Crystdls requiring neither Measurement nor Calculation. 2^9 
Jar polyedrons, but two solids composed of the same number of 
faces, and such that when both are put in position with the pri- 
mitive, all the faces of the one are parallel to all the faces of the 
otlier. Upon this first result of crystallographical researches, 
M. Haliy has built his theory of decrements, supported by so 
many facts, and so much ingenuity. It is well known, that in 
that theory^ a secondary face, such as ABC, Plate VIII. 
Fig. 10. is said to be the result of an intermediary decrement 
upon the s angle o of the primitive, by m rows parallel to the 
edge oa^ n rows parallel to the edges oh, and p rows parallel 
to. the edge oc. If m=zn, then AB is parallel to the diagonal 
ah of the face o ah of the primitive, and the face ABC is then 
the result of a decrement upon the plane angle oah, by m rows 
in breadth, and p in height. Lastly, if p is infinite, then 
ABC becomes parallel to the edge oc, and is said to be the re- 
sult of a decrement upon that edge by m rows in breadth and 
n in height. It will be more simple in what follows, to call m, 
n, p the indices of the face ABC. 
The second important result of Crystallography, is what has 
been called the Law of Symmetry. It consists in this, that, with 
very few exceptions, when any edge or angle of the primitive is 
found to be replaced by a secondary face resulting from a cer- 
tain decrement, all the similar edges or angles are equally re- 
placed by faces resulting from similar decrements. 
This once understood, all questions of crystallography may 
be reduced to problems of solid geometry, and may be resolved 
by plane and spherical trigonometry. The data are the inci- 
dences of the secondary faces with each other and with the pri- 
mitive; the unknown quantities, the linear dimensions of tlie 
primitive, and the indices of the secondary faces. 
To shew in ^ach case how to obtain as many equations as 
there are unknown quantities, or, when this is impossible, tp 
point out the best hypothesis that can be made to replace the 
want of equations, will be the object of another paper, which I 
shall publish, when I have brought iny formulse to that degree 
of simplicity which logarithmic calculation requires. 
My object here, is to explain how the indices of certain se- 
condary faces can be obtained without either measurement or' 
calculation. 
VOL. VI. NO. 12. AmiL 1822. p 
