CHAP. XII, OF STRAIGHT LINES AND PLANES. (285.) THE relations and properties of geometrical figures which have been explained in the preceding chapters are those which belong to straight lines and circles, supposed to be described on the same plane surface. Thus, when several straight lines have been considered, or when straight lines have been viewed in relation to circles, or circles regarded in relation one to another, each line or circle has been understood to be so placed, that the plane surface on which it is formed is the same as that on which the other lines or circles under inquiry are also formed. It is, however, frequently necessary to investigate the geometrical relations of lines which do not lie in the same plane. It has been stated (5.), that one of the properties of a plane surface is, that any two points in it being united by a straight line, every part of that straight line, whether between the two points or beyond them, however far the line be continued, will lie in the plane surface. From this property another immediately flows. If two plane surfaces intersect each other, their line of intersection will be straight. For, let a straight line be imagined to be drawn from any one common point of the two planes to any other; every part of that straight line must, in virtue of the property just referred to, lie at the same time in both planes, and must, therefore, be their common line of intersection. That the intersection of two plane surfaces is a straight line, is a proposition which, when applied to any particular case, becomes so evident, that it can hardly be considered to require proof. The walls of a room being plane surfaces, the corners formed by their intersection are straight lines. In like manner, the lines formed by the junction of the floor with the walls are straight. The surfaces which form the sides of an obelisk are plane, and its corners are straight lines. On the other hand, if a plane surface intersect a curved surface, the line of intersection will be generally curved. Thus the surface forming the side of a bridge intersecting the curved surface forming the arch of the bridge, the line of intersection forming the corners of the arch is curved, and the species of curve depends on the form of the arch. The line of intersection of a plane and curved surface, however, may be straight. Thus, if a circular pillar be cut by a plane along its centre, the lines of intersection of the curved surface of the pillar, and the plane surface formed by the section, will be straight. Thus, the intersection of curved surfaces may be a straight line; but the intersection of plane surfaces must be a straight line. (286.) If a point P (fig. 141.) be assumed any where above a plane, there will be a certain point F upon the plane which is nearer to P than any other point on the plane. The line PF will in that case be perpendicular to every line, such as AFB, drawn through the point F upon the plane. For since PF is the shortest line which can be drawn from P to the plane, and since every point of the line AB is in the plane, B A fig. 141. F P A B PF must be the shortest line which can be drawn from the point P to the line AB, and must therefore be perpendicular to the line AB. (287.) A line, such as PF, which is thus perpendicular to every line that can be drawn in a plane through the point where it meets the plane, is said to be perpendicular to the plane itself. The point F, where the perpendicular meets the plane, is called the foot of the perpendicular. (288.) All lines, such as PA, PB, drawn to a plane from a point P above it, which are equally inclined to the perpendicular to the plane, are equal. For in the triangle PFA and PFB, the side PF is common, the angles at F, being right, are equal, and the angles at Pare supposed equal. Therefore by (61.) the triangles must be in all respects equal, and therefore PA must be equal to PB. (289.) It follows, also, that the points A, B, where such lines meet the plane, are equidistant from F, the foot of the perpendicular. In fact, if a straight line PA revolve round the perpendicular: PF, always making with it the same angle, the part of that straight line between the point P and the plane will continue of the same length, and it will, as it revolves, describe a circle, on the plane of which F will be the centre. (290.) The greater the angle is which the line PA makes with the perpendicular, the greater the line PA will be, and the greater also will be the distance of the point A where it meets the plane from F, the foot of the perpendicular. This may be shown in a manner similar to the proof of the analogous property in (24.). (291.) The perpendicular to a plane is called the axis of all circles described on that plane, round the foot of the perpendicular as a centre. When a circle revolves round its axis, the figure undergoes no real change of position, each point of the circumference taking successively the position deserted by another point. On this geometrical principle is founded the operation of millstones. Two circular stones are placed so as to have the same axis, to which their faces are perpendicular, being therefore parallel to each other and regulated in their distance according to the fineness of the flour intended to be ground. The inferior stone is fixed, while the superior stone is made to revolve by the power which drives the mill. The relative position of the circular faces of the millstones undergoes no real change during the revolution, and their distance being properly regulated, all the corn which passes between them will be ground with the same fineness. The advantage, and even the necessity of great precision in the construction of machinery is strikingly illustrated by the effects of any want of exactitude in the position of millstones. If the parallelism of the faces of the stones be not perfect, - if the axis of the moving stone be not truly at right angles to its circular face, the two grinding surfaces will not be at one uniform distance, and the relative position of the two stones, instead of being uniform, will constantly vary. The grain will be, therefore, differently affected by them, one part not being ground at all, or not sufficiently so, and another part being too much broken, and perhaps heated and spoiled. In the lathe, the axis round which the body to be turned is made to revolve, is the axis of the circles, which the cutting instrument forms by removing the matter which projects beyond the proper distance from that axis. The process of turning, therefore, consists in the formation of a surface, the cross section of every part of which is a circle, all the circles having the same axis. (292.) Since two perpendiculars to the same plane are both perpendicular to the same straight line, in that plane joining their feet, they must be parallel to each other (28.), and hence all perpendiculars to the same plane are parallel to each other. (293.) The plane which the surface of a liquid in a state of quiescence forms is an horizontal plane, and if indefinitely continued in all directions around, is called the plane of the horizon. If a weight be suspended by a flexible string so as to form a plumb line, such string, when the weight is at rest, will have a direction perpendicular to a horizontal plane. The line of direction of such a string is called a vertical line. Since all vertical lines are perpendicular to the same horizontal plane, they will be parallel to each other, provided the distance between them be not so great as to cause a sensible effect to be produced by the curvature of the earth's surface. (294.) When two planes intersect they may be more or less inclined one to the other. The angle which they form with one another, is the angle formed by two straight lines drawn from any one point in their line of intersection perpendicular to that line, one being drawn on the one plane, and the other on the other. Thus, if the straight line A B (fig. 142.) be the line formed by the intersection of the two planes, take P any point on that line, and draw PM in the one plane and PN in the other, both perpendicular to A B. The angle MPN is the angle formed by the planes. It is easy to shew that wherever the point P be taken upon the line A B, the angle MPN will be the same. From any other point P', let P' M' and P' N' be drawn also perpendicular to A B. The angle M' P' N' will be equal to MPN. For take PM equal to P'M', and PN equal to P' N', and draw MM', ÑN', MN, Μ' Ν ́. Since PM and P' M' are equal and parallel, M M' and PP' will be equal and parallel, and for a similar reason NN' and PP' are equal and parallel, and therefore N N' and M M' are equal and parallel, and therefore M N and M'N' are equal and parallel. The triangle MPN has therefore its three sides respectively equal to those |