1821.] Mathematical Principles of Chemical Philosophy. 87 



A the larger angle. Each of these will split into a number of 

 smaller parallelograms, in lines parallel to E F, G H ; and since 

 the smallest visible portion of such a crystal is a parallelogram, 

 equiangular with A B C D, having its sides parallel to the sides 

 A B and B C, either of the crystals A D C, D A B, may be con- 

 sidered as formed of such parallelograms, in which construction 

 we observe the law of decrements of Haliy. If a smaller tri- 

 angle, I B K, be removed, the resulting figure is an irregular 

 pentagon, A E K C D, of which D is the less angle of the paral- 

 lelogram A and C, e ch equal to the greater, A I K, E K C, 

 each equal to the smaller + half the greater : or if a triangle 

 less than D C B be removed, there results a pentagon, of which 

 the angles are the reverse of the above. If the triangles E B K, 

 L D M, be removed, the resulting figure is an irregular hexagon, 

 of which the angles A and C each equal the greater angle of the 

 parallelogram, and the other angles each equal the less + half 

 the greater. Other hexagons may be formed by removing, 

 instead of the above, both the adjacent angles, or one of the 

 above, and one of the adjacent ones. If all the four be removed, 

 we have an irregular octagon ; .and in the same manner, several 

 other figures may be formed, as trapezia, 8cc. The other class 

 of the bases of *^ crystals are the most abundant in nature : of 

 these, the following are some of the prin- 

 cipal forms : A B C D, fig. 5, being the 

 parallelogram, if four triangles, E B H, 

 H C G, See. be removed, so that the 

 straight hnes E H, H G, &c. all meet 

 each other, since the particles are all 

 equal, the resulting figure E F G H is a 

 rectangle, by the nature of the rhombus ; 

 and its directions of splitting will be pa- ^^'^''''^fa^^^^ 

 rallel to I K, L M, as before : this form ^^ 



often occurs in nature. In all the above forms, the sides 

 of the crystal have been parallel either to the sides or the 

 diagonals of the original parallelogram; but there remains a 

 variety of forms, in which this is not the case, of which the fol- 

 lowing is an example : A B C D being the parallelogram, an 

 irregular hexagon, a be clef, may be formed, and besides this, 

 a great variety of quadrilateral, pentagonal, hexagonal, &.c. 

 figures after the same laws of decrement, in all which, every 

 external particle is one of an entire elementary paralblogram ; 

 consequently, prop. 10, in a state of permanent quiescence. In 

 all the above varieties, it is evident that these bases can only be 

 split into equiangular parallelograms ; and hence every form 

 may be derived immediately from that figure, by circumscribing 

 each circle by a parallelogram similar to A B C D, and having 

 their corresponding sides parallel. 



In the same manner we may proceed to examine the structure 



