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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



has led to it may be described in general terras 

 as follows : Some of the properties of space which, 

 on account of their simplicity, theoretical as well 

 as practical, have, in constructing the ordinary 

 system of geometry, been considered as funda- 

 mental, are now seen to be particular cases of 

 more general properties. Thus, a plane surface 

 and a straight line may be regarded as special in- 

 stances of surfaces and lines whose curvature is 

 every where uniform and constant. And it is, per- 

 haps, not difficult to see that, when the special 

 notions of flatness and straightness are aban- 

 doned, many properties of geometrical figures 

 which we are in the habit of regarding as funda- 

 mental will undergo profound modification. Thus, 

 a plane maybe considered as a special case of the 

 sphere, viz., the limit to which a sphere approach- 

 es when its radius is increased without limit. 

 But even this consideration trenches upon an 

 elementary proposition relating to one of the sim- 

 plest of geometrical figures. In plane triangles 

 the interior angles are together equal to two right 

 angles ; but in triangles traced on the surface of 

 a sphere this proposition does not hold good. To 

 this, other instances might be added. 



Further, these modifications may affect not 

 only our ideas of particular geometrical figures, 

 but the very axioms of the science itself. Thus, 

 the idea which, in fact, lies at the foundation of 

 Euclid's method that a geometrical figure may be 

 moved in space without change of size or altera- 

 tion of form, entirely falls away, or becomes only 

 approximate in a space wherein dimension and 

 form are dependent upon position. For instance, 

 if we consider merely the case of figures traced 

 on a flattened globe like the earth's surface, or 

 upon an egg-shell, such figures cannot be made 

 to slide upon the surface without change of form, 

 as is the case with figures traced upon a plane or 

 even upon a sphere. But, further still, these 

 generalizations are not restricted to the case of 

 figures traced upon a surface; they may apply 

 also to solid figures in a space whose very con- 

 figuration varies from point to point. We may, 

 for instance, imagine a space in which our rule 

 or scale of measurement varies as it extends, or 

 as it moves about, in one direction or another 

 — a space, in fact, whose geometric density is not 

 uniformly distributed. Thus we might picture to 

 ourselves such a space as a field having a more 

 or less complicated distribution of temperature, 

 and our scale as a rod instantaneously susceptible 

 of expansion or contraction under the influence 

 of heat ; or we might suppose space to be even 

 crystalline in its geometric formation, and our 



scale and measuring instruments to accept the 

 structure of the locality in which they are applied. 

 These ideas are doubtless difficult of apprehen- 

 sion, at all events at the outset ; but Helmholtz 

 has pointed out a very familiar phenomenon which 

 may be regarded as a diagram of such a kind of 

 space. The picture formed by reflection from a 

 plane mirror may be taken as a correct repre- 

 sentation of ordinary space, in which, subject to 

 the usual laws of perspective, every object ap- 

 pears in the same form and of the same dimen- 

 sions, whatever be its position. In like manner 

 the picture formed by reflection from a curved 

 mirror may be regarded as the representation of 

 a space wherein dimension and form are depend- 

 ent upon position. Thus, in an ordinary convex 

 mirror objects appear smaller as they recede lat- 

 erally from the centre of the picture; straight 

 lines become curved ; objects infinitely distant in 

 front of the mirror appear at a distance only equal 

 to the focal length behind. And by suitable 

 modifications in the curvature of the mirror, rep- 

 resentations could similarly be obtained of space 

 of various configurations. 



The diversity in kind of these spaces is of 

 course infinite ; they vary with the mode in which 

 we generalize our conceptions of ordinary space ; 

 but upon each as a basis it is possible to con- 

 struct a consistent system of geometry, whose 

 laws, as a matter of strict reasoning, have a valid- 

 ity and truth not inferior to those with which we 

 are habitually familiar. Such systems having been 

 actually constructed, the question has not unnatu- 

 rally been asked whether there is anything in 

 Nature or in the outer world to which they cor- 

 respond ; whether, admitting that for our limited 

 experience ordinary geometry amply suffices, we 

 may understand that for powers more extensive 

 in range or more minute in definition some more 

 general scheme would be requisite? Thus, for 

 example, although the one may serve for the solar 

 system, is it legitimate to suppose that it may 

 fail to apply at distances reaching to the fixed 

 stars, or to regions beyond ? Or, again, if our 

 vision could discern the minute configuration of 

 portions of space, which to our ordinary powers 

 appear infinitesimally small, should we expect to 

 find that all our usual geometry is but a special 

 case, sufficient indeed for daily use, but after all 

 only a rough approximation to a truer although 

 perhaps more complicated scheme? Traces of 

 these questions are in fact to be found in the 

 writings of some of our greatest and most origi- 

 nal mathematicians. Gauss, Riemann, and Helm- 

 holtz, have thrown out suggestions radiating, as it 



