U.S. underworld slang sense of "$5 bill" is 1925, from Yiddish finif "five," from German fünf (from PIE root *penkwe- "five") and thus unrelated. The same word had been used in England in 1868 to mean "five pound note" (earlier finnip, 1839).
1890, from French fin de siècle "end of century," phrase used as an adjective. At the time it meant "modern;" now it means "from the 1890s." "App. first in title of a comedy, Paris fin de siècle, produced at the Gymnase, Feb. 1890" [Weekley]. French siècle "century, age" is from Latin saeculum "age, span of time, generation" (see secular).
No proof is needed of the extreme silliness of the term. Only the brain of a child or of a savage could form the clumsy idea that the century is a kind of living being, born like a beast or a man, passing through all the stages of existence, gradually ageing and declining after blooming childhood, joyous youth, and vigorous maturity, to die with the expiration of the hundredth year, after being afflicted in its last decade with all the infirmities of mournful senility. [Max Nordau, "Degeneration," English translation, 1895]
Proto-Indo-European root meaning "five."
It forms all or part of: cinquain; cinque; cinquecento; cinquefoil; fifteen; fifth; fifty; fin (n.) "five-dollar bill;" finger; fist; five; foist; keno; parcheesi; penta-; pentacle; pentad; Pentateuch; Pentecost; pentagon; pentagram; pentameter; pentathlon; Pentothal; Pompeii; Punjab; punch (n.2) "type of mixed drink;" quinary; quincunx; quinella; quinque-; quinquennial; quint; quintain; quintet; quintile; quintessence; quintillion; quintuple.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit panca, Greek pente, Latin quinque, Old Church Slavonic pęti, Lithuanian penki, Old Welsh pimp, Old English fif, Dutch vijf, Old High German funf.
1520s, "fineness" (obsolete); 1530s, "artifice, delicate stratagem," from French finesse "fineness, subtlety," from Old French fin "subtle, delicate" (see fine (adj.)).
The notion is of "ending" (by satisfying) something that is due (compare Greek telos "end;" plural tele "services due, dues exacted by the state, financial means"). The French senses gradually were brought into English: "ransom" (mid-15c.), "taxation" (late 15c.); the sense of "management of money, science of monetary business" first recorded in English 1770.
"a fin-footed mammal," one having feet like fins or flippers, especially of the group of fin-footed aquatic carnivorous quadruped mammals that includes seals, sea-lions, and walruses, 1842, from Modern Latin Pinnipedia, suborder of aquatic carnivorous mammals (seals and walruses), literally "having feet as fins," from Latin pinna in its secondary sense "fin" (see pin (n.)) + pes, genitive pedis "foot" (from PIE root *ped- "foot").
Modern meaning "exaction of money payment for an offense or dereliction" is via sense of "sum of money paid for exemption from punishment or to compensate for injury" (mid-14c., from the same sense in Anglo-French, late 13c.) and from phrases such as to make fine "make one's peace, settle a matter" (c. 1300). Meaning "sum of money imposed as penalty for some offense" is first recorded 1520s.