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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



of sterling literature; and yet in the delineation 

 of outward Nature itself, still more in that of 

 feelings and affections, of the secret springs 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 coun- 

 terpart in recognized speech. And yet which of 

 us cannot find a meaning for these literary figures, 

 an inward response to imaginative poetry, to so- 

 cial fiction, or even to those tales of giant and 

 fairy-land 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 re- 

 marked that our whole experience of space is in 

 three dimensions, viz., of that which has length, 

 breadth, and thickness ; and, if for certain pur- 

 poses 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 Hegel has justly 

 remarked, implies that which is negatived, or, as 

 he expresses it, affirms the opposite. It is by 

 abstraction from previous experience, by a limi- 

 tation 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 prob- 

 lems on plane geometry, although capable of 

 solution on their own account, become much 

 more intelligible, more easy of extension, if 

 viewed in connection 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, al- 

 though we have not the same experimental data 



for the further steps, yet neither the modes of 

 reasoning, nor the validity of its conclusions, are 

 in any way affected by applying an analogous 

 mental process to geometry of three dimensions, 

 and by regarding figures in space of three dimen- 

 sions as sections of figures in space of four in the 

 same way that figures in plane are sometimes 

 considered as sections of figures in solid space. 

 The addition of a fourth dimension to space not 

 only extends the actual properties of geometrical 

 figures, but it also adds new properties, which are 

 often useful for the purposes of transformation or 

 of proof. Thus it has recently been shown that 

 in four dimensions a closed material shell could 

 be turned inside out ' by simple flexure, without 

 either stretching or tearing, 2 and that in such a 

 space it is impossible to tie a knot. 



Again, the solution of problems in geometry 

 is often effected by means of algebra : and as 

 three measurements, or coordinates, as they are 

 called, determine the position of a point in space, 

 so do three letters or measurable quantities serve 

 for the same purpose in the language of algebra. 

 Now, many algebraical problems involving three 

 unknown or variable quantities admit of being 

 generalized, so as to give problems involving 

 many such quantities. And as, on the one hand, 

 to every algebraical problem involving unknown 

 quantities or variables by ones, or by twos, or by 

 threes, there corresponds a problem in geometry 

 of one or of two or of three dimensions, so on 

 the other it may be said that to every algebraical 

 problem involving many variables there corre- 

 sponds a problem in geometry of many dimen- 

 sions. 



There is, however, another aspect under 

 which even ordinary space presents to us a four- 

 fold, or indeed a manifold character. In modern 

 physics space is not regarded as a vacuum in 

 which bodies are placed and forces have play, 

 but rather as a plenum with which matter is co- 

 extensive. And, from a physical point of view, 

 the properties of space are the properties of mat- 

 ter, or of the medium which fills it. Similarly, 

 from a mathematical point of view, space may be 

 regarded as a locus in quo, as a plenum, filled 

 with those elements of geometrical magnitude 

 which we take as fundamental. These elements 

 need not always be the same. For different pur- 

 poses different elements may be chosen ; and 



1 S. Newcomb " On Certain Transformations of Sur- 

 faces," American Journal of Mathematics, vol. i., p. 1. 



*Tait "On Knots," "Transactions of the Roya. 

 Society of Edinburgh," vol. xxviii., p. 145. Klein, 

 Mathematische AnnaCen, ix., p. 478. 



