Pencil of planes of the second order, 59 Pencil of rays, 6, 7, 8, 23; of the second order, 57, 60, 79, 81 Perspective position, 6, 8, 35, 37, 51, 53, 71 Plane system, 16, 23 Planes on space, 17 Point of contact, 87, 88, 89, 90 Point system, 16, 23 Point-row, 6, 7, 8, 9, 23; of the second order, 55, 60, 61, 66, 67, 72 Points in space, 18 Pole and polar, 98, 99, 100, 101, 138, 164, 166 Poncelet (1788-1867), 177, 179, 180, 181, 182, 183, 184 Principal axis of a conic, 157 Projection, 161 Protective axial pencils, 59 Projective correspondence, 9, 35, 36, 37, 47, 71, 92, 104 Projective pencils, 53, 64, 68 Projective point-rows, 51, 79 Projective properties, 24 Projective theorems, 40, 104 Quadrangle, 26, 27, 28, 29 Quadric cone, 59 Quadrilateral, 88, 95, 96 Roberval (1602-1675), 168 Ruler construction, 40 Scheiner, 169 Self-corresponding elements, 47, 48, 49, 50, 51 Self-dual, 105 Self-polar triangle, 102 Separation of elements in involution, 148 Separation of harmonic conjugates, 38 Sequence of points, 49 Sign of segment, 44, 45 Similarity, 106 Skew lines, 12 Space system, 19, 23 Sphere, 21 Steiner (1796-1863), 129, 130, 131, 177, 179, 184 Steiner’s construction, 129, 130, 131 Superposed point-rows, 47, 48, 49 Surfaces of the second degree, 166 System of lines in space, 20, 23 Systems of conics, 125 Tangent line, 61, 80, 81, 87, 88, 89, 90, 91, 92 Tycho Brahe (1546-1601), 162 Verner, 161 Vertex of conic, 157, 159 Von Staudt (1798-1867), 179, 185 Wallis (1616-1703), 162 FOOTNOTES 1 The more general notion of _anharmonic ratio_, which includes the harmonic ratio as a special case, was also known to the ancients. While we have not found it necessary to make use of the anharmonic ratio in building up our theory, it is so frequently met with in treatises on geometry that some account of it should be given. Consider any four points, _A_, _B_, _C_, _D_, on a line, and join them to any point _S_ not on that line. Then the triangles _ASB_, _GSD_, _ASD_, _CSB_, having all the same altitude, are to each other as their bases. Also, since the area of any triangle is one half the product of any two of its sides by the sine of the angle included between them, we have [formula]