42 N. W, Paint of the Lake of the Woods. 



the two points by a straight line on the map, (projected ac- 

 cording to Mercator.) If the angle formed by this Hne, and 

 one of the parallels of the map, be more than an angle of 

 45°, the northern point is the more northwest; if less, the 

 southern is the more northwest ; if 45° exactly, the two 

 points are equally northwest. It will be easily seen, that if 

 through the southern one of the two points, which are thus 

 to be compared, a parallel of latitude be drawn, and through 

 the northern one a meridian, and those lines be continued 

 until their intersection, a right angled triangle will be form- 

 ed, the hypothenuse of which is the straight hne, connecting 

 the two points in question, or the course from the one to the 

 other, and the two cathetes the difference of longitude and 

 the difference of latitude of the two points on the map. The 

 angle formed by the connecting line of the two points with 

 the parallel is exactly 45°, if the difference of longitude on 



the map is equal to the difference of latitude ; it is < . > 



than 45°, if the difference of latitude be <, '^ > than the 



' I less 5 



difference of longitude, both being measured on a map pro- 

 jected on Mercator's principles. If therefore the differences 

 of latitude and longitude be expressed in figures, in the same 

 manner, for example in minutes of a degree, on the principle 

 of Mercator's projection, these numbers will immediately 

 show which is the more northwestern one of the two points. 

 The degrees of longitude are all equal on Mercator's projec- 

 tion, and may be easily expressed in any manner required. 

 The degrees of latitude increase from the Equator to the 

 Pole : the expression of the length of a meridian from the 

 Equator to the latitude L., in minutes of a degree, is 

 = 7915' 7044674 log. tangt. (45° + i L). The numbers 

 resulting from this formula for the different degrees and min- 

 utes of latitude are contained in the common books of navi- 

 gation, under the head of " meridional parts." From all 

 this, it will appear, that in order to ascertain which is the 

 more northwestern one of two points, the longitude and lati- 

 tude of which are given, it will be necessary to find by the 

 above formula the numbers corresponding to the two lati- 

 tudes, and to take the difference of the same ; and likewise 

 to take the diflerence of longitude of the two points, and ex- 

 press it in minutes of a degree, (which numbers thus express 



