298 G. H. KNIBBS. 



merit of a theory of a figure of (n + 1 ) dimensions. This treatment 

 is what may be called relative, as distinguished from absolute; 

 and in applying any deductions of analytic geometry the essential 

 features of the relation must be borne in mind, the deductions 

 being applicable only where analogous relationships exist. The 

 fundamental conceptions of geometry are however an absolute 

 basis, in the sense that any discredit thrown upon them, reduces 

 all geometry to confusion. The relativity of figure we now proceed 

 to consider. 



27. Relativity of geometrical forms and figures. — In the intro- 

 duction, reference was made in a footnote 1, page 249, to the 

 relativity and reciprocal identity of solar and cometary motion in 

 space, the two curves of apparent motion being the same conic 

 sections, that is to say either the sun or the comet could be regarded 

 as moving in the conic, the direction and amount being identical, 

 but the coordinates differing 180°. This is a very elementary 

 case of relativity. We proceed to indicate a more complex case. 

 In the line (or surface) Q 2 0' Fig. 24, suppose successive points to 

 be determined by their distances Q 2 P 2 from the line (or surface) 

 P 2 0, measured vertically to the latter, and the distances OP 2 on 

 the reference line. Then treating this latter as a straight line, 

 the successive points 1, 2, 3, etc., on the absolutely straight 

 line OS become relatively thereto the curve Q 2 0'S Fig. 25, and 

 relatively to this curve the circular arc is a straight line. 1 



That is to say, given only x and y, and that y is always perpen- 

 dicular to x, but no knowledge of the curvature of x, the absolutely 

 straight line would appear as the curve in Fig. 25, and similarly 

 coplanar points, as distributed over a conoidal surface, of which 

 Fig. 25 is the section. In one sense it is indifferent whether OP 

 is regarded as straight and OQ as curved or vice versa, but not 



The equation to the curve is obviously 



x 



y= sec -(p + b)-p 

 P 

 y being the vertical PQ, x the curved line OP, b the distance 00 , and p 

 the radius of the circle. The curve in Fig. 25 has asymptotes at ± \rrp. 



