282 G. H. KNIBBS. 



nature of the theory may be explained as follows: — Let in Fig. 

 13, A, K, L, M, N, B at oo be any series of points on a natural 

 scale, on one line, viz. the heavy one, projected from the point S 

 on to another line — the projections being aklmnb. Calling the 

 heavy one the natural line, and the light one its projection, we 

 observe that although the projection-lengths on the left are greater 

 than the natural lengths, they become equal, 1 and then shorter, till 

 finally the infinite distance NB is projected into the finite distance 

 nb. In the opposite direction the finite distance AP, Fig. 14, is 

 projected into an infinite line, a—p at oo. Assuming that equal 

 stretches of the natural line, are in every way comparable, (con- 

 gruent) and observing that the projective-ratio is continually 

 changing, we reach the idea of a line of non-uniform linear 

 intensity, so that if 8S be a small element on the natural line, 

 and 8s its projection-equivalent, the intensity may be defined 

 by the ratio 8S/8s. Since the anharmonic ratio or its equivalent, 

 or functions thereof are independent of the intensity, that ratio 

 may be utilised to establish a theory of distance which will apply 

 either to lines of uniform, or to lines of non-uniform intensity 

 while ordinary conceptions of distance properly apply only to lines 

 of uniform intensity. 2 Take any pair of points a, b, as reference 

 points, then the anharmonic ratio of any other pairs together with 

 these, viz. (klab), (Imab), (kinab), may be written /o kl , o lm , /o km , 

 then we shall have 3 



1 For the stretches LM, Im. 



2 The subjective idea of a traveller suffering ever-increasing fatigue, 

 as to equal actual distances, takes the form of continual increase of linear 

 value. A crank pin rotating uniformly about a centre, moves a piston- 

 rod at one moment with its own velocity, in about a quarter of a revolu- 

 tion afterwards the rate of motion of the piston-rod has fallen to zero. 

 The uniform recession of a pursued object may cause the pursuit to con- 

 tinually regrede till its actual rate is zero. Illustrations might be 

 indefinitely multiplied as to the idea of intensity of related spatial 

 elements. 



3 Measured by any units (e.g. the distance AK) the points it, I, m may 

 be defined as k, or A, or /^, from a ; and k\ or A', or //, from b (which 

 may be denoted by the symbols «a + k b, Aa + A b, f^a + pb); then the 

 anharmonic ratios of the ranges, and the products of the ratios, are 

 identically _ kA' A//,' _ kjj! _ 



Pkl • Plm 7T • CT - 1 Pkm 



K A A jX K jJ, 



and similarly for any number of ranges. Hence taking logarithms, we 

 have (21) above. The points must be so taken as to give a positive ratio. 



