Relation between Projective and Metrical Scales. 421 



OTy any point F, and join it to go. Join n (the given 

 point) with T y . Let I) be the cross o£ nT y with Fx> and let 

 01) cut Tyco in C. Then the point whose index is —v will 

 be the cross o£ CY with the base line (fig. 2), i. p. their 



Fie- 2. 



actual cross if these two lines do intersect, and the ideal 

 point Tor pencil centre) determined by these two lines, should 

 they not intersect *.- 



Lastly let us recall that the cross-ratio of any tetrad of 

 collinear points whose Staudtian indices are n ly n 2j n g} n^ is 



(?i 1 n 2 , n 2 n 4 ) = 



n B 



(1) 



n l ~ n 4 n 2 ~ n A 



Thus, for instance, (02, lco) = — 1, this being, from the 

 outset, a harmonic range of points. 



2. With this projective or Staudtian scale let us now 

 compare the ordinary metrical scale. 



Let li be the length 01, i. e. the number of metrical units 

 (say, cms.) contained in the segment 01, and l^ that con- 

 tained in the whole segment OUT or in Olco , as in fig. 3. 



Fig. 3. 

 o 4/, t -l-oo 



<l 



oo 



It is required to find the number I of metrical unit steps 

 leading from to any point of our line whose index is n, 

 the sense from through 1 towards co being taken as that of 

 positive I. In short, it is required to find I as a function 

 of t?, such that 



Z(0)=0, 1(1) = l v l(°°)=l„, 

 the latter two being some given finite numbers, such as 

 l 1 = 'd and ^=100. 



** Cf. ' Projective Vector Algebra.' 



