Abū ʿAbdallāh Muḥammad ibn Mūsā alKhwārizmī
Abū ʿAbdallāh Muḥammad ibn Mūsā alKhwārizmī (Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي), earlier transliterated as Algoritmi or Algaurizin, (c. 780, Khwārizm – c. 850) was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom inBaghdad. The word alKhwarizmi is pronounced in classical Arabic as AlKhwarithmi hence the Latin transliteration.
In the twelfth century, Latin translations of his work on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. In Renaissance Europe, he was considered the original inventor of algebra, although we now know that his work is based on older Indian or Greek sources. He revised Ptolemy's Geography and wrote on astronomy and astrology.
Some words reflect the importance of alKhwarizmi's contributions to mathematics. "Algebra" is derived from aljabr, 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 also the origin of (Spanish) guarismo and of (Portuguese) algarismo, both meaning digit.
Life
He was born in a Persian family, and his birthplace is given as Chorasmia by Ibn alNadim.
Few details of alKhwārizmī's life are known with certainty. His name may indicate that he came from Khwarezm (Khiva), then in Greater Khorasan, which occupied the eastern part of the Greater Iran, now Xorazm Province in Uzbekistan. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree".^{}
AlTabari gave his name as Muhammad ibn Musa alKhwārizmī alMajousi alKatarbali (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). The epithet alQutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul),^{[10]} 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 alTabari's second citation should read “Muhammad ibn Mūsa alKhwārizmī and alMajūsi alQutrubbulli,” and that there are two people (alKhwārizmī and alMajūsi alQutrubbulli) between whom the letterwa [Arabic ‘و’ for the article ‘and’] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of alKhwārizmī, 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 alKhwārizmī's religion, Toomer writes:
Another epithet given to him by alṬabarī, "alMajūsī," 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 alKhwārizmī's Algebra shows that he was an orthodox Muslim, so alṬabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.^{}
Ibn alNadīm's Kitāb alFihrist includes a short biography on alKhwārizmī, together with a list of the books he wrote. AlKhwārizmī 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 AlKhwārizmī. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph alMaʾmūn, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts.
D. M. Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā alKhwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā.^{}
Contributions
AlKhwārizmī'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" (alKitab almukhtasar fi hisab aljabr wa'lmuqabalaالكتاب 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 theMiddle East and Europe. It was translated into Latin as Algoritmi de numero Indorum. AlKhwārizmī, 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.
AlKhwārizmī systematized and corrected Ptolemy's data for Africa and the Middle east. Another major book was Kitab surat alard ("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 theMediterranean 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 alMa'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. He introduced Arabic numerals into the Latin West, based on a placevalue decimal system developed from Indian sources.^{}
Algebra
Left: The original Arabic print manuscript of the Book of Algebra by AlKhwarizmi. Right: A page from The Algebra of AlKhwarizmi by Fredrick Rosen, in English.

AlKitāb almukhtaṣar fī ḥisāb aljabr walmuqābala (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 alMa'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 (aljabr, meaning completion, or, subtracting a number from both sides of the equation) 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 subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.^{}
AlKhwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and care 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 aljabr (Arabic: الجبر “restoring” or “completion”) and almuqābala ("balancing"). Aljabr 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} = 40x − 4x^{2} is reduced to 5x^{2} = 40x. Almuqābala 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 alKhwārizmī'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 say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eightyone times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eightyone things. Separate the twenty things from a hundred and a square, and add them to eightyone. 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 fortynine 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" (shay') or "root", is given by the steps,
Let the roots of the equation be 'p' and 'q'. Then , and
So a root is given by
Several authors have also published texts under the name of Kitāb aljabr walmuqābala, including Abū Ḥanīfa alDīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad alʿAdlī, Abū Yūsuf alMiṣṣīṣī, 'Abd alHamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn alṬūsī.
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 alKhwarizmi, 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:
"AlKhwarizmi'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 resolved, 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."^{[20]}
Arithmetic
AlKhwārizmī'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 alKhwārizmī"), orAlgoritmi de numero Indorum ("alKhwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857. The original Arabic title was possibly Kitāb alJamʿ waltafrīq biḥisāb alHind ("The Book of Addition and Subtraction According to the Hindu Calculation")^{}
AlKhwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the HinduArabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with HinduArabic numerals developed by alKhwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of alKhwarizmi's name, Algoritmi and Algorismi, respectively.
Astronomy
AlKhwārizmī's Zīj alSindhind (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, themoon 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 alMajriti (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
AlKhwārizmī's Zīj alSindhind also contained tables for the trigonometric functions of sines and cosine. A related treatise on spherical trigonometry is also attributed to him.^{}
Geography
AlKhwārizmī's third major work is his Kitāb ṣūrat alArḍ (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 Kitāb ṣūrat alArḍ, 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 alKhwārizmī, 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 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.^{}
AlKhwārizmī 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 alKhwarizmi almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic and Indian Oceans as open bodies of water, not landlocked seas as Ptolemy had done." AlKhwarizmi thus set thePrime 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 alKhwarizmi's prime meridian.^{}
Jewish calendar
AlKhwārizmī wrote several other works including a treatise on the Hebrew calendar (Risāla fi istikhrāj taʾrīkh alyahūd "Extraction of the Jewish Era"). It describes the 19year 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 alBīrūnī 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 alKhwārizmī. The Istanbul manuscript contains a paper on sundials, which is mentioned in the Fihrist. 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 almashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat alsamt min qibal alirtifāʿ).
He also wrote two books on using and constructing astrolabes. Ibn alNadim in his Kitab alFihrist (an index of Arabic books) also mentions Kitāb arRukhāma(t) (the book on sundials) and Kitab alTarikh (the book of history) but the two have been lost.