AN ESSAY BY THE UNIQUELY WISE ABEL FATH OMAR BIN AL KHAYYAM ON ALGEBRA AND EQUATIONS pdf

An Essay By The Uniquely Wise Abel Fath Omar Bin Al Khayyam On Algebra And Equations

AN ESSAY BY THE UNIQUELY WISE ABEL FATH OMAR BIN AL KHAYYAM ON ALGEBRA AND EQUATIONS

From the Book

Three of these six equations are mentioned in the books of the algebraists.

 They (the algebraists) said: that the ratio of the object to the square is the ratio of the square to the cube. So equating the square to the cube is like equating the object to the square.

Further, the ratio of the number to the square is the ratio of the root to the cube. So, it follows that the equation of the number and the square is the equation of the root and the cube. They (the algebraists) did not prove that using geometry.

As for the number that equals the (volume) cube, there is no way to determine it (the number) except through mathematical induction. If a geometrical method is to be used to determine the number, then it can only be done through conic sections.

Multi-term equations are of two classes: three-term equations and four-term equations. The three-term equations are of twelve types. The first three of them are:

 (1) Square and a root equal a number

(2) Square and a number equal root

 (3) Root and a number equal square.

These three have been mentioned in the algebraists’ books, including their proofs using geometry, not using numbers.

The second three (of the three-term equations) are: (1) Cube and square equal root

(2) Cube and root equal square

(3) Cube equals root and square. The algebraists said that these three equations are equivalent to the first three, each to the corresponding one.

I mean: cube and root equal square is equivalent to square and number equal root. And the other two are the same.

They (the algebraists) did not prove to them if the topic (unknown) of the problem is the area. But they did solve them if the unknown is a number, as is clear in the book of Elements. I will prove the geometrical ones.