22

CRYSTALLOGRAPHY.

is this second law which gives to Crystallography a mathematicalbasis. It is as follows :—

2. The ratio of the edges removed hy secondary planes is asimple ratio.

In removing the edge A, to produce the planee, parts of the edges U and C are also removed.If then B and C are equal, as in the cube, theparts of B and C removed will, according to theabove law, either be equal, (the edge is then trun-I cated,) or there will be twice as much of one re-

i moved as of the other, or three times as much ;

that is, the ratio of the parts will be either 1 : 1,1:2, 1:3, and also, sometimes, 1 : 4, 2 : 3, 3 : 4. Other ratios some-times occur, but are uncommon. If B and C are unequal, the ratioswill be the same, excepting, that the parts of B and C removed, willbe proportional to the lengths of their edges ; that is, the ratios willbe 1 B : 1 C, or 1 B : 2 C, or 1 B : 3 C, or it may be, 2 B : 1 C, or3 B : 1 C, also, 2 B : 3 C, or 3 B : 2 C. The last expression signi-fies a ratio of three times the length of B to twice the leiisfth of C ;or, if the ed<re B be divided into a certain number of equal parts,and C into the same number, the plane, whose ratio is 3 B: 2 C,cuts off three parts of B, and two of those of C. The figures areused in the same manner in the preceding expressions.

A plane on an angle, (A, B, C, again being equal,) may eithercut off A, B, C, in the ratio of 1:1:1, that is, equal parts fromeach, or in the ratio of 1:1:2, the figures referring to the lettersin the order just given ; or, again, as 1 :1 : 3, 1 : 1 : £, 1: l : $, or1 : 1 : J, in which the part cut from C is only one fourth that cutfrom either A or B. So also, there may occur the ratios 1:1 : §,1:1:2- Others are of occasional occurrence. If A, B, and C,are unequal, the first ratio above, that is, the ratio of equality, be-comes 1 A : 1 B : 1 C ; and the others, 1 A : 1 B : 2 C, 1 A : 1 B : 3 C,1 A : 1 B : i C, 1 A : 1 B : f C, 1 A : 1 B : J C, &c. <fcc. Planes onangles have an equal ratio of A and B, as is observed in the aboveexamples.

Intermediary planes cut off’ unequal parts of the three edges, A,B, C. Some of the occuring ratios are 4:2:1, 6:3:2, that is, ifthese edges are divided into the same number of equal parts, theplane, whose ratio is 4 : 2: 1, is formed by removing 4 of the partson the edge A, 2 on the edge B, and 1 on C.

36. It has been stated, that on these principles depends the appli-cation of mathematics to this science.

A few remarks on this subject may be of interest to the student,who is acquainted with the principles of trigonometry. This sub-ject is treated of in full, in the Appendix , A.

Let BE and BF, in the following figure, represent the two edgesB and C, in the figure on the preceding page, and AC and AD,the intersections of the planes on the edge A with the face M.