The above items were not peculiar to Bligh's Bounty. Their use in the Royal Navy at the time are due to a Scottish doctor named James Lind. Here is an explanation from the Wikipedia article on Lind, under the heading, Prevention and cure of scurvy:
"Since antiquity in various parts of the world, and since the 1600s in England, it had been known that citrus fruit had an antiscorbutic effect, when John Woodall (1570 - 1643), an English military surgeon of the British East India Company recommended them but their use wasn't widespread. Although Lind was not the first to suggest citrus fruit as a cure for scurvy, he was the first to study their effect by a systematic experiment in 1747. It ranks as one of the first clinical experiments in the history of medicine.
"Lind thought that scurvy was due to putrefaction of the body which could be prevented by acids; that is why he chose to experiment with dietary supplements of acidic quality. In his experiment he divided twelve scorbutic sailors into six groups. They all received the same diet, and in addition group one was given a quart of cider daily, group two twenty-five drops of elixir of vitriol (sulfuric acid), group three six spoonfuls of vinegar, group four half a pint of seawater, group five received two oranges and one lemon and the last group a spicy paste plus a drink of barley water. The treatment of group five stopped after six days when they ran out of fruit, but by that time one sailor was fit for duty and the other had almost recovered. Apart from that, only group one also showed some effect of its treatment.
"Shortly after this experiment Lind retired from the Navy and at first practised privately as a physician. In 1753 he published A treatise of the scurvy, which was virtually ignored. In 1758 he was appointed chief physician of the Royal Naval Hospital Haslar at Portsmouth. When James Cook went on his first voyage he carried wort (0.1 mg vitamin C per 100 g), sauerkraut (10-15 mg per 100 g) and a syrup of oranges and lemons (the juice contains 40-60 mg of vitamin C per 100 g) as antiscorbutics, but only the results of the trials on wort were published. In 1762 Lind’s Essay on the most effectual means of preserving the health of seamen appeared. In it he recommended growing salad, i.e. watercress (662 mg vitamin C per 100 g) on wet blankets. This was actually put in practice, and in the winter of 1775 the British Army in North America was supplied with mustard and cress seeds. Lind also continued to advocate citrus fruit. But because he — and most other physicians — still believed that their curative effect was due to the acid, it was reasonable to replace them with cheaper acids."
I shall conclude the precepts relating to the preservation of seamen with showing the best means of obviating many inconveniences which attend long voyages and of removing the several causes productive of this mischief.
The following are the experiments.
On the 20th May, 1747, I took twelve patients in the scurvy on board the Salisbury at sea. Their cases were as similar as I could have them. They all in general had putrid gums, the spots and lassitude, with weakness of their knees. They lay together in one place, being a proper apartment for the sick in the fore-hold; and had one diet in common to all, viz., water gruel sweetened with sugar in the morning; fresh mutton broth often times for dinner; at other times puddings, boiled biscuit with sugar etc.; and for supper barley, raisins, rice and currants, sago and wine, or the like. Two of these were ordered each a quart of cyder a day. Two others took twenty five gutts of elixir vitriol three times a day upon an empty stomach, using a gargle strongly acidulated with it for their mouths. Two others took two spoonfuls of vinegar three times a day upon an empty stomach, having their gruels and their other food well acidulated with it, as also the gargle for the mouth. Two of the worst patients, with the tendons in the ham rigid (a symptom none the rest had) were put under a course of sea water. Of this they drank half a pint every day and sometimes more or less as it operated by way of gentle physic. Two others had each two oranges and one lemon given them every day. These they eat with greediness at different times upon an empty stomach. They continued but six days under this course, having consumed the quantity that could be spared. The two remaining patients took the bigness of a nutmeg three times a day of an electuray recommended by an hospital surgeon made of garlic, mustard seed, rad. raphan., balsam of Peru and gum myrrh, using for common drink barley water well acidulated with tamarinds, by a decoction of wich, with the addition of cremor tartar, they were gently purged three or four times during the course.
The consequence was that the most sudden and visible good effects were perceived from the use of the oranges and lemons; one of those who had taken them being at the end of six days fit four duty. The spots were not indeed at that time quite off his body, nor his gums sound; but without any other medicine than a gargarism or elixir of vitriol he became quite healthy before we came into Plymouth, which was on the 16th June. The other was the best recovered of any in his condition, and being now deemed pretty well was appointed nurse to the rest of the sick …
As I shall have occasion elsewhere to take notice of the effects of other medicines in this disease, I shall
here only observe that the result of all my experiments was that oranges and lemons were the most effectual remedies
for this distemper at sea. I am apt to think oranges preferable to lemons, though it was principally oranges which
so speedily and surprisingly recovered Lord Anson's people at the Island of Tinian, of which that noble, brave and
experienced commander was so sensible that before he left the island one man was ordered on shore from each mess
to lay in a stock of them for their future security. … Perhaps one history more may suffice to put this out of doubt.
James Lind: A Treatise of the Scurvy in Three Parts. Containing an inquiry into the Nature, Causes and Cure
of that Disease, together with a Critical and Chronological View of what has been published on the subject.
A. Millar, London, 1753.