1 63 
of' Weiss , Mohs , and Haiiy. 
result from simple decrements on the edges or angles. The re- 
maining case to be examined, is therefore the determination of 
the indices of a dodecaedron resulting from an intermediary de- 
crement. It has already been proved, that 
cos | (i • i) y z 
cos f {i : i) x-—y 
And if another simple relation may be obtained between a, y , 
and z, the problem will be resolved. It has been proved, that 
the dodecaedron under consideration may be conceived to be de- 
rived from a decrement on the lateral angles of a rhomboid, the 
oblique diagonals of which correspond to the lines db 9 Fig. 1, 
by &+£ — 2A rows in breadth, and that this rhomboid is derived 
x—y 
2z • 
by a decrement of — rows in breadth on the superior angle 
J x q- y r & 
of the primitive. Now, the angle of this rhomboid may be easi- 
ly determined by means of the measured angles ( i * i), ( i : i), and 
the number , which is known since it is equal to 
x 
—y 
2 . 
y — z 
-f- 1, that is to say to 
2 cos 4 ( i * i) 
+ 1 . 
x — y * cos \ (i : i) 
For, in the Number of this Journal before alluded to, it is 
proved that if (P, P) represents the incidence of the faces of 
the rhomboid, the following equation obtains 
n — 2 tang 4 ( e n : e n ) cos \ (P, P). 
From which it will be easy to find, in the present case, the angle 
of the rhomboid, the oblique diagonals of which correspond to 
the lines db , d$. This angle being determined, the law of de- 
crement by which it is derived from the primitive may be cal- 
culated by means of formulas previously explained ; and the in- 
dex of this decrement being made equal to — - — , furnishes a 
& x + ij 
second equation, which, together with the equation 
cos \ (i . i) _ y - — z 
cos \ (i : i) ~~ x — if 
is sufficient to determine the ratio between two of the three 
quantities x, y> z, and the third, that is the indices of the dode- 
