S. L. Penfield — Stereographic Projection. 143 



The stereographic projection is admirably adapted for repre- 

 senting all kinds of relations pertaining to a sphere. It is 

 very difficult, almost impossible, to plot and measure accurately 

 on a spherical surface, and a sphere which may be used for 

 such purposes is rarely at hand ; hence, to be able to plot and 

 measure accurately on a flat surface is a great advantage. The 

 writer hopes, therefore, that as a result of this article the 

 stereographic projection will become more widely known and 

 appreciated, and that it will prove more generally useful. 

 Crystallographers have employed the stereographic projection 

 not only for indicating the distribution of crystal faces, but 

 especially for showing zonal (great circle) relations and the 

 spherical triangles which when solved determine the interfacial 

 angles. The stereographic protractors will now give to the 

 projection a new and far more important significance. Crys- 

 tal forms being plotted according to some fixed scale, desired 

 angles may be measured with sufficient accuracy for most pur- 

 poses by the protractors. Protractor No. IV, page 22, will 

 indicate zonal relations ; and the solution of many other prob- 

 lems follow, which will form the basis of a later communica- 

 tion. 



The application of the stereographic projection to astronom- 

 ical problems are very numerous, and doubtless some astrono- 

 mers will find their work simplified by using scales and 

 protractors similar to the ones described in this article. 



It would seem as though no course in spherical trigonometry 

 could be quite complete without reference to the possibilities 

 which the stereographic projection offers for the solution of all 

 the problems presented. 



Before closing this communication the writer takes pleasure 

 in calling attention to a recent article by Professor E. von 

 Fedorow of Petrowsko-Rasumowskoje, near Moscow, on a 

 " (Jniversalgoniometer mil rnehr als zwei Drehaxen und 

 genaue graphisohe RechnnngP* In this communication, Pro- 

 fessor Fedorow describes a modification of the two-circle goni- 

 ometer, by means of which spherical triangles may be solved 

 accurately by purely mechanical methods. This is accom- 

 plished by having two reflecting surfaces which may be set at 

 any angle {kunstlicher Kry stall), and which take the place of 

 a crystal on the instrument. By obtaining appropriate reflec- 

 tions from the surfaces, which necessitates the turning of cer- 

 tain circles, the problems are solved, the angles being read 

 from verniers accompanying the graduations of the circles. 

 Although it is shown that the instrument is capable of giving 

 exact results, the applications of the Fedorow method of solv- 



*Zeitschr. fur Kryst, xxxii, p. 464, 1890. 



