Tl . PHILOSOPHICAL TRANSACTIONS. [aNNO IGQS-S. 



radius, the proportion will be, as unity to '2.9O8882 &c. so radius to the tan- 

 gent of 71° r 42", whose logarithm is 10.4637'26l 1720718325204 &c. and 

 under that angle is the meridian intersected by that rhumb line, on which the 

 differences of Napier's logarithm tangents of the half complements of the 

 latitudes are the true differences of longitude, estimated in minutes and parts, 

 taking the first four figures for integers. But, for Vlacq's tables, we must 

 say. 



As .2302585 &c. to .29O8882 &c. So radius to 1.26331143874244569212 

 &c. which is the tangent of 51° 38' 9", and its logarithm lO. 1015104285077 

 20941 162 &c. wherefore in the rhumb line, which makes an angle of 51° 

 38' 9" with the meridian, Vlacq's logarithm tangents are the true differences 

 of longitude. And this, compared with our second corollary, may suffice for 

 the use of the tables already computed. 



But if a table of logarithm tangents be made by extraction of the root of 

 the infiniteth power, whose index is the length of the arc you put for unity, 

 as for minutes the .0002908882th &c. power, which we will call a ; such a scale 

 of tangents will be the true meridian line, or sum of all the secants, taken in- 

 finitely many. Here the reader is desired to have recourse to my little Treatise 

 of Logarithms, published in N°2l6, that I may not need to repeat it. By 

 what is there delivered, it will follow, that putting t for the excess or defect 

 of any tangent above or under the radius, or tangent of 45 ; the logarithm 

 of the ratio of radius to such tangent will be 



n into t — \tt -\- \ttt — \utt -[- 4^^*, &c. when the arc is greater than 45°, or 

 i into t ■\- ^it -\- \t^ -\- \t^ -\- il\ &c. when it is less than 45°. And by 

 the same doctrine, putting t for the tangent of any arc, and t for its difference 

 from the tangent of another arc, the logarithm of their ratio will be 



- into — |- 1- r 1 : + 7-; &c. when t is the greater term, or 



m T ' 2tt ' 3t3 ' 4.T* ' 5t' ° 



1 . t it , i3 U , h . , .,11, 



mto \- -— — \- —- &c. when t is the less term : 



m T 2tt 3t3 4t4 5t» 



And if m be supposed .OOO2998882 &c, = a, its reciprocal - will be, 



3437.7467707849392526 &c, which multiplied into the aforesaid series, will 

 give precisely the difference of meridional parts between the two latitudes, to 

 whose half complements the assumed tangents belong. Nor is it material 

 from whether pole you estimate the complements, whether the elevated or 

 depressed ; the tangents being to one another in the same ratio as their comple- 

 ments, but inverted. 



In the same discourse I also showed that the series might be made to con- 



