will coincide with aa".

In the same manner the trace bb" of the other plane upon the third plane may be found; the point d, where it cuts aa", is common to the two planes, and we have merely to determine the projections of this point upon the primitive planes.

Now, if dm be drawn

to a a, its horizontal projection will be m and if dď be drawn to xy, and we take in aa', am dd', its vertical projection will be m'. Join

=

pm and pm: these will be the projections of the intersection required.

4°. Let the planes be parallel to the ground line, then their traces and their intersection are also parallel to that line, and the general construction does not apply. We must in this case also use a third plane, as we have done above; but the mode of proceeding will be precisely the same as that above described, and will therefore be sufficiently intelligible to the learner without further description.

PROBLEM 3.

To find the intersection of a line with a plane.

17. Let a plane pass through the given line and cut the given plane: the point where the given line and the line of the planes' intersection meet will be the point required.

1o. Let the plane passing through the given line be vertical, that is to the horizontrl plane.

Let aa, a'a, be the traces of the given plane; beb'c' the projections of the line. Produce bc to m, draw mm' to

xy, meeting aa' in m'; then bm and mm' will be the hori

B

\n'

m'a'

zontal and vertical traces of the plane through the given line; let bm meet a a in d: then dand m' are the intersection of the traces of the two planes; and if dď be drawn to xy, and m'd' be joined, m'd' will be the vertical projection of the intersection of the two planes. The vertical projection of the required point should be on m'd; but it must also be upon the vertical projection b'c' of the given line; it is .. at o', the point of their intersection.

The horizontal projection o may be found by drawing o'o to xy, and meeting bc in o.

The point o may be found as we have determined o', by supposing the third plane to be to the vertical plane. Its traces are n ́n and n'b', and they cut those of the given plane in n and e; then draw e'exy, join en, it must meet bc in the point o.

2o. Let the plane passing through the given line have any position whatever.

It is obvious that its traces will pass through those of the given line: first, therefore, determine the traces, n and m' of the given line, bc, b'c'; then through those traces, and any point ẞ in ay, draw the lines Bp and Bp'. The given line is in the plane pẞp', since two points of it are in that plane.

Next, to determine the intersection of the two planes. From d and e', where the traces of the planes intersect, draw dd and e'e to xy, draw the lines de and d'e, these are the projections of the intersection of the two planes.

projection, for instance, the horizontal plane,-its horizontal projection will be a single point o, and its vertical projection o'o" is to xy.

The plane cdď passing through it is vertical, and its

trace dď to xy; but its horizontal trace has only one condition, namely, that of passing through the point o.

The other part of the con

d.

a

struction is the same as in the general case, the vertical and horizontal projections of the required point being o' and o.

The preceding construction will enable us to solve the following problem :-Given one of the projections of a point situated in a given plane, to find the other projection.

PROBLEM 4.

To find the points of intersection of three given planes.

18. The planes, taken two and two, give three lines, which pass through the required point. We therefore by Problem 2 find the projections of these lines; and, when the constructions are correctly made, the three horizontal projections intersect in the same point, which is the hori

zontal projection of the required point; the three vertical projections also intersect in the same point, which is the

vertical projection of the same point; and, lastly, the line that passes throught hese two projections ought to be to the ground line. In the figure, the horizontal projections of the intersections of the three planes, taken two and two, are ad, bf, ce; the vertical projections are a'd', b'f', c'é′; and the required point has for its projection o and oʻ.

PROBLEM 5.

Given the projections of two points, to find the projections of the line which passes through them, and the distance between them.

19. Let a a and b b' be the projections of the points a and B. Draw the lines ab a'b'; these, it is evident, will be the projections of the straight line AB.

Next, to find its magnitude. We know that aa' and bbare to xy, also that ra' and sb' are equal to the altitudes of A and в above the horizontal plane; if we conceive two vertical lines to be drawn, one from a and the other from b, and respectively equal to ra' and sb', their extremities will be the points A and B, and the distance between them is the distance required.

Since the two vertical lines are in the same plane with ab, conceive through в a line to be drawn parallel to ab, and terminated at the other vertical, a right-angled triangle will be formed, of which the base is ab, and the altitude = the difference between the verticals, and the hypothenuse is the distance required, AB.

To construct this triangle, through

A

m

b' draw I'm parallel to xy, meeting ra'in m, take mlab, draw a'l'; then a'l' is the distance required.

PROBLEM 6.

Given the projections of a straight line and a point, to find those of a line passing through the point, and parallel to the given line.

20. Let ab a'b' be the projections of the line, cc' of the point; through c draw cd parallel

to a'b'; through c' draw c'd' parallel to a'b': then cd and c'd' are the projections required.

For when two lines are parallel, their projecting planes are also parallel, and

the intersections of these planes

a

with a third plane are also parallel; i. e., the projections of two parallel lines are also parallel.

PROBLEM 7.

Given the traces of a plane, and also one of the projections of a line in the plane, to find the other projection.

21. Suppose the horizontal projection of the line to be given; through it suppose a vertical plane to pass, its inter