Elements of geometry and trigonometry . BOOK IX. 189 Scholium. This proposition is fun-damentally the same as (Book XX.) ; for, O being the centreof the sphere, a sohd angle may beconceived as formed at O by the planeangles AOB, BOC, COD,&c., and thesum of these angles nmst be less thanfour right angles ; which is exactlythe proposition here proved. Thedemonstration here given is different from that of Book XX. ; both, however, suppose that the polygon ABCDEis convex, or that no side produced will cut the PROPOSITIOX V. THEOREM. The poles of a great circle of a spher
Elements of geometry and trigonometry . BOOK IX. 189 Scholium. This proposition is fun-damentally the same as (Book XX.) ; for, O being the centreof the sphere, a sohd angle may beconceived as formed at O by the planeangles AOB, BOC, COD,&c., and thesum of these angles nmst be less thanfour right angles ; which is exactlythe proposition here proved. Thedemonstration here given is different from that of Book XX. ; both, however, suppose that the polygon ABCDEis convex, or that no side produced will cut the PROPOSITIOX V. THEOREM. The poles of a great circle of a sphere, are the extremities of thatdiameter of the sphere which is perpendicular to the circle ;and these extremities are also the poles of all small circlesparallel to it. Let ED be perpendic-ular to the great circleAMB ; then will E andD be its poles ; as alsothe poles of the parallelsmall circles IIPI, FNG. For, DC being per-[)cndicular to the planeAMB, is perpendicularto all the straight linesCA, CM, CB,à:c. drawnthrough its foot in thisplane ; hence all the arcsDA, DM, DB, &c. arequarters of the circumfe-rence. So likewise areall the arcs EA, EM, EB, &c. ; hence the points D and E arceach equally distant from all iIkî points of the circumferenceAMB ; hence, they are tfie poles of that cir.:umfcrence (Def 7.). Again, the radius DC, perpendicular to the [>Iane AMB, isjicrpendicular to its parali<;l FNC ; hence, it passes through Othe centre of the circl- FN(i (Book VIII. Prop. VII. Cor. 1.) ;hence, if the obli
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
