74 PHILOSOPHICAL TRANSACTIONS. [aNNO l6g5-6. 



Where a is the length of any arc, which you design shall be the integer or 

 unity in the meridional parts, whether it be a minute, league, or degree, or any 

 other, s the co-sine of the middle latitude, and s the sine of half the difference 



of latitudes; but the secants being the reciprocals of the co-sines, ■- will be 

 equal to — putting o- for the secant of the middle latitude; and -^^ 

 into — will be = -^. This multiplied by — , that is by — , will give the 



s ar ' -^ 3ss ■' 3r/7T ° 



second step : and that again by — — ~, the third step ; and so forward till you 



have completed as many places as you desire. But the squares of the sines be- 

 ing in the same ratio with the versed-sines of the double arcs, we may, in- 

 stead of — , assume for our multiulicator — -, or the versed-sine of the differ- 

 3ss ' 3v 



ence of the latitudes divided by thrice the versed-sine of the sum of the co- 

 latitudes, &c. which is the utmost compendium I can think of for this purpose, 

 and the same series will become 



mto 1 -f \- — ; — \- — — + -— ;- &c. 



Hence we are enabled to estimate the default of the method of making the 

 meridian line by the continual addition of the secants of equidifFerent arcs, 

 which, as the differences of those arcs are smaller, still nearer and nearer ap- 

 proaches the truth. If we assume, as Mr. Wright did, the arc of one minute 

 to be unity, and one minute to be the common difference of a rank of arcs : 

 it will be in all cases, as the arc of one minute, to its chord :: the secant 

 of the middle latitude, to the first step of our series. This by reason of the 

 near equality between a and 2s, which are to each other in the ratio of unity to 

 1 —0.000000003325(36457713 &c. will only differ from the secant o- in the 

 ninth figure; being less than it in that proportion. The next step being -)- 



-^, will be equal to the cube of the secant of the middle latitude multiplied 

 Sari ' ^ t^ 



into — = 0.00000000705 132Q087 15 : which therefore, unless the secant 



3arr 



exceed ten times radius, can never amount to 1 in the fifth place. These 

 two steps suffice to make the meridian line, or logarithm tangent, to far 

 more places than any tables of natural secants yet extant are computed 



to ; but if the third step be required, it will be found to be -|- <ts into ^—^ = 



0.000000000000000089498. By all which it appears, that Mr. Wright's table 

 no where exceeds the true meridian parts by full half a minute ; which small 

 difference arises by his having added continually the secants of 1', 2', 3', &c. 



