Omar Ibn Khayyam

Syed Ghousullah Husainie

The speed with which the scientific and technological developments are taking place call for sufficient and up-to-date knowledge of many subjects of Arts and Science. Today all positive and social sciences make use of mathematical ideas. No sphere of man’s activity is untouched by Science. Living in the twentieth century is not possible unless one is intelligent enough to his environment and understands the scientific techniques, which in turn requires enough insight in mathematics and basic sciences. In order to develop a technical know how there is felt a persistent need for popularizing such ideas. The evolutions of such ideas is not a separate and isolated activity of human mind, rather it is the result of an impulse in the minds of men to mould their environment. We can not help the feeling of indebtedness and gratitude to the great and powerful Muslim minds. Due to their hard and untiring labour great ideas owe their existence in every branch of basic sciences.

 

History witnesses the fact that the really great minds have always sprung from poor homes. Here we shall study the life and achievements of a great Muslim genius Abul Fath Umar ibn Ibrahim al Khayyami, who, though born in an unknown family, made such great and handy contributions to mathematical ideas and astronomy that they wonder struck the world.

 

Biographical sketch. - Umar ibn Khayyam was born at Naishapur, Kharassan in 1043. Nothing is known about his father, but historians say that he was enthusiastic to see his son well educated. He therefore sent him to a school near his house for early education. At that time, he never imagined that this poor beginning will have a good grand end, and his son will turn out to be a famous mathematician and a great poet of the world. He was the most glamorous personality of the eleventh century who became famous as a poet, astronomer, and mathematician. The love and eagerness for intensive study of Science and Arts did not allow him to leave that country which was known as the cradle of wisdom and knowledge in those days. He remained there and taught mathematics and astronomy to hundreds of students who used to come to him for the sake of education from distant places. Khawaja Nizami of Samarkand who happened to be his well known student writes, “I often used to hold conversations with my teacher Umar in the garden. One day he said to me ‘my tomb shall be in a spot where the north wind may scatter roses over it. Years after when I visited Naishapur I went to his resting place and it was just outside a garden and trees laden with fruits dropped their flowers on his tomb so that the stone was hidden under them. (Refer: Rubaiyat of Umar Khayyam by Fitzgerald). Moritz Cantor, the well known mathematician of the West says that Umar can better claim to immortality as one of the greatest mathe-maticians of all time. The most remark-able piece of work done by Umar ibn Khayyam as a court astronomer to Malik Shah was the Calendar about which Cantor says that the solar year proposed by him is more accurate than any other calendar proposed either before or after his time. The present Gregorian calendar has an error of one day in 3330 days, whereas Umar’s calendar gives an error of one day in 5000 years. These figures will enable the readers to see the accuracy and precision maintained by Umar ibn Khayyam in those days when such delicate and sophisticated, astronomical instruments were not available.

 

Some historians write that Umar’s calendar was known as Tarikh-i-Malik. His written work in the field of astronomy presents him as a mathematical and astronomical genius not only of those days but also of the present century. Before he died at Naishapur he did a lot to widen the realm of mathematical ideas and chose the study of most difficult problems. It is also clear from his written work that he confined himself to the exploration and investigation of the essence of mathematics and then generalized the methods according to the situation. One can also find in his work that he concentrated his attention on the new methods and evolved new codes and from this he tried to determine the scope of their application in astronomy. In order to understand this, one should study his work in a systematic way. For this purpose I would like to give a brief sketch of his contributions for the readers.

 

Umar ibn Khayyam was an extra-ordinary mathematician of his time. He touched several branches of this subject in his writings. For example, one can find his work on Algebra, Geometry, Trigonometry, Astronomy and Astro-physics. He was indeed a versatile scholar. It is not within the scope of my paper to describe the important results of his writings in each subject. I have therefore selected few topics of general interest in order to understand certain fields of investigation where Umar ibn Khayyam appeared as a mathematician:-

a)         Euclid’s Postulates.

b)        Binomial Theorem for positive integral indices.

c)         Theory of Numbers.

d)        Astronomy.

Euclid’s Postulates:- In order to understand the type of work performed by Umar ibn Khayyam on this topic, it is necessary to know something about Euclid who was a Greek mathematician of antiquity. He taught mathematics and founded a school at Alexandria in the days of Ptolemy I, who reigned from 306 to 283 B.C. Euclid’s Elements is a compilation of all his learnings which he had accumulated since Pythagoras. It is in thirteen books and constitute the earliest systematic exposition of elementary geometry. Book I – IV and VI deal with plane geometry, V with proportions in general, VII-IX with properties of numbers, X with irrational quantities and XI-XIII with solid geometry. Euclid’s elements were the common school text books for many generations. It is generally understood that geometry had its origin in Egypt. Geometry as we all know is a subject that deals with points, lines, angles, areas and volumes. Geometry turned to be a deductive science in ancient Arabia. The name of Euclid is worth mentioning because of his considerable contribution towards the postulational procedures in geometry. Euclid assumed a prominent position because he systematized and organized the fragments in terminology and provided proper structures for this subject. His greatness lies in his deductive approach to all the geometry known in those days which was often imitated up to the period of Umar ibn Khayyam but never surpassed. Euclid selected five geometrical statements as the basis of his deductive treatment. These postulates involve a number of technical terms. Euclid’s statements are not very much self-explanatory; it is a fact that in contrast with his explicitly exhibited set of postulates, his Elements contain no list of undefined terms of the subject.

The writings of Umar ibn Khayyam show that these postulates and various other definitions of Euclid were thoroughly investigated by him in 1704-75. He writes that the Elements have a strange collection of statements such as “the whole is greater than the part”. Umar says that Euclid is famous for his postulates and it is his fifth postulate which can rightly be termed as the corner stone on which his greatness as mathematician rests. This idea of Umar ibn Khayyam was a remarkable one because from this moment geometry had taken a new turn. As you know, sometimes we talk about the non-Euclidian geometry, and say that Karl Friedrich Guass (1777-1855) was the first mathematician to work on this branch of mathematics. Others say that Bolyai was the one who developed this subject and published his discoveries quite independently. But the fact is that the first phase of the development of the non-Euclidian geometry actually started with the pioneering work of Umar ibn Khayyam. It was in this period when people started thinking of giving new answers to the questions like, what is geometry? What is the nature of postulate? etc. Naturally new answers to these questions lead to modern view of postulational system. The use of this method in algebra and analysis is a direct consequence of its application. For these successes it was Umar ibn Khayyam who deserved the credit and recognition and then the efforts made by the founders of non-Euclidian geometry need applause.

Binomial Theorem: Umar ibn Khayyam’s work on the formulation of the binomial theorem for the positive integral indices plays an important role in the induction methods of algebra. The triangle used by him for a definite expanded form of a binomial of any given positive index is known as Umar’s Triangle. The knowledge of the sum of difference of the square or cube of two quantities were known to the people before Umar ibn Khayyam, but a systematic way of  writing the coefficients of the terms of the expanded form was not known to them. By the application of Umar ibn Khayyam’s triangle it becomes very easy. (Ref: Mathematics for millions by Lancelot Hogben) Umar’s Triangle has wide applications. The approximate values of various complicated expressions can be easily evaluated by its application. For example the value of (1.01)¹⁰ can be computed by Umar’s method. Write it as (1.0.01)¹⁰ and then write its expansion using Umar’s triangle, and sum up the terms. The correct answer to seven places comes out to be equal to 1.1046221. This shows that Umar ibn Khayyam was keen to get the results as accurate as possible in his computations.

 

Theory of Numbers: - We all know that Fermat’s theorem is of basic importance in the theory of numbers which was established in Europe many years after Umar ibn Khayyam. In order to understand this theorem one should possess sufficient knowledge of real, rational and irrational numbers and their properties. Umar ibn Khayyam’s work shows such problems. This indicates that he had sufficient knowledge of the number theory and was well versed in its applications to various problems of practical importance. He has also asserted the impossibility of finding two cubes whose sum should be a cube.

 

Astronomy:- Umar’s main field of interest was astronomy. He was well known as an astronomer that as a poet in the Western society of educated people. He has done so much work in this field that still there exists a number of results and valuable information in his writings which is used for the preparation of the Nautical Almanacs. His remarkable achievement in astronomy was the calendar which has brought a tremendous reputation for him even in the present century. This calendar is more than enough to testify his proficiency and caliber because for such a master-piece work one should be competent enough to understand and handle the delicate astronomical instruments such as the Transit instruments. Micro-meters, Alt-Azimuth Instruments etc. One should also know the laws of Parallax. Magnitude of Moon, Sun and planets etc. The knowledge of the movements of various planets of the Solar system and the exact time and duration of the lunar and solar eclipses was also known to him. Umar ibn Khayyam did a lot for astronomical development throughout his life which is of tremendous value to us. It is therefore necessary to read the life sketch and work of such people who tried to understand the secrets of nature and benefited the mankind by their useful inventions and discoveries.

 

Umar ibn Khayyam is an unforget-table name even in the field of poetry. In the opinion of some learned men Umar is like a Sufi and even something of a Saint. His work on quatrains is so rich with literature that it does not justify at this stage to write few lines on this aspect. I therefore conclude this paper in the words of two Rubaiyats of Umar ibn Khayyam which were translated by Edward Fitzgerald.

 

“Ah but my computations, peoples say,

Have squared the Year to human compass, eh?

If so, by striking from the Calendar

Unborn to-morrow and dead yesterday”

“Into this Universe, and why not knowing,

Nor whence, like Water willy-nilly flowing:

And out of it, as Wind along the Waste,

I know not whither, willy-nilly blowing”.