Re: Ancient technology ...
Wander the ancient world and you see the difference between Greek and Roman technology. The Greeks carved things from stone (e.g. amphitheatre in Agrigento), where as the Romans used concrete and casting (the Coliseum in Rome).
Gears are not the big deal, as casting was obviously available to the ancients. The mathematics to make real machines however, perhaps not.
One of the most amazing things about Greek philosophy was that they chronicled all of the previous philosophers and what they had all shown. In the 13th century when they rediscovered palimpsests of Euclid's works, they knew there were bits missing partly because of the mathematics(!), but also the chronicles.
But how did we lost the secret of concrete for 10 centuries...?