*Project Gutenberg*. Public domain.

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

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 pointSnot on that line. Then the trianglesASB,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 pointsA,B,C,Dwe should take any other four pointsA’,B’,C’,D’lying on any other line cutting acrossSA,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 ofS. 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 theanharmonic ratioof the four pointsA,B,C,D; the fraction on the right is called theanharmonic ratioof the four linesSA,SB,SC,SD. The anharmonic ratio of four points is sometimes written (ABCD), so that

[formula]