¶ Spherical Triangle Sequence
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Spherical Triangle Sequence:
"add up to 180°, so each of the other corners. . . seems to be 300° so it adds up to 900°. Well you have been taught the wrong way because the sums of the angles never add up to 180°, because you cannot make an absolutely flat plane. There is always something a little more. And this is a case where you really see how much more.
“If you have a very big triangle, for instance, going to the north pole, taking a meridian to the equator, impinging on the equator at 90°; go one-quarter of the way back around the Earth and take a meridian back to the north pole; it leaves the equator at 90° and comes back where the two come together and th angle at the top will be 90°. So 90°, 90°, 90° is 270°-- that is the typical spherical triangle. And the sums of the angles of spherical triangles are approximately never the same. If I make a smaller triangle just inside of the one that I just talked about where there is 90°,90°, 90°, the sums of the angles are going to be a little less. It approaches, but never gets down to 60°, 60°, 60°. It is always a little more. So the smaller they are, the more they approach 180°. And so in all our surveying around our Earth, doing geodesy, we are always dealing with what we call spherical excess,”
