What happened to these advanced civilization of the antiquity? We have drawings of highly advanced siege engines yet it seem that after the fall of Rome we all went back to fighting with sticks for a few centuries. Why is that?
In the case of the Antikythera mechanism, it turns out the knowledge and technology weren't entirely lost. There's a fairly well-established throughline now from Babylonian observation-based astronomical tables to Greek philosophical views and mathematics (and craftmanship) to Arabic astrolabes (retaining some of the gear-train tech) and then back to Europe, to the monastic astronomical clocks. Then the escapement wa…
They never made an overwhelming shift to mathematical / science based civilization. They took over Syracuse with a mandate to keep Archimedes alive, but that failed. Some scholars say the only roman contribution to math was numerals.
Basically they reaped the profits of empire, and fell into the cargo cults of opulent success, abandoning the prior agricultural based common sense by never integrating new ideas in the Aristotelian domains except for to pay homage to the originating culture enough to collect taxes.