122 S. L. Pen field — Stereographic Projection. 



means of duplicate calculations or by applying formulas for 

 checking. Professor Edward S. Dana, in preparing his " Sys- 

 tem of Mineralogy," had all the angles in the book recalculated, 

 using as far as possible the fundamental angles of the original 

 investigators. He has told the writer that the number of errors 

 he detected was astonishing. Some of the errors had been 

 made even before the axial relations of the crystals were deter- 

 mined, and whole tables of angles were, as a consequence, 

 vitiated. Other mistakes had been made early in the century, 

 and had been copied into all the leading text-books of mineral- 

 ogy and crystallography. 



By making use of the graphical stereographic methods, tri- 

 angles can be plotted practically in exact proportions, and, by 

 measuring the unknown parts, a check upon the results of 

 numerical calculations can be made. The importance of hav- 

 ing some simple method of checking can not be overestimated. 

 In the matter of making numerical calculations persons differ 

 greatly. Some have great facility and seldom make mistakes ; 

 others do the work with difficulty, especially, it would seem, if, 

 like the writer, they are only occasionally called upon to make 

 calculations. Some check, therefore, carried out graphically, 

 is a great saving of nervous energy. In the writer's short 

 experience in using graphical methods as applied to the stereo- 

 graphic projection, numerous errors in numerical calculations 

 have been detected. At times the error has not amounted to 

 1°, yet it was evident that some mistake had been made. 

 Then, too, in following out the graphical methods, a map or 

 chart is prepared, which in crystallography, as doubtless also in 

 other departments of science, is of importance. 



In determining geographical distances, the stereographic pro- 

 jection possesses very great advantages. In physical geography, 

 for instance, it is important to find the distance between two 

 points on the earth's surface in order to determine the velocity 

 of wind and ocean currents, seismic waves, tidal waves, etc. 

 In navigation, at least in teaching it from an elementary stand- 

 point, the distance between two points of given longitude and 

 latitude must be determined. To indicate how easily this may 

 be done, the problem of finding the distances of New York 

 and Rio de Janeiro from Queenstown will be presented. The 

 approximate longitudes and latitudes of the places, as taken 

 from an atlas, are as follows : 



Queenstown. New York. Rio de Janeiro. 



8° 15' W. U° 0' W. 43° 0' W. 



51° 50' N. 40° 40' N. 23° 0' S. 



In equatorial stereographic projection, figure 22, the meridian 

 of Greenwich is drawn from the center to 0°. Queenstown is 



