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The semiregular polytopes of the second kind are the polar-reci- 
procal figures of those given in the table with respect to a concentric 
sphere. 
The definition of semiregular polyhedron given above had to be 
modified in order to make it applicable to polydimensional Spaces. 
We say that a polyhedron possesses a “characteristie of regularity”, 
if either all the vertices, or all the edges, or all the faces are equal 
to each other. Equality of vertices signifies that the polyangles 
formed by the edges concurring in each vertex are congruent (or 
symmetric); equality of faces consists in the congruency of the 
limiting polygons. But the equality of edges includes two different 
parts which can present themselves each for itself: equality in length 
of the edges and equality of the angles of position of the faces 
through the-edges. So all the polyhedra of the table have edges of 
the same length but — with exception of the numbers 6 and 7 — 
more than one kind of angles of position, whilst quite the reverse 
presents itself with the corresponding polyhedra of the second kind. 
If the equality of edges is realized only partially — as in tbe case 
of the polyhedra of the table — we speak of a “half characteristic’’ 
so that these polyhedra admit 13 characteristics. By bringing this 
result in connection with the circumstance that a polyhedron can 
admit 3 characteristics, the epitheton “semiregular” obtains a Literary 
signification. As the polyhedra N°. 6 and N°. 7 of the table possess 
both the half characteristics of the edges, these polyhedra must be 
called ‘?/,-regular’” according to our system. 
We remark that the characteristics of a semiregular polyhedron 
of one of the two kinds are lacking in the corresponding polyhedron 
of the other. Moreover that we are obliged to observe a quite 
determinate order of succession in counting the characteristics of a 
polyhedron of defined kind and, beginning at the commencement, to 
count successive characteristics only, i.e. in the case of polyhedra 
of the first kind to take into account successively equality of vertices, 
equality in length of edges, equality of angles of position round 
edges, equality of faces, and reversely in the case of polyhedra of 
the second kind. If this order of succession was not observed e. g. 
with respect to the two half characteristics of the edges a beam 
with different length, breadth, and height would appear as a semi- 
regular polyhedron of the first kind on account of equality of ver- 
tices and angles of position, whilst a double pyramid formed by the 
superposition of two faces of two equal regular tetratredra would 
appear as a semiregular polyhedron of the second kind, to which 
enunciations fundamental objections can be raised, 
