410 The Astronomer Royal on a Projection for Maps 



easily be found for exhibiting, with their assistance, the points 

 of intersection of meridians and parallels which are referred to 

 the pole of the earth. 



3. The two errors, to one or both of which all projections are 

 liable, are, Change of Area, and Distortion, as applying to small 

 portions of the earth's surface. On the one hand, a projection 

 may be invented (to which I shall give the name of " Projection 

 with Unchanged Areas ") in which there is no Change of Area, 

 but excessive Distortion, for parts far from the centre ; on the 

 other hand, the Stereographic Projection has no Distortion, but 

 has great Change of Area for distant parts. Between these lie 

 the projections which have usually been adopted by geographers, 

 with the tacit purpose of greatly reducing the error of one kind 

 by the admission of a small error of the other kind, but without 

 any distinctly-expressed principle (so far as I know) for their 

 guidance in the details of the projection. 



4. My object in this paper is to exhibit a distinct mathema- 

 tical process for determining the magnitudes of these errors, so 

 that the result of their combination shall be most advantageous. 

 This principle I call " The Balance of Errors/'' It is founded 

 upon the following assumptions and inferences : — 



First. The Change of Area being represented by 



projected area 

 original area ' 

 and the Distortion being represented by 

 ratio of projected sides _ projected length x original breadth . 

 ratio of original sides ~" projected breadth x original length 



(where the length of the rectangle is in the direction of the 

 great circle connecting the rectangle's centre with the Centre of 

 Reference, and the breadth is transverse to that great circle), 

 these two errors, when of equal magnitude, may be considered 

 as equal evils. 



Second. As the annoyance produced by a negative value of 

 either of these formulas is as great as that produced by a positive 

 value, we must use some even power of the formulas to represent 

 the real amount of the evil of each. I shall take the squares. 



Third. The total evil in the projection of any small part may 

 properly be represented by the sum of these squares. 



Fourth. The total evil on the entire Map may therefore be 

 properly represented by the summation through the whole Map 

 (respect being had to the magnitude of every small area) of the 

 sum of these squares for every small area. 



Fifth. The process for determining the most advantageous 

 projection will therefore consist in determining the laws ex- 



