Paul Kaufman, Libraries and their Users: Collected Papers in Library History (1969), 222, reports a "Printed label of Lucock's library, with written date 1782." Robin Alston, Circulating Libraries, 1660–1800 offers a different spelling: "Lewcock's Circulating Library." He also names the proprietor—S. Lucock—and adds a question mark to the date.
Lucock's circulating library was in Sheerness, a naval town on the Isle of Sheppey in north Kent. After the Sheerness fort was destroyed during the Second Anglo-Dutch War (1667) the Secretary of the Admiralty, Samuel Pepys, ordered the construction of a naval dockyard at Sheerness as an extension to that at Chatham. The first dry-dock was not completed until 1708. Using materials they were allowed to take from the yard, dockyard construction workers built the first houses in Sheerness. The grey-blue naval paint they used on the exteriors led to their homes becoming known as the Blue Houses. This was eventually corrupted to Blue Town, the modern name of northwest area of Sheerness. This is why the location of Lewcock's Circulating Library is, as you can see, "Blue Houses, Sheerness"
As Wikipedia explains, Blue Town was not a particularly healthy place to live. The town was a small self-contained community built on wet ground reclaimed from marshes. It was very confined, a dense triangle of houses and alleyways compressed between the dockyard wall and Well Marsh; it was also prone to both flood and fire. At one point separated from Sheerness fort by a moat and drawbridge, the area was enclosed by an earthwork bastioned trace at the end of the 18th century amid growing fears of a French invasion (pictures here). In such an unhealthy and dangereous place it is quite likely that Lewcock's copy of George Cheyne, An Essay on Health and Long Life, 8th ed. (1734), was much in demand!
In fact, this book seems to have had a very busy life before it even reached Lewcock's. There are four ownership inscriptions, two dated in the same year. These are, in the order in which they appear: "James Mullett's Book"; "Eleanor Boxer, her book"; "Elizabeth Marsh Jones, Her Book June 22/52"; "John Derby, 1752." How many of these people were denizens of Blue Town only a local historian could tell.
It would be good to know how much of the last two centuries this book spent on the shelves at Lewcock's before it was carried to America. At some point it has been partially re-backed but it is in remarkably good condition for a Circulating Library book, and it was remarkably cheap (USD36) for a book which is so interesting and—for my research—so useful. (But not rare, there are thirty two copies listed under ESTC: t127366, including one at Monash.)
[UPDATE: 2 July 2016: After all my pictures disappeared (again) I decided to give up on external hosts for large versions (1000px) of my image files and, for now on, will stick with the smaller images (500px), which Blogger is prepared to host.]