xlvi 



PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



easily seen that if five such great circles were drawn from the central 

 point of every pentagon, all the sides of all the pentagons would lie 

 on these great circles, which would thus by their mutual intersections 

 form the whole reseau of twelve pentagons. If all these great circles 

 were distinct, they would be sixty in number ; but it will be easily 

 seen, on tracing the course of any one of these, that it passes through 

 the centres of four pentagons ; consequently there are only fifteen 

 of these circles independent of each other, five of them meeting at 

 the central point of each pentagon. 



But a reseau formed by fifteen great circles only was found very 

 insufficient to represent the complicated network of which M. de 

 Beaumont considers his great citxies of reference to form a part. 

 It has, therefore, been necessary to add to the above-mentioned 

 fifteen fundamental great circles of the pentagonal reseau a great 

 number of subordinate circles, each bearing a certain relation of 

 symmetry to the fifteen primitive circles. I have represented in the 

 diagram now exhibited to you (see fig.), those portions of the more im- 

 portant of these subordinate circles, which lie vdthin one of the twelve 

 pentagons, all the lines being supposed projected on a plane surface. 





H, H', &c., are the middle points of the sides of the pentagon. By 

 joining the points H H', H' H", &c., we form a regular pentagon 

 H H' H", &c., the sides of which are projected portions of great 

 circles bearing symmetrical relations to the primitive great circles. 



