( 207 ) 
thereof, as 5n the Gommon Meridian Line. Now the mo- 
mentary augment or fluxion of the Tangent Line at 45 de- 
grees, is exadly double to the fluxion of the arch of the Cir- 
cle, ( as may eafily be proved ) and the Tangent of 45, be- 
ing equal to Radius^ the fluxion alfo of the Logarithm Tan- 
gent will be double to that of the arch, if the Logarithm be 
of Napeirs form : But for Briggs form it will be as the fame 
doubled arch multiplied into, o^ 4i^2^, Sec. or divided by 
2, 50258, &c. Yet this muft beunderftood only of the ad- 
dition of an indivifible arch, for it ceafes to be true if the 
arch have any determinate magnitude. 
Hence it appears, that if one minute be fuppofed 
Unity y the length of the arch of one minute being 
,ooo29o8882o8665'72i5'96i5'4 &c. in parts of the Radius^ 
the proportion will be as Unity to 2,908882 &c. fo Radi- 
us to the Tangent of 71" 1' 42" whofe Logarithm is 
10. 463726ii7207i8325'204 &c. and under that angle is 
the Meridian interfered by that Rumb Line, on which the 
differences of Napeirs Logarithm Tangents of the half Comple- 
ments of the Latitudes are the true differences of Longitude, 
eftimated in minutes and parts, taking the firft Four Figures 
for Integers. But for Vlac^f's Tables we muft fay. 
As .2^02^8^ &c. to 2908882 Sec. So Radius to 
1,263 31143874244 5*69212, &c. which is the Tangent of 
51* 38' 9", and itsLogarithm io,ioi5'io4285'o772094ii62 
&c. wherefore in the Rhumb Line, which makes an angle 
of 51* 38' 9" with the Meridian, Vlac^'s Logarithm Tas» 
gents are the true differences of Longitude. And this com- 
pared with our fecond Ci^r^^^^^r^ may fuffice for the ufe of 
the Tables already computed. 
But if a Table of Logarithm Tangents be made by ex- 
traction of the root of the Infiniteth power, whole Index is 
the length of the arch you put for Unity, ( as for minutes 
the ,ocD29o8882th Sec, powerj which we will call a 5 fuch 
a Scale of Tangents, lhali be the true Meridian Line or 
Kim of all the Secants taken infinitely many. Here the 
^^eader is defired to have recourfe to my little Treatife of 
"Logarithms^ publifhed in zi6. p, 58. that I may not need 
to repeat it. By what is there delivered, it will follow, that 
putting t for the excels or defe<5t of any Tangent above or 
under the Radius or Tangent of 4^ j the Logarithm of the 
i i 2 ratio 
