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]

Now the fraction on the right would be unchanged if instead of the points A, B, C, D we should take any other four points A’, B’, C’, D’ lying on any other line cutting across SA, SB, SC, SD.  In other words, the fraction on the left is unaltered in value if the points _A__, __B__, __C__, __D__ are replaced by any other four points perspective to them._ Again, the fraction on the left is unchanged if some other point were taken instead of S.  In other words, the fraction on the right is unaltered if we replace the four lines _SA__, __SB__, __SC__, __SD__ by any other four lines perspective to them._ The fraction on the left is called the anharmonic ratio of the four points A, B, C, D; the fraction on the right is called the anharmonic ratio of the four lines SA, SB, SC, SD.  The anharmonic ratio of four points is sometimes written (ABCD), so that

[formula]