File:The Quarterly journal of the Geological Society of London (12512212164).jpg

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