66 THE FOURTH DIMENSION. 



ably say, that the definition in question referred to something dif- 

 ferent. A cubical box, or some similar structure, whose angles are 

 all right angles and whose bounding surfaces are consequently all 

 rectangles is the only body of which it* can at all be unmistakably 

 asserted that there are three principal directions distinguishable in 

 it, of which any one can be called the length, any other the breadth, 

 and any third the thickness. We thus see that the notions of length, 

 breadth, and thickness are not sufficiently clear and universal to 

 enable us to derive from them any idea of what is meant when it is 

 said that every body possesses three dimensions, or that the space 

 of the world is three-dimensional. 



This distinction may be made sharper and more evident by the 

 following considerations : We have, let us suppose, a straight line 

 on which a point is situated, and the problem is proposed to deter- 

 mine the position of the point on the line in an unequivocal manner. 

 The simplest way to solve this is, to state how far the point is re- 

 moved in the one or the other direction from some given fixed point; 

 just as in a thermometer the position of the surface of the mercury 

 is given by a statement of its distance in the direction of cold or heat 

 from a predetermined fixed point the point of freezing water. To 

 state, therefore, the position of a point on a straight line, the sole 

 datum necessary is a single number, if beforehand we have fixed 

 upon some standard line, like the centimetre, and some definite 

 point to which we give the value zero, and have also previously de- 

 cided in what direction from the zero-point, points must be situated 

 whose position is expressed by positive numbers, and also in what 

 direction those must lie whose position is expressed by negative 

 numbers. This last-mentioned fact, that a single number is suffi- 

 cient to determine the place of a point in a straight line, is the real 

 reason why we attribute to the straight line or to any part of it a 

 single dimension. 



More generally, we call every totality or system, of infinitely 

 numerous things, 0<r-dimensional, in which one number is all that 

 is requisite to determine and mark out any particular one of these 

 things from among the entire totality. Thus, time is one-dimen- 

 sional. We, as inhabitants of the earth, have naturally chosen as 



