VOL. XLVI.] PHILOSOPHICAL TRANSACTIONS. Ql 



dians and parallels of latitudes at equal angles; consequently between the inter- 

 section of any meridian and parallel, and a rhumb cutting them at 45°, there 

 must be equal parts of the meridian and jjarallel intercepted; now, on tlie equa- 

 tor, or parallels of latitude, are reckoned all the successive differences of longi- 

 tudes; and on the meridians the successive meridional differences of latitudes, or 

 the divisions of the nautical meridian; therefore on the rhumb of 43°, the suc- 

 cessive differences of longitude are equal to the corresponding divisions of the 

 nautical meridian. 



7. The tangents of the angles which different rhumbs make with the meri- 

 dians, are directly proportional to the differences of longitudes made on those 

 rhumbs, when the meridional differences of latitudes are equal ; or, are recipro- 

 cally proportional to unequal meridional differences of latitudes on those rhumbs, 

 when the differences of longitudes are equal. — For the meridional difference of 

 latitude, is to the difference of longitude, as radius is to the tangent of the 

 angle of the course, or of the angle which the rhumb makes with the meridian. 

 Therefore, when the meridional differences of latitudes are equal, the differences 

 of longitudes are as the tangents of the courses ; but when the differences of 

 longitudes are equal, the meridional differences of latitudes are reciprocally as 

 the tangents of the courses. 



8. The logarithmic tangents of the half-complements of the latitudes, are 

 analogous to the lengthened degrees in the nautical meridian line, in a Mercator's 

 chart. — For, in the stereographic projection of the sphere on the plane of the 

 equator, the latitudes of places are projected by the half-tangents of the com- 

 plements of those latitudes; which half-tangents are the rays of a proportional 

 spiral. Now, if a series of successive latitudes be taken on any rhumb, the 

 corresponding differences of longitudes will be logarithms to the rays of the spiral, 

 or to the tangents of the half-complements of those latitudes; therefore the dif- 

 ferences of longitudes are as the logarithmic tangents of the half-complements of 

 the latitudes; but, art. 6, the lengthened degrees on the nautical meridian are 

 as the differences of longitudes on the rhumb of 45°; consequently the loga- 

 rithmic tangents of the half-complements of latitudes are as the lengthened de- 

 grees on the nautical meridian. 



Corol. 1. When the angle between the rhumb line and the meridian is equal 

 to 45°, then the longitudes of places on that rhumb are expressed by logarithms 

 of Napier's kind; whose corresponding numbers are natural tangents of the 

 half-complements of the latitudes to arcs expressed in parts of the radius. 



Carol. 2. Hence, to any two places on a rhumb of 45°, the difference of 

 longitude, or the meridional difference of latitude, is equal to the difference of 

 the Napierian logarithmic tangents of the half-com]jlements of the latitudes of 

 those places, estimated in parts of the radius. 



Corol. 3. As there may be an indefinite variety of rhumbs, and therefore avS 



N 2 



