Page:Carroll - Euclid and His Modern Rivals.djvu/81

. II. § 5.] Euc. But, if you look into the depths of your own consciousness—assuming such depths to exist—you will find, I believe, an eternal distinction maintained, in this respect, between straight and curved Lines: so that Lines of the one kind must, if they approach, ultimately meet, whereas those of the other kind need not.

Min. I will grant it, provisionally, if only to know what you are going to deduce from it.

Euc. I will now ask you to consider this diagram.



Suppose it given that the Lines BD, CE, make with BC two angles together less than two right angles. My object is to show that probably—if not certainly—they will meet, if produced towards D, E.

Let BF be so drawn that the angles FBC, BCE, may be together equal to two right angles.

Now, if any point in BD be nearer to CE than B is, what is required is proved, since BD approaches CE.

But, if this be not so, then F (which is obviously further from CE than some point in BD is) must also be further from CE than B is; i.e. FB must approach EC; i.e. FB and EC must ultimately meet, below BC, and so form a Triangle, whose angles at B and C will be (by my Prop. 17) less than two right angles. Hence the angles FBC, BCE, must be greater than two right angles, since the four angles