with the point r, which is directly below r', and MP', ML are the two projecting lines by which r is represented at p on the horizontal plane, and at L directly below к in the verti cal plane. By the revolution of the vertical plane about EF, the point of projection L directly below x is brought upwards into the point p" on the horizontal plane.

As a farther elucidation of the general principle, let us consider the projections of straight lines.

Let ABCD be the horizontal plane, and EFGH the vertical plane at right angles to the former, and meeting it in their common intersection EF; and let rq be any straight line in space, From Panda any two points of the straight line pa imagine two straight lines PP and Qo' to be drawn at right angles to the hori. zontal plane ABCD and meeting it in P' and '; and from the same points P and a two other straight lines PP" and qq" to be drawn at right angles to the vertical plane EFGH, meeting it in the points P" and a". Draw r'K, q'L at right angles to the ground line EF; join P′′K, Q′′L; and we have as before the rect. angle PP'KP", of which the sides PP" and PP', or their equals P' and "K, are the ordinates of the point P: and in like manner ge" and qo', or their equals q'Lq"L, are the ordinates of the point of space q.

Suppose now a plane to pass through the line in space va, and either of the perpendiculars PP', and Qo'; and it is easy to perceive that it will pass through the other perpendicular, and meet the horizontal plane in the straight r'q' which joins the points r' and q'. It is also evident that all the perpendi.

culars let fall on the horizontal plane from the several points of the line in ra will meet the horizontal plane in the straight line P'q'; the straight line r'q' is therefore called the horizontal projection of the straight line PQ. From this construction it is plain that the projection of a straight line on a plane is a straight line on the plane passing through the projections on the same plane of any two points of the proposed straight line; or which amounts to the same thing, the projection of a straight line on a plane is the common intersection of this plane and another plane at right angles to the former, and passing through the straight line.

From this definition it is manifest that P"Q" is the projection of ra on the vertical plane; so that P'q' and P"Q" are the horizontal and vertical projections of the straight line ro.

Conceive now that after the projections of rQ are thus made, the vertical plane EFGH revolves about the common section EF from a vertical position till it coincide with the ho rizontal plane, the higher part of the vertical plane being supposed to fall behind the common section EF; the straight

lines which are at right angles to EF will fall in continuation of P'K and Q'L: so that the projections of pq will now obtain the positions r'q', P"Q" on the same plane; the ordinate P'K, KP" making one straight line, and Q'L, KQ' also making one straight line.

The various positions of the projections P'a', P"Q" will be fully exemplified in the subsequent problems; it is sufficient

to observe here that a straight line pa is said to be given in space when its projections P'Q and p" Q" are given; and a straight line pq is said to be found when its projections PQ', P"Q" are found. To which we may add that the two planes passing through ro and each of the projections r'Q' and "Q" are called the projecting planes of rq; of course a straight line will also be given in position, when we have the intersection of its projecting planes.

When a plane exists in space it is referred to the planes of projection by means of its two intersections with those two planes. Let ABCD and EгGH be the horizontal and vertical planes; and let KQ'RQ" be any other plane: this plane will in general cut both the planes of projection; the horizontal plane in the straight line Ko' and the vertical plane in the straight line Ko". These intersections Ka', KQ" are called the traces of the plane KQ RQ"; the former Ko' being the horizontal trace of the plane, and Ko", its vertical trace. When the vertical plane EFGH revolves about the ground line from a vertical to a horizontal position, the vertical trace Ko" will be in the horizontal plane, and the two traces will then be in the same horizontal plane, meeting each other in the point K in which the plane KO'RQ" cuts the ground line EF. A plane is said to be given in position when its horizontal and vertical traces are given.

The various positions of the traces of a plane according to the situation of the plane will be exhibited in some of the prob. lems in the following chapter.

CHAPTER II.

Containing Fundamental Problems.

PROBLEM I:

If a straight line be given by its projections, it is required to find its height above the horizontal plane at any point of its projection on that plane.

Let ABCD be the hori D zontal plane, and EFcb the vertical plane, which by revolution about their common intersection or ground line EF, is brought into a horizontal position. E Let r'o' and P"q" be the two projections of the given line, the former P'q on the horizontal plane,

the latter p"q" on the ver

tical plane; it is required A

Q"

C

B

to find the altitude of the given line above any point p' of the horizontal projection p'q'.

Draw P'k at right angles to the ground line EF, and produce P'K if necessary to meet r"q" the vertical projection in p': and KP" will be the height of the given straight line above the point p'.

Because 'o' is the horizontal projection of a straight line, therefore the point p' is the horizontal projection of some point P of that line; but the two projections of a point are always in the same straight line at right angles to the ground line, therefore the vertical projection of P is in P' KP"; and because p"Q" is the vertical projection of the given line, the vertical projection of r must be in "Q", therefore the verti cal projection of P is in the point p", which is the intersection of PKP" and "Q". Therefore r'K, P′′K are the horizontal and vertical ordinates of the point P, and KP" is equal to the height of the given line above the point P'.

If a' be any other point in the horizontal projection r'e'; draw as before o'LQ' at right angles to EF, and LQ" will be the height required. In this second case the point o" falls before

the ground line; and therefore, agreeably to the illustrations of the projections of a point given in the first chapter, the distance Lo" is a depression below the horizontal plane: that is, the point q, of which a' and q" are the horizontal and vertical projections, is directly below the point a' of the horizontal plane, its distance below a' being equal to LQ". Thus it ap pears that the point of the given line, of which and p' are

the projections, is above the horizontal plane and before the vertical plane, but that the point of this line of which q' and Q" are the projectors, is below the horizontal and behind the vertical plane.

Again, let p' a', and p" 'be the horizontal and vertical projections of a straight line, and AB the ground line. In this figure the point of the given line Q, of which a' and q" are the projections, is above the point q' of the horizontal plane, at a height equal to the ordinate L Q"; and the point

of the line r, of which r' and r" are the projections, is below the point p' of the horizontal plane, the depression of the point being equal to the ordinate KP': whence it appears that the point of the given line projected into q' and q" is above the horizontal and behind the vertical plane; and that the point projected into p' and p" is below the horizontal and be. fore the vertical plane.

If the projections, p' q' and "Q" intersect in ', the ordinate RM is common to both the horizontal and vertical projections.

It is evident that if the point P" of the vertical projection were given, we proceed as before to find the point P, and consequently the distance KP' of the point from the vertical plane.

PROBLEM II.

If the projecting planes of a given straight line be supposed to revolve about the projections of the straight line, till they coincide with the planes of projection, it is required to find the positions of the straight line on the horizontal and vertical planes.