8. L. Pen field — Stereographic Projection. 141 



painted on the frame of the black-board, as shown. D is a 

 scale, drawn on cardboard, which may be used for locating 

 degrees between the ten-degree marks of the graduated circle. 

 E is a ruler or straight-edge of light wood, on one side of 

 which are the scales giving the radii of great circles, G.C. and 

 vertical small circles, 8.C., as shown ; while on the other side 

 are the two scales corresponding to Nos. 3 and 4 of figure 3, 

 page 6. For most operations a smaller ruler, which is not 

 shown, is used. It is provided with the same scales as E. A 

 semicircular protractor having the same radius as the circle is 

 shown at E, with stereographically projected degree gradua- 

 tions upon its base line. A beam-compass is shown at G* 

 having one fixed arm ending in a metal point, and a sliding 

 arm carrying a crayon. The sliding arm has two screw nuts, 

 one for clamping the arm to the beam, the other for clamping 

 the crayon. The beam may be graduated, the scale on one 

 side giving -the radii of vertical small circles, 8.C., as shown, 

 while the scale giving the radii of great circles is on the other 

 side. Another compass, exactly like the one shown, except 

 having a shorter arm, is employed in most operations. A 

 large curved ruler is shown at H, provided with a handle h for 

 holding it in position against the black-board. A stereographic 

 protractor, drawn with India ink on transparent celluloid, is 

 shown at /. The face of the protractor is covered with a thin 

 sheet of celluloid to protect the lines, and the two sheets are 

 fastened by screws to a light semicircular frame of wood. K 

 is a long, light blade of wood which may be clamped at any 

 angle to the cross-piece k by means of a thumb-screw. With 

 this instrument, lines of any desired inclination, taken from 

 the graduation of the circle, may be drawn on any part of the 

 black-board. Although the instrument is not needed for con- 

 structing stereographic projections, it has been included, with 

 the others in order to make the description of the black-board 

 instruments complete. It is used for solving problems in 

 crystallography by means of graphical methods which will be 

 described in a later communication. 



Although the black-board implements are graduated only to 

 every fifth degree, quite accurate work can be done with them. 

 For example, in solving the problems given on pages 115 to 120, 

 the maximum error was 77', the average error 30 7 . In making 

 such measurements as from Queenstown to New York or Rio 



* A beam-compass of the form shown is an excellent instrument for black- 

 board demonstrations, far superior to the black-board compasses ordinarily 

 employed. If made with scales giving inches on one side of the beam and centi- 

 meters on the other, so that circles of any desired radius may be drawn, it would 

 be a very useful instrument, not only in the school- or lecture-room, but in offices 

 of architects and designers. Application has been made for a patent for this 

 instrument. 



