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.

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 th t 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.

P

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=OA'P; likewise angle OPB=0B'P, etc.
Therefore angle APB=A'PB', etc.; that is, the angles of the pencil of

four raysP.
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

wholly in the surface which can be drawn between two of its points. Thus, arcs of great circles are the straight lines of the spherical surface.

(b.) Parallel axiom.-Through a given point there is one and only one straight line parallel to a given straight line.

Parallel lines of a surface may be defined as straight lines of that surface which meet at infinity.

(c.) Superposition axiom.-Any figure in a plane may be freely moved about in the plane without change of size or shape.

This axiom as modified would read:

"Any figure of a surface may be freely moved about in that surface without change of size or shape;" that is, would conform to any portion of the surface without stretching.

100. The plane and the surfaces into which it can be bent—the surfaces upon which these axioms hold true-are surfaces of zero curvature.*

101. If the surface or covering of a sphere be detached from the sphere any surface into which it can be bent without stretching is a surface of constant positive curvature. The geometry of such a surface is called spherical geometry.

102. The superposition axiom is true for the spherical surface.

103. The straight-line axiom is true for the spherical surface unless the two points are extremities of a diameter of the sphere, in which case an infinite number of straight lines can be drawn between them.

104. There can be no parallel axiom, for on the sphere any two straight lines meet each other at a finite distance.

105. In Book VIII. the spherical geometry is developed, not from the axioms which are true on the covering of a sphere independent of the sphere itself, but by considering this covering as belonging to the body in space. This is entirely unnecessary; the spherical surface may be regarded as an independent surface which has no relation to the planè, the straight line, or space. Its geometry may be developed entirely from the axioms which apply to it, just as the geometry of the plane is developed from its axioms.

* The geometry of such surfaces is called Euclidean Geometry because Euclid first formally stated the axioms as the basis of a geometry.