Islamic Civilization Heritage: Mathematics

When Europe was entrapped in the darkness of the Middle Ages, other cultures were instead shining in the light of civilization. The medieval Islam in particular was glowing for its numerous achievements in science and technology. One of the subjects the modern world is much indebted is mathematics as the backbone of technological development. 

The relic of the Muslims contributions can be embedded to the very fundamental of mathematics which is the numbering or numeral system. Prior the Middle Ages, numeral systems were mostly developed from letters or alphabetical symbols. Evidence can be seen in the Chinese, or the Roman systems. The other famous system is the Babylonian sexagesimals -numerals based on ‘sixtieth’ as a central- which has been prevalently used in astrology and astronomy (see Figure 1). Along with cultural progress, those systems prove to be impractical in performing advanced mathematical operations. Imagine how complicated the work is, in the Roman systems, when we want to multiply DXXX (530) with XXXIX (39). Even though there is an existing method to do, the use of Roman system in mathematical manipulations is far from being fun. Using a method taught in nowadays schools, most will find it simple to give 20670 as the solution. 

  Figure 1. The sexagesimal presented in Arabic letters. Note the first four letters (reading from right to left); ‘alif’, ba’, ‘jim’ and ‘dal’ construct the word abjad, a term for a less known numeral system which still applies in some Arabic cultures (taken from: Islamic Science and Engineering, by Donald Hill)

The simplification of the numeral system was initiated by Muslim scholars in the 8-9^th century AD during the Abbasid Baghdad period. Having adopted a system from Brahmi culture, Muslim mathematicians, notably Al –Khawarizmi¹ and Al – Kindi², introduced a much simpler system of decimal or numerals based on ten symbols. The Muslims also refined the system with the promotion of ‘zero’ symbol which enables us to distinguish easily between, for instance 5, 50 and 500. The Hindu - Arabic decimal system, later known as the Arabic system however, did not flourish quickly since it was considered strange, even among Muslims at that time because of their more accustomed to the Babylonian sexagesimal. In Europe the system was unrecognized not only due to sentimental prejudice, but also because the Roman system was sufficient to carry out simple calculations in the European daily life. It had taken several years since its introduction by Al – Khawarizmi before Arabic system came into practical applications. In Europe the system was popularized by Leonardo Pisano or Leonardo Fibonacci in the 1200s, after returning home from his learning journey in the Islamic lands. Since 1400s, the Arabic numeral system (see Fig. 2 for evolution description of the system) has become a common choice to represent scientific ideas and theories.

 Figure 2. Illustration of the Hindu – Arabic numeral systems evolution (diagram is adapted from “Pathfinder: The Golden Age of the Arabic Science” by Jim Al – Khalili of the University of Surrey, UK).

Together with the progress of the subject, particularly in arithmetic and algebra due to the swift and simple Arabic numerals, the medieval Muslims contributed further with development of another mathematical branch: trigonometry. The idea of trigonometry was certainly not new as some principles could be traced back to the ancient Greek and Hindu cultures. However, it was the medieval Muslims who complemented it with the invention of trigonometrical function of tangent, cotangent, secant and cosec. Further, the Muslims employed them extensively in various scientific fields.

As an example of integrated application of mathematics, we may pay a visit to a remarkable work of the legendary Muslim scholar Muhammad Al – Biruni³ on the estimation of the earth circumference. The study on the global circle was motivated by an interest to determine the vast empire the Muslims had to administrate, thus enabled them to understand the portion of their realms in the global map. More importantly, the data was essential to determine the qibla, a fixed place Muslims should face in the daily prayers. Before Al – Biruni, a Greek scientist Erastothenes of Cyrene, proposed a method on the subject by measuring the distance between two places which produced few degree of difference on an object’s sun shadow. Once the distance and the angle differences could be determined, full circle circumference could be calculated from the proportional ratio of angle and distance (Figure 3).

 

Figure 3. Simplified illustration of the Erasthotenes method to determine the earth circumference. Syene or Aswan was selected as starting point of observation for its location around the Tropic of Cancer, thus at midday during the summer solstice the sun is relatively vertical /overhead to the town.

The principal idea of Erastothenes was simple but in practical it was too difficult to apply. To produce 1^o difference on sun – ray shadow for instance, an experimenter would need to relocate a distance of hundreds of kilometers from his/her original position. The accurate distance measurement between both points had been problematic as there was no reliable method to calculate such a long range. Scientists then carried out an approach by counting a number of a person’s pace between two observation stations as a method for distance measurement. This practice certainly has large margin of errors due to several uncertainties, such as experimenter physical conditions, weather, and other external factors. No wonder, Erastothenes method produced different results from one experimenter to another. (Erasthothenes suggestion of observation between two places of Alexandria and Syene (now Aswan) in Egypt obtained 7.5^o of difference, turns out to be approximately 800 - 900 km of distance in modern calculation).

Figure 4. The Al – Biruni’s method of the earth circumference measurement:

(I) determination of the height of the mountain from two points of 1 and 2

(II) determination of the earth radius from an angle of observation of ϕ (observer at A position)

Al – Biruni proposal was much simpler in idea as well as implementation. Using a large astrolabe, Al – Biruni picked two places with known distance, approximately 100 m, at the same sea level and measured angle of elevation of both to the top of a mountain. This step provided Al – Biruni the height of the mountain which he used to finalize his calculation. Climbing up to the mountain, he measured the angle of the sight from the mountain top, dipping far down to the horizon. Those four parameters, three angles of observation and one height of the mountain, were obtained and then correlated with trigonometry and simple algebra to determine the radius of the earth. The data of radius eventually enabled Al – Biruni to calculate the earth circumference (See Figure 4). The work of Al – Biruni in the 10th century was phenomenal as his estimation found to be 40233 km (25000 miles) for the earth circumference, less than 1% difference from modern calculation at 40075 km (24902 miles).  

There are still numerous studies in which mathematics was employed as references of the Islamic civilization accomplishments. Those may be found in the excellent works of: Ibn Haytham⁴ on optics and astronomy, Al Tusi⁵ and Ibn Shattr⁶ on celestial movements (by the way, Tusi and Shattr mathematical models were very influential on the development of the 16^th Century AD Copernicus heliocentric theory), Jabr Ibn Hayyan⁷ and Al Razi⁸ on chemical processes (metallurgy, distillation, calcination, crystallization, extraction), Ibn Sina⁹ on astronomy and musical mathematics, Al Jazari¹⁰ and Banu Musa¹¹ on automatic devices, Ibn Mu’adh¹² in estimating the height of the earth atmosphere and Al-Battani¹³ in determining number of days of a solar year. All become factual examples of the civilization milestones from the Islamic world which at one time was very productive in science and technology, not only for the sake of curiosity but also in the effort to embrace the grandeur of the nature.

 Notes:

1) Muhammad Al – Khawarizmi (c.a 750 – c.a 850), was born and brought up as a Zoroastrian. There is no clear record actually, whether he converted to Islam. However, in his most famous book of Al Kitabul Muhtashar fii Hisab Al Jabru wal Muqabbala (The Compendious Book on Calculation by Completion and Balancing) -the first book on mathematical operation of Algebra- Al–Khawarizmi, known also as Algoritmus in the West, began his writing with Bismillahirrahmanirrahiim, or ‘In the Name of The Almighty God, the Most Gracious and the Most Merciful’, a line commonly written by muslim authors. Al – Khawarizmi’s step by step instruction to solve mathematical equations is immortalized in a term called algorithm.

2) Yusuf Ibn Ishaq Al Kindi c. 801 – 873 AD was an Arabic philosopher, born in Kufa, currently a small suburb of Najaf in modern Iraq. Al – Kindi is known as one of great philosophers the World has ever witnessed. Al – Kindi was a polymath, writing at least 200 books on various subjects; philosophy, mathematics, astronomy, medical science, pharmaceutical, psychology, chemistry and zoology. Al - Kindi was also remembered as an exquisite musician. 

3) Abu Rayhan Muhammad Ibn Ahmad Al – Biruni, 973 – 1048 was a Persian polymath mastering mathematics, astronomy, geography, linguistics and philosophy. He is well known as the Father of Geodesy. Al – Biruni was also an avid historian, famous for his book Tarikh Al Hind or the History of the Hindu Land (India).

4) Abu Ali Al Hasan Ibn Hasan Ibn Al – Haytham, born in Basra c.a. 965, was a polymath, a master of mathematics, astronomy, physics, philosophy and optics. Ibn Haytham, sometimes also written in other Romanized Arabic spelling as Ibn Haytsam, is known as Alhazen in the West. He has been famous for his book Al Kitabul Manazhir or the Book of Optics/Visions, and Al – Shukuk ala Batlamyus or The Doubt on Ptolemy’s (theory of celestial movement).

5) Nashiruddin Al – Tusi, or Muhammad Ibn Hassan Al –Tusi, 1201- 1274, was a Persian polymath with expertise in mathematics, astronomy, physics, theology and chemistry. Al – Tusi is well known for his theory of Al–Tusi couple, a significant correction to the hypothesis of Ptolemy on planetary motion.

6) Ala Al-Din Abu'l-Hasan Ali Ibn Ibrahim Ibn al-Shatir (1304 – 1375), was an Arabic mathematician and engineer. Ibn Shattir was a pray – time keeper in the Grand Mosque of Umayyad in Damascus, famous for his versatile sundial to determine prayer-time in the medieval Islam. Ibn Shattir also reformed Ptolemy’s theory on celestial motion. The mathematical model proposed by Al-Tusi and Ibn Shattir has been closely similar to that of Copernicus in 1543.  

7) Jabir Ibn Hayyan, c.a. 721 – 815, was a polymath living in the time of Caliph Harun Al – Rashid of the Abbasid Baghdad. Jabir was among the pioneers to revolutionize alchemy into chemistry through experimental approach. Jabir is remembered as a productive scholar, writing at least 100 books on various topics. At least, another 3000 books and manuscripts were written bearing the name of Jabir (latinized as Geber) as attribution to Jabir’s authoritative knowledge from scholars who came after him.

8) Muhammad Ibn Zakariyya Al – Razi, 854 – 925 AD, was another outstanding Muslim polymath in the time of the Abbasids. He was well known for his expertise in medicine, chemistry, mathematics and philosophy. Al – Razi continued the approach Ibn Hayyan had done several decades earlier, in promoting quantification and experiments to chemistry and chemical processing. Al – Razi was also a renowned medical doctor, a pioneer in the establishment of hospital.  

9) Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Al-Hasan ibn Ali ibn Sina, 980 -1037, was an Uzbek born scholar. Ibn Sina was a genius, authoring approximately 450 books on wide range of subjects notably medicine, mathematics, chemistry, philosophy and arts. His multi-disciplinary expertise is often overshadowed by his famous book  Al-Qanun Fil Tibb or The Canon of Medicine, the primary reference for medical practices in the Middle – East and Europe until around 1700s.  

10) Ismail Ibn Razzaz Al-Jazari, 1136–1206, was a Muslim engineer, mathematician, and artist.  Al – Jazari, commonly attributed as the Father of Robotics, is well-known for his automation concept in water - pump and water - clock. His gigantic elephant water – clock has been remembered as one engineering wonder, up to this day.

11) Banu Musa, c.a. 803 – 875 or the Moses Brothers: Muhammad Ibn Musa, Ahmad Ibn Musa and Hasan Ibn Musa, were prominent engineers in the time of the Caliph Abdullah Al Ma’mun Ibn Harun Al Rashid of the Abbasid dynasty. Banu Musa were among scholars trusted by Al – Ma’mun to develop automatic devices; such as pump, fountains and reading lamp. Banu Musa also wrote dozens books on astronomy.

12) Muhammad Ibn Mu’adh Al–Jayyani, 989 – 1079, was a Qadi, or Islamic Law judge in the time of Andalusian Islam in Spain. Ibn Mu’adh was a prominent scientist in the field of mathematics particularly in spherical trigonometry. 

13) Muhammad Ibn Jabir Ibn Sinan Al-Battani, 858 – 929, was born in Harran, southern part of modern day Turkey. Al – Battani was an expert in astronomy and mathematics, well – known for his expansion of trigonometry function of sines and tangent. Al–Battani was the first person to revise Ptolemy theory on solar apogee. The work of Al-Battani was influential to medieval European scientists such as Copernicus, Brahe and Galileo.