On the Use of the Globe in the Study of Crystallography1

Abstract

IN modern treatises on crystallography, the crystal is imagined projected radially on the surface of a sphere, and the spherical triangles so obtained are dealt with by spherical trigonometry. Problems in astronomy and mathematical geography are also commonly dealt with by the methods of spherical trigonometry. But they can also be dealt with completely by the method of graphical construction on the surface of a sphere where the angles and arcs are directly measured with a divided circle; and the use of spherical trigonometry is dispensed with. Many years ago it occurred to the author that what eliminated the use of spherical trigonometry in the one case might eliminate it in the others: hence the idea of the use of the globe in the study of crystallography. Various arrangements of globe and circles were described and exhibited. The usual method of mounting globes on a polar axis, round which it can revolve inside a metal meridian, supported in its turn at right angles to a horizontal circle or equator, was found to be inconvenient. It is necessary to be able to reach every part of the globe, and to have it steady for drawing, and the fixed circle and axes stand greatly in the way of this. The instrument found most generally useful was a black globe, along with a system of brass circles, divided into degrees, which can be applied directly and exactly to any part of its surface. The system of brass circles is called the mélrosphère, invented by Captain Aved de Magnac, of the French Navy, and published by E. Bertaux, of Paris. With this instrument every problem in the geometry of crystals can be solved with ease and accuracy by graphic construction alone. The various manipulations occurring in the use of the globes were described and illustrated. In the practical determination of a crystal, the inclinations of its faces are observed with the goniometer. From these observations, treated usually by the methods of spherical trigonometry, the elements of the crystal, namely, the inclination of its axes and the proportion of its parameters, are deduced. The process is then reversed, and the elements found are assumed, and from them the inclinations of the faces are calculated. The usefulness of the globe was illustrated by demonstrating how these two processes can be carried out by simple graphical construction. On the globe, the face of a crystal is represented by its pole, or the point where the radius of the sphere, which is perpendicular to the face, pierces the surface of the sphere. The angle between two faces, measured by the goniometer, is the angle contained between their normals. It is therefore ready to be transferred directly to the globe on which it is entered as an arc. In doing so, any point on the globe is taken as the pole of the face from which a start is made. From this a great circle is drawn in any direction. When the first angle has been measured on the goniometer, it is laid off on the globe as an arc, of an equal number of degrees, along this great circle, and from the initial fixed point. The poles of the first pair of faces are situated at the extremities of this arc, which becomes the base line of the survey of the crystal. By triangulation from it, the angles being supplied by the goniometer, the positions of the poles of all the faces are placed as points on the globe.