^ 4- Mijcellanea Curiofa. 



Lemma III. On the Globe, the Rumb Lines 

 make equal angles with every Meridian, and 

 by the aforegoing Lemma, they muft like- 

 wife make equal angles with the Meridians in 

 the Stereographick^Prcjeblion on the plain of the 

 Equator : They are therefore,in that Proje&ion, 

 Proportional Spirals about the Pole Point. 



Xew.IV. Ill thcProportional Spiral (Fig.2.)it is 

 a known property,that the angles BPC, or the 

 arches BD, . are Exponents of the rationes of BP 

 to PC: for if the arch BD be divided into in- 

 numerable equal parts, right lines drawn from 

 them to the Center P, mail divide the Curve B 

 ccC, into an infinity of proportionals ; and all 

 the lines Pc mall be an infinity of proportionals 

 between FB and PC, whofe number is equal to 

 all the points d,d, in the arch BD: Whence 

 and by what I have deliver'd in the next enfu- 

 ing Difcourfe it follows, that as BD to Bd, or 

 as the angle BPC to the angle BPc, fo is the 

 Logarithm of the ratio of PB to PC, to the Lo- 

 garithm of the ratio of PB to Pc. 



From thefe Lemmata our Propofition is very 

 clearly demon ftrated % For by the firfly PB, 

 Pc, PC are the Tangents of half the Comple- 

 ments of the Latitudes in the Stereographies 

 Projection-, and by the laft of them, the diffe- 

 rences of Longitude, or angles at the Pole be- 

 tween them, are Logarithms of the rationes of 

 thofe Tangents one to the other. But the Nau- 

 tical Meridian Line, is no other than a Table 

 of the Longitudes, anfwering to each minute 

 of Latitude, on the Rhumb-line, making an 

 angle of 45 degrees with the Meridian Where- 

 fore the Meridian Line is no other than a Scale 

 of Logarithmick Tangents of the half Comple- 

 ments 



