204 
do not stir, whilst — if we use scales of the same length — the 
others execute a movement enlarging their distance from the centre. 
So the polytopes with a degree of regularity of at least a half 
found by me present themselves quite as well if we use the new 
scale; so in this respect I have not the least objection to accept this 
new scale. *) 
2. However one may not flatter oneself with the hope, that the 
new scale shall not contain superfluous points of division with respect 
to either of the two kinds of polytopes considered for itself. In space 
S, already we find with respect to the polytopes of the first kind 
in this new scale, agreeing with the old one for n= 3, the point 
HEAR 
of division 5 unoccupied. For we have 
0 1 2 3 4 5 d 
6 6 6 6 6 
| | | | SSS SS 
I P e,S(5) e‚„S (5) ce,S(5) R 
where / and R have the same meaning as before, whilst P represents 
a rectangular parallelotope with edges of four different lengths and 
e,5(5), e,S(5), ce, S(5) indicate three polytopes deduced from the 
regular simplex S (5) of S, in the notation given by Mrs. A. Boor 
STOTT ”). 
3. As the new scale contains no unoccupied points of division in 
the case n==8 only, it would not be worth while to substitute it 
for mine, which has the advantage of treating all the groups of 
limiting elements — vertices, edges, faces, etc. and the limits with 
‚the highest number of dimensions — on the same footing, if it did 
not possess a second advantage, in my opinion of great importance. 
We will treat this somewhat in detail. 
In the determination of the semiregular polytopes of the first kind 
I consider of any polytope the corresponding “vertex polytope” ’*). 
In general the vertices of the latter are those vertices of the former 
joined by edges to a vertex of this original polytope. In an appendix 
to my dissertation I state the rule, that a polytope with edges of 
1) Dr. Scnovre requests me to communicate that the primitive idea of this new 
scale for S, presented itself to him in an intercourse with F. Zernike, candidate 
in mathematics and physics at the University of Amsterdam. 
2) “Geometrical deduction of semiregular from regular polytopes and space 
fillings”, Verh. Kon. Akad. v. Wetenschappen, Amsterdam, Ist series, Vol. XI, n°, 1. 
3) Not to be confounded with the polytope of vertex import of Mrs. A. Booze Srort, 
