358 Grinten — Projection of the Whole Earth' 's Surface. 



(1881), augmented by E. Hammer in " Die Netzentwiirfe 

 geogr. Karten, yon A. Tissot, 1887." 



J. H. Lambert (1772) was the first to recognize the circular 

 form as the most natural one for the representation of the 

 whole globular surface, but failed to notice the fact that his 

 conform al projection would admit of a simple geometrical con- 

 struction. Even the most modern treatises on projections 

 reprint his table containing the numerical values of the dis- 

 tances d and s of the parallels and their centers from the 

 equator. Recently Zoppritz, Reclus and others have urgently 

 recommended the circular form of representation, not only for 

 the whole earth, but for parts of it as of the continents. 

 Although Lambert's projection must be considered as a theo- 

 retical " Unicum " its insufficiency for practical purposes 

 becomes evident, in the stereographic arrangement of the 



meridians, and the subsequent primary subdivision lc = tan — l 



of the central meridian, reducing too much the central parts of 

 the map as compared with those near the margin. 



In order to remedy this defect, I am proposing a projection 

 in which the meridians intersect the equator at equal distances, 

 and then the distances and curvatures of the parallels are 

 altered in such a way that no alteration whatever occurs along 

 the equator. Then we readily obtain our distance d, by sub- 



stiti 



iting c = 



d 



• 



-Vi 



c — tan 



in Lambert's formi 



2 



tla 

 -V90- 







1 + tan ~ 



+ \/ 1- 



-tan 



2 , 

 - thus : 



-tan| 



V90 + (£ - 







<f> 

 1 -f- tan - 



-c __ 1- 





d = 



Vl+c 



-4> 



Vl + c + Vl-c c V9O + -f- a/90— 4> 



which can just as easily be constructed geometrically (see 

 figure 2) as that of Lambert. 



It now remains to determine the distance from the equator 

 (y) of the intersecting point of any parallel and any marginal 

 meridian, in such a manner that the distortion of angle 

 ((h) = 90 — t)) formed by them be a minimum ; inasmuch as the 

 requirement of a minimum 2g> would necessitate a maximal & 

 between the equator and the poles, as will be explained later 

 in this paper. 



y can be determined in different ways. If a rectangular 



network is proposed (® = 0), we find y — g = — ; if the paral- 



