ADDRESS. 21 



change in the base of operations has, in fact, been snccessf ully made in 

 geometry of two dimensions, and although we have not the same experi- 

 mental data for the further steps, yet neither the modes of reasoning, nor 

 the validity of its conclnsions, are in any way affected by applying an 

 analogous mental process to geometry of three dimensions ; and hj 

 regarding figures in space of three dimensions as sections of figures in 

 space of four, in the same way that figures in piano are sometimes con- 

 sidered 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 pur- 

 poses 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 simjile flexure, without either stretching or tearing ; 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 co-ordinates as they are called, 

 determine the position of a point in space, so do three letters or measure- 

 able quantities serve for the same purpose in the language of algebra. 

 Now, many algebraical problems involving three unknown or variable 

 quantities admit of being generalised 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 corresponds a problem 

 in geometry of many dimensions. 



There is, however, another aspect under which even ordinary space 

 presents to us a four-fold, or indeed a mani-fold, character. In modern 

 Physics, space is regarded not 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 matter, 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 purposes different elements may be chosen ; and upon the 

 degree of complexity of the subject of our choice will depend the internal 

 structure or mani-foldness of space. 



Thus, beginning with the simplest case, a point may have any siDgly 

 infinite multitude of positions in a line, which gives a one-fold system of 

 points in a line. The line may revolve in a plane about any one of its 

 points, giving a two-fold system of points in a plane ; and the plane may 

 revolve about any one of the lines, giving a three-fold system of points in 

 space. 



Suppose, however, that we take a straight line as our element, and 



