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,
Al-Jabr.
On the Calculation with Hindu Numerals, written about 825, was principally responsible for spreading the
Hindu–Arabic numeral system throughout the Middle East and Europe. When the work was translated into Latin in the 12th century as
Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu art of reckoning), the term "algorithm" was introduced to the Western world. 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 wrote on mechanical devices like the
astrolabe and
sundial. 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. Al-Khwarizmi's contributions to astronomy, which mixed theory and practice, were directly impactful upon astronomers of his epoch and the field as a whole. His meticulous methodological approach and scientific developments in this field directly influenced modern astronomy: "Al-Khwarizmi's astronomical tables, known as Zij, became essential tools for subsequent generations of astronomers. These tables enabled precise calculations for predicting astronomical events, such as eclipses and the positions of celestial bodies, which were crucial for navigation and timekeeping."
Algebra Al-Jabr (The Compendious Book on Calculation by Completion and Balancing, ) is a mathematical book written approximately 820 CE. It 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 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 (, 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 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 method 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 ( "restoring" or "completion") and ("balancing"). is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example,
x2 = 40
x − 4
x2 is reduced to 5
x2 = 40
x. 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 that 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 the 1831 "Rosen" translation) Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician
Brahmagupta,
Carl B. Boyer wrote: It is true that in two respects the work of al-Khowarizmi 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-Khowarizmi 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. 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.
Arithmetic Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text
kitāb al-ḥisāb al-hindī ('Book of Indian computation'), 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 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) 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 Al-Khwārizmī's Zij as-Sindhind|) 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 word Sindhind is a corruption of the
Sanskrit Siddhānta, which is the usual designation of an astronomical textbook. In fact, the mean motions in the tables of al-Khwarizmi are derived from those in the "corrected Brahmasiddhanta" (
Brahmasphutasiddhanta) of
Brahmagupta. The work contains tables for the movements of the
Sun,
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 ) is lost, but a version by the Spanish astronomer
Maslama al-Majriti () has survived in a Latin translation, presumably by
Adelard of Bath (26 January 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). Al-Khwarizmi's contributions to astronomy were both practical and theoretical, impacting the ways that astrolonomical phenomena were understood and applied during his epoch. His meticulous methodological approach and scientific developments in this field directly influenced modern astronomy: "Al-Khwarizmi's astronomical tables, known as Zij, became essential tools for subsequent generations of astronomers. These tables enabled precise calculations for predicting astronomical events, such as eclipses and the positions of celestial bodies, which were crucial for navigation and timekeeping."
Trigonometry Al-Khwārizmī's
Zīj as-Sindhind contained tables for the
trigonometric functions of sines and cosine.
Geography and
Behaim. The general shape of the coastline is the same between
Taprobane and
Cattigara. The
Dragon's Tail, or the eastern opening of the Indian Ocean, which does not exist in Ptolemy's description, is traced in very little detail on al-Khwārizmī's map, although is clear and precise on the Martellus map and on the later Behaim version. of
Ptolemy's
Geography for comparison Al-Khwārizmī's third major work is his (, "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 second-century
Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction. There is one surviving copy of , which is kept at the
Strasbourg University Library. A Latin translation is 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 notes, this 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; 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 deduced 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 did 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 "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. 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 (). 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 () and the determination of the
azimuth from a height (). He wrote two books on using and constructing
astrolabes. == Honours ==