g2' PHILOSOPHICAL TRANSACTIONS. [aNNO 1750. 



many different kinds of logarithms, consequently every species of logarithms has 

 its peculiar rhumb, distinguishable by the angle it makes with the meridian : 

 therefore, among these there are 2 kinds, to which the differences of longitudes 

 are the differences of the logarithmic tangents of the half-complements of lati- 

 tudes, estimated in minutes of a degree; one of them belonging to Napier's 

 form of logarithmic tangents, and the other to Briggs's, or the common loga- 

 rithmic tangents. 



9. The common logarithmic tangents are a table of the differences of longi- 

 tudes to every minute of latitude, on the rhumb line making angles with the 

 meridians of 51° 38' Q". — For, let z represent the meridional difference of lati- 

 tude between 2 places on the rhumb of 45"; or its equal, the difference between 

 the logarithmic tangents of the half-complements of the latitudes of those places, 

 estimated either in parts of the radius, or in minutes of a degree. Then, As 

 the circumference in parts of the radius = 62831,853 &c : To the circum- 

 ference in minutes of a degree = 2160O :: So is a meridional difference of lati- 

 tude in parts of the radius = z : To a meridional difference of latitude in mi- 

 nutes of a degree, = 0,34377468 &c. X z. 



■ Whose corresponding rhumb is different from that which z belonged to; and 

 the angle which this rhumb makes with the meridian, will be found by the fol- 

 lowing analogy from art. 7- — As the meridional difference of latitude on one 

 rhumb = 0,34377^68 &c. z : To the meridional difference of latitude on a 

 rhumb of 45°, = z :: So is the natural tangent of the rhumb of 45°, = 10000 : 

 To the natural tangent of the other rhumb, = 29088,821 &c. 



Which tangent answers to 71° l' 42"; and this is the angle that the rhumb 

 line makes with the meridians, on which the differences of the logarithmic tan- 

 gents of the half-complements of the latitudes, in Napier's form, are the true 

 differences of longitudes estimated in sexagesimal parts of a degree. Now Na- 

 pier's logarithms being to Briggs's, as 2,30258 &c. is to 1 ; therefore, 2,30258 

 &c. : 1 :: 29088,821 &c. : 12633,114 &c.; which is the tangent of 51° 38' Q"; 

 and in this angle are the meridians intersected by that rhumb, on which the dif- 

 ferences of Briggs's logarithmic tangents of the half-complements of the lati- 

 tudes are the true differences of longitudes corresponding to those latitudes. 



10. The difference between Briggs's logarithmic tangents of the half-comple- 

 ments of the latitudes of any two places, is to the meridional difference of lati- 

 tude in minutes between those places, in the constant ratio of 1263,3 Sec. to 1 ; 

 or of 1 to 0,0007915704 &c. — For Briggs's logarithmic tangents are as the dif- 

 ferences of longitudes on the rhumb (a) of 51° 38' Q"; whose natural tangent is 

 1263,3 &c. 



The nautical meridian is a scale of longitudes on the rhumb (b) of 45°, by 

 art. 6, whose tangent being equal to the radius, may be expressed by unity. And 

 the differences of longitude to equal differences of latitudes on different rhumbs, 



