( zo6 ) 
From thefe Lemmata our Propofidon is very clearly de- 
monttrated : For by the firft, PB, Pc, PC are the Tangents 
of half the Complements of the Latitudes in the Stereographick 
FrojeBion: And by thelaftof them, the differences of Lon- 
gitude, or angles at the Pole between them, are Logarithms 
of the rationes of thofe Tangents one to the other. But the 
Nautical Meridian Line is no other than a Table of the Lon- 
gitudes,anrwering to each minute of Latirude,on the Rhumb- 
line making an angle of 45* degrees wich the Meridian. 
Wherefore the Meridian Line is no other than a Scale of 
Logarithmick Tangents of the half Complements of the La- 
titudes, J^od erat demonfirandum. 
CoroU. I. Becaufe that in every point of any Rkumh Line, 
the difference of Latitude is to the Departure^ as the Radius 
to the Tangent of the angle that Rhumb makes wi h the Me- 
ridian j and thofe equal Departures are every where to the 
differences of Longitude, as the Radius to the Secant of the 
Latitude f it follows that the differences of Longitude are, 
on any Rhumb, Logarithms of the fame Tangents, but of a 
differing Species ; being proportioned to one another as are 
the Tangents of the angles made with the Meridian. 
CoroU, 2. Hence any Scale of Logarithm Tangents. ( as 
thofe of the Vulgar Tables made after Brigg's form ; or thofe 
made to Napier Sy or any other form whatfbever ) is a Table 
of the differences of Longitude, to the feveral Latitudes, up- 
on fome determinate Rhumb or other : And therefore, as 
the Tangent of the angle of fuch Rhumb ^ to the Tangent of 
any other Rhumb i So the difference of the Logarithms of 
any two Tangefits, to the difference of Longitude, on the 
propofed Rhumb, intercepted between the two Latitudes, of 
whofe half Complements you took the Logarithm Tan- 
gents. 
And fmce we have a very compleat Table of Logarithm^ 
Tangents of Briggs form, publilhed by Vlacq^ Anno i6j;, in 
his Canon Magnus Triangulorum Legarithmicus, computed to ten 
Decimal places of the Logarithm, and to every ten Seconds 
of the Quadrant^ ( which feems to be more than fufficient 
for the niceft Calculator) I thought fit to enquire the Ob- 
lique angle, with which that Rhumb Line croffes the Meri- 
dian, whereon the faid Canon of Vlac(^ f reciiely anfwers to 
the differences of Longitude, putting Unity for one minute 
thereof^. 
