# Al-Khayyam on algebra and tricks

In Chapter 8 Omar al-Khayyam is quoted to the effect that algebra is not a trick, disagreeing with Avicenna (click on the image to magnify it):

It was pointed out to me today that this may leave the false impression that al-Khayyam had rejected the classical division of mathematics into arithmetic and geometry (and music and astronomy).  The sentence that immediately follows in al-Khayyam’s text is consistent with what is mentioned in passing in Chapter 10, namely that he identified algebra with geometry.  I don’t have the English translation handy so I am relying on the French translation:

Without a doubt algebra and al-muqabala are geometric things, that were demonstrated in Book II of [Euclid’s] Elements, propositions 5 and 6.

Here are the three sentences in the original Arabic.  The word hila’ (trick) is in the red rectangle, and the word handisiya (geometric) in the blue rectangle.

Proposition 5 of Book II of the Elements states

If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.

Proposition 6 is

If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line.

You’ll probably want to look at a diagram to understand what’s going on.  We would interpret Proposition 6 as the formula

(a+b)b + ¼a² = (b + ½a)²

and would definitely count it as algebra.  Does that mean we agree with al-Khayyam?