278 G. H. KNIBBS. 



the same type of equation as (12). 1 The points C and D are 

 conjugate, with respect to the origin O and the unit distance OB, 

 and the distances OC and OD are reciprocals. As the point C 

 moves from B, i.e. + 1, through the origin O to A, i e. - 1, the 

 conjugate point D moves from + oo to - go. This is the basis of 

 the rectilinear system of metrics, that is a system which can be 

 developed by drawing straight lines only, and it is immediately 

 evident from Fig. 7 that a line of any definite ratio to the unit 

 OB, can be so found. 



The anharmonic ratio of any range of four points, (ABCD), is 

 unaffected by projection, 2 consequently if two ranges each of four 

 points are projective they are equianharmonic, 3 and hence if three 

 collinear points A, B, C, are given, a fourth D may be found such 

 that the anharmonic ratio of the range ABOD shall be any given 

 number A. either positive or negative. 4 The conjugacy of points 

 in an harmonic range, and the anharmonic ratio, are the foundation 

 of the theory of distance, to which we shall later refer ; the con- 

 jugacy however can be otherwise established, consequently a 

 system of metrics and therefore a theory of distance can be founded 

 without recourse to the anharmonic ratio, we shall see that con- 

 jugate points are determined by the rotation of a circle round a 

 point on its circumference, the circle having a fixed tangent 

 opposite the point of rotation : the intersections of the circle and 

 the antipodal tangent with a line through the centre of rotation 

 define the specified points. In Fig. 8 let the circles OCA", OC'B" 

 rotate about the point O on the line D'OD. When the diameters 

 of unit length A'O, OB' are perpendicular to" that line the point 

 C is identical with O, as the circle rotates C and D move towards 



1 Putting OB = + a, we get xh, = a 2 not a. Therefore we write I 2 not 

 1. When C is identical with B so also is D identical therewith, hence 

 %£ = lxl. The anharmonic ratio is an abstract number not a unit 

 quantum. 



2 Pappus, Mathematicae Collectiones, Lib vn., prop. 129. 



3 Townsend, Modern Geometry, Art. 278. See also Steiner, Systenia- 

 tische Entwickelung der Abhangigkeit etc., p. 33, § 10, Berlin 1882. 



4 Chasles, Geometrie superieure, p. 10, Paris 1852. 



