Green’s Dictionary of Slang

lesbian sexual intercourse, often as game at/of flats.

1652		Mercurius Democritus 22-29 Dec. 298: A Tryal of Skill Betwixt Jane F— at a Venture [...] and Jone Crab [...] The Names of the Weapons Tippett and Biggin. Flatts. Tooth and Naile .
1655		Mercurius Fumigosus 38 14-–8 Feb. 303: They walk out hand in hand like two disconsolate Virgins to seek Mandrakes to help them make perfect what their lost Sweethearts have left behinde, their concupiscense being so predominant in the House of Venus, that being playing a game at Flatts upon a bed, a young man hearing the bed tell tales, steping softly to the door, discovered the Jogg, and so returned, myuch pittying the extremities poor female mortalls are driven unto by the unkindness of men.
c.1663		‘On the Ladies of the Court’ in J.H. Wilson Court Satires of the Restoration (1976) 3: Mayne and Steward played at flats.
1663		in J.H. Wilson Court Satires of the Restoration (1976) 3: Strangely pleasant were their Chatts, / When Mayne and Steward play’d at flatts [ [...] / Till Charles came there, / And with his ware / Taught how their father got them.
c.1680		‘Queries from Garraways Coffee House’ Harleian Mss. 7315.89: Whether Mall Maynard be best pleasd with the Duke of Monmouth, or to play att Flatts with Lady Betty Howard?
1691		‘Ladys Complaint to Venus’ in Lansdowne Ms. 852.86: You are to blame And have got a new Game Call’d Flatts, with a swinging Clytoris.
1698–9		in Bodleian MS Rawl 159: A New Game / Call’d Flats witha Swinging Clitoris.
1709		N. Ward Compleat and Humorous Account of Remarkable Clubs (1756) 44: The Ladies know, That Flat Things always love long Snouts.
1734		Pretty Doings in a Protestant Nation 24: Not content with our Sex, [Sappho] begins Amours with her own and teaches the Female World a new sort of Sin, call’d the Flats.
1740		Williams in Works I 131: Had I as many lives as twenty cats, I’d give them all for one dear game at [flats].
1749		Satan’s Harvest Home 18: [Sappho] teaches the Female World a new Sort of Sin, call’d the Flats, that was follow’d not only in Lucian’s Time, but is practis’d frequently in Turkey, as well as at Twickenham at this Day.
c.1786		Grose Classical Dict. Vulgar Tongue ms notes n.p.: Game of Flats [no definition offered].
a.1834		in G. Witt Indian Drawings Scrapbook (British Museum Anthropology Library) (1865) n.p.: Chuptee-Bazee ‘Congressus amorosus duarum mulierum’ [i.e. ‘sexual congress between two women’] from Chipee, flat, and Barzee play.
[1894		Mirror of Life 20 Jan. 7/4: Side by side they cooing sat / And stroked each other's curls; / But stupid seemed their love, and flat, / Because they both were girls].