 | Mathematics - 1801 - 658 pages
...sides of a plane triangle are the shortest distances between the angular points on a plane. Therefore the sum of any two sides of a spherical triangle is greater than the third. Q._ ED f DEMONSTRATION. Let the sides AB, AC, containing any angle A, be produced till they meet again... | |
 | John Bonnycastle - Trigonometry - 1806 - 464 pages
...sides are quadrants ; in which case there are an indefinite number of perpendiculars. THEOREM XI. 125. The sum of any two sides of a spherical triangle is greater than the third side ; and the difference of any two sides is less than the third side. A 371 of any two sides AB, A c be... | |
 | Thomas Leybourn - Mathematics - 1817 - 454 pages
...third zq л minimum, which is evidently the case when equal to the difference between qs, zs, because the sum of any two sides of a spherical triangle is greater than the third. Consequently, when the elapsed time is a minimum, gz i* equal to the difference between the altitudes... | |
 | James Mitchell - Mathematics - 1823 - 666 pages
...the less side. 2. Any side or angle of a spherical triangle is less than a semicircle, or 180°. 3. The sum of any two sides of a spherical triangle is greater than the third side; and their difference is less than the third side. 4. The dilference of any two Aides of a spherical... | |
 | James Hayward - Geometry - 1829 - 228 pages
...(220) the sum of the arcs which measure them must be greater than the third arc (113) ; that is— The sum of any two sides of a spherical triangle, is greater than the third. Whence we infer that—The shortest way from one point to another, on the surface of a sphere, is in... | |
 | William Galbraith - Astronomy - 1834 - 454 pages
...angles of another, each to each, their remaining sides and angles will be equal. PROPOSITION XV. 36. The sum of any two sides of a spherical triangle is greater than the third side, and the difference of any two sides is less than the third side. Cor. — The shortest distance between... | |
 | Nathan Scholfield - Conic sections - 1845 - 244 pages
...— cos. «' is always positive. Again, if a', V, c', be the three sides of the polar triangle, since the sum of any two sides of a spherical triangle is greater than the third side : 6' + c' > a' a tan. - ^ / — cos. s' cos. (s' — A) ^ cos. (s'— B)cos. (s'— C) b tan. ¥ =... | |
 | Nathan Scholfield - 1845 - 894 pages
...— cos. *' is always positive. Again, if a', V, c', be the three sides of the polar triangle, since the sum of any two sides of a spherical triangle is greater than the third side : V + c' > a' 138 .-. cos. (s' — A) is always positive, and in like manner, cos. (s1 — B), cos.... | |
 | Nathan Scholfield - Conic sections - 1845 - 542 pages
...— cos. *' is always positive. Again, if a', 6', c', be the three sides of the polar triangle, since the sum of any two sides of a spherical triangle is greater than the third side : I b' + c1 > a1 .-. 180° — B+1800 — C > 180° — A .v B+ C — A< 180° */ — cos. s' cos.... | |
 | Gerardus Beekman Docharty - Geometry - 1867 - 474 pages
...s' is always positive. Again, if af, b', c' be the three sides of the polar triangle, then, because the sum of any two sides of a spherical triangle is greater than the other side, b'+c'>a'; ... 180°— B+180°— C>180°— A; hence B+C— A<180°, and . ^(B+C— A)<90°;... | |
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The sum of any two sides of a spherical triangle is greater than the third side, and their difference is less than the third side. DEM.— Let ABC be any spherical triangle; then l3 BO' < BA + AC, and BC - AC < BA ; and the same is true of the sides in... 
