ANNIVERSARY ADDRESS OF THE PRESIDENT. xlvii 



Again, by joining the points H H", H' H'", H" H"", &c., we form a 

 third pentagon, T T' T", &c., the sides of which are portions of 

 another set of subordinate great circles. It will be observed also 

 that the points a a' a", the middle points of the sides of the pen- 

 tagon T T', &c., and the points b b' b", the middle points of the 

 sides of the pentagon H H' H", &c., are all symmetrically related, 

 by their positions, to the primitive pentagon. A similar observa- 

 tion applies to the points c c' c" and c, c/ c/', &c., the points of 

 intersection of certain lines of symmetry as represented in the 

 figure. There are also many points of intersection in the figure 

 not designated by any letter, and by joining any pairs of these points 

 respectively we might form an immense number of additional sub- 

 ordinate great circles, all bearing some relation of symmetry to the 

 fifteen primitive circles and to those more immediately derived from 

 them. 



The number of subordinate circles which might thus be drawn is 

 really unlimited. All of them properly belong to the pentagonal 

 reseau in the complete acceptation of the term, but it is manifest 

 that, taking them in the order of their formation as above indicated, 

 their symmetrical relation to the fifteen primitive circles becomes 

 more and more obscure. It is therefore reasonable that in any 

 questions in which this symmetry is essentially involved, we should 

 assign different degrees of importance, or weight, to these different 

 orders of subordinate circles. Our author has estimated their re- 

 spective weights numerically, but their absolute numerical values are 

 obviously of no importance except so far as affording some sort of 

 guide for that more general estimate which alone can be of any mo- 

 ment. He has designated the most important circles by names* 

 derived from the immediate relations which they bear to certain of 

 the regular solids. The relative values thus assigned to these prin- 

 cipal circles are proportionate to the following numbers : — for 



The primitive great circles 462 



The Octaedrics 612 



The Regular Dodecaedrics 360 



The Rhomboidal Dodecaedrics 110 



The values assigned to other lines is much smaller, varying from 24 

 to 2. But these numbers must merely be understood as indicating 

 that the great circles comprised in the four classes above named are 

 those in which the characteristic symmetry of the pentagonal reseau 

 is most obviously exhibited. 



To complete the whole reseau of the sphere we must produce all 

 the lines represented in the single pentagon of our diagram into 

 great circles, and complete the same process for every pentagon. 

 Many of the great circles thus drawn commencing with different 

 pentagons will be identical, but still, even if we proceed no farther 

 with the process than is indicated by the actual lines and points of 

 intersection represented in the diagram, the number of great circles 

 which are independent of each other will be immense. 



* The names are attached to the respective lines in the figure. 



