
ISOMETRIC SYSTEM. 23 
e 
from the centre, will meet the other axes at distances expressed by a 
constant ratio, and the expression for the lower right one of the six 
planes will be 8¢: 3): la. By a little study, the expressions for the 
other five adjoining planes can be ovtained, and so also those for all the 
48 planes of the solid. 
In the isometric system the axes a, 5, ¢, are equal, so that in the 
general expressions for the planes these letters may be omitted; the 
expressions for the above mentioned forms thus become —- 
Cube (fig. 1), 7: 1:2. Tetrahexahedron (fig. 5), 7:1: 2. 
Octahedron (fig. 2), 1: Ue : ue Trigonal trisoctahcdron (fig. 5), 
Dodecahedron (fig. 3), 1: Eetleescite 
Trapezohedron (fig, 4), 2 ae 2. Hexoctahedron (fig. 7), 3:1: 3. 
Looking again at fig. 17, representing the cube with planes of the trap- 
ezchedron, 2:1: 2, it will be perceived that there might be a trap- 
ezohedron having the ratios 14:1:14, 8:1:8, 4:1:4, 5:1:5 
and others; and, in fact, such trapezohedrons occur among crystals. 
So also, besides the trigonal trisoctahedron 2:1:1 (fig. 21), there 
might be, and there in fact is, another corresponding to the expression 
%:1:1; and still others are possible. And besides the hexoctahedron 
3:1: 3 (fig. 23), there are others having the ratios 4:1:2, 4:1: 4, 
5 1” ® and so on. ; 
In the above ratios, the number for one of the lateral axes is always 
made a unit, since only a ratio is expressed; omitting this in the ex- 
pression, the above general ratios become: for the cube,72: 7; for the 
octahedron, 1:1; dodecahedron, 1:7; trapezohedron, 2:2; tetra- 
u.exahedron, Ww: Q: trigonal- trisoctahedron, 2:1; and hexoctahedron, 
3) Gaels Wak the lettering of the figures these ratios are put on the planes, 
but itn the second fioure, or that referring to the vertical axis, first. 
Thus the lettering on the hexoctahedron (fig. 23), is 3-3; onthe trigonal 
trisoctahedron (fig. 21) is 2, the figure 1 being unnecessary ; on the 
tetrahexahedron (fig. 31), 7- 2: on the trapezohedron (figs. 4 and 19), 
22; on the dodecahedron (fig. 15), 7; on the octahedron, 1; on the 
cube, 2-2, in place of which // is used, the initial of hexahedron. In the 
printed page these symbols are written with a hyphen in order to avoid 
occasional ambiguity, thus 3-3, 7-2, 2-2, etc. Similarly, the ratios 
for all planes, whatever they are, may be written. The numbers are 
usually small, and never decimal fractions. 
The angle between the planes 7-2 (or?: 1: 2) and O, in fig. 30, page 
21, may be easily ee and the same for any plane of the series 
in (0: : 1). Draw the right-angled triangle, A D C, 
as in ee annexed figure, making the vertical side, 
CD, twice that of AC, the base; that is, give them 
the same ratio as in the axial ratio for the plane. If 
AC=1, CD=2. Then, by trigonometry, making 
ACG the radius, Whee 2 tan DAC ori: Bi: 2: cot 
ADC. Whence tan DAC=cot ADU=2. By ad- 
ding to 90°, the angle of the triangle obtained by work- 
ing the equation, we have the inclination of the basal 
plane O, or the O on the opposite side of the plane 7-2, 
Cc (faces of the cube) on the plane 7-2. So in all cases, 
whatever the value of 72, that value equals the tangent 
of the basal angle of the triangle (or the cotangent of the angle at the 
vertex), and from this the inclination to the cubic faces is directly ob- 

