In the
history of
mathematics,
mathematics in medieval Islam, is
the
mathematics developed in the
Islamic world between 622 and 1600,
during what is known as the
Islamic
Golden Age.
Islamic
science and mathematics flourished under the Islamic caliphate (also known as the ) established across the
Middle East, Central Asia, North
Africa, Sicily, the Iberian
Peninsula, and in
parts of France and the
Indian subcontinent in the 8th
century. The main centres of mathematical activity
were in Iraq, Persia and Egypt, but at its
greatest extent stretched from North Africa and Spain in the west
to India in the
east.
While most scientists in this period were
Muslims and wrote in
Arabic, many of the best known contributors
were
Persians as well as
Arabs,
Berbers,
Moors,
Turks, and
sometimes non-Muslims (
Christian,
Jewish,
Sabian,
Zoroastrian and
irreligious). Arabic was the dominant
language—much like
Latin in
Medieval Europe, Arabic was the written
lingua franca of most
scholars throughout the
Islamic
world.
Use of the term "Islam"
Bernard Lewis writes the following on
the historical usage of the term "
Islam":
In this article, "Islam" and the adjective "Islamic" is used in the
meaning of a
civilization.
Origins and influences
The first century of the
Islamic Arab Empire saw almost no scientific or
mathematical achievements since the Arabs, with their newly
conquered empire, had not yet gained any intellectual drive and
research in other parts of the world had faded. In the second half
of the eighth century Islam had a cultural awakening, and research
in mathematics and the sciences increased. The Muslim
Abbasid caliph al-Mamun (809-833) is said to have had a dream
where Aristotle appeared to him, and as a consequence al-Mamun
ordered that Arabic translation be made of as many Greek works as
possible, including Ptolemy's
Almagest and Euclid's
Elements. Greek works would be given
to the Muslims by the
Byzantine
Empire in exchange for treaties, as the two empires held an
uneasy peace. Many of these Greek works were translated by
Thabit ibn Qurra (826-901), who translated
books written by
Euclid,
Archimedes, Apollonius,
Ptolemy, and Eutocius. Historians are in debt to
many Islamic translators, for it is through their work that many
ancient
Greek texts have survived
only through
Arabic translations.
Greek,
Indian and
Babylonian all played an important
role in the development of early Islamic mathematics. The works of
mathematicians such as Euclid,
Apollonius, Archimedes,
Diophantus,
Aryabhata
and
Brahmagupta were all acquired by the
Islamic world and incorporated into their mathematics.
Perhaps the most
influential mathematical contribution from India was the
decimal place-value Indo-Arabic numeral system, also
known as the Hindu numerals.
The
Persian historian
al-Biruni (c. 1050) in his book
Tariq
al-Hind states that
al-Ma'mun had an
embassy in India from which was brought a book to Baghdad that was
translated into Arabic as
Sindhind. It is generally
assumed that
Sindhind is none other than Brahmagupta's
Brahmasphuta-siddhanta. The
earliest translations from Sanskrit inspired several astronomical
and astrological Arabic works, now mostly lost, some of which were
even composed in verse.
Indian influences were later overwhelmed by Greek mathematical and
astronomical texts. It is not clear why this occurred but it may
have been due to the greater availability of Greek texts in the
region, the larger number of practitioners of Greek mathematics in
the region, or because Islamic mathematicians favored the deductive
exposition of the Greeks over the elliptic Sanskrit verse of the
Indians. Regardless of the reason, Indian mathematics soon became
mostly eclipsed by or merged with the "Graeco-Islamic" science
founded on Hellenistic treatises.
Another likely reason for the declining
Indian influence in later periods was due to Sindh achieving
independence from the Caliphate, thus
limiting access to Indian works. Nevertheless, Indian
methods continued to play an important role in algebra, arithmetic
and trigonometry.
Besides the Greek and Indian tradition, a third tradition which had
a significant influence on mathematics in medieval Islam was the
"mathematics of practitioners", which included the applied
mathematics of "surveyors,
builders,
artisans, in geometric design,
tax and treasury officials, and
some merchants." This applied form of mathematics transcended
ethnic divisions and was a common heritage of the lands
incorporated into the Islamic world. This tradition also includes
the religious observances specific to Islam, which served as a
major impetus for the development of mathematics as well as
astronomy.
Islam and mathematics
A major impetus for the flowering of mathematics as well as
astronomy in medieval
Islam came from religious observances, which presented an
assortment of problems in astronomy and mathematics, specifically
in
trigonometry,
spherical geometry,
algebra and
arithmetic.
The
Islamic law of
inheritance served as an impetus behind the development of
algebra (derived from the
Arabic
al-jabr) by
Muhammad ibn
Mūsā al-Khwārizmī and other medieval Islamic mathematicians.
Al-Khwārizmī's
Hisab
al-jabr w’al-muqabala devoted a chapter on the solution to
the Islamic law of inheritance using algebra. He formulated the
rules of inheritance as
linear
equations, hence his knowledge of
quadratic equations were not required.
Later mathematicians who specialized in the Islamic law of
inheritance included Al-Hassār, who developed the modern symbolic
mathematical notation for
fractions in the 12th
century, and
Abū
al-Hasan ibn Alī al-Qalasādī, who developed an algebraic
notation which took "the first steps toward the introduction of
algebraic symbolism" in the 15th century.
In order to observe holy days on the
Islamic calendar in which timings were
determined by
phases of the moon,
astronomers initially used
Ptolemy's method
to calculate the place of the
moon and
stars. The method Ptolemy used to solve
spherical triangles, however, was a
clumsy one devised late in the first century by
Menelaus of Alexandria. It involved
setting up two intersecting
right
triangles; by applying
Menelaus'
theorem it was possible to solve one of the six sides, but only
if the other five sides were known. To tell the time from the
sun's
altitude, for
instance, repeated applications of Menelaus' theorem were required.
For medieval Islamic astronomers, there was an obvious challenge to
find a simpler
trigonometric
method.
Regarding the issue of moon sighting, Islamic months do not begin
at the astronomical
new moon, defined as
the time when the moon has the same
celestial longitude as the sun and is
therefore invisible; instead they begin when the thin
crescent moon is first sighted in the western
evening sky. The Qur'an says: "They ask you about the waxing and
waning phases of the crescent moons, say they are to mark fixed
times for mankind and
Hajj." This led Muslims
to find the phases of the moon in the sky, and their efforts led to
new mathematical calculations.
Predicting just when the crescent moon would become visible is a
special challenge to Islamic mathematical astronomers. Although
Ptolemy's theory of the complex lunar motion was tolerably accurate
near the time of the new moon, it specified the moon's path only
with respect to the
ecliptic. To predict
the first visibility of the moon, it was necessary to describe its
motion with respect to the
horizon, and this
problem demands fairly sophisticated spherical geometry.
Finding
the direction of Mecca and the time
of Salah are the reasons which led to Muslims
developing spherical geometry. Solving any of these problems
involves finding the unknown sides or angles of a triangle on the
celestial sphere from the known
sides and angles. A way of finding the time of day, for example, is
to construct a triangle whose
vertices are the
zenith, the north
celestial
pole, and the sun's position. The observer must know the
altitude of the sun and that of the pole; the former can be
observed, and the latter is equal to the observer's
latitude. The time is then given by the angle at
the intersection of the
meridian (the
arc through the zenith and the pole) and the
sun's hour circle (the arc through the sun and the pole).
Muslims
are also expected to pray towards the Kaaba in Mecca and orient
their mosques in that direction. Thus
they need to determine the direction of Mecca (
Qibla) from a given location. Another problem is the
time of
Salah. Muslims need to determine from
celestial bodies the proper times
for the prayers at
sunrise, at
midday, in the
afternoon, at
sunset, and in the
evening.
Importance
J. J. O'Conner and E. F. Robertson wrote in the
MacTutor History of
Mathematics archive:
R. Rashed wrote in
The development of Arabic mathematics:
between arithmetic and algebra:
Biographies
- (786 – 833)
- Al-Ḥajjāj translated Euclid's
Elements into
Arabic.
- (c. 780 Khwarezm/Baghdad – c.
850 Baghdad)
- Al-Khwārizmī was a Persian mathematician, astronomer, astrologer
and geographer. He worked most of his
life as a scholar in the House of Wisdom in Baghdad. His
Algebra
was the first book on the systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the
Indian numerals, introduced the
decimal positional number system to the Western world in the 12th century. He revised
and updated Ptolemy's Geography as
well as writing several works on astronomy and astrology.
- (c. 800 Baghdad? – c. 860 Baghdad?)
- Al-Jawharī was a mathematician who worked at the House of
Wisdom in Baghdad. His most important work was his Commentary
on Euclid's Elements which
contained nearly 50 additional proposition and an attempted
proof of the parallel postulate.
- (fl. 830 Baghdad)
- Ibn Turk wrote a work on algebra of
which only a chapter on the solution of quadratic equations has survived.
- (c. 801 Kufa – 873
Baghdad)
- Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in
Baghdad where he wrote commentaries on many Greek works. His
contributions to mathematics include many works on arithmetic and geometry.
- Hunayn ibn
Ishaq (808 Al-Hirah – 873 Baghdad)
- Hunayn (or Johannitus) was a translator who worked at the House
of Wisdom in Baghdad. Translated many Greek works including those
by Plato, Aristotle,
Galen, Hippocrates,
and the Neoplatonists.
- (c. 800 Baghdad – 873+ Baghdad)
- The Banū Mūsā were three brothers who worked at the House of
Wisdom in Baghdad. Their most famous mathematical treatise is
The Book of the Measurement of Plane and Spherical
Figures, which considered similar problems as Archimedes did in his On the Measurement of the
Circle and On the sphere and the cylinder. They
contributed individually as well. The eldest, (c. 800) specialised
in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called
Premises of the book of conics. (c. 805) specialised in
mechanics and wrote a work on pneumatic
devices called On mechanics. The youngest, (c. 810)
specialised in geometry and wrote a work on the ellipse called The elongated circular
figure.
- Al-Mahani
- Ahmed ibn Yusuf
- Thabit ibn Qurra (Syria-Iraq,
835-901)
- Al-Hashimi (Iraq? ca. 850-900)
- (c. 853 Harran – 929
Qasr al-Jiss near Samarra)
- Abu Kamil (Egypt? ca. 900)
- Sinan ibn Tabit (ca. 880 -
943)
- Al-Nayrizi
- Ibrahim ibn Sinan (Iraq,
909-946)
- Al-Khazin (Iraq-Iran, ca.
920-980)
- Al-Karabisi (Iraq? 10th
century?)
- Ikhwan al-Safa' (Iraq, first
half of 10th century)
- The Ikhwan al-Safa' ("brethren of purity") were a (mystical?)
group in the city of Basra in Irak. The group authored a series of
more than 50 letters on science, philosophy and theology. The first
letter is on arithmetic and number theory, the second letter on
geometry.
- Al-Uqlidisi (Iraq-Iran, 10th
century)
- Al-Saghani (Iraq-Iran, ca.
940-1000)
- (Iraq-Iran, ca. 940-1000)
- Al-Khujandi
- (Iraq-Iran, ca. 940-998)
- Ibn Sahl (Iraq-Iran, ca. 940-1000)
- Al-Sijzi (Iran, ca. 940-1000)
- Labana of Cordoba (Spain, ca.
10th century)
- One of the few Islamic female mathematicians known by name, and
the secretary of the Umayyad
Caliph al-Hakem II. She was well-versed in the exact sciences,
and could solve the most complex geometrical and algebraic problems
known in her time.
- Ibn Yunus (Egypt, ca. 950-1010)
- Abu Nasr ibn `Iraq
(Iraq-Iran, ca. 950-1030)
- Kushyar ibn Labban (Iran, ca.
960-1010)
- Al-Karaji (Iran, ca. 970-1030)
- Ibn al-Haytham (Iraq-Egypt, ca.
965-1040)
- (September 15
973 in Kath, Khwarezm –
December 13 1048 in
Gazna)
- Ibn Sina (Avicenna)
- al-Baghdadi
- Al-Nasawi
- Al-Jayyani (Spain, ca.
1030-1090)
- Ibn al-Zarqalluh (Azarquiel,
al-Zarqali) (Spain, ca. 1030-1090)
- Al-Mu'taman ibn Hud (Spain,
ca. 1080)
- al-Khayyam (Iran, ca. 1050-1130)
- (ca.
1130, Baghdad – c. 1180, Maragha)
- Al-Hassār (ca. 1100s, Maghreb)
- Developed the modern mathematical notation for fractions and the digits he uses for
the ghubar numerals also cloesly resembles modern Western
Arabic numerals.
- Ibn al-Yāsamīn (ca.
1100s, Maghreb)
- The son of a Berber father and
black African mother, he was the first
to develop a mathematical notation for algebra since the time of
Brahmagupta.
- (Iran, ca. 1150-1215)
- Ibn Mun`im (Maghreb, ca. 1210)
- (Morocco, 13th century)
- (18 February
1201 in Tus, Khorasan – 26 June
1274 in Kadhimain near Baghdad)
- (c. 1220 Spain – c. 1283 Maragha)
- (c. 1250 Samarqand – c. 1310)
- Ibn Baso (Spain, ca. 1250-1320)
- Ibn al-Banna' (Maghreb, ca.
1300)
- Kamal al-Din Al-Farisi
(Iran, ca. 1300)
- Al-Khalili (Syria, ca.
1350-1400)
- Ibn al-Shatir (1306-1375)
- (1364
Bursa – 1436 Samarkand)
- (Iran, Uzbekistan, ca. 1420)
- Ulugh Beg (Iran, Uzbekistan,
1394-1449)
- Al-Umawi
- Abū
al-Hasan ibn Alī al-Qalasādī (Maghreb, 1412-1482)
- Last major medieval Arab mathematician.
Pioneer of symbolic
algebra.
Algebra
The term
algebra is derived from the Arabic
term
al-jabr in the title of
Al-Khwarizmi's
Al-jabr
wa'l muqabalah. He originally used the term
al-jabr to describe the method of "
reduction" and "balancing",
referring to the transposition of subtracted terms to the other
side of an equation, that is, the cancellation of like terms on
opposite sides of the equation.
There are three theories about the origins of Islamic algebra. The
first emphasizes Hindu influence, the second emphasizes
Mesopotamian or Persian-Syriac influence, and the third emphasizes
Greek influence. Many scholars believe that it is the result of a
combination of all three sources.
Throughout their time in power, before the fall of Islamic
civilization, the Arabs used a fully rhetorical algebra, where
sometimes even the numbers were spelled out in words. The Arabs
would eventually replace spelled out numbers (eg. twenty-two) with
Arabic numerals (eg. 22), but the
Arabs never adopted or developed a syncopated or symbolic algebra,
until the work of
Ibn
al-Banna al-Marrakushi in the 13th century and
Abū
al-Hasan ibn Alī al-Qalasādī in the 15th century.
There were four conceptual stages in the development of algebra,
three of which either began in, or were significantly advanced in,
the Islamic world. These four stages were as follows:
- Geometric stage, where the concepts of algebra
are largely geometric. This dates back to
the Babylonians and continued
with the Greeks, and was revived
by Omar Khayyam.
- Static equation-solving stage, where the
objective is to find numbers satisfying certain relationships. The
move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn't decisively move
to the static equation-solving stage until Al-Khwarizmi's
Al-Jabr.
- Dynamic function stage, where motion is an
underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī,
but algebra didn't decisively move to the dynamic function stage
until Gottfried Leibniz.
- Abstract stage, where mathematical structure
plays a central role. Abstract
algebra is largely a product of the 19th and 20th
centuries.
Static equation-solving algebra
- Al-Khwarizmi and Al-jabr wa'l muqabalah
The MuslimPersian mathematician (c. 780-850) was a faculty member
of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was
established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D.,
wrote more than half a dozen mathematical and astronomical works;
some of which were based on the Indian
Sindhind.One of
al-Khwarizmi's most famous books is entitled
Al-jabr wa'l
muqabalah or
The
Compendious Book on Calculation by Completion and
Balancing, and it gives an exhaustive account of solving
polynomials up to the second degree. The book also introduced the
fundamental method of "
reduction" and "balancing",
referring to the transposition of subtracted terms to the other
side of an equation, that is, the cancellation of like terms on
opposite sides of the equation. This is the operation which
Al-Khwarizmi originally described as
al-jabr. "It is not
certain just what the terms
al-jabr and
muqabalah
mean, but the usual interpretation is similar to that implied in
the translation above. The word
al-jabr presumably meant
something like "restoration" or "completion" and seems to refer to
the transposition of subtracted terms to the other side of an
equation; the word
muqabalah is said to refer to
"reduction" or "balancing" - that is, the cancellation of like
terms on opposite sides of the equation."
Al-Jabr is divided into six chapters, each of which deals
with a different type of formula. The first chapter of
Al-Jabr deals with equations whose squares equal its roots
(ax² = bx), the second chapter deals with squares equal to number
(ax² = c), the third chapter deals with roots equal to a number (bx
= c), the fourth chapter deals with squares and roots equal a
number (ax² + bx = c), the fifth chapter deals with squares and
number equal roots (ax² + c = bx), and the sixth and final chapter
deals with roots and number equal to squares (bx + c = ax²).
J. J. O'Conner and E. F. Robertson wrote in the
MacTutor History of
Mathematics archive:
The
Hellenistic
mathematician
Diophantus was
traditionally known as "the father of algebra" but debate now
exists as to whether or not
Al-Khwarizmi
deserves this title instead. Those who support Diophantus point to
the fact that the algebra found in
Al-Jabr is more
elementary than the algebra found in
Arithmetica and that
Arithmetica
is syncopated while
Al-Jabr is fully rhetorical. Those who
support Al-Khwarizmi point to the fact that he gave an exhaustive
explanation for the algebraic solution of quadratic equations with
positive roots, introduced the fundamental methods of reduction and
balancing, and was the first to teach algebra in an
elementary form and for its own sake,
whereas Diophantus was primarily concerned with the
theory of numbers. In addition, R. Rashed and
Angela Armstrong write:
- Ibn Turk and Logical Necessities in Mixed
Equations
'Abd al-Hamīd ibn Turk
(fl. 830) authored a manuscript entitled
Logical Necessities in
Mixed Equations, which is very similar to al-Khwarzimi's
Al-Jabr and was published at around the same time as, or
even possibly earlier than,
Al-Jabr.The manuscript gives
the exact same
geometric demonstration as
is found in
Al-Jabr, and in one case the same example as
found in
Al-Jabr, and even goes beyond
Al-Jabr by
giving a geometric proof that if the determinant is negative then
the quadratic equation has no solution. The similarity between
these two works has led some historians to conclude that Islamic
algebra may have been well developed by the time of al-Khwarizmi
and 'Abd al-Hamid.
- Abū Kāmil and al-Karkhi
Arabic mathematicians were also the first to treat
irrational numbers as
algebraic objects.
The Egyptian mathematician Abū Kāmil Shujā ibn
Aslam (c. 850-930) was the first to accept irrational
numbers (often in the form of a
square
root,
cube root or
fourth root) as solutions to
quadratic equations or as
coefficients in an
equation. He was also the first to solve three
non-linear
simultaneous
equations with three unknown
variables.
Al-Karkhi (953-1029), also known as
Al-Karaji, was the successor of
Abū al-Wafā'
al-Būzjānī (940-998) and he was the first to discover the
solution to equations of the form ax
^{2n} + bx
^{n}
= c. Al-Karkhi only considered positive roots. Al-Karkhi is also
regarded as the first person to free algebra from
geometrical operations and replace them with the
type of
arithmetic operations which are
at the core of algebra today. His work on algebra and
polynomials, gave the rules for arithmetic
operations to manipulate polynomials. The
historian of mathematics F.
Woepcke,
in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed
Ben Alhacan Alkarkhi (Paris, 1853),
praised Al-Karaji for being "the first who introduced the theory of
algebraic calculus". Stemming from
this, Al-Karaji investigated
binomial coefficients and
Pascal's triangle.
Linear algebra
In
linear algebra and
recreational mathematics,
magic squares were known to
Arab mathematicians, possibly as early as the 7th
century, when the Arabs got into contact with Indian or South Asian
culture, and learned Indian mathematics and astronomy, including
other aspects of
combinatorial
mathematics. It has also been suggested that the idea came via
China.
The first magic squares of order 5 and 6
appear in an encyclopedia from Baghdad circa 983 AD, the Rasa'il Ihkwan
al-Safa (Encyclopedia of the Brethren of Purity);
simpler magic squares were known to several earlier Arab
mathematicians.
The Arab mathematician
Ahmad al-Buni,
who worked on magic squares around 1200 AD, attributed mystical
properties to them, although no details of these supposed
properties are known. There are also references to the use of magic
squares in astrological calculations, a practice that seems to have
originated with the Arabs.
Geometric algebra
Omar Khayyám (c. 1050-1123) wrote
a book on Algebra that went beyond
Al-Jabr to include
equations of the third degree.Omar Khayyám provided both arithmetic
and geometric solutions for quadratic equations, but he only gave
geometric solutions for general cubic equations since he mistakenly
believed that arithmetic solutions were impossible. His method of
solving cubic equations by using intersecting conics had been used
by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám
generalized the method to cover all cubic equations with positive
roots. He only considered positive roots and he did not go past the
third degree. He also saw a strong relationship between Geometry
and Algebra.
Dynamic functional algebra
In the 12th century,
Sharaf al-Dīn al-Tūsī
found algebraic and
numerical
solutions to cubic equations and was the first to discover the
derivative of
cubic polynomials. His
Treatise on
Equations dealt with
equations up to
the third degree. The treatise does not follow
Al-Karaji's school of algebra, but instead
represents "an essential contribution to another algebra which
aimed to study curves by means of equations, thus inaugurating the
beginning of
algebraic geometry."
The treatise dealt with 25 types of equations, including twelve
types of
linear equations and
quadratic equations, eight types
of
cubic equations with positive
solutions, and five types of cubic equations which may not have
positive solutions. He understood the importance of the
discriminant of the cubic equation and used an
early version of
Cardano's formula
to find algebraic solutions to certain types of cubic
equations.
Sharaf al-Din also developed the concept of a
function. In his analysis ofthe
equation \ x^3 + d = bx^2 for example, he begins by changing the
equation's form to \ x^2 (b - x) = d. He then states that the
question of whether the equation has a solution depends on whether
or not the “function” on the left side reaches the value \ d. To
determine this, he finds a maximum value for the function. He
proves that the maximum value occurs when x = \frac{2b}{3}, which
gives the functional value \frac{4b^3}{27}. Sharaf al-Din then
states that if this value is less than \ d, there are no positive
solutions; if it is equal to \ d, then there is one solution at x =
\frac{2b}{3}; and if it is greater than \ d, then there are two
solutions, one between \ 0 and \frac{2b}{3} and one between
\frac{2b}{3} and \ b. This was the earliest form of dynamic
functional algebra.
Numerical analysis
In
numerical analysis, the
essence of
Viète's method was
known to
Sharaf
al-Dīn al-Tūsī in the 12th century, and it is possible that the
algebraic tradition of Sharaf al-Dīn, as well as his predecessor
Omar Khayyám and successor
Jamshīd al-Kāshī,
was known to 16th century European algebraists, or whom
François Viète was the most
important.
A method algebraically equivalent to
Newton's method was also known to Sharaf
al-Dīn. In the 15th century, his successor al-Kashi later used a
form of Newton's method to numerically solve \ x^P - N = 0 to find
roots of \ N. In
western Europe, a
similar method was later described by Henry Biggs in his
Trigonometria Britannica, published in 1633.
Symbolic algebra
Al-Hassār, a mathematician from the
Maghreb
(
North Africa) specializing in
Islamic inheritance
jurisprudence during the 12th century, developed the modern
symbolic
mathematical notation
for
fractions, where the
numerator and
denominator are separated by a horizontal bar.
This same fractional notation appeared soon after in the work of
Fibonacci in the 13th century.
Abū
al-Hasan ibn Alī al-Qalasādī (1412-1482) was the last major
medieval
Arab algebraist, who improved on the
algebraic notation earlier
used in the
Maghreb by
Ibn al-Banna in the 13th century and by Ibn
al-Yāsamīn in the 12th century. In contrast to the syncopated
notations of their predecessors,
Diophantus and
Brahmagupta, which lacked symbols for
mathematical operations,
al-Qalasadi's algebraic notation was the first to have symbols for
these functions and was thus "the first steps toward the
introduction of algebraic symbolism." He represented
mathematical symbols using
characters from the
Arabic
alphabet.
The
symbol \mathit{x} now commonly
denotes an unknown
variable. Even though any letter can
be used, \mathit{x} is the most common choice. This usage can be
traced back to the
Arabic word
šay' شيء = “thing,” used in Arabic algebra texts such as
the
Al-Jabr,
and was taken into
Old Spanish
with the pronunciation “šei,” which was written
xei, and
was soon habitually abbreviated to \mathit{x}. (The
Spanish pronunciation of “x” has changed since). Some
sources say that this \mathit{x} is an abbreviation of
Latin causa, which was a translation of
Arabic شيء. This started the habit of using letters to represent
quantities in
algebra. In mathematics, an
“
italicized x” (x\!) is often used to
avoid potential confusion with the multiplication symbol.
Arithmetic
Arabic numerals
The
Indian numeral system came to be
known to both the
Persian
mathematician
Al-Khwarizmi, whose book
On the Calculation with Hindu Numerals written
circa 825, and the
Arab mathematician
Al-Kindi, who wrote four volumes,
On
the Use of the Indian Numerals (Ketab fi Isti'mal al-'Adad
al-Hindi)
circa 830, are principally responsible for the
diffusion of the Indian system of numeration in the
Middle-East and the West
[211642]. In the 10th century,
Middle-Eastern mathematicians extended the
decimal numeral system to include
fractions using
decimal point notation, as recorded in a
treatise by
Syrian
mathematician
Abu'l-Hasan
al-Uqlidisi in 952-953.
In the
Arab world—until early modern
times—the Arabic numeral system was often only used by
mathematicians.
Muslim
astronomers mostly used the
Babylonian numeral system, and
merchants mostly used the
Abjad numerals.
A distinctive
"Western Arabic" variant of the symbols begins to emerge in ca. the
10th century in the Maghreb and Al-Andalus, called the ghubar ("sand-table" or
"dust-table") numerals, which is the direct ancestor to the modern
Western Arabic numerals now used throughout the world.
The first mentions of the numerals in the West are found in the
Codex Vigilanus of 976
[211643]. From the 980s,
Gerbert of Aurillac (later, Pope
Silvester II) began to spread knowledge of the
numerals in Europe.
Gerbert studied in Barcelona in his youth, and he is known to have requested
mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had
returned to France.
Al-Khwārizmī, the
Persian scientist, wrote in 825 a treatise
On the Calculation with Hindu Numerals, which was
translated into
Latin in the 12th
century, as
Algoritmi de numero Indorum, where
"Algoritmi", the translator's rendition of the author's name gave
rise to the word
algorithm (Latin
algorithmus) with a meaning "calculation method".
Al-Hassār, a mathematician from the
Maghreb
(
North Africa) specializing in
Islamic inheritance
jurisprudence during the 12th century, developed the modern
symbolic
mathematical notation
for fractions, where the
numerator and
denominator are separated by a horizontal bar.
The "dust
ciphers he used are also nearly
identical to the digits used in the current Western Arabic
numerals. These same digits and fractional notation appear soon
after in the work of
Fibonacci in the 13th
century.
Decimal fractions
In discussing the origins of
decimal
fractions,
Dirk Jan Struik
states that (p. 7):
"The introduction of decimal fractions as a
common computational practice can be dated back to the Flemish pamphelet De Thiende, published at
Leyden in 1585,
together with a French translation, La Disme, by the
Flemish mathematician Simon Stevin
(1548-1620), then settled in the Northern Netherlands. It is true that decimal fractions were used
by the Chinese many centuries
before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his
Key to arithmetic (Samarkand, early fifteenth
century)."
While the
Persian mathematician
Jamshīd al-Kāshī
claimed to have discovered decimal fractions himself in the 15th
century, J.
Lennart Berggrenn notes that he was
mistaken, as decimal fractions were first used five centuries
before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as
the 10th century.
Real numbers
The
Middle Ages saw the acceptance of
zero,
negative,
integral and
fractional numbers, first by
Indian mathematicians and
Chinese mathematicians, and then by
Arabic mathematicians, who were also the first to treat
irrational numbers as algebraic objects,
which was made possible by the development of algebra. Arabic
mathematicians merged the concepts of "
number" and "
magnitude" into a more general idea
of
real numbers, and they criticized
Euclid's idea of
ratios, developed the theory
of composite ratios, and extended the concept of number to ratios
of continuous magnitude. In his commentary on Book 10 of the
Elements, the
Persian
mathematician
Al-Mahani (d. 874/884)
examined and classified
quadratic
irrationals and cubic irrationals. He provided definitions for
rational and irrational magnitudes, which he treated as irrational
numbers. He dealt with them freely but explains them in geometric
terms as follows:
In contrast to Euclid's concept of magnitudes as lines, Al-Mahani
considered integers and fractions as rational magnitudes, and
square roots and
cube roots as irrational
magnitudes. He also introduced an
arithmetical approach to the concept of
irrationality, as he attributes the following to irrational
magnitudes:
The
Egyptian mathematician Abū Kāmil Shujā ibn
Aslam (c. 850–930) was the first to accept irrational
numbers as solutions to
quadratic
equations or as
coefficients in an
equation, often in the form of square
roots, cube roots and
fourth roots.
In the
10th century, the Iraqi
mathematician Al-Hashimi provided general proofs (rather than
geometric demonstrations) for irrational numbers, as he considered
multiplication, division, and other arithmetical functions.
Abū Ja'far al-Khāzin
(900-971) provides a definition of rational and irrational
magnitudes, stating that if a definite
quantity is:
Many of these concepts were eventually accepted by European
mathematicians some time after the
Latin translations of the
12th century. Al-Hassār, an Arabic mathematician from the
Maghreb (
North
Africa) specializing in
Islamic inheritance
jurisprudence during the 12th century, developed the modern
symbolic
mathematical notation
for fractions, where the
numerator and
denominator are separated by a horizontal bar.
This same fractional notation appears soon after in the work of
Fibonacci in the 13th century.
Number theory
In
number theory,
Ibn al-Haytham solved problems involving
congruences using what is now
called
Wilson's theorem. In his
Opuscula, Ibn al-Haytham considers the solution of a
system of congruences, and gives two general methods of solution.
His first method, the canonical method, involved Wilson's theorem,
while his second method involved a version of the
Chinese remainder theorem. Another
contribution to number theory is his work on
perfect numbers. In his
Analysis and
synthesis, Ibn al-Haytham was the first to discover that every
even perfect number is of the form
2
^{n−1}(2
^{n} − 1)
where 2
^{n} − 1 is
prime, but he was not able to prove this result
successfully (
Euler later proved it
in the 18th century).
In the early 14th century,
Kamāl al-Dīn
al-Fārisī made a number of important contributions to number
theory. His most impressive work in number theory is on
amicable numbers. In
Tadhkira al-ahbab
fi bayan al-tahabb ("Memorandum for friends on the proof of
amicability") introduced a major new approach to a whole area of
number theory, introducing ideas concerning
factorization and
combinatorial methods. In fact, al-Farisi's
approach is based on the unique factorization of an
integer into powers of
prime
numbers.
Geometry
The successors of
Muhammad ibn
Mūsā al-Khwārizmī (born 780) undertook a systematic application
of arithmetic to algebra, algebra to arithmetic, both to
trigonometry, algebra to the
Euclidean theory
of numbers, algebra to
geometry, and
geometry to algebra. This was how the creation of polynomial
algebra, combinatorial analysis, numerical analysis, the numerical
solution of equations, the new elementary theory of numbers, and
the geometric construction of equations arose.
Al-Mahani (born 820) conceived the idea of
reducing geometrical problems such as duplicating the cube to
problems in algebra.
Al-Karaji (born 953)
completely freed algebra from geometrical operations and replaced
them with the
arithmetical type of
operations which are at the core of algebra today.
Early Islamic geometry
- See also Applied
mathematics
Thabit ibn Qurra (known as
Thebit in
Latin) (born 836) contributed to a
number of areas in mathematics, where he played an important role
in preparing the way for such important mathematical discoveries as
the extension of the concept of number to (
positive)
real numbers,
integral calculus, theorems in
spherical trigonometry,
analytic geometry, and
non-Euclidean geometry. An important
geometrical aspect of Thabit's work was his book on the composition
of ratios. In this book, Thabit deals with arithmetical operations
applied to ratios of geometrical quantities. The Greeks had dealt
with geometric quantities but had not thought of them in the same
way as numbers to which the usual rules of arithmetic could be
applied. By introducing arithmetical operations on quantities
previously regarded as geometric and non-numerical, Thabit started
a trend which led eventually to the generalization of the number
concept. Another important contribution Thabit made to
geometry was his generalization of the
Pythagorean theorem, which he extended
from
special right triangles
to all
right triangles in general,
along with a general
proof.
In some respects, Thabit is critical of the ideas of Plato and
Aristotle, particularly regarding motion. It would seem that here
his ideas are based on an acceptance of using arguments concerning
motion in his geometrical arguments.
Ibrahim ibn Sinan ibn Thabit (born
908), who 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
Ibn al-Haytham
(Alhazen), studied
optics and investigated
the optical properties of mirrors made from
conic sections (see
Mathematical physics).
Astronomy, time-keeping and
geography
provided other motivations for geometrical and trigonometrical
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 pioneered
spherical
geometry in order to solve difficult problems in
Islamic astronomy. For example, to predict
the first visibility of the moon, it was necessary to describe its
motion with respect to the
horizon, and this
problem demands fairly sophisticated spherical geometry.
Finding
the direction of Mecca (Qibla) and the time for Salah
prayers and Ramadan are what led to Muslims
developing spherical geometry.
Algebraic and analytic geometry
In the early 11th century,
Ibn
al-Haytham (Alhazen) was able to solve by purely algebraic
means certain cubic equations, and then to interpret the results
geometrically. Subsequently,
Omar
Khayyám discovered the general method of solving
cubic equations by intersecting a parabola
with a circle.
Omar Khayyám (1048-1122) was a
Persian mathematician, as well as a
poet. Along with his fame as a poet, he was also famous during his
lifetime as a mathematician, well known for inventing the general
method of solving
cubic equations by
intersecting a parabola with a circle. In addition he discovered
the
binomial expansion, and
authored criticisms of
Euclid's theories of
parallels which made their way to
England, where they contributed to the eventual development of
non-Euclidean geometry. Omar
Khayyam also combined the use of trigonometry and
approximation theory to provide methods
of solving algebraic equations by geometrical means. His work
marked the beginnings of
algebraic
geometry and
analytic
geometry.
In a paper written by Khayyam before his famous algebra text
Treatise on Demonstration of Problems of Algebra, he
considers the problem:
Find a point on a quadrant of a circle
in such manner that when a normal is dropped from the point to one
of the bounding radii, the ratio of the normal's length to that of
the radius equals the ratio of the segments determined by the foot
of the normal. Khayyam shows that this problem is equivalent
to solving a second problem:
Find a right triangle having the
property that the hypotenuse equals the sum of one leg plus the
altitude on the hypotenuse. This problem in turn led Khayyam
to solve the cubic equation x
^{3} + 200x = 20x
^{2}
+ 2000 and he found a positive root of this cubic by considering
the intersection of a rectangular hyperbola and a circle. An
approximate numerical solution was then found by interpolation in
trigonometric tables. Perhaps even more remarkable is the fact that
Khayyam states that the solution of this cubic requires the use of
conic sections and that it cannot be solved by compass and
straightedge, a result which would not be proved for another 750
years.
His
Treatise on Demonstration of Problems of Algebra
contained a complete classification of cubic equations with
geometric solutions found by means of intersecting conic sections.
In fact Khayyam gives an interesting historical account in which he
claims that the Greeks had left nothing on the theory of cubic
equations. Indeed, as Khayyam writes, the contributions by earlier
writers such as al-Mahani and
al-Khazin
were to translate geometric problems into algebraic equations
(something which was essentially impossible before the work of
Muḥammad ibn Mūsā al-Ḵwārizmī). However, Khayyam himself seems
to have been the first to conceive a general theory of cubic
equations.
Omar Khayyám saw a strong relationship between geometry and
algebra, and was moving in the right direction when he helped to
close the gap between numerical and geometric algebra with his
geometric solution of the general
cubic
equations, but the decisive step in
analytic geometry came later with
René Descartes.
Persian mathematician
Sharafeddin
Tusi (born 1135) did not follow the general development that
came through
al-Karaji's school of algebra
but rather followed Khayyam's application of algebra to geometry.
He wrote a treatise on cubic equations, entitled
Treatise on
Equations, which represents an essential contribution to
another algebra which aimed to study curves by means of equations,
thus inaugurating the study of
algebraic geometry.
Non-Euclidean geometry
In the early 11th century,
Ibn
al-Haytham (Alhazen) made the first attempt at proving the
Euclidean parallel postulate, the fifth
postulate in
Euclid's
Elements, using a
proof by contradiction, where he
introduced the concept of
motion
and
transformation into
geometry. He formulated the
Lambert quadrilateral, which Boris
Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert
quadrilateral", and his attempted proof also shows similarities to
Playfair's axiom.
In the late 11th century,
Omar
Khayyám made the first attempt at formulating a
non-Euclidean postulate as an alternative to the
Euclidean parallel postulate, and he was the first
to consider the cases of
elliptical
geometry and
hyperbolic
geometry, though he excluded the latter.
In
Commentaries on the difficult postulates of Euclid's
book Khayyam made a contribution to non-Euclidean geometry,
although this was not his intention. In trying to prove the
parallel postulate he accidentally proved properties of figures in
non-Euclidean geometries. Khayyam also gave important results on
ratios in this book, extending Euclid's work to include the
multiplication of ratios. The importance of Khayyam's contribution
is that he examined both Euclid's definition of equality of ratios
(which was that first proposed by ) and the definition of equality
of ratios as proposed by earlier Islamic mathematicians such as
al-Mahani which was based on
continued fractions. Khayyam proved that
the two definitions are equivalent. He also posed the question of
whether a ratio can be regarded as a number but leaves the question
unanswered.
The
Khayyam-Saccheri
quadrilateral was first considered by Omar Khayyam in the late
11th century in Book I of
Explanations of the Difficulties in
the Postulates of Euclid. Unlike many commentators on Euclid
before and after him (including of course Saccheri), Khayyam was
not trying to prove the
parallel
postulate as such but to derive it from an equivalent postulate
he formulated from "the principles of the Philosopher" (
Aristotle):
- Two convergent straight lines intersect and it is impossible
for two convergent straight lines to diverge in the direction in
which they converge.
Khayyam then considered the three cases right, obtuse, and acute
that the summit angles of a Saccheri quadrilateral can take and
after proving a number of theorems about them, he (correctly)
refuted the obtuse and acute cases based on his postulate and hence
derived the classic postulate of Euclid. It wasn't until 600 years
later that
Giordano Vitale made an
advance on the understanding of this quadrilateral in his book
Euclide restituo (1680, 1686), when he used it to prove
that if three points are equidistant on the base AB and the summit
CD, then AB and CD are everywhere equidistant.
Saccheri himself based the whole of his long,
heroic and ultimately flawed proof of the parallel postulate around
the quadrilateral and its three cases, proving many theorems about
its properties along the way.
In 1250,
Nasīr
al-Dīn al-Tūsī, in his
Al-risala al-shafiya'an al-shakk
fi'l-khutut al-mutawaziya (
Discussion Which Removes Doubt
about Parallel Lines), wrote detailed critiques of the
Euclidean parallel postulate and on
Omar Khayyám's attempted proof a century
earlier. Nasir al-Din attempted to derive a
proof by contradiction of the
parallel postulate. He was one of the first to consider the cases
of
elliptical geometry and
hyperbolic geometry, though he
ruled out both of them.
His son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a
book on the subject in 1298, based on al-Tusi's later thoughts,
which presented one of the earliest arguments for a
non-Euclidean hypothesis equivalent
to the parallel postulate.
Sadr al-Din's work was published in Rome in 1594 and
was studied by European geometers. This work marked the
starting point for
Giovanni
Girolamo Saccheri's work on the subject, and eventually the
development of modern
non-Euclidean geometry.Victor J. Katz
(1998),
History of Mathematics: An Introduction, p.
270-271,
Addison-Wesley, ISBN
0321016181:
"But in a manuscript probably written by his son Sadr
al-Din in 1298, based on Nasir al-Din's later thoughts on the
subject, there is a new argument based on another hypothesis, also
equivalent to Euclid's, [...] The importance of this latter work is
that it was published in Rome in 1594 and was studied by European
geometers. In particular, it became the starting point for the work
of Saccheri and ultimately for the discovery of non-Euclidean
geometry."
A proof from Sadr al-Din's work was quoted by John Wallis and Saccheri in the 17th and 18th
centuries.
They both derived their proofs of the parallel
postulate from Sadr al-Din's work, while Saccheri also derived his
Saccheri quadrilateral from
Sadr al-Din, who himself based it on his father's
work.
The theorems of Ibn al-Haytham
(Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the
first theorems on elliptical
geometry and hyperbolic
geometry, and along with their alternative postulates, such as
Playfair's axiom, these works
marked the beginning of non-Euclidean geometry and had a
considerable influence on its development among later European
geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and
Giovanni Girolamo
Saccheri.
Trigonometry
The early Indian works on
trigonometry were translated and
expanded in the Muslim world by
Arab and
Persian
mathematicians. They enunciated a large number of theorems which
freed the subject of trigonometry from dependence upon the complete
quadrilateral, as was the case in
Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S.
Kennedy, it was after this development in Islamic mathematics that
"the first real trigonometry emerged, in the sense that only then
did the object of study become the spherical or plane triangle, its sides and angles."
In the early 9th century, (c. 780-850) produced tables for the
trigonometric functions of
sines and cosine, and the first tables for tangents. He was also an
early pioneer in spherical
trigonometry. In 830, Habash al-Hasib al-Marwazi
produced the first tables of cotangents as well as tangents.
Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) discovered
the reciprocal functions of secant and cosecant, and produced the
first table of cosecants, which he referred to as a "table of
shadows" (in reference to the shadow of a gnomon), for each degree from 1° to 90°. He also
formulated a number of important trigonometrical relationships such
as:
- \tan a = \frac{\sin a}{\cos a}
- \sec a = \sqrt{1 + \tan^2 a }
By the 10th century, in the work of Abū al-Wafā'
al-Būzjānī (959-998), Muslim mathematicians were using all six
trigonometric functions, and had sine tables in 0.25° increments,
to 8 decimal places of accuracy, as well as tables of tangent values. Abū al-Wafā' also
developed the following trigonometric formula:
- \sin 2x = 2 \sin x \cos x \
Abū al-Wafā also established the angle addition identities, e.g.
sin (a + b), and discovered the law of sines for spherical trigonometry:Jacques
Sesiano, "Islamic mathematics", p. 157, in
- \frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin
C}{\sin c}.
Also in the late 10th and early 11th centuries, the Egyptian
astronomer Ibn Yunus performed many
careful trigonometric calculations and demonstrated the following
formula:
- \cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}
Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a
sphere, which is considered "the first treatise on spherical trigonometry" in its modern
form, although spherical trigonometry in its ancient Hellenistic
form was dealt with by earlier mathematicians such as Menelaus of Alexandria, who developed
Menelaus' theorem to deal with
spherical problems. However, E. S. Kennedy points out that
while it was possible in pre-lslamic mathematics to compute the
magnitudes of a spherical figure, in principle, by use of the table
of chords and Menelaus' theorem, the application of the theorem to
spherical problems was very difficult in practice. Al-Jayyani's
work on spherical trigonometry "contains formulae for right-handed triangles, the general
law of sines, and the solution of a spherical triangle by means of the polar
triangle." This treatise later had a
"strong influence on European mathematics", and his "definition of
ratios as numbers" and "method of solving a
spherical triangle when all sides are unknown" are likely to have
influenced Regiomontanus.
The method of triangulation, which was
unknown in the Greco-Roman world, was
also first developed by Muslim mathematicians, who applied it to
practical uses such as surveying and
Islamic geography, as described by
Abū Rayhān
al-Bīrūnī in the early 11th century. In the late 11th century,
Omar Khayyám (1048-1131) solved
cubic equations using approximate
numerical solutions found by interpolation in trigonometric tables.
All of these earlier works on trigonometry treated it mainly as an
adjunct to astronomy; the first treatment as a subject in its own
right was by Nasīr al-Dīn
al-Tūsī in the 13th century. He also developed spherical
trigonometry into its present form, and listed the six distinct
cases of a right-angled triangle in spherical trigonometry. In his
On the Sector Figure, he also stated the law of sines for
plane and spherical triangles, discovered the law of tangents for spherical triangles, and
provided proofs for these laws.
Jamshīd al-Kāshī
(1393-1449) provided the first explicit statement of the law of cosines in a form suitable for
triangulation. In France, the law of
cosines has in recent years been called the théorème
d'Al-Kashi. He also gives trigonometric tables of
values of the sine function to four sexagesimal digits (equivalent to 8 decimal
places) for each 1° of argument with differences to be added for
each 1/60 of 1°. In one of his numerical approximations of
π, he correctly computed 2π to 9 sexagesimal digits. In order to determine
sin 1°, al-Kashi discovered the following triple-angle formula often attributed
to François Viète in the
16th century:
- \sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,.
Al-Kashi's colleague Ulugh Beg (1394-1449)
gave accurate tables of sines and tangents correct to 8 decimal
places.
(1526-1585) contributed to trigonometry in his Sidrat al-Muntaha, in which he was the first mathematician to compute a highly accurate numeric value for sin 1°. He discusses the values given by his predecessors, explaining how Ptolemy (ca. 150) used an approximate method to obtain his value of sin 1° and how Abū al-Wafā, Ibn Yunus (ca. 1000), al-Kashi, Qāḍī Zāda al-Rūmī (1337-1412), Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din then solves the problem to obtain the value of sin 1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):
- \sin 1^\circ = 1^P 2' 49 43' 11 14'
44'16 \ (= 1/60 + 2/60^2 + 49/60^3 +
\cdots)\,.
Calculus
Integral calculus
Around 1000 AD, Al-Karaji, using mathematical induction, found a
proof for the sum of integral cubes. The
historian of mathematics, F. Woepcke,
praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham (known as Alhazen in the
West), an Iraqi
mathematician working in Egypt, was the first mathematician to
derive the formula for the sum of the fourth powers, and using an early proof by mathematical induction, he developed
a method for determining the general formula for the sum of any
integral powers. He used his result on sums of integral
powers to perform an integration, in order
to find the volume of a paraboloid. He
was thus able to find the integrals for
polynomials up to the fourth degree, and came close to
finding a general formula for the integrals of any polynomials.
This was fundamental to the development of infinitesimal and integral calculus.
His
results were repeated by the Moroccan mathematicians Abu-l-Hasan ibn Haydur (d.
1413) and Abu Abdallah ibn Ghazi (1437-1514), by Jamshīd al-Kāshī (c.
1380-1429) in The Calculator's Key, and by the Indian mathematicians of the Kerala school of
astronomy and mathematics in the 15th-16th centuries.
Differential calculus
In the 12th century, the Persian
mathematician Sharaf al-Dīn al-Tūsī
was the first to discover the derivative
of cubic polynomials, an important
result in differential
calculus. His Treatise on Equations developed concepts
related to differential
calculus, such as the derivative
function and the maxima and minima
of curves, in order to solve cubic equations which may not have
positive solutions. For example, in order to solve the equation \
x^3 + a = bx, al-Tusi finds the maximum point of the curve \ bx -
x^3 = a. He uses the derivative of the function to find that the
maximum point occurs at x = \sqrt{\frac{b}{3}}, and then finds the
maximum value for y at 2(\frac{b}{3})^\frac{3}{2} by substituting x
= \sqrt{\frac{b}{3}} back into \ y = bx - x^3. He finds that the
equation \ bx - x^3 = a has a solution if a \le
2(\frac{b}{3})^\frac{3}{2}, and al-Tusi thus deduces that the
equation has a positive root if D = \frac{b^3}{27} - \frac{a^2}{4}
\ge 0, where D is the discriminant of
the equation.
Applied mathematics
Geometric art and architecture
Geometric artwork in the form of the
Arabesque was not widely used in the
Middle East or Mediterranean Basin until the golden age of Islam came into full bloom,
when Arabesque became a common feature of Islamic art. Euclidean geometry as expounded on by
Al-Abbās ibn Said
al-Jawharī (ca. 800-860) in his Commentary on Euclid's
Elements, the trigonometry of
Aryabhata and Brahmagupta as elaborated on by Muhammad ibn
Mūsā al-Khwārizmī (ca. 780-850), and the development of
spherical geometry by Abū al-Wafā'
al-Būzjānī (940–998) and spherical trigonometry by Al-Jayyani (989-1079) for determining the
Qibla and times of Salah
and Ramadan, all served as an impetus for
the art form that was to become the Arabesque.
Recent discoveries have shown that geometrical quasicrystal patterns were first employed in
the girih tiles found in medieval
Islamic architecture dating
back over five centuries ago. In 2007, Professor Peter
Lu of Harvard
University and Professor Paul
Steinhardt of Princeton University published a paper in the journal Science
suggesting that girih tilings possessed properties consistent with
self-similar fractal quasicrystalline tilings such as the
Penrose tilings, predating them by
five centuries.
Mathematical astronomy
An impetus behind mathematical astronomy
came from Islamic religious observances, which presented a host of
problems in mathematical astronomy, particularly in spherical geometry. In solving these
religious problems the Islamic scholars went far beyond the Greek
mathematical methods. For example, predicting just when the
crescent moon would become visible is a special challenge to
Islamic mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was
tolerably accurate near the time of the new moon, it specified the
moon's path only with respect to the ecliptic. To predict the first visibility of the
moon, it was necessary to describe its motion with respect to the
horizon, and this problem demands fairly
sophisticated spherical geometry.
Finding
the direction of Mecca and the time
of Salah are the reasons which led to Muslims
developing spherical geometry. Solving any of these problems
involves finding the unknown sides or angles of a triangle on the
celestial sphere from the known
sides and angles. A way of finding the time of day, for example, is
to construct a triangle whose vertices are the zenith, the north celestial
pole, and the sun's position. The observer must know the
altitude of the sun and that of the pole; the former can be
observed, and the latter is equal to the observer's latitude. The time is then given by the angle at
the intersection of the meridian (the arc through the zenith and the pole) and the
sun's hour circle (the arc through the sun and the pole).
The Zij treatises were astronomical
books that tabulated the parameters used for astronomical
calculations of the positions of the Sun, Moon, stars, and planets.
Their principal contributions to mathematical astronomy reflected
improved trigonometrical, computational and observational
techniques. The Zij books were extensive, and typically
included materials on chronology,
geographical latitudes and longitudes, star tables,
trigonometrical
functions, functions in spherical astronomy, the equation of time, planetary motions,
computation of eclipses, tables for first
visibility of the lunar crescent,
astronomical and/or astrological
computations, and instructions for astronomical calculations using
epicyclic geocentric models. Some zījes go beyond
this traditional content to explain or prove the theory or report
the observations from which the tables were computed.
In observational astronomy,
Muhammad ibn
Mūsā al-Khwārizmī's Zij al-Sindh (830) contains
trigonometric tables for the movements of the sun, the moon and the
five planets known at the time. Al-Farghani's A compendium of the science of
stars (850) corrected Ptolemy's
Almagest and gave revised values
for the obliquity of the ecliptic, the
precessional movement of the apogees of the
sun and the moon, and the circumference of the earth.
Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) discovered
that the direction of the Sun's eccentric was changing, and studied the
times of the new moon, lengths for the
solar year and sidereal year, prediction of eclipses, and the phenomenon of parallax. Around the same time, Yahya Ibn Abi
Mansour wrote the Al-Zij al-Mumtahan, in which he
completely revised the Almagest values. In the 10th
century, Abd al-Rahman al-Sufi
(Azophi) carried out observations on the stars
and described their positions, magnitude, brightness, and colour and drawings for each constellation in his
Book of Fixed Stars
(964). Ibn Yunus observed more than 10,000
entries for the sun's position for many years using a large
astrolabe with a diameter of nearly 1.4
meters. His observations on eclipses were
still used centuries later in Simon
Newcomb's investigations on the motion of the moon, while his
other observations inspired Laplace's
Obliquity of the Ecliptic and Inequalities of Jupiter
and Saturn's.
In the late 10th century, Abu-Mahmud al-Khujandi accurately
computed the axial tilt to be 23°32'19"
(23.53°), which was a significant improvement over the Greek and
Indian estimates of 23°51'20" (23.86°) and 24°, and still very
close to the modern measurement of 23°26' (23.44°). In 1006, the Egyptian astronomer Ali ibn
Ridwan observed SN 1006, the brightest
supernova in recorded history, and left a
detailed description of the temporary star. He says that the
object was two to three times as large as the disc of Venus and about one-quarter the brightness of the
Moon, and that the star was low on the southern
horizon. In 1031, al-Biruni's Canon
Mas’udicus introduced the mathematical technique of analysing
the acceleration of the planets, and
first states that the motions of the solar
apogee and the precession are not
identical. Al-Biruni also discovered that the distance between the
Earth and the Sun is larger than Ptolemy's
estimate, on the basis that Ptolemy disregarded the annual solar eclipses.
During
the "Maragha
Revolution" of the 13th and 14th centuries, Muslim astronomers
realized that astronomy should aim to describe the behavior of
physical bodies in mathematical
language, and should not remain a mathematical hypothesis, which would only save the phenomena. The Maragha astronomers also
realized that the Aristotelian view
of motion in the universe being
only circular or linear was not true, as the Tusi-couple showed that linear motion could also
be produced by applying circular motions only. Unlike the ancient
Greek and Hellenistic astronomers
who were not concerned with the coherence between the mathematical
and physical principles of a planetary theory, Islamic astronomers
insisted on the need to match the mathematics with the real world
surrounding them, which gradually evolved from a reality based on
Aristotelian physics to one
based on an empirical and mathematical physics after the work of Ibn al-Shatir. The Maragha Revolution was thus
characterized by a shift away from the philosophical foundations of
Aristotelian cosmology and Ptolemaic astronomy and towards a
greater emphasis on the empirical observation and mathematization
of astronomy and of nature in general, as
exemplified in the works of Ibn al-Shatir, Ali Qushji, al-Birjandi and al-Khafri. In particular, Ibn
al-Shatir's geocentric model was
mathematically identical to the later heliocentric Copernical
model.
Mathematical geography and geodesy
The
Muslim scholars, who held to the spherical Earth theory, used it in an
impeccably Islamic manner, to calculate the distance and direction
from any given point on the earth to Mecca. This
determined the Qibla, or Muslim direction of
prayer. Muslim mathematicians developed spherical trigonometry which was used
in these calculations.
Around
830, Caliph al-Ma'mun commissioned a group
of astronomers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They
found the cities to be separated by one degree of latitude and the
distance between them to be 66 2/3 miles and thus calculated
the Earth's circumference to be 24,000 miles. Another estimate
given by Al-Farghānī was 56
2/3 Arabic miles per degree, which corresponds to 111.8 km per
degree and a circumference of 40,248 km, very close to the
currently modern values of 111.3 km per degree and 40,068 km
circumference, respectively.
In mathematical geography, Abū Rayhān
al-Bīrūnī, around 1025, was the first to describe a polar
equi-azimuthal
equidistant projection of the celestial sphere. He was also regarded as
the most skilled when it came to mapping cities
and measuring the distances between them, which he did for many
cities in the Middle East and western
Indian subcontinent. He often
combined astronomical readings and mathematical equations, in order
to develop methods of pin-pointing locations by recording degrees
of latitude and longitude. He also developed similar techniques
when it came to measuring the heights of mountains, depths of valleys,
and expanse of the horizon, in The
Chronology of the Ancient Nations. He also discussed human geography and the planetary habitability of the
Earth. He hypothesized that roughly a quarter of
the Earth's surface is habitable by humans,
and also argued that the shores of Asia and
Europe were "separated by a vast sea, too
dark and dense to navigate and too risky to try" in reference to
the Atlantic
Ocean and Pacific
Ocean.
Abū Rayhān
al-Bīrūnī is considered the father of geodesy for his important contributions to the
field, along with his significant contributions to geography and
geology. At the age of 17, al-Biruni calculated the latitude of Kath, Khwarazm,
using the maximum altitude of the Sun. Al-Biruni also solved a
complex geodesic equation in order to
accurately compute the Earth's circumference, which were close to modern
values of the Earth's circumference. His estimate of 6,339.9 km for
the Earth radius was only 16.8 km less
than the modern value of 6,356.7 km. In contrast to his
predecessors who measured the Earth's circumference by sighting the
Sun simultaneously from two different locations, al-Biruni
developed a new method of using trigonometric calculations based on the angle
between a plain and mountain top which yielded more accurate
measurements of the Earth's circumference and made it possible for
it to be measured by a single person from a single location.
Mathematical physics
Ibn al-Haytham's work on geometric
optics, particularly catoptrics, in "Book V" of the Book of Optics (1021) contains the
important mathematical problem known as "Alhazen's problem"
(Alhazen is the Latinized name of Ibn
al-Haytham). It comprises drawing lines from two points in the
plane of a circle meeting at a point on the circumference and making equal angles with the
normal at that point. This leads to an equation of the fourth degree. This
eventually led Ibn al-Haytham to derive the earliest formula for
the sum of the fourth powers, and using
an early proof by mathematical induction, he developed
a method for determining the general formula for the sum of any
integral powers, which was fundamental to the
development of infinitesimal and
integral calculus.
Ibn al-Haytham eventually solved "Alhazen's problem" using conic sections and a geometric proof, but
Alhazen's problem remained influential in Europe, when later
mathematicians such as Christiaan
Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to
find an algebraic solution to the problem, using various methods,
including analytic methods of
geometry and derivation by complex
numbers. Mathematicians were not able to find an algebraic
solution to the problem until the end of the 20th century.
Ibn al-Haytham also produced tables of corresponding angles of incidence and refraction of light passing
from one medium to another show how closely he had approached
discovering the law of constancy of ratio of
sines, later attributed to Snell. He also correctly accounted for
twilight being due to atmospheric refraction, estimating
the Sun's depression to be 19 degrees below the horizon during the commencement of the phenomenon in
the mornings or at its termination in the evenings.
Ibn al-Haytham systematically endeavoured to mathematize physics in
the context of his experimental research and controlled testing,
which was oriented by geometric models of the structural
mathematical principles that governed physical phenomena,
particularly in relation to the explication of the behaviour and
nature of vision and light. Ibn al-Haytham also advanced in his
Discourse on Place (Qawl fi al-makan) a
geometrical understanding of place as mathematical space
that is akin to the 17th century conceptions of extensio
by Descartes and analysis situs by Leibniz. Ibn al-Haytham
established his geometrical thesis about place as space in
the context of his mathematical refutation of the Aristotelian
physical definition of topos as a boundary surface of
a containing body (as argued in Book delta [IV] of Aristotle's
Physics).
Abū Rayhān
al-Bīrūnī (973-1048), and later al-Khazini (fl. 1115-1130), were the first to
apply experimental scientific methods to the statics and dynamics fields of mechanics, particularly for determining specific weights, such as those based on the
theory of s and weighing. Muslim
physicists applied the mathematical theories of ratios and infinitesimal
techniques, and introduced algebraic and
fine calculation techniques into the
field of statics.Mariam Rozhanskaya and I. S. Levinova (1996),
"Statics", p. 642, in
Abu 'Abd
Allah Muhammad ibn Ma'udh, who lived in Al-Andalus during the second half of the 11th century, wrote a
work on optics later translated into Latin as Liber de
crepisculis, which was mistakenly attributed to
Alhazen. This was a "short work containing an estimation of
the angle of depression of the sun at the beginning of the morning
twilight and at the end of the evening
twilight, and an attempt to calculate on the basis of this and
other data the height of the atmospheric moisture responsible for
the refraction of the sun's rays." Through his experiments, he
obtained the accurate value of 18°, which comes close to the modern
value.
In 1574, estimated that the stars are millions
of kilometres away from the Earth and that the
speed of light is constant, that if
light had come from the eye, it would take too long for light "to
travel to the star and come back to the eye. But this is not the
case, since we see the star as soon as we open our eyes. Therefore
the light must emerge from the object not from the eyes."
Other fields
Cryptography
In the 9th century, al-Kindi was a pioneer
in cryptanalysis and cryptology. He gave the first known recorded
explanation of cryptanalysis in A
Manuscript on Deciphering Cryptographic Messages. In
particular, he is credited with developing the frequency analysis method
whereby variations in the frequency of the occurrence of letters
could be analyzed and exploited to break ciphers (i.e. crypanalysis by frequency analysis).
This was detailed in a text recently rediscovered in the Ottoman
archives in Istanbul, A Manuscript on Deciphering Cryptographic
Messages, which also covers methods of cryptanalysis,
encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter
combinations in Arabic. Al-Kindi also had knowledge of polyalphabetic ciphers centuries
before Leon Battista Alberti.
Al-Kindi's book also introduced the classification of ciphers,
developed Arabic phonetics and syntax, and described the use of
several statistical techniques for cryptoanalysis. This book
apparently antedates other cryptology references by several
centuries, and it also predates writings on probability and statistics by Pascal
and Fermat by nearly eight centuries.
Ahmad al-Qalqashandi
(1355-1418) wrote the Subh al-a 'sha, a 14-volume
encyclopedia which included a section on cryptology. This
information was attributed to Taj ad-Din Ali ibn ad-Duraihim ben
Muhammad ath-Tha 'alibi al-Mausili who lived from 1312 to 1361, but
whose writings on cryptology have been lost. The list of ciphers in
this work included both substitution and transposition, and for the first time,
a cipher with multiple substitutions for each plaintext letter. Also traced to Ibn al-Duraihim
is an exposition on and worked example of cryptanalysis, including
the use of tables of letter
frequencies and sets of letters which can not occur together in
one word.
Math and the Astrolabe
The astrolabe is a mathematical tool that could be used to solve
all the standard problems of spherical astronomy in five different
ways.
Mathematical induction
The first known proof by mathematical induction was introduced
in the al-Fakhri written by Al-Karaji around 1000 AD, who used it to prove
arithmetic sequences such as
the binomial theorem, Pascal's triangle, and the sum formula for
integral cubes.Victor J. Katz (1998), History of
Mathematics: An Introduction, p. 255-259, Addison-Wesley, ISBN 0321016181:
"Another important idea introduced by al-Karaji and continued by al-Samaw'al and
others was that of an inductive argument for dealing with certain
arithmetic sequences.
Thus al-Karaji used such an argument to prove the
result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state
a general result for arbitrary n.
He stated his theorem for the particular integer 10
[...] His proof, nevertheless, was clearly designed to be
extendable to any other integer.
His proof was the first to make use of the two basic components of
an inductive proof, "namely the truth of the
statement for n = 1 (1 = 1^{3}) and the deriving
of the truth for n = k from that of n =
k - 1."Katz (1998), p. 255:
"Al-Karaji's argument includes in essence the two basic
components of a modern argument by induction, namely the truth of
the statement for n = 1 (1 = 1^{3}) and the
deriving of the truth for n = k from that of
n = k - 1. Of course, this second component is
not explicit since, in some sense, al-Karaji's argument is in
reverse; this is, he starts from n = 10 and goes down to 1
rather than proceeding upward. Nevertheless, his argument in
al-Fakhri is the earliest extant proof of the sum formula
for integral cubes."
Shortly afterwards, Ibn al-Haytham
(Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any
integral powers, which was an
important result in integral calculus. He only stated it for particular
integers, but his proof for those integers was by induction and
generalizable.
Ibn Yahyā
al-Maghribī al-Samaw'al came closest to a modern proof by
mathematical induction in pre-modern times, which he used to extend
the proof of the binomial theorem and Pascal's triangle previously
given by al-Karaji. Al-Samaw'al's inductive argument was only a
short step from the full inductive proof of the general binomial
theorem.Katz (1998), p. 259:
"Like the proofs of al-Karaji and ibn al-Haytham,
al-Samaw'al's argument contains the two basic components of an
inductive proof. He begins with a value for which the result is
known, here n = 2, and then uses the result for a given
integer to derive the result for the next. Although al-Samaw'al did
not have any way of stating, and therefore proving, the general
binomial theorem, to modern readers there is only a short step from
al-Samaw'al's argument to a full inductive proof of the binomial
theorem."
See also
Notes
- O'Connor 1999
- Joseph A. Schumpeter, Historian of Economics: Selected Papers
from the History of Economics Society Conference, 1994, Laurence S.
Moss, Joseph Alois Schumpeter, History of Economics Society.
Conference, Published by Routledge, 1996, ISBN 041513353X, p.64.
Excerpt: A great portion (and most of the best) of medieval
Muslim philosophers, physicians, ethicists, scientists, Islamic
jurists, historians, and geographers were Persian-speaking
Iranians
- Ibn Khaldun,
Franz Rosenthal, N. J. Dawood (1967), The Muqaddimah: An
Introduction to History, p. x, Princeton University Press, ISBN
0691017549. page 430: Only the Persians engaged in the task of
preserving knowledge and writing systematic scholarly works. Thus,
the truth of the following statement by the Prophet becomes
apparent:"If scholarship hung suspended in the highest parts of
heaven, the Persians would attain it."
- Hogendijk 1999
- Bernard
Lewis in What Went Wrong? Western Impact and Middle
Eastern Response
- , in
- "Diophantus sometimes is called "the father of algebra," but
this title more appropriately belongs to al-Khwarizmi. It is true
that in two respects the work of al-Khwarizmi represented a
retrogression from that of Diophantus. First, it is on a far more
elementary level than that found in the Diophantine problems and,
second, the algebra of al-Khwarizmi is thoroughly rhetorical, with
none of the syncopation found in the Greek Arithmetica or
in Brahmagupta's work. Even numbers were written out in words
rather than symbols! It is quite unlikely that al-Khwarizmi knew of
the work of Diophantus, but he must have been familiar with at
least the astronomical and computational portions of Brahmagupta;
yet neither al-Khwarizmi nor other Arabic scholars made use of
syncopation or of negative numbers."
- "Diophantus, the father of algebra, in whose honor I have named
this chapter, lived in Alexandria, in Roman Egypt, in either the
1st, the 2nd, or the 3rd century CE."
- "The six cases of equations given above exhaust all
possibilities for linear and quadratic equations having positive
root. So systematic and exhaustive was al-Khwarizmi's exposition
that his readers must have had little difficulty in mastering the
solutions."
- Gandz and Saloman (1936), The sources of al-Khwarizmi's
algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more
entitled to be called "the father of algebra" than Diophantus
because Khwarizmi is the first to teach algebra in an elementary
form and for its own sake, Diophantus is primarily concerned with
the theory of numbers".
- Jacques Sesiano, "Islamic mathematics", p. 148, in
- Swaney, Mark. History of Magic Squares.
- :
- "The chief difference between Diophantine syncopation and the
modern algebraic notation is the lack of special symbols for
operations and relations, as well as of the exponential
notation."
- D.J. Struik, A Source Book in Mathematics 1200-1800
(Princeton University Press, New Jersey, 1986). ISBN
0-691-02397-2
- P. Luckey, Die Rechenkunst bei Ğamšīd b. Mas'ūd
al-Kāšī (Steiner, Wiesbaden, 1951).
- Aydin
Sayili (1960), "Thabit ibn Qurra's Generalization of the
Pythagorean Theorem", Isis 51 (1): 35-37
- Kline, M. (1972), Mathematical Thought from Ancient to
Modern Times, Volume 1, p. 193, Oxford
University Press
- Kline, M. (1972), Mathematical Thought from Ancient to
Modern Times, Volume 1, pp. 193-5, Oxford
University Press
- R. Rashed (1994). The development of Arabic mathematics:
between arithmetic and algebra. London.
- Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician",
The Journal of the American Oriental Society
123.
- :
- Victor J. Katz (1998), History of Mathematics: An
Introduction, p. 270, Addison-Wesley, ISBN 0321016181:
- Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 447-494 [469],
Routledge, London
and New York:
- Boris Abramovich Rozenfelʹd (1988), A History of
Non-Euclidean Geometry: Evolution of the Concept of a Geometric
Space, p. 65. Springer, ISBN 0387964584.
- Boris A Rosenfeld and Adolf P Youschkevitch (1996),
Geometry, p.467 in Roshdi Rashed, Régis Morelon (1996),
Encyclopedia of the history of Arabic science, Routledge,
ISBN 0415124115.
- Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 447-494 [469],
Routledge, London
and New York:
- Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 447-494 [469],
Routledge, London
and New York:
- Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 447-494 [470],
Routledge, London
and New York:
- (cf. , in )
- "Book 3 deals with spherical trigonometry and includes
Menelaus's theorem."
- (cf. , in )
- Donald Routledge Hill (1996),
"Engineering", in Roshdi Rashed, Encyclopedia of the History of
Arabic Science, Vol. 3, pp. 751-795 [769]
- Al-Kashi, author: Adolf P. Youschkevitch, chief
editor: Boris A. Rosenfeld, p. 256
- Victor J. Katz (1998). History of Mathematics: An
Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
- F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par
Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
- Victor J. Katz (1995), "Ideas of Calculus in Islam and India",
Mathematics Magazine 68 (3): 163-174
[165-9 & 173-4]
- J. L. Berggren (1990), "Innovation and Tradition in Sharaf
al-Din al-Tusi's Muadalat", Journal of the American Oriental
Society 110 (2): 304-9
- Supplemental figures
- Kennedy, Islamic Astronomical Tables, p. 51
- Benno van Dalen, PARAMS (Database of parameter values occurring
in Islamic astronomical sources), "General background of the parameter database"
- Kennedy, Islamic Astronomical Tables, pp. 17-23
- Kennedy, Islamic Astronomical Tables, p. 1
- (cf. )
- (cf. )
- (cf. )
- (cf. )
- David A. King, Astronomy in the Service of Islam,
(Aldershot (U.K.): Variorum), 1993.
- Gharā'ib al-funūn wa-mulah al-`uyūn (The Book of
Curiosities of the Sciences and Marvels for the Eyes), 2.1 "On the
mensuration of the Earth and its division into seven climes, as
related by Ptolemy and others," (ff. 22b-23a)[1]
- Edward S. Kennedy, Mathematical Geography, pp. 187-8,
in
- David A. King (1996), "Astronomy and Islamic society: Qibla,
gnomics and timekeeping", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 1, p. 128-184 [153].
Routledge, London
and New York.
- Akbar S. Ahmed (1984). "Al-Beruni: The First Anthropologist",
RAIN 60, p. 9-10.
- H. Mowlana (2001). "Information in the Arab World",
Cooperation South Journal 1.
- James S. Aber (2003). Alberuni calculated the Earth's
circumference at a small town of Pind Dadan Khan, District Jhelum,
Punjab, Pakistan. Abu Rayhan al-Biruni, Emporia
State University.
- Lenn Evan Goodman (1992), Avicenna, p. 31,
Routledge, ISBN
041501929X.
- Victor J. Katz (1995). "Ideas of Calculus in Islam and India",
Mathematics Magazine 68 (3), p.
163-174.
- John D. Smith (1992). "The Remarkable Ibn al-Haytham", The
Mathematical Gazette 76 (475), p.
189-198.
- Bradley Steffens (2006), Ibn al-Haytham: First
Scientist, Chapter Five, Morgan Reynolds Publishing, ISBN
1599350246
- George
Sarton, Introduction to the History of Science, "The
Time of Al-Biruni"
- (cf. )
- Simon Singh. The Code Book. p. 14-20
- Ibrahim A. Al-Kadi (April 1992), "The
origins of cryptology: The Arab contributions”, Cryptologia
16 (2): 97–126
- Victor J. Katz (1995), "Ideas of Calculus in Islam and India",
Mathematics Magazine 68 (3), p.
163-174:
- Katz (1998), p. 255-259.
Further reading
- (Reviewed: ; )
- (Reviewed: )
- Sowjetische Beiträge zur Geschichte der Naturwissenschaft
pp. 62–160.
External links