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As the leaders of the United States and China met in Beijing on Thursday, Xi Jinping had a much older rivalry on his mind.

The Chinese president invoked a warning from the Classical world, when the Greek city-states of Athens and Sparta went to war, saying that the United States and China should beware the “Thucydides Trap” in their own relations.

Mr. Xi cited the concept, popularized in recent decades, as he warned that Beijing and Washington could enter an “extremely dangerous place” if President Trump sought to impede China as it asserted itself over Taiwan.

The trap referred to by Mr. Xi was named for Thucydides, the ancient Athenian general, whose account of the Second Peloponnesian War (431 B.C. to 404 B.C.) is considered one of the first written military histories.

In it, Thucydides argued that the war between Athens and Sparta was driven by the threat posed to an established power by one gaining strength. “The rise of Athens frightened Sparta and forced them into war,” wrote Thucydides. (The precise translation is contested among classicists).

For some scholars, the war — and the explanation offered for it in that ancient passage — presaged nearly every major conflict to follow. The international relations theorist Graham Allison dubbed it the “Thucydides Trap” in the early 2010s.

“The idea is that when an established, great power is met with a rising power, conflict between the two is certainly likely if not inevitable,” said Daniel Sutton, a classicist at the University of Cambridge who studies Thucydides, on Thursday.

In Mr. Xi’s version of the analogy, an emboldened China is the Athens to an American Sparta.

To demonstrate his theory, Professor Allison identified 16 times in history that a rising power threatened to displace a ruling one. According to his tally, which is subjective, 12 of the 16 rivalries ended in a conflict.