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 17th 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 5o off-centre of ours = it's 27.21222 days long
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.