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 (836- 901). 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: n Theorem (Thabit Ibn Qurra) Let n>1 be any integer and let pn = 3" 2 # 1 and 2n # 1 qn = 9" 2 # 1. If pn " 1, pn, and qn are all prime numbers then n n a = 2 " pn " pn # 1 and b = 2 " qn are amicable numbers, with a being an abundant number and b being a deficient number. ! Let us! t ry some initial exampl!e s of Thabit’s formula. ! 2 3 p2 = 3" 2 # 1 = 11, p1 = 3" 2 # 1 = 5 and q2 = 9" 2 # 1 = 71 so a = 2(5)(11) = 220 and b = 4(71) = 284. Here is the old Pythagorean pair of amicable numbers. ! 3 ! 5 p3 = 3" 2 # 1 = 23, p2 = 11 as before and q3 = 9" 2 # 1 = 287 = 7(41) which is not a prime.