There are actually many primary classical sources which describe these mechanisms. There was more than one mechanism. Some were larger scale (like say a "modern" town or church clock) for public display and some were smaller like the Antikythera Mechanism. Cicero goes into a fair bit of detail in describing them and states that, when what he called "the bronze device" was turned it showed the various positions of the sun, the moon and the five planets, eclipses etc, through "unequal and diversified progressions in dissimilar motions" with "a single turn" of the device.
Here in South Lanarkshire our (and Scotland's) oldest public timepiece is the Douglas Clock perched in St Bride's Church Tower. At a mere 500 years old it still keeps perfect time, chiming the hour three minutes before the hour, as it was designed to do - referring to the Douglas family motto of "Jamais Arriere" ie. "never behind".