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" THEOREM. The sum of the sides of a spherical polygon, is less than the circumference of a great circle. Let... "
Elements of Geometry, Conic Sections, and Plane Trigonometry - Page 177
by Elias Loomis - 1877 - 443 pages
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Elements of Geometry: Including Plane, Solid, and Spherical Geometry

George Washington Hull - Geometry - 1807 - 408 pages
...sphere at equal distances from the centre of a sphere are equal. PROPOSITION IV. THEOREM. 579. TJie sum, of the sides of a spherical polygon is less than the circumference of a great circle. Given — ABCD a spherical polygon on the sphere whose centre is 0. To Prove — AB + BC + CD + DA...
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A Treatise on Spherics: Comprising the Elements of Spherical Geometry, and ...

Daniel Cresswell - Geometry - 1816 - 294 pages
...of a great circle of the sphere. (75.) COR. In the same manner, it may be shewn, that the aggregate of the sides of a spherical polygon, is less than the circumference of a great circle of the sphere, if the polygon be bounded by arches of great circles, each of which is less than the...
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An Analytical Treatise on Plane and Spherical Trigonometry, and the Analysis ...

Dionysius Lardner - Plane trigonometry - 1828 - 438 pages
...sum of the three sides is less than the circumference of a great circle. (130.) Cor. The periphery of a spherical polygon is less than the circumference of a great circle. ( 131.) Eight triangles formed by three great circles. , (132.) Cor. 1. Given any one of these eight...
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A System of Plane and Spherical Trigonometry: To which is Added a Treatise ...

Richard Wilson - Logarithms - 1831 - 372 pages
...polygon is a portion of the surface of a sphere contained by several arcs of great circles. 32. PROP. The sum of the sides of a spherical polygon is less than the circumference of a great circle. Let a, , y, S, &c. be the sides of the polygon subtending plane angles at the centre of the sphere ;...
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An Elementary Treatise on Plane and Solid Geometry

Benjamin Peirce - Geometry - 1837 - 216 pages
...the arcs AB, ACj BC; and, therefore, each of these arcs is less than the sum of the other two. 439. Theorem. The sum of the sides of a spherical polygon...is less than the circumference of a great circle. Polar Triangle. Demonstration. From the centre O (fig. ISO) of the sphere draw the radii OA, OB, OC,...
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An Elementary Treatise on Plane and Solid Geometry

Benjamin Peirce - Geometry - 1847 - 150 pages
...the arcs AB, AC, BC ; and, therefore, each of these arcs is less than the sum of the other two. 439. Theorem. The sum of the sides of a spherical polygon...is less than the circumference of a great circle. Sum of the Sides of a Spherical Polygon. Proof. From the centre O (fig. 180) of the sphere draw the...
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Elementary Course of Geometry ...

Charles William Hackley - Geometry - 1847 - 248 pages
...Show how to construct a spherical triangle with any three parts given. 10. Prove that the sum of all the sides of a spherical polygon is less than the circumference of a great circle. 11. Make a sphere pass through four given points, or prove that every tetrahedron may he circumscribed...
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Elements of Geometry: With Practical Applications ...

George Roberts Perkins - Geometry - 1847 - 308 pages
...circumference. add AB+ AC to eacli ; we PROPOSITION xvirr. THEOREM. The sum of all the sides of any spherical polygon is less than the circumference of a great circle.. Let us take, for example, jr. _c the pentagon ABCDE. Produce the sides AB, DC till they meet in F ; then...
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A treatise on plane and spherical trigonometry

Thomas Grainger Hall - Trigonometry - 1848 - 194 pages
...the triangle, is always less than the circumference of a great circle. COR. Hence it is obvious that the sum of the sides of a spherical polygon is less than the circumference of a great circle. The Polar TriangU. 12. Let ABC be a spherical triangle. With points, A, В, С, as poles, describe...
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Elements of Geometry and Conic Sections

Elias Loomis - Conic sections - 1849 - 252 pages
...shortest path from A to B, can be out of the arc of a ijreat circle ADB. Therefore, the shortest path, &c. PROPOSITION IV. THEOREM. The sum of the sides of a...the circumference of a great circle. Let E be the center of the sphere, and join AE, BE, CE, DE. The solid angle at E is contained by the plane angles...
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