Page:Carroll - Euclid and His Modern Rivals.djvu/100

62 a nugatory principle.' Now let us see, mein Herr, whether Mr. Cooley has done anything worthy of the writer of such 'brave 'orts' (as Shakespeare has it): and first let me ask how you define Parallel Lines.

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 to say, if we see a Pair of Lines cut by a certain transversal, and are told that they make equal angles with it, we say 'these Lines are parallel'; and conversely, if we are told that a Pair of Lines are parallel, we say 'then there is a transversal, somewhere, which makes equal angles with them'?

Nie. Surely, surely.

Min. But we have no means of finding it? We have no right to draw a transversal at random and say 'this is the one which makes equal angles with the Pair'?

Nie. Ahem! Ahem! Ahem!

Min. You seem to have a bad cough.

Nie. Let us go to the next subject.

Min. Not till you have answered my question. Have we any means of finding the particular transversal which makes the equal angles?

Nie. I am sorry for my client, but, since you are so exigeant, I fear I must confess that we have no means of finding it.

Min. Now for your proof of Euc. I. 32.

Nie. You will allow us a preliminary Theorem?