41 
Ordinary Meeting, April 6th, 1858. 
W. Fairbairn, F.ll.S., &c.. President, in the Chair. 
The Rev. T. P. Kirkman, M.A., F.R.S., communicated 
the following theorem — “ In every Polyedron whose summits 
are all triedral, and whose 'simplest face is pentagonal, the 
number of pentagons =12 + that of the heptagons + twice 
that of the octagons + three times that of the nonagons + 
&c. Or, putting a. &c. for these numbers, 
— 12 = rt, + 2a« + 3«9 + dcfio + 
a formula independent of the number of the hexagons.” 
F. Grace Calvert, M.R. A. of Turin, F.C.S. &c., and Mr. 
Richard Johnson communicated a paper “On the Hardness 
of Metals and Alloys.” — The process at present adopted for 
determining the comparative degree of hardness of bodies con- 
sists in rubbing one body against another, and that which 
indents or scratches the other is admitted to be the harder of 
the two bodies experimented upon. Thus for example. 
Diamond, 
Glass. 
This method is not only very unsatisfactory in its results, 
but it is also inapplicable for determining with precision the 
various degrees of hardness of the different metals and their 
alloys. 
We therefore thought that it would be useful and interesting 
if we were to adopt a process which would enable us to repre- 
sent by numbers the comparative degrees of hardness of various 
metals and their alloys, and tUe following are some of the 
results arrived at. 
Prochtdings No. 13. 
