On Crystallography. 8 19 



assume the form of the dodecahedron with rhoiTihic planes, 

 will also be met with under that of" the regular octahedron. 

 Now it ajipears, on the first general view, that it is possible 

 to refer the structure of this octahedron to a decreirieiit oii 

 the edi^esof a cube : f ^r, if we confine ourselves to dtcreas- 

 ing the lannnae of supi-i position, nierelv on the cd>ies of 

 two opposite faces of thi> culje, fur example, on those of 

 the <u|)enor base AhlOI (fig. 2o), and of the inferior 

 A' E' O' I', we shall have in general two pyramids placed 

 on tliese same bases ; and if ue suppose besides that ihe 

 faces of each pvraniid are prolonge^d until they meet those 

 «)f llie other pvramid, which does nothing more than con- 

 tinue the efiect of the law of decrements in the space situ- 

 ated between the bases of the cube, we shall arrive at ati 

 octahedron, the angles of which will vary according as the 

 law shall deternnne a more or less considerable number of 

 subtracted rows. But theory demonstrates, that there i.s 

 no law, however complex we may supi)0se it, which is ca- 

 pable of giving equilateral triangles for the faces of this oc- 

 tahedron. 



On the other hand, if we divide a regular octahedron 

 originating from the cube, we perceive that the cubical nu- 

 cleus is situated in this octahedron, in such away (hat each 

 of the six solid angles of the first answers to the centre of 

 one of the faces of the second, which could not take place 

 in the hypothesis of a decrement on the edges. Fig. 20 re- 

 presents this arrangement ; and we may conceive from a 

 simple inspection, that in order to obtain the nucleus, we 

 must successively lay down the six solid angle; of the oc- 

 tahedron by perpendicular sections on the axes that pass by 

 these same angles, which would necessarily be paralit-l to the 

 faces of the cube. 



I fiave coneludecl from the statement of the position just 

 mentioned, added to the impossibility of here t-pplyino; 

 theoretical calculation, that the law of decrements ha^ 

 attained its object in these cases, by a route different from 

 that which leads to the forms previously described ; and tlie 

 inquiries relative to this object have developed a new order 

 of facts greatly contributing to the fecundity of crysialhiju- 

 tion, and at the same time to that of the theory. 



Let O 1 V O' (fig. 21) be one of the faces of the cubical 

 nucleus, sub'dividi-d into a midlitudc of small squares, 

 which shall be the bases of as many molecules. We may 

 couaider rows or files of molecules in two different direc- 

 tions, namglv, in the direiaion of the edges, like the row 

 tlesjeualed by the lellers a, n, </, /, v', b~:<-., oi in the direc- 

 tion 



