Zeno of Elea, a Greek philosopher of the fifth century B.C., was a follower of Parmenides, who believed that the world of the senses is an illusion, and the universe is literally singular and unchangeable. Zeno supported these ideas by claiming that motion is impossible, and he used four paradoxes to "prove" his claim.
One of these paradoxes involves a race between the great warrior Achilles and a tortoise. Because the tortoise is very slow, he is given a head start. Now, before Achilles catches up to the tortoise, he must first pass a point that the tortoise has already passed. When he reaches that point, though, the tortoise will have moved on to another point, which Achilles must also pass before overtaking him. But again, once Achilles reaches that next point, the tortoise will have moved on again, and so on. Achilles will never catch up to the tortoise!
Another paradox can be illustrated by this exciting race. The tortoise takes his head start and travels distance x before Achilles begins. Before the warrior can reach distance x, he must traverse half of x. But before he can go that far, he must traverse one fourth of x. And before he can go that far, he must first traverse one eighth of x, and so on. At this rate, he will never even get started!
From these paradoxes (and a few others), Zeno drew the conclusion that motion was impossible: an infinite number of finite distances cannot be traversed in a finite amount of time.
However, the invention of calculus by Newton and Leibniz over 1000 years later led to a different conclusion: the sum of an infinite number of infinitesimals can be finite. Therefore, motion is possible, after all.