Mifcettanta Cmiofa. 5 \ 



PROPOSITION III. 



JnswS °3 feups *tt* tot , 



If through the remtte Poles of Fig. 9. 

 two Circles, two other Circles 

 be drawn upon the Surface of the Sphere, they 

 will cut off equal Arches in thofe great Circles ^ 

 and likewife in the fmall Circles equi-diftanu 

 from thofe Poles through which they pafs. 

 Let p be the Pole pf the Circles md and rw, c 

 the Pole 6f the Circles ab and qo ; pdnob and 

 pmrqa the Reprefentation of the two Circles 

 paffing through the Poles p and c. I fay, ' 



1. Of the great Circles md and ah, the Arches 

 md and ab cut off are equal. 



For the Arch pdb is equal to cbd, and pma e- 

 qual to cam, and the Angle apb equal to the 

 Angle bca ; wherefore the Triangle apb and tned 

 are equal, and confequently ab equal to ntJ. 

 q. e. d. 



aoHc2. Of the teller Circles, qo and rn equally di- 

 ftantfrom 'the Poles c and p. I fay, the Arches 

 qo and rn are equal: For the Archc^ is equal 

 to pmr, and cbo equal to pdn> and the Angle 

 qco equal to rph \ wherefore the Triangles aco 

 and rpn are equal, and confequently the Arches 

 qo and rn are equal; q. e. d. 



Whence it follows, 



1. That if through the Pole of any proje&ed 

 great Circle, and any Part or Segment thereof, 

 Lines be drawn to the Periphery of the Plain 

 of Proje&ion, they will cut off an Arch equal 

 to the Segment of the proje&ed Arch. 



a. That 



