428 Proceedings of the Royal Physical Society. 



are brought into the same position. The first set are normal 

 to the faces of the cube, the second normal to those of the 

 Dodecahedron, and the third normal to the faces of the 

 Octahedron. The first and the third are easily enough 

 drawn; but suppose that we desire to get a standard for 

 the Dodecahedron, the method of procedure is not quite so 

 clear, and is not given in any book I have seen, except in 

 the Eev. Walter Mitchell's "Crystallography," in Orr's 

 Circle of the Sciences (1856), from which, in 1863-64,1 gained 

 my first insight into that science, and which, nearly forty 

 years later, I value still. The methods given here are some- 

 what different from his, and are even simpler. 



In the case of the DoDECAHEDRON,measureoff half the length 

 of each of the Triad Axes, and join the points by lines drawn 

 to the extremities of each of the Tetrad Axes, and the figure 

 is drawn. If it is desired to draw a standard for the Three- 

 faced Octahedron, first draw the Octahedron in the cus- 

 tomary manner, then measure off from anything more than 

 half to two-thirds of the Triad Axes, according to the form 

 desired, and join these points by lines drawn to the extre- 

 mities of the Tetrad Axes. Dana records twenty-one of these 

 Trigonal Trisoctahedrons, as he terms them. As example may 

 be given, 40.40.1 ; 661; 552; 774; 332; 331; 221; 65.65.64. 

 To draw any of the twenty-one forms, measure oft' from o 

 along the Triad Axes a fractional length of each whose 

 numerator is the first figure of the index, and whose de- 

 nominator is the sum of all three. Thus for (221) take two- 

 fifths of the Triad Axes, for (332) take three-eighths, and so 

 on. From the points so determined draw lines to the ex- 

 tremities of the Tetrad Axes, and the figure is completed. 

 In the forms known collectively as the Twenty-four-faced 

 Trapezohedron, sometimes called the Icositetrahedron, 

 Leucitoid, or Tetragonal Trisoctahedron, the second and 

 third indices are equal, and are less than the first. Dana 

 gives twenty-seven of these, ranging from 40.1.1 to 655, 

 211 being the most common. We will consider how they 

 may be drawn. The Tetrad Axes are all at normal length ; 

 the Triad Axes are cut at a fractional length whose numera- 

 tor is the first figure of the index, and whose denominator is 



