424 Haiiy on the Measuring of the Angles of Crystals. [Junz, 
the faces, the axes, the lines drawn perpendicular to these axes, 
either from the centre of the faces, or from the solid angles. If 
the formula, for example, relates to a rhomboid, it will contain 
the expressions g and p, half of the horizontal diagonal and of 
the oblique one of each rhomb, the expression a of the axis and 
that of the measure of the decrement to be determined, or the 
number of ranges subtracted, which is denoted by n. This last 
expression is always simple, or deviates very httle from simpli- 
city. With regard to the other expressions, they are equally 
simple in the forms, which have a particular character of symme- 
try and regularity, or those derived from these forms. Thus in 
the rhomboid which represents the subtractive molecule of the 
thomboidal dodecahedron, the ratio between the semidiagonals 
of each rhomb is that of »/ 2 to 1; this is also the ratio between 
the perpendicular drawn from the middle of each face upon the 
axis, and the portion of the axis which it intercepts. In the 
rhomboid which I consider as the subtractive molecule of the 
regular octahedron, the first ratio is that of 1 to / 3, and the 
second that of 1 to ./8. In the cube there exists equality 
between the two terms of the first ratio, and the second is that 
of 1 to .f 2. The cosine, either of the small plane angle, or of 
the smallest incidence of the faces, has this remarkable, that its 
ratio with radius is rational; and to confine myself here to that 
which concerns the incidence of the faces, it is half the radius 
in the rhomboid of garnet; it is the third in that which belongs 
to the regular octahedron, and in the eube it becomes zero. — 
A part of the laws of decrement on which the secondary varie- 
ties of the forms under consideration, are in the same case as the 
ratios between the dimensions of these forms; that is to say, 
that their measurement is considered as given @ priori. Thus 
the passage of the cube to the rhomboidal dodecahedron, in 
aplome, that of the same solid to the regular octahedron in sul- 
phuret of iron, and that of the last solid into the two preceding 
m fluate of lime, takes place evidently in consequence of a 
decrement by one range on the edges or angles of the form which 
performs the function of the primitive. The same consideration 
may be applied to the trapezoidal solid, taken as a secondary 
form, either of the cube as in analcime, or of the regular octahe- 
dron as in sal ammoniac, or of the rhomboidal dodecahedron as 
in the garnet. When the law of decrement is not indicated 
immediately by the aspect of the form, it may be determined 
with certainty from the reason of the greatest simplicity. Thus 
when we measure by the common goniometer the respective 
inclination of the pentagons of dodecahedral sulphuret of iron 
where their bases meet, [ find it nearly equal to 127°. Further, 
calculation informs me that on the hypothesis that the decre- 
ments producing these pentagons take place by two rows in 
breadth on the edges of the primitive cube from which they set 
