Relating to or characteristic of Demosthenes, an ancient Greek orator, or his style of powerful, persuasive speeches.
From Demosthenes, the famous Athenian statesman and orator (384-322 BCE) + -ean (relating to). Demosthenes was known for overcoming a speech impediment to become the greatest orator of ancient Greece.
Demosthenes practiced his speeches while running along beaches and speaking with pebbles in his mouth to improve his delivery—so if someone's being 'demosthenean,' they're using that same kind of powerful, hard-hitting oratory to convince people.
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