fO PHILOSOPHICAL TRANSACTIONS. [aNNO ]666. 



pretty nearly with the meridian-line of the sea chart ; both of them growing, 

 as it were, after the same proportion. But the table of meridional degrees, 

 being calculated only to every sexagesimal minute of a degree, shows some small 

 difference from the said logarithmical tangent-line. Hence it may be doubted, 

 whether that difference does not arise from that little error which is committed 

 by calculating the table of meridional degrees only to every minute. 



Mr. Oughtred, in chap. vi. of his Navigation, mentions a method disco- 

 vered by himself, by which it may be proved that the small parts of the meri- 

 dian may not be one minute (which on the face of the earth answers to above an 

 English mile) nor the hundred-thousandth, or, if necessary, the millionth part 

 of a minute, scarce exceeding one fifteenth part of an inch : which thing, he 

 says, he is able to perform in tables, to the radius 10000000; yet nothing at 

 all differing either in their form or manner of working from those that are now 

 commonly in use. 



How this is to be done, this author has not made known to the public. 

 And, though such tables to the radius 10000000 had been brought to light, 

 yet would they not be sufficient to prove the identity or sameness of the said two 

 lines, as to continue the comparison between them as far as the one of them, 

 viz. the logarithmical tangent-line, is already calculated, that is, to ten places, 

 besides the characteristic. 



Now, therefore, if a certain rule could be produced, by which the agreement 

 or disagreement of the said two lines might be shewn, not only to that extent 

 of places to which that tangent line is already calculated, but also to as many 

 more as the same may be yet further extended to in infinitum, surely that rule 

 would not only save us the labour of making tables to the radius 10000000, but 

 also the helix or spiral line of the ship's course would be reduced to a more 

 precise exactness than ever was pretended by him : and this most noble and 

 useful science (as he justly calls it) which is the bond of most distant countries, 



printed in l653, where he teaches, from this property, how to resolve all the cases of Mercator's or 

 Wright's sailing by the logarithmic tangents, independent of the table of meridional parts. This 

 analogy had only been found to be nearly true by trials, but not demonstrated to be a strict mathe- 

 matical property. Such demonstration seems to have been first discovered by Mercator, the author 

 of the above memoir, who, wishing to make the most advantage of this and another concealed in- 

 vention in navigation, in the above paper invites the public to enter into a wager with him, on his 

 ability to prove the truth or falsehood of the supposed analogy. But tliis mercenary proposal seems 

 not to have been taken up by any one, and Mercator reserved his demonstration. The proposal 

 however excited the attention of mathematicians to the subject itself, and a demonstration was not 

 long wanting. The first was published about two years after, by James Gregor}% in his Exercitationes 

 Geometricae, and from thence, and other similar properties, there demonstrated, he shows how the 

 tables of logarithmic tangents and secants may easily be computed from tlie natural tangents and 

 secants. 



