The history of mathematics: A brief course (2013)
Part V. Islamic Mathematics, 800-1500
The next three chapters constitute a sampling of mathematical advances from a civilization that flourished over a huge region from Spain to India during the period known in Europe as the Middle Ages.
Contents of Part V
1. Chapter 25 (Overview of Islamic Mathematics) establishes the cultural and historical context of the subject and introduces the major figures and their works to be discussed in the two chapters that follows.
2. Chapter 26 (Islamic Number Theory and Algebra) discusses number theory and algebra from al-Khwarizmi and Thabit ibn-Qurra through Omar Khayyam.
3. Chapter 27 (Islamic Geometry) is devoted to Islamic advances in geometry.
Chapter 25. Overview of Islamic Mathematics
As mentioned in the introduction to this part of the present history, the most important advances in science and mathematics in the West from 700 to 1300 CE came in the lands under Muslim rule.
25.1 A Brief Sketch of the Islamic Civilization
Starting as a small and persecuted sect in the early seventh century, by mid-century the Muslims had expanded by conquest as far as Persia. They then turned West and conquered Egypt, all of the Mediterranean coast of Africa, and the island of Sicily.
25.1.1 The Umayyads
A palace revolution among the Islamic leaders led to the triumph of the first dynasty, the Umayyad (sometimes spelled Ommiad) in the year 660. Under the Umayyads, Muslim expansion continued around the Mediterranean coast and eastward as far as India. This expansion was checked by the Byzantine Empire at the Battle of Constantinople in 717. In the West a Muslim general named Tarik led an army into Spain, giving his name to the mountain at the southern tip of Spain—Jabal Tarik, known in English as Gibraltar. The Muslim expansion in the West was halted by the Franks under Charles Martel at the Battle of Tours in 732. In 750 another revolution resulted in the overthrow of the Umayyad Dynasty and its replacement in the East by the Abbasid Dynasty. The Umayyads remained in power in Spain, however, a region known during this time as the Caliphate of Cordoba.
From the early ninth century on, scholars working under the rule of the caliphs formed a unique tradition within the story of mathematics, sharing a common literature of mathematical classics, communicating with one another, and working to extend the achievements of their predecessors. Their achievements were considerable, and Europeans from the eleventh century on were eager to learn about them and apply them. Because the origin of Islam lies in the Arabic-speaking world, and its holy text is written in Arabic, most of the documents produced within this tradition were written in Arabic by scholars for whom the Arabic language was either native or learned at school. Some non-Arabic writers, especially in the early years, adopted Arabic names. As with Mesopotamian and Greek mathematics, there is some inaccuracy in any name one might choose to refer to this tradition. Should it be called Arabic mathematics because of the language most commonly used to write it, or Islamic mathematics because Islam is the most obvious feature that most of the writers had in common? Whichever name the reader prefers, the important thing is to grasp what the name signifies—that is, the specific sets of questions and problems the mathematicians studied and the approaches they had for solving them.
25.1.2 The Abbasids
Al-Mansur, the second of the Abbasid caliphs, built the capital of the new dynasty, the city of Baghdad, on the Tigris River. Both the Abbasids and the Umayyads cultivated science and the arts, and mathematics made advances in both the Eastern and Western parts of the Islamic world. The story of Islamic mathematics begins in the city of Baghdad in the reign of two caliphs. The first of these was Harun al-Raschid (786–809), a contemporary of Charlemagne. The second is the son of Harun al-Raschid, al-Mamun (813–833), whose court life provided the setting of the Thousand and One Nights.
25.1.3 The Turkish and Mongol Conquests
Near the end of the tenth century a group of Turkish nomads called Seljuks migrated from Asia into the Abbasid territory and converted to Islam. Gradually the Seljuks began to seize territory from the Abbasids, and in 1055 they occupied Baghdad. It was their advance into Palestine that provoked the First Crusade in 1096. The Crusades, which established a Christian-ruled enclave in Palestine, were another source of continuing disruption throughout the twelfth century and even later. The Seljuks left the Abbasids as the nominal rulers of the empire, but in the thirteenth century both Abbasids and Seljuks were conquered by the same Mongols who had earlier overrun Russia and China. The Mongol conquest of Iraq was particularly devastating, since it resulted in the destruction of the irrigation system that had supported the economy of the area for thousands of years. As in China, the Mongol rule was short-lived and was succeeded by another conquest, this time by the Ottoman Turks, who also conquered Constantinople in 1453 and remained a threat to Europe until the nineteenth century. While it lasted, the vast Mongol Empire transmitted mathematical works and ideas over prodigious distances. In particular, astronomical treatises came into China from Persia, along with Arabic numerals (Li and Du, 1987, pp. 171–174).
25.1.4 The Islamic Influence on Science
The portion of the Islamic empire around the Mediterranean Sea was secure from invasion for three hundred years in the East and six hundred in Spain. During this period, Islamic mathematicians assimilated the science and mathematics of their predecessors and made their own unique additions and modifications to what they inherited. For many centuries they read the works of Archimedes, Apollonius, and Euclid and advanced beyond the work of these illustrious Greek mathematicians. The Greek mathematicians, however, were not the only influence on them. From earliest times the Caliph was in diplomatic contact with India, and one of Harun Al-Raschid's contributions was to obtain translations from Sanskrit into Arabic of the works of Aryabhata, Brahmagupta, and others. Some of the translators took the occasion to write their own mathematical works, and so began the Islamic contribution to mathematics.1
In addition to the Arabic translations that preserved many Greek works of which the originals have been lost, the modern world has inherited a considerable amount of scientific and mathematical literature in Arabic. This language has given us many words relating to science, such as alcohol, alchemy, almanac, zenith, and the mysterious names of the stars such as Altair, Aldebaran, Algol, and Betelgeuse. In Spain, the libraries were incomparably richer than those in northern Europe until well past the year 1000, and many scholars from the Christian countries of Europe came there to translate Arabic works into Latin.2
Thus, from the end of the eighth century through the period referred to as Medieval in European history, the Umayyad and Abbasid Caliphates, centered in what is now Spain and Iraq respectively, produced an artistically and scientifically advanced culture, with works on mathematics, physics, chemistry, and medicine written in Arabic, the common language of scholars throughout the Muslim world. Persian, Hebrew, and other languages were also used by scholars working in this predominantly Muslim culture. The label Islamic mathematics that we are going to use has one important disadvantage, since we certainly have no wish to imply that mathematical results valid in one religion are not valid in another. Yet the alternative, Arabic mathematics, also does not seem to fit as well as the corresponding label Greek mathematics, in which the majority of the major authors had Greek as their native language.
25.2 Islamic Science in General
The religion of Islam calls for prayers facing Mecca at specified times of the day. That alone would be sufficient motive for studying astronomy and geography. Since the Muslim calendar is lunar rather than lunisolar, religious feasts and fasts are easy to keep track of. Since Islam forbids representation of the human form in paintings, mosques are always decorated with abstract geometric patterns (see Özdural, 2000). The study of this ornamental geometry has interesting connections with the theory of transformation groups. Unfortunately, we do not have space to pursue this interesting topic, nor the equally fascinating subject of the astrolabe, which was highly developed as an almanac and surveying tool by Muslim scholars.
25.2.1 Hindu and Hellenistic Influences
According to Colebrooke (1817, pp. lxiv–lxv), in the year 773 CE, al-Mansur, the second caliph of the Abbasid Dynasty, who ruled from 754 to 775, received at his court a Hindu scholar bearing a book on astronomy referred to in Arabic as Sind-hind (most likely, Siddhanta). Al-Mansur had this book translated into Arabic. No copies survive. It was once conjectured that this book was the Brahmasphutasiddhanta mentioned in Chapter 21, but Plofker (2009, p. 256) cites two papers of Pingree (1968, 1970) in rejecting this conjecture. This book was used for some decades, and an abridgement was made in the early ninth century, during the reign of al-Mamun (caliph from 813 to 833), by Muhammad ibn Musa al-Khwarizmi (ca. 780–850), who also wrote his own treatise on astronomy based on the Hindu work and the work of Ptolemy. Al-Mamun founded a “House of Wisdom” (Bait al-Hikma) in Baghdad, the capital of his empire. This institution was much like the Library at Alexandria, a place of scholarship, analogous to a modern research institute.
In the early days of this scientific culture, one of the concerns of the scholars was to find and translate into Arabic as many scientific works as possible. The effort made by Islamic rulers, administrators, and merchants to acquire and translate Hindu and Hellenistic texts was prodigious. The works had first to be located, a job requiring much travel and expense. Next, they needed to be understood and adequately translated; that work required a great deal of labor and time, often involving many people. The world is much indebted to the scholars who undertook this work, for two reasons. First, some of the original works have been lost, and only their Arabic translations survive.3Second, the translators, inspired by the work they were translating, wrote original works of their own. The mechanism of this two-part process has been described by Berggren (1990, p. 35):
Muslim scientists and patrons were the main actors in the acquisition of Hellenistic science inasmuch as it was they who initiated the process, who bore the costs, whose scholarly interests dictated the choice of material to be translated and on whom fell the burden of finding an intellectual home for the newly acquired material within the Islamic d
r al-‘ilm (“abode of learning”).
The acquisitions were extensive, and we have space for only a partial enumeration of them. Some of the major ones were listed by Berggren (2002). They include Euclid's Elements, Data, and Phænomena, Ptolemy's Syntaxis(which became the Almagest as a result) and his Geography, many of Archimedes' works and commentaries on them, and Apollonius' Conics. The development process as it affected the Conics of Apollonius was described by Berggren (1990, pp. 27–28). This work was used to analyze the astrolabe in the ninth century and to trisect the angle and construct a regular heptagon in the tenth century. It continued to be used down through the thirteenth century in the theory of optics, for solving cubic equations and to study the rainbow. To the two categories that we have called acquisition and development, Berggren adds the process of editing the texts to systematize them, and he emphasizes the very important role of mathematical philosophy or criticism engaged in by Muslim mathematicians. They speculated on and debated Euclid's parallel postulate, for example, thereby continuing a discussion that began among the ancient Greeks and continued for 2000 years until it was finally settled in the nineteenth century.
The scale of the Muslim scientific schools is amazing when looked at in comparison with the populations and the general level of economic development of the time. Here is an excerpt from a letter of the Persian mathematician al-Kashi (d. 1429) to his father, describing the life of Samarkand, in Uzbekistan, where the great astronomer Ulugh Beg (1374–1449), grandson of the conqueror Timur the Lame, had established his observatory (Bagheri, 1997, p. 243):
His Royal Majesty had donated a charitable gift. . .amounting to thirty thousand. . .dinars, of which ten thousand had been ordered to be given to students. [The names of the recipients] were written down; [thus] ten thousand-odd students steadily engaged in learning and teaching, and qualifying for a financial aid, were listed. . .Among them there are five hundred persons who have begun [to study] mathematics. His Royal Majesty the World-Conqueror, may God perpetuate his reign, has been engaged in this art. . .for the last twelve years.
25.3 Some Muslim Mathematicians and their Works
We now survey some of the more important mathematicians who lived and worked under the rule of the caliphs.
25.3.1 Muhammad ibn Musa al-Khwarizmi
This scholar, who lived from approximately 790 to 850, translated a number of Greek works into Arabic but is best remembered for his Hisab al-Jabr w'al-Mugabalah (Book of Completion and Reduction). The word completion (or restoration) here (al-jabr) is the source of the modern word algebra. It refers to the operation of keeping an equation in balance by adding or subtracting the same terms on both sides of an equation, as in the process of completing the square. The word reduction refers to the cancelation of a common factor from the two sides of an equation. The author came to be called simply al-Khwarizmi, which may be the name of his home town (although this is not certain); this name gave us another important term in modern mathematics, algorithm.
The integration of intellectual interests with religious piety that we saw in the case of the Hindus is a trait also possessed by the Muslims. Al-Khwarizmi introduces his algebra book with a hymn of praise of Allah and then dedicates his book to al-Mamun:
That fondness for science, by which God has distinguished the Imam al-Mamun, the Commander of the Faithful. . ., that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties—has encouraged me to compose a short work on Calculating by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts. . .My confidence rests with God, in this as in every thing, and in Him I put my trust. . .May His blessing descend upon all the prophets and heavenly messengers. [Rosen, 1831, pp. 3–4]
25.3.2 Thabit ibn-Qurra
The Sabian (star-worshipping) sect centered in the town of Harran in what is now Turkey produced an outstanding mathematician/astronomer in the person of Thabit ibn-Qurra (826–901). Being trilingual (besides his native Syriac, he spoke Arabic and Greek), he was invited to Baghdad to study mathematics. His mathematical and linguistic skills procured him work translating Greek treatises into Arabic, including Euclid's Elements. He was a pioneer in the application of arithmetic operations to ratios of geometric quantities, which is the essence of the idea of a real number. The same idea occurred to René Descartes (1596–1650) and was published in his famous work on analytic geometry. It is likely that Descartes drew some inspiration from the works of the fourteenth-century Bishop of Lisieux Nicole d'Oresme (1323–1382); Oresme, in turn, is likely to have read translations from the Arabic. Hence it is possible that our modern concept of a real number owes something to the genius of Thabit ibn-Qurra. He also wrote on mechanics, geometry, and number theory.
25.3.3 Abu Kamil
Although nothing is known of the life of Abu Kamil (ca. 850–930), he is the author of certain books on algebra, geometry, and number theory that influenced both Islamic and European mathematics. Many of his problems were reproduced in the work of Leonardo of Pisa (Fibonacci, 1170–1250).
25.3.4 Al-Battani
Another Sabian from Harran, Abu Abdallah Muhammad al-Battani, known in Latin translation as Albategnius, seems to have abandoned the Sabian beliefs of his parents and converted to Islam. That, at least, is what has been inferred from his Muslim name. Since he himself reported making astronomical observations in the year 877, he must have been born some time during the 850s. He died around 929. He worked in al-Raqqa in what is now Syria, on the Euphrates River. His best-known work is the Kitab al-Zij (Book of Astronomy). The word zij apparently comes from Persian, where it means a certain strand in a rug.
The first three of the 57 chapters of al-Battani's book contain a development of trigonometry using sines, one that has been claimed to be independent of the work of Aryabhata I. Obviously, however, he must have known something about Aryabhata's works, or else he would have invented an Arabic name for the sine, instead of borrowing the Sanskrit j-y-b that will be discussed in Chapter 27.
25.3.5 Abu'l Wafa
Muhammad Abu'l Wafa (940–998) was born in Khorasan (now in Iran) and died in Baghdad. He was an astronomer–mathematician who translated Greek works and commented on them. In addition he wrote a number of works on practical arithmetic and geometry. According to R¯ashid (1994), his book of practical arithmetic for scribes and merchants begins with the claim that it “comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know” in relation to taxes, business transactions, civil administration, measurements, and “all other practices. . .which are useful to them in their daily life.”
25.3.6 Ibn al-Haytham
Abu Ali al-Hasan ibn al-Haytham (965–1039), known in the West as Alhazen, was a natural philosopher who worked in the tradition of Aristotle. He continued the speculation on the parallel postulate, offering a proof of it that was, of course, flawed. He is famous for Alhazen's problem in optics, which is to determine the point on a reflecting spherical surface at which a light ray from one given point P will be reflected to a second given point Q.
25.3.7 Al-Biruni
Abu Arrayhan al-Biruni (973–1048), was an astronomer, geographer, and mathematician who as a young man worked out the mathematics of maps of the earth. Civil wars in the area where he lived (Uzbekistan and Afghanistan) made him into a wanderer, and he came into contact with astronomers in Persia and Iraq. He was a prolific writer. According to the Dictionary of Scientific Biography, he wrote what would now be well over 10,000 pages of texts during his lifetime, on geography, geometry, arithmetic, and astronomy.
25.3.8 Omar Khayyam
The Persian mathematician Omar Khayyam, also known as Umar al-Khayyam, was born in 1044 and died in 1123. He is thought to be the same person who wrote the famous skeptical and hedonistic poem known as the Rubaiyat(Quatrains), but not all scholars agree that the two are the same. Since he lived in the turbulent time of the invasion of the Seljuk Turks, his life was not easy, and he could not devote himself wholeheartedly to scholarship. Even so, he advanced algebra beyond the linear and quadratic equations discussed in al-Khwarizmi's book and speculated on the foundations of geometry. He explained his motivation for doing mathematics in the preface to his Algebra. Like the Japanese wasanists, he was inspired by questions left open by his predecessors. As with al-Khwarizmi, this intellectual curiosity is linked with piety and with gratitude to the patron who supported his work:
In the name of God, gracious and merciful! Praise be to God, Lord of all Worlds, a happy end to those who are pious, and ill-will to none but the merciless. May blessings repose upon the prophets, especially upon Mohammed and all his holy descendants.
One of the branches of knowledge needed in that division of philosophy known as mathematics is the science of completion and reduction, which aims at the determination of numerical and geometrical unknowns. Parts of this science deal with certain very difficult introductory theorems, the solution of which has eluded most of those who have attempted it. . .I have always been very anxious to investigate all types of theorems and to distinguish those that can be solved in each species, giving proofs for my distinctions, because I know how urgently this is needed in the solution of difficult problems. However, I have not been able to find time to complete this work, or to concentrate my thoughts on it, hindered as I have been by troublesome obstacles. [Kasir, 1931, pp. 43–44]
25.3.9 Sharaf al-Tusi
Sharaf al-Din al-Tusi (ca. 1135–1213) is best remembered for work on cubic equations. Judging from the name al-Tusi, he must have been born near the town of Tus in northeastern Iran. Like Omar Khayyam, he lived in turbulent times. The Seljuk Turks had captured Damascus in 1154 and established their capital in that city. Sharaf al-Tusi is known to have taught there around 1165 and to have moved from there to Aleppo (also in Syria).
25.3.10 Nasir al-Tusi
Nasir al-Din al-Tusi (1201–1274) had the misfortune to live during the time of the westward expansion of the Mongols, who subdued Russia during the 1240s and then went on to conquer Baghdad in 1258. Al-Tusi himself joined the Mongols and was able to continue his scholarly work under the new ruler Hulegu, grandson of Genghis Khan. Hulegu, who died in 1265, conquered and ruled Iraq and Persia over the last decade of his life, taking the title Ilkhanwhen he declared himself ruler of Persia. A generation later the Ilkhan rulers converted from Buddhism to Islam. Hulegu built al-Tusi an observatory at Maragheh, a city in the Azerbaijan region of Persia that Hulegu had made his seat of government. Here al-Tusi was able to improve on the earlier astronomical theory of Ptolemy, in connection with which he developed both plane and spherical trigonometry into much more sophisticated subjects than they had been previously, including the statement that the sides of triangles are proportional to the sines of the angles opposite them. Because of his influence, the loss of Baghdad was less of a blow to Islamic science than it would otherwise have been. Nevertheless, the constant invasions had the effect of greatly reducing the vitality and the quantity of research. Al-Tusi played an important role in the flow of mathematical ideas back into India after the Muslim invasion of that country; it was his revised and commented edition of Euclid's Elements that was mainly studied (De Young, 1995, p. 144).
Questions
Historical Questions
25.1 Describe the general history of Muslim expansion and political decline over the period from the eighth to fifteenth centuries.
25.2 Who were the major mathematicians working within the world of Islamic scholarship during this time, and what topics did they develop?
25.3 What justifications do al-Khwarizmi and Omar Khayyam give in the prefaces to their work for the algebra that they develop?
25.4 In what way was Nasir al-Tusi's trigonometry an advance on the subject as inherited from the Hindu mathematicians?
Questions for Reflection
25.5 How did the conquests by different groups of Muslims affect the course of scholarship in the conquered areas (Spain, Mesopotamia, India, China)?
25.6 How did the Islamic injunction against representation of the human body in art influence art and architecture in the Islamic countries?
25.7 If one needs to pray facing Mecca while living in (say) Chicago, how is “facing Mecca” to be interpreted? How can one work out how to face Mecca from Chicago? This problem is not difficult to solve using spherical trigonometry. To find out how al-Biruni solved it, see the book by Berggren (1986, pp. 182–186).
25.8 An expository short book (Brett, Feldman, and Sentlowitz, 1974) giving some history of mathematics contains the following statement (p. 41) about Islamic mathematics:
It is often said that the Arabs were learned but not original; thus, they played the role of preservation rather than invention of knowledge. Even if we believe this description of them, we must be forever grateful for the benevolent custody by the Moslems of the world's intellectual possessions which might otherwise have been lost forever in the mire of the Dark Ages.
It is to their credit that the authors do not endorse what they report as a popular impression of the Islamic world. Most Western historians have given that culture credit for outstanding achievements in art, literature, and science. The charge of a lack of creativity is also sometimes made against Byzantine Empire contemporaneous with the Islamic—again unfairly, since its wealth and geographical range were tiny by comparison with the world of Islam. Here, for example, is what the British philosopher Bertrand Russell (1872–1969) said about it (1945, p. xvi):
In the Eastern Empire, Greek civilization, in a desiccated form, survived, as in a museum, till the fall of Constantinople in 1453, but nothing of importance to the world came out of Constantinople except an artistic tradition and Justinian's Codes of Roman law.
Russell did not disparage Islamic science, but in his own area of philosophy, he did tend to look down on Islamic scholarship, saying (p. 417)
Arabic philosophy is not important as original thought. Men like Avicenna [ibn Sina (980–1037), Persian physician] and Averroes [ibn Rushd (1126–1198), Spanish philosopher], are essentially commentators.
Whether this negative opinion is justified or not is a matter for philosophers to discuss, and any opinion by a non-philosopher would be rash. Let it be said, however, that important is a word whose meaning may vary from one philosopher to another.
Western writers, it is true, sometimes overlook Islamic contributions and slight them with silence. For example, Kline (1953, p. 93) in discussing the Medieval period in Europe, says:
The progress that was made during this period was contributed by the Hindus and Arabs. . .
He goes on to list a number of Hindu mathematical discoveries, and then finishes with this comment:
These and other Hindu contributions were acquired by the Arabs who transmitted them to Europeans.
Kline was simply writing carelessly here. He knew better, and he gave more detailed discussions of the Islamic contributions in his later, encyclopedic work (Kline, 1972).
Giving these authors the benefit of the doubt, since no one can discuss every single meritorious deed in any history, how is it possible to write concisely, yet with fairness to the subject? If you were editing the works just quoted, how would you advise the authors to recast these sentences?
Notes
1. Plofker, (2009, p. 258), however, cautions against assuming that algebra among the Muslims had its roots in these Sanskrit works, pointing out that the works written in Arabic do not use negative numbers.
2. Constantinople, which had preserved its independence, continued a mathematical tradition until the fifteenth century, and it also was an important source of ancient works for the Europeans. Unfortunately, we do not have space to discuss the details of that recovery effort.
3. Toomer (1984b) points out that in the case of Ptolemy's Optics the Arabic translation has also been lost, and only a Latin translation from the Arabic survives. As Toomer notes, some of the most interesting works were not available in Spain and Sicily, where medieval scholars went to translate Arabic and Hebrew manuscripts into Latin.