lii PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



it here seems most reasonable to take all those sets, the importance 

 of which, according to the author's own numerical estimate, is not 

 less than that of any of the circles which he has selected as theoreti- 

 cal representatives of his great circles of reference of the European 

 systems. Many of these woidd not pass through D, but I will first 

 restrict myself to those which do so, and also pass through any of 

 the sets of points H, H' &c., I, I' &c., T, T' &c., a, a' &c., b, V &c., 

 e, e' &c., and h, K &c., because these are points through some one 

 of which in each set M. de Beaumont has taken a great circle to 

 represent one of his European great circles of reference. It will be 

 svifficient to take the portion of the sphere lying between two great 

 circles, such as DH'" and DI'", since everything is symmetrical in the 

 other similar portions of the spherical surface. The angle between 

 two such circles is 36°, and when they are produced to their other 

 point of intersection diametrically opposite to D, the number of 

 points between them, such as those above specified, is about thirty. 

 According to the rough calculations I have made, the smallest angle 

 which a great circle from D through one of these points makes with 

 DH'" is nearly 4^, which is also the angle made with DI'", by a great 

 circle through D and another of these specified points. The angle 

 between any tw"o contiguous circles intermediate to the two just men- 

 tioned never exceeds 2\°, and is generally considerably less. Suppose 

 now Ave draw at random a great circle on the sphere, or a straight 

 line in the diagram, through D and lying within the angle H'" D I'". 

 If tbe line so drawn bisect the angle of 4.^° either between DH'" and 

 the nearest line of the pentagonal reseaii drawn as above mentioned, 

 or between DI'" and the line of the reseau nearest to that line, it will 

 then deviate by about 2^° from either of the contiguous lines of the 

 reseau ; but if the great circle, or line, drawn at random, lie in any 

 other position within the angle II'" D I'", it cannot deviate from the 

 nearest line of the reseau by more than I-}" (since the angle between 

 two contiguous lines does not exceed 2-§°), and the deviation will 

 generally be much less. Now, if we refer to the table above men- 

 tioned of the angles between the European great circles of reference, 

 as provisionally determined, and their respective representatives in 

 the pentagonal reseau, we find them quite equal to those just men- 

 tioned. Where then is the proof that the great circles of reference 

 which ma}^ pass through D, may be better represented by the circles 

 of the pentagonal reseau through that point, than any other set of 

 circles drawn through it entirely at random 1 



I have here admitted the assumption that certain great circles of 

 reference pass accurately through I) ; but let us now suppose them 

 to pass near to D, i. e. within a few degrees of it, a supposition very 

 far more probable than the former. Then we bring in many other 

 circles of the reseau with which our great circles of reference may 

 be compared, as M. de Beavimout has done with respect to those 

 circles of reference which do not pass through D. In such case the 

 angular space becomes still more subdivided by circles of the reseau 

 exactly similar to those which our author has put into requisition. 

 This only shows more obviously the impossibility of describing any 



