October 5, 1883.] 



SCIENCE. 



481 



clidian geometry: there would be intersecting lines, 

 which, produced a few miles or so, would seem to go 

 on diverging, and apparently parallel lines, which 

 would exhibit no tendency to approach each other; 

 and the inhabitants might very well conceive that 

 they had by experience established the axiom that 

 two straight lines cannot enclose a space, and the 

 axiom as to parallel lines. A more extended expe- 

 rience and more accurate measurements would teach 

 them that the axioms were each of them false; and 

 that any two lines, if produced far enough each way, 

 would meet in two points: they would, in fact, arrive 

 at a spherical geometry, accurately representing the 

 properties of the two-dimensional space of their ex- 

 perience. But their original Euclidian geometry 

 would not the less be a true system; only it would 

 apply to an ideal space, not the space of their expe- 

 rience. 



Secondly, consider an ordinary, indefinitely ex- 

 tended plane; and let us modify only the notion of 

 distance. We measure distance, say, by a yard meas- 

 ure or a foot rule, any thing which is short enough to 

 make the fractions of it of no consequence (in mathe- 

 matical language, by an infinitesimal element of 

 length). Imagine, then, the length of this rule con- 

 stantly changing (as it might do by an alteration of 

 temperature), but under the condition that its actual 

 length shall depend only on its situation on the plane, 

 and on its direction; viz., if for a given situation and 

 direction it has a certain length, then whenever it 

 comes back to the same situation and direction it 

 must liave the same length. The distance along a 

 given straight or curved line between any two points 

 could then be measured in the ordinary manner with 

 this rule, and would have a perfectly determinate 

 value; it could be measured over and over again, and 

 would always be the same: but of course it would be 

 the distance, not in the ordinary acceptation of the 

 term, but in quite a different acceptation. Or in a 

 somewhat different way: if the rate of progress from 

 a given point in a given direction be conceived as 

 depending only on tlie configuration of the ground, 

 and the distance along a given path between any two 

 points thereof be measured by the time required for 

 traversing it, then in this way, also, the distance would 

 have a perfectly determinate value ; but it would be a 

 distance, not in the ordinary acceptation of the term, 

 but in i|uile a different acceptation; and, correspond- 

 ing to the new notion of distance, we should have a 

 new non-Euclidian system of plane geometry. All 

 theorems involving the notion of distance would be 

 altered. 



We may proceed farther. Suppose that as the rule 

 moves away from a fixed central point of the plane it 

 becomes shorter and shorter: if this shortening take 

 place with sufficient rapidity, it may very well be that 

 a distance which in the ordinary sense of the word is 

 finite will in the now sense be infinite. No number 

 of repetitions of the length of the ever-shortening rule 

 will be sufficient to cover it. There will be surround- 

 ing the central point a certain finite area, such that 

 (in the new acceptation of the term ' distance ') each 

 point of the boundary thereof will be at an infinite 



distance from the central point. The points outside 

 this area you cannot by any means arrive at with 

 your rule: they will form a terra incognita, or, rather, 

 an unknowable land (in mathematical language, an 

 imaginary or impossible space); and the plane space 

 of the theory will be that within the finite area, that 

 is, it will be finite instead of infinite. 



We thus, with a proper law of shortening, arrive at 

 a system of non-Euclidian geometry which is essen- 

 tially that of Lobatschewsky ; but, in so obtaining it, 

 we put out of sight its relation to spherical geometry. 

 The three geometries (spherical, Euclidian, and Lo- 

 batschewsky's) should be regarded as members of a 

 system : viz., they are the geometries of a plane (two- 

 dimensional) space of constant positive curvature, 

 zero curvature, and constant negative curvature, re- 

 spectively ; or, again, they are the plane geometries 

 corresponding to three dilTerent notions of distance. 

 In this point of view, they are Klein's elliptic, para- 

 bolic, and hyperbolic geometries respectively. 



Next as regards solid geometry : we can, by a mod- 

 ification of the notion of distance (such as has just 

 been explained in regard to Lobatschewsky's system), 

 pass from our present system to a non-Euclidian sys- 

 tem. For the other mode of passing to a non-Euclidi- 

 an system, it would be necessary to regard our space 

 as a flat three-dimensional space existing in a space 

 of four dimensions (i.e., as the analogue of a plane 

 existing in ordinary space), and to substitute for 

 such flat three-dimensional space a curved three-di- 

 mensional space, say, of constant positive or negative 

 cur\ature. In regarding the physical space of our 

 experience as possibly non-Euclidian, Riemann's idea 

 seems to be that of modifying the notion of distance, 

 not that of treating it as a locus in foiur-dimeusional 

 space. 



I have just come to speak of four-dimensional 

 space. What meaning do we attach to il ? or can we 

 attach to it any meaning? It may be at once ad- 

 mitted that we cannot conceive of a fourth dimen- 

 sion of space ; that space as we conceive of it, and 

 the physical space of our experience, are alike three- 

 dimensional. But we can, 1 think, conceive of space 

 as being two- or even one-dimensional; we can im- 

 agine rational beings living in a one-dimensional 

 space (a line) or in a two-dimensional space (a sur- 

 face), and conceiving of space accordingly, and to 

 whom, therefore, a two-dimensional space or (as the 

 case may be) a three-dimensional space would be as 

 inconceivable as a four-dimensional space is to us. 

 And very curious speculative questions arise. Sup- 

 pose the one-dimensional space a right line, and that 

 it afterwards becomes a curved line: would there be 

 any indication of the ch.ange "? or, if originally a 

 curved line, would there be any thinglo suggest to 

 them that it was not a right line ? rrob:ibly not; for 

 a one-dimensional geometry hardly e.\ists. But let 

 the space be two-dimensional, and im.igine it origi- 

 nally a plane, and afterwards bent (converted, that 

 is, into some form of developable surface), or con- 

 verted into a curved surface; or imagine it originally 

 a developable or curved surface. In the former case 

 there should be an indication of the change, for tlie 



