One theory about witches taking to the sky on broomsticks involved something called a "flying ointment." The idea comes from a magician, Abramelin, who in the 13th century wrote about witches (presumably) rubbing a blend of herbs and oil or fat on a staff and then riding it. Abramelin also explained that sometimes the women would rub the mixture "under the arms and in other hairy places." The salve might have included the hallucinogenic Mandragora plant, which might have given the women the impression that they were flying on their staffs (via Psychology Today). Perhaps once word of these experiences spread, the idea may have stuck.
A book from the era seems to back up this idea that women accused of being witches rubbed something on a staff-like object. "Irish Witchcraft and Demonology" includes a section (posted at the Internet Archive) about Dame Alice Kyteler, who was accused of sorcery around 1324. When her house was raided, authorities found "a pipe of ointment, wherewith she greased a staffe, upon which she ambled and galloped through thicke and thin."
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