38 



for the further steps, yet neither the modes of reasoning, 

 nor the vahdity 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 dimensions as sections of figures in space of 

 four, in the same way that figures in piano are some- 

 times 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 ; and that 

 in such a space it is impossible to tie a knot. 



Again, the solution of problems in geometry is often 

 eff'ected by means of algebra ; and as three measure- 

 ments, or co-ordinates 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 corre- 

 sponds 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, hoAvever, another aspect under which even 

 ordinary space presents to us a four-fold, or indeed a 

 mani-fold, character. In modern Physics, space is 



