In 1901, divers lifted a corroded lump of bronze from a Roman-era shipwreck off the Greek island of Antikythera. Inside were more than thirty gears. Once researchers could X-ray it, they found one dial that ended in a tiny inscription: the number 223, written in Greek letters (ΣΚΓ). That number let a machine built in the second century BCE say when the Moon would hide the Sun.
A new Moon passes between us and the Sun every 29.53 days, but most months it slips above or below the Sun and nothing happens. The Moon's orbit is tilted about 5° to the Earth's, so the two paths cross at just two points, the nodes. An eclipse is possible only when a new (or full) Moon happens near a node. The Sun returns to the same node every 346.62 days, an interval called the eclipse year.
So the sky runs two clocks: one that ticks every 29.53 days (new Moons) and one every 346.62 days (node crossings). An eclipse pattern repeats only when a whole number of one clock lines up with a whole number of the other.
Pick a gap in months. The top strip shows the first 30 new Moons from today, lit when the Moon is within about 18° of a node, which is when an eclipse is possible. The bottom strip shows the 30 new Moons after the gap you chose. When the strips match, the sky has reset.
Slide around and most gaps fail. A few work: 223, 135, 47 and 358 months all put the new Moons back in step with the nodes. So why did the Babylonians, and then the Greeks, build their machine around 223? Because 135 and 358 land on the opposite node, and 47 is a day and a half off. Only 223 months bring back the same node, to within half a day: 223 new Moons last 6,585.32 days, almost exactly 242 node passes.
It has a second advantage the strips cannot show. 223 months also equal almost exactly 239 trips of the Moon from its closest point to Earth and back (238.992, to be exact). A Moon at the same distance looks the same size, so the repeat eclipse is the same kind: similar length, similar look.
The Saros is not a whole number of days. It is 18 years and 10 or 11 days, plus about eight hours. The Earth turns another third of a turn during that time, so each repeat eclipse is seen about 120° of longitude further west. Three Saros periods make about 54 years, and the Sun is then back over your side of the world. That is why the Antikythera mechanism has a second, smaller dial that adds the eight hours: the Exeligmos dial.
The Saros period was found by Babylonian astronomers centuries before the machine was built. The Greek mechanism turned it into a gear train. A pointer winds along a four-turn spiral on the back, one cell for each of the 223 months. Cells where an eclipse is possible carry a glyph telling you whether it is a Moon or Sun eclipse and, from an index letter, other details of its timing. Turn the crank and the pointer walks forward through months and years. The mechanism itself is dated to about 150–100 BCE, and one study of its eclipse scheme puts the design as early as about 205 BCE.
Try 235 months in the tool above, which is the 19-year Metonic cycle the same machine uses for its calendar. It lines up the Moon's phases with the date, but not the nodes. Only 223 does both. Even the Saros drifts: each repeat is about half a degree off, so after 70 to 80 repeats (roughly 1,200 to 1,500 years) the eclipse series fades out. Series after series overlap, which is why we always have some repeat from some series.
The idea is as simple as it is old: if the sky is a pair of clocks, find where they agree. Everything else is just keeping count.