The regular polytopes in two and three dimensions (polygons and polyhedra) and the “Archimedean solids ” have been known since ancient times. To these, KEPLER and POINSOT added the regular star-polyhedra. About the middle of last century, L. SCHLAFLI* discovered the (convex) regular polytopes in more dimensions. As he was ignorant of two of the four KEPLER-POINSOT polyhedra, his enumeration of the analogous star-polytopes in four dimensions remained to be completed by E. HESS. Recently, D. M. Y. SOMMERVILLE interpreted the (convex) regular polytopes as partitions of elliptic space, and considered the analogous partitions of hyperbolic space. Some particular processes, for constructing “ uniform ” polytopes analogous to the Archimedean solids, were discovered by Mrs. BOOLE STOTT and discussed in great detail (with the help of co-ordinates) by Prof. SCHOUTE. Further, E. L. ELTE completely enumerated all the uniform polytopes having a certain “ degree of regularity,” these including seven new ones (in six, seven and eight dimensions).