258 G. H. KNIBBS. 



a point generates a line, 1 of a line a surface, of a surface a volume, 

 etc.; or adopting Henrici's definitions the path of a point is a line, 

 of a line a surface, and of a surface a volume. The qualification 

 'in general,' logically fatal to a definition, is avoided by properly 

 characterising the essential nature of the generative motion, viz. 

 that it shall be into a new spatial dimension. A point moving in 

 its O-dimensional space does not generate aline or 1 -dimensional 

 geometrical figure, similarly a line moving in its own path or the 

 continuation of its path, i.e., in its own 1-dimensional space, 

 generates only a 1-dimensional figure, viz. a line and not a surface, 

 and, identically an (n - l)-dimensional geometrical figure will 

 generate a figure of only the same dimension unless its motion 

 take place into the nth dimension itself, when it will generate an 

 w-dimensional figure. 



Whether finite, infinite or infinitesimal, the scheme of generation 

 is completely defined by the following system, in which the symbol 



%~ denotes an operator, translating the quantity on its left, 



the amount and direction specified by the quantity on its right, 

 the axes, to which the motions are to be parallel, being x,y,z,w, etc. 



1 ^Oi^Oi ; 1 — >0 y ^0 y ; 1 — ; >O z eeeO z ■ etc. 



X — > y = 0% eee y -- >0* = 2 x ; y — > o; = 01 j etc. 



2 xy — > 0\ = 3 xyz ee XZ > y eee xzy = 2 yz — > X eee 3 yz ; etc. 



rv3 s, m = 4 = 3 i> 1 = 3 > 1 = 4 • pto 



"xyz ^ U w u xyzw u xyw ^ W z u yzw "* u x u xyzw ^ t 5 ^ ' 



etc. etc. etc. etc. etc. etc. 2 



proves that the ratio or relation of quantities may be conceptually retained 

 up to point of their evanishing, that is to say, their zeros have definite 

 relations, if the law of relation is susceptible of being continuously 

 expressed. It is only when the facts are represented as being sensuously 

 perceptible that each point is surrounded by a "circle of confusion/' 

 through which it fuses with its neighbours. 



1 In Henrici's definitions the qualification f in general' is not applied 

 to the motion of a point. The conception of motion into a new spatial 

 dimension, hereinafter mentioned, shews that logically this case should 

 not be excepted. 



a The idea of successive infinitesimal motions, and infiaitesimal paths 

 produced thereby, may be clearly apprehended by conceiving it as the 

 actual path of the generatrix during the time, l/oo multiplied into the 

 unit of time, the last being understood as the interval of time necessary 

 for the generatrix to move over a unit in any particular axial direction. 

 Let It denote such a time unit, then l/oo x It = Ot, a zero of duration, but 



