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also eye tooth, 1570s, so called for its position immediately under or next to the eye.
eyetooth eye·tooth (ī'tōōth')
A canine tooth of the upper jaw.
(Variations: left ball or left nut may replace eyeteeth) To pay a very high price; sacrifice much; give anything: She said she'd give her eyeteeth for the role/ I'd give my left nut to get into NET/ I'd give my left ball for a case like that (entry form 1905+, variants 1940s+)