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" ABC (fig. 224) be any spherical triangle ; produce the sides AB, AC, till they meet again in D. The arcs ABD, ACD, will be... "
Elements of Geometry - Page 160
by Adrien Marie Legendre - 1841 - 235 pages
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1819 - 208 pages
...hypothesis, the shortest ; 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...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...
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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...
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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,...
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 224 pages
...N to B. But the first way is, by hypolhesis, the shortest; consequently the distance from A to ./If is less than the distance from A. to JV", which is...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...
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 280 pages
...JVto B. But the first way is, by hypothesis, the shortest ; consequently the distance from A to Mis less than the distance from A to JV, which is absurd,...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...
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Elements of Geometry

Adrien Marie Legendre - 1825 - 570 pages
...hypothesis, the shortest ; 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...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...
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Elements of Geometry...: Translated from the French for the Use of the ...

Adrien Marie Legendre, John Farrar - Geometry - 1825 - 280 pages
...distance from Jl to M is less than the distance from A. to JV, which is absurd, since the arc AJ\1 is greater than AN; whence no point of the shortest...is less than the circumference of a great circle. Kg. 224. Demonstration. Let ABC (fig. 224) be any spherical triangle ; produce the sides AB, AC, till...
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An Analytical Treatise on Plane and Spherical Trigonometry, and the Analysis ...

Dionysius Lardner - Plane trigonometry - 1828 - 438 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...
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Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1828 - 346 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...
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Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1830 - 344 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,...
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