Arabic Mathematics and History

Everybody would agree that Modern Algebra, Geometry, Astronomy, and even Geography, owes the great deal to the Arab mathematics. Just as George Sarton, the famous Harvard professor of history and science wrote in his famous book “Introduction to the History of Science”:

“From the second half of the eighth to the end of the eleventh century, Arabic was the scientific, the progressive language of mankind. When the West was sufficiently mature to feel the need of deeper knowledge, it turned its attention, first of all, not to the Greek sources, but to the Arabic ones.” (A al’Daffa p.65).

Before we proceed, it is worth trying to define the period that this essay will cover and give an overall description to the Arab mathematicians who contributed. The period we will be covered is easy to describe: it stretches from the end of the Eigth century to about the middle of the Fifteenth century. Most of the Arab mathematicians were Muslims, but some of them were Jews, Christians, and some of other faiths. Not all of these mathematicians were Arabs, but for convenience we will call our topic “Arab Mathematics”.

The regions from which the “Arab mathematicians” came was centered on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.

The background to the mathematical developments, which began in Baghdad around 800, is not well understood. Certainly there was an important influence, which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important.

In the 12th century, Europe learned about the scientific progress of the Arabs and took much of the achievement, transformed them to the modern mathematics. A special college for translators was founded in Toledo, Spain, and it was there, and in other centers, that some of the great Christian scholars translated most of the Arabic works on mathematics and astronomy. In most European universitie5 Arab treatises formed the basis of mathematical studies.

Muhammad ibn Musa al-Khawarazmi started the history of Arab Mathematics in the 9th century. He went to the East India to take the courses of sciences of that time. Muhammad introduced the Hindu numerals; thanks to him we know and understand the concept of Zero. This number system was later transmitted to the West. Prior to the use of” Arab” numerals, as we know them today, the West relied upon the somewhat clumsy system of Roman numerals. Whereas in the decimal system, the number 1948 can be written in four figures, eleven figures were needed using the Roman system: MDCCCXLVIII. It is obvious that even for the solution of the simplest arithmetical problem, Roman numerals called for an enormous expenditure of time and labor. The Arab numerals, on the other hand, rendered even complicated mathematical tasks relatively simple.

The scientific advances of the West would have been impossible had scientists continued to depend upon the Roman numerals and been deprived of the simplicity and flexibility of the decimal system and its main glory, the zero. Though the Arab numerals were originally a Hindu invention, it was the Arabs who turned them into a workable system; the earliest Arab zero on record dates from the year 873, whereas the earliest Hindu zero is dated 876. For the subsequent four hundred years, Europe laughed at a method that depended upon the use of zero, “a meaningless nothing.”

Have the Arabs given us nothing but the decimal system, their contribution to progress would have been considerable. In actual fact, they gave us infinitely more. While religion is often thought to be an impediment to scientific progress, we can see, in a study of Arab mathematics, how religious beliefs actually inspired scientific discovery.
Because of the Quran’s very concrete prescriptions regarding the division of an estate among children of a deceased person, it was incumbent upon the Arabs to find the means for very precise delineation of lands. For example, let us say that the father left an irregularly shaped piece of land-seventeen acres large-to his six sons. Each OAA~ of whom had to receive precisely one-sixth of his legacy. The Arab mathematics was inherited from the Greeks, who made such division extremely complicated, if not possible. It was the search for a more accurate, more comprehensive, and more flexible method that led Khawarazmi to the invention of algebra. According to Professor Sarton, Khawarazmi “influenced mathematical thought to a greater extent than any other medieval writer.,” Both algebra, in the true sense of the term, and the term itself (al-jabr) we owe to them. Apart from Arab mathematics, Khawarazmi also did pioneer the work in the fields of astronomy, geography and theory of music.

Omar Khvyym, who lived in twelfth century, was the other man whose achievement helped the Arab civilization and mathematics make a huge leap forward, two centuries after Khawarazmi. Known in the West, as the author of Rubayat, a poem made famous by Edward Fitzgerald’s translation, he was admired in the East mainly as a mathematician. In his use of analytical geometry, he anticipated the geometry of Descartes.

Commissioned by the Seijuk Sultan Halikshah to reform the Persian calendar, he prepared a calendar said to be more accurate than the Gregorian one in use to the present day. Whereas the latter leads to an error of one day in 3,300 years, in Omar Khayyam’s calendar that error is one day in 5,000 years.

Because of their Islamic faith, it was essential for the Arabs to obtain a more precise knowledge of astronomy and geography than was already available: a Muslim is obliged to perform a number of religious observances with distinctly astronomic geographical implications. When he prays, he must face Mecca; if he wishes to perform the pilgrimage to Mecca, he must first know in what direction and what distance he will have to travel.

Yet a thousand years ago such a journey might take months or even years, for the would-be pilgrim might have been living in Spain, Sicily or Asia Minor-all of these forming parts of the medieval Arab Empire. During Ramadan, the month of fast, when between sunrise and sunset he has to abstain from food and drink, he must know in advance the precise moment at which the moon rises and sets. All these functions required a detailed knowledge of astronomy and geography.

It was, thus, under the great Caliph Ma’mun (813-833) that the Arabs set out upon their astronomical investigations. Ma’mun-a son of Harun al-Rashid of Arabian Nights fame-built a special observatory in Palmyra, Syria, and gradually, his scientists determined the length of a degree, thus establishing longitude and latitude.

Among the Arabs who laid the foundations for modern astronomy were Battani (858-929) and Biruni (973-1048). Battani’s astronomical tables were not only adopted enthusiastically by the West, but were in use there until the Renaissance. He was the first to replace the Greek chord by the sine, in trigonometry. His works were translated and published in Europe from the twelfth until the mid-sixteenth century.

Professor Sarton considers Biruni “One of the very greatest scientists of all time.” It was he who gave, finally, an accurate determination of latitude and longitude, and who, six hundred years before Galileo, discussed the possibility of the earth’s rotation around its own axis. He also investigated the relative speeds of sound and light.

However much astronomy depends upon Arab mathematics, equally vital to it are instruments, and in that field, also, the Arabs proved themselves the chief pioneers. In the early middle Ages, measurements had to be made with purely mechanical instruments, such as the quadrant, the sextant, or the astrolabe. To reduce the margin of error, the Arabs made their instruments larger than any known before and, consequently, obtained remarkably accurate results. The most famous observatory at which these instruments were being used was at Maragha, in the thirteenth century, where distinguished astronomers from many countries collaborated-not only Muslim, Christian and Jew, but even Chinese. It was the latter who were responsible for the otherwise surprising appearance of Arab trigonometry in China.

It has already been indicated that, in the hands of the Arabs, mathematics acquired a new “dynamic” quality. We find this in Biruni’s trigonometry, where numbers became elements of function, and in Khawarazmi’s algebra, where the algebraic symbols contain within themselves potentialities for the infinite. What is significant about this development is that it reveals an intuitive correspondence between mathematics and religion. The Quran does not present the universe as finally created or as a finished “article.” Rather, God keeps re-creating it at every moment of existence. In other words, creation is an ever-living process, and the world is not static but dynamic. This dynamic character, inherent in Islam, is amply manifested in Arab mathematics.
We should emphasize that the translations into Arabic at this time were made by scientists and mathematicians such as those named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. It is important to realize that the translating was not done for its own sake, but was done as part of the current research effort. The most important Greek mathematical texts, which were translated, are listed below:

“Of Euclid’s works, the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes’ works only two – Sphere and Cylinder and Measurement of the Circle – are known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century. On the other hand, virtually all of Apollonius’s works were translated, and of Diophantus and Menelausone book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy’s Almagest furnished important astronomical material”. (R Rashed p.115)

Although the Arabic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan who was born in 908, introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.

Astronomy, time keeping and geography provided other motivations for geometrical and trigonometrically research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu’l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan. Al-Biruni (born 973) used the sin formula in both astronomy and in the calculation of longitudes and latitudes of many cities. Again both astronomy and geography motivated al-Biruni’s extensive studies of projecting a hemisphere onto the plane.

Thabit ibn Qurra undertook both theoretical and observational work in astronomy. Al-Battani (born 850) made accurate observations which allowed him to improve on Ptolemy’s data for the sun and the moon. Nasir al-Din al-Tusi (born 1201), like many other Arabic mathematicians, based his theoretical astronomy on Ptolemy’s work but al-Tusi made the most significant development of Ptolemy’s model of the planetary system up to the development of the heliocentric model in the time of Copernicus.

Many of the Arabic mathematicians produced tables of trigonometric functions as part of their studies of astronomy. These include Ulugh Beg (born 1393) and al-Kashi. The construction of astronomical instruments such as the astrolabe was also a speciality of the Arabs. Al-Mahani used an astrolabe while Ahmed (born 835), al-Khazin (born 900), Ibrahim ibn Sinan, al-Quhi, Abu Nasr Mansur (born 965), al-Biruni, and others, all wrote important treatises on the astrolabe. Sharaf al-Din al-Tusi (born 1201) invented the linear astrolabe. The importance of the Arabic mathematicians in the development of the astrolabe is described in:

“The astrolabe, whose mathematical theory is based on the stereographic projection of the sphere, was invented in late antiquity, but its extensive development in Islam made it the pocket watch of the medieval. In its original form, it required a different plate of horizon coordinates for each latitude, but in the 11th century the Spanish Muslim astronomer az-Zarqallu invented a single plate that worked for all latitudes. Slightly earlier, astronomers in the East had experimented with plane projections of the sphere, and al-Biruni invented such a projection that could be used to produce a map of a hemisphere. The culminating masterpiece was the astrolabe of the Syrian Ibn ash-Shatir (1305-75), a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.” (J L Berggren p.90).

Recent research paints a new picture of the debt that we owe to Arabi mathematics. Certainly many of the ideas, which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth, and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, Arab mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.

In conclusion, it is clear that Arab mathematicians, besides passing on to the West the Hindu and Greek legacies, have developed most branches of trigonometry and astronomy, have given us algebra, have invented many astronomical instruments, and have shown that science, instead of being a denial of faith, can be its instrument if not its affirmation.

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