Grinten — Projection of the Whole Earth? s Surface. 357 



Art. XXXVIII. — New Circular Projection of the Whole 

 Earth 's Surface ; by Alphons J. van dee Gteinten, Chicago. 



The representation of the surface of a sphere upon a plane 

 is a problem which has occupied the attention of cartographers 

 for centuries. The problem has been solved in many ways, 

 but always at some sacrifice of form or relation of parts, de- 

 pending upon the requirements of conformity or equivalence. 



Conformal projections necessarily exaggerate the areas toward 

 the margin (Eisenlohr, August, et al.), while the equivalent 

 ones (Werner, Mollweide, Sanson) reduce the angles formed 

 by the intersection of parallels and meridians considerably. 

 Each of these devices introduces such errors of representation 

 that a comparison of areas and places in the different parts of 

 the globe becomes rather difficult. A third principle, intro- 

 duced in an attempt to distribute these errors over the map, 

 also fails to obtain the most favorable result, an increase of 

 distortion in tropical latitudes hardly being offset by an 

 increase in accuracy in the less important polar regions. 



These conditions, therefore, make desirable a new method of 

 projection, by which all the deformations shall increase regu- 

 larly from a zero value at the equator to the least possible 

 maxima at the poles. The network may be formed exclu- 

 sively by straight lines and circular arcs, as the polar flattening 

 can be neglected as unimportant in a map of so small a scale as 

 is required to represent the whole earth's surface. 



The device of using circular arcs for parallels and meridians 

 results in the production of an apple-shaped marginal meridian, 

 having the central meridian and the equator as straight lines. 

 If o represents the ratio of 

 the lengths of these main 1 



lines of the projection, a 

 mathematical investigation 

 shows that the most favor- 

 able expression of the total 

 d ef ormation , dependent 

 upon the elements h, k and 

 O, as will be shown later, 

 is obtained when the two 

 circles, the marginal me- 

 ridians of the two hemispheres, cover each other, so fusing 

 into one true circle : b = 1. The mathematical deduction as 

 given here, for this case, is based upon Tissot's theories of 

 deformation, as given in his famous " Memoire sur la represen- 

 tation des surfaces et les projections des cartes geographiques " 



