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♦ KNOWLEDGE ♦ 



[Nov. 2, 1883. 



anything which is short enough to make the fractions of it 

 of no consequence (in mathematical language by anintini- 

 tesimal element of length) ; imagine, then, the length of 

 this rule constantly changing (as it might do by an altera- 

 tion 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 lengtli, then whenever it comes 

 ba,ck to the same situation and direction it must have 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 dill'erent acceptation. Or in a some- 

 what dill'erent way ; if the rate of progress from a given 

 point in a given direction be conceived as depending only 

 on the 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 quite a diti'erent accepta- 

 tion. And corresponding to the new notion of distance we 

 should have a new, non-Euclidian system of plane geo- 

 metry ; all theorems involving the notion of distance would 

 be altered. 



We may proceed further. Suppose that as the rule 

 moves away from a fixed central point of the plane it 

 becomes shorter and shorter ; if this shortening takes 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 new sense be infinite ; no number of repetitions of the 

 length of the ever-shortening rule will be sufficient to cover 

 it. There will be surrounding the central point a certain 

 finite area such that (in tlie 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 un- 

 knowable 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 

 essentially that of Lobatschewsky. But in so obtaining it 

 we put out of sight its relation to spherical geometry ; the 

 three geometries (spherical, Euclidian, and Lobatschewsky's) 

 should be regarded as members of a system — viz., they are 

 the geometries of a plane (two-dimensional) space of con- 

 stant positive curvature, zero curvature, and constant 

 negative curvature respectively ; or again they are the 

 plane geometries corresponding to three ditierent notions of 

 distance ; in this point of view they are Klein's elliptic, 

 parabolic, and hyperbolic geometries respectively. 



Next, as regards solid geometry, we can by a modifica- 

 tion of the notion of distance (such as has just been ex- 

 plained in regard to Lobatschewsky's system) pass from 

 our present system to a non-Euclidian system ; for the 

 other mode of passing to a non-Euclidian 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 fiat three-dimensional space a curved 

 three-dimensional space, say of constant positive or nega- 

 tive curvature. 



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 four-dimensional space. 



I have just come to speak of four-dimensional space. 

 What meaning do we attach to it ? Or can we attach to it 

 any meaning 1 It may be at once admitted that we can- 

 not conceive of a fourth dimension of space ; the 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 

 imagine rational beings living in a one dimensional space 

 (a line) or in a two-dimensional space (a surface) and con- 

 ceiving 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. Suppose the one-dimensional space a right 

 line, and that it afterwards becomes a curved line ; would 

 there be any indication of the change 1 Or, if originally a 

 curved line, would there be anything to suggest to them 

 that it was not a right line ? Probably not, for a one- 

 dimensional geometry hardly exists. 



But let the space be two-dimensional, and imagine it 

 originally a plane and afterwards bent (converted, that is, 

 into some form of developable surface) or converted into a 

 curved surface ; or imagine it originally a developable or 

 curved surface. In the former case there should be an in- 

 dication of the change, for the geometry originally applic- 

 able to the space of their experience (our own Euclidian 

 geometry) would cease to be applicable ; but the change 

 could not be apprehended by them as a bending or defor- 

 mation of the plane, for this would imply the notion of a 

 three-dimensional space in which this bending or deforma- 

 lion could take place. In the latter case their geometry 

 would be that appropriate to the developable or curved 

 surface which is their space — viz., this would be their 

 Euclidian geometry ; would they ever have arrived at our 

 more simple system '? 



But take the case where the two-dimensional space is a 

 plane, and imagine the beings of such a space familiar 

 with our own Euclidian plane geometry ; if, a third 

 dimension being still inconceivable by them, they were 

 by their geometry or otherwise led to the notion of 

 it, there would be nothing to prevent them from forming 

 a science such as our own science of three-dimensional 

 geometry. 



Evidently all the foregoing questions present themselves 

 in regard to ourselves, and to three-dimensional space as 

 we conceive of it, and as the physical space of our ex- 

 perience. And I need hardly say that the first step is the 

 difficulty, and that granting a fourth dimension we may 

 assume as many more dimensions as we please. But what- 

 ever answer be given to them, we have, as a branch of 

 mathematics, potentially, if not actually, an analytical 

 geometry of vt-dimensional space. 



Coming now to the fundamental notion already referred 

 to, that of imaginary magnitude in analysis and imaginary 

 space in geometry, I connect this with two great discoveries 

 in mathematics made in the first half of the seventeenth 

 century, Harriot's representation of an equation in the 

 form f (.'■) = 0, and the consequent notion of the roots of 

 an equation as derived from the linear factors of f {x), 

 and Descartes' method of co-ordinates, as given in the 

 " Geometric. " By these we are led analytically to the 

 notion of imaginary points in geometry; for instance, we 

 arrive at the theorem that a straight line and a circle in 

 the same plane intersect always in two points, real or 

 imaginary. The conclusion as to the two points of inter- 

 section cannot be contradicted by experience ; take a sheet 

 of paper and draw on it the straight line and circle, and 



