106 Mr. J. Y. Buchanan on the Use of the Globe 



circles, the diagram produced on the globe is the one which 

 we have been considering. As was pointed out in the paper, 

 the corners of the cube when radiating from the centre of the 

 sphere delineate the projection of the octahedron on it. This 

 fact can be expressed by saying that the central representation 

 of the cube is identical with the radial projection of the 

 octahedron. 



Models exhibiting the central grouping of corners, and the 

 forms thereby produced, are quite easily made out of card- 

 board or stiff paper, and are very instructive. 



Let us consider the regular tetrahedron. The normals to 

 its faces, four in number, are found to be inclined to each 

 other at an angle of 10 ( J^° across an edge. The poles of 

 these faces when placed on the globe form a group of four 

 points symmetrically arranged, each being separated from its 

 neighbour by an arc of 109^°. Draw the great circles of 

 which these points are the poles, and call any one of them 

 No. 0. Take its intersection with any other and call it 

 node (0), the opposite node is (0, 1/), and the great circle 

 which intersects No. in these nodes is No. 1, diameter (0, 1) 

 being parallel to and representing the edge made by Nos. 

 and 1. Another pair of nodes are found in azimuth 120° 

 and 300°. Their diameter represents the edge (0, 2), and the 

 great circle represents face No. 2. Similarly the nodes of 

 circle No. 3 are in azimuths 240° and 60°. The inclination 

 of the faces is equal, and we find that the altitude of circles 

 1, 2, and 3 is 70i°. 



The edges (1,2), (2,3), and (3,1) are found by measure- 

 ment of the positions of nodes (1, 2), (2, 3), and (3, 1) ; 

 node (1,2) in (330° and 54±°) ; node (2,3) in (90°, 54 J°) ; 

 and node (3, 1) in (150°, 54^°). The theoretical figures are 

 here given because the observed ones have been mislaid. 

 The errors, which did not exceed 1^°, occurred in the quarter 

 sphere of azimuth over 180°. This is always the case, be- 

 cause the metrosphere is never exactly true to the globes, 

 and the errors show in proportion as it is shifted from its 

 original position. 



If we consider, as we did in the case of the cube, what 

 form gives in radial projection the same diagram on the globe 

 as the central representation of the tetrahedron, we find that 

 it is the combination of the cube and octahedron " in equili- 

 brium." The radial projection of each of its edges is an arc 

 of 60°, and its principal sections are regular hexagons. 



In the central representation of the tetrahedron let one of 

 the great circles be suppressed. There remain three, and 

 they constitute the central representation of the rhombo- 



