We heard from a listener curious about any connection between the Sufi mystic and poet Rumi and the English verb ruminate. She theorized there might be a link between whirling dervishes and turning something over in one's mind, but after consulting some reference books (and taking a quick whirl through the barnyard) we can report no such relationship exists.
Jalal ad-Din ar-Rumi is celebrated for his lyrical poems that were produced while the 13th century poet was in a state of ecstasy. Rumi's ecstasy—and his outpouring of words—were commonly accompanied by wild dancing. So how did this rapturous poetry recited while dancing come to be quoted and admired more than 700 years later? Because Rumi's words were recorded by his disciple (and successor) Husam ad-Din.
Rumi is honored as the greatest poet in the Persian language and as the man who inspired the order of whirling dervishes, but his words did not give rise to the verb ruminate. That 16th-century coinage may have kin in the Sanskrit term meaning "act of chewing the cud," but its more direct Latin ancestor named the gullet, "the compartment from which food is regurgitated for rumination or rechewing."

The same Latin term that gave our language ruminant, the name for a cud-chewing mammal, also gave us ruminate the verb, with its prosaic senses "to chew repeatedly for an extended period"; or "to chew a cud", and its more pensive meanings "to muse upon"; "contemplate over and over"; and "ponder."