276 G. H. KNIBBS. 



to zero of any particular order, yet to higher orders, the zeros have 

 successive differences. Thus we realize that the method of finite 

 differences leads to identical results with the infinitesimal methods, 

 only because a continuous curve is supposed to be drawn through 

 the successive points determined by the scheme of differences. 

 When in equation (16), r)/£ becomes dy/dx, we see that the principle 

 of continuity demands a recognition, that no matter how far the 

 order of infinitesimal is carried back in the conception of differenti- 

 ation, the infinitesimal stretch of the curve ds = V (dx 2 + dy 2 ) 

 approximately contains deviations from the chord, identical in 

 character with that represented by the equation itself, but of 

 course reduced in scale. 



The same remark applies to curves of 'double curvature,' the 

 continuity of tortuosity as well as of curvature, is conceptually 

 carried back indefinitely. 1 We see therefore that a curve of 

 discrete points, is essentially different from a continuous curve; 

 and the chord drawn through two points infinitesimally distant, 

 is clearly different from the tangent to the absolutely continuous 

 curve through either, no matter what the order of the infinitesimal. 



It may be said that the necessity for admitting an indefinitely 

 great order of infinity will appear just as unequivocally, in the 

 theory of metrics, of projective distance, of curved surfaces, and 

 of parallels. We shall now refer to these. 



15. The theory of metrics. — By von Staudt's theorem, 2 all quadri- 

 laterals KLMN in planes containing the line or range ACBD, 

 so constructed that pairs of opposite sides, KL, MN, shall meet 

 in the point A, while the other pairs of opposite sides LM, KN 

 shall meet in B, and the diagonals KM shall pass through C, will 



1 A curve of any character may be regarded as a one-dimensional region 

 or ' space/ defined in relation to other spatial elements by its equation : 

 treating it as a closed region can lead to no defect in mere physical 

 applications, if the path be infinite. To treat plus and minus infinity as 

 identical is none the less mere artifice, and it is only in the artificial 

 sense, that " a one-dimensional region is to be conceived as a closed 

 region, such that two elements divide it into two parts." See Whitehead, 

 Universal Algebra, Vol. i., p. 168. 



a Geometrie der Lage, Art 93. 



