( 643 ) 



the axis upon whicli we iiiuist projee't to determine the projection 

 of the hounding elements. Now it is clear that the projection of the 

 cube with AB as a diagonal is obtained by in-qjecting tirst this 

 bounding body on the [)rqjection AB of the axis ()(' on its space 

 ,^1 = 1 which furnishes with regard to the vertices the strati tication 

 1, 3, 3, 1 and by determining then the projection on OC of these 

 new ])oints lying on AB. Now, angle BiJC is a right one, for out 



of the coordinates (1,^1, — 1, — 1) and ( ^, ^, ^> -^ ) of B and 



C follows immediately OB'- -\- OC" = BC-. So B projects itself on 

 OC in and so this of course is also the case with the vertex 

 ( — J, 1,1, J) of the eighteen lying opposite B. So we find — see 

 the first plate under head (3, 1, 1, 1) C, — the stratification of 

 the 16 vertices by causing the group of points 1, 3, 3, 1 laid 

 upon the axis of projection at equal intervals to be followed by 

 a second group of the same structure in such a way that the first 

 1 of this second group coincides with the last 1 of the first group. 

 It is from this that this projection has its type, as is indicated at 

 the foot. One really finds without any difficulty all that is given on 

 the scheme by representing to oneself the two bounding cubes indi- 

 cated in the typical image — here lying in the spaces ,?•, = 4: 1 — 

 and to suppose that their corresponding vertices, edges, faces are 

 united by edges, faces and bounding bodies. 



If again we do not take the isolated point .4 into consideration, 

 then we have to deal here with six different forms of the section, three 

 intermediary forms and three forms of transition ; these are given with 



6 5 1 

 the addition of the coi'responding fractional symbols — , — ,..., — 



^ Ö .r 12' 12 12 



on the second plate. We shall indicate somewhat in details how 

 these diagrams are deduced by drawing, independently of the results 

 of the first plate, and to this end we immediately notice that the space 

 through .1 perpendicular to OC is represented by 3.i\-)-.i'3-|— 1'3+'?'4 = 6 

 and that this space after a slight parallel displacement to O truncates 

 from the edges of the eightcell passing through A segments which 

 are in the ratio to each other of 1:3:3:3. If now the edge AB' 

 drawn horizontally is parallel to 0A\, we begin to set off, in order 

 to obtain the first intermediary form, on the other edges through 

 A — see the last of the six diagrams — segments AP^ , AP^ , AF^ 

 to the length of half the edge, i. e. of the unit, on the edge AB 

 a segment AP^ with a length of a third of the unit, which causes 



the tetrahedron P^J\PJ\ corresponding to the symbol — to be gene- 



44* 



