PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 277 



have their remaining diagonals LN so determined as to pass 

 through the one point D, see Fig. 7. These points ABOD taken 

 in that order are called an harmonic range, 1 and the cross- or 

 anharmonic ratio 2 thereof, viz., 



AC/BO : AD/BD = (ABCD)= -1 (17) 



that is to say the anharmonic ratio of four harmonic elements is 

 always — I. 3 An harmonic range possesses the property that if 

 it be projected through any point S on to any other straight line, 

 the projection A'B'C'D' will also be harmonic; (Fig 7.). If 

 A"B"C"D" be drawn absolutely parallel to DS, DS will never meet 

 A"C": or as it is usually put, it will 'meet at infinity.' 4 If AC be 

 a unit distance, + 1, A being the origin, and AO = CB, then D 

 will be 'at infinity,' the anharmonic (or cross-) ratio being 



1/-1 : oo/(oo-2)='-l+ — +etc (18) 



GO 



Thus for a first order infinity the anharmonic ratio would differ 



infinitesimally from - 1 ; and it cannot have the value - 1 even if 



the infinity is absolute. Geometrically this may be illustrated by 



drawing A"C" = B"C" = 1, and the line C"D" being oo 11 , the defect 



from parallelism is the infinitesimal angle, say 2/ oo 11 , of the same 



order as the infinity. Hence NL is always inclined to AD by 



infinitesimal angles, and D can never be indifferently on either 



side of C, as is generally assumed. This result may be stated in 



another form. Let be a point midway between A and B so 



that OA= — 1; OB= + 1. As C approaches O, the harmonically 



related point D moves away from B, in such a manner that 



OD . OC = 1. For let OD = x, and 00 = f , then (17) may be written 



1+f 05+1 , , 



.. — ~ : = — 1, whence 



-l+£ x-l 



x£ = ¥ (19) 



1 And are determined by the symbol (ABCD). Mobius, Barycentrische 

 Calcul, § 183. If two circles cut one another orthogonally any diameter 

 of either is divided harmonically. Poncelet, Traite des proprietes pro- 

 jectives, Art. 79. Paris, 1822. 



2 Clifford and Henrici prefer the term "cross-ratio" to Chasles* term 

 anharmonic ratio. 



3 Mobius, loc. cit. 



4 That is never in an absolutely homaloidal space. 



