Crystals requiring neither Measurement nor Calculatkyn, ^38 
will be considerably simplified, and frequently dispensed with. 
They might ha-ve been obtained in a very different way, by a se- 
ries of similar triangles. The preceding method has been thought 
more simple and more direct. However, as the other may be 
used with advantage in some cases, it will not perhaps be useless 
briefly to indicate it. 
By the extremity c of the edge oc of the primitive. Fig. 12., 
let the planes cHG, cKL, cMN, cPQ, be drawn parallel to 1, 
2, 9, 4, the distances oH, oG, &c. will be known, and equal to 
~ 5, — a, &c. The intersection of 1 and 2 will be parallel to 
Pi Pi 
oE, that of 3 and 4 to oF. The new plane 5 will therefore be pa- 
rallel to the plane oEF ; and if the line EF is drawn, oA will be 
equal to — , and oB to — . Now, to find the values of these, let 
Ps Ps 
ER, F^, be drawn parallel to ob. It is very obvious, that 
knowing oG, oH, oK, oL, the values of ER, oR may easily be 
found by similar triangles. Those of F<s and os are not more 
difficult to obtain, and from these four, oR, RE, os, s¥, the 
values of oA and oB are easily deduced. This would' lead tq 
the same formula. 
These formulas, although they are complicated, are remark- 
able. They contain neither a, nor b, nor c. Therefore, the li- 
near dimensions of the primitive form are not requisite to deter- 
mine the laws of decrements of the faces under consideration. 
This remark proves, also, that it is not possible to find the va- 
lues of the linear dimensions of the primitive, from any obser- 
vation of parallelism between the edges of a secondary crystal. 
The values of m^, n^, &c. being simple numbers, it is ob- 
vious, from the nature of the functions of them, which repre- 
sent the values m^, n^, that the values of these last quan- 
tities will never be very large numbers. It is necessary tq 
add, that when these formulae are used, m ^ , for instance, must 
be taken negatively, if the face whose indices are m^, n^, p^, 
cuts the edge oa, Fig. 10., in its prolongation oa'. The same 
observation applies to any other index. 
To shew at once the importance and use of what precedes, I 
shall apply it to the determination pf the several modifications of 
