160 
EEV.  T.  P.  KIEKMAN  ON  THE  PAETITIONS  OP  THE  E-PTEA:MID, 
+ {R”'(6, 1+6,  2+6,  3)P(6,  3)+E'“(6, 1+6,  2 + 6,  3).P(6,  3)}.|j  [=  18] 
+ {R»-(6,  3)1(6, 2)+I(6,2)R“(6,  3)}|j+{R”'(6,  3)P(6,  3)+P(6,  3).E*“(6,  3)}^  [=  6] 
+ {l(6,2),P(6,3)+P(6,3).I6,2}.2.^j  [=  12] 
-(N6, 1+N6,  2 + N6,  3)  [=-2] 
But  I see  no  advantage  that  can  arise  from  the  construction  of  these  partitions  of  the 
5-based  and  6-based  pyramids,  which  is,  however,  to  be  effected  with  the  greatest  ease. 
XXII.  It  may  be  worth  while  to  observe,  that  the  regular  12-edron,  and  of  course 
its  polar  syntyp  the  regular  20-edron,  are  8-gonous  polyedra  of  the  second  class.  The 
12-edron  is  made  by  laying  a 5-partitioned  8-ace  having  only  two  triangular  faces  on  an 
8-gon  in  which  four  lines  are  drawn  each  cutting  two  edges,  and  passing  through  no 
summit.  Two  of  these  cutting  lines  terminate  in  the  base  of  each  of  the  triangles  just 
named.  This  sympolar  pair  can  be  exhibited  in  a paradigm  thus ; for  closed  polygons 
can  be  drawn  through  the  20  summits  and  on  the  12  faces  of  the  12-edron; — 
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