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LI. On Maps of the World. By George Darwin, M.A., Fel- 

 low of Trinity College, Cambridge*. 

 [With a Plate.] 



THE ordinary stereographic projection of the world in two 

 hemispheres is utterly worthless as giving a true impres- 

 sion of the whole ; for the linear scale at the margins of the 

 circles is twice that at their centres. Its only merit is that 

 there is no angular distortion. Mercator's projection gives a 

 still more fallacious impression, except as regards the equatorial 

 regions. 



It appears to me therefore that there is a want, in the school- 

 room and lecture-room, of some map which shall give a more 

 truthful representation of the globe than the above, and which 

 yet shall not be so expensive and cumbrous as a globe. 



A gnomonic projection on to the faces of a regular icosahedron 

 is but very slightly distorted, although a slight amount of 

 angular distortion is here introduced. I have been told that at 

 the recent Geographical Congress at Paris, some such projec- 

 tions as this were exhibited, and that they were of old date. 

 Mr. Proctor has also made star-maps by projection on to the 

 faces of a regular dodecahedron ; but in 1872, when the idea 

 occurred to me of using this projection, I w r as not aware of the 

 fact. 



If the icosahedral projection be developed and arranged as a 

 band of ten triangles round the equator, with saw-like edges of 

 five triangles in the north and five in the south, a very fair 

 representation of the globe is given. And the interstices 

 between the teeth of the saws may be arranged so as not to 

 damage the continents very severely. 



In this map the meridians are straight lines, but are broken 

 in direction at the junction of two triangles. The parallels of 

 latitude become ellipses, which may be easily laid out by aid of 

 a property of conic sections ; viz. if a circular cone be placed 

 with its vertex at the centre of a sphere, and a section made by 

 a tangent plane to the sphere, the radius of curvature at the 

 vertices of this conic section is constant for all tangent planes, 

 and varies as the tangent of the semiangle of the cone. 



Now in our map the ellipses are represented with sufficient 

 accuracy by the circles of curvature at their vertices ; and the 

 radii of these circles may be taken direct with the compasses 

 from a sector, as the cotangents of the corresponding latitudes. 



Besides a map of this kind, I have also constructed a portable 

 quasi-globc with this method of projection. The faces of the 



* Communicated by the Author. 



