254 G. H. KNIBBS. 



separated by dimensionless points, i.e. not really separated at all. 1 

 Algebraically we may say that if m and n denote abstract numbers, 

 and p and q characteristic suffixes denning the kind of units to 

 which they are appended, no relation can exist between 2 



m x l p and n x l q 

 unless it can be established between l p and l q . It matters not 

 what magnitude m and n have, if l p denotes a point, and l q a line, 

 no direct relation can exist between ra p and n q even if in be an 

 infinity, or an infinity of infinities. 



Hence no number of pure points can constitute a line, and 

 similarly since a line is without breadth, no number of lines can 

 constitute a surface, and for the same reason no number of surfaces 

 a solid. 



3. The straight line. — Before considering other modes of gener- 

 ating geometrical figures, the nature of the second important 

 conception of geometry, viz. the straight line, may be examined. 

 The definition that this is the shortest distance between two points* 

 is unsatisfactory, inasmuch as that property is deducible rather 

 than immediately evident; and furthermore when restricted to a 

 surface, the geodesic also answers to the definition. There is still 

 another objection, the most cogent of all, viz. that the definition 



1 Clifford, [1845 - 1879] referring to Pliicker's mode of generating curves 

 says : — ' now a point is absolutely no line, as a line is no surface and a 

 surface no space/ — Math. Papers, p. 40. Thus Kant: 'Raum ist ein 

 solches Ganze, dessen Teile bei aller Dekoinposition immer wiederum 

 Raume sind, und ist daher ins Unendliche teilbar/ . . . ' Die unendliche 

 Teilung bezeichnet nur die Erscheinung als quantum continuum und ist 

 von der Erfiillung der Eaumes unzertrennlich ; weil eben in derselben 

 der Grund der unendlichen Teilbarkeit liegt. Sobald aber etwas als 

 jpunctum discretum angenommen wird : so its die Menge der Einheiten 

 darin bestimmt/ — Op. cit., Elementarlehre, Buch n., 2, 9, i. 



2 In the language of vector algebra and quaternions m and n are scalars, 

 ] p and l q unit vectors of different kinds. 



8 This quantitative definition has been strongly approved. A. M. 

 Legendre, Elements de Geometrie. B. Pierce, An elementary treatise on 

 plane and solid Geometry : Helmholtz, Henrici, and many others. Wilson 

 (Elementary Geometry) defines a straight line as " that which has the 

 same direction at all parts of its length." The Assoc. Impt. Geom. 

 Teaching (1878) "A straight line is such that any part will, however 

 placed, lie wholly in any other part, if its extremities are made to fall on 

 that other part." Chrystal, Proc. Roy. Soc, Edin. x., 642, says "Two 

 points in general determine a straight line." 



