5. The projection of a straight line AB is a line; for the perpendiculars let fall from the different points of A B are all in the same plane. produced, will be the projection required. A plane drawn through AB to MN is called the projecting plane of the line AB. Hence to find the projecting plane of AB, we draw from any point a, a a, perpendicular to the plane MN; and suppose a plane to pass through the two lines AB and Aα, it will be the plane required. If AB be perpendicular to MN, its projection is the point where AB or AB produced meets MN. 6. A point is determined in space, when its projections, upon two planes which intersect, are given. For the point should be found upon each of the perpendiculars drawn to the planes from the given projections; and these perpendiculars can only intersect in one point, which is the point required. The two planes to which reference is here made, and which are generally at right angles to each other, are called the planes of projection, or co-ordinate planes. The following proposition will determine when two points in the co-ordinate planes are the projections of the same point. 7. When two points, situated in two co-ordinate planes, are the projections of the same point in space, the per pendiculars drawn from these points upon the line of intersection of the planes, must meet at the same point in that line. Let xyz, xyu, be the two co-ordinate planes; xy the line of intersection. From any point A, draw the u P a x perpendiculars, Aa, Aa': and through a a, sa', draw the plane Aa'pa, cutting xy in p, and the two planes in ap and a'p; then the line ay shall be to the plane Aapa' and to ap and a'p; and thus thes, from the projections a and a' of the point A, meet in the same point. Next, the points a and a' are always the projection of the same point in space. The line ay being to ap and a'p, is also to the plane apa'; and the planes xyz, xyu, are to plane apa'; . conversely apa' is to xyz, and xyu. .. if from a and a perpendiculars be drawn to the planes xyz, xyu, these lines will be in the plane apa'; whence it is evident that they ought to intersect in the point a, of which the projections are a and a'. 8. A line is known, when its projections upon two planes which intersect are given. Let ab, a'b', be the projections of a line; through ab conceive a plane to xyz, and through a'b', a plane to xyu; the intersection AB of these two planes is the line of which the projections are ab and a'b'. If the two lines ab, a'b' are to xy, but intersect it in different points m, n, they cannot be the projections of the same line, since they are obviously in parallel planes, pma and qna'. But if the projections ab, a'b', be both to xy, and meet xy in the same point m, then the plane ama' of these lines is both to xyz, and xyu, so that every line situated in ama', will have the same projections ab and a'b'. In this case we must intro duce a third plane tas, which B m a does not pass through xy, and the line may be determined from its projection a"b" upon tas. 9. The projections of any curve upon two intersecting planes will determine the curve. For abc, a'b'c', being the projections of ABC upon xyz and ayu, suppose perpendicu W lars to these planes to be drawn from the different points of a bc, a'b'c'; these will form two cylindrical surfaces, acca, a'c'ca, both of which will contain the curve ABC, and which will de termine that curve by their intersection. If the cylindrical surfaces do not intersect, the projec tions do not belong to the same curve. If ABC be a plane curve, and its plane beto xy, the two projections are merely the intersections of the plane through ABC, and the curve remains undetermined. DEF. The intersections of a plane with the co-ordinate planes are called the traces of the plane. 10. When the traces of a plane are given, the plane itself is given. We have already seen (Proposition II. of the Introduction), that two lines which intersect determine a plane; we might, indeed, make use of the projections of three of its points upon the two co-ordinate planes, but it is more convenient to use the traces. In general, the plane meets the line of intersection of the two co-ordinate planes, and it is manifest that the point of intersection a, must belong to the two traces a a, a'a. If the plane be parallel to xy, its traces are also parallel a through xy, its two traces coincide with xy, and a third plane of projection must be introduced to determine its position. Hitherto we have made no hypothesis with regard to the angle between the two planes of projection. But to render the constructions the more simple, we shall hereafter consider the two planes to be at right angles to each other, and that one of them, xyz, is horizontal, and the other, xyu, vertical. Their intersection, xy, is called the ground 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: 1o. 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. 2o. 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 Aaap, and ▲a=a'p. (See fig., p. 21.) 11. In order that all the constructions may be made in one plane; the vertical plane ayu, is supposed to turn round the intersection xy, as if xy were a hinge, until it coincides with the plane xyu', which is ayu produced upon the other side of xy. The lines in the plane of xyz undergo no change; but after the revolution round xy, Q |