Luckily, the answer to this conundrum was provided by the philosopher Zeno in the fifth century b.c. Achilles, a man of action and urgency, trained to measure his exertions to the tenth of a second, should be able to quickly dispense with a twenty-yard dash. But in order to advance a yard, the hero must first advance eighteen inches; and in order to advance eighteen inches, he must first advance nine; but to advance nine, he must first advance four and a half, and so on. Thus, on his way to completing the twenty-yard dash, Achilles must traverse an infinite number of lengths—which, by
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