Saros

It's nice when things work out; when you find a good match. The problem is that science deals with the real / natural world and there's a lot out there that is incommensurate: things that are occur on a different scale so you can never get an exact match. There are, for example, Imperial/US measurements for length (inch foot yard furlong mile) and there are SI ditto [cm metre km] and these systems are internally consistent: 8 furlongs to the mile; 12 inches to the foot vs the far more convenient and memorable 10s, 100s 1000s for all the SI units. But you can never get an exact measure of inches in cm; WolframAlpha says 1cm = 0.3937 inches which for most practical purposes [cutting cheese, short bits of timber, measuring shoes] we can call 40% or 2.5 cm to the inch. When driving a car, 1 km is 5/8^thishes of a mile.

I've reflected on the annoying inconsistencies in the lunar and solar orbits, both as to cycle length and to eccentricity, which mess up such important occurrences as equinoxes and solstices;  the christian calendar; and the apparent size of the sun and moon.  If June 17^th won't stay in the same place w.r.t. the seasons without adding a day to the calendar now and then, then it seems an Impossible Dream to predict when eclipses are going to occur. Perhaps most extraordinary is that people would care enough to do the calculations, because if we live in the same country all our lives we're only going to experience a total solar eclipse once a lifetime. If we live in the same village all our lives, we'd have to wait about 300 years for the next one. Of course, you'd be able to twig that a total eclipse was a subset of [partial] eclipses and, if it was important enough, you'd start gathering data and making and winning bets with your math-anxious neighbours. Having made your brain bleed with the calculations, you'd surely tend to believe in your immortality.

Eclipses happen because of orbital and gravitational resonances between the earth moon and sun; which are complicated. Any fule kno that the moon orbits the earth once-a-month [moonth! after all]. But it turns out that there are four/five months each of slightly different lengths - and I'm not talking about the wholly artificial man-made 28, 29, 30 and 31 day months.

• the sidereal month is the time it takes for the moon to line up with some arbitrary star in the firmament having gone once round in its orbit. It is approximately 27.32166 days (27 days, 7 hours, 43 minutes, 11.6 seconds)
• the synodic month is a reg'lar month: the time between successive full moons = 29.53059 days. It's two days longer than the sidereal month because, to be full again, the moon has to get back to the same place w.r.t. to the sun and in a month the earth has made two days progress in its own orbit.
• the tropical month is the time between the moon's passes through the same equinox point. It is for all practical purposes [3 parts in a million diff]  the same as the sidereal month = 27.32158 days
• the anomalistic month is the interval between the moon's closest approach (perigee) to the earth = 27.55455 days
• finally the draconic month is the interval between the moon's crossing of the plane of the earth's orbit - the moon's orbit is about 5^o off-centre of ours = it's 27.21222 days long

To predict an eclipse you have keep track of all these similar but not identical cycles and the ancient Chaldeans noticed a curious thing about their inter-relationships. If you have a solar eclipse in a particular spot on the globe then 6,585 days, 8 hours and 11 minutes later the sun moon and earth will be almost in the same relative positions and you'll have another eclipse. 6585.3211 days = 241* sidereal months = 223 * synodic months = 242 * draconic months = 239 * anomalistic months. The Chaldeans called this extraordinary coincidence intersection something in their own dead language; probably the equivalent of "well scutter me pink, mates". We call it a Saros because Edmond "Comet" Halley took the word, possibly incorrectly,  from an 11^thC Byzantine Greek dictionary.  The extra 8 hours is a bummer because if it was light enough to see an eclipse in one Saros cycle, it will be dark over the next two and so the local eclipse is easily detectable only after about 54 years: a lifetime. Hope it's not cloudy!

Matt "Numberphile" Cooper explains the Saros cycle much better than me because a) he understands what he's talking about and b) he's got moving graphics. 15 minutes well spent - go on, you'll learn something.