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regarded not as a vacuum in which bodies are placed 

 and forces have play, but rather as a plenum with which 

 matter is coextensive. And from a physical point of 

 view the properties of space are the properties of matter, 

 or of the medium which fills it. Similarly from a mathe- 

 matical point of view, space may be regarded as a locus 

 in quo, as a plenum, filled with those elements of geo- 

 metrical magnitude which we take as fundamental. 

 These elements need not always be the same. For 

 different purposes different elements m^ be chosen ; and 

 upon the degree of complexity of the subject of our 

 choice will depend the internal structure or mani-fold- 

 ness of space. 



Thus, beginning with the simplest case, a point may 

 have any singly infinite multitude of positions in a line, 

 which gives a one-fold system of points in a line. The 

 line may revolve in a plane about any one of its points, 

 giving a two-fold system of points in a plane ; and 

 the plane may revolve about any one of the lines, giving 

 a three-fold system of points in space. 



Suppose, however, that we take a straight line as our 

 element, and conceive space as filled with such lines. 

 This will be the case if we take two planes, e.g. two 

 parallel planes, and join every point in one with every 

 point in the other. Now the points in a plane form 

 a two-fold system, and it therefore follows that the 

 system of lines is four-fold ; in other words, space 

 regarded as a plenum of lines is four-fold. The same 

 result follows from the consideration that the lines in 

 a plane, and the planes through a point, are each 

 two-fold. 



Again, if we take a sphere as our element we can 

 through any point as a centre draw a singly infinite 

 number of spheres, but the number of such centres is 



