The 1.898 degrees is the angle at the Moon between the POSITIONS of the observers separated by the full diameter of the Earth? Y/N

Yes. The image, as I am using it here, is assuming two observers on the equator. At 45 N and 45 S, the circle that the earth turns on is smaller than the equator, and it will be less... about ~1 degree instead of ~2 degrees.

Updated Image, Top-Down View:

IMG

So ... Tom has taken two observer points on opposite sides of the globe, and formed a triangle to a point representing the Moon, some 240k miles away, and calculated the angle formed by the two sightlines at the Moon.

If we really want to establish the angle between each observer's sightlines at 45N and 45S, we need to do better than calculating a single angle at the Moon from a single triangle drawn to their locations.

1. From this source, the angular size of the Moon is 31 arc mins, or approx 0.5 degrees.

https://lco.global/spacebook/using-angles-describe-positions-and-apparent-sizes-objects/2. From the single points on Earth of these two observers, at 45N and 45S, mirroring this picture, we can draw in these angles (the Orange circular dots). These are 0.5 degrees. I've marked in the upper and lower limits of this 0.5 degrees with the orange lines.

3. A straight line connecting the two observers through the Earth is the length of a Chord, where the angle of Arc covered is 90 degrees (one observer at 45N, one at 45S, 45+45 = 90)

We have already established that for an Earth radius of 3959 miles, this chord has length 5,598 miles (Green line in above)

4. If the two observers were truly looking along sightlines parallel to each other, they would be looking along the blue lines. They are separated by 5,598 miles at Earth, and true parallel sightlines would arrive at the Moon's distance still separated by 5,598 miles. The Moon has a diameter of 2,159 miles (Purple square), so the 'space' left each side of the Moon where these imaginary sightlines pass it by is (Blue square) 1,720 miles on each side (5,598 - 2,159 = 3,439, div by 2 = 1,720)

5. This gives us a right-angle triangle, where we need to solve for the small angle indicated by the blue circle. The intermediate side is equal to 240k miles (distance to Moon, rounded, and taken as the distance from observer's eyeball to a vertical drawn through the geometric centre of the Moon), and the small side (Blue square) we have just derived as 1,720. Put this into a right triangle calculator (

http://www.cleavebooks.co.uk/scol/calrtri.htm), and this gives us a blue circle angle of 0.411 degrees.

This is entirely consistent with the angular size of the Moon being 0.5 degrees. (Angular size of 1720 miles = 0.411, angular size of 2159 miles = 0.5)

6. Adding this to the 0.5 degrees of the Orange circle, this gives an absolute MAXIMUM deviation from a true parallel sightline of 0.9 degrees approx.

Therefore, to all intents and purposes, the two observers, if looking at the Moon at the same instant, are looking at it along the

same sightline. Neither observer would be able to tell, with the naked eye, if they were looking at any different angle from the other.

The above shows a side-on view. From a position behind Earth and the observers, looking toward the Moon, it would look something like this;

Agreed? Tom?