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" The sum of all the sides of any spherical polygon is less than the circumference of a great circle. "
Higher Geometry and Trigonometry: Being the Third Part of a Series on ... - Page 11
by Nathan Scholfield - 1845 - 232 pages
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Military Schools and Courses of Instruction in the Science and Art of War ...

Henry Barnard - Military education - 1862 - 399 pages
...of a great circle which joins the two given points. The sum of the sides of a spherical triangle, or of any spherical polygon, is less than the circumference of a great circle. Poles of an arc of a great or small circle.—They serve to trace arcs of circles on the sphere. Measure...
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Elements of Geometry: With Practical Applications to Mensuration

Benjamin Greenleaf - Geometry - 1863 - 504 pages
...is measured by the arc of a great circle joining the two points. PROPOSITION IV. — THEOREIH. 525. The sum of all the sides of any spherical polygon is less than the circumference of a great circle. Let ABCDE be a spherical polygon ; then the sum of the sides AB, BC, CD, &c. is less than the circumference...
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Elements of Geometry: With Practical Applications to Mensuration

Benjamin Greenleaf - Geometry - 1868 - 338 pages
...is measured by the arc of a great circle joining the two points. PROPOSITION IV. — THEOREM. 525. The sum of all the sides of any spherical polygon is less than the circumference of a great circle. Let ABCDB be a spherical polygon ; then the sum of the sides AB, BC, CD, &c. is less than the circumference...
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Elements of Geometry and Trigonometry: With Practical Applications

Benjamin Greenleaf - 1869 - 516 pages
...is measured by the arc of a great circle joining the two points. PROPOSITION IV. — THEOREM. 526. The sum of all the sides of any spherical polygon is less than the circumference of a great circle. Let ABCDE be a spherical polygon ; then the sum of the sides AB, BC, CD, <fec. is less than the circumference...
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Elements of Geometry: Plane and Solid

John Macnie - Geometry - 1895 - 386 pages
...as its limit; hence ACB must be greater than AMB. 'E 313 PROPOSITION XII. THEOREM. 628. The sum of the sides of any spherical polygon is less than the circumference of a great circle. Given: A spherical polygon ABCD, on a sphere with center O; To Prove: AB+BC+CD+DA < a circumf. of a...
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Plane and Solid Geometry: Inductive Method

Arthur A. Dodd, B. Thomas Chace - Geometry - 1898 - 492 pages
...sides and the center, С, of the sphere. What is formed by these planes ? Can you prove that the sum of the sides of any spherical polygon is less than the circumference of a great circle? Give demonstration, and write Prop. XV. 658. PROPOSITION XVI. SPHERICAL TRIANGLES. Write an expression...
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Plane and Solid Geometry

Isaac Newton Failor - Geometry - 1906 - 440 pages
...Then Z AOC < Z BOA + Z BOC. .-.AC < AB + BC. PROPOSITION X. THEOREM 590 788 QED 794 The sum of the sides of any spherical polygon is less than the circumference of a great circle. HYPOTHESIS. ABCD is a spherical polygon. CONCLUSION. AB + BC + CD + AD < 360. PROOF. Draw the radii...
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Plane and Solid Geometry

Isaac Newton Failor - Geometry - 1906 - 431 pages
...radii OA, OB, 00. Then Z AOC < Z BOA + Z BOG. /.AC < AB + BC. PROPOSITION X. THEOREM 794 The sum of the sides of any spherical polygon is less than the circumference of a great circle. 590 788 QED HYPOTHESIS. ABCD is a spherical polygon. CONCLUSION. AB + BC + CD + AD < 360. PROOF....
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Handbook of Mathematics for Engineers and Engineering Students

Joseph Claudel - Mathematics - 1906 - 758 pages
...638, 658) apply as well to spherical triangles as to plane triangles. 863. The sum of the sides ol any spherical polygon is less than the circumference of a great circle. 864. The angle of two arcs of great circles (856) is equal to the plane angle of the dihedral angle...
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Plane and Solid Geometry

Edward Rutledge Robbins - Geometry - 1907 - 428 pages
...by arc AB (?), etc. .-. arc AB + arc BC + arc CD + arc AD < 360 (Ax. 6). QBD 756. COR. The sum of the sides of any spherical polygon is less than the circumference of a great circle. 757. THEOREM. If one spherical triangle is the polar of a second triangle, then the second is the polar...
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