4IO 



NATURE 



[August 15, 1878 



mitted to speak. For to the fertile imagination of the late 

 Astronomer-Royal for Ireland we are indebted for that mar- 

 vellous calculus of quaternions, which is only now beginning to 

 be fully understood, and which has not yet received all the 

 applications of which it is doubtless capable. And even 

 although this calculus be not coextensive with another (the 

 Ausdehnungslehre of Grassmann)^ which almost simultaneously 

 germinated on the Continent, nor with ideas more recently 

 developed in America (Pierce's "Linear Associative Algebras") ;'^ 

 yet it must always hold its position as an original discovery and 

 as a representative of one of the two great groups of generalised 

 algebras (viz., those the squares of whose units are respectively 

 negative unity and zero) the common origin of which must still 

 be marked on our intellectual map as an unknown region. Well 

 do I recollect how in its early days we used to handle the method 

 as a magician's page might try to wield his master's wand, 

 trembling as it were between hope and fear, and hardly knowing 

 whether to trust our own results until they had been submitted 

 to the present and ever ready coimsel of Sir W. R. Hamilton 

 himself. 



To fix our ideas, consider the measurement of a line, or the 

 reckoning of time, or the performance of any mathematical 

 operation. A line may be measured in one direction or in the 

 opposite ; time may be reckoned forward or backward ; an 

 operation may be performed or be reversed, it may be done or 

 may be undone ; and if having once reversed any of these processes 

 we reverse it a second time, we shall find that we have come 

 back to the original direction of measurement or reckoning, or 

 to the original kind of operation. 



Suppose, however, that at some stage of a calculation our 

 forrnulse indicate an alteration in the mode of measuremen such 

 that if the alteration be repeated, a condition of things ntot the 

 same as, but the reverse of the original, will- be produced. Or 

 suppose that, at a certain stage, our transformations indicate that 

 time is to be reckoned in some manner diflferent from future or 

 past, but still in a way having definite algebraical connection 

 with time which is gone and time which is to come.3 It is clear 

 that in actual experience there is no process to which such 

 measurements correspond. Time has no meaning except as 

 future or past ; and the present is but the meeting point of the 

 two. Or, once more, suppose that we are gravely told that all 

 circles pass through the same two imaginary points at an infinite 

 distance, and that every line drawn through one of these points 

 is perpendicular to itself. On hearing the statement we shall 

 probably whisper, with a smile or a sigh, that we hope it is not 

 true, but that in any case it is a long way off, and perhaps, after 

 all, it does not very much signify. If, however, we are not 

 satisfied to dismiss the question on these terms, the mathe- 

 matician himself must admit that we have here reached a defi- 

 nite point of issue. Our science must either give a rational 

 account of the dilemma, or yield the position as no longer 

 tenable. 



ExJ>lanation of them. 



Special modes of explaining this anomalous state of things 

 have occurred to mathematicians. But, omitting details as un- 

 suited to the present occasion, it will, I think, be sufficient to 

 point out in general terms that a solution of the difficulty is to 

 be found in the fact that the formulce which give rise to these 

 results are more cojipreheasive than the signification which has 

 been given to them ; and when we pass out of the condition of 

 things first contemplated they cannot (as it is obvious they ought 

 not) give us any results intelligible on that basis. But it does 

 not_ therefore by any means follow that upon a more enlarged 

 basis the formulae are incapable of interpretation ; on the con- 

 trary, the difficulty at which we have arrived indicates that 

 there must be some more comprehensive statement of the pro- 

 blem which will include cases impossible in the more limited, 

 but possible in the wider view of the subject. 



' Grunert's Archiv. , vol. vl. p. 337 ; also separate work, Berlin, 1862. 



» "Linear Associative Algebra," by Benjamin Pierce, Wash.ngton City, 

 1S70. 



3 Sir W. Thomson, Cambridge Matliematical Journal, vol. iii. p. 174. 

 Jevons' "Principles of Science," vol. ii. p. 438. But an explanation of! the 

 difficulty seems to me to be found in the fact that the problem, as stated, is 

 one of the conduction of heat, and that the " impossib.lity " which attaches 

 Itself to the expression for the ' ' time ' ' merely means that previous to a certain 

 epoch the conditions which gave rise to the phenomena were not those of 

 conduction, but those of some other action of heat. If, therefore, we desire 

 to comprise the phenomena of the earher as well as of the later period in one 

 problem, we must find some more general statement; viz., that of physical 

 th" tVS"^ ^liich at the critical epoch will issue in a case of conduction. I 

 tiunK.tiiat Prof. Clifford has somewhere given a similar explanation. 



A very simple instance will illustrate the matter. If from a 

 point outside a circle we draw a straight line to touch the curve, 

 the distance between the starting point and the point of con- 

 tact has certain geometrical properties. If the starting point 

 be shifted nearer and nearer to the circle the distance in 

 question becomes shorter, and ultimately vanishes. But as 

 soon as the point passes to the interior of the circle the notion 

 of a tangent and distance to the point of contact cease to have 

 any meaning; and the same anomalous condition of things 

 prevails as long as the point remains in the interior. But if 

 the point be shifted still further until it emerges on the other 

 side, the tangent and its properties resume their reality; and 

 are as intelligible as before. Now the process whereby we 

 have passed from the possible to the impossible, and again re- 

 passed to the possible (namely the shifting of the starting point) 

 is a perfectly continuous one, while the conditions of the pro- 

 blem as stated above have abruptly changed. If, however, we 

 replace the idea of a line touching by that of a line cutting the 

 circle, and the distance of the point of contact by the distances 

 at which the line is intercepted by the curve, it will easily be 

 seen that the latter includes the former as a limiting case, when 

 the cutting line is turned about the starting point until it coin- 

 cides with the tangent itself. And further, that the two inter- 

 cepts have a perfectly distinct and intelligible meaning whether 

 the point be outside or inside the area, ITie only difTerence 

 is that in the first case the intercepts are measured in the same 

 direction ; in the latter in opposite directions. 



The foregoing instance has shown one purpose which these 

 imaginaries may serve, viz., as marks indicating a limit to a 

 particular condition of things, to the application of a particular 

 law, or pointing out a stage where a more comprehensive law is 

 required. To attain to such a law we must, as in the inst^ce 

 of the circle and tangent, reconsider our statement of the ^^o- 

 blem ; we must go back to the principle from which we set dVit, 

 and ascertain whether it may not be modified or enlarged. And 

 even if in any particular investigation, wherein imaginaries have 

 occurred, the most comprehensive statement of the problem of 

 which we are at present capable fails to give an actual repre- 

 sentation of these quantities ; if they must for the present be 

 relegated to the category of imaginaries ; it still does not follow 

 that we may not at some future time find a law which will endow 

 them with reality, nor that in the meantime we need hesitate to 

 employ them, in accordance with the great principle of con- 

 tinuity, for bringing out correct results. 



Illustration from Art and Literature. 



If, moreover, both in geometry and in algebra w'e occasion- 

 ally make use of points or of quantities which from our present 

 outlook have no real existence, which can neither be delineated 

 in space of which we have experience, nor measured by scale as 

 we count measurement ; if these imaginaries, as they are termed, 

 are called up by legitimate processes of our science ; if they 

 serve the purpose not merely of suggesting ideas, but of actually 

 conducting us to practical conclusions ; if all this be true in 

 abstract science, I may perhaps be allowed to point out, at all 

 events in illustration, that in art unreal forms are frequently 

 used for suggesting ideas, for conveying a meaning for which 

 no others seem to be suitable or adequate. Are not forms un- 

 known to biology, situations incompatible with gravitation, 

 positions which challenge not merely the stability but the very 

 possibility of equilibrium — are not these the very means to 

 which the artist often has recourse in order to convey his mean- 

 ing and to fulfil his mission ? Who that has ever revelled in the 

 ornamentation of the Renaissance, in the extraordinary tran- 

 sitions from the animal to the vegetable, from faunic to floral 

 forms, and from these again to almost purely geometric curves, 

 who has not felt that these imaginaries have a claim to recog- 

 nition very similar to that of their congeners in mathematics ? 

 How is it that the grotesque paintings of the middle ages, the 

 fantastic sculptmre of remote nations, and even the rude art of 

 the prehistoric past, still impress us, and have an interest over 

 and above their antiquarian value ; unless it be that they are 

 symbols which, although hard of interpretation when taken 

 alone, are yet capable from a more comprehensive ^xjint of view 

 of leading us mentally to something beyond themselves, and to 

 truths which, although reached through them, have a reality 

 scarcely to be attributed to their outward forms ? 



Again, if we turn from art to letters, truth to nature and to 

 fact is undoubtedly a characteristic of sterling literature ; and 

 yet in the delineation of outward nature itself, still more in that 



