Geographical Maps and Sailing Charts. 375 



Conclusions. — From a mathematical standpoint there is per- 

 haps nothing in the present communication that was not 

 known hundreds of years ago; hence it is a matter of some 

 interest to have shown that the stereographic is to-day the best 

 projection for geographical purposes, in spite of the fact that 

 geographers have not made much use of it. The principle 

 of making a stereographic map on a plane tangent at any 

 desired point has long been known, but the present writer is 

 not aware that any one has ever before pointed out that maps 

 of limited area, thus made, are probably more accurate, that is, 

 show less distortion, than those made by other methods of pro- 

 jection. Certainly it is a most important feature, and why it 

 has not been made use of before, if such is really the case, is 

 difficult to explain. Possibly the construction of the very flat 

 circular arcs have seemed to present too great difficulties ; and, 

 if so, the discovery of the Wulff curved ruler* has overcome 

 this obstacle. Without such a ruler for constructing some of 

 the flat circular arcs, it would have been difficult to have 

 drawn the maps shown in this article. The possibility of solv- 

 ing problems in spherical trigonometry by means of a stereo- 

 graphic protractor, which seems to have been first discovered 

 by Chauvenet, has never become generally known. This is 

 doubtless because it was necessary to have some such material 

 as celluloid in order to make the protractor a successful working 

 instrument. 



It has been pointed out that in maps based upon the stereo- 

 graphic projection the shapes of countries are admirably 

 shown, and that, when stereographic protractors are used with 

 them, the courses of great circles may be traced and measure- 

 ments made, both of distances and directions. Thus the 

 requirements, mentioned at the beginning of this article as be- 

 ing especially desirable for geographical maps, are satisfied by 

 the use of the stereographic projection, while no other projec- 

 tion combines these valuable properties. 



From an educational standpoint the more general use of 

 the stereographic projection would have many advantages, and 

 to any one who delights in accuracy it should appeal most 

 strongly. 



Note. — In the writer's previous paper on the stereographic projection, the 

 use of certain engraved scales and protractors was suggested, which have 

 proved to be in every way satisfactory. For map work, however, it is nec- 

 essary to vary the size of the projection frequently, consequently somewhat 

 different methods must be resorted to. In place of the engine divided 

 scales, previously suggested, tables have been prepared, giving (1) the 

 degrees of a great circle stereographically projected upon a diameter ; (2) the 

 radii of arcs of great circles ; (3) the radii of arcs of certain small circles 



*This Journal (4), xi, p. 138, 1901; Zeitschrift fur Kryst., xxi, p. 253, 

 1892. 



