Books Books The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle; produce the sides AB, AU, till they meet again in D. Elements of Geometry - Page 136
by Adrien Marie Legendre - 1825 - 224 pages ## Elements of Geometry

Adrien Marie Legendre - Geometry - 1819 - 208 pages
...consequently the distance from A to M is less than the distance from A to N, which is absurd, since the arc AM is greater than AN ; whence no point of the shortest...sum of the three sides of a spherical triangle is lets Hum the circumference of a great circle. Fig. 224. Demonstration. Let ABC (fig. 224) be any spherical... ## A Treatise on Plane and Spherical Trigonometry

Robert Woodhouse - Geometrical optics - 1819 - 264 pages
...angle, as AC p *, is less than the two others AC q, pCq; .•• A p is less than Aq +pq. PROP. III. The sum of the three sides of a spherical triangle is less than the circumference of a great circle (2тг). Let ACB be the spherical triangle ; then, CB < CD + BD J> by the former Proposition, and AC... ## Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1822 - 367 pages
...is itself the shortest distance between its two extremities. PROPOSITION IV. THEOREM. The sum of all the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle ; produce the sides AB, AC till they meet again in D. The arcs ABD,... ## Elements of Geometry

Adrien Marie Legendre - 1825 - 224 pages
...consequently the distance from A to M is less than the distance from A to N, which is absurd, since the arc AM is greater than AN; whence no point of the shortest...is less than the circumference of a great circle. Fig. 524. Demonstration. Let ABC (Jig. 224) be any spherical triangle ; produce the sides AB, AC, till... ## Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 224 pages
...distance from A to ./If is less than the distance from A. to JV", which is absurd, since the arc AM is greater than AN; whence no point of the shortest...is less than the circumference of a great circle. Fig. 224, Demonstration. Let ABC (fig. 224) be any spherical triangle ; produce the sides AB, AC, till... ## An Analytical Treatise on Plane and Spherical Trigonometry, and the Analysis ...

Dionysius Lardner - Plane trigonometry - 1828 - 317 pages
...difference of any two sides of, a spherical triangle is less than the third side. PROP. XLVII. (129.) The sum of the three sides of a spherical triangle is less than the circumference of a great circle. For let any two of the sides a, b, be produced through the third side c until they meet again. The... ## Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1828 - 316 pages
...A to B can lie out of the arc ANB ; hence this arc is itself the shortest distance between its two extremities. THEOREM. 461. The sum of the three sides of a spherical' triangle is leas than the circumference of a great circle. Let ABC be any spherical triangle ; produce the sides... ## Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1830 - 316 pages
...A to B can lie out of the arc ANB ; hence this arc is itself the shortest distance between its two extremities. THEOREM. • 461 . The sum of the three sides of a spherical triangle is less tha n the circumference of a great circle. Let ABC be any spherical triangle ; produce the sides AB,... ## A System of Plane and Spherical Trigonometry: To which is Added a Treatise ...

Richard Wilson - Logarithms - 1831 - 330 pages
...than . y ; 7Г 7Г that is, a + ß is greater than y : similarly, a + y ß, ß + У a• 30. PROP. The sum of the three sides of a spherical triangle is less than the circumference of a great circle of the sphere. By (Eue. XI. 21.) the sum of the plane angles at О is less than four right angles ;... 