182 



SCIENCE, 



[N. S. Vol. XI. No. 



(m = arc <!). No projection can be devised 

 which gives all distances correctly ; Postel's 

 gives the correct distance from one point, the 

 center of the map ; but for a limited area, even 

 one as large as the United States which requires 

 zenith distances of 21°, the error in distances 

 is much less than in any of the other projec- 

 tions in common use, not excepting the poly- 

 conic, and it is to be greatly recommended for 

 such maps as railroad and post-route maps. 

 The most important one of these projections 

 however is Lambert's equivalent, which gives 

 distances l)y the chord (»i = 2 sin Ji5), because 

 it gives true areas which for most ordinary pur- 

 poses is the most desirable requisite. It would 

 be most admirably suited for census purposes ; 

 for accuracy of distances it is inferior to the pre- 

 ceding one but superior to the polyconio. [ For 

 20° zenith distance Blundau gives the elements 

 of linear distortion for Postel's as follows : Tan- 

 gential direction a=1.021 ; central direction b= 

 1.000, and for Lambert's a = 1.015, 6=0.985. ] 

 In the gnomonic projection radial distances are 

 given by the tangents of the zenith distances 

 { TO = tang (!) it has the valuable property pos- 

 ses.sed by no other projection, that all great 

 oircles are represented by straight lines. For 

 this reason it is a valuable adjunct to sailing 

 charts, and the Hydrographic Office has pub- 

 lished charts of all the great oceans on tliis pro- 

 jection. This is also a perspective projection 

 with the point of view at the center of the 

 earth. The orthographic projection, in which 

 the sines of the zenith distances are taken as 

 radii (?n = sin (5) may also be regarded as a per- 

 spective one, it interests us only in so far as all 

 lunar charts are constructed on it, in fact, can- 

 not be constructed on any other. The stereo- 

 graphic projection which has the formula to = 

 2 tang J/iJ deserves mention on account of its 

 antiquity, having been already used by Hip- 

 parchus (160-125 B. C), and it is the only 

 azimuthal projection which has no angular 

 distortion or in which every circle is projected 

 as a circle. It may also be treated as a per- 

 spective projection if the point of view is taken 

 on the surface and the earth is assumed to be 

 transparent ; the map will then appear reversed 

 like the type of a print. Amongst the conven- 

 tional azimuthal projections I wish to call 



attention to one proposed by Hammer in Peter- 

 mann's Mitt, of 1892, which consists of a Lam- 

 bert's azimuthal hemisphere converted into a 

 full sphere by a manipulation suggested by 

 Aitow. This projection appears to be well 

 adapted to replace the Mercator projections in 

 atlases of Physical Geography, and has the ad- 

 vantage over Mollweide's, so often used for that 

 purpose, that angular distortions are greatly re- 

 duced, besides being, like the latter, equivalent. 

 Conical Projections. — The transition of azi- 

 muthal projections into conical projections is 

 eiTected by substituting the apex of the devel- 

 oping cone in the place of the center of the 

 map, and by reducing tlie azimuthal angles in a 

 common ratio, i. e., multiplying them by a con- 

 stant factor, which in the ordinary conical pro- 

 jection is the cosine of the polar distance of the 

 tangent parallel. This ordinary conical projec- 

 tion, in which the central parallel and the dis- 

 tances between the parallels only preserve their 

 relative values, while all other parallels and 

 areas are exaggerated, should be used only 

 where facility of construction is the principal, 

 and accuracy a subordinate consideration. Two 

 different methods are in use for compromis- 

 ing the exaggeration of the parallels. In the 

 first one two parallels instead of one are given 

 their true dimensions, one at half the distance 

 between the lowest parallel and the middle 

 one, the other at half the distance between 

 the middle and highest parallel. The radii of 

 the concentric parallels are prescribed by the 

 condition that the latter shall retain their true 

 distance from each other. This method is 

 usually called that by an intersecting cone, which 

 designation is misleading for the reason that the 

 cone thus constructed is an ideal one which can- 

 not be directly applied to the sphere ; Blundau 

 proposes to call it the De U Isle conic projection 

 after the French astronomer who made the first 

 use of it. The second modification of the con- 

 ical projection which is also frequently used in 

 atlases, was devised by Blercator and should 

 be called after him. Here also two parallels 

 are given in their true dimensions, just as in the 

 preceding projection, but the radii of the con- 

 centric parallels are not those of an ideal cone, 

 but those furnished by a cone tangent to the 

 middle parallel ; the meridians no longer cross 



