THE FOURTH DIMENSION. 



6 9 



The question might be asked, In what way and to what extent 

 in this case is the specification of two numbers requisite and suffi- 

 cient to determine amid all the rays which pass through the speci- 

 fied point a definite individual ray? To get a clear idea of the 

 problem here involved, let us imagine the ray produced far into the 

 heavens where some quite definite point will correspond to it. Now, 

 the position of a point in the heavens depends, as does the position 

 of a point on all spherical surfaces, on two numbers. In the heavens 

 these two numbers are ordinarily supplied by the two angles called 

 altitude, or the distance above the plane of the horizon, and azi- 

 muth, or the angular distance between the circle on which the alti- 

 tude is measured and the meridian of the observer. It will be seen 

 thus that the totality of all the luminous rays that an eye, conceived 

 as a point, can receive from the outer world is two-dimensional, and 

 also that a luminous point emits a two-dimensional group of luminous 

 rays. It will also be observed, in connexion with this example, that 

 the two-dimensional totality of all the rays that can be drawn through 

 a point in space is something different from the totality of the rays 

 that pass through a point but are required to lie in a given plane. Such 

 a group of objects as the last-named one is a one-dimensional totality. 



Now that we have sufficiently discussed the attributes that are 

 characteristic of one and two-dimensional aggregates, we may, 

 without any further investigation of the subject, propose the follow- 

 ing definition, that, generally, an n-dimensional totality of infinitely 

 numerous things is such that the specification of n numbers is necessary 

 and sufficient to indicate definitely any individual amid all the infinitely 

 numerous individuals of that totality. 



Accordingly, the point-aggregate made up of the world-space 

 which we inhabit, is a three-dimensional totality. To get our bear- 

 ings in this space and to define any determinate point in it, we have 

 simply to lay through any point which we take as our zero-point 

 three axes at right angles to each other, one running from right to 

 left, one backwards and forwards, and one upwards and downwards. 

 We then join each two of these axes by a plane and are enabled 

 thus to specify the position of every point in space by the three per- 

 pendicular distances by which the point in question is removed in a 



