Sept. 20, 1883] 



NATURE 



493 



shilling in purchasing power, although they may not be abso- 

 lutely of the same weight and fineness : but it is hardly necessary ; 

 one coin + one coin = two coins, even if the one be a shilling 

 and the other a half-crown. In fact, whatever difficulty be 

 raisable as to geometry, it seems to me that no similar difficulty 

 applies to arithmetic ; mathematician or not, we have each of us, 

 in its most abstract form, the idea of a number ; we can each of 

 us appreciate the truth of a proposition in regard to numbers ; 

 and we cannot but see that a truth in regard to numbers is some- 

 thing different in kind from an experimental truth generalised 

 from experience. Compare, for instance, the proposition that 

 the sun, having already risen so many times, will rise to-morrow, 

 and the next day, and the clay after that, and so on ; and the 

 proposition that even and odd numbers succeed each other alter- 

 nately ad infinitum : the latter at least seems to have the cha- 

 racters of universality and necessity. Or, again, suppose a 

 proposition observed to hold good for a long series of numbers, 

 one thousand numbers, two thousand numbers, as the case may 

 be : this is not only no proof, but it is absolutely no evidence, 

 that the proposition is a true proposition, holding good for all 

 numbers whatever ; there are in the Theory of Numbers very 

 remarkable instances of propositions observed to hold good for 

 very long series of numbers and which are nevertheless untrue. 



I pass in review certain mathematical theories. 



In arithmetic and algebra, or say in analy is, the numbers or 

 magnitudes which we repre-ent by symbols are in the first 

 instance ordinary (that h, positive) numbers or magnitudes. We 

 have also in analysis and in analytical geometry negative magni- 

 tudes ; there has been in regard to these plenty of philosophical 

 discussion, and I might refer to Kant's paper, " Ueber die 

 negativen Grossen in die Weltweisheit " (1763), but the notion of 

 a negative magnitude has become quite a familiar one, and has 

 exteuded itself into common phraseology. I may remark that 

 it is used in a very refined manner in bookkeeping by double 

 entry. 



But it is far otherwie with the notion which is rea'ly the 

 fundamental one (and I cannot too strongly emphasise the asser- 

 tion) underlying and pervading the whole of modern analysis and 

 geometry, that of imaginary magnitude in analysis and of 

 imaginary space (or space as a locus in quo of imaginary points 

 and figures) in geometry : I use in each case the word imaginary 

 as including real. This has not been, so far as I am aware, a 

 subject of philosophical discussion or inquiry. As regards the 

 older metaphyseal writers, this would be quite accounted for 

 by saying that they knew nothing, and were not bound to know 

 anything, about it ; but at present, and considering the pro- 

 minent position which the notion occupies — say even that the 

 conclusion uere that the notion belongs to mere technical 

 mathematics, or has reference to nonentities in regard to which 

 no science is possible, still it seems to me that (as a subject of 

 philosophical discussion) the notion ought not to be thus 

 ignored ; it should at least be shown that there is a right to 

 ignore it. 



Although in logical order I should perhaps now speak of the 

 notion just referred to, it will be convenient to speak first of 

 some other quasi-geometrical notions ; those of more-than three- 

 dimensional space, and of non-Euclidian two- and three- 

 dimensional space, and also of the generalised notion of 

 distance. It is in connection with these that Riemann comi- 

 dered that our notion of space is founded on experience, or 

 rather that it is only by experience that we know that our space 

 is Euclidian space. 



It is well known that Euclid's twelfih axiom, even in Playfair's 

 form of it, has been considered as needing demonstration ; and 

 that Lobatschewsky constructed a perfectly consistent theory 

 wherein this axiom was assumed not to hold good, or say a 

 system of non-Euclidian plane geometry. There is a like sys- 

 tem of non-Euclidian solid geometry. My own view is that 

 Euclid's twelfth axiom in Playfair's form of it does not need 

 demonstration, but is part of our notion of space, of the physical 

 space of our experience — the space, that is, which we become 

 acquainted with by experience, but which is the representation 

 lying at the foundation of all external experience. Riemann's 

 view before referred to may I think be said to be that, having 

 in intellectu a more general not ion of space (in fact a notion of 

 non-Euclidian space), we learn by experience that space (the 

 physical space of our experience) is, if not exactly, at least to 

 the highest degree of approximation, Euclidian space. 



But, suppose the physical space of our experience to be thus 

 only approximately Euclidian space, what is the consequence 



which follows ? Not that the propositions of geometry are only 

 approximately true, but that they remain absolutely true in re- 

 gard to that Euclidian space which has been so long regarded as 

 being the physical space of our experience. 



It is interesting to consider two different ways in which, with- 

 out any modification at all of our notion of space, we can arrive 

 at a system of non-Euclidian (plane or two-dimensional) 

 geometry ; and the doing so will, I think, throw some light on. 

 the whole question. 



First, imagine the earth a perfectly smooth sphere ; understand 

 by a plane the surface of the earth, and by a line the apparently 

 straight line (in fact an arc of preat circle) drawn on the surface; 

 what experience would in the first instance teach would be 

 Euclidian geometry ; there would be intersecting lines which 

 produced a few miles or so would seem to go on diverging, ai 65 

 apparently parallel lines which would exhibit no tendency to 

 approach each other ; and the inhabitants might very well con- 

 ceive that they had by experience established the axiom that two 

 straight lines cannot inclose a space, and the axiom as to parallel 

 lines. A more extended experierce and more accurate measure- 

 ments 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 spheri- 

 cal geometry, accurately representing the properties of the two- 

 dimensional space of their experience. But their original 

 Euclidian gerrnetry would not the les be a true system; only 

 it would apply to an ideal space, not the space of their 

 experience. 



Secondly, consider an ordinary, indefinitely extended place; 

 and let us modify only the notion of distance. We measure 

 distance, say, by a yard measure or a foot rule, anything which 

 is short enou?h to make the fractions of it of no consequence (in 

 mathematical language by an infinitesimal element of length) ; 

 imagine, then, the length of this rule constantly changing (as it 

 might do by an alteration of temperature), but under the condi- 

 tion 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 have the same length. 

 The distance along a given straight or curved line between any 

 two points could then he 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, hut in quite a different ac- 

 ceptation. 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 the configuration of the ground, and the 

 distance along a given path between anv 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 different acceptation. And corresponding 

 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 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 in- 

 finite ; 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 

 the ne'v 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 'he plane space of the theoi y 

 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 Lobat- 

 schewsky. But in so ohtainingit we put out of sight its relation 

 to spherical geometry : the three geometries (spherical, Euclidian, 

 and Lobatschewsky's) should he 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 respectively ; or, again, they are the plane 

 geometries corresponding to three different notions of distance ; 



