456 



CRYSTALLOGRAPHY. 



Mathema- 

 tical 

 Theory. 



to the plane of the divided circle, or that the edge or 

 common section of the surfaces of the crystal, points 

 to the bar of the window, the index is set to the be- 

 _ Y " ginning of the scale by means of a stop at the 180th 

 degree, and the image of the vertical window bar, or 

 any rectilineal object formed by reflection from the 

 first or right hand surface of the crystal, is brought to 

 coincide Avith the direct image by the vertical motion 

 of the cock. The whole graduated circle is then made 

 to revolve by the toothed pinion, till the reflected image 

 of the vertical bar again coincides with the direct 

 image when examined in the other surface of the crys- 

 tal. When this position is obtained, the index of the 

 vernier will point out, on the divided arch, the angle 

 of the crystal. In order that the instrument may be 

 used merely when held in the hand, a vertical frame 

 HK is attached to it by the arm DH, and the parallel 

 silver wires stretched across it are used instead of the 

 window bar. See the article Goniometers ; and Brew- 

 ster's Treatise on New Philosophical Instruments, p. 89. 

 We have never had an opportunity of seeing any de- 

 scription of the repeating goniometer of Malus. But it 

 is easy to see how the goniometer of Wollaston might 

 be made a repeating circle. In most cases, however, 

 the use of such an instrument implies a degree of ac- 

 curacy which cannot be attained in the mensuration of 

 the angles of crystals. 



Chap. II. 



Mathematical Theory of the Structure of Crystals. 



Mathema- In the preceding Chapter we have given a popular 

 tical theo- view of the structure of crystals, which we conceive 

 T" will be easily understood by any attentive reader, even 



though he should not be conversant with mathematics. 

 If we were to confine ourselves to this popular view of 

 the subject, however, this article would have little utili- 

 ty, because it would be out of the power of readers to 

 judge of the accuracy of the principles which have been 

 laid down, or to understand how these principles were 

 discovered. Far less would they be able to prosecute 

 the subject themselves, to investigate the structure of 

 new crystals, and to carry the theory of crystallization 

 to a state of perfection. To bring this valuable bi-anch 

 of knowledge within their power, is the object of the 

 present Chapter. 



The whole mathematical theory of crystals belongs 

 to the Abbe Hauy. For what has been clone by others 

 is, comparatively speaking, so trifling, that we may over- 

 look it altogether. He has prosecuted the subject with 

 indefatigable industry for more than 30 years. His 

 first essays on it, appeared in the Memoirs of the French 

 Academy. He afterwards published an Essay on the 

 subject, in which he developed the mathematical theory. 

 From that period to the year 1801, numerous papers of 

 his appeared in the Journal dc Mines, investigating the 

 crystals belonging to different species of minerals. In 

 the year 1801, his Traite de Mineralogie appeared. In 

 this work he has inserted a complete view of the sub- 

 ject, so luminous and well arranged, that we shall have 

 little more to do than to extract the essential parts of 

 that treatise, unless there happen to be one or two cases 

 in which subsequent improvements have been made. 

 We ought to mention, that since the year 1801, numer- 

 ous papers on the same subject have appeared in the 

 Journal de Mines; and the Annates de Museum d'His- 

 ioire Naturelle, drawn up by the Abbe Hauy, and con- 

 taining much new and valuable matter. Count Bour- 

 ■Bon has likewise published an important work on cal- 



Mathema. 1 



tical 



Theory. 



careous spar, in which he has given a theory of crystal- 

 lization of his own, which has been animadverted on in 

 a masterly manner by the Abbe Hauy, in one of the 

 numbers of the Annates de Museum d'Histoire Naturelle. 

 We are not aware that any thing has been published in 

 Great Britain upon the mathematical theory of crystal- 

 lization, though we are acquainted with several per- 

 sons who have studied the subject. A valuable paper 

 on the crystals of tinstone, by Mr Philips, has been pre- 

 sented to'the Geological Society, and inserted in then- 

 transactions. The want of any English treatise on the 

 subject, will oblige us to be more particular than would 

 otherwise be proper for an article published in an En- 

 cyclopaedia. 



I. Preliminary Notions. 



1 . The object of the theory is to determine all the Prelimina- 

 diff'erent forms which can be produced from the super- ry actions, 

 position of plates diminishing in size according to given 



laws, and in given directions, upon the different faces 

 of a solid, the form of which is likewise given. 



2. This solid, called the nucleus or primitive form, is 

 always one of the six following : 1 . The parailelopiped 

 2. The regular six-sided prism. 3. The rhomboidal 

 dodecahedron. 4. The octahedron. 5. The regular 

 tetrahedron. 6. The triangular dodecahedron, con- 

 sisting of two six-sided pyramids, applied base to 

 base. 



3. By subdividing these primitive forms in the way 

 described in the last Chapter, we obtain the shape of 

 the integrant molecules. These are either, 1. Paral- 

 lelopipeds. 2. Triangular prisms ; or, 3. Tetrahe- 

 drons. 



4. When the integrant molecules are tetrahedrons or 

 triangular prisms, they are always so grouped together 

 in the crystal, as to compose parallelopipeds. And the 

 decrements which produce the secondary faces are al- 

 ways made by the abstraction of ranges of these paral- 

 lelopipeds. Hauy gives to these parallelopipeds the 

 name of subtractivc molecules. As far as the theory of 

 crystallization is concerned, we have to do only with 

 molecules of the form of parallelopipeds, Indeed, the 

 whole doctrine of the shape of the integrant molecules 

 is entirely hypothetical. 



5. When the nucleus is not a parailelopiped, we may 

 always substitute in place of it a solid of that form, ei- 

 ther by abstracting some of the faces if there are more 

 than six, or by multiplying the subdivisions in the di- 

 rection of the natural joints if it is a tetrahedron. But 

 simpler results are often obtained by giving the prefe- 

 rence to the true nucleus. 



6. The decrements which the plates of superposition 

 undergo, may take place in every possible direction. 

 The limits of these directions are the edges and the 

 diagonals of the faces of the nucleus. Between these 

 two limits there are an infinity of intermediate direc- 

 tions, according as the molecules, the ranges of which 

 determine the decrement, are conceived to be single, 

 double, triple, &c. When the decrements are parallel 

 to the edges, they are called decrements on the edges ; 

 when parallel to the diagonals, they are called decre- 

 ments on the angles ; and when they are parallel to lines 

 intermediate between the edge and diagonal, they are 

 called intermediate decrements. 



Let us now run over all the primitive forms, giving, 

 with respect to each, the method of calculating the re- 

 sults of all the laws of decrement of which they are sus- 

 ceptible ; and beginning with the parailelopiped, which 

 is the term of comparison to which the other forms are 

 referred. 



