hy Means of Graphical Methods. 261 



ing the divided circle at 90° from the poles 010 and 100. A 

 line parallel to the face m (at right angles to the normal to m) 

 is next drawn. To do this easil}^ use may be made of the 

 principle of geometry that the angle between two lines meet- 

 ing at the circumference of a circle is measured by half the 

 arc intercepted on the periphery ; hence the line may be drawn 

 from X to 91° 46^ (twice ^Am, iS"" 53^) on the graduation of 

 the divided circle, measured from —x. This line determines a 

 point y on the second line of reference, and establishes the 

 ratio of Ox to Oy, the numerical value of which, however, 

 need not be determined. Designating the distances Ox and Oy 

 as X and y^ respectively, the law of definite mathematical ratio 

 demands that all faces in the zone under consideration must 

 intercept the two lines of reference at definite multiples or 

 fractions of x and y. In figure 11 the angle of h/\fi^ 18° 4^, 

 and by drawing a line from a?. to 36° 8' (double the angle) on 

 the periphery it is found that it intersects at -J- 2/ ; the symbol 

 of the face is therefore 130. Likewise by drawing lines from 

 X to —-J y, — y, and —3 y, _the angles of the forms 130, 110 

 and 310 (measured from 010) are found, the values being most 

 easily obtained by halving the arcs intercepted on the per- 

 iphery, as indicated in the figure. The case just cited, 

 although a simple one, is general, for it is wholly a matter of 

 choice in the orientation of a rhodonite crystal that the forms 

 designated as h,f^ m, «, etc. are taken as parallel to the vertical 

 axis. 



Cases may arise where reliable measurements may be had 

 from three known forms, for example, J, f and m, figure 11, 

 and it is desired to find the angle of a fourth form a, 100, 

 which may either be absent, or be of such poor quality as to 

 yield no satisfactory measurement. To solve such a problem 

 construct the line x^ — x parallel to h, and from x draw lines 

 parallel to f and on. The direction of a is then found by 

 constructing a line through the center which wdll be intersected 

 by the lines previouslj^ drawn in the ratio ^ : 1. AVith the aid 

 of a straight edge and a pair of dividers such a line is easily 

 found. 



Again, it will often happen that the zone under considera- 

 tion is of such a nature that the ratio of the intercepts on the 

 line of reference Oy can not be told by simple inspection, in 

 which case the following procedure may be resorted to. Any 

 four faces taken in succession may be designated as ^, ^, C 

 and D^ and their indices, respectively, as hM ; mno ; jpqr^ 

 and xyz. Taking the first line of reference (corresponding 

 to x^ — x, figure 11) as parallel to the face A^ the ratio of the 

 intercepts on a line parallel to the fourth face D may be found 

 from a system of left to right, right to left, cross-multiplica- 



