@Joe Nation,
I can see why someone would not necessarily know that it's "weigh." It's a chanty, though, which would be sung "call and response." Often a man physically disabled, especially someone blinded, at sea would be hired to sing and play music. So, when weighing anchor, the men put the bars in the capstan and push to wind up the "messenger," the continuous loop cable with which the heavy hawser of the anchor is brought home. The singer actually sits on the capstan head, going 'round and 'round. The call and response goes like this:
(Blind singer):
King Louis was the king of France, before the rev-oh-loo-shy-un
(Crewmen):
Weigh haul away, we'll haul away Joe
(Blind singer):
Then he got his head cut off, it spoilt his con-sti-too-shy-un
(Crewmen):
Weigh haul away, we'll haul away Joe
"To me" is the call for the response. It would only be called two or three times.
"Weigh haul away" is a slow, lugubrious chanty, which would be used for weighing anchor, or setting up a very heavy cable, such as the mainstay.
Others were more lively, and would be used when the men could move the capstan bars more quickly, such as hauling in a large, heavy sail, or setting up small cables such as the forestays or the backstays.
In South Australia i was born
Heave away, haul away
In South Australia 'round Cape Horn
We're bound for South Australia
Haul way you rollin' boys
Heave away, haul away
Haul away you rollin' boys
We're bound for South Australia
Most of that is "call," the "heave away haul away" being the only response. When the boys are almost running 'round the capstan, they won't have much breath for singing. "What shall we do with a drunken sailor" is probably the best known chanty in English, and also one in which everybody can sing, if they are so inclined.