.. Ah, here it is! I think we do know the sweet Roman hand.' Here's the Proposition, as large as life, and proved by I. 19. Now, infidel, I have thee on the hip!' You shall have such a sweet thing to do in vivávoce, my very dear friend! You shall have the two Propositions together, and take them in any order you like. It's my profound conviction that you don't know how to prove either of them without the other. They'll have to introduce each other, like Messrs. Pyke and Pluck. But what fearful confusion the whole subject is getting into! (Knocking heard.) Come in!

Enter RHADAMANTHUS.

Rhad. I say! Are we bound to mark an answer that's a clear logical fallacy?

Min. Of course you are—with that peculiar mark which cricketers call 'a duck's egg,' and thermometers 'zero.' Rhad. Well, just listen to this proof of I. 29.

Reads,

'Let EF meet the two parallel Lines AB, CD, in the points GH. The alternate angles AGH, GHD, shall be equal.

For AGH and EGB are equal because vertically opposite, and EGB is also equal to GHD (Definition 9); therefore AGH is equal to GHD; but these are alternate angles.'

Did you ever hear anything like that for calm assumption?

Min. What does the miscreant mean by Definition 9'? Rhad. Oh, that's the grandest of all! You must listen to that bit too. There's a reference at the foot of the page to 'Cooley.' So I hunted up Mr. Cooley among the heaps of Geometries they've sent me-(by the way, I wonder if they've sent you the full lot? Forty-five were left in my rooms to-day, and ten of them I'd never even heard of till to-day!)—well, as I was saying, I looked up Cooley, and here's the Definition.

Reads.

'Right Lines are said to be parallel when they are equally and similarly inclined to the same right Line, or make equal angles with it towards the same side.'

Min. That is very soothing. So far as I can make it out, Mr. Cooley quietly assumes that a Pair of Lines, which make equal angles with one Line, do so with all Lines. He might just as well say that a young lady, who was inclined to one young man, was 'equally and similarly inclined' to all young men!

Rhad. She might make equal angling' with them all, anyhow. But, seriously, what are we to do with Cooley? Min. (thoughtfully) Well, if we had him in the Schools, I think we should pluck him.

Rhad. But as to this answer?

Min. Oh, give it full marks! What have we to do with logic, or truth, or falsehood, or right, or wrong? 'We are but markers of a larger growth'-only that we have to mark foul strokes, which a respectable billiardmarker doesn't do, as a general rule!

Rhad. There's one thing more I want you to look at. Here's a man who puts Wilson' at the top of his paper, and proves Euc. I. 32 from first principles, it seems to me, without using any other Theorem at all.

Min. The thing sounds impossible.

Rhad. So I should have said. Here's the proof.

'Slide / DBA along BF into position GAF, GA having same direction as DC (Ax. 9); similarly slide / BCE along AE into position GAC. Then the ext. Zs=CAF, FAG, GAC one revolution=two straight Zs. But the ext. and int. s=3 straight Zs. Therefore the int. <s= one straight Q. E. D.'

=

2 right angles.

I'm not well up in Wilson': but surely he doesn't beg the whole question of Parallels in one axiom like this!

Min. Well, no. There's a Theorem and a Corollary. But this is a sharp man: he has seen that the Axiom does just

as well by itself. Did you ever see one of those conjurers bring a globe of live fish out of a pocket-handkerchief? That's the kind of thing we have in Modern Geometry. A man stands before you with nothing but an Axiom in his hands. He rolls up his sleeves. 'Observe, gentlemen, I have nothing concealed. There is no deception!' And the next moment you have a complete Theorem, Q. E. D. and all!

Rhad. Well, so far as I can see, the proof's worth nothing. What am I to mark it?

Min. Full marks: we must accept it. Why, my good fellow, I'm getting into that state of mind, I'm ready to mark any thing and any body. If the Ghost in Hamlet came up this minute and said 'Mark me!' I should say 'I will! Hand in your papers!'

Rhad. Ah, it's all very well to chaff, but it's enough to drive a man wild, to have to mark all this rubbish! Well, good night! I must get back to my work. [Exit. Min. (indistinctly) I'll just take forty winks, and

(Snores.)

ACT I

SCENE II.

Οὐκ ἀγαθὸν πολυκοιρανίη· εἷς κοίρανες ἔστω,
Εἷς βασιλεὺς.

[MINOS sleeping: to him enter the Phantasm of EUCLID. MINOS opens his eyes and regards him with a blank and stony gaze, without betraying the slightest surprise or even interest.]

§ I. A priori reasons for retaining

Euclid's Manual.

Euc. Now what is it you really require in a Manual of Geometry?

Min. Excuse me, but-with all respect to a shade whose name I have reverenced from early boyhood-is not that rather an abrupt way of starting a conversation? Remember, we are twenty centuries apart in history, and consequently have never had a personal interview till now. Surely a few preliminary remarks—