Etymology
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furlong (n.)
measure of distance of roughly 660 feet, from Old English furlang, originally the length of a furrow in a common field of 10 acres, from furh "furrow" (see furrow (n.)) + lang "long" (see long (adj.)). The "acre" of the common field being variously measured, the furlong varied but eventually was fixed by custom at 40 rods. Used from 9c. to translate Latin stadium (625 feet), one-eighth of a Roman mile, and so the English word came to be used for "one-eighth of an English mile," though this led to a different measure for the English mile than the Roman one. Furlong being so important in land deed records (where mile hardly figures) it was thought best to redefine the mile rather than the furlong, which was done under Elizabeth I.
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stadium (n.)
late 14c., "a foot race; an ancient measure of length," from Latin stadium "a measure of length; a course for foot-racers" (commonly one-eighth of a Roman mile or a little over 600 English feet; translated in early English Bibles by furlong), from Greek stadion "a measure of length; a race-course, a running track," especially the track at Olympia, which was one stadion in length. The meaning "running track," recorded in English from c. 1600, was extended to mean in modern-day context "large, open oval structure with tiers of seats for viewing sporting events" (1834).

"Originally the distance between successive stations of the shouters and runners employed to estimate distances" [Century Dictionary]. According to Barnhart, the Greek word might literally mean "fixed standard of length" (from stadios "firm, fixed," from PIE root *sta- "to stand"), or it may be from spadion, from span "to draw up, pull," with form influenced by stadios.
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