while Achilles is running one, the tortoise is running one-tenth of a yard; and so on forever. This sophism has been considered insoluble even by Dr. Thomas Brown, since it actually leads to an absurd conclusion by a sound argument. The fallacy lies in the concealed assumption that what is infinitely divisible is also infinite.

But a paradox which looks like it at first sight is absolutely irrefragable. A man who owes a dollar starts by paying half a dollar, and every day thereafter pays one-half of the balance due,-twenty-five cents the third day, twelve and a half the fourth day, and so on. Suppose him to be furnished with counters of infinitesimal value, so as to be able to pay fractions of a cent when the balance left is less than a cent, he would never pay the full amount of his debt, even though, Tithonus-like, he were endued with immortality; there would always be some outstanding fraction of a cent to his debt.

The famous "Syllogismus Crocodilus" is not Zeno's, but dates from an unknown antiquity. A crocodile seizes an infant playing on the banks of a river. The mother rushes to its assistance. The crocodile, an intelligent animal, promises to restore the child if she will tell him truly what will happen to it. "You will never restore it," cries the mother, somewhat rashly. The crocodile astutely rises to the occasion. "If you have spoken truly," he says, "I cannot restore the child without destroying the truth of your assertion. If you have spoken falsely, I cannot restore the child, because you have not fulfilled the agreement; therefore I cannot restore it whether you have spoken truly or falsely.”

But the mother, too, exhibits logical powers that are rare indeed in her

sex.

"If I have spoken truly," she says, "you must restore the child by virtue of your agreement. If I have spoken falsely, that can only be when you have restored the child. Therefore, whether I have spoken truly or falsely, the child must be restored."

Mother and crocodile may still be arguing out that question. History at It is one of the unsolved problems, like that

least is silent as to the issue.

of "The Lady or the Tiger?"

Another paradox equally astute is closely parallel. Young Euathlus received lessons in rhetoric from Protagoras, who was to receive a certain fee if his client won his first cause. Euathlus, however, being lazy, neglected to accept any cause. Then Protagoras brought suit. Euathlus defended himself, and it was consequently his first cause. The master argues thus: "If I be successful in this cause, O Euathlus, you will be compelled to pay by vir tue of the sentence of the court; but should I be unsuccessful, you will then have to pay me in fulfilment of your contract." "Nay," replies the apt pupil, "if I be successful, O master, I shall be free by the sentence of the court; and if I be unsuccessful, I shall be free by virtue of the contract."

The judges were completely staggered by the convincing logic on each side, and postponed the judgment sine die.

A similar dilemma puzzled Aristotle half out of his wits, and drove Philetas, the celebrated grammarian and poet of Cos, into an untimely grave. It is known as "The Liar," and is stated as follows: "If you say, ‘Í lie,' and in so saying tell the truth, you lie; but if you say, 'I lie,' and in so saying tell a lie, you tell the truth."

The sophism of The Liar reappears in another form in the argument of the lying Cretians. St. Paul says (Titus i. 12, 13), “One of themselves, even a prophet of their own, said, The Cretians are always liars, evil beasts, slow bellies. This witness is true." Now, this witness cannot be true: the Cretians being always liars, the prophet, as a Cretian, must be a liar, and lied when he said they were always liars. Consequently, the Cretians are not

always liars. And yet, again, the witness may be true.

For if the Cretians

are not always liars, then the Cretian prophet was not always a liar, and told the truth when he said that they were always liars.

And are not these sophisms identical in essence with the famous legal case of the Bridge, which was decided by His Excellency Sancho Panza, when governor of the island of Barataria ?

Here are some more paradoxes of Attic origin:

"The Veiled Man."-There is a man standing before you with his face and form entirely hidden by a veil. Do you know who this man is? No. Do you know who your father is? You say you do. But this cannot be so, for the veiled man happens to be your father, and you just said you did not know who he was.

"The Horns."-What you have not got rid of you still have. You agree to that. But you have not got rid of horns: therefore you have horns.

"The Bald Man."-You say that you call a man bald when he has only a few hairs. What is the difference between few and many? Would ten be a few and eleven not? Where shall the line be drawn? You say that there are such things as few and many, and that there is a difference between them. Define the difference, then. Such an examination makes it plain that the difference between few and many is not anything in particular, which is as much as to say that it has no particular existence.

In one of Plato's dialogues, Euthydemus, a skilful hand at this sort of work, tangles up a young man named Ktesippus in this fashion: "Have you a dog?"

"Yes."

"Is he yours?"

"Yes."

Is the dog their father, then?"

"To my certain knowledge."

"Then the dog is a father and is yours, therefore he is your father."

This unexpected revelation fairly takes away Ktesippus's breath, and before he can recover Euthydemus goes on:

"Do you ever thrash that dog?"

"Yes."

"Then you are in the habit of thrashing your own father!"

But as the talk goes on, Ktesippus gets even with Euthydemus. For the purpose of his argument he wants to make Euthydemus confess that men like to have gold.

"No," says Euthydemus, "you can't lay that down as a general principle. Men don't always like to have gold; they only want it under certain special conditions. No one would want to have gold in his skull, for instance.'

"Oh, yes," answers Ktesippus. "You know that the Scythians use skulls for drinking-cups, and inlay them with gold. Now, these are their skulls in just the same way that you said the dog was my father. So the Scythians want to have gold in their skulls."

Euthydemus has no answer ready for this, and Ktesippus carries off the honors.

A modern dilemma of a somewhat similar sort proves that the much-used maxim, “All rules have their exception," is self-contradictory, for if all rules have exceptions, this rule must have its exceptions. Therefore the proverb asserts in one and the same breath that all rules have exceptions and that some rules do not,- -a clear case of proverbial suicide.

Every school-boy, to use Macaulayese, is familiar with the good old paradox

which proves that one cat has three tails: No cat has two tails; one cat has one tail more than no cat; consequently one cat has three tails.

A famous old problem opens out a fertile but somewhat hopeless subject of inquiry: "If an irresistible force strikes an immovable body, what will be

the result?"

There are a number of more or less familiar problems which are not catchquestions, and which at first sight seem extremely simple, yet require considerable ingenuity to arrive at a correct result. And the correct result, when arrived at, proves to be the exact opposite of the simple prima facie answer that had sprung immediately to mind.

Can a ship sail faster than the wind? Undoubtedly. Ice-boats, especially, which meet with little or no frictional resistance, can, with a very light wind, be sent ahead of a fast express-train,—an experiment frequently seen in action on the Hudson River. But even an ordinary yacht can be propelled twelve or fifteen knots an hour by a breeze blowing only ten knots an hour.

Of course this cannot happen when the ship sails straight before the wind. In that case it must travel more slowly than the wind, on account of the resistance made by the water. "But," you may say, "that is the only way to get the full effect of the wind. If the ship sails at an angle with the wind, the wind must act with less effect, and the ship will sail more slowly."

Plausible. Yet every yachtsman and every mathematician knows it is not

true.

Suppose we illustrate. You put a ball on a billiard-table, and, holding the cue lengthwise from side to side of the table, push the ball across the cloth. Here, in a rough way, the ball represents the ship, the cue the wind, only, as there is no waste of energy, the ball travels at the same rate as the cue; evidently it cannot go any faster. Now, let us suppose that a groove is cut diag. onally across the table, from one corner-pocket to the other, and that the ball rolls in the groove. Propelled in the same way as before, the ball will now travel along the groove (and along the cue) in the same time as the cue takes to move across the table. The groove is much longer than the width of the table,-double as long, in fact. The ball, therefore, travels much faster than the cue which impels it, since it covers double the distance in the same time. Just so does the tacking ship sail faster than the wind.

Both

When a wheel is in motion, does the top move faster than the bottom? Nine people out of ten would cry "Nonsense !" at the mere question. the top and bottom of the wheel must of necessity, it would seem, be moving forward at one and the same rate,-i.e., the speed at which the carriage is travelling. Not so, however, as a little reflection would convince you. The top is moving in the direction of the wheel's motion of translation, while the bottom is moving in opposition to this motion. In other words, the top is moving forward in the direction in which the carriage is progressing, while the bottom is moving backward, or in an opposite direction.

That is why an instantaneous photograph of a carriage in motion shows the upper part of the wheel a confused blur, while the spokes in the lower part are distinctly visible.

You want more proof? Very well; try a practical experiment. Take a wheel, or, if none is convenient, a silver dollar, which you are sure to have about your person. Mark points at the top and bottom, as A and B. Make a mark at the starting-point, directly beneath A and B, upon whatever surface the wheel or dollar is rolled. Roll the wheel forward a quarter revolution, which brings A and B upon the dividing line between the upper and lower halves of the wheel. It will be seen that A moves upon a radius equal to the diameter of the circle, and, by actual measurement, that A has moved a much greater distance and described a greater curve than B.

Consequently it must have moved faster.

To clinch the matter, make another quarter revolution, or, in other words, a half revolution entire. A and B have now changed places. B is at the top of the wheel, A at the bottom. It will be found that in the second quarter revolution B has travelled the greater distance and described the greater

curve.

The following proposition is left for the reader to think about:

If there are more people in the world than any one person has hairs upon his head, then there must exist at least two persons who possess identically the same number of hairs, to a hair.

This same proposition may be applied to the faces of human beings in the world. If the number of perceptible differences between two faces be not greater than the total number of the human race, then there must exist at least two persons who are to all appearances exactly alike. When it is considered that there are about one billion five hundred million persons in the world and that the human countenance does not vary, except within comparatively narrow limits, the truth of the proposition becomes obvious, without applying the logical reasoning of it.

You remember the egg-problem: "If a hen and a half lay an egg and a half in a day and a half, how many eggs will six hens lay in seven days?" The proposition is really as easy as the familiar one which every school-boy has puzzled over the first time he heard it, and wondered at himself ever after that it was not absolutely self-evident: "If a herring and a half cost a cent and a half, how much will six herrings cost?"-the answer to which is six cents, of course, for if a herring and a half cost a cent and a half, one herring will cost one cent.

Now, if the egg-problem were stated in this way, "If a hen and a half lay an egg and a half in thirty-six hours, how many eggs will six hens lay in seven days?" probably every one would see that the proposition can be simplified by saying that one hen lays one egg in thirty-six hours, and then it becomes a mere question of rudimentary mathematics to ascertain that six hens will lay twenty-eight eggs in seven days.

But many people are bewildered by the third fraction, and insist that, if it requires a day and a half for a hen and a half to lay an egg and a half, one hen will lay one egg in one day, and six hens will lay six eggs in one day; hence in seven days six hens will lay forty-two eggs. They do not see that although the first two fractions balance each other, and may be both cancelled, the last must remain as the measurement of time in which it takes either one hen or one hen and a half to perform a given feat.

Many ingenious casuists insist on twenty-four as the right answer, arguing that, as hens are never known to lay two-thirds of an egg, the six hens, having laid twenty-four eggs at the end of the six days, must patiently wait thirty-six hours before laying again. This is mere quibbling. The object of the problem is to find out how many eggs may be expected, week by week, from six hens under given conditions. To the mathematical mind there is no absurdity in saying that each hen lays two-thirds of an egg per day, and therefore six hens lay four eggs per day.

Of course, a mere humorist, who has no mathematical instincts, might assert that the entire proposition, as originally stated, is an absurdity, since half a hen cannot lay an egg, or any fractional part thereof, unassisted by the other half. The egg end of a hen only, he might assert, is constructed for that purpose. The other end merely announces the result of the hen's efforts and takes in the materials from which the egg is formed. A hen doing business with one-half of itself and trying to run a branch establishment with the other half would be a dismal failure.

But mathematics was not made for humorists.

The above are illustrations of paradoxes in which it requires a certain in genuity to arrive at the correct answer. Here is a paradox of another sort, in which the answer given is an obvious and barefaced fallacy, and yet in which it requires considerable ingenuity to expose the falsehood:

A Dublin chambermaid is said to have put a round dozen of travellers into eleven bedrooms, and yet to have given each a separate bedroom. Here is a diagram of the eleven bedrooms:

"Now," said the quick-witted Irish girl, "if two of you gentlemen will go into No. 1 bedroom, I'll find a spare room for one of you as soon as I've shown the others to their rooms."

So, having put two gentlemen into No. 1, she put the third in No. 2, the fourth in No. 3, the fifth in No. 4, the sixth in No. 5, the seventh in No. 6, the eighth in No. 7, the ninth in No. 8, the tenth in No. 9, the eleventh in No. 10. Then, going back to No. 1, where you will remember that she left the twelfth gentleman along with the first, she said,

"I have now accommodated all the rest, and have still a room to spare; so, if one of you will step into Room 11 you will find it empty."

Thus the twelfth man got his bedroom.

Now, every one sees at a glance that there is a flaw somewhere; but not every one recognizes immediately that the flaw lies in rolling two single gentlemen (No. 2 and No. 12) into one, like the hero of Peter Pindar's poem. Here is another semi-mathematical puzzle:

"A train starts daily from San Francisco to New York, and one daily from New York to San Francisco, the journey lasting seven days. How many trains will a traveller meet in journeying from San Francisco to New York?"

The same nine people out of our mythical ten, unless they have been warned by their former lapses, will answer off-hand, "Seven." But they overlook the fact that every day during the journey a fresh train is starting from the other end, while there are seven on the way to begin with. The traveller will therefore meet, not seven trains, but fourteen.

Here is a question which was seriously and gravely considered in the late R. A. Proctor's ponderous paper, Knowledge:

"A man walks round a pole on the top of which is a monkey. As the man moves, the monkey turns round on the top of the pole so as still to keep face to face with the man. Query: When the man has gone round the pole, has he or has he not gone round the monkey?"

Some correspondents held that the man had not gone round the monkey, since he had never been behind it. But Knowledge decided that the man had gone round the monkey in going round the pole.

Parallel. None but himself can be his parallel, a persistent misquotation of a famous line in "The Double Falsehood, or Distrest Lovers." Act iii., Sc. 1. The line and its context run as follows:

O my good Friend, methinks I am too patient.
Is there a treachery like this in baseness
Recorded anywhere? It is the deepest:
None but itself can be its parallel:
And from a friend professed!

The play is taken from a novel in "Don Quixote," and according to tra dition was written by Shakespeare and presented to one of his natural