79 
qf fFem^ Mohs^ and Hauy. 
remarks, will necessarily meet the first the edge EB, above or be- 
low the point E, and the othpr the oblique diagonal CB, above 
or below the point C. It is the on relative position of these points 
K and G that depends the division of the dodecahedrons, in- 
to dodecahedrons produced by decrements on the lateral angles 
of the primitive, those resulting from decrements on the superior 
edges, from decrements on the inferior edges, and finally those re- 
sulting from intermediate decrements. The first case corresponds 
to the coincidence of the points K and E, and we shall first inves- 
tigate the formulae which relate to it. The sign of such dode- 
cahedrons will be placing the index below to distinguish it 
from the sign relative to rhomboid, produced on the inferior 
angle of the primitive, and it is said to be the result of a decre- 
ment by n rows in breadth on the lateral angle of the primitive ; 
so that if E and G are joined, and the line EG produced to F, 
CA is equal to n times CF. The angle of two faces of this 
dodecahedron meeting in a line parallel to the oblique diagonal 
AK, will be designated by ; that of two faces meeting 
in a line in the same vertical plane, as AC, by {e^ . ; and the 
incidence of one of the faces of the upper pyramid upon the 
corresponding face below, in the usual way, This un- 
derstood, we shall have successively, 
sin CAF r= cos CAF = - ^ 1 ^ . CAD), 
AF ’ AF ^ 
hence cot CAF = ; from A as a centre describe 
sin CAD 
a sphere, and let L, M, N be the points, where it meets the lines 
AC, AF, AK. In the spherical triangle LMN we shall have 
4. 7 Ti/rxT <^ot LM sin LN — cos NLM cos LN , . t xTT./r 
cot LMN = ; but LNM 
000 ___ 
sin NLM 
and NLM = (P, P), substituting, we obtain 
w-l- cosCAD . 1 ^ ^ ' 1 A 
^ _ sin CAD • i i 
^ 2 sin (P, F) 
lS^^cos(P,P)cos|CAD 
sin (P, P) 
