35 



tion of a particitlar law, or pointing out a stage wliere 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 Illustration 

 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 ex- 

 perience, nor measured by scale as we count measure- 

 ment ; 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 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 



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