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”.
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