Simpler Methods in Crystallography. 415 



pyramid cuts both a and b at full leugth, and also c. 

 Lines of indefinite length passing through a and h are 

 accordingly drawn. But there is usually a set of pyramids 

 in the unit zone which cut the vertical axis at a fraction of 

 its normal length, e.g., (112), (223), (334), etc., which cut 

 the vertical axis at a half, a third, a fourth, etc., of its unit 

 length. Now, by the principles of this projection, all the 

 faces must pass through the vertical axis at unit length. 

 We therefore, in such cases, simply multiply all the three 

 intercepts by the denominator of last of the three, con- 

 verting the first into f f f , the second into f f f, the third 

 ^ 1^ 1^. That is to say, in the first instance we take both a 

 and h at twice the unit length, and the vertical axis at twice 

 half, or, in other words, the whole. To draw the trace of 

 this, we therefore measure twice the length of a, and twice 

 that of h, and draw lines connecting both points. In like 

 manner, we take one and a half times the length of a and of 

 h, and three-thirds, or the whole, of c, and proceed as before. 

 In the third case we take four-thirds of each, and so on. The 

 same principle is followed in drawing the macrodomes ; thus 

 (102), (305), (403), (503) are represented respectively by 

 lines cutting twice the length of a and parallel to b ; five- 

 thirds the length of a, three-fourths the length of a, or three- 

 fifths, as in the last case. In the case of the brachydomes 

 (012), (023), (021), (032), all those figures are multiplied by 

 the last, and we proceed as usual. The second and fourth 

 examples are taken respectively at one and a half times the 

 length of b, parallel to a, and two-thirds the length of b, 

 and parallel to a, and so on. 



We might take one or two other cases by way of further 

 elucidation. Suppose it is desired to lay down the trace of 

 (375). In this we have one-third of a, one-seventh of b, and 

 one-fifth of c. c must be a whole number. We therefore 

 multiply all three by five, and get 5/3 of a, 5/7 of b, and one 

 of c. These proportions are, accordingly, taken either by the 

 proportional compasses, or, better still, by means of the pro- 

 portional scale, whose construction was described in Part I. 

 p. 343. 



If the drawing is correctly done, it will soon be evident 



