Meton's approximation is off by just 2 hours 4.4 minutes! And it's bettered only by comparing 4,131 months to 334 years. Now consider the successive fractions (number of months divided by the corresponding number of years): 12/1, 25/2, 37/3, 99/8, 136/11, 235/19, 4,131/334. Those fractions, in turn, can be written in the following manner:



Such expressions are known as continued fractions. They can be used in designing gear trains, including those that might be found in a planetarium to simulate the relative motion of the sun and moon around Earth.

The so-called Antikythera mechanism, apparently constructed in the first century B.C., recovered in 1900 from a Mediterranean shipwreck, and analyzed just a few decades ago, is one of the most striking examples of such engineering in the ancient world. It contained a system of gears whose gear ratios corresponded to well-known astronomical cycles involving the moon, including the Metonic cycle. The mechanism was clearly a type of analog computer, using fixed gear ratios to make calculations displayed as pointer readings on a dial.

The Antikythera mechanism -- the sole survivor of what was undoubtedly a long tradition of astronomical automata -- served primarily as an elegant simulation of the heavens. It was a tabletop monument to Greek and Alexandrian astronomy. Such ingenious devices also illuminated the intimate link between mathematics and astronomy, especially the role of number in astronomical prediction.

link to Ivars Peterson's Math Trek at Science News Online furnished by the astonishing mimi smartypants who also shares with the world the 'anal' game, wherein cars on the road have their given names prefixed with 'anal'. Anal Cruiser, Anal Explorer, Anal Excursion, Anal Fiesta...