ANNIVERSARY ADDRESS OF THE PRESIDENT. li 



which rests on the approximate coincidence of these two sets of great 

 circles would be invalidated in a degree proportionate to the uncer- 

 tainty with which the positions of the great circles of reference are 

 determined. If the European systems should be proved to extend 

 over much wider regions than that of Western and Middle Europe, 

 there would no longer be the same uncertainty respecting the centres 

 of reduction of these great circles of reference, and the evidence de- 

 duced from them would become proportionally more determmate. 

 At present the great uncertainty respecting these centres appears to 

 me to invalidate, in a great degree, the existing evidence in favour of 

 our author's theory. 



Suppose,- however, that we admit the fact of ten of the European 

 circles of reference passing exactly through a single point, the pen- 

 tagonal centre D. Let us next examine the evidence deducible from 

 the approximate coincidence of these great circles with the lines of 

 the reseau which our author has taken for their true theoretical re- 

 presentatives. 



I must here again refer you to the excellent map which accom- 

 panies M. de Beaumont's work, for the graphical representation of 

 his great circles of reference. The table in page 1123 will enable 

 you to appreciate accurately the deviation of each great circle of the 

 European systems from the line of the pentagonal reseau which is 

 its theoretical representative. To the twenty-one systems recognized 

 as European, are added the great circles of the Volcanic axis, that of 

 the Ural, and that of the Azores. 



We observe, in this table, that in two cases out of the ten central 

 systems there is a difference of nearly 2\° between the great circles 

 of reference, as at present determined, and their respective represen- 

 tatives, the corresponding differences in the other cases being for the 

 most part about 1°, except in that of Tenara, which, as I have before 

 stated, is assumed, in fixing the position of the reseau, to coincide 

 with one of the primitive circles. We must also recollect that the 

 determinations of the angular positions of the great circles of refer- 

 ence are liable to a possible error of 3^ or 4°. It is not that an error 

 to that amount is likely to exist simultaneously in every one of these 

 great circles, but any one of them may be so affected. Now the 

 question is— whether, if we should draw ten lines at random thi'ough 

 I), the angular differences between these lines and the lines of the 

 reseau nearest to each of them, would be generally much greater than 

 those above given in the table. If not, then the approximate coin- 

 cidences between the great circles of reference and their represen- 

 tatives in the pentagonal reseau, indicated by the table, afford no 

 evidence that those great circles constitute a system having any 

 especial and necessary relation to the pentagonal reseau. 



In deciding this question, we must first determine what circles of 

 the pentagonal reseau must be admitted as those from which we are 

 to measure the angular distances of the great circles which are to be 

 compared with them. The whole number of great circles is indefi- 

 nitely great ; but we have seen that M. de Beaumont has assigned 

 different degrees of importance to chfferent sets of these circles, and 



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