of the triangle M'P'N'; and therefore (62.) the angle MPN will be equal to the angle M' P' N'. (295.) If a straight line be perpendicular to a plane, any plane drawn through that straight line will be also perpendicular to the plane. fig. 143. P C Let the line PF (fig. 143.) be perpendicular to the plane ABC, and let the plane A'B'C' be drawn through the line PF, then the plane A' B'C' will be perpendicular to the plane ABC for let FG be drawn in the plane ABC perpendicular to A'B', then, since PF is perpendicular to the plane ABC, the angle GFP will be a right angle (286.), and the lines GF and PF, being both perpendicular to A'B', the right angle GFPis the inclination of the two planes (294.), and the planes are therefore perpendicular. G A A' F B C If the plane A'B'C' be supposed to turn round on the line FP as an axis, it will be perpendicular to the plane ABC in all its positions. Thus, a door turning on its hinges is perpendicular to the floor and ceiling of the room in every position which it can assume. As it turns, it changes its inclination to the wall in which it is constructed, the angle of inclination being that which is formed by the edge of the top of the door, and of the corresponding edge of the top of the door-frame. (296.) If a vertical line be drawn from any point in an horizontal plane, all planes passing through that vertical line will be perpendicular to the horizontal planes. Such planes are called vertical planes. (297.) If two lines be drawn on a plane at right angles to each other, and a third line be drawn from the point where these two lines cross each other perpendicular to the plane, then these three right lines will be perpendicular to each other. If the straight lines XX and YY' (fig. 144.) be supposed to be drawn at right angles on a hori zontal plane, and Z Z' be drawn vertically, or perpendicular to that plane, through the point O, where X X'and YY' in- x' tersect, the angles formed by each pair of these lines at O will be right angles. fig. 144. The three planes through each pair of these lines will be also at right angles. Thus the horizontal plane through XX and Y Y' will be perpendicular to the vertical plane through Z Z' and X X', and also to the vertical plane through Z Z' and Y Y'. The two vertical planes through ZZ', XX', and through ZZ', Y Y', will also be at right angles. The right lines which intersect at O form round that point twelve right angles, four being formed on each of the three rectangular planes. Three rectangular planes, and no more, can therefore be always drawn through the same point. be parallel to B'A'. All perpendiculars drawn from C'D', or from its continuation in either direction, will be equal to the perpendiculars C'B' and D'A'; and they will be perpendicular to the plane ABC, and will therefore be the distances of the points in C'D', or its continuation from the plane A B С. (299.) A line is said to be parallel to a plane when all its points are thus equally distant from the plane. (300.) If a plane A'B'C'D' (fig. 146.) pass through a line, such as C'D', parallel to another plane ABCD, then the intersection A' B' of these two planes will be parallel to C'D', whatever be the angle the two planes make with each other. D fig. 146. C D For, since C'D' is parallel to the plane ABCD, it can never meet that plane, however it may be prolonged; and therefore cannot meet any line drawn in that plane. It cannot, therefore, meet the line A' B', formed by the intersection of the two planes. The two lines A' B' and C'D' can never, there A A' B B fore, meet; and since they are both in the same plane, they must be parallel. (301.) It may be here observed, that the conditions under which two straight lines are parallel are twofold: first, they must be both in the same plane; and, secondly, their directions must be such, that, however they may be prolonged in either direction, they can never meet. It is easy to conceive two lines differing very much in direction, and therefore not parallel, but which nevertheless can never meet, however they may be prolonged: thus, if from any point in a horizontal plane a vertical line be drawn, and from another point in the same plane, lying north of the former point, a line be drawn east and west; these two lines will evidently not be parallel, and yet however prolonged, they can never meet. (302.) Three points, however they may be placed, must always lie in the same plane. For if a straight line be drawn, uniting two of them, and a plane be drawn through that line, and be made to revolve upon it as an axis, it must, at some point of its revolution, pass through the third point; in that position therefore of the plane, the third point will be in it. (303.) If more than three points be considered, they may or may not be in the same plane, since the fourth may be above or below the plane through the other three. (304.) It is on this geometrical principle that stability in practice is more readily obtained by three supports than by a greater number. A three-legged stool must be steady if placed on a plane surface, since the ends of its legs, being in the same plane, will always accommodate themselves to the surface which supports it; but if the stool have four legs, the end of one of these may not be in the same plane with the ends of the other three, in which case it will be unstable, since the ends of the four legs cannot possibly at the same time rest on the surface which supports the stool. In well constructed furniture, the ends of the legs are formed in the same plane, and therefore four or more legs are used; but in rudely constructed stools and tables it is not unusual to form them with three legs, the inequality of length being then not a cause of instability. (305.) The use of three rectangular planes, such as those described in (297.), is very frequent in the arts, and especially in architecture, carpentering, and the other departments of art relating to buildings. The floor and walls of a room present an obvious example of a system of such planes; beams of wood, bricks, blocks of stone, and almost all the materials used in building, afford like examples. In architectural and mechanical drawing, it is usual to represent buildings and machines by views taken of them in the direction of three rectangular planes: the view taken in the horizontal plane, is called the ground plan, in addition to which a view is taken in two vertical planes at right angles to each other: if these views exhibit the exterior of the object, they are called elevations; if they show its interior, they are called sections. (306.) If three points be taken at equal distances above a plane, the plane which passes through these three points will be parallel to the former plane. Let the three points be A', B', C', taken at equal distances above the plane ABC (fig. 147.); and from them let three perpendiculars A A', B B', and C C', be fig. 147. B drawn to the plane, these meet the other plane, and therefore the planes themselves can never meet; for if they did, one or other of the three lines joining the three given points must meet the line of intersection of the planes, since all the three lines could not be parallel to that line, and therefore one of them would meet the other plane, contrary to what has been proved. (307.) If two planes be parallel one to the other, they will be every where equally distant from one another. For if any two points in the one be at unequal distances from the other, let perpendiculars be drawn from these points to the other plane. The line joining the tops of these perpendiculars in the one plane, will therefore not be parallel to the line joining their feet in the other plane; these two lines would therefore meet if continued, and therefore the planes in which they are drawn would meet, and could not be parallel; all points, |