1821.] Mathematical Principles of Chemical Philosophy. 85 



heat alone, or by the agency of a liquid menstruum ; and it i& 

 when these particles again approach to contact with each other 

 that they arrange themselves in some particular and determinate 

 order, in consequence of the operation of those forces which 

 have been investigated. It is well known that crystals of pre- 

 cisely the same substance assume a greut variety of external 

 forms, arising from the external circumstances under which 

 they are formed. We are perfectly ignorant of the causes which 

 determine the external figure, and we can only expect to be able 

 to show what external forms may result from a given arrange- 

 ment of particles ; and it is manifest that any external figure 

 may result, in which every external particle is in a state of per- 

 manent equihbrium, or quiescence, which may easily be inves- 

 tigated by help of the foregoing propositions. There are, 

 however, many pecuHarities which cannot be examined until the 

 nature of chemical union, and the structure of compound bodies^ 

 have been investigated. 



Prop. 12. — If any number of equal and similar particles, 

 influenced only by their inherent forces, mutually approaching 

 each other, ultimately come into contact, they will be symmetri- 

 cally arranged. 



Let A, B, C, D, fig. 2, be four equal 

 spherical particles of the same kind, 

 which are preserved in contact by the 

 force of cohesion, they will form a 

 parallelogram (prop. 10). Let the two 

 E, F, approach to D, C; they will join 

 themselves in such a direction that a 

 parallelogram will result, which is equian- 

 gular with the former ; if not, let them 



unite differently; and by prop. 10, they must form the paral- 

 lelogram C B E F, whose major diagonal D F is not parallel 

 to that of A B C D : join A E, produce A D, and from E 

 draw E G, perpendicular to A G. If A E represent the 

 force with which A and E mutually attract each other, it 

 may be resolved into the two A G, G E, of which, A G tends to 

 bring E into contact with D, and G E to bring its centre to the 

 line A G ; and the same may be proved of the particle F; where- 

 fore the particles E and F will, when they join D and C, have 

 their centres in the straight hnes AD, B C, produced, or the 

 two parallelograms being equal will have their major and minor 

 diagonals respectively parallel to each other. 



Cor. — Hence crystals are always terminated by straight lines 

 and plane surfaces. 



Prop. 13. — Crystals will spHt so as to have a smooth fracture 

 only in the directions of the rows of particles, and these direc- 

 tions are either parallel to, or make a determinate and constant 

 angle with, each other. 



