600 



CRYSTALLOGRAPHY 



same substance may differ much in general appear- 

 ance, but when the angles between their faces are 

 measured these angles are found constant. Thus 

 the crystals A and 1J (fig. 1 ), when cut through in 



Fig. 1. 



Drawings of two crystals differing much in appearance, but 

 with angles at a shown to be constant when similar sections 

 are made. 



the direction xy at right angles to the prism, give 

 the sections shown at A', B' ; and in each section 

 the angles a will be found the same viz. 120 ; or 

 again, if the angles between the faces ab, be, or 

 ac, be measured, they will be found identical in both 

 crystals. 



(2) Law of Symmetry. Suppose we cut a crystal 

 in two, and then place the two parts with their cut 

 surfaces on a mirror. The mirror will reflect each 

 part, and may or may not produce the appearance 

 of the original crystal. If the mirror will produce 

 the appearance of the original crystal, we have 

 severed the crystal in a plane of symmetry. Thus 

 with a cube, if we cut it in either of the planes 

 abc, def, ghk, Imn, opq, 

 rhm, nhg, Ikn, gmk, and 

 place iii each case the 

 two severed parts on a 

 mirror in the way de- 

 scribed, the reflection 

 together with the object 

 Avill reproduce a cube. 

 There are then in the 

 cube nineplanes of sym- 

 metry. The octahedron 

 and dodecahedron simi- 

 larly have nine planes of 

 symmetry. With such 

 a form as a common 

 brick there are three 

 planes of symmetry, 



Fig. 2. 



Each of the planes represented 

 by dotted lines is a plane of 

 . symmetry. 



while with other forms varying numbers "of planes 

 of symmetry may be found, until with a sphere 

 there are an infinite number of planes of symmetry, 

 for it is obvious that if a sphere be cut anywhere 

 by a plane passing through its centre, and the 

 half thus obtained be laid upon a mirror, the 

 appearance of a complete sphere will be produced. 

 NOAV examining all (holohedral) crystals, it is 

 found that they fall into one of the following six 

 categories or systems: (1) Anorthic System. No 

 plane of symmetry examples, copper sulphate 

 and anorthite. (2) Oblique System. One plane of 

 symmetry gypsum and washing-soda. (3) Pris- 

 matic System. Three planes of symmetry at right 

 angles to each other barytes, saltpetre, and native 

 sulphur. (4) Rhombohcdral System. Three planes 

 of symmetry at 120 to each other calcite, quartz, 

 and ice. (5) Pyramidal System. Five planes of 

 symmetry cassiterite, zircon, and idocrase. (6) 



Cubic System. Nine planes of symmetry fluor- 

 spar, galena, and alum. 



(3) Law of nationality of Indices. The various 

 planes of crystals, as explained below, are indicated 

 in the Millerian system by three numbers, which 

 together form the symbol of the plane. Thus we 

 have planes represented by 1 2 3, by 1 1 1, by 

 110, &c. Now the law of rationality asserts 

 that the symbol of a plane must be represented by 

 numbers which are rational i.e. numbers which 

 can be expressed exactly, not those like \ 2) </4, 

 &c., which can only be obtained approximately. 

 Thus by the law of rationality, no plane of a crystal 

 can have such a symbol as 1 \/3 5, 1 v2 0, &c. 



only 



Miller's notation and Naumann's notation. In 

 both systems the planes are referred to three axes 

 corresponding in direction to three edges of the 

 crystal. 



Let abc (fig. 3) represent parts or parameters 

 cut off from three axes xyz, then in Miller's 



- x 



Y' 



z' 



Fig. 3. 



The plane 1 1 1 in Miller's 

 notation. 



Fig. 4. 



The plane 1 2 3 in Miller's 

 notation. 



system the plane 111 represents a plane which 

 cuts the x axis at one-oneth of a, the y axis at 

 one-oneth of b, and the 

 z axis at one-oneth of c. 

 Such a plane is indicated 

 by pqr. The plane 123 

 means a plane which cuts 

 the x axis at one-oneth of 



a, the y axis at one-half of 



b, and the z axis at one- 

 third of c. Such a plane 

 is represented by stu, fig. 

 4. The plane 110 means 

 a plane which cuts the x 

 axis at one-oneth of a, the 

 y axis at one-oneth of 6, 

 and the z axis at one- 

 noughth of c i.e. does not 

 cut c at all, or is parallel 



to it. Such a plane is represented by uvxs in fig. 5, 

 In Naumann's system some form is selected as 

 the fundamental pyra- 

 mid of the crystal, and 

 his pyramid, which cor- 

 responds to Miller's 

 form, 1 1 1, is represented 

 by the letter P in all 

 systems but the cubic ( in 

 this system it is called 

 O) and the rhombohed- 

 ral (in this system it is 



Fig. 5. 



The plane 1 1 in Miller 1 * 

 notation. 



called R). Thus the 



Fig. 6. 



planes marked P ( fig. 6 ) A crystal with the faces marked 

 form the fundamental in Naumann's notation. 

 pyramid, the planes 4P 



are those of a pyramid one-half the height, while 

 the basal plane is represented by oP or a pyramid 



