| John Radford Young - Astronomy - 1833 - 286 pages
...B'OC', it follows that A'OD + B'OD = A' OB' < A'OC' + B'OC' ... AB < AC + CB. (39.) 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, then the arcs... | |
| Adrien Marie Legendre - Geometry - 1836 - 394 pages
...shortest distance between B and A can lie out of the arc of the great circle BDA. PROPOSITION HI. THEOREM. 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, AC, till they meet again in D. The arcs ABD,... | |
| Adrien Marie Legendre - Geometry - 1841 - 288 pages
...the distance from A to M 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.ssi. Demonstration. Let AB C (fig. 224) be any spherical triangle, produce the sides AB, AC, till... | |
| Andrew Bell - Mathematics - 1842 - 402 pages
...is less than 180°. 348. It is to be observed, in forming examples in spherical trigonometry, that the sum of the three sides of a spherical triangle is less than the circumference of a circle; and the sum of any two sides is greater than the third ; also the greater angle is opposite... | |
| Nathan Scholfield - Geometry - 1845 - 506 pages
...hence this arc is itself the shortest distance between its two extremities. PROPOSITION IV. THEOREM. v 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, A(J, till they meet again in D. The arcs... | |
| Nathan Scholfield - Conic sections - 1845 - 542 pages
...hence this arc is itself the shortest distance between its two extremities. PROPOSITION IV. THEOREM. 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. The arcs ABD,... | |
| Nathan Scholfield - 1845 - 894 pages
...hence this arc is itself the shortest distance between its two extremities. PROPOSITION IV. THEOREM. 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. The arcs ABD,... | |
| Charles William Hackley - Geometry - 1847 - 248 pages
...the arcs AB, AC, BC, which measures these angles, is less than the sum of the other two. PROP. VIII. 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, AC to / /^* iioset in D. Then, since two great... | |
| George Roberts Perkins - Geometry - 1847 - 326 pages
...itself the shortest distance between its two extremities. PROPOSITION XVII. 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,... | |
| Thomas Grainger Hall - Trigonometry - 1848 - 192 pages
...BC; hence any two sides of a spherical triangle are together greater than the remaining side. 1 1. The sum of the three sides of a spherical triangle is less than the circumference of a great circle. For the solid angle at 0 is contained by plane angles, the sum of which is always less than four right... | |
| |