VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. . J I 



gitude, or angles at the pole between them, are logarithms of the rationes of 

 those tangents to each other. But the nautical meridian line is no other than 

 a table of the longitudes, answering to each minute of latitude, on the rhumb 

 line making an angle of 45 degrees with the meridian. Therefore the meridian 

 line is no other than a scale of logarithmic tangents of the half complements of 

 the latitudes. Quod erat demonstrandum. 



Corol. 1. — Because that, in every point of any rhumb line, the difference of 

 latitude is to the departure, as the radius to the tangent of the angle that rhumb 

 makes with the meridian : and those equal departures are every where to the 

 differences of longitude, as the radius to the secant of the latitude ; it follows, 

 that the differences of longitude are, on any rhumb, logarithms of the same 

 tangents, but of a different species, being proportioned to one another as are 

 the tangents of tlie angles made with the meridian. 



Corol. 2. — Hence any scale of logarithm tangents, (as those of the common 

 tables made after Briggs's form, or those made to Napier's, or any other form 

 whatever,) is a table of the differences of longitude, to the several latitudes, 

 upon some determinate rhumb or other ; and therefore, as the tangent of the 

 angle of such rhumb to the tangent of any other rhumb : so the difference of 

 the logarithms, of any two tangents, to the difference of longitude, on the 

 proposed rhumb, intercepted between the two latitudes, of whose half com- 

 plements the logarithm tangents were taken. 



And since we have a very complete table of logarithm tangents of Briggs's 

 form, published by Vlacq, Anno l633, in his Canon Magnus Triangulorum 

 Logarithmicus, computed to 10 decimal places of the logarithm, and to every 

 10 seconds of the quadrant which seems to be more than sufficient for the 

 nicest calculator, I thought fit to inquire the oblique angle, with which that 

 rhumb line crosses the meridian, on which the said canon of Vlacq precisely 

 answers to the differences of longitude, putting unity for one minute of it, as 

 in the common meridian line. Now the momentary augment or fluxion of the 

 tangent line, at 4 5 degrees, is exactly double to the fluxion of the arch of the 

 circle, as may easily be proved ; and the tangent of 45 being equal to radius, 

 the fluxion also of the logarithm tangent will be double to that of the arch, if 

 the logarithm be of Napier's form ; but for Briggs's form, it will be as the same 

 doubled arc multiplied into 0.43429 &c. or divided by 2.30258 &c. Yet this 

 must be understood only of the addition of an indivisible arc ; for it ceases to 

 be true if the arc have any determinate magnitude. 



Hence it appears, that if one minute be supposed unity, the length of the 

 arc of one minute being .0002g088820865572l596l54 &c. in parts of the 



