The “Cautions To Learners, and Advice to Bathers, by the Late Celebrated Dr. Benj. Franklin” are “a pastiche of pieces of two of the good Doctor’s letters, one to Oliver Neave written some time before 1769 and the other to Jacques Barbeu-Dubourg of March 1773”
These letters were available in published form during the 18th century: that to Neave, published in Franklin’s Experiments and Observations on Electricity. (4th edition, London, 1769), pp. 463-8; and that to Barbeu-Dubourg appeared in an English translation published in the Works of the late Doctor Benjamin Franklin (London, 1793, 1794, 1796, 1799; Dublin, 1793, Dundee, 1796, 1800) (per Edwin Wolf 2nd, Annual Report of the Library Company of Philadelphia for the Year 1980, p.47).
This pastische was first published as such in London by Ann Lemoine in 1798 (ESTC N64777, T216540, and N50903) and was reprinted many times thereafter.
The Princeton copy illustrated here is one such reprint. The date ‘1819’ was assigned by a bibliographer of the literature of swimming, Ralph Thomas, in his Swimming; With Lists of Books Published in English, German, French and Other European Languages , p. 224.
Thomas also notes that this reprint has the ‘objectional interpolation’ – a moniker for nonsense advice thus explained by Thomas: “Unfortunately about 1812 some ignoramus in one of the catchpenny reprints after ‘Then plunge under it with your eyes open’ added ‘which must be kept open before going under, as you cannot open the eyelids for the weight of water above you.’ This nonsense, which at once stamps the writer, and all those who quote it, as ignorant of diving, because it is perfectly easy to open the eyes under water, has been copied from one publication to another, right down to the present day : nobody ever thinking of verifying the passage, but some of the later writers have refuted the idea.” (p. 188)
Full text of this Princeton copy is available at Hathi Trust.