78 THE FOURTH DIMENSION. 



of a number which stands for the length of its sides, leads to the 

 notion of four-dimensioned structures, and, consequently, to the 

 notion of a four-dimensioned point-space. When we note that a? 

 stands for the contents of a square, and a 9 for the content of a cube, 

 we naturally inquire after the contents of a structure which is pro- 

 duced from the cube as the cube is produced from the square and 

 which also will have the contents a 4 . We cannot, it is true, clearly 

 picture to ourselves a structure of this description, but we can, 

 nevertheless, establish its properties with mathematical exactness.* 

 It i% bounded by 8 cubes just as the cube is bounded by 6 squares ; 

 it has 1 6 corners, 24 squares, and 32 edges, so that from every cor- 

 ner 4 edges, 6 squares, and 4 cubes proceed, and from every edge 

 3 'squares and 3 cubes. . 



Yet despite the great service to algebra of this idea of multi- 

 dimensioned space, it must be conceded that the conception al- 

 though convenient is yet not indispensable. It is true, algebra is 

 in need of the idea of multiple dimensions, but it is not so abso- 

 lutely in need of the idea of point- aggregates of multiple dimensions. 



This notion is, however, necessary and serviceable for a pro- 

 found comprehension of geometry. The system of geometrical 

 knowledge which Euclid of Alexandria created about three hundred 

 years before Christ, supplied during a period of more than two 

 thousand years a brilliant example of a body of conclusions and 

 truths which were mutually consistent and logical. Up to the pres- 

 ent century the idea of elementary geometry was indissolubly bound 

 up with the name of Euclid, so that in England where people ad- 

 hered longest to the rigid deductive system of the Grecian mathe- 

 matician, the task of "learning geometry" and /'reading Euclid" 

 were until a few years ago identical. Every proposition of this 

 Euclidean system rests on other propositions, as one building-stone 

 in a house rests upon another. Only the very lowest stones, the 

 foundations, were without supports. These are the axioms or fun- 



* Victor Schlegel, indeed, has made models of the three-dimensional nets of all 

 the six structures which correspond in four-dimensioned space to the five regular 

 bodies of our space, in an analogous manner to that by which we draw in a plane 

 the nets of five regular bodies. Schlegel's models are made by Brill of Darmstadt. 



