Prof. Miller’s Crystallographic Notices. 329 
of the circle, corresponding to the observations by reflexion from 
any two faces of a crystal, gives the angle between normals to 
the faces, from a point within the crystal. The data employed 
in calculating dihedral angles, and the results of the calculations, 
are expressed in the same measure. The supplement of the 
angle between normals to two faces is nowhere used as the 
measure of a dihedral angle, except in the lists of angles which 
accompany the descriptions of mineral species. An angle taken 
from one of these lists cannot be compared with the direct result 
of observation or of calculation, without first subtracting one or 
the other of them from 180°. In order to avoid the needless 
trouble of subtracting angles from 180°, which from its frequent 
occurrence becomes extremely irksome, the editors of the last 
edition of Phillips’s ‘Mineralogy’ ventured to measure a dihedral 
angle by the angle between normals to the faces containing it, 
from a point within the crystal. That they have not been over 
hasty in breaking through an inconvenient and unphilosophical 
convention, may be gathered from the fact that the same defini- 
tion has since been adopted by Beer, Dauber, Grailich, Guiscardi, 
Handl, Hess, v. Lang, Murmann, Rotter, Schréder, Sella, 
de Sénarmont, and v. Waltershausen. 
The use of the angle between normals to the faces as the 
measure of the dihedral angle they make with each other, is 
attended by some incidental advantages. It enables the reader 
to apprehend more clearly the relative positions of the faces by 
inspection of the recorded angles; employs fewer figures; and 
in the descriptions of twin crystals, marks re-entering angles by 
giving them negative values. 
On the Cleavages of Rutile. 
In Breithaupt’s ‘Mineralogy’ rutile is described as having 
cleavages parallel to the faces of the forms 100 and 110, with 
traces of cleavage parallel to the faces of the form 111. The 
last of these has been overlooked in the mineralogical treatises 
which have appeared since 1847, the date of the last volume 
published of Breithaupt’s work. Three crystals of rutile forming 
part of the Brooke Collection, now in the Mineralogical Museum 
of Cambridge, exhibit the cleavage 111 very distinctly. In two 
of them it is interrupted by traces of cleavage parallel to the 
faces of the form 321. I have also observed the cleavage 111 
in two crystals of rutile in my own possession, and in one of 
them, rather obscure traces of the cleavage 321. The symbol 
111 is used to denote the simple form in which 111, 111 
=95° 20; 111, 111=56° 52’. 321, 001=66° 42’; 
$21, 231=20° 46; 821, 111=26° 0. 
