202 
Now the definition of “degree of regularity” extended to higher 
spaces runs as follows: 
“The degree of regularity of an 7-dimensional polytope is a fraction 
with m2 as numerator and the number p of the successive charac- 
teristics of regularity as denominator, this number p being counted 
in the case of a polytope of the first kind from the vertex end, in 
the case of a polytype of the second kind from the end of the limiting 
n—1-dimensional polytope.” 
In my dissertation 1 have contined myself to polytopes of the 
first kind, the degree of regularity of which is } at least. For the 
methods employed in unearthing these polytopes I must refer to 
that memoir. 
In discussing my dissertation my promotor Dr. P. H. Scnourr 
remarked that if all the fractions representing possible degrees of 
regularity of an n-dimensional polytope are reduced to the denomi- 
nator 2n the numerators 1 and 2n—1 will be lacking, on account 
of the fact that the first and the last characteristic have not been 
subdivided into two halves; so in this sense my scale contains 
something superfluous. é 
Indeed the classification of the polyhedra according to my scale 
is indicated in the diagram . 
u 2 
0 ai 
— | ox 
6 
I Js 1—5, 6,7 R 
where the numbers 1—5, 8-—15 at the midpoint and 6,7 at the 
right designate the polyhedra bearing these numbers in the table, 
whilst / and Ff stand for quite irregular and regular polyhedra and 
P either for the beam or for the double pyramid mentioned above, 
according to the scale corresponding either to polyhedra of the first 
or to polyhedra of the second kind. Indeed the points of division 
Id 
1 5) Ens 
= and ra are unoccupied and in $S,„ the analogous characteristic 
| i 
property presents itself with respect to the points of division = 
n 
2n—1 
on 
It goes without saying that we can take away the superfluity 
indicated (of the two points of division adjacent on either side to 
the extremities) either by counting each of the two extreme charact- 
and 
