INAUGURAL ADDRESS BEFORE THE BRITISH ASSOCIATION. 505 



things first contemplated they cannot (as it is 

 obvious they ought not) give us any results in- 

 telligible on that basis. But it does not there- 

 fore by any means follow that upon a more en- 

 larged basis the formulae are incapable of inter- 

 pretation ; on the contrary, the difficulty at which 

 we have arrived indicates that there must be some 

 more comprehensive statement of the problem 

 which will include cases impossible in the more 

 limited but possible in . the wider view of the 

 subject. 



A very simple instance will illustrate the mat- 

 ter. If from a point outside a circle we draw a 

 straight line to touch the curve, the distance be- 

 tween the starting-point and the point of contact 

 has certain geometrical properties. If the start- 

 ing-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 ceases 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 farther 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 repassed to the possible (namely, the shift- 

 ing of the starting-point), is a perfectly continuous 

 one, while the conditions of the problem 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 coincides with 

 the tangent itself. And, further, that the two in- 

 tercepts have a perfectly distinct and intelligible 

 meaning, whether the point be outside or inside 

 the area. The only difference is, that in the first 

 case the intercepts are measured in the same di- 

 rection ; 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 instance of the circle and tangent, re- 

 consider our statement of the problem ; we must 

 go back to the principle from which we set out, 



and ascertain whether it may not be modified or 

 enlarged. And even if in any particular inves- 

 tigation, wherein imaginaries have occurred, the 

 most comprehensive statement of the problem of 

 which we are at present capable fails to give 

 an actual representation 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 mean time we need hesitate to employ them, 

 in accordance with the great principle of continu- 

 ity, for bringing out correct results. 



If, moreover, both in geometry and in algebra 

 we occasionally make use of points or of quan- 

 tities which, from our present outlook, have no 

 real existence, which can neither be delineated 

 in space of which we have experience, nor meas- 

 ured 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 frequent- 

 ly used for suggesting ideas, for conveying a 

 meaning for which no others seem to be suitable 

 or adequate. Are not forms unknown 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 meaning and to fulfill his 

 mission ? Who that has ever reveled in the 

 ornamentation of the Renaissance, in the extraor- 

 dinary transitions from the animal to the vege- 

 table, 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 recognition very similar to that of their con- 

 geners in mathematics ? How is it that the gro- 

 tesque paintings of the middle ages, the fantastic 

 sculpture 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 point 

 of view, of leading us mentally to something be- 

 yond 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 



