where L 3 denotes a 3-axis, &c.; this is in accordance with the notation of M. Bravais 
in the memoir subsequently referred to. 
8. The regular polyhedra may be exhibited in connexion with each other as 
follows: Imagine the polyhedron projected on a concentric sphere by lines through the 
centre; so that the summits become points on the sphere, the edges arcs of great 
circles, and the faces spherical polygons. Starting from the dodecahedron, the centres 
of the pentagonal faces are the summits of the icosahedron, and conversely for the 
icosahedron the centres of the triangular faces are the summits of the dodecahedron: 
moreover each edge of the dodecahedron cuts at right angles an edge of the icosahedron 
and the two edges have the same mid-point. Again if in any face of the dodecahedron 
we draw one of the five diagonals (arcs through two non-adjacent summits) there is 
in the face a single edge not met by this diagonal; and in the other face through 
this edge a single diagonal not met by the edge; joining the extremities of the twn 
diagonals we have a spherical square, the face of the cube; it is to be observed that 
the summits of the cube are eight out of the twenty summits of the dodecahedron,, 
and that the centres of the faces of the cube are the mid-points of six out of the 
thirty edges of the dodecahedron or the icosahedron. The cube given by the foregoing 
construction is of course one out of five different cubes. The centres of the faces of 
the cube are the summits of the octahedron; and conversely the centres of the faces 
of the octahedron are the summits of the cube; moreover each edge of the cube cuts 
at right angles an edge of the octahedron; and the two edges have the same mid 
point. Finally, taking four non-adjacent summits of the cube (which can be done in 
two different ways), these are the summits of the tetrahedron, and the mid-points of 
the edges of the tetrahedron are the summits of the octahedron. 
9. Considering the polyhedra in the foregoing mutual connexion, all the axes of 
the tetrahedron are axes of the cube and octahedron, viz. the 2-axes of the tetrahedron 
are the 4-axes of the cube and octahedron; and the 3-axes of the tetrahedron are 
the 3-axes of the cube and octahedron; moreover the 3-axes of the cube and 
octahedron are included among the 3-axes of the dodecahedron and icosahedron and 
the 4-axes of the cube and octahedron are included among the 2-axes of the dodeca 
hedron and icosahedron; but the 2-axes of the cube and octahedron are not included 
among the axes of the dodecahedron and icosahedron. The 4-axes of the cube and 
67—2