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SCIENCE. 



[Vol. II., No. 35. 



have a likeness to, and are represented more or less 

 imperfectly, — and, from a geometer's point of view, 

 no matter how imperfectly, — by corresponding phys- 

 ical points, lines, circles, etc. I shall have to re- 

 turn to geometry, and will then speak of Riemann; 

 but I will first refer to another passage of the ' Logic' 



Speaking of the truths of arithmetic, Mill says (p. 

 297) that even here there is one hypothetical element: 

 " In all propositions concerning numbers, a condition 

 is implied without which none of them would be 

 true; and that condition is an assumption which may 

 be false. The condition is, that 1 = 1; tliat all the 

 numbers are numbers of the same or of equal units." 

 Here, at least, the assumption may be absolutely 

 true: one shilling=one shilling in purchasing-power, 

 although they may not be absolutely of the same 

 weight and fineness. But it is hardly necessai-y: 

 one coin-|-one coin=two coins, even if the one be a 

 shilling and the other a half-crown. In fact, what- 

 ever 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 something different in 

 kind from an experimental truth generalized 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 day after 

 that, and so on, and the proposition that even and 

 odd numbers succeed each other alternately ad infini- 

 tum : the latter, at least, seems to have the charac- 

 ters 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 propo- 

 sition is a true proposition, holding good for all num- 

 bers 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 analysis, the 

 numbers or magnitudes which we represent by sym- 

 bols are, in the first instance, ordinary (that is, posi- 

 tive) 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 weltweis- 

 heit' (1763); but the notion of a negative magni- 

 tude has become quite a familiar one, and has 

 extended itself into common phraseology. I may re- 

 mark that it is used in a very refined manner in 

 book-keeping by double entry. 



But it is far otherwise with the notion which is 

 really the fundamental one (and I cannot too strong- 

 ly emphasize the assertion ), underlying and pervading 

 the whole of modern analysis and geometry, — that 

 of imaginary magnitude in analysis, and of imagi- 

 nary 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 meta- 

 physical writers, this would be quite accounted for 

 by saying that they knew nothing, and were not 

 bound to know any thing, about it. But at present, 

 and considering the prominent position which the 

 notion occupies, — say, even, that the conclusion were 

 that the notion belongs to mere technical mathemat- 

 ics, or has reference to nonentities in regard to which 

 no science is possible, — still it seems to me, th.at, 

 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 con- 

 venient to speak first of some other quasi-geometri- 

 cal notions, — those of more-than-three-dimensional 

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

 sional space, and also of the generalized notion of dis- 

 tance. It is in connection with these, that Riemann 

 considered 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 twelfth axiom, even 

 in Playfair's form of it, has been considered as need- 

 ing demonstration, and that Lobatschewsky con- 

 structed 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 

 system 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, with which we be- 

 come acquainted by experience, but which is the rep- 

 resentation lying at the foundation of all external ex- 

 perience. Riemann's view, before referred to, may, I 

 think, be said to be, that, having in intellectv a more 

 general notion of space (in fact, a notion of non-Eu- 

 clidian 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 regard to that 

 Euclidian space which has been so long regarded as 

 being tlie physical space of our experience. 



It is interesting to consider two different ways in 

 which, without 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 ques- 

 tion. 



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 are 

 of great circle) drawn on the surface. What experi- 

 ence would in the first instance teach would be Eu- 



