Prior to the invention of the telescope many astronomers worked out models of the motion of the Moon to predict the position of the Moon in the sky. These geometrical models implied a certain range of distances of the Moon from Earth. Ptolemy’s most quoted model predicted that the Moon was nearly twice as far away at apogee than at perigee. Measurements of the angular size of the Moon were within the capabilities of pretelescopic astronomers. Such measurements could have helped refine the models of the motion of the Moon, but hardly anyone seems to have made any measurements that have come down to us. We use a piece of cardboard with a small hole in it which slides up and down a yardstick to show that it is possible to determine the eccentricity ϵ0.039±0.006 of the Moon’s orbit. A typical measurement uncertainty of the Moon’s angular size is ±0.8 arc min. Because the Moon’s angular size ranges from 29.4 to 33.5 arc min, carefully taken naked eye data are accurate enough to demonstrate periodic variations of the Moon’s angular size.

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. The “mean Moon diameter” (2×1738.2km) divided by the mean distance (384401 km) times 180/π=0.5182deg=31.09arcmin. Given the extreme range of the Moon’s distance (356400–406700 km), its geocentric angular size ranges from 29.39 to 33.53 arc min. The mean distance of the Moon divided by the equatorial radius of the Earth, 6378 km, gives a mean geocentric distance of 60.27 Earth radii. Note that the maximum distance is 5.8% greater than the mean distance, and the minimum distance is 7.3% less than the mean distance.
For more information on the variation of the time between successive occurrences of new/full Moon with respect to the mean value of 29.53 days, see
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. In Sec. 5.13 Ptolemy gives the mean distance of the Moon at syzygy (new/full Moon) of 59 Earth radii. The mean distance at quadrature (first or third quarter) is 384360 Earth radii, and the radius of the epicycle is 51060 Earth radii. It follows that the greatest distance occurs at syzygy and equals 59+51060=64.17 Earth radii. The minimum distance occurs at quadrature and is 384360 minus 51060=33.55 Earth radii. See also Ref. 17.
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