270 G. H. KNIBBS. 



ever the higher orders of infinities and infinitesimals be considered, 

 the zeros and infinities become also of higher order, and the dis- 

 continuity correspondingly augmented. 



The importance therefore of distinguishing between the dimen- 

 sionless point, and the infinitesimal of any dimension, and also the 

 necessity of admitting the conception of different orders of 

 infinitesimals and infinities now more fully appears. 1 The matter 

 may be brought into a still clearer light by observing that if we 

 regard the cc-axis as a mere assemblage of points separated by 

 infinitesimal distances of the first order, and the y terminals as 

 the corresponding or dependent assemblage P, then the graph of 

 xy = 2 , is to the first order infinitesimal indistinguishable from 

 the x-axis itself. But if we interpolate between this assemblage 

 a second infinite series, this is no longer true. The conception of 

 absolute continuity, requires that there be no limit to this interpo- 

 lative process; hence, if x vary in an absolutely continuous manner, 

 viz., in the way an absolutely dimensionless point may be conceived 

 as moving upon or generating a line, then it follows that the sur- 

 face necessary to completely define the graph must admit of absolute 

 extension, for the order of the infinity, must coincide with the 

 order of the infinitesimal. Conceptual space 2 is subject to no 

 limits except those which limit the operation of human thought, 

 and hence the "dreary infinities of homaloidal space," 3 must include 

 the concept of what may be called pure homaloidal space, viz., 



1 And the significance of successive differentiation is correspondingly 

 enhanced. 



2 It is remarkable to find the phrase the ' space of experience ' employed 

 as if it were a thing to be investigated, instead of an a priori concept in 

 terms of which we are obliged to explain phenomena. This is a matter 

 which we shall farther discuss. See Whitehead, Universal Algebra, Vol. 

 i., footnote p. 499. 



3 Clifford, Lectures and Essays, Vol. i., pp. 322-3, Mathematical Works 

 xlvi. Pure homaloidal space, may be popularly defined as that in which 

 a point travelling rectilinearly never returns upon its path. Imagine 

 two points so moving, one vastly faster than the other. Then in infinite 

 time, the journey of the faster would be vastly greater than that of the 

 more slowly moving point ; if infinitely faster, then though the first point 

 would have travelled an infinite distance the latter would have travelled 

 an infinity of such infinities. Conceptually there is no limit to the 

 "dreary infinities" of such space. 



