(c. 780,
Khwārizm – c.
850) was a Persian mathematician, astronomer and geographer, a scholar in the House of
Wisdom in Baghdad.
His
Kitab
al-Jabr wa-l-Muqabala presented the first systematic
solution of
linear and
quadratic equations. He is considered the
founder of
algebra, a credit he shares with
Diophantus. In the twelfth century,
Latin translations of
his work on the
Indian numerals, introduced the
decimal positional
number system to the
Western
world. He revised
Ptolemy's
Geography and wrote on astronomy
and astrology.
His contributions had a great impact on language. "Algebra" is
derived from
al-jabr, one of the two operations he used to
solve
quadratic equations.
Algorism and
algorithm stem from
Algoritmi, the
Latin form of
his name. His name is the origin of (
Spanish)
guarismo and of (
Portuguese)
algarismo, both meaning
digit.
Life
Few details of al-Khwārizmī's life are known with certainty, even
his birthplace is unsure.
His name may indicate that he came from
Khwarezm (Khiva), then in Greater Khorasan, which occupied the
eastern part of the Persian Empire,
now Xorazm
Province in Uzbekistan. Abu Rayhan
Biruni calls the people of Khwarizm "a branch of the
Persian tree".
Al-Tabari gave his name
as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic:
).
The
epithet al-Qutrubbulli could
indicate he might instead have come from Qutrubbul (Qatrabbul), a
viticulture district near Baghdad.
However, Rashed points out that:
Regarding al-Khwārizmī's religion, Toomer writes:
In
Ibn al-Nadīm's
Kitāb
al-Fihrist we find a short biography on al-Khwārizmī, together
with a list of the books he wrote. Al-Khwārizmī accomplished most
of his work in the period between 813 and 833.
After the Islamic conquest of Persia,
Baghdad became the centre of scientific studies and trade, and many
merchants and scientists from as far as China and India traveled to this city, as did
Al-Khwārizmī. He worked in Baghdad as a scholar at the
House of Wisdom established by
Caliph , where he studied the sciences and
mathematics, which included the translation of
Greek and
Sanskrit
scientific manuscripts.
Contributions
Al-Khwārizmī's contributions to
mathematics,
geography,
astronomy, and
cartography established the basis for innovation
in
algebra and
trigonometry. His systematic approach to
solving
linear and
quadratic equations led to
algebra, a word derived from the title of his 830 book on
the subject, "The Compendious Book on Calculation by Completion and
Balancing" (
al-Kitab al-mukhtasar fi hisab al-jabr
wa'l-muqabalaالكتاب المختصر في حساب الجبر والمقابلة).
On the Calculation with Hindu Numerals written about 825,
was principally responsible for spreading the
Indian system of numeration
throughout the
Middle East and
Europe. It was translated into Latin as
Algoritmi
de numero Indorum. Al-Khwārizmī, rendered as (Latin)
Algoritmi, led to the term "
algorithm".
Some of
his work was based on Persian and Babylonian astronomy,
Indian numbers, and Greek mathematics.
Al-Khwārizmī systematized and corrected
Ptolemy's data for
Africa and
the
Middle east.
Another major book was
Kitab surat al-ard ("The Image of the Earth"; translated
as Geography), presenting the coordinates of places based on those
in the Geography of Ptolemy but with
improved values for the Mediterranean Sea, Asia, and Africa.
He also wrote on mechanical devices like the
astrolabe and
sundial.
He assisted a project to determine the circumference of the Earth
and in making a world map for
al-Ma'mun,
the caliph, overseeing 70 geographers.
When, in the 12th century, his works spread to
Europe through
Latin
translations, it had a profound impact on the advance of
mathematics in Europe.
Algebra
A page from al-Khwārizmī's
Algebra
(Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious
Book on Calculation by Completion and Balancing”) is a
mathematical book written approximately 830 CE.
The book was written with the encouragement of the
Caliph Al-Ma'mun as a popular work on calculation
and is replete with examples and applications to a wide range of
problems in trade, surveying and legal inheritance . The term
algebra is derived from the name of
one of the basic operations with equations (
al-jabr)
described in this book.
The book was translated in Latin as Liber
algebrae et almucabala by Robert
of Chester (Segovia, 1145) hence
"algebra", and also by Gerard of
Cremona. A unique Arabic copy is kept at Oxford and was
translated in 1831 by F. Rosen. A Latin translation is kept in
Cambridge.
The
al-jabr is considered the foundational text of modern
algebra. It provided an exhaustive account of solving polynomial
equations up to the second degree, and introduced the fundamental
methods 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. "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-Khwārizmī's method of solving linear and quadratic equations
worked by first reducing the equation to one of six standard forms
(where
b and
c are positive integers)
- squares equal roots (ax^{2} =
bx)
- squares equal number (ax^{2} =
c)
- roots equal number (bx = c)
- squares and roots equal number (ax^{2} +
bx = c)
- squares and number equal roots (ax^{2} +
c = bx)
- roots and number equal squares (bx + c =
ax^{2})
by dividing out the coefficient of the square and using the two
operations (Arabic: الجبر “restoring” or “completion”) and
al-muqābala ("balancing"). is the process of removing
negative units, roots and squares from the equation by adding the
same quantity to each side. For example,
x^{2} =
40
x − 4
x^{2} is reduced to
5
x^{2} = 40
x. Al-muqābala is the process
of bringing quantities of the same type to the same side of the
equation. For example,
x^{2} + 14 =
x + 5 is reduced to
x^{2} + 9 =
x.
The above discussion uses modern mathematical notation for the
types of problems which the book discusses. However, in
Al-Khwārizmī's day, most of this notation
had not yet been invented,
so he had to use ordinary text to present problems and their
solutions. Forexample, for one problem he writes, (from an 1831
translation)
In modern notation this process, with 'x' the "thing" (shay') or
"root", is given by the steps,
- (10-x)^2=81 x
- x^2+100=101 x
Let the roots of the equation be 'p' and 'q'. Then
\tfrac{p+q}{2}=50\tfrac{1}{2}, pq =100 and
- \tfrac{p-q}{2} = \sqrt{\tfrac{p+q}{2} -
pq}=\sqrt{2550\tfrac{1}{4} - 100}=49\tfrac{1}{2}
So a root is given by
- x=50\tfrac{1}{2}-49\tfrac{1}{2}=1
Several authors have also published texts under the name of
Kitāb al-ğabr wa-l-muqābala, including .
J. J. O'Conner and E. F. Robertson wrote in the
MacTutor History of
Mathematics archive:
R. Rashed and Angela Armstrong write:
Page from a Latin translation,
beginning with "Dixit algorizmi"
Arithmetic
Al-Khwārizmī's second major work was on the subject of arithmetic,
which survived in a
Latin translation but was
lost in the original
Arabic. The
translation was most likely done in the twelfth century by
Adelard of Bath, who had also translated the
astronomical tables in 1126.
The Latin manuscripts are untitled, but are commonly referred to by
the first two words with which they start:
Dixit algorizmi
("So said al-Khwārizmī"), or
Algoritmi de numero Indorum
("al-Khwārizmī on the Hindu Art of Reckoning"), a name given to the
work by
Baldassarre
Boncompagni in 1857. The original Arabic title was possibly
("The Book of Addition and Subtraction According to the Hindu
Calculation")
Al-Khwarizmi's work on arithmetic was responsible for introducing
the
Arabic numerals, based on the
Hindu-Arabic numeral
system developed in
Indian
mathematics, to the
Western world.
The term "
algorithm" is derived from the
algorism, the technique of performing
arithmetic with Hindu-Arabic numerals developed by al-Khwarizmi.
Both "algorithm" and "algorism" are derived from the
Latinized forms of al-Khwarizmi's
name,
Algoritmi and
Algorismi,
respectively.
Astronomy
Corpus Christi College MS 283
(Arabic: زيج "astronomical tables of Sind and Hind") is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind. The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge. Al-Khwarizmi's work marked the beginning of non-traditional methods of study and calculations.
The original Arabic version (written c. 820) is lost, but a version
by the Spanish astronomer
Maslamah Ibn Ahmad al-Majriti
(c. 1000) has survived in a Latin translation, presumably by
Adelard of Bath (January 26, 1126).
The four surviving manuscripts of the Latin translation are kept at
the Bibliothèque publique (Chartres), the Bibliothèque Mazarine
(Paris), the Bibliotheca Nacional (Madrid) and the Bodleian Library
(Oxford).
Al-Khwarizmi made several important improvements to the theory and
construction of
sundials, which he inherited
from his
Indian and
Hellenistic predecessors. He made
tables for these instruments which considerably shortened the time
needed to make specific calculations. His sundial was universal and
could be observed from anywhere on the Earth. From then on,
sundials were frequently placed on mosques to determine the
time of prayer.
The shadow square, an
instrument used to determine the linear height of an object, in
conjunction with the alidade for angular
observations, was also invented by al-Khwārizmī in ninth-century
Baghdad.
The first
quadrants and
mural instruments were invented by
al-Khwarizmi in ninth century Baghdad. The sine quadrant, invented
by al-Khwārizmī, was used for astronomical calculations. The first
horary
quadrant for specific
latitudes, was also invented by
al-Khwārizmī in Baghdad, then center of the development of
quadrants. It was used to determine time (especially the times of
prayer) by observations of the Sun or stars. The
Quadrans
Vetus was a universal horary quadrant, an ingenious
mathematical device invented by al-Khwarizmi in the ninth century
and later known as the
Quadrans Vetus (
Old
Quadrant) in medieval Europe from the thirteenth century. It
could be used for any
latitude on Earth and
at any time of the year to determine the time in hours from the
altitude of the Sun. This was the second
most widely used astronomical instrument during the
Middle Ages after the
astrolabe. One of its main purposes in the Islamic
world was to determine the times of
Salah.
Geography
Al-Khwārizmī's third major work is his (Arabic: كتاب صورة الأرض
"Book on the appearance of the Earth" or "The image of the Earth"
translated as
Geography), which was finished in 833. It is
a revised and completed version of
Ptolemy's
Geography, consisting
of a list of 2402 coordinates of cities and other geographical
features following a general introduction.
There is only one surviving copy of , which is kept at the
Strasbourg University Library.
A Latin
translation is kept at the Biblioteca Nacional de
España in Madrid. The
complete title translates as
Book of the appearance of the
Earth, with its cities, mountains, seas, all the islands and
rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī,
according to the geographical treatise written by Ptolemy the
Claudian.
The book opens with the list of
latitudes
and
longitudes, in order of "weather
zones", that is to say in blocks of latitudes and, in each
weather zone, by order of longitude. As Paul Gallez
points out, this excellent system allows us to deduce many
latitudes and longitudes where the only document in our possession
is in such a bad condition as to make it practically
illegible.
Neither the Arabic copy nor the Latin translation include the map
of the world itself, however Hubert Daunicht was able to
reconstruct the missing map from the list of coordinates. Daunicht
read the latitudes and longitudes of the coastal points in the
manuscript, or deduces them from the context where they were not
legible. He transferred the points onto
graph paper and connected them with straight
lines, obtaining an approximation of the coastline as it was on the
original map. He then does the same for the rivers and towns.
Al-Khwārizmī corrected Ptolemy's gross
overestimate for the length of the Mediterranean SeaEdward S. Kennedy, Mathematical Geography,
p. 188, in (from the Canary Islands to the eastern shores of the Mediterranean);
Ptolemy overestimated it at 63 degrees of longitude, while al-Khwarizmi almost correctly
estimated it at nearly 50 degrees of longitude. He "also depicted the
Atlantic and Indian
Oceans as open bodies of water, not
land-locked seas as Ptolemy had done."
Al-Khwarizmi thus set the Prime Meridian of the Old World at the eastern shore of the
Mediterranean, 10–13 degrees to the east of Alexandria (the prime meridian previously set by Ptolemy) and
70 degrees to the west of Baghdad. Most
medieval Muslim geographers continued to use al-Khwarizmi's prime
meridian.
Jewish calendar
Al-Khwārizmī wrote several other works including a treatise on the
Hebrew calendar ( "Extraction of the
Jewish Era"). It describes the
19-year
intercalation cycle, the rules for determining on what day of
the week the first day of the month
Tishrī
shall fall; calculates the interval between the
Jewish era (creation of Adam) and the
Seleucid era; and gives rules for determining
the mean longitude of the sun and the moon using the Jewish
calendar. Similar material is found in the works of
al-Bīrūnī and
Maimonides.
Other works
Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and
Paris contain further material that surely or with some probability
comes from al-Khwārizmī. The Istanbul manuscript contains a paper
on sundials, which is mentioned in the
Fihirst.
Other
papers, such as one on the determination of the direction of
Mecca, are on the spherical astronomy.
Two texts deserve special interest on the
morning width (
Maʿrifat saʿat al-mashriq
fī kull balad) and the determination of the
azimuth from a height (
Maʿrifat al-samt min
qibal al-irtifāʿ).
He also wrote two books on using and constructing
astrolabes.
Ibn
al-Nadim in his (an index of Arabic books) also mentions (the
book on
sundials) and (the book of
history) but the two have been lost.
The shaping of our mathematics can be attributed to Al-Khwarizmi,
the chief librarian of the observatory, research center and library
called the House of Wisdom in Baghdad. His treatise, "Hisab al-jabr
w'al-muqabala" (Calculation by Restoration and Reduction), which
covers linear and quadratic equations, solved trade imbalances,
inheritance questions and problems arising from land surveyance and
allocation. In passing, he also introduced into common usage our
present numerical system, which replaced the old, cumbersome Roman
one.
See also
Notes
- There is some confusion in the literature on whether
al-Khwārizmī's full name is or . Ibn Khaldun notes in his
encyclopedic work: "The first who wrote upon this branch (algebra)
was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ
ibn Aslam." (MacGuckin de Slane). (Rosen 1831, pp. xi–xiii)
mentions that "[Abu Abdallah Mohammed ben Musa] lived and wrote
under the caliphat of Al Mamun, and must therefore be distinguished
from Abu Jafar Mohammed ben Musa, likewise a mathematician and
astronomer, who flourished under the Caliph Al Motaded (who reigned
A.H. 279-289, A.D. 892-902)." Karpinski notes in his review on
(Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks
of the author as Abū Gaʿfar M. b. M., instead of Abū Abdallah M. b.
M."
- Gandz 1936
- Abu Rahyan Biruni, "Athar al-Baqqiya 'an al-Qurun
al-Xaliyyah"(Vestiges of the past: the chronology of ancient
nations), Tehran, Miras-e-Maktub, 2001. Original Arabic of the
quote: "و أما أهل خوارزم، و إن کانوا غصنا ً من دوحة الفُرس" (pg.
56)
- "Iraq After the Muslim Conquest", by Michael G.
Morony, ISBN 1593333153 (a 2005 facsimile from the original
1984 book), p. 145
- Ruska
- Neugebauer
- David A. King (2002), "A Vetustissimus Arabic Text on the
Quadrans Vetus", Journal for the History of Astronomy
33: 237-255 [238-9]
- David A.
King, "Islamic Astronomy", in Christopher Walker (1999), ed.,
Astronomy before the telescope, p. 167-168. British Museum Press.
ISBN 0-7141-2733-7.
- Daunicht
Further reading
- Biographical
- Fuat Sezgin. Geschichte des arabischen Schrifttums.
1974, E. J. Brill, Leiden, the Netherlands.
- Sezgin, F., ed., Islamic Mathematics and Astronomy,
Frankfurt: Institut für Geschichte der arabisch-islamischen
Wissenschaften, 1997–9.
- Algebra
- Barnabas Hughes. Robert of Chester's Latin translation of
al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F.
Steiner Verlag Wiesbaden (1989). ISBN 3-515-04589-9.
- Arithmetic
- Astronomy
- Suter, H. [Ed.]: Die astronomischen Tafeln des Muhammed ibn
Mûsâ al-Khwârizmî in der Bearbeitung des Maslama ibn Ahmed
al-Madjrîtî und der latein. Übersetzung des Athelhard von Bath auf
Grund der Vorarbeiten von A. Bjørnbo und R. Besthorn in Kopenhagen.
Hrsg. und komm. Kopenhagen 1914. 288 pp. Repr. 1997 (Islamic
Mathematics and Astronomy. 7). ISBN 3-8298-4008-X.
- Van Dalen, B. Al-Khwarizmi's Astronomical Tables Revisited:
Analysis of the Equation of Time.
- Jewish calendar
- Geography
General references
- For a more extensive bibliography see: History of mathematics, Mathematics in medieval Islam,
and Astronomy in medieval
Islam.
- Roshdi Rashed, The development of Arabic mathematics:
between arithmetic and algebra, London, 1994.