VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 75 



instead of 0^-', U', 1>' , 3^', &c. But as it is, it is abundantly sufficient for 

 nautical uses. That in Sir Jonas Moor's New System of the mathematics is 

 much nearer the truth, but the difference from Wright is scarcely sensible, till 

 you exceed those latitudes where navigation ceases to be practicable, the one 

 exceeding the truth by about half a minute, ihe other being a very small matter 

 deficient. 



For an example, easy to be imitated by any person, I have added the true 

 meridional parts to the first and last minutes of the quadrant ; not so much that 

 there is any occasion for such accuracy, as to show that I have obtained, and 

 here laid down, the full doctrine of these spiral rhumbs, which are of so great 

 concern in the art of navigation. 



The first minute is l.O0OO0OOI4]O'265862178 



The second 2.0000000564 IO63806707 



The last, or 89"" 59', is 30374.963431 1414228643 



and not 32348.5279, as Mr. Wright has it, by the addition of the secants of 

 every whole minute : nor 30249-8, as Mr. Oughtred's rule makes it, by adding 

 the secants of every other half minute. Nor 30364.3, as Sir Jonas Moor had 

 concluded it, by I know not what method, though in the rest of his table he 

 follows Oughtred. 



And this may suffice to show how to derive the true meridian line from the 

 sines, tangents, or secants, supposed ready made ; but we are not destitute of a 

 method for deducing the same independently, from the arc itself. If the lati- 

 tude from the equator be estimated by the length of its arc a, radius being 

 unity, and the arc put for an integer be a, as before ; the meridional parts 

 answering to that latitude will be 



i into A + iA= + ^A* + ^pJ or T|4TrA' + 4|i|A'or ^I-H^a", &c. 



which converges much swifter than any of the former series, and besides has 

 the advantage of a increasing in arithmetical progression, which would be of 

 great ease if any should undertake de novo to make the logarithm tangents, or 

 the meridian line, to many more places than now we have them. The logarithm 

 tangent to the arc of 45 + ^a being no other than the aforesaid series a -f 

 \p^ -\- -Va' &c. in Napier's form, or the same multiplied into 0.43429 ^c. for 

 Briggs's. 



But because all these series, towards the latter end of the quadrant, converge 

 exceeding slowly, so as to render this method almost useless, or at least very 

 tedious; it will be convenient to apply some other artifice, as by assuming the 

 secants of some intermediate latitudes ; and you may for s, or the sine of a, 

 the arc of half the difference of latitudes, substitute a — \x + tt-u-^' — 



L 2 



