ISOMETRIC SYSTEM. 



23 



from the centre, wr'll 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 3c : f£ : la. By a little study, the expressions for the 

 other five adjoining planes can be obtained, and so also those for all the 

 48 planes of the solid. 



In the isometric system the axes a, b, c, 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), i : 1 : i. 

 Octahedron (fig. 2), 1 : 1 : 1. 

 Dodecahedron (fig. 3), 1 : 1 : i. 

 Trapezohedron (fig. 4), 2 : 1 : 2. 



Tetrahexahedron (fig. 5), i : 1 : 2. 

 Trigonal trisoctahedron (fig. 6), 



2:1:1. 

 Hexoctahedron (fig. 7), 3 : 1 : $. 



Looking again at fig. 17, representing the cube with planes of the trap- 

 ezohedron, 2 : 1 : 2, it will be perceived that there might be a trap- 

 ezohedron having the ratios H : 1 : l-£, 3:1:3, 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 

 3:1:1; and still others are possible. And besides the hexoctahedron 

 3 : 1 : f (fig. 23), there are others having the ratios 4:1:2, 4 : 1 : f, 

 5 : 1 : tK 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, % : i\ for the 

 octahedron, 1:1; dodecahedron, 1 : i\ trapezohedron, 2:2; tetra- 

 hexahedron, i : 2 ; trigonal-trisoctahedron, 2:1; and hexoctahedron, 

 3 : f. In the lettering of the figures these ratios are put on the planes, 

 but with the second figure, or that referring to the vertical axis, first. 

 Thus the lettering on the hexoctahedron (fig. 23), is 3-f; on the trigonal 

 trisoctahedron (fig. 21) is 2, the figure 1 being unnecessary ; on the 

 tetrahexahedron (fig. 31), i-2 ; on the trapezohedron (figs. 4 and 19), 

 2-2 ; on the dodecahedron (fig. 15), i ; on the octahedron, 1 ; on the 

 cube, i-i, in place of which H 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-f, a-2, 2-2, etc. Similarly, the ratios 

 for all planes, whatever they are, may be written. The numbers are 

 U3ualiy small, and never decimal fractions. 



The angle between the planes i-2 (or i : 1 : 2) and 0, in fig. 30, page 

 2i , may be easily calculated, and the same for any plane of the seriea 

 i-n (ill: n). Draw the right-angled triangle, ADC, 

 as in the annexed figure, making the vertical side, 

 CD, twice that of AG, the base; that is, give them 

 the same ratio as in the axial ratio for the plane. If 

 AO=l, CD = 2. Then, by trigonometry, making 

 AC the radius, 1 : R::2 : tan DAC\ oil: R::2: cot 

 ADC. Whence tan DAC = cot ADC — 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 0, or the on the opposite side of the plane i-2, 

 (faces of the cube) on the plane i-2. So in all cases, 

 whatever the value of 71, 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« 



