Your question is not correct as the correct name is Archimedes. He was indeed Greek, not Carthaginian. He was a citizen of Syracuse, a Greek city in Sicily that allied itself to the Carthaginians in the Second Punic War. In 212 BC Syracuse was besieged by the Romans and Archimedes contributed much to its defence. He is credited with inventing the arrow slit (ie making it wider on the inside than the outside, to give the defenders a greater range of fire), and of re-designing the defensive catapults, much to the consternation of the Romans. His greatest weapon was basically a hook on the end of a rope, hanging from a beam with a counterweight. With this, the Syracusans hooked Roman ships (Syracuse was a port the siege was by land and sea) out of the sea, then dropped them.
Syracuse was eventually stormed and, though the Roman commander had ordered that Archimedes should be taken alive, he was not recognised in the confusion and killed.