 The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180° and less than 540°. (gr). If A'B'C' is the polar triangle of ABC...  Elements of Geometry - Page 379by Andrew Wheeler Phillips, Irving Fisher - 1896 - 540 pagesFull view - About this book
 | American Mathematical Society - Mathematics - 1905 - 1032 pages
...the triangle can be read off as L. e., p. 595. pure spherics. The proof of the theorem (§ 567) — the sum of the angles of a spherical triangle is greater than two and less than six right angles — assumes that a spherical triangle is always positive. The theorem can be proved in the usual way... | |
 | Ephraim Miller - Plane trigonometry - 1894 - 222 pages
...90°. REMARK II. The functions of £a, $b, and $c, in [57] and [59], are real quantities. For since the sum of the angles of a spherical triangle is greater than 180°, and less than six right angles, then S, or $(A + В + C)- in [57] and [59], is greater than... | |
 | John Macnie - Geometry - 1895 - 386 pages
...the side of A'B'C' that is opposite ZA, etc., we have the relations : PROPOSITION XV. THEOREM. 633. The sum, of the angles of a spherical triangle* is...than two, and less than six, right angles. Given: A, B, C, the angles of a spherical triangle ABC; To Prove : Z A+^ B+Z C> 180° and < 540°. Construct... | |
 | John Macnie - Geometry - 1895 - 390 pages
...the side of A'B'C' that is opposite ZA, etc., we have the relations : PROPOSITION XV. THEOREM. 633. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles. B.. Given: A, B, C, the angles of a spherical triangle ABC; To Prove : ZA+ZB+ZC> 180° and < 540°.... | |
 | Wooster Woodruff Beman, David Eugene Smith - Geometry - 1895 - 344 pages
...Geometry. EXERCISES. 736. Show that a trirectangular triangle is its own polar. 737. From step 7 show that the sum of the angles of a spherical triangle is greater than a straight angle*. 738. A spherical triangle is to the surface of the sphere as the spherical excess... | |
 | George Cunningham Edwards - Geometry - 1895 - 330 pages
...the area of any spherical polygon, the angles of which are given. NOTE. — It has been shown that the sum of the angles of a spherical triangle is greater than 180°: The amount, in degrees, by which the sum of the angles exceeds 180°, is called the spherical... | |
 | Wooster Woodruff Beman, David Eugene Smith - Geometry - 1895 - 346 pages
...Geometry. EXERCISES. 736. Show that a trirectangular triangle is its own polar. 737. From step 7 show that the sum of the angles of a spherical triangle is greater than a straight angle. 738. A spherical triangle is to the surface of the sphere as the spherical excess... | |
 | Andrew Wheeler Phillips, Irving Fisher - Geometry - 1896 - 554 pages
...AB and AC. In a similar manner the remaining relations are proved. QED PROPOSITION XXX. THEOREM 878. The sum of the angles of a spherical triangle is greater...opposite in the polar triangle by a', b', c'. To PROVE We have C=i8o°-c'. Adding these equations we get Hence Also, A + B+C<$4O°. since a' + b' + c'< 360°,... | |
 | Andrew Wheeler Phillips, Irving Fisher - Geometry - 1896 - 570 pages
...AB and AC. In a similar manner the remaining relations are proved. QED PROPOSITION XXX. THEOREM 878. The sum of the angles of a spherical triangle is greater...the spherical triangle ABC. Denote its angles by A, /,, C, and the sides opposite in the polar triangle by a', b', c'. To PROVE A + B + C> 1 80° and <... | |
 | Henry Dallas Thompson - Geometry, Solid - 1896 - 226 pages
...less than six right angles. (This is a restatement of § 241). 243. DEFINITION. The angle by which the sum of the angles of a spherical triangle is greater than two right angles is called the angle of the spherical excess or the spherical excess of the triangle. The... | |
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