traced upon the Surface of the Sphere. 343 



, Let e = a ; then the plane of projection is the upper polar tan- 

 gent plane/the same as used by JOEDANUS NEMORARIUS in his Construc- 

 tion of the Planisphere *. 



(a). Let b = 0, or the gnomonic be taken. Its equation is, 



___ f _ 

 v ~ 2ak<> 2 a 



(b). Let b = a, then we have the stereographic of JORDANTJS, and its 

 equation is, 



..(8.) 



4 a 2 



where f is, as it should be, double its former value. 



(c). Let 6 , then we have the Jordanian stereographic of the re- 

 versed branch of the logarithmic spiral, viz. 



v = =p2aAr-" ............ (9.) 



(d). Let b = infinity, then a vanishes in comparison of b, and we have 

 again the equation of the orthographic projection of the loxodrome, viz. 



1 Te- 1 + 



-- 



v 



, as round in (b.) 



3tio, Let e = a, then the projection is on the lower polar tangent 

 plane, and we shall have in a similar manner, when we put 



(a) ... b = 0, the gnomonic, or 



* 





(b) ... b = a, then 



1 "Half- 6 OTV - 







* See the two editions of this tract, Basle, 1536, p. 280, edited by ZIEGLER, and the 

 other edited by COMMANDINE, Ven. 1558, p. 30. I have given some account of these in 

 the Mathematical Repository, vol. vi. pt. ii. p. 42, in treating of the History of the Ste- 

 reographic Projection of the Sphere. 



xx2 



