418 The Astronomer Royal on a Projection for Maps 



The distortion in Sir H. James's is expressed by 2-9; and this, 

 in my opinion, is a more serious inconvenience than the exagge- 

 ration in the Balance of Errors, namely 4-8. On the whole, I 

 think that, for this value of 0, the Balance of Errors is preferable 

 to Sir H. James's. The nearest in merit, I think, is that of Equal 

 Radial Degrees; but the distortion of Balance of Errors 1*4 is 

 preferable to that of Equal Radial Degrees 2*2, while the exagge- 

 ration 4*8 is not much more injurious than 2-2. I prefer the 

 Balance of Errors. 



17. For 0= 135°, Sir H. James's is inapplicable. The remain- 

 ing systems stand in these orders : — 



Distortion. 



1. Balance of Errors . 1*4 



2. Equal Radial Degrees 3*3 



3. Unchanged Areas . 6*8 



Exaggeration. 



1. Unchanged Areas . . 1*0 



2. Equal Radial Degrees . 3-3 



3. Balance of Errors . . 14*9 



In my opinion, the Balance of Errors is here the best. A square 

 whose sides are 1, 1, is projected into a parallelogram whose sides 

 are 3*3, 4'6 \ and this is better than the parallelogram in the 

 Equal Radial Degrees whose sides are 1, 3*3. This is on the sup- 

 position that we desire to preserve an intelligible representation 

 of every part of the earth depicted in the map. I will shortly 

 state under what circumstances this opinion may perhaps be 

 modified. 



18. If we take for Centre of Reference the point defined by 

 longitude 3 h 30 m east of Greenwich, latitude 23° north, the circle 

 whose radius is 135° will contain every continent and large island, 

 including Australia and New Zealand, omitting only the South 

 Pacific Ocean. If we take for Centre of Reference the point de- 

 fined by longitude l h east of Greenwich, latitude 10° south, the 

 circle whose radius is 135° will contain every continent and 

 large island as before, omitting only the North Pacific Ocean. 

 In such maps, the countries which are found on the borders of 

 the map are sufficiently extensive and important to require to be 

 exhibited without much distortion ; and all constructions are 

 equally troublesome. For these maps, therefore, I should use 

 the system of Balance of Errors. 



19. If, however, we take for Centre of Reference the point de- 

 fined by longitude 16 h 20 m eastof Greenwich, latitude 77° north, 

 the circle whose radius is 135° will contain the same continents 

 and large islands as before, including also the small islands of 

 the Pacific Ocean, omitting only the Antarctic Seas. Such a 

 map is of extraordinary value, because it not only contains all 

 the known lands, but may also exhibit all the sea-courses between 

 the southern capes. But where, as in this case, the boundary is 

 touched by little more than the headlands, distortion is less im- 



