Demonstrating Spherical Excess, Spherical Angles on the Surface of the Earth

Mick West

Staff member
"Jon Leighton, post: 208106, member: 10216"]I think spherical excess is also worthy of a mention. In general, the internal angles of any large surveyed triangle will sum to more than 180 degrees.[/QUOTE]

I was wondering if there was a good place we could try this in real life, three mountains where the sum of the angles is more than 180°. We could do the math and see what distances would be needed.

Something like the Sierra mountains, or some other place where we have good visibility, probably need at least 50+ miles, but the problem is how to measure the angles if you don't have a theodolite?
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Mick West

Staff member
Thinking a little more on this, it's one of those proofs of a globe earth whereby in measuring it you'd actually prove it earlier, but in a different way.

If you think about the triangle draw between the peaks of three mountains (assume they are of equal height above sea level for convenience) then the lines between the peaks ARE straight lines, and the angles of that triangle actually DO add up to 180°

What does not add up to 180° is the differences in bearing between the peaks. This can be a little confusing. Consider an extreme example. The triangle formed by the North pole (point A) and two points (B and C)
on the equator: one at 0° and one at 90° west. Imagine if at each point there was an enormous tower, tall enough so you could see the tops of the other towers from it. This is about 2,7000km in this example:


The red lines are the great circles, they indicate the shortest path, and the bearings, between any two points. Where they meet they form an angle of 90 degrees.


But if you were to stand on the top of the tower at point B, and measured the visual angle between point A and point C, it would come out at 60°, and not 90°


The reason is pretty obvious if you think about the geometry. The heading is measured in the horizontal plane, the visual angle between A and C is measured looking down.

So if you wanted to actually measure the spherical excess, then in the course of doing it you would also note that distant hills are below the horizontal plane (at the level of the camera), and hence by noticing this you would already have proved curvature.


New Member
Hi Mick,

I would suggest getting three observers underneath stars, and use those stars to measure the angles of a big triangle. A random example I checked a while ago (using Stellarium, not IRL):

Julian Day JD 2458474.457368

A: +46°00'51",+4°25'10 (46.01417,4.419444)
B: -16°44'44",+26°23'18" (-16.74556,26.38833)
C: +11°52'18",+77°14'18" (11.87167,77.23833)

The Azimuths and Altitudes of the stars as seen from these locations are:

From A (Az/Alt):
Capella +153°09'06.2"/+89°59'59.8"
Sirius +156°52'45.9"/+24°12'21.7"
Regulus +93°59'03.5"/+20°27'23.6"

From B (Az/Alt):
Capella +343°27'16.9"/+24°12'21.5"
Sirius +47°53'45.4"/+89°59'59.2"
Regulus +63°42'12.8"/+32°11'33.1"

From C (Az/Alt):
Capella +314°55'55.1"/+20°27'23.5"
Sirius +241°18'49.6"/+32°11'33.3"
Regulus +245°04'09.1"/+89°59'59.4"

A is underneath Capella. De horizontal angle between Sirius en Regulus is +156°52'45.9" - 93°59'03.5" = 62.90°
B is underneath Sirius. De horizontal angle between Cappella en Regulus is 360 - 343°27'16.9" + 63°42'12.8" = 80.25°
C is underneath Regulus. De horizontal angle between Capella en Sirius is +314°55'55.1" - 241°18'49.6" = 73.62°

62.90° + 80.25° + 73.62° = ~217°

Mick West

Staff member
I would suggest getting three observers underneath stars, and use those stars to measure the angles of a big triangle. A random example I checked a while ago (using Stellarium, not IRL):
Something I discuss in my book is how you can actually use Stellarium instead of going outside:
The idea being that if you can prove that Stellarium models the earth as a sphere (and the stars essentially as an infinitely distant celestial sphere) then the Flat Earth argument fails if they can't show Stellarium differing from real world observations.

Of course the math/geometry is a bit beyond the average person, and the point is easily rejected if you don't understand it, but it's another tool.