CRYSTALLOGRAPHY. 9 



equally inclined severally to the adjoining faces. Only edges that are 

 formed by the meeting of two similar planes can be truncated or bev- 

 elled. The angle between the truncating plane and the plane adjoining 

 it on either side always equals 90° plus half the interfacial angle ovei 

 the truncated edge. When a rectangular edge, or one of 90°, is trun- 

 cated, this angle is accordingly 135° (=90° v45°) ; when an edge of 70°, 

 it is 125° (=90° + 35°) ; when an edge of 140°, it is 160° (=90° + 70°). 



7. Z?ne. — A zone of planes includes a series of planes having the 

 edges between them, that is, their mutual intersections, all parallel. 

 Thus in Fig. 14, on page 6, at top of figure, i2, *f, in front, and 

 two planes below, and others on the back of the crystal are in one zone, 

 a vertical zone. Again, in the same figure, O at top, 42, 3f , 22, 42, i2, 42, 

 22, 3f. and the continuation of this series below and over the back of 

 the crystal lie in another vertical zone. And so in other cases, in 

 other directions. All planes in the same zone may be viewed as on the 

 circumference of the same circle. The planes of crystals are generally 

 all comprised in a few zones, and the study of the mathematics of 

 crystals is largely the study of zones of planes. 



Axes. — Imaginary lines in crystals intersecting one another at their 

 centres. Axes are assumed in order to describe the positions of the 

 planes of crystals. In each system of crystallization there is one verti- 

 cal axis, and in all but hexagonal forms there are two lateral axes. 



Diametral sections. — The sections of crystals in which lie any two of 

 the axes. In forms having two lateral axes, there are two vertical 

 diametral sections and one basal. 



Diametral 'prisms. — Prisms whose sides are parallel to the diametral 

 sections. 



Measurement of Angles, 



The angles of crystals are measured by means of instruments called 

 goniometers. These instruments are of two kinds, one the common 

 goniometer, the other, the reflecting goniometer. 



The common goniometer depends for its use on the very simple prin- 

 ciple that when two straight lines cross one an- 

 other, as A E, C D, in the annexed figure, the parts 

 will diverge equally on opposite sides of the point 

 of intersection (O) ; that is in mathematical lan- 

 guage, the angle A O D is equal to the angle CUE, 

 and A O C is equal to D O E. 



A common form of the instrument is represented in the figure on 

 page 10. 



The two arms a b, c d, move on a pivot at o, and their divergence, 

 or the angle they make with one another, is read off on the graduated 

 arc attached. In using it, press up between the edges a o and c o, 

 the edge of the crystal whose angle is to be measured, and con- 

 tinue thus opening the arms until these edges lie evenly against the 

 faces that include the required angle. To insure accuracy in this 

 respect, hold the instrument and crystal between the eye and the light, 

 and observe that no light passes between the arm and the applied faces 

 of the crystal. The arms may then be secured in position by tighten- 

 ing the screw at o ; the angle will then be measured by the distance on 

 the arc from k to the I ft. ox outer edge of the arm c d, this edge being in 

 the line of o. the centre of motion. As the instrument stands in the 



