Etymology
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rummy (n.1)

card game, by 1905, rhum, rhummy, a word of unknown origin, perhaps from the drink, by analogy with whisky poker, or from card-playing terms in German (Rum) or Dutch (roem), which are related to German Ruhm "glory, fame." Gin rummy is attested by 1941.

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rummy (n.2)

 "habitual drunkard," by 1851, from rum (n.) + -y (3); the meaning "opponent of temperance" in U.S. politics is from 1860. 

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gin (n.1)

type of distilled drinking alcohol, 1714, shortening of geneva, altered (by influence of the name of the Swiss city, with which it has no connection) from Dutch genever "gin," literally "juniper" (because the alcohol was flavored with its berries), from Old French genevre "the plant juniper" (12c.), from Vulgar Latin *jeniperus, from Latin juniperus "juniper" (see juniper).

[I]t was not till about 1724 that the passion for gin-drinking appears to have infected the masses of the population, and it spread with the rapidity and the violence of an epidemic. Small as is the place which this fact occupies in English history, it was probably, if we consider all the consequences that have flowed from it, the most momentous in that of the eighteenth century—incomparably more so than any event in the purely political or military annals of the country. [W.E.H. Lecky, "A History of England in the Eighteenth Century," 1878]

Gin and tonic is attested by 1873; gin-sling by 1790; gin-fizz (with lemon juice and aerated water) is from 1878. Gin-mill, U.S. slang for "low-class tavern or saloon where spirits are drunk" (1872) might be a play on the senses from gin (n.2). British gin-palace "gaudily decorated tavern or saloon where spirits are drunk" is from 1831.

The card game gin rummy first attested 1941 (described in "Life" that year as the latest Hollywood fad); OED lists it with the entries for the liquor, but the sense connection seems obscure other than as a play on rummy.

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