37 



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 ? 



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

 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 

 iin 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 Hegel has justly remarked, implies that 

 \ which is negatived, or, as he expresses it, affirms the 

 : opposite. It is by abstraction from previous experience, 

 I by a limitation of its results, and not by any independent 

 I process, that we arrive at the idea of space whose dimen- 

 sions are less than three. 



It is doubtless on this account that problems in plane 

 geometry which, although capable of solution on theil* 

 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 change in the base of operations has, in fact, 

 been successfully made in geometry of two dimensions, 

 and although we have not the same experimental data 



