144 ANNUAL OF SCIENTIFIC DISCOVERT. 



sphere. Each arc is now bisected. These twelve points of bisection are the 

 poles of the rhombic dodecahedron; the lines joining the opposite pairs of 

 these poles are the rhombic axes, each of these axes being perpendicular to 

 two faces of the rhombic dodecahedron inscribed in the spheres, or inscribed 

 in the cube inscribed within the sphere. Let each of the eight equilateral 

 spherical triangles be divided into six equal and similar spherical triangles 

 by arcs, joining the angle of each triangle -with the centre of its opposite 

 side. The armillary portion of the sphere is now completed. The point 

 within each of the eight equilateral spherical triangles, formed by the inter- 

 section of the three arcs by which it is divided, is the octahedral pole. There 

 arc, of course, eight of these; the lines joining the opposite pairs of these 

 poles are the octahedral axes, each one being perpendicular to two opposite 

 faces of the regular octahedron inscribed in the sphere, or in the cube in- 

 scribed within the sphere. If we now join each pole of the octahedron with 

 the three poles of the octahedron in the three adjacent equilateral spherical 

 triangles by straight wires, and do this symmetrically for the eight poles, 

 we shall then have the edges of the cube inscribed within our armillary 

 sphere, the octahedral axes joining the opposite solid angles of this cube, 

 and the rhombic axes passing through the centres of each opposite edge. 

 Within this skeleton cube we now inscribe a regular octahedron, using elastic 

 strings for its edges, by uniting the point where each cubical axis passes 

 through the face of the cube, with the similar points on the two adjacent 

 faces. Each face of the octahedron is therefore represented by an equilateral 

 triangle of elastic cord. Y\ T e now suppose each side of the eight equilateral 

 triangles to be bisected. Every angle of the eight equilateral triangles is 

 joined to the bisection of its opposite edge by another scries of elastic cords. 

 We have now an octahedron inscribed in the cube inscribed within .our 

 armillary sphere, every face of the octahedron having marked upon it the 

 traces formed by an imaginary plane passing through the zones of its sphere 

 and its centre. It will now be seen that the cubical axes join the opposite 

 solid angles of the octahedron ; the rhombic axes the bisections of its oppo- 

 site edges ; while the octahedral axes pass through the intersections of the 

 elastic cords, which join each solid angle of the octahedron with the centres 

 of the edges opposite to it. The points where the elastic cords meet, and the 

 octahedral axes pass through the faces of the octahedron, are now fastened 

 to cords. These cords are made to run round pulleys, and arc united to- 

 gether, so that by pulling them simultaneously, the points uniting, every one 

 of the three elastic cords which are described on the face of tt.c inscribed 

 octahedron, can be made to travel uniformly and symmetrically along each 

 of the octahedral axes, from the face of the octahedron to the solid angle of 

 the circumscribing tube. Another series of cords are united to each of the 

 four elastic cords, which meet at the point bisecting each of the edges of the 

 inscribed octahedron. These, by a similar contrivance, are made to draw 

 these points along the rhombic axes. The instrument is now completed. 

 By simply pulling the eight cords united together, which cause the clastic 

 cords to ascend the octahedral axes, the inscribed octahedron passes through 

 every form of the three-faced octahedron, till it reaches the limiting form of 

 the rhombic dodecahedron, each three-faced octahedron being inscribed 

 within the cube inscribed within the sphere. In a similar manner, by pulling 

 the cords running along the rhombic axes in combination with those running 

 along the octahedral axes, all the other forms are shown as passing within 

 then; prescribed limits. As soon as the cords are loosened, the elastic bands 



