THE 



AMERICAN JOURNAL OF SCIENCE 



[FOURTH SERIES.] 



Art. I. — The Stereographic Projection and its Possibili- 

 ties, from a Graphical Standpoint; by S. L. Penfield. 

 (With Plate I.) 



Introduction. — The results which are given in the present 

 paper are the outgrowth of a desire on the part of the writer 

 to simplify some of the processes of plotting and determining 

 crystal forms. The whole subject of stereographic projection, 

 as it has gradually unfolded itself to him during the past two 

 years, has revealed so many possibilities, and seems so important 

 and of such general interest, that it has been decided to pre- 

 sent first a paper treating of the stereographic projection alone, 

 leaving for a later communication its applications to special 

 problems of crystallography. 



As far as the mathematical principles of the projection are 

 concerned, the writer lays claim to no new facts. The projec- 

 tion is treated, in more or less detail (usually very briefly), in 

 most text-books of crystallography, and instructions are given 

 for making stereographic projections. The processes recom- 

 mended, however, are generally tedious, and one of the objects 

 of the present paper is to indicate how projections may be 

 constructed easily and very accurately. Moreover, no mathe- 

 matical formulas nor equations have been used in developing 

 the subject, neither have tables been employed other than one 

 of natural tangents for calculating a certain scale. The prin- 

 ciples of the projection, as set forth in this article, are abso- 

 lutely exact; while the errors involved in solving problems by 

 graphical methods are dependent upon one's ability to locate 

 points and read scales correctly, the errors generally diminish- 

 ing as the size of the projection increases. It is also true of 

 numerical calculations that the processes are limited. Given 

 exact data, results accurate to the minute or to the second are 



Am. Jour. Sci. — Fourth Series, Vol. XI, No. 1. — January, 1901. 



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