THE FOURTH DIMENSION 389 



The equation 



ax -\-byc=zc 



represents a line in 2-space, but has no meaning in 1-space. 

 The equation 



ax -\- by -[• cz ^ d 



represents a plane in 3-space, but has no meaning in 2-space or 1-space. 



So, by analogy, the equation 



ax -}- by -^ cs -{- dw = e 

 would have a meaning in 4-space, — say a 3-space section of 4-space — 

 but has no meaning in 3-space. 



In general, an algebraic equation of Jc variables has no meaning in a 

 space of lower dimension than k, but has a meaning in 7i-space, where 



Discarding experience and reasoning wholly from analogy, we intro- 

 duce some properties of the fourth dimension as follows. 



Four-dimensional measure depends upon length, breadth, height and 

 a fourth dimension all multiplied together. In the graphical representa- 

 tion of 3-space, points are referred to three mutually perpendicular 

 planes formed by three lines mutually at right angles. In a similar 

 way, to represent 4-space we must assume another axis at right angles 

 to each of the other three. In the present development of human 

 thought, this is purely subjective, a mere mental conception, and it is 

 upon this conception that the theory of hyperspace is built. 



The position of a point in a plane may be determined, as we have 

 seen, by its distance from each of two perpendicular right lines; in 

 3-space, by its distance from each of three mutually perpendicular 

 planes ; and in 4-space, by its distance from each of four mutually per- 

 pendicular 3-spaces, for there are four arrangements of the four axes 

 taken three at a time, and each independent set of three perpendicular 

 axes define a 3-space, for example, wxy, wxz, wyz, xyz. Just as in our 

 space it requires at least three points to determine a plane (2-space), 

 so in 4-space four points are necessary to determine a 3-space. 



As portions of our space are bounded by surfaces, plane or curved, 

 so portions of 4-space are bounded by hyperspace (three-dimensional). 



In our space, a point moving in an unchanging direction generates 

 a straight line. 



This straight line (say of a units in length), moving perpendicular 

 to its initial position through the distance a, generates a square. 



This square, moving perpendicular to its initial position through the 

 distance a, generates a cube. 



This cube, we will suppose, moving perpendicular to our space for 

 a distance equal to one of its sides (that is, equal to a), will generate a 

 hypercube. 



Now the line contains a units, the square a- units, the cube a^ units, 

 the hypercube a* units. 



