a 4 Mtjcellanea Curio fa. 



the TranfaBions. But neither Dr. iVallis^nov 

 Dr. Barrow? in their faid Treatifes, have any 

 where touched upon the aforefaid relation of 

 the Meridian-line to the LogarithmickfT mgent ; 

 nor hath any one, that I know of, yet difcover- 

 ed the Rule for computing independently the 

 interval of the Meridional farts anfwering to 

 any two given Latitudes. 



Wherefore having attained, as I conceive, a 

 very facile and natural demonftration of the 

 faid Analogy, and having found out the Rule 

 for exhibiting the difference of Meredional parts, 

 between any two parallels of Latitude, without 

 finding both the Numbers whereof they are the 

 difference: I hope I may be entituled to a 

 lhare in the Improvements of this ufeful part 

 of Geometry. And firft, let us demonftrate 

 the following Proportion. 



The Meridian Line is a Scale of Logarithmick^ 

 Tangents of the half Complements of the Latitudes. 



For this Demonftration, it is requifite to 

 premife thefe four Lemmata. 



Lemma. I. In the Stereographich^ProJeBion of 

 the Sphere upon the plain of the Equinodial, 

 the diftances from the Center, which in this 

 cafe is the Pole, are laid down by the Tan- 

 gents of half thofe diftances, that is, of half 

 the Complements of the Latitudes. This is 

 evident from Eucl. 3. 20. 



Lem. II. In the Stereographick, Projection, the 

 Angles under which the Circles interfed each 

 other, are in all cafes equal to the Spherical 

 Angles they reprefent : Which is perhaps as 

 valuable a property of this ProjeBion^ as that 

 of all the Circles of the Sphere thereon appear- 



