Muḥammad ibn Mūsā al-Khwārizmī (Persian: Muḥammad
Khwārizmī محمد بن موسى خوارزمی; c. 780 – c. 850), Arabized as
al-Khwarizmi with al- and formerly Latinized as Algorithmi,
was a Persian scholar who produced works
in mathematics, astronomy, and geography. Around 820 AD he was
appointed as the astronomer and head of the library of the House of
Wisdom in Baghdad.
Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on
Calculation by Completion and Balancing, c. 813–833 CE) presented the first
systematic solution of linear and quadratic equations. One of
his principal achievements in algebra was his demonstration of how to solve
quadratic equations by completing the square, for which he provided
geometric justifications. Because he was the first to treat algebra as an
independent discipline and introduced the methods of "reduction" and
"balancing" (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), he has been described as the father or
founder of algebra. The term algebra itself comes
from the title of his book (specifically the word al-jabr meaning
"completion" or "rejoining"). His name gave rise to
the terms algorism and algorithm. His name is
also the origin of (Spanish) guarismo and of
(Portuguese) algarismo, both meaning digit.
In the 12th
century, Latin translations of his textbook on arithmetic (Algorithmo
de Numero Indorum) which codified the various Indian numerals,
introduced the decimal positional number system to the Western
world. The Compendious Book on Calculation by Completion and Balancing,
translated into Latin by Robert of Chester in 1145, was used until
the sixteenth century as the principal mathematical text-book of European
universities.
In addition to his
best-known works, he revised Ptolemy's Geography, listing the
longitudes and latitudes of various cities and localities. He further
produced a set of astronomical tables and wrote about calendaric works, as well
as the astrolabe and the sundial. He also made important contributions
to trigonometry, producing accurate sine and cosine tables, and the first
table of tangents.
Life
Few details of
al-Khwārizmī's life are known with certainty. He was born into
a Persian family and Ibn al-Nadim gives his birthplace
as Khwarezm in Greater
Khorasan (modern Khiva, Xorazm Region, Uzbekistan).
Muhammad ibn Jarir
al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy
al-Majūsiyy al-Quṭrubbaliyy (محمد
بن موسى الخوارزميّ المجوسـيّ القطربّـليّ).
The epithet al-Qutrubbulli could indicate he might
instead have come from Qutrubbul
(Qatrabbul), a viticulture district near Baghdad. However, Rashed suggests:
There is no need to be
an expert on the period or a philologist to see that al-Tabari's second
citation should read "Muhammad ibn Mūsa
al-Khwārizmī and al-Majūsi al-Qutrubbulli," and that there are
two people (al-Khwārizmī and al-Majūsi al-Qutrubbulli) between whom the
letter wa [Arabic 'و' for the conjunction 'and'] has been omitted in an early copy. This would
not be worth mentioning if a series of errors concerning the personality of
al-Khwārizmī, occasionally even the origins of his knowledge, had not been
made. Recently, G.J. Toomer ... with naive confidence constructed an
entire fantasy on the error which cannot be denied the merit of amusing the
reader.
Regarding
al-Khwārizmī's religion, Toomer writes:
Another epithet given
to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was
an adherent of the old Zoroastrian religion. This would still have been
possible at that time for a man of Iranian origin, but the pious preface to
al-Khwārizmī's Algebra shows that he was an orthodox Muslim, so
al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he
in his youth, had been Zoroastrians.
Ibn al-Nadīm's Kitāb
al-Fihrist includes 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 Muslim 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 al-Ma’mūn, where he studied the sciences and
mathematics, which included the translation
of Greek and Sanskrit scientific manuscripts.
Douglas Morton
Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā
al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the
eldest of the three Banū Mūsā.
Contributions
A page from
al-Khwārizmī's Algebra
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 book
on the subject, "The Compendious Book on Calculation by Completion and
Balancing".
On the Calculation with
Hindu Numerals written about 820, was principally responsible for spreading
the Hindu–Arabic numeral system 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
Main article: The Compendious Book on Calculation by Completion and
Balancing
|
Further information: Latin translations of the 12th
century and Science in the medieval Islamic world
|
Left: The original
Arabic print manuscript of the Book of Algebra by
Al-Khwārizmī. Right: A page from The Algebra of Al-Khwarizmi by
Fredrick Rosen, in English.
The Compendious Book on
Calculation by Completion and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala) is a
mathematical book written approximately 820 CE. The book was written with the
encouragement of 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,
meaning "restoration", referring to adding a number to both sides of
the equation to consolidate or cancel terms) 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.
It provided an
exhaustive account of solving polynomial equations up to the second
degree, and discussed the fundamental methods of "reduction" and
"balancing", referring to the transposition of terms to the other
side of an equation, 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 (ax2 = bx)
·
squares equal number (ax2 = c)
·
roots equal number (bx = c)
·
squares and roots equal
number (ax2 + bx = c)
·
squares and number
equal roots (ax2 + c = bx)
·
roots and number equal
squares (bx + c = ax2)
by dividing out the
coefficient of the square and using the two operations al-jabr (Arabic: الجبر "restoring"
or "completion") and al-muqābala ("balancing"). Al-jabr is
the process of removing negative units, roots and squares from the equation by
adding the same quantity to each side. For example, x2 = 40x − 4x2 is reduced to 5x2 = 40x. Al-muqābala is
the process of bringing quantities of the same type to the same side of the
equation. For example, x2 + 14
= x + 5 is reduced to x2 + 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. For example, for one problem he writes, (from an 1831 translation):
If some one says:
"You divide ten into two parts: multiply the one by itself; it will be
equal to the other taken eighty-one times." Computation: You say, ten less
a thing, multiplied by itself, is a hundred plus a square less twenty things,
and this is equal to eighty-one things. Separate the twenty things from a
hundred and a square, and add them to eighty-one. It will then be a hundred
plus a square, which is equal to a hundred and one roots. Halve the roots; the
moiety is fifty and a half. Multiply this by itself, it is two thousand five
hundred and fifty and a quarter. Subtract from this one hundred; the remainder
is two thousand four hundred and fifty and a quarter. Extract the root from
this; it is forty-nine and a half. Subtract this from the moiety of the roots,
which is fifty and a half. There remains one, and this is one of the two parts.
In modern notation this process, with x the "thing" (شيء shayʾ) or "root", is given by the steps,
Let the roots of the equation be x = p and x = q. Then , and
So a root is given by
Several authors have
also published texts under the name of Kitāb al-jabr wal-muqābala,
including Abū Ḥanīfa Dīnawarī, Abū Kāmil Shujāʿ ibn Aslam, Abū
Muḥammad al-‘Adlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn
‘Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī.
J.J. O'Conner and E.F.
Robertson wrote in the MacTutor History of Mathematics archive:
Perhaps one of the most
significant advances made by Arabic mathematics began at this time
with the work of al-Khwarizmi, namely the beginnings of algebra. It is
important to understand just how significant this new idea was. It was a
revolutionary move away from the Greek concept of mathematics which was
essentially geometry. Algebra was a unifying theory which allowed rational
numbers, irrational numbers, geometrical magnitudes, etc., to all be
treated as "algebraic objects". It gave mathematics a whole new
development path so much broader in concept to that which had existed before,
and provided a vehicle for future development of the subject. Another important
aspect of the introduction of algebraic ideas was that it allowed mathematics
to be applied to itself in a way which had not happened before.
R. Rashed and Angela
Armstrong write:
Al-Khwarizmi's text can
be seen to be distinct not only from the Babylonian tablets, but also
from Diophantus' Arithmetica. It no longer concerns a series
of problems to be solved, but an exposition which starts with
primitive terms in which the combinations must give all possible prototypes for
equations, which henceforward explicitly constitute the true object of study.
On the other hand, the idea of an equation for its own sake appears from the
beginning and, one could say, in a generic manner, insofar as it does not
simply emerge in the course of solving a problem, but is specifically called on
to define an infinite class of problems.
According to
Swiss-American historian of mathematics, Florian Cajori, Al-Khwarizmi's
algebra was different from the work of Indian mathematicians, for Indians
had no rules like the ''restoration'' and ''reduction''. Regarding the
dissimilarity and significance of Al-Khwarizmi's algebraic work from that of
Indian Mathematician Brahmagupta, Carl Benjamin Boyer wrote:
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. Nevertheless, the Al-jabr comes
closer to the elementary algebra of today than the works of either Diophantus
or Brahmagupta, because the book is not concerned with difficult problems in
indeterminant analysis but with a straight forward and elementary exposition of
the solution of equations, especially that of second degree. The Arabs in
general loved a good clear argument from premise to conclusion, as well as systematic
organization – respects in which neither Diophantus nor the Hindus excelled.
Page from a Latin
translation, beginning with "Dixit algorizmi"
Arithmetic
Algorists vs. abacists,
depited in a sketch from 1508 CE
Al-Khwārizmī's second
most influential work was on the subject of arithmetic, which survived in Latin
translations but lost in the original Arabic. His writings include the
text kitāb al-ḥisāb al-hindī ('Book on computation with Indian
numerals'), and perhaps a more elementary text, kitab al-jam'
wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian
arithmetic'). These texts described algorithms on decimal numbers
(Hindu–Arabic numerals) that could be carried out on a dust board. Called takht in
Arabic (Latin: tabula), a board covered with a thin layer of dust
or sand was employed for calculations, on which figures could be written with a
stylus and easily erased and replaced when necessary. Al-Khwarizmi's algorithms
were used for almost three centuries, until replaced by Al-Uqlidisi's
algorithms that could be carried out with pen and paper.
As part of 12th century
wave of Arabic science flowing into Europe via translations, these texts proved
to be revolutionary in
Europe. Al-Khwarizmi's Latinized name, Algorismus,
turned into the name of method used for computations, and survives in
the modern term "algorithm". It gradually replaced the previous
abacus-based methods used in Europe.
Four Latin texts
providing adaptions of Al-Khwarizmi's methods have survived, even though none
of them is believed to be a literal translation:
·
Dixit Algorizmi (published in
1857 under the title Algoritmi de Numero Indorum)
·
Liber Alchoarismi de
Practica Arismetice
·
Liber Ysagogarum
Alchorismi
·
Liber Pulveris
Dixit Algorizmi ('Thus spake
Al-Khwarizmi') is the starting phrase of a manuscript in the University of
Cambridge library, which is generally referred to by its 1857 title Algoritmi
de Numero Indorum. It is attributed to the Adelard of Bath, who had
also translated the astronomical tables in 1126. It is perhaps the closest to
Al-Khwarizmi's own writings.
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-Khwārizmī. Both "algorithm" and "algorism" are derived
from the Latinized forms of al-Khwārizmī's name, Algoritmi and Algorismi,
respectively.
Astronomy
Page from Corpus
Christi College MS 283. A Latin translation of al-Khwārizmī's Zīj.
Al-Khwārizmī's Zīj
al-Sindhind (Arabic: زيج
السند هند, "astronomical tables of Siddhanta") 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.
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
Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).
Trigonometry
Al-Khwārizmī's Zīj
al-Sindhind also contained tables for the trigonometric
functions of sines and cosine. A related treatise on spherical
trigonometry is also attributed to him.
Al-Khwārizmī produced
accurate sine and cosine tables, and the first table of tangents.
Geography
Daunicht's
reconstruction of the section of al-Khwārizmī's world map concerning
the Indian Ocean.
A 15th-century
version of Ptolemy's Geography for comparison.
A stamp issued
September 6, 1983 in the Soviet Union, commemorating al-Khwārizmī's
(approximate) 1200th birthday.
Statue of Al-Khwārizmī
in his birth town Khiva, Uzbekistan.
Al-Khwārizmī's third
major work is his Kitāb Ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض, "Book of the
Description of the Earth"), also known as his Geography,
which was finished in 833. It is a major reworking of Ptolemy's
2nd-century 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 Kitāb Ṣūrat al-Arḍ, which is kept at
the Strasbourg University Library. A Latin translation is kept at
the Biblioteca Nacional de España in Madrid. 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 the deduction of many
latitudes and longitudes where the only extant document 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
Sea from the Canary Islands to the eastern shores of the
Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while
al-Khwārizmī 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-Khwārizmī's Prime Meridian at
the Fortunate Isles was thus around 10° east of the line used by
Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use
al-Khwārizmī's prime meridian.
Jewish
calendar
Al-Khwārizmī wrote
several other works including a treatise on the Hebrew calendar,
titled Risāla fi istikhrāj ta’rīkh al-yahūd (Arabic: رسالة في إستخراج تأريخ اليهود, "Extraction of
the Jewish Era"). It describes the Metonic cycle, a 19-year
intercalation cycle; the rules for determining on what day of the week the
first day of the month Tishrei shall fall; calculates the interval
between the Anno Mundi or Jewish year and the Seleucid era; and
gives rules for determining the mean longitude of the sun and the moon using
the Hebrew calendar. Similar material is found in the works of Abū Rayḥān
al-Bīrūnī and Maimonides.
Other
works
Ibn al-Nadim's Kitāb
al-Fihrist, an index of Arabic books, mentions al-Khwārizmī's Kitāb
al-Taʾrīkh (Arabic: كتاب
التأريخ), a book of annals. No direct manuscript survives; however, a copy had
reached Nusaybin by the 11th century, where its metropolitan
bishop, Mar Elyas bar Shinaya, found it. Elias's chronicle quotes it
from "the death of the Prophet" through to 169 AH, at which point
Elias's text itself hits a lacuna.
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; the Fihrist credits
al-Khwārizmī with Kitāb ar-Rukhāma(t) (Arabic: كتاب الرخامة). 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 (Ma‘rifat sa‘at al-mashriq
fī kull balad) and the determination of the azimuth from a height
(Ma‘rifat al-samt min qibal al-irtifā‘).
He also wrote two books
on using and constructing astrolabes.
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