248 G. H. KNIBBS. 



that their analogues 1 on surfaces, that to us may be physically 

 undistinguishable from planes, 2 may do so, does not in any way 

 qualify our assurance. Poincare goes to the root of the matter 

 when he says that "geometry is not an experimental science" 3 All 

 ratiocination concerning spatial conceptions would assuredly be 

 futile unless the straight lines of homaloidal space were the con- 

 ceptual background for comparison with other geometrical forms 

 or figures. The straight line of geometry is in fact not a physical 

 entity to be examined, tested, and experimented upon in order 

 that its properties may be disclosed. To attempt to physically 

 delineate a straight line, as by optical methods 4 would, apart from 

 the consequences of the exceedingly complex nature of the earth's 

 motion in space, 5 be involved in difficulty and uncertainty owing 

 to our ignorance of the physical (?) substances distributed therein' 

 and their effect on the delineation, and would require an ante- 

 cedent knowledge of what was meant by such a line. In drawing 

 a representation of any conception, our ratiocination is always 

 guarded against both error of drawing and the limitations of 

 drawing itself. 



Nor is the straight line a thing to which position in the external 

 world has to be assigned in order to discover whether in particular 

 directions it may change its properties. It and the other elements 

 of space, i.e. surface, volume, etc., viz. the entities with which 

 geometry operates, are conceptual, so that geometrical ratio- 

 cination depends, not upon the physical constitution of the 

 universe, nor on the contents and properties of stellar space, but 

 rather upon our mental constitution, and the terms in which we 

 are compelled by that constitution to translate experience. Hence 

 its relation to the physical world is indirect. It is only when we 



1 Geodesies. 2 That is homaloidal plane-surfaces. 



3 The Monist, Vol. ix., p. 41, 1898. 



4 .An idea which has presented itself to Lobatchewsky, Hoiiel, Chrystal 

 and others. 



5 Though to the beholder the path of a moving point may be straight, 

 the actual motion is, even so far as we know complex almost beyond 

 description. 



