From the notes to the Penguin Book (1959):

"The Portugese Ballad  A Nau Caterineta  and the French ballad  La Courte Paille  tell much the same story.  The ship has been long at sea, and food has given out.  Lots are drawn to see who shall be eaten, and the captain is left with the shortest straw.  The cabin boy offers to be sacrificed in his stead, but begs first to be allowed to keep lookout till the next day.  In the nick of time he sees land ("Je vois la tour de Babylone, Barbarie de l'autre côté") and the men are saved.  Thackeray burlesqued this song in his  Little Billee.  It is likely that the French ballad gave rise to The Ship in Distress, which appeared on 19th. century broadsides.  George Butterworth obtained four versions in Sussex (FSJ vol.IV [issue 17] pp.320-2) and Sharp printed one from James Bishop of Priddy, Somerset (Folk Songs from Somerset, vol.III, p.64) with "in many respects the grandest air" which he had found in that county.  The text comes partly from Mr. Bishop's version, and partly from a broadside."  -R.V.W./A.L.L.

This version was collected by George Butterworth from Mr. Harwood of Watersfield, Sussex, in 1907, and was first published in the Folk Song Journal, vol.IV, [issue 17], p.321.

On the DT:

The Ship in Distress     From the Oxford Book of Sea Songs, with tune.
Sept Ans sur Mer         Canadian version of La Courte Paille, with tune.
Il était un Petit Navire  Canadian version of La Courte Paille, with tune.

Le Petit Navire is a later development of the song, dated by Henri Davenson (Le Livre des Chansons, 1955) to the mid-19th. century.   It was re-made in the music-halls, becoming a comic song in which the cabin-boy is actually eaten, in a variety of interesting sauces.  In this form it re-entered tradition -largely as a chidren's song- and continued alongside its more serious-minded ancestor.

See also:

The Banks of Newfoundland  (no tune).
The Silk Merchant's Daughter   From the Oxford Book of Sea Songs, with tune.

These two songs share the central motif, though the surrounding stories are different.

In the Forum:

The Silk Merchant's Daughter  Discussion of variants and broadside sources.
Il était un petit navire  Discussion of variants

@sailor @cannibal

There are entries at  The Traditional Ballad Index:

The Ship in Distress

La Courte Paille

See Also:

The Banks of Newfoundland (II)

The Silk Merchant's Daughter  Laws N10

There is a version of A Nau Catarineta, and links to others, at:   Barcarolas e Marinhas   I am not in a position to provide an adequate translation from the Portugese, so have not included a text at this time.  If there's anybody out there who could do one, that would be great.

LITTLE BILLEE

(William Makepeace Thackeray)

There were three sailors in Bristol City,
Who took a boat and went to sea.
But first with beef and captain's biscuit,
And pickled pork they loaded she.

There was gorging Jack and guzzling Jimmy,
And the youngest he was little Billee.
Now when they'd got as far as the Equator
They'd nothing left but one split pea.

Says gorging Jack to guzzling Jimmy,
"I am extremely hungaree."
To gorging Jack says guzzling Jimmy,
"We've nothing left, us must eat we."

Says gorging Jack to guzzling Jimmy,
With one another we shouldn't agree!
There's little Bill, he's young and tender,
We're old and tough, so let's eat he."

"O Billy! we're going to kill and eat you,
So undo the button of your chemie."
When Bill he heard this information,
He used his pocket-handkerchie.

"First let me say my catechism
Which my poor mother taught to me."
"Make haste! make haste!" says guzzling Jimmy,
While Jack pulled out his snickersnee.

Billy went up to the main top-gallant mast,
And down he fell on his bended knee,
He scarce had come to the Twelfth Commandment
When up he jumps - "There's land I see!"

"Jerusalem and Madagascar
And North and South Amerikee,
There's the British flag a-riding at anchor,
With Admiral Napier, K.C.B"

So when they got aboard of the Admiral's,
He hanged fat Jack and flogged Jimmee,
But as for little Bill, he made him
The Captain of a Seventy-three.

There are apparantly no broadside texts at  Bodleian Library Broadside Ballads,  but there are variants of the Silk Merchant's Daughter.  These are the most legible:

New York Streets  Printed between 1828 and 1842 by Batchelar, 14, Hackney Road Crescent, London.

The Silk Merchant's Daughter  Printer & Date unknown

There is also a sheet of two broadsides relating to an incident of cannibalism at sea:

The Loss of the Francis Spaight/ Loss of the Ship Francis Spede J. Scott, Pittenweem; Sold at 49 N. Richmond St., and 6 High Riggs, Edinburgh.

The Francis Spaight, (I presume the one wrecked in 1846), was also the subject of a 1908 short story by Jack London, which may be found here: The Francis Spaight: A True Tale Retold.

These kinds of incident, though not common, were generally accepted as legitimate in extreme circumstances.  There is an article about the 1821 incident involving the Nantucket whaleship, the "Essex", here;  The Terrible Truth behind the Legend of Moby Dick.  The "Mignonette" incident of 1884, however, in which ailing cabin boy Richard Parker was consumed by his shipmates, resulted in the trial of Captain Dudley and the other two survivors.  Though the public was shocked, it was generally sympathetic to the accused; they were found guilty of murder, and sentenced to death with a recommendation to clemency.  There was an immediate appeal and a re-trial,"using a classic defence which is still quoted in courts today". The seamen were found guilty of manslaughter, and served six months hard labour.

Malcolm