of Minerals in the Thin Section. 319 



therefore, on these different methods of projection, their 

 essential characteristics, and their application to microscopic 

 work, are inserted at this point because of the important role 

 they play in the theoretical consideration and elucidation of 

 the optical methods to be given below. 



The spacial relations between the optic and morphologic 

 properties of a crystal are frequently complicated and extremely 

 difficult of correct conception and description without the aid 

 of special models or projections of the same. Both observation 

 and theory have shown that for any crystal the optical phe- 

 nomena which it presents can be ascertained for a particular 

 color of light, both in direction and length, by reference to an 

 ellipsoid, either triaxial ellipsoid or ellipsoid of rotation, or 

 sphere in the limiting case. Having once determined the 

 exact position and character of this ellipsoid within the crystal 

 and the relative lengths of its axes, it is possible to figure 

 mathematically and to represent graphically the optical proper- 

 ties of the crystal for light waves of the specified length 

 transmitted in any given direction. In the general treatment 

 of the optical properties of crystals, the source of light is 

 considered to be located at a central point within the crystal 

 and the light waves to emerge in all directions from that central 

 point along the radii of an inscribed sphere of unit radius. In 

 space any radius can be represented accurately by its intersec- 

 tion with the surface of the unit sphere, and like any point on 

 the earth's surface, its position can be fixed accurately by two 

 angles equivalent to those of latitude and longitude. 



structed with great accuracy and measurements can be made directly by their 

 use with errors of less than J". Having given the stereographic and ortho- 

 graphic plats of Plate 1 and fig. 1, the process of plotting angles from them 

 graphically and measuring angular distances is rendered easy and certain by 

 the use of tracing paper, as first suggested by Professor Wulff. With the 

 plat as a base, the measured angles, obtained from the microscope, are 

 plotted directly on tracing paper from the projection plat underneath, and 

 the various necessary operations of passing great circles through given points 

 and measuring angular distances, etc., accomplished by revolving the sheet 

 of tracing paper about the center of the projection plat as a pivot, until the 

 required great or small circle of the projection plat beneath is found. The 

 circle is then sketched with sharp pencil point on tracing paper and is suffi- 

 ciently accurate for practical purposes, the slight errors produced by using 

 tracing paper and stereographic plat being of the same order of magnitude as 

 those of observation and therefore negligible. The amount of time and energy, 

 moreover, expended in this method of free-hand sketching, is much less than 

 that necessary for constructing accurately the required small and great 

 circles. In actual practice it has been found expedient to draw with colored 

 ink, on both stereographic and orthographic projection plats, radii and circles 

 2° apart, and thus greatly facilitate the ease with which observed data can be 

 plotted. To incorporate these radii and circles in the original plats did not 

 seem advisable because of the great complexity of lines which arises when 

 radii, circles and great and small projection arcs are all superimposed in 

 black ink. By the use of colored ink, no confusion arises and the radii and 

 circles can be added at any time with little effort by the aid of rule and 

 compass. 



