NavList:

Alex Eremenko, you wrote:
"Assuming that you already know your latitude, you can take a sight of any star and find your local time from the formula
  sin h=sin(phi)sin(delta)+cos(phi)cos(delta)cos(t),
where h is the altitude, phi is the latitude, delta is the declination, and t is the local hour angle."

True. However I think I will needle you just a wee little bit for introducing "variables" that you immediately toss out the window. :) I know, I know, there's lots of traditional theoretical literature in positional astronomy that insists on using φ for latitude, δ for declination, etc., but who needs 'em! :)

You continued:
"You find the local hour angle, and knowing the right ascension  of the star, this gives you the local time. If I do not mistake this is called a 'time sight'."

It might be safer to say that it gives us "a" local time since we can go in a couple of directions from here. Sidereal Time? Mean Time? Anything else?? And, yes, that is commonly called a time sight, and that's what I call it regularly...

And yet, something to consider: in a way, there's really no such thing as a time "sight". Or we could say that the "sight" in this is misleading, a misnomer. It's just an altitude of a star, a "sight" like any other, that we work up using "time math". 

Where we go next depends in part on whether we're trying to get a difference in longitude between two observers of the same eclipse, or if we're trying to get one observer's longitude from the known, predicted UT of events in an eclipse. Both are interesting cases. Not fundamentally different, of course, but some details are distinct to each case.

Alex E, you added:
"There is no record that Ptolemy actually used this method, or of any determination of longitude in antiquity using eclipses."

Right. Ptolemy did not employ the method himself, as far as we know. He was more of a "stay at home" sort. But he did recommend it.

Ptolemy also lacked one critical asset that might have put this "longitude by eclipse" method into practice almost 1900 years ago. The asset he lacked was assets ...or a budget! He certainly advocated, in his Geography, for the general principle of using eclipses for longitude though he never spelled out the details of getting local time, and he never had the resources to send an expedition of dozens of eclipse observers across the known world in his day. There was no practical merit in better longitudes at that time. It was a cart before the horse problem. Or a chart before the horse problem.

Ptolemy did describe an example of longitude by lunar eclipse in the Geography, but it was one of those typical Greek pre-science examples pulled from posterity, centuries before Ptolemy, the eclipse in 331 BC. It was one of the few ever recorded in different longitudes (and poorly, by the way), but it was more famous as an omen before the Battle of Gaugamela. 

Frank Reed