Page:Carroll - Euclid and His Modern Rivals.djvu/155

. § 1.] Nie. And yet this Axiom is the converse of the preceding, which you granted so readily.

Min. The technical converse, my good sir, not the logical! I will not suspect you of so gross a logical blunder as the attempt to convert a universal affirmative simpliciter instead of per accidens. The only converse, as you are no doubt aware, to which you have any logical right, is 'Some Lines, which have "different directions," would meet if produced'; and that I grant you. It is true of intersectional Lines, and I would limit the Proposition to such Lines, so that it would be equivalent to 'Lines, which would meet if produced, would meet if produced'—an indisputable truth, but not remarkable for novelty! You may proceed.

Nie. I beg to hand in this diagram, and will read you our explanation of it:—



'Thus A and B in the figure have the same direction; and C and D, which meet, have different directions; and E and F, which have different directions, would meet if produced far enough.'

Min. I grant the assertion about C and D; but I am wholly unable to guess on what grounds you expect me to grant that A and B 'have the same direction,' and that E and F 'have different directions.' Do you expect me to judge by eye? How if the lines were several yards apart? Is this what Geometry is coming to? Proceed.