Page:Carroll - Euclid and His Modern Rivals.djvu/106

68 no further difficulty in proving Euc. I. 32 and all other properties of Parallels. How do you proceed?

Nie. We prove (p. 34. Lemma) that, if two Lines have a common perpendicular, each is equidistant from the other.

Min. What then?

Nie. Next, that any Line intersecting one of these will intersect the other (p. 35).

Min. That, I think, depends on Deduction G, at p. 33?

Nie. Yes.

Min. A short, but not very easy, Theorem; and one containing a somewhat intricate diagram. However, it proves the point. What is your next step?

p. 34. Lemma. 'Through a given point without a given straight Line one and only one straight Line can be drawn in the same Plane with the former, which shall never meet it. Also all the points in each of these straight Lines are equidistant from the other.'

Min. I accept all that.

Nie. We then introduce Euclid's definition of 'Parallels. It is of course now obvious that parallel Lines are equidistant, and that equidistant Lines are parallel.

Min. Certainly.

Nie. We can now, with the help of Euc. I. 27, prove I. 29, and thence I. 32.

Min. No doubt. We see, then, that you propose, as a substitute for Euclid's 12th Axiom, a new Definition, two new Axioms, and what virtually amounts to five