The LPOD Lunar Photo of the Day for May 26, 2007 shows the parallactic shift in the Moon's position as seen from two locations about 1400 miles apart. Amateur astronomers Anthony Ayiomamitis and Pete Lawrence took pictures of the Moon from Greece and England at the same moment on the afternoon of May 23. They combined the photos so that the Moon images overlap, revealing an apparent shift of about a third of a degree in the Moon's position relative to Regulus, a bright star in the constellation Leo.

I added my two cents to the discussion of this image on sci.astro.amateur, offering a refined estimate of the Moon's distance. Here, I'm exhibiting several LightWave renderings showing the geometry of the image. These use my star plotter along with a short program I wrote specifically to analyze the Moon/Regulus image, which gave me the positions and orientations of the Earth and Moon, the sunlight angle, and the sight lines to the Moon from Selsey and Athens.

Selsey and Athens are in sunlight because the photos were taken during the day, near the time of a lunar occultation of Regulus (actually a near miss in these two locations).

Most diagrams of lunar parallax (including the one further down on this page) are wildly out of scale. This is necessary in order to make the angles visible, but I think the iconography of diagrams like these has ingrained in people a distorted sense of the scale of the solar system. The renderings below and along the left margin show the true scale of the Earth-Moon system.

The two images below simulate the view of the Moon from Selsey (left) and Athens. Try these as a stereo pair: cross your eyes until the two Moons overlap. If you have the knack for this, the two Moons will appear to fuse into a single 3D image.

With a little geometry, the parallax in these views can be used to estimate the distance to the Moon.

Selsey, Athens, and the Moon (S, A, and M in the schematic diagram) form a long, thin triangle. The angle at M is the parallax, the angular separation of the two images of Regulus. We also know the length of side AS, which is just the chord distance between Athens and Selsey.

If we make the simplifying assumption that triangle ASM is isosceles, then the distance to the Moon is just

( length of AS ) / ( 2 tan( parallax / 2 ))

But this gives us an estimate of the lunar distance that's too large by about 10%. We can do much better by  finding the length of the line AC and using that instead of AS in the formula. One way to estimate AC's length is to create unit direction vectors for AS and AM (or SM or Earth-M, the minor difference isn't important). The dot product of these two vectors is the cosine of the angle at A (or at S). The sine of this angle is the foreshortening of AS in the direction of M, or the length of the projection of AS onto the image plane, and this is (very nearly) the length of AC.

Another approach is to use the angles at S (or A), the length of AS, and the parallax (the angle at M) to solve the scalene triangle directly.

length of MS = ( length of AS ) sin( angle at A ) / sin parallax
length of MA = ( length of AS ) sin( angle at S ) / sin parallax

Either method is capable of producing a very accurate estimate of the Moon's distance from Earth.

We don't know Hipparchus's exact procedure, but we do know that he used an estimate of the lunar parallax derived from a solar eclipse. The eclipse was total at the Hellespont. At Alexandria, the Moon covered 4/5 of the Sun's diameter. The parallax was therefore 1/10 of a degree, and the baseline was the distance from Alexandria to the Hellespont.