xlviii PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



Such, Gentlemen, is the reseau pentagonal of the distmguished 

 French geologist. In my explanation of it I have omitted all the 

 numerous and complete details given by its author, but I trust that 

 what I have stated will enable you to understand his theory, and to 

 judge for yourselves of the degree of importance which may attach 

 to it. The theory of the pentagonal reseau asserts the approximate 

 coincidence of each of the author's great circles of reference with 

 some great circle comprised in the reseau ; so that these great circles 

 of reference, instead of being drawn, as it were, at random, on the 

 surface of the globe, are connected by a well-defined geometrical 

 law, and belong to a system of which the symmetry is as perfect as 

 that of any system of great circles which can be described on the 

 surface of a sphere. The number of great circles of the reseau with 

 which, great circles of reference of known existing systems coincide, 

 is very small, but it is considered that other systems may yet be 

 formed, though the whole number that ever could be formed by ex- 

 isting causes is probably limited by the nature of those causes and 

 the conditions under which they operate. 



The manner in which M. Elie de Beaumont first proceeded to test 

 his theory was to calculate the angles included between the principal 

 circles of the pentagonal reseau*, and to compare them with the 

 series of angles above mentioned as included between different pairs 

 of his great circles of reference^. In this comparison he observed 

 an approximate accordance sufficient, as he conceived, to aiford a 

 strong sanction to his theory. Still the inferences from these ap- 

 proximate coincidences of angular values were necessarily somewhat 

 vague. The real question to be decided was — coidd the pentagonal 

 reseau be so applied to a terrestrial globe, on which all the great 

 circles of reference were delineated, so that those great circles should 

 actually coincide with great circles forming a part of the reseau 

 itself? 



Our author's first mode of answering this question was a tentative 

 graphical one. He constructed a pentagonal reseau with its princi- 

 pal great circles, of fine thread, and of such dimensions as to fit as 

 accurately as possible on a common terrestrial globe. Thus applied 

 it might be moved into any required position, and the first object 

 was to find that position in which some of its great circles should, if 

 possible, coincide, at least approximately, with the great circles of 

 reference. For this purpose the central point of one of the penta- 

 gons was assumed to coincide with a point in a given latitude and 

 longitude. This was not sufficient, however, to fix completely the 

 position of the reseau, for it might still be turned about a diameter 

 terminating in the point just mentioned. Another condition was, 

 therefore, necessary. The one selected was, that one of the great 

 circles of the reseau passing through the assumed point of given 

 latitude and longitude should make a given angle with the meridian 

 at that point. 



When our author came to represent all his great circles of refer- 

 ence for his European systems at once, he found that a considerable 

 * Op. cit. p. 933. t 02). cit. p. 1128. 



