The words Fae/Fay originate from the 12th century in old French. It likely stems from the word Fata meaning "spirits of fate". It was possibly used as a term for Celtic goddesses, before evolving into a word for enchanting or something of enchanting nature. In old English, the term transformed into Fay-erie or even has been seen as "Fayereye" in The Wife of Bath's Tale.
Modern English has since turned this into 'Fairy'
Source: Fairy The Otherworld By Many Names by Morgan Daimler


















