Revised Jun 12 2021
From Carteret's Voyage, at page 561, in the 1st volume of Dr. Hawkesworths's Voyages in the Southern Hemisphere.
We continued our course westward till the evening of Thursday the 2d of July, when we discovered land to the northward of us. Upon aproaching it the next day, it apeared like a great rock rising out of the sea: it was not more than five miles in circumference, and seemed to be uninhabited; it was however convered with trees, and we saw a small stream of fresh water running down one side of it. I would have landed upon it, but the surf, which at this season broke upon it with great violence, rendered it impossible. I got soundings on the west side of it, at somewhat less than a mile from shore, in twenty five fathom, with a bottom of coral and sand; and it is probable that in fine summer weather, landing here may not only be practicable, but easy. We saw a great number of sea birds hovering about it, at somewhat less than a mile from the shore, and the sea here seemed to have fish. It lies in latitude 20° 2' S., longitude 133° 21' W, and about a thousand leagues to the westward of the continent of America. It is so high that we saw it at the distance of more than fifteen leagues, and it having been discovered by a young gentleman, son to Major Pitcairn of the marines, who was unfortunately lost in the Aurora, we called it PITCAIRN'S ISLAND.
[The longitude was off by 3°, which represented ~190/304 mi/km. This proved fortunate for the mutineers because the island wasn't where everyone thought it was.]