( 493 ) 



which therefore bisects the diagonal CD of this J/,._i. This i)„-ois 

 the section of Da-\ with the space Spnl^, through parallel to 

 the spaces /S/^S,*^!?, which are in ^i and A' normal to the axis and 

 which Iruncate the rhombotope. So we find : 



"Each space Spn-i through the centre parallel to a bounding 

 space Spn-i intersects Dn-\ according to a Dn-i of which is 

 again the centre." 



From this follows again more generally: 



"Each space /S/)Jf^ (0 < 7> < ^z — 1) through the centre parallel 

 to a bounding space /S/V ii^tersects Da according to a Z)^_i, of which 

 is again the centre". 



Thus we find ascending from below : 



"Each chord of D,i-\ through parallel to an edge has a 

 length y 1, each plane through parallel to a face intersects Bn~\ 



according to a regular hexagon with sides -|/2, each space through 



parallel to a bounding body intersects Bn-\ according to a regular 

 octahedron with edges |/2, etc." 



6. We retrace our steps and determine of the above mentioned rhombo- 

 tope the length of the axis before and after the truncation. Out of 

 the similitude of the triangles AOB and POC follows in connection 



1 1 1 



with the length - \^n — 1, - V^n, -- of OC, OP, OB for OA the value 



i/^ {n _ 1) and so for half of the unmutilated axis which 



t{n-l) ^ ' 



1 



is n — 1 times as large - y^n [n — 1). If we represent by Rh^J [_q_, r] 



Li 



a rhombotope with p dimensions of which q is the length of the axis, 

 r are the parts of the axis removed by the truncation, the section Z)„_i 



r n — 2 " 



has to be represented by the symbol Bh^-x Vn [n — 1), ^ _ ■^^, 



So the theorem holds: 



"We obtain the section A_i, if we allow the measure-polytope 

 Mn-\ to pass in the indicated way by stretching in the direction of 

 a diagonal as far as y^n times the original length into a rhombotope 

 with a length of axis [/n {n — 1) and if we truncate this rhombotope 

 by two spaces Spn-2 normal to the axis to a 



