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How can the date of lunar eclipses be calculated? Especially without the aid of a computer.

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What sorts of eclipses are you interested in? Solar? Lunar? Eclipses of the moons of Jupiter? Etc. Each has its own opportunities for simplifications. (Ahh - I see the tag "moon". I'll add that to the text for clarity....) Next, do you want just the date, or the time also, or more information than that? –  nealmcb Jan 16 '12 at 2:16

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Lunar eclipses are much easier to calculate than solar ones, and it has been done over the ages via a variety of methods. E.g. there is evidence that the design of Stonehenge embodied some aspects of calculating eclipses.

If we can assume that you have information on previous lunar eclipses at hand, you'll observe patterns. One of the most useful for this purpose is the pattern of near-repetition of eclipses after a Saros (Wikipedia), which is a period of approximately 6585.3213 days, or nearly 18 years 11 days. One saros after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle. That is because a saros is a nearly whole number of repeats of each of three cycles of the lunar orbit: the synodic month, the draconic month, and the anomalistic month. See the article for details.

After decades of observation (or via access to tables like the List of Saros series for lunar eclipses or the Five Millennium Canon of Lunar Eclipses: -1999 to +3000), you'll have a list of the currently active saros series. It is then easy to extend any particular saros series by hand, using the information in the saros article. If there is a particular date you're interested in, we could work through an example here.

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Perhaps this pdf will help. Which starts:

The calculation of eclipses is chiefly a combination of geometry and orbital mechanics. Hence, it is necessary to understand the orbital mechanics involved and also the underlying geometry.

The end of the pdf warns:

I assume that you are not trying to do an eclipse calculation exactly. If so, the problem becomes much more complicated, as you have to take into account the perturbations of all planets, the exact relation between θp, θu and the radii R1, R2, etc. In addition, you will need the initial condition of the Earth-Moon-Sun system, which you can obtain over the internet.

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The ancient Greeks were able to predict eclipses long before Kepler formulated his descriptions of planetary motion, i.e., orbital mechanics for the purpose of discussion. Nor did they have the data that the PDF mentions is necessary. –  dotancohen Jan 17 '12 at 8:39
    
@dotancohen exactly. The time span between lunar and solar eclipses seem to be so long that it is amazing that the ancients ever remembered the exact date of three of them to realise that there was a pattern. –  Stuart Woodward Jan 22 '12 at 12:23
    
@Stuart: Additionally, the eclipses happen at 120 degree differences over the face of the Earth, not in the same place. I have a hard time believing that that information could be used to construe any correlations. –  dotancohen Jan 22 '12 at 22:04

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