72 3. L. Penfield — Crystal Drawing. 



tions with the A and B axis, e and e\ are noted and points 

 corresponding to this are projected down on the clinographic 

 axes beneath. The line 11, through e and e' on the lower axes, 

 is the linear projection of p. The point i 9 the intersection of 

 II and VV of the orthographic projection above, may now be 

 transferred to the line 11 of the lower axes by projecting down 

 parallel to the vertical axis : the intersection between p and 

 q is parallel to the line from C to i. If the point i on the 

 upper axes has been determined by means of the cotangent 

 scale, without the use of the linear projection, the correspond- 

 ing point i on the lower axes may be found as follows : On 

 both the upper and lower axes draw lines from A to — B, 

 and on the upper axes note the point h where the lines 

 A to — B and Ci cross ; on the lower axes find the corresponding 

 point h on the lineal to — B by projecting down from h above, 

 draw a line from through h and find i by projecting down 

 from i above. 



In following out the methods just described, two conditions 

 may be encountered which give rise to difficulties ; (1) the 

 pole of a certain crystal face may be located within a few 

 degrees of the center of the stereographic projection, in which 

 case the line representing its linear projection would be so far 

 removed from the center that it is difficult to construct it, and 

 (2), two lines of a linear projection may happen to be so nearly 

 parallel that their intersection falls too far from the center of 

 the figure for convenience of drawing. Such difficulties may 

 be overcome easily by making the linear projection either on 

 the plane of the A and Caxes, supposing the faces to pass 

 through unity on B ; or on the plane of the B and axes, 

 supposing that the faces intersect unity on A. To illustrate 

 how a linear projection may be made on the plane of the A 

 and 6 Y axes: — The polej?, figure 55, is on the meridian 55° 20' 

 from B, and a crystal face corresponding to^> would intersect 

 the plane of the equator at right angles to a radius drawn to a 

 point on the equator 55° 20' from B ; such a plane if shifted 

 so as to intersect B at unity would intersect the A axis at the 

 point marked x, cot. 55° 20' (best laid off with the cotangent 

 scale), which is projected down upon the A axis beneath. 

 The great circle stereographic protractor is next centered over 

 the projection, and it is found that the great circle passing 

 through B and p makes an angle of 45° 50' with the equator 

 at B\ hence it follows that all the possible faces in the zone 

 Bp, if made to intersect A at unity, would intersect the verti- 

 cal axis at a distance equal to the cotangent of 45° 50' measured 

 from the center. By means of the cotangent scale the point 

 cot. 45° 50 / is laid off from on the vertical axis and the 

 linear projection of p is the line nn, drawn through x, previ- 



