NAVIGATION CHART, &C. 309 



CASE IX. 



Given the Distance, Difference of Latitude, and Difference of 

 Longitude, to find the Course and both Latitudes 



RULE. 



Make the distance on the Index subtend the difference of lati- 

 tude. N. or S. between any parallels of latitude, and the Course 

 will be found; and if the difference of longitude on the Chart 

 exceed that given, move the Index to a less latitude: or if it 

 be too little, to a greater latitude, until the given differences 

 of longitudes and latitudes are subtended by the given dist- 

 ance; when both latitudes will be known. Q. E. I. 



N. B. This case is indeterminate in the same circumstance, 

 as are the cases 7 and 8, which are only given, to shew that 

 every case can be readily solved by this Chart; but the three 

 last cases can seldom be of use in practice. 



To demonstrate the principles on which this Chart is con- 

 structed, and to shew its application to Sailing by the Arc of a 

 Great Circle. 



Let A, B, be two places on the Chart, whose 

 difference of longitude is equally divided by the 

 central meridian E D which is the part of it be- 

 tween the two latitudes; draw the line A B; and 

 D F parallel to it; also the parallels of latitude E F, 

 A D; then EF = EB + AD; but E B and A D 

 liy the construction of the Chart arc = the cosines A D 

 of their respective latitudes x by half tiie difference of longitude 

 between A and B; and therefore E F = half the sum oj the co- 

 sines x by the different e of longitude. 



In the right angled triangle E D F, E F : tangent E D F : : 

 E D: radius. Or half the sum of the cosines of the latitude*: tan- 

 gent of the course :: difference oj latitude: the difference of Imigitude, 

 which is a well known theorem in navigation ; half the sum of 

 the cosines of the livo latitudes being generally* more accurate 

 than the cosine of the middle latitude commonly used. Q. E. D. 



1 The exceptions we of no consequence in practice. Sec Emerson's Navigation, Page 71. 



