Whence 'draw a line in the sand'?

There are two explanations of this phrase on Word Detective:

http://www.word-detective.com/backidx.html

(You have to page down to the index & look for 'line in the sand'.)

Neither seems applicable to its current usage, which is roughly equivalent to 'draw a line under' (itself presumably an accounting reference).

Ancient mathematicians (Euclid for instance) drew lines in sand when studying geometry, but this doesn't seem relevant.

Does anyone know the origins of this common phrase?

Here's what the Word-detective website has to say:

In his book "In Love With Norma Loquendi" (a collection of his Sunday New York Times Magazine columns, published by Random House in 1994), Mr. (William) Safire provides two possible origins for "drawing a line in the sand."

The more recent possible origin for the phrase is an incident said to have taken place during the siege of the Alamo in 1836, when William Barret Travis drew a line in the sand with his sword and urged those willing to stay and defend the fort to step across it. Unfortunately, this heroic story seems to have been invented by a 19th century promoter long after the fall of the Alamo. But the myth itself probably greatly popularized the phrase, so it does count as an origin of sorts even if the incident itself was apocryphal.

Another possible origin dates back to the time of the Roman Empire. It seems that one of the Macedonian kings, a bit short of cash, decided to invade Egypt, then a Roman protectorate. His army was met at the border by a lone Roman senator named Popillius Laenas, who ordered the king to withdraw. The king began to stall for time, so Popillius Laenas drew a circle in the sand around the king and demanded that the king agree to withdraw his army before he stepped out of the circle. The king, apparently impressed by the senator's nerve (or, more likely, by the Roman Empire in general), withdrew. Incidentally, not only is this account verified by contemporary historians, but it also may be the only known instance of a line drawn in the sand actually stopping someone.

"To draw a line in the sand" is a "tongue-in-cheek" expression not meaning to conclude an affair. It refers to a conclusion which can be washed away as quickly as a wave can literally obliterate a line drawn in the sand of a beach.

I have no idea whether this has any connection whatever, but when Francisco Pizarro's underfed and sickly fellows wanted to get back to Panama rather than keep trying to find a new empire to conquer, he actually drew a line in the sand (in the beach of Isla del Gallo) an said more or less, 'on this side (south = on to Peru) you'll find toil, hunger, nakedness, and storms, and death; on the other  (north = back to Panama) ease and pleasure (...) por aquí se ba à Panamá à ser pobres, por alla al Perú à ser ricos. Now choose as a brave Castilian should. For me, I'll go southward'. (Montesinos, Annales, MS. year 1527)

As to the meaning, according to the current Webster,

draw the line or draw a line 1 : to fix an arbitrary boundary between things that tend to intermingle 2 : to fix a boundary excluding what one will not tolerate or engage in

in other several other languages there are similar expressions, usually restricted to "cross this line (that is, ignore what I'm telling you) and you'll have to pay for it"

Anonymous

Alamo?

The battle at the Alamo is famous in the U.S. and in Texas in particular. You can Google it.

Didn't this expression get popular during the first Gulf war?