ANNIVERSARY ADDRESS OF THE PRESIDENT. xliii 



S.Wales, Bingerlocli on the Rhine, and Corinth, — and reduced all the 

 great circles of reference of the European systems to each of these 

 centres. All the lines thus reduced are represented radiating from 

 these centres on three small maps, which thus exhibit at once to the 

 eye the arrangement and grouping of the lines. If the lines to be 

 reduced to these centres were straight lines on a jilane surface, in- 

 stead oi great circles on a spherical surface, they would manifestly 

 radiate from each centre so as to make exactly the same angles with 

 each other as at the other centres ; but, after the reduction of the 

 great circles to these centres, such is not exactly the case, as may be 

 seen by inspection of the maps above mentioned. The differences 

 however are not great. M. de Beaumont was not satisfied to estimate 

 these angles by a graphical method, but calculated, by trigonometri- 

 cal formulae, the angles which every radiating line makes with every 

 other for his twenty-one recognized European systems. These angles 

 are tabulated in three separate columns corresponding to the three 

 centres*. He has also added a fourth column, giving the means of 

 the three preceding, and a fifth containing the angles at which every 

 great circle of reference intersects every other. These angles do not 

 differ much from the former f. 



Having obtained all these angles, our author again arranged them 

 in each column according to order of magnitude, in a manner of 

 which a specimen is given, p. 876, the whole table thus formed being 

 too long for insertion. He was thus the better able to observe the 

 intervals between successive angles so arranged, and remarked that 

 for sevei'al successive angles together, in certain parts of each series, 

 the intervals were smaller, and in other parts considerably larger, 

 the former presenting closer groups, and the latter wider open spaces 

 (lacwnes) . These groups and lacunes were again graphically repre- 

 sented (in a manner of which a specimen is given in Plate iv.) by 

 hoi'izontal lines drawn across vertical columns, the closer representing 

 those parts of each series of angles contained in the vertical columns 

 of the table just mentioned, in which the groups occur, and the open 

 spaces those in which the lacunes occur. 



It is in these groups and lacunes, whether studied numerically in 

 the table or graphically in the plate, that our author recognizes, as 

 he conceives, indications of law in contradistinction to arrangements 

 which might have resulted from mere chance. He especially recog- 

 nized the fact, that several pairs of great circles of reference are in- 

 clined to each other respectively at very nearly equal angles. Thus 

 the angle between the system of the Pyrenees and that of the Ne- 

 therlands is about 22° 14' ; between that of Vercors and that of the 



* Op. cit. p. 840. 



+ This is due to the circumstance of the great circles of reference for the Eu- 

 ropean systems having been so chosen that they must necessarily intersect each 

 other at points in or near the European region, with the exception perhaps of 

 some which intersect at small angles. Consequently the above centres are not 

 generally far from the points of intersection, and hence the differences of direction 

 at those centres do not differ much from the angles of intersection to which they 

 would, of course, be exactly equal if the lines were straight lines on a plane 

 surface. 



