On the Drawing of Figures of Crystals. 47 
tices of the solid angles. If therefore these points (the extremities 
of the interaxes,) can be determined in the several crystalline forms, 
it will only remain to connect them, in order to form a projection of 
these solids. ‘The principles of analytical geometry afford the means 
of determining how much the interaxes of the octahedron must be 
increased or diminished to equal the interaxes in these different forms. 
It is thus found that each half of a trigonal interaxis must be increas- 
ed by that portion expressed in the fraction 
2 mn—(m+n) 
mn-+ ~ mn+(m+ n) } 
and for each half of a rhombic interaxis, we have the corresponding 
fraction 
n—1 
« n+l 
_ By giving mand x different values from 0 toa, the values of 
these interaxes for any monometric form may be obtained. The 
following values are thus deduced for several occurring forms ; 
Trig. interaxes. Rhombic interax. 
‘ 
Trigonal trisoctahedron (fig. 20.)* 20 il 0 
Dodecahedron (fig. 7.) «oO a 0 
Hexoctahedron (fig. 25.) 302 2 1 
hi ! 402 é = 
ce 002 = a 
Tetrahexahedron (fig. 11.) wO2 1 1 
i 003 = 1 
Tetragonal trisoctahedron (fig. 16.) 202 + 1 
c 303 4 1 
Cube, ~  @Oa 2 1 
To construct the form 402, the octahedron is-first to be projected, 
and its axes and interaxes drawn. ‘Then add to each half of each 
trigonal interaxis, five sevenths of its length; and to each half of 
each rhombic interaxis, one third of its length. ‘The extremities of 
the lines thus constructed, are situated in the vertices of the solid 
angles of the hexoctahedron 402, and by connecting them, the pro- 
jection of this form is completed. 
22. In the inclined hemihedral monometric forms—that is, those 
hemihedral forms whose opposite faces are inclined to one another 
and not parallel, as the tetrahedron, &c.—the rhombic interaxes do 
* For these and the following references to figures, the reader is referred to the 
copperplates in my system of Mineralogy. 
