1859.] visible some of the abstruse problems of Crystallography. 87 



also limiting forms. Between the octahedron and the rhombic dodeca- 

 hedron we may conceive an infinite number of varieties of the three- 

 faced octahedron, passing from the form of the octahedron to that of 

 the rhombic dodecahedron ; similarly, the octahedron and the cube are 

 limiting forms of an infinite series of twenty-four-faced trapezohedrons, 

 and the cube and rhombic dodecahedron of a series of four-faced cubes. 

 The forty-eight-faced scalenohedron or the six-faced octahedron is a 

 form varying within the limits of all the others. 



To represent to the eye the passage of all the varieties of these 

 forms between their respective limits is the object of the mechanical 

 contrivance which is the subject of this paper. A skeleton or armillary 

 sphere is constructed of iron wire, so as to mark out the principal 

 zones of the sphere of projection of the forms of the cubical system ; 

 three circles are united at right angles to each other, so as to represent 

 eight equilateral spherical triangles, each of whose sides are arcs of 

 90°. The six points where the arcs cross each other are the poles of 

 the six faces of the cube ; the lines joining each pair of opposite poles 

 represent the cubical axes, each axis being perpendicular to two faces 

 of the cube which can be inscribed in the 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 intersection of the three arcs by which it is divided, is the 

 octahedral pole. There are 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 inscribed 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. We 

 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 series 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 



