ANNIVERSARY ADDRESS OF THE PRESIDENT. llll 



system of great circles on the surface of a sphere which might not be 

 just as well represented by the pentagonal reseau, as that particular 

 system formed by the gi^eat circles of reference of the different parallel 

 systems of elevation, whether determined provisionally, or with all 

 the accuracy with which, from the nature of the phfenomena, Vv'e can 

 consider it possible to determine them. 



It might appear from certain incidental expressions of our author, 

 that he regards the indefinite number of circles which may be drawn 

 in his pentagonal reseau as one of the great resources of his theory ; 

 and had it been a question of accurate instead of ajjproximafe coin- 

 cidence of two systems, it might perhaps have been so regarded. 

 But in the actual case it appears to me to be this very circumstance 

 which invalidates any proof which can possibly be given of this 

 theory of the pentagonal reseau. The great uncertainty also which 

 attaches to the present determination of the provisional r/reat circles 

 of reference appears to me to render it impossible to make a compa- 

 rison between them and the great circles of the reseau sufficiently 

 satisfactory to allow us to deduce from it any proof of our author's 

 theory. The only circumstance which could at present have afforded, 

 in my estimation, a presumption of its truth, would have been that 

 of the approximate coincidence of the great circles of reference with 

 those comparatively few circles onl}^ of the reseau which present to 

 us the most obvious relations of symmetry. This, however, is far 

 from being the case, as appears at once from the vast number of 

 circles belonging to those classes from ¥/hich the author has selected 

 his representative circles. 



M. de Beaumont has not explicitly tested his theory exactly in 

 the manner I have now done. He has insisted on the approximate 

 accordance of two series of angles, as I have already stated, the one 

 series containing the angles between pairs oi great circles of reference, 

 and the other containing angles between certain pairs of lines of the 

 pentagonal reseau. But this accordance is simply a consequence of 

 the approximate coincidence between each circle of reference and its 

 representative circle in the reseau, which, as I have endeavoured to 

 show, appears to me no greater than that which might exist if we 

 should substitute lines drawn at random for the actual circles of re- 

 ference. The final test of the theory must, I conceive, be that which 

 I have applied to it. The inference which I should draw from the 

 approximate accordance of the two above-mentioned series of angles, 

 as well as from an inspection of the map on which the great circles 

 of reference are delineated, is merely that the physical cause to which 

 lines of elevation are to be referred has so acted as to distribute those 

 lines pretty equally with reference to different points of the compass ; 

 but, I confess that I can see no solid grounds for the induction that 

 the author's great circles of reference have any necessary connexion 

 with the pentagonal reseau. 



Both these theories of M. Elie de Beaumont — that which asserts 

 the parallelism of synchronous lines of elevation, and that which 

 asserts the approximate coincidence of the great circles of reference 

 of such parallel systems with circles of the pentagonal reseav. — are 



