S. L. Penfield — Stereographic Projection. 121 



ing the angle A, is first plotted. Then, taking protractor No. 

 IV, page 22, the great circle corresponding 'most nearly to B 

 is selected, and the protractor, centered on the drawing, is 

 turned until the great circles A p of the drawing and B p' of 

 the protractor cross at an angle approximating to C, which is 

 told by means of an ordinary protractor, page 19. Thus an 

 approximate solution may be quickly obtained. 



Some Practical Applications of the Stereographic Projec- 

 tion. — Some mathematicians may contend, and justly, that the 

 seven problems previously presented might be solved easier by 

 numerical calculations than by plotting; and that by using 

 four- or five-place logarithm tables the results would be correct 

 to minutes, while those obtained by plotting are only approxi- 

 mately correct. In spite of the truth of this, graphical methods 

 still have their advantages. Most persons who have occasion 

 to make calculations, the writer included, use formulas and 

 tables, as a rule, wholly in a mechanical way. With graphi- 

 cal methods, on the other hand, formulas and tables are not 

 needed, and every operation is clearly understood, for graphi- 

 cal methods can scarcely be applied otherwise. In the majority 

 of cases, numerical calculations are laborious, while graphical 

 solutions appeal to one like pictures which, to a certain extent, 

 tell their own story. Although it may be known that for some 

 problems there are two solutions, it is safe to assume that of those 

 persons who are accustomed to solve spherical triangles by the 

 use of formulas, only a few could satisfactorily explain why 

 two solutions apply to the conditions shown in figure 21 where 

 a is less than &, while only one solution is possible when a is 

 greater than 5, figure 20. The two figures, however, being 

 accurately constructed, not only show the point in question 

 clearly, but present the problems in such a manner that desired 

 parts can be measured. 



Most persons have never studied spherical trigonometry, 

 and those who have studied it usually regard the solution of 

 a spherical triangle as at least a laborious, if not a difficult, 

 matter. This is especially true of those who for a long time 

 have not made numerical calculations, and who have thus lost 

 their familiarity with the use of formulas and of logarithmic 

 tables. Perhaps no one can appreciate the difficulties presented 

 by the subject more than one who has had occasion to teach 

 crystallography to advanced students. It is generally the case 

 that students desiring to take up crystallography (chemists 

 especially) have not made numerical calculations, other than 

 very simple arithmetical ones, for a number of years. As a con- 

 sequence, the problems in spherical trigonometry are regarded 

 as a great trial. Mistakes of all kinds arise, and the only way 

 to make absolutely sure of one's work is to check, either by 



