1819.] Haiiy on the Measuring of the Angles of Crystals. 427 
tatic dodecahedron it is only 10’ and 4’ for the two respective 
inchnations of the faces situated towards the same summit. In 
another dodecahedron, which results from a decrement whose 
exponent is ¢ on the same edges of the primitive rhomboid, it is 
reduced to 2’ and 1’ 2”; and in a third dodecahedron, produced 
by an intermediate decrement on the lower angle, and which 
belongs to the variety which I have called euthetic, it falls 
between 1’ 50” and 26”. 
Now it is evident that the ordinary goniometer employed to: 
verify these different results, is of a precision which may be 
considered as rigorous. The angles of the crystals of quartz, of 
oxide of tin, and of sulphate of lead, have presented conver- 
gences of the same nature, though rather less sensible. 
I add that the forms of the integrant molecule, being the 
geometrical types of the species, the ratios which I have 
adopted have, in consequence of their simplicity, the advantage 
of offering neat conceptions, and easy to take up from that which 
characterizes these types, and the lines of demarcation between 
the different species deduced from them, while the mind perceives 
only through a mist, as it were, these distinguishing characters 
obscured by the great numbers in which they are enveloped. 
We perceive at once and we remember the result which informs 
us that the cosine of the smallest incidence of the faces in the 
primitive rhomboid of quartz is the thirteenth of the radius. But 
the other result, according to which it is only the -93,, is not 
easily understood, and cannot be remembered. 
I have advanced above, that the ratios between the dimensions 
of the primitive solids, such as | have chosen them, are sufficient 
to determine without ambiguity the laws of decrement from 
which the secondary forms are derived. This I shall render 
sensible by an example drawn from the forms produced by decre- 
ments on the inferior edges, D, D (fig. 11), of the primitive rhom- 
boid of calcareous spar. This decrement produces dodecahedrons 
with scalene triangular faces, more or less elongated, which If 
represent in general by that represented in fig. 12. When two 
ranges are abstracted, we obtain the metastatic variety in which 
the incidence of N on N is 144° 20’ 26”, that of N on N’ 
104° 28’ 40”, and that of N on N” 133° 26’. Among the other 
known dodecahedrons, that which approaches most nearly to the 
i 
preceding has for its sign D. This law gives 
For the incidence of N on N , 139° 52’ 50”. Diff. 4° 27’ 36”, 
of Non N’, 106 13 30. Diff. 1 44 50. 
of N on N”, 141 12 24. Diff. 7 46 24. 
Hence it is obvious that we can easily avoid mistaking this 
last dodecahedron for the metastatic. 
Let us suppose a dodecahedron much nearer than the last, 
