250 G. H. KNIBBS. 



surfaces are spatial in the ordinary sense, and their geometries 

 may, like the geometries of regular surfaces of various kinds, be 

 regarded as systems of theorems and problems, restricted to the 

 several surfaces : and generally it may be said that the non- 

 euclidean geometries are really geometries of surfaces, or of specially 

 constituted regions of space, rather than geometries which have 

 some peculiar applicability to a supposed actual (!) space itself. 



A region of absolutely homogeneous homaloidal space of three 

 dimensions, may be supposed to contain some physical entity 

 subject to a definite variation dependent upon its position therein : 

 the physical element after it has been mathematically specified 

 may be disregarded, and the region treated as a specially con- 

 stituted region of space, for which a special geometry may be 

 created : such a geometry might be of the same number of dimen- 

 sions but would have special features. And further, a region 

 of space may involve the idea of such special constitution as has 

 been indicated, and be moreover affected in some other way, also 

 susceptible of mathematical specification. In such a case the 

 specialised geometry would need a further dimension : that is to 

 say it would be four-dimensional as well as special in its other 

 features. There is no limit to the development of dimensions, or 

 the intricacy of a conceptual constitution of space, and the 

 analogies derived through surfaces of positive and negative curva- 

 ture, and from variations of linear intensity in projections, are 

 from this point of view easily seen to be after all only the simplest 

 developments of pangeometry, the possibilities of which are 

 doubtless absolutely inexhaustible. 



It is proposed in the following paper to discuss the generation 

 of geometrical figures in homaloidal space of w-dimensions, with a 

 view to elucidating what is involved in the conception of absolute 

 continuity, and of shewing that there need be no confusion in 

 respect of the infinitesimal, n say, and the absolute zero, nought. 

 In doing this, the purview will be extended to the regions of non- 

 euclidean geometry with a view of shewing that after all it may 



