VII Persian and Islamic Mathematics We begin with the story of the Prophet, Muhammad. He was born in what is now Saudi Arabia, in 571 A.D. His father died before his birth, his mother soon after. He was raised an orphan, by his uncle. When he was about 40 years old, while meditating in a cave in the hills above Mecca, he received his call from the Almighty. He began preaching a religious monotheism, in a polytheistic Arabian society. His first followers were from poorer classes, but his wealthy father-in-law, Abu Bakr and a powerful Arabian chieftan, Umar, also were among his first followers. The growth of his group began to threaten the city fathers in Mecca who depended economically on pagan pilgrims coming to Mecca to worship the many gods. Trouble began and Muhammad withdrew with his followers to Medina in September of 622, in a flight known as the Hegira, the start of the Muslim calendar. He grew stronger in Medina, returned to take control of Mecca, and by about 630 controlled most of the Arabian peninsula. When Muhammad died on June 28, 632 there was already a question of who his successor would be. Would his family continue his legacy as would usually have been the case in Arabian society of the time, or would his idealistic teachings, as are written in the Quran, be followed? That is, would the most able leader take control, and would the word “able” refer to the religious or the political arena, or both together? Muhammad’s cousin and son-in –law, Ali, was the family candidate and the powerful Umar was the more secular candidate. Muhammad avoided this argument by indicating that Abu Bakr, his elderly father-in-law should succeed him. Thus Abu Bakr was the first Caliph, or successor to Muhammad in the leadership of Islam. He was Caliph for two years, from 632 to 634, during which time he united the Arabian peninsula into the Islamic fold. Abu Bakr was then followed by Umar, who assimilated Sassanid Persia, Egypt, Syria and northern Africa into Islam. In 640 the Library of the Museum in Alexandria was destroyed. Umar was Caliph from 634 to 644 and was succeeded by Uthman. Uthman, who led Islam from 644 to 656, then extended the empire to Morocco in the west, to southeastern Pakistan in the east and to Armenia and Azerbaijan in the north Uthman was, in turn, succeeded by Muhammad’s son-in-law, Ali, who led Islam from 656 to 661. He was married to Fatima, the daughter of Muhammad. These first four Caliphs are known as the Orthodox Caliphate of Islam, based in Mecca and Medina. When Ali died his two sons were the beloved grandsons of Muhammad, Hasan the older and Husayn the younger. Hasan came to be the next caliph but gave up the secular side of the position to a more powerful governor of Damascus, in Syria, Mu’awiya. Mu’awiya is the founder of the Ummayad Caliphate. Hasan kept the spiritual leadership of Islam, the Imamate. When Hasan died in 669 his brother, Husayn became the Imam of Islam. About ten years later Mu’awiya died and left the Caliphate to his drunken son, Yazid. Muslim people in Kufa, in southern Iraq, asked Husayn to leave Mecca and come to be the Caliph in Kufa, and to help them overthrow Yazid. Husayn went to Kufa, where one local army joined him, but they were opposed by a larger army with orders from Yazid. Husayn did not want to have a battle between Muslims and tried to go back to Mecca. He and his family were slaughtered on the plains at Karbala, and his head sent to Yazid. The followers of Husayn are known as the Shi at Ali or Shia ( followers of Ali), while the other side of this split are called the Sunnah, (trodden path) or Sunni, the followers of the word or example of Muhammad.

From 661 to 750 the center of Islam shifted from Mecca and Medina to Damascus in Syria where the Ummayad Caliphs ruled. The Ummayads soon were confident enough in their own beliefs, and had need for more broad knowledge which

other cultures had developed so that Christianity, Judaism and Zoroastrianism were all tolerated if not encouraged. However the Ummayads created a tradition which kept the central power entirely in the hands of Arabs. This created resentment and by 750 a revolt led to the overthrow of the Ummayads and the rise of the Abbasid dynasty During this era the Muslim armies conquered all of Spain and surged deep into France. At Tours, on the Loire River on October 10, 732, 100 years after Muhammad’s death, a French army under Charles Martel (The Hammer) defeated the large Muslim cavalry under Abd ar Rahman al Ghafiqi. It was a first time that infantry held their ground before the onslaught of the Muslim cavalry. The forested landscape played into the foot soldiers tactics and the French killed Abd ar Rahman al Ghafiqi. The Muslims withdrew and did not return. When the eastern Ummayads were overthrown in 750 by the Abbasid Caliphs the Ummayads remained in power in Cordoba, in Spain until 1031. The second Abbasid Caliph, Al Mansur founded Baghdad in 762 and the capital of Islam now shifted to Baghdad. This city is situated on a fertile plain with access to good irrigation and is relatively safe from mosquito borne diseases. It lay along the caravan roads from China and India to the west. Its location was in the area of the Sassanid Persian culture and capital city of Ctesiphon which Umar had conquered. The Abbasid Caliphs soon allowed Persian administrators into the top levels of their governments. Persian knowledge and culture became mixed into the Islamic culture. Also, in 751, during a battle by the Talas River in what is now Kyrgistan where the Islamic armies defeated a T’ang Chinese army, some Chinese papermakers were taken prisoner. The Muslims then learned to make paper which the Chinese had been keeping secret from others. The Islamic societies in turn kept this a secret. The science and medicine which grew in the world’s first hospital in Gundishapur, migrated to Baghdad. The center of science and medicine in Gundishapur was the creation of the Sassanid Persians around 550. Events such as the closing down of Plato’s academy in Athens by the Christian ruler of Rome, Justinian, in 529, had drawn the Mediterranean intellectuals to Gundishapur. Older Persian and Indian styles of learning also contributed to this center of medical knowledge. Now the heritage of Greece, Babylonia, Persia, India and Egypt flowed into Baghdad. The fifth Abbasid Caliph, Harun al-Rashid (ruled 786-809), encouraged science and education for their own sake. He established diplomatic relations with Charles Martel’s grandson, Charlemagne, in order to try and keep the Ummayad caliphs in Cordoba in check. The “Tales of The Arabian Nights” and of Scheherezade are set in the time of Harun al-Rashid. The seventh Abbasid caliph, al-Mamun, the oldest son of Harun al-Rashid ruled from 813 to 833. He founded the House of Wisdom in Baghdad in 830. This was an institute for advanced study along the lines of the center in Gundishapur and the Museum in Alexandria. Workers there were dedicated to the advancement of purely theoretical and speculative knowledge as well as the careful geography and astronomy necessary to find the true direction of Mecca, towards which Muslims must face to pray five times each day. Greek logic was used to fabricate legal codes and to create Islamic theological arguments. The first mathematicians we look at in the House of Wisdom are the three brothers, the Banu Musa, Muhammad, Ahmad and al-Hasan. They worked with translations of Euclid and Archimedes, and offered different proofs of some of the results

in the work of those Greek mathematicians. They differed from the Greeks in their treatment of both rational and irrational numbers together and freely calculated with areas, lengths and volumes as we do. They brought a brilliant pagan mathematician and astronomer to Baghdad from Harran in what is now Turkey, Thabit Ibn Qurra (836901). Thabit grew up as a Sabian star-worshipper in a culture which valued astronomy and mathematics. He converted to Islam after coming to Baghdad. He translated Archimedes and fully grasped the use of Eudoxus’ method of exhaustion and Archimedes way of using it to do integral calculus problems. Thabit used these methods to calculate volumes of revolved parabolas. He made two different efforts to prove the parallel postulate from the other postulates. But, whereas the Greeks had the definitions of perfect and amicable numbers from the Pythagoreans, and a formula for all even perfect numbers by the time of Euclid, there was no such formula for amicable numbers. Enter Thabit: Theorem (Thabit Ibn Qurra) Let n>1 be any integer and let pn = 3" 2 n # 1 and qn = 9 " 2 2n # 1 # 1. If pn " 1, pn , and qn are all prime numbers then a = 2 n " pn " pn # 1 and b = 2 n " qn are amicable numbers, with a being an abundant number and b being ! a deficient number. ! ! Let us try some initial examples of Thabit’s formula. ! 3" 2 2 # 1 = 11, p = 3" 2 # 1 = 5 and q = 9 " 2 3 # 1 = 71 p2 = 1 2 so a = 2(5)(11) = 220 and b = 4(71) = 284. Here is the old Pythagorean pair of amicable numbers.

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p3 = 3" 2 3 # ! 1 = 23, p2 = 11 as before and q3 = 9 " 2 5 # 1 = 287 = 7(41) which is not a prime.

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p4 = 3" 2 4 # 1 = 47, p3 = 23 as before and q4 = 9 " 2 7 # 1 = 1151 a prime. !

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So we have pay dirt, a = 16(47)(23) = 17,296 and b = 16(1151) = 18,416. Knowledge has gone beyond the Pythagoreans in this area of number theory after about 1200 years. ! Thabit’s numbers probably got too big too fast for the mathematicians of his time to check very many of the p’s and q’s for primality. For instance the next pair of amicable numbers to be given by his formulas are from p6 = 3" 2 6 # 1 = 191, p7 = 3" 2 7 # 1 = 383 and q7 = 9 " 213 # 1 = 73,727 and the latter number is a prime so a = 128(191)(383) = 9,363,584 and b = 128(73,727) = 9,437,056 is a third pair of amicable numbers.

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! It is somewhat discouraging to know that modern methods have shown that the above three pairs of amicable numbers are the only ones that Thabit’s formulas give for n < 1,000. Whether there are more for larger values of n is an unknown problem in number theory.

Thabit’s formulas do miss a lot of amicable numbers. Here are all the amicable numbers less than 100,000: (220, 284) (1184, 1210) ( 2620, 2924) (5020, 5564) (6232, 6368) (10,744, 10,856) (12,285, 14,595) (17,296, 18,416) (63,020, 76,084) (66,928, 66,992) (67,095, 71,145) (69,615, 87,633) (79,750, 88,730) Thabit also solved quadratic equations using proofs and began a program of trying to solve cubic equations using algebra. The Chinese methods focused on numerical, iterative methods, and less on algebraic manipulations. One of the first mathematical scholars in the House of Wisdom in Baghdad was Muhammd Ibn Musa al Khwarizmi (780-847) who was from Khwarizm. This was an area south of the Aral Sea in middle Asia in what is now Uzbekistan and Turkmenistan, above Iran and Afghanistan. You can find this region on the map of Alexander the Great’s Empire in the Hellenistic section of these notes, or in the map on the next page. Our word “algorithm” is from his name and place of origin. He is known as the father of algebra, as he wrote the first and most influential early book on the subject. The book is the “Kitab al-Jabr Waal- Muqabala,” first published in 830 and dedicated to alMamun. The word algebra comes from the phrase “al-jabr” in the title of the book. This was a medical, scientific term for setting and restoring broken bones, perhaps a reference to the scientific prowess of Gundishapur. Al-Khwarizmi uses it to refer to the algebraic manipulation of eliminating negative terms from equations by adding the corresponding positive quantities to both sides of the equation. x 2 = 23 " 4 x Thus Add 4x 4x 2 x + 4 x = 23 To get He writes to teach the methods of using algebraic manipulations, separate from ! any geometric justification. The other manipulation is the waal-muqabala, which amounts to a sort of balancing by collecting like terms as a positive quantity on one side ! 75 + 3x 2 = 17 + 50x of the equation. Thus Subtracting -17 -17 2 58 + 3x = 50x To get ! Al Khwarizmi was an orthodox Muslim who worked for al-Mamun. In his book he shows that quadratic equations can have two roots, but sticks to positive roots and ! coefficients. He adopted the nine Indian numerals, the zero and the place-value notation and spread them among the Islamic intellectuals. Al Khwarizmi was quite influenced by the work of Brahmagupta in India. He was not a highly original mathematician, but organized the subject and pushed for the acceptance of his methods in a forceful way.

He also wrote on geography and astronomy and understood Ptolemy’s work which he improved in some spots. It was important to know the direction of Mecca. Such societal imperatives gave momentum to the development of trigonometry and astronomy among the Muslim nations.

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A next mathematician in the tradition of working on the program to solve cubic equations algebraically was Abu Ja’far al-Khazin, who died about 971. He was successful in solving the class of cubic equations of the form x 3 + a = bx 2 . From 973 to 1048 another mathematician emerged from Khwarizm, Abu AlRayhan al Biruni. He wrote 146 treatises, of which 15 are on mathematics, 62 on astronomy and chronology and 19 on geography and geodesy. He worked at more ! accurate observational techniques in order to improve his ideas. he criticized Ptolemaic astronomy, considered that the earth rotated on an axis, and thought of the solar system as heliocentric, but thought of this as a purely philosophical question. He made the most accurate measurement up to that time of the length of a degree of longitude along a meridian. This is not the same everywhere, due to the flattening of he earth as you get far from the equator. In 1055 Tugril Beg and his Seljuk Turkish army conquered Baghdad and Iraq. Support for mathematics and science became less in eastern Islam, then. But one Seljuk ruler, Malik Shah supported science, mathematics and culture in general. An important astronomical observatory was founded in Isfahan, in Iran. The Persian mathematician, poet and astronomer Omar Khayyam (The tentmaker) was born in 1048 in the city of Nishapur in Khorasan, in northeastern Persia. Earlier than 1070 he went to Samarkand where a judge became his supporter, financially. While he worked in Samarkand he wrote a famous treatise on cubic equations, the Risala fi’l-Barahim ‘ala mas ‘il al-Jabr wa’l Muqabala. His work on classifying and using different conics to solve different types of cubic equations carried forward the program initiated by Thabit ibn Qurra centuries before. For instance, to solve the equation x 3 + bx = r he rewrites the equation in the form x 3 + a 2 x = a 2c, so a = b . He then considers the algebra of the two conics, the parabola ay = x 2 and the circle x 2 " cx + y 2 = 0 . The x-coordinate of their intersection solves the cubic equation as follows: (in our algebraic language) ! In the figure, c = 2, a = 2 and so b = 4 and r = 8.

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If ( x 0 , y 0 ) is the common point on the two conics, then x 02 = ay 0 and since y 02 = x 0 (c " x 0 ) we have that x 04 = a 2 y 02 = a 2 x 0 (c " x 0 ) so x 03 = a 2 (c " x 0 ) or x 03 + bx 0 = r , the equation we wanted to solve. One problem remaining here is to solve for the value of this point, numerically. In our simple case x 0 = 1.364655608 , ! ! approximately. ! ! Omar does not seek three roots for all cubic equations but recognizes that some cubics do have three real roots. He also investigated Euclid’s parallel postulate. By ! 1070 he became known and soon after that the Seljuk ruler, Malik Shah invited him to direct the astronomical observatory in Isfahan. Omar went to Isfahan and worked there for 18 years. He directed the work of the observatory and created the Jalali calendar which is more accurate than our own Gregorian calendar. He was an active intellectual in the cultural life of Isfahan and wrote the book of poetry which is still in paperback editions in our bookstores, The Rubaiyat. He also was interested in philosophy and was a suspected Faylasuf, one who did not always think in strict orthodox Muslim theological principles. In 1092 Malik Shah died and more conservative Muslims took control of the court in Isfahan. Omar was marginalized, moved to Marv in his native Khorasan, and was buried in a tomb in Isfahan in 1131. Sharaf al-Din al-Tusi came from the city of Tus, which was also the home of the great Persian poet, Ferdowsi, (935-1020) who wrote the Persian epic poem, the Shah Nameh, The Book of Kings. This poem is the analog of the Iliad and the Odyssey in Persian literature. Tus was in northern Khorasan near the current city of Mashad. It was overrun by the Mongols and never rebuilt. Al-Tusi wrote a work called Treatise on Equations which covers linear, quadratic and cubic equations. He classifies cubics into 18 different types, and is the first mathematician to predict the number of positive roots using the coefficients of the equation. He also discovers the discriminant of a cubic b3 a2 equation, in the case of the cubic equation x 3 + a = bx . We shall soon see " 27 4 this discriminant again in the work of the Italian Renaissance mathematicians. Sharaf alDin al-Tusi works in the years around 1170. ! from Tus was Nasir al-Din al-Tusi who A second mathematician and astronomer ! lived from 1201 to 1274. He was educated in Nishapur and lived during turbulent times for the central Asian Islamic countries. The Mongols under the grandson of Genghis Khan, Hulagu, conquered Baghdad, Iran and Iraq during the years from 1255 to 1260. Nasir al-Din al-Tusi became a close and valued intellectual in the court of Hulagu, whose mother and wife were Nestorian Christians. In his role as science advisor he was put in charge of the construction of an astronomical observatory in Maragha, a city in northwestern Iran, near the current city of Tabriz. Very accurate astronomy was practiced here, and a model of astronomy created, which at least one scholar has claimed is the basis of Copernicus’ heliocentric model of our solar system. The building still remains. He also put trigonometry on its own feet as a separate field of study from astronomy, of which it had always been a part. He attempted to prove the parallel postulate of Euclid. The final mathematician we consider is Jamshid al-Kashi who worked from 1406 to 1437. He was an astronomer, first in his native city of Kashan, a city half way

between Tehran and Isfahan in Iran. Jamshid al-Kashi was the first mathematician to use a full positional decimal system of numbers in which the fractional parts of the numbers are treated in the same way as the whole number parts, as we do today. The Babylonians had this for sexagesimal arithmetic, but subsequent cultures had never picked up on it. In Europe, Simon Stevin was the first to do this, in 1585. Al-Kashi clearly sees that the calculations give equally true answers whether you work in base ten or in base sixty. Jamshid al-Kashi became the director of a school in Samarkand in 1417 and constructed an astronomical observatory there under the patronage of Ulugh Beg, the ruler in Samarkand and grandson of Tamerlane, the mongol conqueror of central and western Asia who lived from 1336 to 1405. Ulugh Beg was an educated scholar of mathematics and science himself. Al-Kashi supervised the creation of a table of coordinates for the positions of 1,018 stars. He calculated π correctly to 16 decimal places for astronomical uses. He also writes about the Chinese triangle of binomial coefficients and the Chinese algorithms which are called Horner’s numerical methods in the West. A final note. A Muslim scholar, Abdul al-Hassan al-Marrakushi in Cairo in 1230, created a time schedule of 24 hours per day, all of equal length. This had been proposed by Hipparchus in ancient Greece, but the time was not right for people to listen to Hipparchus. Of course the ancient Egyptians had 12 decans of time each day and 12 decans of time each night, with varying lengths of time in a decan, depending on the season of the year. The key in Abdul’s case was that mechanical clocks were invented in 1270. There were public clocks in London in 1292, in Paris in 1300 and in Padua in 1344.