318 G. H. KNIBBS. 



contradictory, 1 for it is on this essentially simple homaloidal con- 

 cept that every other must be grounded. One may say further 

 that this conception is our mental standard of reference, by means 

 of which the differences of other conceptions are to be discerned, 

 and is not affected by the fact that in physical applications, or in 

 actual lines in the objective world, we never know how far the 

 relativity of things limits our interpretations. In mathematical 

 science the doctrine of relativity is of no moment in so far as the 

 fundamental conceptions are concerned. 



Ordinary space then may be regarded as really the locus of all 

 possible determinate geometrical figures, and the domain of all 

 spatial specialisations, susceptible of geometrical definition. The 

 variety not only of these possible figures, but also of the possible 

 types of figures, is probably illimitable, or at least is limited only 

 by our failing to perceive them. And similarly the number of 

 modes in which space may be specialised is also probably illimitable. 

 The great reach of projective geometry as compared with metric, 

 suggests that an extension in that direction will not be unfruitful. 

 A geometry — which will aim at treating broadly the generation 

 of complex geometrical figures by means of simpler ones, which 

 will exhibit the limits of theorems respecting plane figures when 

 those figures are constructed upon surfaces, which will shew the 

 relationships of plane or solid figures when referred to rectilinear 

 or to curved axes, which will in fact generalize geometry to the 

 last possible extent — would be well worthy of the name "pan- 

 geometry." Many of the splendid results obtained by the great 

 mathematicians who have given some support to the remarkable 



1 Newcomb states that "there is nothing within our experience which 

 will justify a denial of the possibility that the space in which we find 

 ourselves may be curved in the manner here supposed/' i.e. in the 4th 

 dimension. See Crelle's Journ. f. reine und Angewandte Mathematik, 

 Bd. lxxxiii., 1877. The title of Newcomb's treatise is "Elementary 

 theorems relating to the Geometry of a space of three dimensions, and of 

 uniform positive curvature in the 4th dimension. Space is evidently 

 regarded as an object, not as the locus of objects. If a region were dis- 

 covered in which the angles A + B + C = 7r±e, we could define a curved 

 surface in it, in which A' + B'+ C, the vertices of the triangle being the 

 same, would be 7r. Which are we to rely on, the straightness of the line 

 or the sum of the angles ? 



