STEREOGRAPHIC, ORTHOGRAPHIC, GNOMONIC, AND ANGLE PROJECTIONS. 63 
ing lens surfaces of the objective on the plane of polarization of the trans- 
mitted light-waves can be eliminated by using the Bertrand lens, which 
together with the ocular forms a low-power microscope for observing the 
crystal plate. With this arrangement there are no lens surfaces between 
the two nicols and the disturbing rotatory effects of the oblique glass sur- 
faces on the plane of polarization of transmitted light-waves are not present. 
On cloudy days, when the sun is not available, the adjustment of the 
nicols can be made with a fair degree of accuracy by means of the Bertrand 
ocular or the bi-quartz wedge plate*, while for the adjustment of the cross- 
hairs a twinned plate of selenitef or an artificially twinned quartz plate or 
wedgej renders good service. 
THE STEREOGRAPHIC, ORTHOGRAPHIC, GNOMONIC, AND 
ANGLE PROJECTIONS. 
In the study of crystal optics, as well as in the description and application 
of petrographic microscopic methods, the phenomena considered often 
involve special relations and require the concepts of solid geometry for 
their solution. In order to represent these adequately on a plane, different 
projections have been devised and aid the observer materially in forming 
correct conceptions of the morphological and optical relations in crystals, 
which are frequently complicated and difficult to describe accurately. 
Actual models might be used to represent these special phenomena, but 
usually it is neither convenient nor feasible to make such models and the 
observer is forced to use some form of projection. In all types of projec- 
tion the relation of the object to its projection is one of definite construction 
and is dependent on the method of projection adopted. In each projection 
the directions (optical or crystallographic) within the crystal are considered 
to pass through the center of a sphere of unit radius. In space any radius 
can be represented accurately by its point of intersection with the surface 
of the unit sphere and, like any point on the earth's surface, its position 
can be definitely fixed by two angles equivalent to those of latitude and 
longitude. In the projection, such points on the unit sphere are pictured 
in a fixed plane and represent definite directions within the crystal. 
The kinds of projection required in optical work are different from those 
which serve in ordinary map projections where the effort is made to indicate 
on a plane surface the relative positions of points and lines on the earth's 
surface. Such representation can only be approximately correct, as it is 
geometrically impossible to develop a spherical surface on a plane; and 
different projections have been devised to meet the different requirements 
which may arise. Map projections which represent the different areas on 
the globe in correct relations are called equal Surface or equivalent projections, 
while those which preserve the angular relations are called orthomorphic 
projections. 
Compare F. E. Wright. Amer. Jour. Sci. (4). 2*. 386, 1008. 
tE. Soramerfeldt. Zeitschr. fiir wisscnschaftliche Mikroskopic. 24. 24-35. 1907. 
JF. E. Wright. Amer. Jour. Sci. (4). 26, 387. 1908. 
JFor comparative studies on projection see A. A. C. Germain. Traiti 4 des Projections. Paris. 1865; H. F. 
Gretschel. Lehrbuch dcr Kartcnprojektion, Weimar. 1873; C. A. Schott. Appendix 15. Ann. Report 1880. 
U. S. Coast and Geodetic Survey; Thomas Craig. A Treatise on Projections. U. S. Coast and Geodetic 
Survey. 1882; V. Goldschmidt. Ueber Projektion und graphische Kristallberechnung. Berlin. 1887; Ueber 
Winkelprojektion. Z. Kryst. 36, 388-402. 1902. 
