THE FOURTH DIMENSION. 87 



Thus, for example, on this plane the points 1-2, 1-3. and 2-3 will lie in 

 a straight line, for through the three spatial points i, 2, 3, a plane can 

 be drawn which will cut the plane of a drawing in a straight line. 

 The reason, therefore, that the three points 3-4, 2-4, 2-3, also must 

 ultimately lie in a straight line, consists in the simple fact that the 

 plane of the three points 2, 3, 4, must cut the plane of the drawing 

 in a straight line. The figure here considered consists of ten points 

 of which sets of three so lie ten times in a straight line that con- 

 versely from every point also three straight lines proceed. 



Now, just as this figure is a section of a complete three-dimen- 

 sional pentagon, so another remarkable figure, of similar proper- 



Fig. 37. 



ties, may be obtained from the section of a figure of four-dimen- 

 sioned space. Imagine six points, i, 2, 3, 4, 5, 6, situated in 

 this four-dimensioned space, and every three of them connected by 

 a plane, and every four of them by a three-dimensioned space. We 

 shall obtain thus twenty planes and fifteen three-dimensioned spaces 

 which will cut the plane in which the figure is to be produced in 

 twenty points and fifteen rays which so lie that each point sends out 

 three rays and every ray contains four points. (See Fig. 37.) Fig- 

 ures of this description, which are so composed of points and rays 

 that an equal number of rays proceed from every point and an equal 



