( 6-i7 ) 



systeni of cooidiiiales with tlie edges passing' llirongli this point as 

 axes, the edge PQ as axis corresponding to the figure 2 of (2, 1,1), 

 then the centre F' of tlie up|)er plane of that cube is the point 

 (2, i, 1) and l*F' is therefore the axis normal to the series of intersecting 

 planes ^). Now it follows from the rectangle APQE with the sides 

 AE='2, .l/'=2l 2, that AQ is normal to PF' and that the points 

 A and Q project themselves on FF' in the same point. Thus we 

 lind the projection of the eigiit vertices of the cube under considera- 

 tion on PF' hv placing the |)rqjections (1, 2, 1) of the faces with 

 PA and QE as diagonals so side by side that the last 1 of the 

 first coincides with the first 1 of the last, by which the stratification 

 1, 2, 2, 2, 1 is arrived at, which, with a view to upper and lower 

 cube, passes by doubling into 2, 4, 4, 4, 2. From this ensue then 

 the results given on the first plate. If we now — returning to the 

 second plate -set off on the three edges of the cube passing thi-ough 

 F, in the assumed sup|)Ositiou that l^Q agrees with the 2 of (2, 1, 1), 



from P segments , 1, 1 then — see the last diagram — the triangle 



F^FJ\ appears forming the up|)er plane of the triangular prism cor- 



j 2 3 4 



responding to the fraction and out of this the sections -^, -, -arede- 



8 8 8 8 



veloped in the same w^ay as was pointed out above. Of triangle 

 P^P^F^ the line connecting F^ with the middle of P^P^ is an axis 

 with the period two, or to express it more simply a line of symmetry, 

 and this line is parallel to the diagonal AQ of the first diagram. In 

 each position of the intersecting plane the section has the line of 

 intersection of this plane with the plane AFQE as line of symme- 

 try ; in connection with this die lozenge, unmutilated for the case 



4 . 1 



—, which when following the re\erse way to the case moves pa- 



8 ' 8 



rallel to itself thi'ough the cube in such a way that the vei'tex Q 

 describes the edge QP, is cut by the groundplane of the cube accord- 

 ing to a perpendicular on the line of symmetry. If we imagine in 

 the chosen space of the upper cube of the eightcell the threefold 

 net of cubes and if we I'emove before passing to the intersection by 

 the series of parallel planes the partitions parallel to the endplanes, 

 w^e obtain in the intersecting plane a net of lozenges which are cut 

 by the removed partitions into segments of the found form, etc. 



In the ensuing j>arts we shall pass on to the intersection of the 

 itets ((.;,) and (6;j. 



1) It is really inaccurate to speak of an upper plane of the upper cube ; of 

 course the plane is meant, which appears in the diagram as upper plane to the eye. 



