76 PHILOSOPHICAL TRANSACTIONS. [aNNO IGqS-S. 



•5-oVo«^ + ■s-b-tVtto-'^^ &c. according to Mr. Newton's rule for given the sine 

 from the arc ; and if a. be no more than a degree, a very iew steps will suffice 

 for all the accuracy that can be desired. 



And if a be commensurable to a, this is, if it be a certain number of those 



arcs which you make the integer, then will — be that number ; which if v/f 



call n, the parts of the meridional line will be found to be 



&c. 



&c. 



, &c. 



5040;-"' "■^' 



In this, the first two steps are generally sufficient for nautical uses, especially 

 when neither of the latitudes exceed 60 degrees, and the difference of latitudes 

 does not pass 30 degrees. 



But I am sensible I have already said too much for the learned, though too 

 little for the learner ; to such I can recommend no better treatise, than that of 

 ■ Dr. Wallis, in N° 1 76, where he has, with his usual brevity, and that perspi- 

 cuity peculiar to himself, handled this subject from the first principles, which 

 here for the most part we suppose known. 



I need not show how, by regressive work, to find the latitudes from the 

 meridional parts, the methods being sufficiently obvious. I shall only conclude 

 with the proposal of a problem which remains to make this doctrine complete, 

 and that is this. A ship sails from a given latitude, and having run a certain 

 number of leagues, has altered her longitude by a given angle. It is required 

 to find the course she steered. The solution of this would be very acceptable, 

 if not to the public, at least to the author of this tract, being likely to open 

 some further light into the mysteries of geometry. 



To conclude, I shall only add, that, unity being radius, the cosine of the 

 arc A, according to the same rules of Mr. Newton, will be 



1 _ xh- A- ' A^ I a" _l __j ^A** J- a'° See 



from which and tl>e former series, exhibiting the sine by the arc, by division it 

 is easy to conclude, that the natural tangent to the arc a is 



A + J-a' + -iVA^ + VtVa' + t4^A° &C. 



and the natural secant to the same arc 



1 + ^A'^ + -.VA^ + T^Va" + ^VVA* &C. 



and from the arithmetic of infinites, the number of these secants being the 



