S. L. Penfield — Stereographic Projection. 5 



deciding upon an appropriate scale on which to make stereo- 

 graphic projections, it was necessary carefully to consider the 

 two following points : 



If the construction is too small, it is difficult to locate given 

 points with sufficient accuracy, by means of a pencil and ordi- 

 nary drawing instruments ; while, on the other hand, if the con- 

 struction is on too large a scale, the drawing becomes unwieldy 

 and presents many difficulties, although the degree of accuracy 

 is greatly increased. After some experimenting it was decided 

 to make the projections on a circle of 14 cm diameter, and after 

 nearly two years' experience, during which time almost all 

 kinds of crystallographic problems have been under considera- 

 tion, it may be stated that this scale has proved very satis- 

 factory. 



The Graduated Circle. — As one of the first aids for the 

 quick and accurate construction of stereographic projections, 

 a circle of 14 cra diameter, graduated into degrees, was engraved. 

 Every tenth degree of the graduation is accentuated, so that it 

 may be quickly caught by the eye, but it was thought best not 

 to number the entire graduation, as degrees can be easily 

 counted off. The exact center of the circle is indicated by a 

 small cross. The circle was engraved by means of a dividing 

 engine, which is equivalent to a guarantee of its accuracy. 

 (See page 23.) The circle and some scales which will be 

 described later are shown, much reduced and with only 5° 

 graduation, in figure 3. They are printed together on large 

 sheets of paper of excellent quality, the idea being that the 

 sheets may be purchased at a trifling expense, and used for 

 plotting all kinds of problems in stereographic projection. 



Protractor No. I for plotting Stereographic Projections. — 

 In order to facilitate the work of locating known points on a 

 diameter of the graduated circle, a special protractor, desig- 

 nated as No. I, has been devised, which is shown without 

 reduction in figure 4. The semicircle, divided into degrees, 

 has a diameter of 14 cm , thus corresponding to the diameter of 

 the graduated circle, figure 3, of the printed sheets. Holding 

 the protractor in a vertical position, and regarding the upper 

 0°-90° point of the semicircle as the north pole, lines drawn 

 from degree points on the semicircle to an imaginary south 

 pole would cross the diameter at correspondingly numbered, 

 stereographically projected, degree points. The distance from 

 the center to each degree line of the graduation was deter- 

 mined by calculation, and the scale was then engraved by 

 means of a dividing engine. The graduation has been num- 

 bered in both directions in order that angles given from either 

 the pole or the equator can be conveniently located. The pro- 

 tractor is printed on cardboard and is inexpensive. It may be 



