CRYSTALLOGRAPHY. 



473 



Mathema- culaT to c£ ; lastly, draw A n. The angle i> A ?r will be 

 the supplement of that which measures the inclination 

 wanted. Let us find > A and » s. 



Theory. 



PHT( 



CCXXV. 

 Fig. C. 



(l.)ForvA; , te ^'. 

 ' Cy 



We have already found c y = \/2. And cv = 

 V(c^) 2 -}-(^) 2 and we have found, c £=</£. Farther, 

 y £=V|7 B ut y C : C » : ■ V5j «/3. Hence £ »- y^. 

 Therefore c »=v / f+^=/I|=y'*. 



y»=y / (y0 2 +(C>) g = v4+^=:v4^ Hence 



(2.) For , *•; ,!«=£.!*&, Apd ^ _ ^ . ^ _ 



Cl = 



-vA v _»- 



^/ T V; and c£ 

 V 25 o 



-vAy- Therefore » ?r = 



</i 



rr g . 2. 



Primitive 

 forms dif- 

 ferent from 

 ilie paral- 

 lelopiptd. 



Hence .>:» r: : ^± : J 3b ■ :% /5 : </3. 



~ 5 5 

 Now this is the ratio between the side of the primi- 

 tive rhomb and half the horizontal diagonal Conse- 

 quently the angle i^= — — — = 50° 46' 6". 



Hence it follows that the incidence of c-yfti (Fig. 2.) 

 on cypr, is 129° 13' 54". 



Of the Primitive Forms different from the Parallele- 

 piped. 



When the primitive form is a cube, the investigation 

 of tire secondary crystals admits »f certain modifica- 

 tions, which in some cases considerably shorten the 

 calculus. But as there is no new principle in these in- 

 vestigations different from what has been already ex- 

 plained while treating of the rhomboid ; we do not 

 consider it as necessary to introduce the peculiar me- 

 thods here. Those readers who are interested in the 

 subject, will find it amply discussed in Hauy's Miner- 

 alogie, vol. i. p. 410. 



All the other primitive forms, namely, the rhomboi- 

 dal dodecahedron, the tetrahedron, the octahedron, 

 the six-sided prism, and the bipyramidal dodecahedron, 

 may, by very simple analogies, be brought under the 

 case of parallelopipeds. Indeed we ma}', without in- 

 juring the theory, substitute instead of them a paralle- 

 lopipedal nucleus, and refer all the decrements to it. 

 The most difficult to manage in that way is the bipy- 

 ramidal dodecahedron ; but it so very seldom occurs in 

 the mineral kingdom as a primitive form, that we do 

 not think it necessary to enter upon the subject here. 

 We again refer the reader to Hauy {Miner alogie, vol. i. 

 p. 451.) for all the elucidations necessary to beginners. 



Method of determining the ratio between the principal 

 dimensions of the integrant molecules. 



On tlie ra. Tms is an element in all the calculations respecting 

 tio by ween secondary crystals; of course, the consideration of it 

 the princi- cannot be omitted. Some forms furnish us at once with 

 'loL^tT' thCSe rati ° S ' in consCl i l!! ' nte of the perfect regularity 

 integrant '* which they a PP ear to possess. For example, we cannot 

 Molecules, doubt that the form of the integrant molecule of com- 

 mon salt is a cube, and that the primitive crystal of 

 blende is a dodecahedron, with rhomboidal faces equal 



VOL, VII. PART II. 



and similar ; from which it follows, that the integrant Mathema 

 molecules are tetrahedrons, having equal and similar t,cal 

 triangular faces. From this it follows, that the ratio be- "~ "' y " 

 tween the two diagonals of each rhomb is that of ^/2 ~~* , <~ m 

 to 1. 



But in certain cases (as when the primitive form is 

 a rhomboid) there is nothing which indicates the size 

 of the angles. In such cases, peculiar methods must 

 be employed to obtain the requisite ratios. Hauy, to 

 whom we are indebted for every thing relating to this 

 subject, has been guided in his investigations by this 

 maxim, that two quantities are to be considered as equal, 

 when observation points out no difference between them. 



To give an example : When the regular hexahednd 

 prism of carbonate of lime is mechanically divided, we 

 observe, that each section has the same inclination both 

 to the base and to the adjacent face of the prism. If 

 we suppose that this holds rigorously, it is easy to see, 

 that in the rhomboid of calcareous spar, the triangle 

 a c n (Fig. 7.)> formed by the oblique demi-diagonal p LlTE 

 a c, by the demi-pendicular c n on the axis, and by a n, CCXX1 V. 

 the third of the axis, is at the same time rectangu- F 'g- <• 

 lar and isosceles. Hence it follows, that cnz=.an, or 

 V \ g*z=$V9p* — g % . Taking away the radicals, get- 

 ting rid of the denominators, and simplifying, the equa- 

 tion becomes g z =z3jJ 2 — g*- Hence 2g 2 = 3p* and 

 g:p:-W3 :x /<2. 



As a second example, we shall make choice of the 

 tourmaline. Crystals of this mineral are known, which 

 have the form represented in Fig. 7. When we mea- Pure 

 sure the inclination of o to I, we find it sensibly the CCXXV. 

 same as that of the edge x to the face P'. But the la- P'ff- 7 - 

 teral edges i/, y' of the faces o being parallel to each 

 other, and to the oblique diagonal of the primitive face 

 P, it is evident, from simple inspection, that these faces 

 result from the decrement I E I (Fig. 8.) We see, like 

 wise, that the faces / (Fig. 7.) are produced by a de 



crement e. 



Let gads (Fig. 18.) be the section of the nucleus of Pi,ate 

 the tourmaline, and tg a line situated as the apotheme of COXXIV. 

 the triangle o ( Fig. 7, Plate CCXXV.) It follows, from Fi S- l8 - 

 what has just been stated, that the inclination of o to I 

 is equal to that of tg (Fig. 18.) to a line drawn through 

 the point g parallel to the axis. Or, which is the same 

 thing, it is the supplement of the angle gtn, on the 

 hypothesis that gt is the oblique angle of a rhomboid 

 resulting from the law "E*. Again, the angle formed 

 by x with P' (Fig. 7.) is equal to the angle gad 

 (Fig. 18.); or, which comes to the same thing, it is 

 the supplement of the angle ags. Hence gtn—ags. 

 Therefore gn:tn:: Sin. ags : Cos. ags. Substituting 



the algebraic values, we get -v/igH — rp— V^p'' og 2 : : 



V3g 2 p 2 — g 4 : g 2 — /5 2 . Getting rid of the denomina- 

 tors, and dividing the two first terms by 2, the propor- 

 tion becomes Vg i : 2\/3p* — g z : : M8 p* g- — g 4 ~: g 2 — p*. 

 Dividing the two antecedents by g, and multiplying 



the extremes and means, we get 2 (3/; 2 — g 2 ) =rg 2 tt 2 . 



Therefore 7 p'=3 g 2 , and g : p:: a/7 : </3. 



When such analogies are wanting, Hauy arrives at 

 the ratio of the dimensions by adopting the simplest 

 ratio which agrees with the measurement of the incli- 

 nations of the faces, and considering it as exact. An 

 example will make the method obvious. Let us make 

 choice of sulphate of iron, the primitive form of which 

 is an acute rhomboid. By the goniometer we find that 

 the smallest inclination of the faces of this rhomboid in 

 nearly 81 ° 30'. If the ratio between the cosine of tins 

 3o 



Fig. «. 



