What did the men tell Alice to do when they started dancing in Through the Looking-Glass?

In Through the Looking-Glass, the two men, Tweedledum and Tweedledee, tell Alice to shake hands, then started dancing round in a ring with her.

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In chapter 4 of Through the Looking-Glass, Alice encounters Tweedledum and Tweedledee, who are constantly arguing with one another. Tweedledum tells Alice that the first thing one should do on a visit is to say “How d’ye do?” and shake hands, whereupon the two of them, apparently agreeing for once, hold out their hands for her to shake.

Alice is worried that if she shakes the hand of either one of them first, she will offend the feelings of whichever one she neglects. She therefore takes both hands at the same moment. Tweedledum and Tweedledee take this as a sign to dance round in a circle. This seems perfectly natural to Alice at the time, and she even thinks that she hears music coming from the tree under which they are dancing.

As with all the incidents in Through the Looking-Glass, the absurdity of this situation comes not from randomness or even whimsy, but from a strict chain of logical reasoning. No single link in the chain is unreasonable. Alice's motivation for clasping both hands at once is perfectly comprehensible, and once they are in this position, with hands linked, it seems natural to dance round in a ring. Regarded as a complete sequence, however, the events take on the absurdity which always results when logic is applied too strictly to an illogical world.

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