ADDRESS. 1 9 



reality, and are as intelligible as before. Now tlie process whereby we 

 have passed from the possible to the impossible, and again repassed to the 

 possible (namely, the shifting of the starting point) is a perfectly con- 

 tinuons 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 cut- 

 ting line is turned about the starting point until it coincides vith the tangent 

 itself. And further, that the two intercepts 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 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 instance of the circle and tangent, reconsider 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 investigation, 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 meantime we need hesitate to 

 employ them, in accordance with the great principle of continuity, for 

 bringing out correct results. 



If, moreover, both in Geometry and in Algebra we occasionally 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, in illustration 

 of my argument, 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 unknown to Biology, situations incompatible 

 with gravitation, positions which challenge not merely the stability but 

 even the possibility of equilibrium,— are not these the very means to which 

 the artist often has recourse in order to convey his meaning and to fulfil 

 his mission ? Who that has ever revelled in the ornamentation of the 

 Renaissance, in the extraordinary transitions from the animal to the 

 vegetable, from faunic to floral forms, and from these again to almost 



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