User:Xashaiar/Khayyam the mathematician

Untitled paper on cubic equation Treatise on Demonstration of Problems of Algebra Problems of Arithmetic Commentaries on the difficult postulates of Euclid's book

Geometric Algebra

Main article: Treatise on Demonstrations of Problems of 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

This philosophical view of mathematics (see below) has had a significant impact on Khayyam's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. In this regard khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations.

"Cubic equation and intersection of conic sections" the first page of two-chaptered manuscript kept in Tehran University

In an untitled writing on cubic equation by Khayyam discovered in 20th century, where the above quote appears, Khayyam 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" (see the image on the left). Again in solving this problem, he reduce 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. To solve this geometric problem, he specialize a parameter and reaches at the cubic equation $x^{3}+200x=20x^{2}+2000$. 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.

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. A proof of this impossibility was plausible only 750 years after Khayyam passed away. In this paper Khayyam 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."

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. In particular, he derived general methods for solving cubic equations and even some higher orders.

Binomial theorem and extraction of roots

See also: History of binomial theorem

From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 1², 2², 3² (etc.) and their respective products, i.e. 2 × 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic, founded on the arithmetic of The Elements.

Omar Khayyam Treatise on Demonstration of Problems of Algebra

This particular remark of Khayyam and certain propositions found in his Algebra book has made some historian of mathematics believe that Khayyam had indeed a binomial theorem up to any power. The case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians. Omar was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Omar had a general binomial theorem is based on his ability to extract roots.

postulate of parallels

Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami. Khayyam studied philosophy at Naishapur. He lived in a time that did not make life easy for learned men unless they had the support of a ruler at one of the many courts. However Khayyam was an outstanding mathematician and astronomer and he did write several works including Problems of Arithmetic, a book on music, and one on algebra before he was 25 years old.

In the latter, Khayyam considered the problem of finding a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years.

In 1070 he moved to Samarkand in Uzbekistan, one of the oldest cities of Central Asia. There Khayyam was supported by a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra. This contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact, Khayyam gives an interesting historical account in which he claims the contributions by earlier writers such as Al-Mahani and Al-Khazin were to translate geometric problems of the Greeks into algebraic equations, something which was essentially impossible before the work of Al-Khwarizmi. However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions.

Also in his algebra book, Khayyam refers to another work of his which is now lost. In the lost work, Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.

In another book, Khayyam made a contribution to non-euclidean geometry, although this was not his intention. In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios.

A powerful sultan then invited Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work. During this time, Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. Khayyam measured the length of the year as 365.24219858156 days. This shows an incredible confidence to attempt to give the result to this degree of accuracy, and it is amazingly accurate. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. The length of the year in 1900 was 365.242196 days, while in 2000 it was 365.242190 days.

In 1092 political events ended Khayyam's period of peaceful existence. Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold. Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith. Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour. He wrote a work in which he described former rulers in Iran as men of great honour who had supported public works, science and scholarship. Another empire rose in 1118, this time with Merv, Turkmenistan as its capital. The shah created a great center of Islamic learning in Merv where Khayyam wrote further works on mathematics.

Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Of all the verses that can be attributed to him with certainty, the best known is the following:

The Moving Finger writes, and, having writ, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a Line, Nor all thy Tears wash out a Word of it.

Omar Khayyam was famous during his times as a mathematician. He wrote the influential Treatise on Demonstration 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. In particular, he derived general methods for solving cubic equations and even some higher orders:

From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 1², 2², 3² (etc.) and their respective products, i.e. 2 × 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic, founded on the arithmetic of The Elements. - Omar Khayyam: Treatise on Demonstration of Problems of Algebra

His method for solving cubic equations worked by intersecting a conic section with a circle (examples). Although this approach had been used earlier by Menaechmus and others, Khayyám provided a generalization extending it to all cubics with positive roots. In addition he discovered the binomial expansion. His method for solving quadratic equations is also similar to that used today.

Also, he was the first Persian mathematician to call the unknown factor of an equation (i.e., the x) shay (meaning thing or something in Arabic). This word was transliterated to Spanish during the Middle Ages as xay, and, from there, it became popular among European mathematicians to call the unknown factor either xay, or more usually by its abbreviated form, x, which is the reason that unknown factors are usually represented by an x.

In the Treatise he also wrote on the triangular array of binomial coefficients known as Pascal's triangle. In 1077, Omar wrote Sharh ma ashkala min musadarat kitab Uqlidis (Explanations of the Difficulties in the Postulates of Euclid). An important part of the book is concerned with Euclid's famous parallel postulate, which had also attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Omar's attempt was a distinct advance, and his criticisms made their way to Europe, and may have contributed to the eventual development of non-Euclidean geometry.

Omar Khayyám also had other notable work in geometry, specifically on the theory of proportions.

Khayyam-Saccheri quadrilateral

The Khayyam-Saccheri quadrilateral was first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):

Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.

Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.

It wasn't until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

References

Other references

• Nasr, S. H. (2006). Islamic Philosophy from Its Origin to the Present: Philosophy in the Land of Prophecy. SUNY Press. ISBN 0791467996.
• KnoebelNasr, Arthur; Laubenbacher, Reinhard; Lodder, Jerry; Pengelley, David (2007). Mathematical Masterpieces: Further Chronicles by the Explorers. Springer. ISBN 0387330615.
• The Cambridge History of Iran (5): The Saljug and Mongol Periods. Cambridge University Press. ISBN 052106936X.
• E.G. Browne. Literary History of Persia. (Four volumes, 2,256 pages, and 25 years in the writing). 1998. ISBN 0-700-70406-X
• Jan Rypka, History of Iranian Literature. Reidel Publishing Company. 1968 OCLC 460598. ISBN 90-277-0143-1