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Publishing history:
v1.0: 01/10/24
feòirling f. [ˈfjɔːrliŋʲɡ̊ʲ], [ˈfjɔːɭiŋʲɡ̊ʲ], gen. idem, ‘farthing; farthingland’ is derived by Henderson (1910, 132–33) from ON fjórðungr m. (leg. fjórðung acc.) ‘a fourth part’. MacBain (1906, 351; 1907, 77) takes the same view, 
So also McDonald (2009, 353).
Pers. comm. Arne Kruse. (ON jó for the most part yields ý via i-mutation (Iversen 1973, 19).)
OEng. fēorþling yields EG feóirling regularly, hence SG feòirling, Ir. feoirling (Dinneen 1947; Ó Dónaill 1977) and, with a ~ o alternation, Mx farling (Y Kelly 1866). However, final -ing has a number of variant reflexes. While O’Begly’s 1732 English-Irish dictionary cites feoirling, Lhuyd (1707) already lists feoirlinn, and, while Kneen (1978) gives Mx [farl.ing], Broderick (1984 II, 158) records [faːrl´ən], [faːrlən]. SG feòirling (Shaw 1780: feoirling) yields feòirlig (cf. SG (Lewis) farsaig from farsaing ‘wide’, tarsaig from tarsaing ‘across’ (Cox 2022, 711–14)) and feòirlinn (MacDomhnuill 1741, 189: feoirlin), either via back-formation on the analogy, say, of tarsaing from tarsainn, or on the analogy of SG sgillinn ‘penny’ 
SG sgillinn ‘1 English penny; 12 Scots pence’, occasionally sgillig (Oftedal 1956, 356, s.v. sgillinn: Lewis), occasionally sgilling (Faclan bhon t-Sluagh˄: Cape Breton); cf. EG scilling, scillic ‘shilling’ < OEng. scilling (eDIL˄). (SG tastan (formerly also tasdan) ‘shilling’ goes back to Eng. testoon, first applied in England to the shilling of Henry VII (1543–), and MScots testan(e), first applied in Scotland to a silver coin of Mary (1553–), both of which go back to French and Italian usage for coins bearing a portrait or head (It. testa ‘head’) (OED˄; DOST˄). SG tastan appears in a late 17th-century song by Roderick Morison (An Clàrsair Dall, c.1656–1713/14; in Matheson 1970, 68, line 956). The same song refers to the (SG) ducatùn (ibid., line 952), from MScots ducatoon (ultimately from It. ducatone (DOST˄)), a large silver coin whose presence in Scotland reflects the increased continental element in Scottish coinage during the last quarter of the 17th century (Stevenson and Porteous 1972, 136).)
Cf. also SG (Perthshire) bòrlaig ~ bòrlainn ‘a strip of arable land’ (also bòrlum (Dwelly 1911)), from MScots bordland ‘?land providing supplies for the lord’s table’ (DOST˄) (Robertson in King 2019, 280, 302–03, 306, 317, 321; MacKenzie 1931, 203, 214–15). Further on the development of historical -ng in Irish, Scottish Gaelic and Manx, see O’Rahilly 1976, 184, 270.
Y Kelly (1866, s.v. Mx farling ‘farthing’) notes Ir. fardain, but this is probably from O’Reilly’s Irish dictionary (1817: fardin; 1864: idem ‘vulg[ar/arly]’), where it may have been adopted from Shaw’s 1780 Scottish Gaelic dictionary (fardin). OEng. fēorþling yields MScots farding (DOST˄), modern Scots farden (SND˄), which in turn yields a number of reflexes in Scottish Gaelic: (1) fàrdan and (2) fàirdean (AFB˄, s.vv.), which reflect dialectal realisations of medial rd (cf. SGDS Items 55 and 56); so also (3) fàrtainn, with its alternative ending based on feòirlinn, as in (4) àrdainn, with initial f- lost via back-formation; 
From use of the radical and/or dative feminine article: fàrdainn f. > an fhàrdainn (in which fh = zero) > an àrdainn.