AB.CD AD.BC AC.DB AD.CB AC.BD AB.DC AB.CD AD.BC AC.DB is an anharmonic ratio of the points A, B, C, D. 78. If a pencil of four rays cuts two transversals, each anharmonic ratio of the four points of intersection with one transversal is equal to the corresponding ratio of the four points of intersection with the other transversal. 79. Cor. 1.-Anharmonic ratios are preserved in perspective. 80. Def.-It follows from § 78 that the anharmonic ratios of a pencil of four rays may be defined as the anharmonic ratios of its four points of intersection with a transversal. $78 81. Cor. 2.—If the corresponding rays of two pencils meet on a common transversal, the pencils are equal, that is, have equal anharmonic ratios. F 82. Cor. 3.-If two pencils are equal, have a common vertex, and three rays of the first coincide with three rays of the second, the fourth ray of the first coincides with the fourth ray of the second. 83. Exercise.-If two pencils have their vertices on a circle and their corresponding rays intersect in points on the circle, the pencils are equal. 84. If two equal pencils have a common ray, the intersections of the three remaining pairs of corresponding rays are in a straight line. Hint.-Employ the method of reductio ad absurdum. 85. Exercise.-Prove by means of § 84 that if two triangles are in plane perspective, the intersections of their corresponding sides are in a straight line. 86. (PASCAL'S THEOREM.) If a hexagon is inscribed in a circle, the intersections of the opposite sides are in a straight line. E FIG. 86 (1) M FIG. 86 (2) B Hint.-The opposite sides are the 1st and 4th, 2d and 5th, 3d and 6th. Let L, M, N be the intersections of the opposite sides. § = { Pencil N.AEDL}={A.NEDL by $ 81, C.FEDB by § 83, = {N.AEDM by § 81. Therefore L, M, N are in a straight line. $82 Remark. This theorem is true of any of the sixty hexagons which can be constructed with six given points as vertices. 87. Exercise. If six points are three by three on two straight lines, the intersections of the opposite sides of a hexagon of which these points are the vertices are in a straight line. 88. (BRIANCHON'S THEOREM.) If a hexagon is circumscribed about a circle, the three lines joining the opposite vertices meet in a point. Hint. The vertices of the circumscribed hexagon are the poles of the sides of an inscribed hexagon. Therefore this theorem may be inferred from 86 by the principle of duality. 89. Exercise.—If four points are in a straight line, their anharmonic ratio is equal to the anharmonic ratio of their four polars. Hint.-Compare with § 55. INVOLUTION 90. Def.-If the distances of several points, A, A', etc., in a straight line from a point O in that line, are connected by the relation OA.OA'=OB.OB'=OC.OC'= the points form a range in involution. 91. If six points form a range in involution, the anharmonic ratios of any four of the points are equal to the anharmonic ratios of their four conjugates. A B C C B' FIG. 91 Hint.-At O erect a perpendicular OP=√OA.OA'. Then OP is tangent to the circle described through A, A', P. $ 321, p. 145 Hence angle OPA=0A'P; likewise angle OPB=0B'P, etc. four rays{P.AA'BC are equal to the angles of the pencil { P.A'AB'C' } The anharmonic ratios of the points A, A, B, C are consequently equal to the anharmonic ratios of the points A', A, B', C'. 92. Cor.-The anharmonic ratios of four points in a straight line are equal to the anharmonic ratios of their inverses, if the centre of inversion is on this line. 93. Def.-A pencil of which the rays pass through the points of a range in involution is a pencil in involution. ANTIPARALLELS 94. Def.—If two lines are such that the inclination of the first to one side of an angle is equal to the inclination of the second to the other side of the angle, the lines are antiparallel to each other with respect to the angle. K 95. An antiparallel to a side of a triangle with respect to the opposite angle is parallel to the tangent to the circumscribing circle drawn at the vertex of that angle. Hint-Angle YCB=CAB=CB'A'. 96. Exercise. The lines joining the feet of the perpendiculars of a triangle are antiparallel to the sides with respect to the opposite angles. THE GEOMETRICAL AXIOMS PLANE, SPHERICAL, AND PSEUDO-SPHERICAL GEOMETRIES 97. The geometrical axioms in the Introduction of this Geometry really define the surface on which the theorems of plane geometry are true. This surface is the plane. The axioms also hold true of any surface into which the plane can be bent without stretching, such as the cylinder or cone, provided the definitions of a straight line and parallel lines be modified to apply to these surfaces. 98. A sheet of paper may be wrapped about a pencil to form a cylindrical surface; every layer of the paper forms a different part of the surface, and two points that lie in different layers one above the other are separated by the distance which must be traversed to get from one to the other without piercing the paper-that is, by the distance they would be separated in the plane if the paper were unrolled. 99. The geometrical axioms are— (a.) Straight-line axiom.-Through every two points there is one and only one straight line. A straight line of any surface may be defined as the shortest line lying |