156 TJ. S. COAST AND GEODETIC SURVEY. 



longed if necessary; then let O'F'M'Q' be the rectangle 

 corresponding to OPMQ', let us call iV the point of inter- 

 section of the lines OM' and PM. We can consider this 

 point as the orthogonal projection of the point that M 

 would be if we should turn the plane of the rectangle 

 OPMQ through a suitable angle with OP as an axis. But 

 this angle, which depends only upon the ratio of the two 

 lines NP and MP, is the same whatever point M may be ; 

 for denoting, respectively, by c and d the ratios of the 

 lengths in the directions OP and OQ — that is, on setting 



OP' .OQ' . 



-^ = cand-g^=cZ, 



we should have 



NP^_0P__1 . MP _0Q _1 

 M'P'~OP'~c' ^^^ MT'~OQ'~d' 



and, consequently, 



MP~c' 



Thus if M moves on an infinitesimal curve traced around 

 0, we shall obtain the locus described by N by turning this 

 curve through a certain angle around OP as an axis and 

 by then projecting orthogonally upon the plane tangent 

 at 0. On the other hanji, we have 



OM' _0P' 

 0N~ 0P~^' 



so that the locus of the points M' is homothetic to that of 

 the points N] the center of similitude is 0, and the ratio of 

 similitude has the value c. The representation of the 

 infinitesimal figure described by the point M is then in 

 reality an orthogonal projection of this figure made on a 

 suitable scale, or the figure formed by the pouits N and 

 that formed by the points M' are formed by parallel sec- 

 tions of the same cone. Any geo^aphic map can, there- 

 fore, be considered as produced by juxtaposition of orthog- 

 onal projections of all the surface elements of the country, 

 provided that we vary from one element to the other both 

 the scale of the reduction and the position of the element 

 with respect to the plane of the map. 



