Line in the Sand

Where does this phrase come from? Why, the encounter between Gaius Popillius Laenas and Antiochus IV Epiphanes of the Seleucid Empire and Ptolemaic Egypt. Livy:

"After receiving the submission of the inhabitants of Memphis and of the rest of the Egyptian people, some submitting voluntarily, others under threats, [Antiochus] marched by easy stages towards Alexandria. After crossing the river at Eleusis, about four miles from Alexandria, he was met by the Roman commissioners, to whom he gave a friendly greeting and held out his hand to Popilius. Popilius, however, placed in his hand the tablets on which was written the decree of the senate and told him first of all to read that. After reading it through he said he would call his friends into council and consider what he ought to do. Popilius, stern and imperious as ever, drew a circle round the king with the stick he was carrying and said, "Before you step out of that circle give me a reply to lay before the senate." For a few moments he hesitated, astounded at such a peremptory order, and at last replied, "I will do what the senate thinks right." Not till then did Popilius extend his hand to the king as to a friend and ally. Antiochus evacuated Egypt at the appointed date, and the commissioners exerted their authority to establish a lasting concord between the brothers, as they had as yet hardly made peace with each other." Ab Urbe Condita, xlv.12.

What interests me here is that this was a circular, surrounding line in the sand. Surely the phrase is generally taken to be a straight line, drawn to demarcate enemies, with the implication: 'step over this line, towards me, and we will fight.' But the circular line is a much more interesting thing.