HIS PHILOSOPHICAL POSITION 



are diametrically opposite one another, in which case 

 an infinite number of shortest lines (t.e. t great 

 circles) can be drawn between them. Sphere- 

 dwellers, as Helmholtz calls them, would know 

 nothing of parallel lines ; they would say that two 

 'straight' (portions of great circles) lines, sufficiently 

 produced, must cut in two points, or form parts of the 

 same 'straight' line. The sum of the angles of a 

 triangle would always be greater than two right 

 angles, increasing as the surface of the triangle became 

 greater. Helmholtz also imagined the geometry of 

 intelligent beings living on the surface of an egg- 

 shaped body, or rather we must think of egg-shaped 

 space. They would find it impossible to construct 

 two congruent triangles at different parts of the 

 surface. A triangle constructed near the pole would 

 not be congruent with one at the equator. The 

 periphery of a circle described at the blunt end of 

 the body would be greater than that at the narrower, 

 although the radii were equal, the radii being always 

 measured by shortest lines on the surface. He then 

 discusses the geometry of what is called a pseudo- 

 spherical surface (a curious surface, of which a strip 

 may be represented by the inner surface turned 

 towards the axis of a solid arch or ring, the curves 

 of which are infinitely continued in all directions). 

 Such a surface was first investigated by Beltrami. 

 The straightest lines of the pseudo-sphere may be 

 infinitely produced, but they do not, like those 

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