If a line be parallel to the horizontal plane, it is said to be horizontal; and to be vertical when it is perpendicular to the horizontal plane. But a line is not necessarily vertical when it is parallel to the vertical plane. Each projection of a line, or each trace of a plane, takes its name from the plane which contains it; thus the projections and traces situated in the vertical plane, are the vertical projections and traces. In consequence of the planes of projection being at right angles to each other, we deduce: 1°. If a point or a line be in one of the two planes, its projection upon the other plane will be a point in the ground line. 2°. If a line be parallel to one of the two planes of projection, its projection on the other plane will be parallel to the ground line. 3°. If a plane be perpendicular to one of the planes of projection, its trace upon the other plane will be perpendicular to the ground line. 4°. The perpendiculars to the ground line, from the projections of a point, are respectively equal to the distances of that point from the planes of projection; for it is manifest that ad'=ap, and ad=a'p. (See fig., p. 21.) 11. In order that all the constructions may be made in one plane ; the vertical plane xyu, is supposed to turn round the intersection xy, as if xy were a hinge, until it coincides with the plane xyu', which is xyu produced upon the other side of xy. The lines in the plane of xy: undergo no change ; but after the revolution round xy, с the projections a and a' have a connexion which must be carefully observed and remembered. Let a and a' be the original projections of a, before the motion of the plane xyu: we have seen that the perpendiculars from a' and a, upon xy, meet in the same point p. Now, while xyu revolves round xy, a'p is still to wy; and when xyu coincides with xyu, a"p, which is the position of a'p, is in the same plane with ap, and to the same line XY, and at the same point p in xy: :: ap and a"p must be in one and the same straight line; whence we see that the straight line joining the horizontal and certical projections of the same point A is perpendicular to the ground line xy; and that the part a'p gives the elevation of the point a above the horizontal plane, and that ap is the distance of it from the vertical plane. 12. We now proceed to give problems relative to the straight line and the plane. The drawing that contains the construction of the problem is called the Draught. The given quantities and the results of the problems are generally expressed by continuous lines, and the lines of construction by dotted lines. And that the explanations of the drawings may be readily understood, a uniformity in the notation is very necessary. We therefore indicate points in space by capital letters, A, B, C, &c., but these rarely appear in the drawings. The small letters abc, &c., belong to the projections in the horizontal plane; the accentuated letters a'b'c', &c., indicate projections in the vertical plane. The ground line is always designed by xy, and the Greek letters a, B, Y, &c., generally indicate points situated in that line. The following abbreviations are also used :-The point [a, a'], designates the original point a, which has a and a' for its projections; the line sab, a'b'], that line which has for its projections ab and a'b'; and the plane aaa', that of which the traces are a a and aa'. And when a point, a line, or a plane, is said to be known, it is meant that the projections of these are known. And conversely, when a point, a line, or a plane, is to be determined, it is sufficient to find the projections of the point, those of the line, or the traces of the plane. PROBLEM 1. Given the projections of a line to find its traces, or the points where the line meets the planes of projection. 13. The straight line in space is the intersection of its projecting planes ; and the point of intersection of the horizontal traces of these planes is evidently in both planes, and thus is a point of the line required : similarly, the intersection of the vertical traces of the two projecting planes is the vertical trace of the line. Let ab and a'B' be the two given projections, meeting the ground line in a and B A vertical plane through ab will have ab for its horizontal, and or', + xy at , for its vertical, trace. A plane through a'b' + to the vertical plane will be the other projecting plane of the line, and a'b' will be its vertical, and b'b + to xy in the horizontal plane its horizontal trace. The point b, where the horizontal traces of the projecting planes intersect, and d', where the vertical traces intersect, are two points of the line in space, and, being also in the co-ordinate planes, are the traces required. V Hence, we obtain this practical rule to find the vertical trace:-Produce the horizontal projection of the line to meet the ground line ; from o, the point of intersection, draw voo + xy, meeting the vertical projection a'b' in o': and is the vertical trace; and, to find the horizontal trace, draw from b', where a'b' meets xy, b'b + xy, to meet the horizontal projection ab in b: b is the horizontal trace. 14. Hence, conversely, given the traces of a line, its horizontal and vertical projections, may be found. For, given o' and b, we have merely to draw o'o and 66' + to the ground line xy: then join ob and v'b': these lines produced will be the projections required. 15. If we change the projections ab and a'b', the traces • and v can take a variety of different positions. In fig. 1, the horizontal trace is in front of the ground line, and the vertical trace above it. ข! In fig. 2, the horizontal trace is still before the ground line, but the vertical trace is in the lower part of the vertical plane. In fig. 3, the horizontal trace is behind the ground line, and the vertical trace above it. In fig. 4, the horizontal trace is behind the ground line, and the vertical trace is in the lower part of the vertical plane. PROBLEM 2. Given the traces of two planes, to find the projections of the common section of the planes. 16. Let m be the point of intersection of the two horizontal traces ap and bq of the planes, n' that of the vertical traces, a'p and b'q. 1 These points are common to the two planes, and the line which joins them in space is evidently the intersection of the planes. The point m being in the horizontal plane is the horizontal trace of the line; and if mm' be drawn to to xy, m' will be the vertical projection of m. In the same way, n' is the vertical trace of the line, and n the horizontal projection of n'. If, therefore, we draw the lines mn and m'n', these are the projections required. The following cases require attention: 1°. Let one of the planes have one of its traces, the horizontal trace, for instance, perpendicular to the ground line; this plane is perpendicular to the vertical plane, and its vertical trace is the vertical projection of its intersection with the other given plane. The rest of the construction is the same as above. 2°. Suppose two of the traces, viz., the horizontal traces to be parallel; the intersection of the two planes will be |