section_path: History/Zeno: Achilles and the tortoise
section_name: Zeno: Achilles and the tortoise
paragraph_text: Zeno: Achilles and the tortoise
  
  Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
  
  Achilles races a tortoise, giving the latter a head start.
  
  Etc.
  
  Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
  
  Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
  
  Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1, a + a x + a x 2 + a x 3 + a x 4 + a x 5 + ⋯ = a 1 − x. {\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}
  
  Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him