20 REPORT — 187 S. 



purely geometric curves, who lias not felt that these imaginaries have 

 a claim to recognition very similar to that of their congeners in mathe- 

 matics ? How is it that the grotesque paintings of the Middle Ages, the 

 fantastic sculpture of remote nations, and even the rude art of the pre- 

 historic 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 com- 

 prehensive point 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 ? 



Ao-ain if we turn from Art to Letters, truth to nature and to fact is un- 

 doubtedly a characteristic of sterling literature ; and yet in the delinea- 

 tion of outward nature itself, still more in that of feelings and affections, 

 of the secret parts of character and motives of conduct, it frequently 

 happens that the writer is driven to imagery, to an analogy, or even to 

 a paradox, in order to give utterance to that of which there is no direct 

 counterpart in recognised speech. And yet which of us cannot find 

 a meaning for these literary figures, an inward response to imaginative 

 poetry, to social fiction, or even to those tales of giant and fairyland 

 written, it is supposed, only for the nursery or schoolroom ? But in order 

 thus to reanimate these things with a meaning beyond that of the mere 

 words, have we not to reconsider our first position, to enlarge the ideas with 

 which we started ; have we not to cast about for some thing which is 

 common to the idea conveyed and to the subject actually described, and 

 to seek for the sympathetic spring which underlies both ; have we not, 

 like the mathematician, to go back as it were to some first principles, or, 

 as it is pleasanter to describe it, to become again as a little child ? 



Passing to the second of the three methods, viz., that of Manifold 

 Space, it may first be remarked that our whole experience of space is in 

 three dimensions, viz., of that which has length, breadth, and thickness; 

 and if for certain purposes we restrict our ideas to two dimensions as in 

 plane geometry, or to one dimension as in the division of a straight line, 

 we do this only by consciously and of deliberate purpose setting aside, 

 but not annihilating, the remaining one or two dimensions. Negation, as 

 Hecel has justly remarked, implies that which is negatived, or, as he 

 expresses it, affirms the opposite. It is by abstraction from previous ex- 

 perience, by a limitation of its results, and not by any independent process, 

 that we arrive at the idea of space -whose dimensions are less than three. 



It is doubtless on this account that problems in plane geometry which, 

 although capable of solution on their own account, become much more 

 intelligible more easy of extension, if viewed in connexion with solid 

 space, and as special cases of corresponding problems in solid geometry. 

 So eminently is this the case, that the very language of the more general 

 method often leads us almost intuitively to conclusions which, from the 

 more restricted point of view, require long and laborious proof. Such a 



