746 _ MR DUNCAN M. Y. SOMMERVILLE ON 
determine a simple hyperbolic network for all values of m and m for which the network 
is neither Euclidean nor elliptic. The networks are all infinite. 
As regards composite types, we can apply the same remarks as were made in con- 
nection with the elliptic networks. The angle of a polygon is determined by the 
particular combination in which it occurs, and the multiplicity of composite types is 
thus limited. But, at the same time, it is infinite. For, consider the regular network 
3, (p>6) (fig. 46). Any group of p covertical triangles can be replaced by a p-gon, so 
that from this network alone we obtain an infinite number of composite types. 
Note added on July 29, 1905.—Since writing the above, I have come across some 
of the previous work on the subject. The semi-regular polyhedra have long been 
known. It appears, from the works of Pappus of Alexandria and Keppusr, that they 
were described in a lost work of ARcHIMEDES.* Pappus} enumerates the series of 
thirteen (7.e. excluding the two infinite groups, figs. 24 and 25), with the numbers 
of their faces, edges, and vertices, for which he gives the general formule of § 22. 
KeppLeR{ establishes them by taking the different possible combinations, first binary 
and then ternary, containing triangles, squares, and pentagons successively. More 
recently, accounts of them have been given by Meter Hirscu§ and R. Batrzer.|| An 
elaborate article, containing numerous calculations relating to the radius of the cir- 
cumscribed sphere, inclinations of the faces, etc., was presented by M. VauarT to the 
French Institute in 1854.1 He refers to other writings, in particular to one by 
LiponneE (1808), but gives no details of them. He shows also how the semi-regular 
polyhedra are obtained by truncating the Platonic solids. The connection between 
these polyhedra was also expressed by KEPPLER in an ingenious nomenclature which 
he employed to describe them. The following list of names corresponds to the table 
on p. 743; the numbers refer to the diagrams :— 
I. Tetrahedron truncum (23). his er 
II. Cubus truncus (20). Rhombicuboctahedron (26), Cubus simus (33). 
Octahedron truncum (22). Cuboctahedron (30). sf 
Cuboctahedron truncum (35). = 
III. Dodecahedron truncum (19). 
Icosihedron truncum (21). Icosidodecahedron (28). 
Icosidodecahedron truncum (34). Rhombicosidodecahedron (36). 
Dodecahedron simum (32). 
In none of these writings is any notice taken of possible varieties, the reason 
being probably that these varieties do not exhibit the same symmetry as the funda- 
mental varieties. KrppLER gives this as the reason for excluding the two infinite 
series. 
* See also T, L. Heatu, The Works of Archimedes (Camb, 1897), p. xxxvi. 
+ Collectio, lib. v. pars 2. 
t Harmonices Mundi (1619), lib. ii. pp. 61-65. 
§ Sammlung geometrischer Aufyaben (Berlin, 1805-7), vol. ii. pp. 189-185. 
|| Elemente der Mathematik (1862), Bd. ii., Buch v. § 7. 
{ Published 1867, under the title “Des Polyédres semi-réguliers, dits solides d’Archiméde,” Mém. de la Soc. des 
Sciences phys. et nat. de Bordeaux, v. 319-369. 
