JS'ov. 9, lb83.] 



♦ KNOWLEDGE * 



287 



MATHEMATICS OF THE IMAGINARY. 



By Richard A. Proctor. 



PROFESSOR CAYLEY, President of the British 

 Association, has delivered to that body (for the first 

 time in its history) an address devoted entirely to mathe- 

 matics — first with regard to the development of the mathe- 

 matical theories by which science has been advanced, 

 secondly (and principally) with reference to the strange 

 fancies about imaginary geometries and space of more than 

 three dimensions on which some of the ablest mathema- 

 ticians of our time have squandered their powers. Certain 

 journals, not too celebrated for profundity, but rather 

 known for the stupendous blunders about scientific matters 

 into which they have occasionally fallen, find much to 

 commend in the profundity of Professor Cayley's address. 

 The British Association was established, they tell us, 

 for the advancement, not for the popularisation of 

 science, and therefore the mere fact that out of 2,000 

 persons who collected to hear Cayley's opening address 

 not a hundred even knew what he was talking 

 about, and of these not ten could follow his semi-meta- 

 physical semi-transcendental reasoning, was not open to 

 exception. Just here, without considering the very doubt- 

 ful question, or rather the question not open to any sort of 

 doubt, whether discussions about non-Euclidian geometry 

 and dimensions outside length, breadth, and thickness, in 

 space, tend very greatly to the advancement of science, I 

 may note that whatever the purpose of the British Associa- 

 tion itself, the annual meetings of the Association have for 

 their object, distinctly and emphatically, the popularisation 

 of science. They even cater for this object in ways which 

 are little consistent in my opinion with the dignity of 

 science. Excursions are good things in their way, but they 

 certainly do not tend in any marked degree (at least as 

 arranged by the authorities of the British Association) to 

 advance scientific research. Evening assemblies, in which 

 the local folk, mayor, aldermen, and common councihnen, 

 with their wives and mothers, sisters, cousins and aunts, 

 form a motley gathering, in company with professors and 

 their kinsfolk, and many by no means scientific representa- 

 tives of fogeydom, are not calculated to excite a very pro- 

 found feeling of respect for science. 



No one who has attended meetings of the British Asso- 

 •ciation can have failed to notice how large a proportion of 

 the Members and Associates know scarce anything about 

 science. For such folk certainly a discourse on imaginary 

 points in space was wanting in interest, to say the least. 

 For my own part, I think all such discussions not only 

 wanting in value, but mischievous. All the powers of 

 mathematics as now developed, and as ever likely to be 

 developed, are insufficient for the work we want mathe- 

 matics to do for science ; and yet some of the ablest pro- 

 fessors of mathematics are not ashamed to waste their 

 energies in mere dreaming, and wild and foolish dreaming 

 at that. 



What, after all, do these notions about non-Euclidian 

 geometry and a fourth dimension in space, amount to ? In 

 mathematics we deal with conceptions, more or less illus- 

 trated by things within our experience and subject to 

 observation. All our conceptions, and all our experiences 

 so far as they can be pushed, agree in presenting certain 

 doctrines to us as axiomatic. It is not supposed, nor even 

 regarded as conceivable, by the mathematicians of the new 

 school, that our conceptions respecting mathematical 

 axioms will ever change, or our experiences be ever corrected 

 by others inconsistent with these conceptions. Straight 

 lines which diverge from a point we never can conceive 



as converging again, without losing their straightness, after 

 reaching a certain remoteness from a point of divergence. 

 Nor can we suppose or even imagine (" which is else ") that 

 were we only able to carry on two such lines far enough we 

 should find — say somewhere about a billion times the dis- 

 tance of the remotest star seen by the great Rosse telescope — 

 divergence beginning to be transformed into convergence. 

 So of other axiomatic properties. We cannot conceive a 

 fourth dimension in space besides length and breadth and 

 depth : if we take any plane whatever in which to measure 

 length and breadth, we cannot conceive that plane, while 

 remaining plane, to fail to divide all space into two portions, 

 one on one side of it the other on the other, and this being 

 so we cannot conceive any point as existing outside the 

 plane which has not a certain distance from the plane, a 

 distance which we may call depth or height as the case may 

 be. It would be absolutely essential to our conception of a 

 fourth dimension in space that we should be able to conceive 

 of such points. Go to, then, say the mathematicians of 

 the fancifully useless school, since we cannot conceive 

 diverging lines as meeting again let us pretend we can 

 conceive them so doing, and make a new geometry based 

 on such a conception (a.% Lobatschewsky has done) ; and 

 since we cannot conceive a fourth dimension in space let us 

 act as though we could, and try to find what would be the 

 order of things in universes of four, five, six, or n dimen- 

 sions. So long as we measure as we do measure, and con- 

 ceive perfect measurement as simply the kind of measure- 

 ment which we aim at but cannot attain, our geometry is 

 such as Euclid and his fellows have dealt with. Let us 

 therefore imagine another and entirely different state of 

 things, — let us pretend that every measuring line we use 

 varies in lenL^th according to the place in space where it is 

 used, increasing it may lie towards some central point 

 where it becomes absolutely infinite though it was finite 

 where we began to use it, or instead diminishing down to 

 nothing at the centre, though it was originally not to be 

 distinguished from a respectable foot-rule. Or instead 

 suppose it of a certain definite length at the centre, but 

 either increasing to infinity or diminishing to nothing at 

 some given distance from the centre. 



The folly of all this is manifest enough. It may suit 

 mathematicians of a certain school to suppose that l)y 

 talking about the inconceivable as if they could grasp and 

 understand it, they convince the world of their profundity. 

 But the world is not altogether wanting in common sense, 

 ■not ready to sing in chorus about the professors of the 

 unintelligible — 



If that will not suit tliem which will very well suit " we," 



Why what very very wise wise men these wise wise men must be. 



But is is not altogether upon the unwisdom of incon- 

 ceivable mathematics that I would comment. There is a 

 more serious aspect of the matter. Science wants for her 

 advancement the assistance of all who are competent to wort 

 well in her service. In mathematical research, especially, 

 science needs the lielp of every one of the few who caa 

 really advance our knowledge in that direction. Science, 

 then, has just reason to be angry when those who are best 

 able to assist in the development of mathematics waste 

 their time and scjuander their power in attempting to dis- 

 cover what would be the state of things in a world where 

 all things were unlike what we know and can alone con- 

 ceive, — where things equal to the same thing were not 

 equal to each other, where two and two made three or five, 

 where two straight lines might enclose a space or many 

 spaces, where lines and curves not intersecting in 02(r world 

 might have imaginary points of intersection, and where 

 finally besides length and breadth and depth there might 



