Elements of geometry and trigonometry . 200 GEOMETRY. PROPOSITION XVII. THEOREM. The surface of a htne is to the surface of the sphere, as the anghof this hcne, is to four right angles, or as tJie arc which mea-sures thai angle, is to the circumference. Let AMBN be a lune ; then will itssurface be to the surface of the sphereas the angle NCM to four right angles,or as the arc NM to the circumferenceof a great circle. Suppose, in the first place, the arcMN to be to the circumference MNPQas some one rational number is to ano-ther, as 5 to 48, for example. The cir-cumference MNPQ being divided in
Elements of geometry and trigonometry . 200 GEOMETRY. PROPOSITION XVII. THEOREM. The surface of a htne is to the surface of the sphere, as the anghof this hcne, is to four right angles, or as tJie arc which mea-sures thai angle, is to the circumference. Let AMBN be a lune ; then will itssurface be to the surface of the sphereas the angle NCM to four right angles,or as the arc NM to the circumferenceof a great circle. Suppose, in the first place, the arcMN to be to the circumference MNPQas some one rational number is to ano-ther, as 5 to 48, for example. The cir-cumference MNPQ being divided into48 equal parts, MN will contain 5 of them ; and if the pole Awere joined with the several points of division, by as manyquadrants, we should in the hemisphere AMNPQ have 48 tri-angles, all equal, because all their parts are equal. Hence thewhole sphere must contain 96 of those partial triangles, the luneAMENA will contain 10 of them ; hence the lune is to thesphere as 10 is to 96, or as 5 to 48, in other words, as the arcMN is to the
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
