Page:Carroll - Euclid and His Modern Rivals.djvu/57

. II. § 3.] Geometry, but rather in connection with the Theorems with which they are essentially related.'

Euc. It seems rather a strange proposal, to print the Propositions in one order and read them in another. But a stronger objection to the proposal is that several of the Problems are Theorems as well—such as I. 46, for instance.

Min. How is that a Theorem?

Euc. It proves that there is such a thing as a Square. The definition, of course, does not assert real existence: it is merely provisional. Now, if you omit I. 46, what right would you have, in I. 47, to say 'draw a Square'? How would you know it to be possible?

Min. We could easily deduce that from I. 34.

Euc. No doubt a Theorem might be introduced for that purpose: but it would be very like the Problem: you would have to say 'if a figure were drawn under such and such conditions, it would be a Square.' Is it not quite as simple to draw it?

Then again take I. 31, where it is required to draw a Parallel. Although it has been proved in I. 27 that such things as parallel Lines exist, that does not tell us that, for every Line and for every point without that Line, there exists a real Line, parallel to the given Line and passing through the given point. And yet that is a fact essential to the proof of I. 32.

Min. I must allow that I. 31 and I. 46 have a good claim to be retained in their places: and if two are to be retained, we may as well retain all.

Euc. Another argument, for retaining the Problems