FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 97 



conceivably be, for example, to beings living on the surface of a 

 solid body and without the ability to perceive anything outside of this 

 surface. For us, dwellers in space of three dimensions, it is possible 

 to conceive what manner of perceptions of space beings limited to two 

 dimensions would have, though we can not, on the other hand, con- 

 ceive of space of more than three dimensions, because all our means of 

 perception extend only to tridimensional space. What would in the 

 case of dwellers in two dimensions become of these axioms of our 

 geometry: (a) Between two points only one shortest liue, the straight 

 line, can be drawn ; (b) through three points not lying in a straight line a 

 surface called a plane can be passed, such that all straight lines joining 

 any pairs of points upon it lie wholly within the surface; and finally, 

 (c) if two straight lines lying in the same plane, which never meet, how- 

 ever far produced, be defined as parallel, only one line parallel to these 

 can be drawn through a point not lying upon either of them. What 

 would become of these and of all the other axioms that require the con- 

 tinuity of geometrical figures'? The surface dwellers would in general 

 draw shortest lines between two points, which, however, would not nec- 

 essarily be straight lines, and which Helmholtz called straightest lines. 

 But in the simplest case, a sphere, an infinite number of straightest 

 lines might be drawn between the two poles, though parallel straightest 

 lines could not be drawn, and the sum of the angles in a triangle would 

 be different from two right angles. These beings would, like us, find 

 space endless though of finite extent, and in the development of a geom- 

 etry they would have other axioms than we, but they could still move 

 their figures about the sphere at will without altering their dimensions. 

 Yet this peculiarity would in general be lost upon surfaces of other 

 forms, since only those surfaces possess it, which have at all points a 

 constant curvature, and of these, surfaces of a constant positive or 

 negative curvature other than spherical are such as may be unwrapped 

 from a sphere without bending or tearing, and may be called pseudo- 

 spherical. An analytical investigation of surfaces of this latter kind 

 shows that it is possible for straightest lines to be infinitely extended 

 without ever returning to the surface, and that, as in the plane, only 

 one shortest line is possible between two points. But the validity of 

 the parallel axiom fails here, since through a point outside a straightest 

 line an infinite number of other straightest lines may be drawn, which 

 when infinitely extended do not cut the first. Thus the three axioms 

 mentioned above are necessary and sufficient in order to characterize 

 the surface for which Euclid's geometry holds in distinction from all 

 other figures having two dimensions, as plane. Passing now to space 

 of three dimensions and considering it as a domain of quantities in 

 which the situation of any point may be determined by three measure- 

 ments, we may compare it with other threefold extended subjects 

 of consideration, such, for example, as the arrangement of systems of 

 colors affords, in order to investigate whether we may discover special 

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