for Maps applying to a large extent of the Earth's Surface. 311 



is here at the distance of ^ of the radius from the surface 

 instead of \ the radius. 



The expression then for r is 



1-66261 sin 



1-36763 + cos 

 When 0=113° 30', this becomes It = 1*5737, which is very near 

 to the size of the Balance of Errors development, viz. 11 = 1*5760. 

 The values of r are as follows : — 



e. 



r. 



e. 



r. 











0-0000 







60 



0-7710 



*) 



00613 



65 



0-8417 



10 



01227 



70 



0-9138 



15 



0-1844 



75 



0-9874 



20 



0-2464 



80 



1-0623 



25 



0-3090 



85 



1-1385 



30 



0-3722 



90 



1-2157 



35 



0-4361 



95 



1-2935 



40 



0-5009 



100 



1-3713 



45 



9-5666 



105 



1-4484 



50 



0-6335 



110 



1-5233 



55 



0-7016 



115 



1-5945 



A. R. C. 



I have had the projection by Balance of Errors and my pro- 

 jection of two-thirds of the surface of the sphere drawn of the 

 exact same size to facilitate the comparison of their relative 

 merits (Plate IV.) ; and I have drawn circles in their centres, that 

 the extent to which figures are distorted in form and exaggerated 

 in area may be seen, by comparing them with the elliptical figures 

 into which circles are projected towards the limits of the map*. 

 My projection of two-thirds of the surface of the sphere is 

 described in the Corps' Papers of the Royal Engineers in 1858, 

 and in the Mittheilungen for the same year. It is a true geome- 

 trical or optical projection, in which the sphere is supposed to be 

 hollow, the plane of projection drawn parallel to and at the 

 distance of 23° 30' from the plane of any great circle, and the 

 point of sight or projection is at the distance of half the radius 

 from the surface of the sphere. In my published maps the plane 

 of projection is drawn parallel to the plane of the ecliptic. 



Maps drawn on this projection have consequently a true 

 perspective effect, and all the circles are represented by true 

 elliptical arcs. 



But in the projection, or, to speak more correctly, in the 



* The diagrams (Plate IV.) have been reduced from larger diagrams, and 

 printed by photo-zincography at the Ordnance Survey Office, Southampton, 



