VOL. XIX.] PHILOSOPHICAL TRANSACTIOKS. 73 



verge twice as swift, all the even powers being omitted : and that putting t for 

 the sum of the two tangents the same logarithm would be 



— or — - into - + h \- \- — &c. 



but the ratio of t to t, or of the sum of two tangents to their difference, is the 

 same as that of the sine of the sum of the arcs, to the sine of their differ- 

 ence. Therefore, if s be put for the sine complement of the middle latitude, 

 and s for the sine of half the difference of latitudes, the same series will be 



2r . * «3 , «5 ,7 j9 



IT '"^° T + lil + 5jr + 7i7- + 9^ &c- 

 where, as the differences of latitude are smaller, fewer steps will suffice. And if 

 the equator be put for the middle latitude, and consequently s = r, and s = 

 the sine of the latitude, the meridional parts reckoned from the equator will be 



S 53 iS s7 



a ' '6rra ' or^a ' 7r''a 

 which coincides with Dr. Wallis's solution in N° 176. And this same series, 

 being half the logarithm of the ratio of r + ^ to r — s, that is, of the versed 

 sines of the distances from both poles, agrees with what Dr Barrow had shown 

 in his 1 1th lecture. 



The same ratio, of t to t, may be also expressed by that of the sum of the 

 CO- sines of the two latitudes, to the sine of their difference : as also by that of 

 the sine of the sum of the two latitudes, to the difference of their co-sines : or 

 by that of the versed-sine of the sum of the co-latitudes, to the difference of 

 the sines of the latitudes ; or as the same difference of the sines of the latitudes, 

 to the versed-sine of the difference of the latitudes ; all which are in the same 

 ratio of the co-sine of the middle latitude, to the sine of half the difference of 

 the latitudes. As it would be easy to demonstrate, if the reader were not sup- 

 posed capable to do it himself, on a bare inspection of a scheme duly represent- 

 ing these lines. 



This variety of expression of the same ratio, I thought not fit to be omitted, 

 because by help of the rationality of the sine of 30°, in all cases where the 

 sum or difference of the latitudes is 30°, 60", C)0°, 120° or 150°, some one of 

 them will exhibit a simple series, in which great part of the labour will be saved : 

 and besides, I am willing to give the reader his choice, which of these equi- 

 pollent methods to make use of; but for his exercise shall leave the prosecution 

 of them, and the compendia thence arising, to his own industry. Content- 

 ing myself to consider only the former, which for all uses seems the most 

 convenient, whether we design to make the whole meridian line, or any 

 part of it, viz. 



— mto V- y ^-r A &c. 



VOL. IV, L 



