Does the English word "eye" derive from the Arabic word عين 'ayn /ʕajn/ ?
The words sound very similar, and have the same meaning.

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Yes maybe, I'm not sure for this one but I know lot of similar words.
Some words of Arabic origin:

Amber “العنبر”  (anbar)
    Crimson “قرمزي”  (karmazi)
    Elixir “إكسير” (ikseer)
    Carat “قيراط” (kirat)
    Cottom “القطن” (al-kutn)
    Sherbet “شربات” (sharabat)
    Tahini “الطحينة” (tahina)
    Chipher / zero “صفر” (sifr)
    Mask / mascara “المسكرة” (mascara)
    Alcohol “الكحول” (al-kohool)
    Hazard “زهر” (zahr)
    Caliber “قالب” (ka’leb)
    Jar “جرة” (jarra)
    Chemistry “الكيمياء” (al-keemya’a)

and so forth

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check this one
http://www.zompist.com/arabic.html

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Just because two words look or sound alike it doesn't mean they have the same origins, for example, "ojo" and "eye" have the same roots from proto-indo european, and they have nothing in common.

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No it's not derived from Arabic .

English is mostly derived from German or and Latin languages .

http://en.wiktionary.org/wiki/eye#Etymology