Page:Carroll - Euclid and His Modern Rivals.djvu/89

. II. § 7.] draw CF parallel to AD or BE, cutting BD at G; and through G draw HK parallel to AB or BE.

∵ BD cuts Parallels AD, CF,

∴ exterior angle CGB&thinsp;=&thinsp;interior opposite angle ADB. also ∵ AD&thinsp;=&thinsp;AB,

∴ angle ADB&thinsp;=&thinsp;angle ABD;

∴ angle CGB&thinsp;=&thinsp;angle ABD;

∴ CG&thinsp;=&thinsp;CB;

but BK&thinsp;=&thinsp;CG, and GK&thinsp;=&thinsp;CB;

∴ CK is equilateral. also, ∵ angle CBK is right,

∵ CK is rectangular;

∴ CK is a Square.

Similarly HF is a Square and&thinsp;=&thinsp;square of AC, for HG&thinsp;=&thinsp;AC.

Also, ∵ AG, GE are equal, being complements,

∴ AG and GE&thinsp;=&thinsp;twice AG; &thinsp;=&thinsp;twice rectangle of AC, CB.

But these four figures make up AE.

Therefore the square of AB &c. Q. E. D.'

That is just 128 words, counting from 'On AB describe' down to the words 'rectangle of AC, CB.' What author shall we turn to for a rival proof?

Min. I think Wilson will be best.

Euc. Very well. Do the best you can for him. You may use all my references if you like, and if you can do so legitimately.

Min. 'Describe Square ADEB on AB. Through C draw CF parallel to AB, meeting——'