ACHILLES

, a name given by the schools to the principal argument alleged by each sect of philosophers in behalf of their system. In this sense we say this is his Achilles; that is, his master-proof: alluding to the strength and importance of the hero Achilles among the Greeks.

Zeno's argument against motion is peculiarly termed Achilles. That philosopher made a comparison between the swiftness of Achilles, and the slowness of a tortoise, pretending that a very swift animal could never overtake a slow one that was before it, and that therefore there is no such thing as motion: for, said he, if the tortoise were one mile before Achilles, and the motion of Achilles 100 times swifter than that of the tortoise, yet he would never overtake it; and for this reason, namely, that while Achilles runs over the mile, the tortoise will creep over one hundredth part of a mile, and will be so much the foremost; again while Achilles runs over this 1/100th part, the tortoise will creep over the 100th part of that 1/100th part, and will still be this last part the foremost; and so on continually, according to an infinite series of 100th parts: from which he concluded that the swifter could never overtake the slower in any finite time, but that they must go on ap- | proaching to infinity. But this sophism lay in their considering as an infinite time, the sum of the infinite series of small times in which Achilles could run over the infinite series of spaces, 1 + 1/100 + 1/10000 + 1/1000000 &c, not knowing that the sum of this infinite series is equal to the quantity 1 1/99 of a mile, and that therefore Achilles will overtake the tortoise when the latter has crawled over 1/99th of a mile.