70 THE FOURTH DIMENSION. 



positive or negative sense from these three planes. It is customary 

 to denote the numbers which are the measures of these three dis- 

 tances by x, y, and z t the positive x, positively, and positive z ordi- 

 narily being reckoned in the right hand, the hitherward, and the 

 upward directions from the origin. If now, with direct reference to 

 this fundamental axial system, any particular specification of x, y, 

 ind z be made, there will, by such an operation, be cut out and iso- 

 lated from the three-dimensional manifoldness of all the points of 

 space a totality of less dimensions. If, for example, z is equal to 

 seven units or measures, this is equivalent to a statement that only 

 the two-dimensional totality of the points is meant, which consti- 

 tute the plane that can be laid at right angles to the upward-passing 

 z-axis at a distance of seven measures from the zero-point. Conse- 

 quently, every imaginable equation between x, y, and z isolates and 

 defines a two dimensional aggregate of points. If two different equa- 

 tions obtain between x, y, and z, two such two-dimensional totalities 

 will be isolated from among all the points of space. But as these 

 last must have some one-dimensional totality in common, we may 

 say that the co-existence of two equations between x,y, and z defines 

 a one-dimensional totality of points, that is to say a straight line, a 

 line curved in a plane, or even, perhaps, one curved in space. It 

 is evident from this that the introduction of the three axes of refer- 

 ence forms a bridge between the theory of space and the theory of 

 equations involving three variable quantities, x, y, z. The reason 

 that the theory of space cannot thus be brought into connection 

 with algebra in general, that is, with the theory of indefinitely nu- 

 merous equations, but only with the algebra of three quantities, x, 

 y, z, is simply to be sought in the fact that space, as we picture it, 

 can have only three dimensions. 



We have now only to supply a few additional examples of n- 

 dimensional totalities. All particles of air are four-dimensional in 

 magnitude when in addition to their position in space we also con 

 sider the variable densities which they assume, as they are expressed 

 by the different heights of the barometer in the different parts of the 

 atmosphere. Similarly, all conceivable spheres in space are four- 

 dimensional magnitudes, for their centres form a three-dimensional 



