515 PERSPECTIVE 61. Def.-Two figures are in perspective if the lines joining their corresponding points meet in a common point, the centre of perspective. If the figures are in the same plane, and if, when the lines of the figures are indefinitely produced, the lines joining the corresponding points of intersection meet in a common point, the figures are in plane perspective. Thus if in Fig. (1) lines Aa, Bb, Cc meet in a point 0, the triangles ABC, abc are in perspective. 62. If two triangles are in perspective their corresponding sides intersect in points which are in a straight line. Hint.-Let O be the centre of perspective of ABC, abc.* Since AB and ab are both in the plane 40B they must meet; since AB is in the plane MN and ab in the plane M'N, the point of meeting must be in LN the line of intersection of these planes. (2.) If the two triangles are in the same plane. Hint.-Draw any line 00'0" not in the plane of the triangles through the centre of perspective. From any two points O', O" on this line draw lines through the vertices of the triangles. O'A and O'a meet in a point A' because both are in the plane O'OA; Thus both the triangles ABC and abc are projected into A'B'C'; hence, their corresponding sides meet on the line of intersection of the plane M.V with the plane of A'B'C'. 63. Exercise. If two polygons are in perspective their corresponding sides meet in points which are in a straight line. * If MN be a transparent plane and a point of light be at O, the shadow cast upon the plane M'N by the triangle ABC is the triangle abc. 64. Def.-The line on which the corresponding lines of two figures in perspective meet is the axis of perspective of the figures. 65. Conversely, if the corresponding sides of two plane triangles intersect in points on a straight line, the triangles are in perspective. (1.) If the triangles are not in the same plane. Hint.-If AB and ab meet at X, Aa and Bb are both in the plane AXa, and must therefore meet. Hence Aa, Bb, Cc intersect in pairs, and since they are not all three in the same plane, must therefore meet in a point. (2.) If the triangles are not in the same plane. Hint-Pass any plane through the line in which the corresponding sides meet and construct in it a triangle in perspective with each of the given triangles [§ 62 (2)]. The line through the centres of perspective, O', O", thus found will meet Aa, Bb, Cc. Therefore Aa, Bb, Cc meet in a point. 66. If three triangles are in perspective two by two, and have the same axis of perspective, their three centres of perspective are in a straight line. Hint.-Let ABC, A'B'C', A" B"C" be the triangles, and X, Y, Z the points in which their corresponding sides meet. The triangles AA'A", BB'B" are in perspective from the centre X. Hence the intersections of their corresponding sides are in a straight line. But these intersections are the centres of perspective of the original triangles. 67. Cor.-If three triangles are in perspective two by two and have the same axis of perspective, the three triangles formed by joining the corresponding vertices of these triangles are also in perspective two by two and have the same axis of perspective; and the axis of perspective of either system of triangles passes through the centres of perspective of the other system. 68. If three triangles are in perspective two by two and have the same centre of perspective, their three axes of perspective meet in a point. Hint.-Let ABC, A'B'C', A"B"C" be the triangles and O their centre of perspective. The triangles formed by the lines AB, A'B', A"B" and by the lines AC, A'C', A"C" are in perspective, since their corresponding sides meet on the line AA'. Therefore the lines joining their corresponding vertices meet in a point. 69. Cor.-If three triangles which are in perspective two by two have the same centre of perspective, the three triangles formed by the corresponding sides of these triangles are also in perspective two by two and have the same centre of perspective; and the three axes of perspective of either system meet in the centre of perspective of the other system. 70. Exercise.-Extend the theorems of §§ 66 and 68 to figures other than triangles. DUALITY 71. If the polar of each point and the pole of each line of a figure be taken, a second figure is formed having a peculiar relation to the first and called its reciprocal. Thus the triangle A'B'C' is the reciprocal of ABC. The sides of A'B'C' are the polars of the vertices of ABC; the vertices of A'B'C' are the poles of the sides of ABC. To a point of the first, corresponds a line of the second. To points in a straight line in the first, correspond lines through a point in the second. § 53 To lines through a point in the first, correspond points in a straight line in the second. $53 It follows from these relations, that from a theorem concerning the points and lines of a figure, a reciprocal theorem concerning the lines and points of the reciprocal figure can be inferred. 72. Def. The principle upon which these relations between a figure and its reciprocal depend is called the principle of duality. 73. The principle of duality in a plane is not necessarily derived from the consideration of poles and polars. A plane figure may be looked upon as composed either of points and the lines joining them, or of lines and their points of intersection, so that the point and line are elements correlative to each other; the relations between reciprocal figures which have already been obtained would follow from this conception. 74. Neither is the principle confined to plane figures; in the same way figures in space may be considered as composed either of points or of planes, so that in the geometry of space the point and plane are elements correlative to each other. It follows, that for reciprocal figures in space : To a point in the first, corresponds a plane in the second. To a plane in the first, corresponds a point in the second. To points in a plane in the first, correspond planes through a point in the second, and vice versa. To points in a straight line in the first, correspond planes through a straight line in the second, and vice versa. Remark. In the geometry of space the straight line is correlative to itself. 75. Examples of reciprocal theorems of plane geometry. 1. Two points determine a straight line. 2. If the points of intersection of the corresponding sides of two triangies are in a straight line, the lines joining the corresponding vertices of the triangles meet in a point. $ 65 3. If three triangles are in perspective two by two and have the same centre of perspective, their three axes of perspective meet in a point. § 68 I. Two straight lines determine a point, their point of intersection. 2. If the lines joining the corresponding vertices of two triangles meet in a point, the corresponding sides of the triangles intersect in points which are in a straight line. § 62 3. If three triangles are in perspective two by two and have the same axis of perspective, their three centres of perspective are in a straight line. § 66 76. Examples of reciprocal theorems of the geometry of space. 1. A straight line and a point determine a plane. 2. Three points not in the same straight line determine a plane. 3. Two straight lines which meet in a point are in the same plane. I. A straight line and a plane determine a point, the point in which the line meets the plane. 2. Three planes which do not pass through the same straight line determine a point. 3. Two straight lines which are in the same plane meet in a point. ANHARMONIC SECTION 77. Def.-If A, B, C, D are four points taken in order on a straight line, any one of the following six ratios, |