Omer Khayyam
The discovery of "Cartesian" coordinates, central to modern mathematics, is attributed to French mathematician Rene Descartes -- However: It may sound strange, but Descartes never used "the Cartesian coordinates" in his treatise of 1637 nor ever in his life; he wrote about "coordinates" in the sense of distances to describe the locus of a curve, and even more, he didn't used negative distances.
In fact, the real originator of analytic geometry, the attempt to study geometry by the use of Algebra, is Omar Khayyam. He explicitly describes the philosophy of mathematics, which he developed, and which led him to the use of these methods which became the basis for modern mathematics. Integration of Algebra with geometry is mentioned as follows by Omar Khayyam:
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by propositions five and six of Book two of Elements.
Omar Khayyam[17]
Geometric algebra
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by propositions five and six of Book two of Elements.
Omar Khayyam[17]
Omar Khayyám's geometric solution to the cubic equation x3 + 200x = 20x2 + 2000.
This philosophical view of mathematics (see below) has influenced Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations. His solution is not a direct path to a numerical solution, and his solutions are not numbers but line segments. Khayyám's work can be considered the first systematic study and the first exact method of solving cubic equations.[18]
In an untitled writing on cubic equations by Khayyám discovered in the 20th century,[17]where the above quote appears, Khayyám works on problems of geometric algebra. First is the problem of "finding a point on a quadrant of a circle such that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal." Again in solving this problem, he reduces it to another geometric problem: "find a right triangle having the property that the hypotenuse equals the sum of one leg (i.e. side) plus the altitude on the hypotenuse ".[19] To solve this geometric problem, he specializes a parameter and reaches the cubic equation x3 + 200x = 20x2 + 2000.[17] Indeed, he finds a positive root for this equation by intersecting a hyperbola with a circle.
This particular geometric solution of cubic equations has been further investigated and extended to degree four equations.[20]
Regarding more general equations he states that the solution of cubic equations requires the use of conic sections and that it cannot be solved by ruler and compass methods.[17] A proof of this impossibility was only plausible 750 years after Khayyám died. In this paper Khayyám mentions his will to prepare a paper giving full solution to cubic equations: "If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art, will be prepared."[17]
This refers to the book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, part of the body of Persian mathematics that was eventually transmitted to Europe.[18] In particular, he derived general methods for solving cubic equations and even some higher orders.
