What sources are you basing this distinction on? I've never heard the falx described or depicted as anything other than a two-handed weapon (unless you want to argue that large sicae were called falxes). The difference between the romphaia and the falx, it seems, was entirely in the shape of the blade (straighter versus strongly curved) and in the primary people who used it (Thracians versus Dacians).
And from Herodian IV.14.3 - Nisibis, AD 217:Herodian of Antioch, book IV, chapter XIV, line 3:
"Macrinus thus received the office of emperor not so much because of the soldiers' affection and loyalty as from necessity and the urgency of the impending crisis.
While these events were taking place, Artabanus was marching toward the Romans with a huge army, including a strong cavalry contingent and a powerful unit of archers and those mail-clad soldiers who hurl spears from camels."
"Meanwhile Artabanus was upon them with his vast and powerful army
composed of many cavalry and an enormous number of archers and
kataphrakts who fought on camels, jabbing with long spears (possible
corrupt text, but the camels and the long spears are secure).
15.2-3:
"The barbarians caused heavy casualties with their rain of arrows and
with the long spears of the kataphraktoi on horses and camels, as the
wounded the Romans with downward thrusts. But the Romans had easily
the better of those who came to close-quarters fighting. And when the
size of the cavalry andthe numbers of the camels began to cause them
troublem they pretended to retreat and then threw down caltrops and
other iron devices with sharp spikes sticking out of them. They were
fatal to the cavalry and the camel-riders as they lay hidden in the
sand, not seen by them. The horses and the camels trod on them and
(this applied particularly to the camels with their tender pads) fell
onto their knees and were lamed, throwing the riders off their backs.
As long as the eastern barbarians are riding on horses or camels they
fight bravely; if they dismount or are thrown they are easily taken
prisoner because they do not resist in close-quarters fighting. And
further, they are hindered from running away (if this were necessary)
by the loose fodls of their clothes hanging around their legs."
Needless to say, both of these mentions postdate the EB timeframe by several centuries.
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