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NATURE 



[Aiig:ist 15, 1 878 



common focus ? Or once more, when space already filled with 

 material substances is mentally peopled with immaterial beings, 

 may not the imagination be regarded as having added a new 

 element to the capacity of space, a fourth dimension of which 

 there is no evidence in experimental fact ? 



Non-Euclid Geometry. 



The third method proposed for special remark is that which 

 has been termed non-Euclidean geometry, and the train of reason- 

 ing which has led to it may be described in general terms as 

 follows : some of the properties of space which on account of 

 their simplicity, theoretical as well as practical, have, in con- 

 structing the ordinary system of geometry, been considered as 

 fundamental, are now seen to be particular cases of more 

 general properties. Thus a plane surface and a straight line 

 may be regarded as special instances of surfaces and lines whose 

 curvature is everywhere uniform or constant. And it is, perhaps, 

 not difficult to see that, when the special notions of flatness and 

 straightness are abandoned, many properties of geometrical 

 figures which we are in the habit of regarding as fundamental, 

 will undergo profound modification. Thus a plane may be 

 considered as a special case of the sphere, viz., the limit to 

 which a sphere approaches when its radius is increased without 

 limit. But even this consideration trenches upon an elementary 

 proposition relating to one of the simplest 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 par- 

 ticular geometrical figures, but the veiy 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 alteration of form, entirely falls away, or be- 

 comes 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 figiures cannot be 

 made to slide upon the siu-face without change of form, as is 

 the case with figures traced upon a plane or even upon a sphere. 

 But, further still, these generalisations are not restricted to the 

 case of figures traced upon a surface ; they may apply also to 

 solid figures in a space whose very configuration 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 

 apprehension, 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 appears in the same form and of the 

 same dimensions, whatever be its position. In like manner the 

 pictiu-e formed by reflexion from a curved mirror may be re- 

 garded as the representation of a space wherein dimension and 

 form are dependent upon position. Thus in an ordinary convex 

 mirror objects appear smaller as they recede laterally 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, representations could similarly be 

 obtained of space of various configurations. 



Its Meaning and Use, 



The diversity in kind of these spaces is of course infinite ; 

 they vary with the mode in which we generalise our conceptions 

 of ordinary space ; but upon each as a basis it is possible to 

 construct a consistent system of geometry, whose laws, as a 

 matter of strict reasoning, have a validity and truth not inferior 

 to those with which we are habitually familiar. Such systems 

 havmg been actually constructed, the question has not un- 

 naturally been asked, whether there is anything in nature or in 



the outer world to which they correspond ; 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 con- 

 figuration 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 vsritings of 

 some of our greatest and most original mathematicians. Gauss, 

 Riemann, and Helmholtz have thrown out suggestions radiating, 

 as it were, in these various directions from a common centre ; 

 while Cayley, Sylvester, and Clifford in this country, Klein in 

 Germany, Lobatscheffsky in Russia, Bolyai in Hungary, and 

 Beltrami in Italy, with many others, have reflected similar ideas 

 with all the modifications due to the chromatic dispersion or 

 their individual minds. But to the main question the answer 

 must be in the negative. And, to use the words of Newton, 

 since "geometry has its foundation in mechanical practice,' 

 the same must be the answer until our experience is dif- 

 ferent from what it now is. And yet, all this notwithstanding, 

 the generalised conceptions of space are not without their prac- 

 tical utility. The principle of representing space of one kind 

 by that of another, and figures belonging to one by their ana- 

 logues in the other, is not only recognised as legitimate in pure 

 mathematics, but has long ago found its application in carto- 

 gi-aphy. In maps or charts, geogi-aphical positions, the contour 

 of coasts, and other features, belonging in reality to the earth's 

 surface, are represented on the flat ; and to each mode of repre- 

 sentation, or projection as it is called, there corresponds a special 

 correlation between the spheroid and the plane. To this might 

 perhaps be added the method of descriptive geometry, and all 

 similar processes in use by engineers, both military and civil. 



It has often been asked whether modern research in the field 

 of pure mathematics has not so completely outstripped its phy- 

 sical applications as to be practically useless ; whether the 

 analyst and the geometer might not now, and for a long time to 

 come, fairly say, "hie artem remumque repono," and turn his 

 attention to mechanics and to physics. That the Pure has out- 

 stripped the Applied is largely true ; but that the former is on 

 that account useless is far from true. Its utility often crops up 

 at unexpected points : witness the aids to classification of physi- 

 cal quantities, furnished by the ideas (of Scalar and Vector) 

 involved in the " Calculus of Quaternions ; " or the | advantages 

 which have accnied to physical astronomy from Lagrange's 

 "Equations," and from Hamilton's "Principle of Varying 

 Action;" on the value of complex quantities, and the proper- 

 ties of general integrals, and of general theorems on integration 

 for the theories of electricity and magnetism. The utility of 

 such researches can in no case be discounted, or even imagined 

 beforehand; who, for instance, would have supposed that the 

 calculus of forms or the theory of substitutions would have 

 thrown much light upon ordinary equations ; or that Abolian 

 functions and hyperelliptic transcendents would have told us any- 

 thing about the properties of curves ; or that the calculus of 

 operations would have helped us in any way towards the figure 

 of the earth ? But upon such technical points I must not now 

 dwell. If, however, as I hope, it has been sufficiently shown 

 that any of these more extended ideas enable us to combine 

 together, and to deal with as one, properties and processes 

 which from the ordinary point of view present marked distinc- 

 tions, then they will have justified their own existence ; and in 

 using them we shall not have been walking in a vain shadow, 

 nor disquieting our brains in vain. 



Mathematical Symmetry. 

 These extensions of mathematical ideas would, however, be | 

 overwhelming, if they were not compensated by some simplifi- 

 cations in the processes actually employed. Of these aids to 

 calculation I will mention only two, viz., symmetry of form and 

 mechanical appliances ; or, say, mathematics as a fine art, and 

 mathematics as a handicraft. And first, as to symmetry of 

 form. There are many passages of algebra in which long pro- 

 cesses of calculation at the outset seem unavoidable. Results 

 are often obtained in the first instance through a tangled maze of 



