xiiv PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



Longmynd 22° 10' ; between that of Forez and that of Vercors 22° 24', 

 all these systems behig referred to Cormth as the centre of reduction ; 

 and for several other pairs the interA-ening angle was also found of 

 very nearly the same value * . Hence our author was led to the m- 

 ference that the great circles of his different systems must probably 

 belong to some symmetrical figure, because in any such figure, as is 

 manifest, it will frequently happen that different pairs of lines are 

 inclined to each other at equal angles. Such evidence of symmetry 

 will doubtless be felt to be extremely vague. It is not, however, to 

 be regarded as a part of the final evidence on which this part of our 

 author's theory is made to rest, but rather as affording him some 

 indistinct and distant view of that symmetry which he appears to 

 have been persuaded must really characterize some system of lines 

 of which his great circles of reference formed a part. It was thus 

 that he seems to have been encouraged to proceed with that elaborate 

 series of calculations by which alone his views could be adequately 

 tested. 



With these impressions, his first object was to consider the differ- 

 ent ways in which the surface of the sphere may be symmetrically 

 divided into a number of parts by great circles. Now every one ac- 

 quainted with the elements of crystallography will be aware of the 

 usual manner of representing the characteristic lines and angles of 

 crystals by means of the projection of the crystalline edges into great 

 circles of an imaginary sphere, whose centre is symmetrically situated 

 with respect to the crystalline faces. In the same manner the edges 

 and angles of any solid figure may be represented. A regular tetra- 

 hedron is a triangular pyramid with three triangular faces and 

 a triangular base, or, in more general terms, a figure bounded by 

 four triangular faces, each of which is an equilateral triangle and all 

 of them equal. If we describe a sphere having its centre equidistant 

 from each side of the tetrahedron, and make sections of the solid by 

 planes passing respectively through each of its edges and the centre 

 of the sphere, these planes will intersect the surface of the sphere in 

 great circles, which will divide it into four equal and similar spheri- 

 cal triangles. The spherical surface will thus be symmetrically di- 

 vided into four equal portions. Again, we may draw other great 

 circles bisecting each angle of each spherical triangle. Each triangle 

 will thus be divided into six equal right-angled triangles, and conse- 

 quently the whole sphere will be symmetrically divided into twenty- 

 four such triangles. We might in like manner proceed to subdivide 

 the whole spherical surface symmetrically into a great number of 

 portions, by joining points of intersections of lines previously drawn, 

 or by drawing other great circles, having certain relations of sym- 

 metry with the primitive ones ; and thus we might form a network, 

 the number of lines in which would only be limited by the limitation 

 which we should impose on ourselves in the degree to which the 

 subdivision of the spherical surface should be carried. Such a net- 

 work of great circles is termed by M. de Beaumont a reseau trian- 

 gulaire. 



* See op. cit. p. 863. 



