TOL. XLVI.] PHILOSOPHICAL TBANSACTIONS. 93 



being to each other as the tangents of the angles those rhumbs make with the 

 meridians. Tlierefore, as the tangent of a (51° 38' Q") = 1,2633 &c : To the 

 tangent of b (45°) = 1,0000 : So is the difference of longitudes on a, or the 

 ditt'erence between the logarithmic tangents of the half co-latitudes of two places : 

 To the difference of longitudes on b, or the meridional difference of latitudes of 

 those places. 



And hence arise the rules which are given in nautical works, for finding the 

 meridional parts by a table of common logarithmic tangents. 



This curious discovery of Dr. Halley's, joined to that excellent thought of his, 

 of delineating the lines, showing the variation of the compass, on the nautical 

 chart, are some of the very few useful additions made to the art of navigation 

 within the last 130 years; for if, beside these, we except the labours of that in- 

 genious artist Mr. Richard Norwood, who improved the art by adding to it the 

 manner of sailing in a current, and "by finding the measure of a degree on a 

 great circle, the theory of navigation will be found nearly in the same state in 

 which it was left by that eminent mathematician Mr, Edward Wright; who, 

 about the year l600, published the principles on which the true nautical art is 

 founded; and showed, what does not appear to have been known before, how to 

 estimate a ship's true place at sea, as well in longitude as in latitude, by the use 

 of a table of meridional parts, first made by himself, and constructed by the 

 constant addition of the secants, and which differs almost insensibly from such a 

 table made on Dr. Halley's principles, contained in the preceding articles. 



Mr. R. concludes this discourse with an article which, though it be somewhat 

 foreign to the preceding subject, yet, as it was discovered while he was contem- 

 plating some part of it, and perhaps is not exhibited in the same view by others, 

 it is annexed in this place; which is, to demonstrate this common logarithmic 

 property, that the fluxion of a number divided by that number, is equal to the 

 fluxion of the Napierian logarithm of that number. 



Let BEG be a logarithmic spiral, cutting its ^^ 



rays at angles of 45°: then, if ae be taken as a » •■ ''/V M 



number, bc will be its Napierian or hyperbolic \ -^ ^f.// \ '■■ 



logarithm. Also, let CD express the fluxion of tVc'"' ''X/^ / \ \ 



the logarithm bc ; then the corresponding \/r' / I ■ 



fiiixion of the number ae, will be represented ^^v\ / I '■ 



by FG, or its equal fe; as the angles feg and Bv""*"^ I "• 



FGE are equal. Now, ac : cd :: ae : (ef =) fg. \ ^ tcT'^D 



Therefore cd = — X ab. And if ab be taken \ | • 



AE \ I ■• 



as the unit or term from whence the numbers \l .". _ 



En. ' I'F 

 begin: then CD = — . Q. E. d. >v": 



*' AE ^V 



