ACHILLES AND THE TORTOISE.

(Vol. ii., p. 154.)

This paradox, whilst one of the oldest on record (being attributed by Aristotle to Zeus Eleates, B.C. 500), is one of the most perplexing, upon first presentation to the mind, that can be selected from the most ample list. Its professed object was to disprove the phenomenon of motion; but its real one, to embarrass an opponent. It has always attracted the attention of logicians; and even to them it has often proved embarrassing enough. The difficulty does not lie in proving that the conclusion is absurd, but in showing where the fallacy lies. From not knowing the precise kind of information required by [Greek: Idiotaes], I am unwilling to trespass on your valuable space by any irrelevant discussion, and confine myself to copying a very judicious note from Dr. Whateley's Logic, 9th edit. p. 373.

"This is one of the sophistical puzzles noticed by Aldrich, but he is not happy in his attempt at a solution. He proposes to remove the difficulty by demonstrating that in a certain given time, Achilles would overtake the tortoise; as if any one had ever doubted that. The very problem proposed, is to surmount the difficulty of a seeming demonstration of a thing palpably impossible; to show that it is palpably impossible, is no solution of the problem.

"I have heard the present example adduced as a proof that the pretensions of logic are futile, since (it was said) the most perfect logical demonstration may lead from true premises to an absurd conclusion. The reverse is the truth; the example before us furnishes a confirmation of the utility of an acquaintance with the syllogistic form, in which form the pretended demonstration in question cannot be exhibited. An attempt to do so will evince the utter want of connection between the premises and the conclusion."

What the Archbishop says is true, and it disposes of the question as one of "Formal Logic:" but yet the form of the sophism is so plausible, that it imposes with equal force on the "common sense" of all those who repose their conclusions upon the operations of that faculty. With them a different procedure is necessary; and I suspect that if any one of the most obstinate advocates of the sufficiency of common sense for the "balancing of evidence" were to attempt the explanation of a hundred fallacies that could be presented to him, he would be compelled to admit that a more powerful and a more accurate machine would be of advantage to him in accomplishing his task. This machine the syllogism supplies.

The discussion of Gregory St. Vincent will be found at pages 101-3. of his Opus Geometricum, Antw., 1647 fol. The principle is the same as that which Aldrich afterwards gave, as above referred to by Dr. Whateley. I can only speak from memory of the discussion of Leibnitz, not having his works at hand; but I am clear in this, that his principle again is the same. [Greek: Idiotaes] is in error, however, in calling St. Vincent's "a geometrical treatment" of it. He indeed uses lines to represent the spaces passed over; and their discussion occurs in a chapter on what is universally (but very absurdly) called "geometrical proportion." It is yet no more geometrical than our school-day problem of the basket and the hundred eggs in Francis Walkinghame. Mere names do not bestow character, however much philosophers as well as legislators may think so. All attempts of the kind have been, and must be, purely numerical.

T.S.D.

Shooter's Hill, August 3.

Achilles and the Tortoise.—Your correspondent will find references in the article "Zeno (of Elea)" in the Penny Cyclopædia. For Gregory St. Vincent's treatment of the problem, see his Quadratara Circuli, Antwerp, 1647, folio, p. 101., or let it alone. I suspect that the second is the better reference. Zeno's paradox is best stated, without either Achilles or tortoise, as follows:—No one can go a mile; for he must go over the first half, then over half the remaining half, then over half the remaining quarter; and so on for ever. Many books of logic, and many of algebra, give the answer to those who cannot find it.