200 
Mathematics. — “The scale of regularity of polytopes’. By Dr. 
E. L. Eure (Meppel). (Communicated by Prof. P. H. Scouts). 
In my dissertation’) it was my aim to determine the semiregular 
polytopes, i.e. the polytopes analogous to the semiregular polyhedra. 
So this investigation had to be based on a definition of the notion 
“semiregular polytope’. Now ordinarily a semiregular polyhedron is 
defined as follows: “A semiregular polyhedron has either congruent 
(or symmetric) vertices and regular faces or congruent faces and 
regular vertices. So there are two kinds of semiregular polyhedra 
which we will call with CataLan’*) “semiregular of the first kind” 
and “semiregular of the second kind”; those of the first kind are 
enumerated in the following table. For any of these polyhedra this 
table gives the numbers of vertices, edges, faces and indicates which 
faces pass through each vertex and which couples of faces pass 
through each kind of edges. Here p, denotes a regular polygon 
with ” vertices. 
Pe Po Pe »P3 
Ps ‚Ps | Ps yP3 
Pe ‚Po Po ‚Pa 
2P10 Pio Pr0 | Pio P3 
2p6 Pe ‚Pe | P6 ‚Ps 
2p4 Pa »P3 
2p5 Ps »P3 
Spa Pa ‚Pa \Pa,P3 
2p4 , lps [Pa ‚Ps |Pa,P3 
Ipe , Ibs} Po ‚Ps | Ps ‚Ps 
Ip6 « 1ProfPicrPe | Pio Pas 
Ip, , 4P3 P., Pa \P3,P3 
Ips , 4p3 | Ps »P3 |P3,P3 
lpn » Ps PniPs | Pa» Pa 
Ipn , 3p3 Pn» P3 | 3 ,P3 
1) “The semiregular polytopes of the hyperspaces”, Groningen, 1912. 
2) “Mémoire sur la théorie des polyèdres”, Journal de I’ Ecole Polytechnique, 
Cahier 47. 
