316 G. H. KNIBBS. 



37. Illimitability of operative schemes for the generation of geo- 

 metrical figures. — It must now be evident that no limit can be 

 assigned to operative schemes for the generation of specialised 

 regions of space, or of geometrical figures either therein, or in 

 ordinary, i.e. homaloidal space. The bizarre idea of the curvature 

 of the latter, scarcely touches the fringe of the subject. Every 

 type of geometrical figure, and every variation of it, can be made 

 the subject of a special geometry, having its own peculiar features; 

 and space can be so specialised as to be analogous thereto : that 

 is to say a region of ordinary space can be constituted so that its 

 analytical treatment, from some particular point of view, will be 

 analogous to the special geometry referred to. To fix our ideas, 

 suppose Fig. 34 x to represent a double surface of both positive 

 and negative curvature:' 2 its geometry would present all the 

 features of "elliptic" and "hyperbolic space," passing continuously 

 from the one to the other. 3 The surfaces cross one another at 

 right angles at the curved line ADCEB. 4 It will be at once 

 recognised that the geometry of this continuous interpenetrating 

 surface will present remarkable features. Still more remarkable 

 would be the geometry of the type of surface represented per- 

 spectively in Fig. 35, which may be generated as follows: — 



On the straight line AB let a quadric surface perpendicular to 

 the line (say an ellipse) move along it from A to B, so that the 

 terminal of one diameter remains on the line, this diameter how- 

 ever, both turning round AB and altering its dimensions as the 

 surface in which it lies moves along. 5 The successive values of 

 the principal diameter might be represented by the ordinates of 

 either of the curves, Figs. 35 (b) or 35 (c); or by those of curves 

 of higher degree. A surface of this type can be so constructed 



1 Only one half of the figure is shaded. 



2 The section perpendicular to AB shews two semi-lemniscates of differ- 

 ent parameters except at C where they are equal. 



3 Consequently Helmholtz's sphere-dwellers would, according to his 

 view, conclude that parallel lines, both converged and diverged! 



4 Also that straight lines may intersect at right-angles ! 



6 The surface may be supposed to increase its area, and the ratio of the 

 diameters to change. 



