ANNIVERSARY ADDRESS OF THE PRESIDENT. xlv 



If we take a cube and describe a sphere having its centre in the 

 middle point of the cube, it is manifest that each of the six faces of 

 the sohd might be projected as before into six equal quadrilateral 

 spaces on the surface of the sphere, which would thus be divided 

 symmetrically into six portions, and the number of spaces might be 

 indefinitely increased by the addition of other great circles in a man- 

 ner similar to that above indicated in the case of the reseau trian- 

 gulaire. This would be our author's reseau quadrilateral. 



By cutting off the angles of a cube we may form a regular solid, 

 an octahedron, with eight equal faces, each of which is an equilateral 

 triangle, and each face might be projected as before, but the reseau 

 which might be thus formed might be equally deduced from the cube, 

 and is therefore included in the reseau quadrilateral. 



A fourth regular solid is the dodecahedron, which has twelve equal 

 faces, each of which is a regular equilateral pentagon. These pen- 

 tagons may be projected on the surface of a sphere into twelve equi- 

 lateral pentagonal spaces, bounded by equal arcs of as many great 

 circles ; and the twelve spaces into which the sphere is thus divided 

 may be again divided and subdivided by other great circles in the 

 manner already indicated. The network thus formed is the reseau 

 'pentagonal, the pentagonal network of our author. 



In consequence of the greater number of elementary spaces into 

 which the spherical surface is thus divided, the greater number of 

 sides bounding each such space, and the greater number of angles 

 contained in it, it will be easily seen that a greater number of lines 

 of which the symmetry is at once obvious, may be dravra in this 

 reseau than in either of the preceding. It appears to have been for 

 this reason that M. de Beaumont selected this reseau as the one 

 most likely to afford the geometrical type of the network of lines 

 formed by his great circles of reference of all the systems of lines of 

 elevation existing on the face of our planet. 



To form a conception of the positions of the twelve equilateral 

 pentagons which occupy the whole surface of the sphere, conceive 

 one pentagon with its middle point (that equidistant from each an- 

 gular point) in a given position on the sphere. Another pentagon 

 will have its middle point diametrically opposite to the former, and 

 so situated that the half of a great circle drawn between the centres 

 of these pentagons, and bisecting a side of one of them, shall bisect 

 an angle of the other. Round each of these pentagons are five 

 others, each of which has one common side with it ; and these two 

 series, consisting each of five pentagons, so interlace with each 

 other, that each pentagon of one series has a side in common with 

 one pentagon of the other series. The twelve pentagons thus oc- 

 cupy the surface of the whole sphere. 



The symmetry of this arrangement is complete, for any one pen- 

 tagon is surrounded by five others, precisely as in the manner above 

 described. 



We may also observe that a great circle drawn from the centre of 

 any pentagon to one of its angular j)oints, passes, when prolonged, 

 along the common side of two adjacent pentagons. Hence it is 



