Arab And Muslim Contributions To All Sciences And Progress Of Humanity

Docteur Ben Salem Ezzedine Islamic Medicine

Part III: Diseases of the Middle Ages
A. Life expectancy (how long a person was expected to live on the average) during the Middle Ages was very much lower than it is today. [Today, for example, males born in the United States are expected to live to the age of 73.6 years and females to the age of 79.4 years. - U.S.C.D.C. - 1997 statistics.]

People of the Middle Ages suffered from many diseases and problems that we no longer worry about. They had high infant mortality rates (babies often died at or soon after birth). There were few medicines to treat many illnesses and poorly trained doctors who worked without good hospitals. Health care was not very advanced in many places, but it was especially bad in Europe. Diet was generally poor. Famine (no food), war, and epidemics (rapid spread of disease) were much more common. One of the most feared of all the problems people faced was the Plague.

B. The Bubonic Plague (The Black Death)

Flea greatly magnified. . . .

The Plague began about 1331 in the grasslands of Central Asia. The Plague is found in rodents like ground squirrels and rats, but it is spread to humans through the bites of fleas living on infected rodents. The fleas had found their way into the caravans of the traders. It spread rapidly as people tried to escape along the trade routes of the steppe. The same Mongol law and order that made possible a century of trade and intense human exchange between China and the Atlantic coast, now quickened the progress of the plague across Eurasia. In China the outbreaks of the plague caused massive death rates and economic chaos. Italian ships carried infected rats and fleas in their cargo to the major European cities of the Mediterranean. The plague reached Egypt in 1347. One Egyptian historian tells of a ship: out of a total of 332 on board, only 45 arrived at the port of Cairo alive. All of those who had survived died soon after in the port. [Dols, The Black Death in the Middle East, p. 69]
From the sea ports caravans unknowingly transmitted the disease throughout Europe, northern Africa, and the Middle East. Estimates of the death tolls vary between 1/2 to 1/3 of the populations.

European painting showing the Plague.

Ibn Battuta, the 14th century Muslim traveler from Morocco, told about the effects of the plague in Damascus, Syria where the death toll was 2,000 people a day! The business of the city had come to a halt. The people begged God for the plague to stop.

"The people fasted for three successive days... [Then all the people] assembled in the Great mosque until it was filled to overflowing... and spent the night there in prayers... Then, after performing the dawn prayer..., they all went out [barefoot] together... carrying Qur'ans in their hands. The entire population of the city joined... The Jews went out with their book of the law and the Christians with the Gospel... [all] of them in tears... imploring (begging) the favor of God through His Books and His Prophets." [Gibb, Ibn Battuta, p. 143-144]

The people of the 14th century were uneducated and susceptible to superstitions. Some early treatments in Europe included:
bathing in human urine
wearing of excrement
placing dead animals in homes
use of leeches (a worm-like animal that sucked out blood)
drinking molten gold (gold heated until it melted) and powdered emeralds (a green jewel)

As plague epidemics occurred regularly after 1350, preventive measures emerged. Plague patients were placed in pesthouses, isolated from the general population. Ships coming from plague infested areas were forced to stay out of port for a 40 day quarantine until the disease died out. [Treatments from Janis' website.]

Learn more about the Bubonic Plague or "Black Death" which wiped out about 1/3 to 1/2 of parts of Europe and Asia.

See a famous historian's (Ibn Khaldun's) account of the plague in the Islamic world during the 14th century:
"...in the middle of the fourteenth century, civilization both in the East and the West was visited by a destructive plague which devastated nations and caused populations to vanish. It swallowed up many of the good things of civilization and wiped them out. It overtook the dynasties at the time of their senility (time of extreme old age ready to die], when they had reached the limit of their duration. It lessened their power and curtailed (limited) their influence. It weakened their authority. Their situation approached the point of annihilation (total destruction) and dissolution. Civilization decreased with the decrease of mankind. Cities and buildings were laid waste, roads and way signs were obliterated, settlements and mansions became empty, dynasties and tribes grew weak. The entire inhabited world changed. ... It was as if the voice of existence in the world had called out for oblivion and restriction, and the world had responded to its call. ... a world brought into existence anew. . . ."

Docteur Ben Salem Ezzedine C. Al-Zahravi is called the "father of surgery". Abul Qasim al-Zahrawi (963-1013 AD) is known as Albucasis to the West. He was a famous surgeon at the court of Caliph al-Hakam II in Baghdad. Students and patients flocked to him from the Muslim world and Europe. He wrote a Medical Encyclopedia which contained 30 sections of surgical knowledge and illustrations of 200 surgical instruments, most of which he designed himself. The Encyclopedia was required reading for physicians, and even five centuries later it was being used as the standard textbook on surgery in universities in Europe.

Al-Zahravi also performed many delicate operations such as caesareans (by cutting open the mother's belly for the delivery of the baby) and was also the first to use silk thread for stitching wounds. He developed many surgical tools that were used for centuries throughout the world of Islam and in Europe.

D. Ibn Zuhr (known as Avenzoar in the West), was born at Seville, Spain in 1091 and graduated from Cordova Medical University. After a brief stay in Baghdad and Cairo, he returned to Spain and worked for the Muslim rulers as a physician. He died in 1161 C.E.

Ibn Zuhr was a skillful doctor and surgeon. He emphasized observation and experiment in his work. He was also excellent in the art of dissecting (cutting open for studying) dead human bodies and knew anatomy in detail. Ibn Zuhr was the first to test different medicines on animals before giving them to humans. Also, he was the first to describe diseases caused by parasites (small animals that live off of other animals) and is known as the first parasitologist (one who studies parasites). He was also the first to describe the operation of tracheotomy (cutting a hole in the wind pipe so the patient can breathe) and practiced direct feeding through a tube in the throat when normal feeding was not possible. He described intestinal problems, inflammation (redness) of the middle ear, and some kinds of tumors (growths like cancer).

Ibn Zuhr wrote many important books for medical specialist and for the common people. Several of his books were translated into Latin and Hebrew and were in great demand in Europe until the late Eighteenth century.

Ibn Zuhr's influence on the development of medical science was felt for several centuries throughout the world. Ibn Zuhr's descendants included five generations of Spanish physicians, and included two women physicians who served the household of the rulers of Islamic Spain.

E. Biruni (Abu Raihan Muhammad al-Biruni - 973 - 1048 AD) was a contemporary of (lived at the same time as) the famous doctor Ibn Sina and they corresponded (sent letters to each other). Biruni was best known for his study astronomy and science, but he was also a skilled physician and wrote about medicinal plants. Below is a painting which illustrates one of his books. It shows a caesarean (where a baby is taken out from the the mother's belly).

Other Famous Doctors
Ibn Nafis (1210 - 1288) is credited for the discovery of the pulmonary circulation system (circulation of blood throughout the body as pumped by the heart and through the lungs). (Many history texts credit European physicians about three centuries later.) See the first paragraphs of this article.
Al-Tabari (teacher of Al-Razi) was also an outstanding physician and wrote medical texts.

Part II: Dentists

A. Dental Care

Dentists made false teeth out of bone which was recommended in many books on dental care. Dentists also pulled teeth that had become decayed or damaged.

B. A Famous Dentist and Orthodontist (one who straightens teeth)

Al-Zharavi, listed above as the Father of Surgery, was also an expert in oral surgery and dentistry. In one of his books he discussed the problem of non-aligned or deformed teeth and gave procedures to correct these problems. In addition, he developed the procedure for preparing and setting false teeth made from animal bones.

Docteur Ben Salem Ezzedine Medieval Medicine, Health and Hygiene

Introduction: Medicine and health care were perhaps the highest scientific achievement of the Muslims during the Middle Ages. The reason that they achieved such heights was directly related to Islam itself. The Qur'an and the sayings of the Prophet Muhammad encouraged the gaining of medical knowledge. The Prophet Muhammad said, "Make use of medical treatment, for Allah has not made a disease without appointing a remedy for it, with the exception of one disease, namely old age." [Hadith on Health.] This motivated Muslim scientists to find cures. Islam encouraged health in other ways, too. The Third Pillar of Islam was that of Charity - to help the poor and sick. The wealth of the empire was often directed to health care, too. Muslim sultans (kings) and caliphs, showing their people that they were good Muslims, built hospitals throughout the Empire. By the 900s almost every city boasted a fine hospital where all people were treated regardless of their ability to pay. Another important idea of Islam was cleanliness. Good hygiene (keeping clean) was important to the Muslims for health, as well as for religious reasons. Another reason for the high achievement in medical science was that the Muslim scientists and scholars received the vast knowledge of science from the classical Greeks (such as Hippocrates and Galen), Egyptians, and other ancient cultures. Muslim scientists and scholars translated them into Arabic, tested, and built upon this knowledge with their own research and discoveries. Finally, because the Empire was so large and easily traveled, scientists benefited from an exchange of ideas and knowledge of such far away places as China, India, the Middle East, as well as Greece and the Mediterranean Lands and Africa. Because of the Arab science of medicine, the whole world has benefited.

What does the Qur'an and some Hadith (sayings of the Prophet) say about health and medicine? See "Medicine (Kitab Al-Tibb)" (from U.S.C.).

Part I: Famous Doctors
A. Al-Razi (known in Europe as Rhazes) 864-930 A.D.
This stained glass window of Al-Razi is found in Cambridge University's Medical School (in London, England).

Al-Razi was born in Iran. At an early age he became an expert in medicine and patients and students flocked to him from distant parts of Asia. He was eventually put in charge of the most famous hospital in Baghdad.

His contributions were very important to the science of medicine. His writings were translated into Latin and many European languages and used in European medical schools. One was ten volumes and told about Greek-Arab medicine. Another work (Al-Hawi) became the largest medical encyclopedia ever written. It told about each medical subject and gave all important information that was available from Greek and Arab sources, and he gave his own remarks about each topic from his own experiences and opinions.

He is especially famous for his study of smallpox and chicken-pox. He found a treatment for kidney and gall stones, and explained the nature of various infectious diseases. He was the first to introduce the use of alcohol for medical purposes. He was also an expert surgeon and was the first to use o***m for anesthesia (a drug to put patients to sleep so they won't feel pain during an operation).

Al-Razi thought diet was important to health. He also emphasized the importance of psychological factors on health (such as having a positive attitude, avoiding stress, etc.). He tested treatments scientifically, first on animals to evaluate in their effects and side effects.

B. Ibn Sina (980 - 1037 A.D.) - the Boy Genius

Ibn Sina's portrait is in the hall of the Faculty of Medicine in the University of Paris. He is known in the West as Avicenna. [Modern image.]

Ibn Sina was born in 980 C.E.. He showed exceptional intellectual powers even as a child. At the age of ten, he already had memorized the Qur'an. During the next six years he studied Muslim law, philosophy, natural science, logic, geometry, and advanced mathematics.

At the age of 17, he started to study medicine and found it "not difficult". By the age of 18 he had a reputation as a great physician (doctor) and was summoned (ordered to come) to take care of the king. In exchange he asked only to be allowed to use the royal library. He eagerly read the contents of the library. At the age of 21 he wrote his first book.

After many years of working for different rulers, he moved to Ray, Iran and established a busy medical practice. When Ray was attacked, Ibn Sina fled to Hamadan where he cured that city's amir (prince, or the military ruler) and was made Prime Minister (high government position). Soldiers rebelled against him and he was imprisoned (put in jail). But then the amir again became sick, so Ibn Sina was released and returned to his position and cured him again!

After the amir died a few years later, Ibn Sina fled (ran away) to Isfahan. He spent his final years in the service of the ruler of that city. Friends advised him to slow down and take life easier, but this was not in his character. "I prefer a short life with width to a narrow one with length," he replied. Worn out by hard work and hard living, Ibn Sina died in 1036 at the age of 58 years.

Ibn Sina's most important medical work is the al-Qanun al-Tibb ("Canon" or Encyclopedia of Medicine) which represents the final bringing together of Greek and Arabian thoughts on Medicine. He wrote about such matters as fatal (deadly) illnesses, ideas about cleanliness and hygiene, remedies and cures, anatomy (the study of the human body), and cardiac (heart) drugs. The Canon, or Encyclopedia contains about one million words and is divided into five books.

Ibn Sina first recognized the contagious nature (the way it is spread from one person to another) of tuberculosis - T.B. - a disease of the lungs, and the spread of disease by water and soil. He described diseases caused by intestinal worms. He pointed out the importance of diet (what someone eats), climate, and environment on health, and the surgical use of oral (taken by mouth) anesthetics (pain killers). Ibn Sina advised surgeons to treat cancer in its earliest stages and to remove all the diseased tissue. The Canon, or Encyclopedia identifies 760 drugs, with comments on their application (how to take the drugs and how often) and effectiveness. He recommended the testing of new drugs on animals and humans prior to (before) general use.

Ibn Sina noted the close relationship between emotions and the physical condition and felt that music had a definite effect on patients. Of the many psychological disorders that he described in the Canon, or Encyclopedia, one is "love sickness"! (Ibn Sina diagnosed this condition in a Prince who lay sick and whose illness had baffled (confused) local doctors. Ibn Sina noted a fluttering in the Prince's pulse when the name of his beloved was mentioned. The great doctor had a simple remedy: unite the sufferer with the beloved.)

The Encyclopedia was translated into Latin. It became the textbook for medical education in the schools of Europe from the 12th-17th century.

Ibn Sina is known as the 'doctor of doctors'.

Docteur Ben Salem Ezzedine Omar Khayyam
Born: May 1048 in Nishapur, Persia (now Iran)
Died: Dec 1122 in Nishapur, Persia (now Iran)
Omar Khayyam's full name was Abu al-Fath Omar ben Ibrahim al-Khayyam. A literal translation of his name means 'tent maker' and this may have been his fathers trade. 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 was a poet as well as a mathematician. He discovered a geometrical method to solve cubic equations by intersecting a parabola with a circle but, at least in part, these methods had been described by earlier authors such as Abu al-Jud.

Consider the circle and parabola

Substitute and simplify to get
which factored gives
So, the intersection x is the solution of the cubic:

Khayyam was an outstanding mathematician and astronomer. His work on algebra was known throughout Europe in the Middle Ages, and he also contributed to a calendar reform. Khayyam refers in his algebra book to another work of his which is now lost. In that lost work Khayyam discusses Pascal's triangle but the Chinese may have discussed triangle slightly before this date.

The algebra of Khayyam is geometrical, solving linear and quadratic equations by methods appearing in Euclid's Elements.

Khayyam also gave important results on ratios giving a new definition and extending Euclid's work to include the multiplication of ratios. He poses the question of whether a ratio can be regarded as a number but leaves the question unanswered.

Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and few of its verses can be attributed to him with certainty.

Ghiyath al'Din Jamshid Mas'ud al'Kashi
Born: 1390 in Kashan, Iran
Died: 1450 in Samarkand (now Uzbek)

Al-Kashi worked at Samarkand, having partron Ulugh Beg.

He calculated to 16 decimal places and considered himself the inventor of decimal fractions. In fact, he gives as

which was the best until about 1700.

He wrote The Reckoners' Key which summarizes arithmetic and contains work on algebra and geometry.

In another work, al'Kashi applied the method now known as fixed-point iteration to solve a cubic equation having as a root.

Generally, for an equation of the form

we define the iteration

where is some initial ``guess". If the iterations converge, then it is a solution of the equation. Such a method is called a fixed point iteration. Another more famous fixed point iteration is Newton's Method

He also worked on solutions of systems of equations and developed methods for finding the root of a number - Horner's method today. [Note. This method also appeared in Chinese mathematics in 1303 in the Ssu-yüan-yü-chien (Precious Mirror of the Four Elements)]

Horner's Method
Example. Solve

First determine that a solution lies between x=19 and x=20. Now apply the transformation
to obtain
We know there is a root between y=0 and y=1. Thus there are two ways to approximate the solution for y:

If then is even closer to zero, and this term may be taken as zero, giving the approximate solution

so that . We may also factor the equation as

Letting the y in the parentheses be 1, solve for the other to get hence the approximation

so that .

Clearly the first is slightly too large, while the second is slightly too small. Which should be selected? al'Kashi selects the second, . Why?

After Al-Kashi, Arabic mathematics closes as does the whole Muslim world. But scholarship in Europe at this time was on the up-swing.

Docteur Ben Salem Ezzedine Arab Contributions

Within a century of Muhammad's conquest of Mecca, Islamic armies conquered lands from northern Africa, southern Europe, through the Middle East and east up to India. Within a century of that the Caliphate split up into several parts. The eastern segment, under the Abbasid caliphs, became a center of growth, of luxury, and of peace. In 766 the caliph al-Mansur founded his capitol in Baghdad and the caliph Harun al-Rashid, established a library. The stage was set for his successor, Al-Ma'mum.

In the 9 century Al-Ma'mum established Baghdad as the new center of wisdom and learning. He establihed a research institute, the Bayt al-Hikma (House of Wisdom), which would last more than 200 years. Al-Ma'mum was responsible for a large scale translation project of as many ancient works as could be found. Greek manuscripts were obtained through treaties. By the end of the century, the major works of the Greeks had been translated. In addition, they learned the mathematics of the Babylonnians and the Hindus.

What follows is a brief introduction to a few of the more prominent Arab mathematicians, and a sample of their work

Abu l'Hasan al-Uqlidisi
c.950

In al-Uqlidisi's book Kita b al-fusul fi-l-hisab al-Hindii (The book of chapters on Hindu Arithmetic), two new contributions are significant: (1) an algorithm for multiplication on paper is given, and (2) decimal fractions are used for the first time. Both methods do not resemble modern ones, but the methods are easily understood using modern terminology.

sometimes called the ``Father of Algebra''.

Al-Khwarizmi most important work Hisab al-jabr w'al-muqabala written in 830 gives us the word algebra . This treatise classifies the solution of quadratic equations and gives geometric methods for completing the square. No symbols are used and no negative or zero coefficients were allowed.

Al-Khwarizmi also wrote on Hindu-Arabic numerals. The Arabic text is lost, but a Latin translation, Algoritmi de numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title.

To him we owe the words AlgebraAlgorithm

His book Al-jabr wál Mugabala, on algebra, was translated into Latin and used for generations in Europe.

It isstrictly rhetorical - even numerals and more elementary than Arithmetica by Diophantus
It is a practical work, by design, being concerned with straightforward solutions of deterministic problems, linear and especially quadratic.
Chapters I-VI covers cases of all quadratics with a positive solution is a systematic and exhaustive way.
It would have been easy for any student to master the solutions. Mostly he shows his methods using examples - as others have done.

Al-Khwarizmi then establishes geometric proofs for the same solutions of these quadratics. However, the proofs are more in the Babylonnian style.
He dealt with three types of quantities: the square of a number, the root of the square (i.e. the unknown), and absolute numbers. He notes six different types of quadratics:

The reason: no negative numbers -- no non-positive solutions
He then solves the equation using essentially a rhetorical form for the quadratic equation. Again note: he considers examples only. There are no ``general" solutons.

Other Arab mathematicians

Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja
Born: about 850 in (possibly) Egypt
Died: about 930

Abu Kamil Shuja is sometimes known as al'Hasib and he worked on integer solutions of equations. He also gave the solution of a fourth degree equation and of a quadratic equations with irrational coefficients.

Abu Kamil's work was the basis of Fibonacci's books. He lived later than Al-Khwarizmi; his biggest advance was in the use of irrational coefficients (surds).

Abu'l-Hasan Thabit ibn Qurra
Born: 826 in Harran, Mesopotamia (now Turkey)
Died: 18 Feb 901 in Baghdad, (now in Iraq)

Thabit was a native of Harran and inherited a large family fortune which enabled him to go to Baghdad where he obtained his mathematical training. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad.

Thabit generalized Pythagoras's theorem to an arbitrary triangle (as did Pappus. He also considers parabolas, angle trisection and magic squares.
He was regarded as Arabic equivalent of Pappus, the commentator on higher mathematics.

He was also founder of the school that translated works by Euclid, Archimedes, Ptolemy, Eutocius but Diophantus and Pappus were unknown to the Arabs until the 10 century. Without his efforts many more of the ancient books would have been lost.

Perhaps most impressive is his contribution to amicable numbers, that is two numbers who are each the sum of the divisors of the other.

Theorem. , then and are amicable.
Theorem. (Generalization of Pythagorean Theorem.) From the vertex A of , construct B' and C' so that Then

Proof. Apply similarity ideas

Note: If , this is the Pythagorean Theorem.

This is the third generalization of the Pythagorean Theorem.

Mohammad Abu'l-Wafa al'Buzjani
Born: 10 June 940 in Buzjan (now in Iran)
Died: 15 July 998 in Baghdad (now in Iraq)

Abu'l-Wafa translated and wrote commentaries, since lost, on the works of Euclid, Diophantus and Al-Khwarizmi. For example, he translated Arithmetica by Diophantus.

He is best known for the first use of the tangent function and compiling tables of sines and tangents at 15' intervals. This work was done as part of an investigation into the orbit of the Moon.

His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places!!

Abu Bakr al-Karaji
( al-Karkhi)
early 11 century)

Arabic disciple of Diophantus - without Diophantine analysis.

Gave numerical solution to equations of the form

(only positive roots were considered).

He proved
in such a way that it was extendable to every integer. The proof is interesting in the sense that it uses the two essential steps of mathematical induction. Nevertheless, this is the first known proof.

al-Karkji's mathematics, more that most other Arab mathematics, pointed to the direction of Renaissance. mathematics.

Docteur Ben Salem Ezzedine Portraits of Twelve Muslim Scientists in Various Fields

Jabir Ibn Haiyan (Geber) - Chemistry - (Died 803 C.E).
Ali Ibn Rabban Al-Tabari - Medicine, Mathematics, Calligraphy - (838-870)
Al-Razi (Rhazes) - Medicine, Ophthalmology, Smallpox , Chemistry, Astronomy - (864-930)
Al-Farabi (Al Pharabius) - Sociology, Logic, Philosophy, Political Science, Music -(870-950)
Abu Al-Qasim Al-Zahravi (Albucasis) - Surgery, Medicine - (936-1013)
Muhammad Al-Buzjani - Mathematics, Astronomy - (940-997)
Ibn Al-Haitham (Alhazen) - Physics,Optics, Mathematics - (965-1040)

Abu Raihan Al-Biruni - Astronomy, Mathematics, determined Earth's circumference - (973-1048)
Ibn Sina (Avicenna) - Medicine, Philosophy, Mathematics, Astronomy - (986-1037)
Omar Al-Khayyam - Mathematics, Poetry - (1044-1123)
Nasir Al-Din Al-Tusi - Astronomy, Non-Euclidean Geometry - (1201-1274)

Ibn Al-Nafis Damishqui - Medicine - (1213-1288)

Docteur Ben Salem Ezzedine C. Geometry

The scholars at the House of Wisdom in Baghdad and at universities in Cairo, Egypt also contributed to geometry. Geometry was highly developed by the Greeks, and the Muslims translated such great Greek thinkers as Euclid. Muslims used their understanding of geometry into designing wheels of all kinds, especially waterwheels and other systems for drawing up water, in improving farming equipment, and in designing devices of war such as catapults and crossbows. Geometry was also put to work in art, with beautiful geometric designs. Muslims further defined Euclidian geometry, and pointed the way toward the discovery of independent, non-Euclidean geometry developed in the most recent centuries.

D. Trigonometry is also mostly a Muslim creation. It is a branch of mathematics which studies plane and spherical triangles. It developed from the need of astronomers to map points in the sky on a heavenly sphere. Trigonometry's functions, involving ratios such as sine and cosine, tangent and cotangent, were greatly developed and refined in the Islamic lands.

E. Famous Muslim Mathematicians of the Middle Ages

1. Al-Khwarizmi (770 - 840 C.E.) was one of the greatest mathematicians who ever lived and is called the "Father of Algebra". He also helped to bring "Arabic numerals" into use into the Islamic Empire, as well as later into Europe. He also demonstrated operations with fractions for the first time. Khwarizmi influenced the growth of science and mathematics. Several of his books were translated into many other languages, and were used as university textbooks until the 16th century. His approach was systematic and logical. He brought together the knowledge of his time on various branches of science, especially mathematics, and also added his original contributions.

2. Omar Khayyam (1044 - 1123 C.E.): Another great Muslim mathematician was Omar Khayyam. He is best known today for his poetry, but his contribution to mathematics was great. He showed how to express roots of cubic equations by line segments obtained by intersecting conic sections. Khayyam was an outstanding poet, mathematician, and astronomer. His work on algebra was known throughout Europe in the Middle Ages, and he also contributed to a calendar reform. Khayyam refers in his algebra book to Pascal's triangle. The algebra of Khayyam is geometrical, solving linear and quadratic equations by methods appearing in Euclid's Elements. Khayyam also gave important results on ratios giving a new definition and extending Euclid's work to include the multiplication of ratios. He poses the question of whether a ratio can be regarded as a number but leaves the question unanswered.

3. Al-Khashi was born in 1390 in Kashan, Iran and died in 1450 in Samarkand (now Uzbek). He calculated 1 (pi) to 16 decimal places which was the best until about 1700. He considered himself the inventor of decimal fractions. He wrote The Reckoners' Key which summarizes arithmetic and contains work on algebra and geometry.

4. Al-Biruni (973 - 1048 C.E.) was a philosopher, astronomer, pharmacologist (one who studies drugs and herbs used for health), botanist (one who studies plants), geologist and mathematician. He translated Euclid's work into Sanskrit (an Indian language), and calculated the earth's circumference (distance around the earth) and radius (distance to the center) with an accuracy that is close to today's measurements.

5. Nasir Al-Din Al-Tusi (1201 - 1274 C.E.) pioneered spherical trigonometry which includes six fundamental formulas for the solution of spherical right-angled triangles. One of his most important mathematical contributions was the treatment of trigonometry as a new mathematical discipline. He wrote on binomial coefficients which Pascal later introduced. (He can be called the "Father of Trigonometry".) He was also an astronomer philosopher, and medical scholar as well as a mathematician.