The contributions of Arabic cultures to the history of mathematics is often underrepresented. In particular, scientific activity in the near east between 900 and 1500a.d. was vital in maintaining the continuum of knowledge between ancient and modern scholars. The story of the preservation, expansion, and transmission of trigonometry is just one example of the accomplishments of the medieval Moslem people.

 The World Context

After the fall of the Western Roman Empire in the late 5th century, the Byzantine and Persian Empires dominated the world scene. The Byzantine Empire, centered at Constantinople, maintained complex relations with the Eastern Orthodox Catholic Church. Within this reign, Greek remained a common language among the native tongues of the Syrians and Arabs. Because Greek continued to be spoken and scholars still had a direct access to many Hellenistic libraries, there was little concern for the preservation or translation of ancient documents (Neugebauer). In fact, the era following the fall of Rome and preceding the rise of Islam was marked by extreme intellectual neglect. The famous library at Alexandria was already in terrible disrepair when it was burned by invading Arabs in the 8th century.

Beginning in the 7th century, the Islamic faith culturally unified a vast region from Spain to India. While the conquering Arab empire did not force people to adopt the Moslem religion, it did require the Arabic language to be spoken universally. One consequence of this trend was the motivation for scholars to translate ancient Greek and contemporary Indian works into Arabic (Neugebauer). Initially, the new Islamic nation was not an educated civilization, but within a few hundred years of their exposure to Greek and Hindu scholarship, the Arab peoples had begun to quickly assimilate the knowledge and intellectual curiousity of their captors. Translations represented both the cause and the result of much of Arabic scholarship. Such labors, begun even prior to the beginning of the second millenium, were essential in preserving the text of manuscripts that are no longer available in the original. The ninth and tenth centuries marked the golden age of Arab-Moslem culture. Baghdad replaced Alexandria as the center of mathematical activity. Even though the Islamic state had begun to disintegrate at the turn of the millenium, the impact of the Islamic culture continued to unify the activity of scientists from Spain to India for centuries to come.

The shift of powers at the turn of the millenium is far more complex than the scope of this paper can begin to cover. Hopefully, a few general remarks can suffice to set the background. The Byzantine Empire which arose in the 5th century had passed its peak and would officially end when Constantinople fell to the Turks in 1453. Persia experienced a period of significant academic and artistic activity in the 10th century, even though they were successively dominated by the Arabic and Ottoman (Turkish) empires. The Arabic Empire, after rapid, expansive growth, declined as a result of internal religious and political divisions, but still left behind permanent cultural influences. And looking toward the future, it is important to keep in mind that the Ottoman Empire (culturally Islamic) and the Holy Roman Empire (Western Christian) were in the midst of respectively consolidating power.

This series of changing powers was significant in exposing different peoples to on another and fostering interational trade and communication. They also reflect which cultures dominated the intellectual world. The rise of European power coincided with its cultural renaissance just as Islamic advances reigned in the ninth and tenth century. Unfortunately, this simplistic perspective often neglects the accomplishments of individual scholars, but it does help to keep in mind how the bulk of knowledge passed from one culture to the next.

The eleventh through thirteenth centuries were highlighted by the European Crusades to regain control of the Holy Land from the Turks. The significance of these wars to European history is the transfer of knowledge as a result of the proximity of Western and Arabic peoples. Histories of science often attribute a great deal of honor to Western scribes working during this period to translate the ancient works into Latin. However, as we have noted already, it was the activity of the Arabic scholars, beginning centuries earlier and continuing even after the crusades, which was most essential to the preservation and transfer of knowledge from both ancient and contemporary texts.

The Scientific Context

Any mathematician at this time concerned with trigonometry was an astronomer first. Like artisans in Europe, Arabic scientists and scribes were employed by the ruling class, primarily the sultans or caliphs, the religious and political rulers of Islam. The greatest astronomers were provided observatories by their rulers and "schools" became associated with these courts where astronomers worked together.

Moslem work in astronomy seemed to begin suddenly in Baghdad in the ninth century at the Abbasid caliphate's court (Neugebauer). Their scholarship arose from the study of foreign works, beginning with Indian and Persian manuscripts, like the Surya Siddhanta, and later from the study of Greek works, especially in the Ptolemaic tradition. The fact that Indian astronomy, itself built upon Babylonian and Greek traditions, was known to the Moslem astronomers before Ptolemy's system is interesting because often when the Hindu and Greek methods or notation were different, the Arabs used the Hindu form, thereby making it more standard in the mathematical community (Kennedy). This tendency may also have had a political intention to undermine Greek culture during the spread of Islam (Boyer). The two primary examples are the fact that Hindu numerals were chosen over Greek, and the sine function took precedence over chords.

The Arabs contributed to the knowledge of trigonometry through the translation and study of Greek and Hindu texts as well as through their own observations and studies. It is difficult to separate the two activities, since much original work seemed to be inspired in the very process of translation. Perhaps this was merely because of the exposure to interesting problems. More likely, damaged texts left gaps in proofs and translators took it upon themselves to reconstruct the missing pieces. And of course, most discoveries were sought primarily for furthering astronomical observations. Nevertheless, many of the astronomers which will be discussed later engaged in both the review of literature and the contribution of original material to the mathematical body of knowledge.

Here, however, it is relevant to mention two men who were primarily known for their translations. Hunain ibn Ishaq (808-873) was a Christian physician trained in both Baghdad and Greece. He held the post of chief physician to the court of Caliph al-Mutawakkil for most of his life, and had an excellent familiarity with many languages. Most of his textual research was spent collating various copies and sources of works for corrections, after which he translated first from Greek to Syriac and then to Arabic (Al-Daffa, O’Connor).

Thabit ibn Qurra, born in 826 in Harran, was able to obtain mathematical training in Baghdad as a result of family fortunes. Aftwerwards, he returned home, but his philosophies had him branded a 'heretic'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad. Like Hunain, he also had a fluid command of languages, part of the skills which gained him the appointment in Baghdad. There he translated Appolonius' Conics, Euclid's Elements, Ptolemy's Almagest, and Nicomachus' Arithmetic. He also offered several original contributions including a treatise on cylindrical sections and theories for the composition of ratios. Both mens' works were later translated into Latin by Gerard of Cremona (1114-1187). Gerard, while a prolific translator, did not have the fluent command of language that his arabic predecessors had. He learned Arabic only for the sake of his work, which was a very literal, word-for-word translation (Al-Daffa, O’Connor).

These different styles of translation are significant. Scribes like Hunain and Thabit were more likely to translate meanings rather than an author's own words. For mathematics this is especially important since a literal translation may or may not always give a clear description of the concept at hand. A modern reader of ancient texts often comes away thinking too much about the awkardness of the language used. It is important to remember how many sources of translation have altered the words from their original expression.

Trigonometry

The study of the relationships of parts of triangles goes back as far as any recorded mathematical activity. The significant milestones are noted here for context. There are two very different sources of development of our six modern trigonometric functions. Sine and its complement cosine originated with the Greek studies of chords of a circle whose length could be calculated relative to a central angle by constructing inscribed polygons. The Hindu peoples were the first to introduce the modern sine function which was equivalent to the "half-chord" of the Greeks (Eves).

Tangents and cotangents were known as shadow functions. Heights and lengths which were not easily measured directly could be calculated using the shadow an object cast. Secant and cosecants represented the hypotenuse of these shadow functions and were among the last functions to be tabulated (Kennedy).

Many tables of function values of varying accuracies from both Greek and Indian scholars had been computed by the time Arab mathematicians joined the discussion. Several Indian works were used in early Arab studies, primarily the Sindhind and the Surya Siddhanta (Kennedy). Ptolemy's Almagest, introduced later, was the definitive Greek text on astronomy and trigonometry.

Much of the work of astronomers in medieval Arabia was the mundane, but astronomically important work of computing increasingly accurate tables of trigonometric values. It is interesting to note, that while tables were often written sexigesimally, only the greatest mathematicians actually computed in sexigesimals. Most computed in decimals, a method learned earlier with the reception of the Hindu numeral system (Kennedy).

The Men

Abu'l-Wafa

Mohammad Abu'l-Wafa al'Buzjani was born in June 940 in the Khorasan region known today as Iran. He is one of the most prolific and well-known astronomer-mathematicians of medieval Arabia. He worked at the observatory of Sharaf al Daula in Baghdad and wrote many commentaries, now lost, on the works of Euclid, Diophantus, and al-Khwarizmi. Like his peers, his emphasis for studying trigonometry was for the sake of astronomy. Several of his trigonometric results were applied by al'Biruni to the main problem in mathematical geography, namely the determination of latitude and longitude (O’Connor).

Abu'l Wafa's proof of the "rule of four quantities" is exemplary of a transitional work in trigonometry. Formerly, triangle relations were studied using Menelaus' theorem which required six quantities. Where Menelaus used spherical quadrilaterals to establish relationships, Abu'l Wafa actually used triangles to reduce the number of quantities needed. The relationship between sides a, a', c, and c' of two similar triangles was found to be sin(a)/sin(a')=sin(c)/sin(c'). This work is considered transitional because it does not yet utilize angles, only the sides of triangles. The proof uses a common technique of constructing plane objects inside a sphere to prove relationships on the surface of the sphere (Kennedy).

Not long afterward, three astronomers, including Abu'l Wafa simultaneously proved the law of sines. Ptolemy had discussed it and Brahmagupta seems to have had some familiarity with the rule, but it was the Arabic formulation and proof which presented the idea clearly to the larger mathematical community (O’Connor). The formulation made explicit the relationship among spherical triangles. The proof used only great circles working on the surface of the sphere and explicitly used functions of angles. This use of angles marks a significant development in trigonometry, as does Abu'l Wafa and al-Biruni's move to use a radius length of one in their calculations (Kennedy).

By the end of the ninth century, Abu’l Wafa had tabulated all six modern trigonometric functions at 15 degree intervals to eight decimal places of accuracy versus Ptolemy's three digits (Boyer). He also made explicit the complementary relations between functions. It was evident even then that the varying values for radii were a great inconvenience in computation and collaboration. Habash (c.830) used R=60 which is equal to 1,0 in the sexagesimal system. This offered many advantages and was widely used. It is interesting, however, that Abu’l Wafa’s initiative to use one was not given significant attention until relatively recently (Kennedy). Abu'l-Wafa also proved the equivalents of many important trigonometric identities, including: 1.) sin2x = 2sinx cosx ; 2.) tanx = sinx/cosx = 1/cotx ; and 3.) secx = sqrt(1+tan²x).

Al-Biruni

Abu Arrayhan Muhammad ibn Ahmad al'Biruni was born in September of 973 in Khwarazm which is today known as Uzbekistan. He is the most widely travelled of the mathematicians discussed here and represents one way in which Hindu scholarship travelled to Arabia. Al'Biruni, whether by force or voluntarily is uncertain, accompanied his ruler, Mahmud of Ghazna, on military campaigns to India during the expansion of the Arabic empire. Al'Biruni's most famous book is a survey of Indian life, language, religion and culture. He also authored an extremely useful compilation of oriental studies in trigonometry, or shadow lore. These works and others provided insight into relationships between Greek and Hindu scholars. They also conveyed the accomplishments of both cultures to Arabia, and thus to the West (Kennedy, O’Connor).

Jabir ibn Aflah ("Geber")

Jabir's work is exemplary of the Arabic contribution to the transfer of knowledge to the Latin west. Known most commonly in Europe by the name "Geber," his true identity was obscured until very recently, and even now, little is known of his life. Scholars estimate that his most famous work, the Islah al-majasti, an extensive exposition and critique of the Almagest, was written in the 12th century.

The Islah al-majasti was translated within the century that it was written by Gerard of Cremona. It was through this singular work that the most significant details of Arabic and Greek trigonometry were initially conveyed to the Latin West. Jabir can not take credit for originating much of the mathematics in his treatise, but like Euclid, his axiomatic versus procedural organization of the theorems made his text popular among scholars.

The Islah al-majasti is essentially a variation of Ptolemy's Almagest. The original information is organized into nine books in such a way that theorems build upon each other. Originally, material was presented and presented again as necessary for individual problems. Jabir's re-organization significantly reduces the text to a more easily comprehensible size. In addition, however, Jabir includes a great deal of improvements and critiques of Ptolemy's work. Much of this knowledge is gleaned from Hindu scholarship, like the use of sines versus chords, or from his Arabic predecessors Abu'l Wafa and al'Biruni who contributed the "rule of four quantities" and the law of sines.

Jabir reduces all of Ptolemy's problems to the task of solving triangles versus quadrilaterals, usually by making two right triangles. It is interesting that he uses sines for solving spherical triangles, but uses chords for solving plane triangles. There is no use of the shadow functions, which had been discovered by that time. For this reason, Jabir's work is sometimes criticized as being less sophisticated than his predecessors (Lorch).

"Nasir Eddin"/ "Al-Tusi"

Nasir al-din al-Tusi was born in 1201 in the area of present day Iran, but like most of his peers, lived and worked in Baghdad. Dissatisfied with his position as astrologer to the Isma'ili governor, he attempted to join the caliph's court, but was imprisoned by terrorists in the castle of Alamut. In 1256, invading Mongols offered al-Tusi the perfect opportunity for revenge. He joined the enemy and betrayed the city's defences. Al-Tusi was rewarded by Ilkhanid Hulagu with the opportunity to design and construct his own observatory.

Al-Tusi contributed to the transfer of ancient texts with improved translations of Euclid, Ptolemy, and others. One of his most important contributions to mathematics, however, was the a text in which trigonometry was treated as a mathematical discipline in its own right rather than as a tool for astronomy (O’Connor).

Ulugh Beg

Ulugh Beg was born in present day Iran in 1393. The son of political and military leaders, he inherited the city of Samarkand and became the ruler of Turkestan despite the fact that he was primarily an astronomer. One of the greatest observatories of that time was Ulugh's immense three level, circular structure. Many Moslem mathematicians and astronomers spent their time at the observatory in Samarkand, including Al'Kashi and Kadizada. Work at the observatory resulted in corrections to errors in Ptolemy's computations. Accurate data on the length of the year, positions of stars, calendar calculations, and table of sines and tangents to one degree intervals and eight decimal places, were all computed at the observatory by groups of astronomers in the 15th century.

Unfortunately, Ulugh Beg did not inherit his family's political prowess. Upon his father's death he was unable to maintain control of his kingdom and was put to death by his own son (O’Connor).

Others

Abu Allah Mohammad ibn Jabir al'Battani was born around 850 in Syria. An astronomer first, he was among the new group of mathematicians who used trigonometric instead of geometric methods to solve problems. His book, On the motion of the stars, gives both astronomical tables and trigonometric formulas.

Abu Nasr Mansur ibn Ali ibn Iraq, born in 970 in Khwarazm, now Uzbekistan, was a student of Abu'l Wafa and a teacher to Al'Biruni, although all three were contemporaries. He was among the discoverers of the law of sines.

Ghiyath al'Din Jamshid Mas'ud al'Kashi was born in 1390 in Iran and served at the court of Ulugh Beg in Samarkand. He is most famous for accurately computing sin(1) as a root of a cubic equation. He also considered himself the inventor of decimal fractions (O’Connor).

Significance for Secondary Mathematics

Unlike generations past, today’s youth seem to have little interest in the mysteries of the stars. Perhaps because so much is known that little mystery seems to remain. One of the great purposes of teaching is to arouse the instinctive curiosities within students. The world of astronomy is just one example of a field which may hold more innate interest for students than mathematics, and thus can serve as a powerful intermediary. Just as these mathematicians rarely pursued mathematics for its intrinsic value, so also we should provide students with a legitimate for learning new skills.

The history of trigonometry, perhaps more than any other field in mathematics was directed by the need for tools of astronomy and the availability of mathematical discoveries like analysis. As teachers, I believe it is valuable to adopt a frame of reference for ourselves and our students that is similar to that of the medieval Arabic astronomers. They were studying, learning, re-learning, improving, and inventing mathematics as needed and as they were inspired by the work of others. Incorporating the historical references of mathematical development should be used to create and reinforce an environment which reflects this genuine nature of mathematics.

Bibliography