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 8-9th 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 1o 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
7.5o 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 16th 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.
