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had overtaken the Tortoise, and had seated himself comfortably on its back.

"So you've got to the end of our race-course?" said the Tortoise. "Even though it does consist of an infinite series of distances? I thought some wiseacre or other had proved that the thing couldn't be done?"

"It can be done," said Achilles. "It has been done! Solvitur ambulando.  You see the distances were constantly diminishing; and so—"

"But if they had been constantly increasing?" the Tortoise interrupted. "How then?"

"Then I shouldn't be here," Achilles modestly replied; "and you would have got several times round the world, by this time!"

"You flatter me—flatten, I mean," said the Tortoise; "for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?"

"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. "Proceed! And speak slowly, please!  Short-hand isn't invented yet!"

"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily. "You admire Euclid?"

"Passionately! So far, at least, as one can admire a treatise that wo'n't be published for some centuries to come!"

"Well, now, let's take a little bit of the argument in that First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them in your note-book.  And in order to refer to them conveniently, let's call them A, B, and Z:—

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"

"Undoubtedly! The youngest child in a High School—as soon as High Schools are invented, which will not be till some two thousand years later—will grant that."

"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"