So, about the antikythera... (SPOILERS)
So, according to Helena, the antikythera was actually only ever going to find time rifts that brought people to Syracuse at the time of the siege. Which begs the following questions:
1. Just how many rifts in time lead back to the Siege of Syracuse, anyway? One assumes there are more rifts in time, but the antikythera only predicts the ones Archimedes cared about.
2. Indy, while present at the siege, says to Helena, "reverse these numbers, they'll get you back home." How would that work, if the mechanism only ever finds rifts that bring you to the time of the siege? If you can find other rifts by reversing the numbers, shouldn't the anikythera be able to take you to any time you'd like? In the end, it appears they simply went back through the same rift before it closed.
3. If the purpose of the antikythera was to bring help against the Romans, why hide it with obscure clues and the like? It's not like it's a distress signal. "Please come to our aid, and bring troops!" would have been helpful. As it is, what are the odds that anyone travelling through time would A) even be able to help at all, or B) side with Syracuse in the first place? After all, it could be the Romans themselves who unlocked the secrets of the antikythera and travelled back in time to the Siege of Syracuse. They'd be baffled, as Indy et al. were, and take the opportunity to make their victory even more complete, and less costly. It's a really dumb idea by Archimedes, no matter how you look at it.