`Just so,' said Uncle Joe. `Then the Hypothetical "If Allen is out Brown is out" is always in force, isn't it?"

`I suppose so,' said Uncle Jim. (He seemed to be getting a little nervous himself now.)

`Then, if Carr is out, we have two Hypotheticals, "if Allen is out Brown is in", 'and "if Allen is out Brown is out", in force at once. And two incompatible Hypotheticals, mark you! They can't possibly be true together!'

`Can't they?' said Uncle Jim.

`How can they?' said Uncle Joe. `How can one and the same protasis prove two contradictory apodoses? You grant that the two apodoses, "Brown is in" and "Brown is out", are contradictory, I suppose?'

`Yes, I grant that,' said Uncle Jim.

`Then I may sum up,' said Uncle Joe. `If Carr is out, these two Hypotheticals are true together. And we know that they cannot be true together. Which is absurd. Therefore Carr cannot be out. There's a nice Reductio ad Absurdum for you!'

Uncle Jim looked thoroughly puzzled; but after a bit he plucked up courage, and began again. `I don't feel at all clear about that incompatibility. Why shouldn't those two Hypotheticals be true together? It seems to me that would simply prove "Allen is in". Of course, it's clear that the apodoses of those two Hypotheticals are incompatible--"Brown is in" and "Brown is out". But why shouldn't we put it like this? If Allen is out Brown is out. If Carr and Allen are both out, Brown is in. Which is absurd. Therefore Carr and Allen can't be both of them out. But, so long as Allen is in, I don't see what's to hinder Carr from going out.'

`My dear, but most illogical brother!' said Uncle Joe. (Whenever Uncle Joe begins to "dear" you, you may make pretty sure he's got you in a cleft stick!) `Don't you see that you are wrongly dividing the protasis and the apodosis of that Hypothetical? Its protasis is simply "Carr is out"; and its apodosis is a sort of sub-Hypothetical, "If Allen is out, Brown is in". And a most absurd apodosis it is, being hopelessly incompatible with that other Hypothetical, that we know is always true, "If Allen is out, Brown is out". And it's simply the assumption "Carr is out" that has caused this absurdity. So there's only one possible conclusion--Carr is in!'

HALF of the world, or nearly so, is always in the light of the sun: as the world turns round, this hemisphere of light shifts round too, and passes over each part of it in succession.

Supposing on Tuesday, it is morning at London; in another hour it would be Tuesday morning at the west of England; if the whole world were land we might go on tracing¹ Tuesday morning, Tuesday morning all the way round, till in twenty-four hours we get to London again. But we know that at London twenty- four hours after Tuesday morning it is Wednesday morning. Where, then, in its passage round the earth, does the day change its name? Where does it lose its identity?

Practically there is no difficulty in it, because a great part of the journey is over water, and what it does out at sea no one can tell: and besides there are so many different languages that it would be hopeless to attempt to trace the name of any one day all the year round. But is the case inconceivable that the same land and the same language should continue all round the world? I cannot see that it is: in that case either¹ there would be no distinction at all between each successive day, and so week, month, etc., so that we should have to say, `The Battle of Waterloo happened to-day, about two million hours ago', or some line would have to be fixed where the change should take place, so that the inhabitants of one house would wake and say, `Heigh-ho,² Tuesday morning!' and the inhabitants of the next (over the line), a few miles to the west would wake a few minutes afterwards and say, `Heigh-ho! Wednesday