With Java animations by Bill Casselman
The most economical explanation is that in keeping records, early astronomers were struck by the almost exact duplication of the pattern of equinoxes and solstices (sun) and phases of the moon in a 19-year cycle. Nineteen years almost exactly matches 235 lunar-phase cycles ("synodic months"), which correspond to 235+19=254 revolutions of the moon with respect to the stars. It picks up an extra one each year from its trip with us around the sun.
But part of the answer comes from the astronomical ratio itself, which turns out to be one of those numbers that can be very well approximated by rationals. This is manifest in its continued fraction expansion:
13.368267.. = [13, 2, 1, 2, 1, 1, 17, ...] 1 = 13 + ------------------------ 1 2+ --------------------- 1 1+ ------------------ 1 2+ --------------- 1 1+ ------------ 1 1+ -------- 1 17+ ---- etcStopping the process after the last "1+" gives the "continuant" 254/19 used in the Antikythera Mechanism. Continuing with the the 17 gives the next continuant, 4465/334. The large increase in the denominator comes from the 17.
Here is a useful fact from the theory of continued fractions:
1 pn 1 -------- < |L - --- | < ------- . 2 qnqn+1 qn qnqn+1
4. Gear ratios and continued fractions