17G HERSCHEL'S NEW PROJECTION OF THE SPHERE. [April 11, 1859. 



about a fixed centre on the plane." The author then shows, that 

 on such a supposition, " if j9 be the polar distance of any parallel of 

 latitude, and 6 the radius of the circular segment representing that 

 parallel, we have (taking 1 for the equatorial radius in the pro- 

 jection) d = ( tan ^ j , from which it is easy to calculate 6 for each 



polar distance from 0° to 180^. This expression, it will be observed, 

 still contains one arbitrary quantity, n. By giving specific values 

 to w, we have various projections including and analogous to the 

 stereographic. The author then calculates a table of values for the 

 radii in the projection for the polar distances 0^, 10^, 20^, &c. — 160^ ; 

 for the values of n, 1, f , -|, ^. 



" The first series of numbei:s," he concludes, " exhibits the pro- 

 gression of the radii of the successive projected parallels in the 

 stereographic projection. The second, in that which occupies a 

 section of 240°, such as by cutting out the unoccupied portion, 

 would roll into a cone, well adapted for a transparent map on a 

 lamp-shade. The third is that which occupies a semicircle — a con- 

 venient form for a reference-chart, but which becomes too much 

 dilated beyond the 55th parallel of south latitude ; and the last, 

 that comprised in a sector of 120°, which is preferable to either, 

 and seems to me not unlikely to supersede all other projections for 

 a general chart." 



Numerical Values of 6 in the above Equation, when »i = ^. 



* The general question has been also discussed by Gauss in an answer to the prize 

 question proposed by the Royal Society of Copenhagen in 1822. His method is, in some 

 respects, still more general than that of the present paper ; but, although he arrives at, 

 amongst others, an equivalent to the formula given above, or, rather, to that from which 



it was obtained, viz., r = 2.^2 . 1 tan — ^ j, he does not particularly specify the " lamp- 

 shade" projection, nor those represented by the third and the fourth cases. 



In two memoirs, published in the Comm. Gott., vols. ii. and iii., the same author 

 examines in considerable detail the projection of a spheroid on a sphere. — Note communi- 

 cated by F.G. ., 



