Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi, also known as al-Khwarizmi (c. 780 – c. 850), was a distinguished mathematician of the Islamic Golden Age whose scholarly contributions encompassed Arabic-language treatises in mathematics, astronomy, and geography. Approximately 820 CE, he was affiliated with the House of Wisdom in Baghdad, which served as the capital of the Abbasid Caliphate during that era. As a preeminent scholar of his time, his extensive body of work significantly impacted subsequent generations of authors across both the Islamic world and Europe.

Muhammad ibn Musa al-Khwarizmi, or simply al-Khwarizmi (c. 780 – c. 850) was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe.

His influential treatise on algebra, titled Al-Jabr (The Compendious Book on Calculation by Completion and Balancing) and composed between 813 and 833, introduced the inaugural systematic approach to solving linear and quadratic equations. A notable algebraic accomplishment was his elucidation of solving quadratic equations through the method of completing the square, supported by geometric proofs. Al-Khwarizmi is often recognized as the 'father' or 'founder' of algebra because he was the first to establish it as a distinct mathematical discipline and to introduce the fundamental methods of 'reduction' and 'balancing'. The method of 'balancing' involves transposing subtracted terms to the opposite side of an equation, effectively canceling identical terms on both sides. The English word algebra derives from the abbreviated title of his aforementioned work (الجبر Al-Jabr, transl. 'completion' or 'rejoining'). Furthermore, his name is the etymological source for the English terms algorism and algorithm, as well as the Spanish, Italian, and Portuguese term algoritmo, the Spanish term guarismo, and the Portuguese term algarismo, all of which signify 'digit'.

During the 12th century, Latin translations of al-Khwarizmi's treatise on Indian arithmetic, titled Algorithmo de Numero Indorum, played a pivotal role in introducing the decimal-based positional number system to the Western world. This work systematically codified the diverse Indian numerals. Similarly, his work Al-Jabr, rendered into Latin by the English scholar Robert of Chester in 1145, served as the primary mathematical textbook in European universities until the 16th century.

Al-Khwarizmi undertook a revision of Ptolemy's 2nd-century Greek treatise, Geography, meticulously cataloging the longitudes and latitudes of various cities and geographical locations. His contributions also included the compilation of astronomical tables and scholarly writings on calendric systems, the astrolabe, and the sundial. Furthermore, al-Khwarizmi significantly advanced trigonometry by generating precise sine and cosine tables.

Life

Precise biographical details concerning al-Khwārizmī remain largely uncertain. Ibn al-Nadim identifies his birthplace as Khwarazm, a region from which he is widely believed to have originated. He was of Persian ancestry; his name itself signifies 'from Khwarazm', an area historically part of Greater Iran and currently encompassing parts of Turkmenistan and Uzbekistan. Notwithstanding his Persian heritage, all of his scientific treatises were composed exclusively in Arabic.

Al-Tabari recorded his full name as Muḥammad ibn Musá al-Khwārizmī al-Majūsī al-Quṭrubbullī (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). The appellation al-Qutrubbulli potentially suggests an origin from Qutrubbul (Qatrabbul), a locality situated near Baghdad. However, this assertion is contested by Roshdi Rashed, who states:

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.

Conversely, David A. King corroborates al-Khwārizmī's nisba to Qutrubbul, explaining that the designation 'al-Khwārizmī al-Qutrubbulli' arose from his birth in the immediate vicinity of Baghdad.

Concerning al-Khwārizmī's religious affiliation, Toomer observes:

Another epithet bestowed upon him by al-Ṭabarī, "al-Majūsī," suggests that he adhered to the ancient Zoroastrian religion. Such adherence was plausible for an individual of Iranian descent during that era; however, the devout preface to al-Khwārizmī's Algebra demonstrates his adherence to orthodox Islam. Consequently, al-Ṭabarī's epithet likely signifies that his ancestors, and potentially al-Khwārizmī himself in his early life, practiced Zoroastrianism.

Ibn al-Nadīm's Al-Fihrist contains a concise biographical account of al-Khwārizmī, alongside a catalog of his literary works. Al-Khwārizmī produced the majority of his scholarly output between 813 and 833 CE. Following the Muslim conquest of Persia, Baghdad emerged as a prominent hub for scientific inquiry and commerce. Approximately 820 CE, he received an appointment as an astronomer and the chief librarian at the House of Wisdom, an institution founded by the Abbasid Caliph al-Ma'mūn. Al-Khwārizmī pursued studies in various sciences and mathematics, notably engaging in the translation of Greek and Sanskrit scientific manuscripts. Furthermore, he functioned as a historian, whose writings are referenced by scholars such as al-Tabari and Ibn Abi Tahir.

During the reign of al-Wathiq, he reportedly participated in the first of two embassies to the Khazars. Douglas Morton Dunlop posits the possibility that Muḥammad ibn Mūsā al-Khwārizmī is identical to Muḥammad ibn Mūsā ibn Shākir, who was the eldest among the three Banū Mūsā brothers.

Contributions

Al-Khwārizmī's contributions to mathematics, geography, astronomy, and cartography laid the groundwork for advancements in algebra and trigonometry. His methodical framework for resolving linear and quadratic equations gave rise to the discipline of algebra, a term originating from the title of his seminal work on the subject, Al-Jabr.

On the Calculation with Hindu Numerals, composed around 820 CE, played a pivotal role in disseminating the Hindu–Arabic numeral system across the Middle East and into Europe. Upon its translation into Latin during the 12th century as Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu art of reckoning), the word "algorithm" became known in the Western world.

Elements of his work were founded on Persian and Babylonian astronomical traditions, Indian numerical systems, and Greek mathematical principles.

Al-Khwārizmī systematized and refined Ptolemy's geographical data for Africa and the Middle East. A significant work, Kitab surat al-ard ("The Image of the Earth," also rendered as Geography), provided geographical coordinates derived from Ptolemy's Geography, yet incorporated enhanced values for the Mediterranean Sea, Asia, and Africa.

He authored treatises concerning mechanical instruments such as the astrolabe and the sundial. He contributed to a project aimed at calculating the Earth's circumference and creating a world map for Caliph al-Ma'mun, supervising a team of 70 geographers. The dissemination of his works into Europe via Latin translations during the 12th century profoundly influenced the progression of mathematics across the continent.

Algebra

Al-Jabr (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 treatise composed around 820 CE. Authored at the behest of Caliph al-Ma'mun, it served as an accessible guide to calculation, featuring numerous examples and practical applications relevant to commerce, land surveying, and legal inheritance. The word "algebra" originates from one of the fundamental operations involving equations (al-jabr, signifying "restoration," which denotes the addition of a quantity to both sides of an equation to consolidate or eliminate terms) detailed within this text. The work was subsequently translated into Latin as Liber algebrae et almucabala by Robert of Chester in Segovia in 1145, thus giving rise to the term "algebra," and also by Gerard of Cremona. A singular Arabic manuscript is preserved at Oxford and was translated by F. Rosen in 1831, while a Latin translation resides in Cambridge.

The treatise offered a comprehensive exposition on the resolution of polynomial equations up to the second degree. It also elucidated the foundational principles of "reduction" and "balancing," which involve transposing terms across an equation and canceling identical terms on opposing sides, respectively.

Al-Khwārizmī's method for solving linear and quadratic equations involved initially simplifying the equation into one of six canonical forms, wherein b and c represent positive integers.

• Squares equal roots (ax² = bx)
• Equations where squares are equivalent to a numerical value (ax² = c).
• Equations where roots are equivalent to a numerical value (bx = c).
• Equations where squares and roots are equivalent to a numerical value (ax² + bx = c).
• Equations where squares and a numerical value are equivalent to roots (ax² + c = bx).
• Equations where roots and a numerical value are equivalent to squares (bx + c = ax§67§).

These equations were solved by normalizing the coefficient of the square term and applying two fundamental operations: al-jabr (Arabic: الجبر, meaning "restoring" or "completion") and al-muqābala ("balancing"). Al-jabr involves eliminating negative terms (units, roots, and squares) from an equation by adding an equivalent positive quantity to both sides. For instance, the expression x§1617§ = 40x − 4x§2223§ simplifies to 5x§2627§ = 40x. Conversely, al-muqābala is the procedure of consolidating like terms on the same side of the equation. An example is the reduction of x§3637§ + 14 = x + 5 to x§4243§ + 9 = x.

The preceding discussion employs contemporary mathematical notation to describe the problem types addressed in the text. However, during al-Khwārizmī's era, much of this symbolic representation was undeveloped, necessitating the articulation of problems and their corresponding solutions using natural language. As an illustration, one problem is presented as follows (excerpted from the 1831 "Rosen" translation):

If some one say: “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 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.

Expressed in contemporary mathematical notation, this procedure, where x represents the "thing" (شيء, or shayʾ) or "root", unfolds through the following steps:

${\displaystyle (10-x)^{2}=81x}$

${\displaystyle 100+x^{2}-20x=81x}$

${\displaystyle x^{2}+100=101x}$

Assuming the equation's roots are x = p and x = q, then the following relationships hold: p + q §2324§ = 50 §3536§ §3738§ {\displaystyle {\tfrac {p+q}{2}}=50{\tfrac {1}{2}}} , ${\displaystyle pq=100}$ and

p −<!-- − --> q §1718§ = ( p + q §3940§ ) §4748§ −<!-- − --> p q = 2550 §6970§ §7172§ − 100 = §8586§ §9091§ §9293§ {\displaystyle {\frac {p-q}{2}}={\sqrt {\left({\frac {p+q}{2}}\right)^{2}-pq}}={\sqrt {2550{\tfrac {1}{4}}-100}}=49{\tfrac {1}{2}}}

Consequently, one root is determined as follows:

x = 50 §1516§ §1718§ − 49 §3031§ §3233§ = §3940§ {\displaystyle x=50{\tfrac {1}{2}}-49{\tfrac {1}{2}}=1}

Numerous scholars, including Abū Ḥanīfa Dīnawarī, Abū Kāmil, 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, have authored works titled Kitāb al-jabr wal-muqābala.

Solomon Gandz characterized Al-Khwarizmi as the progenitor of Algebra, stating:

Al-Khwarizmi's algebraic contributions are considered the fundamental basis and essential element of scientific disciplines. In a broader sense, Al-Khwarizmi holds a stronger claim to the title "father of algebra" than Diophantus, primarily because Al-Khwarizmi systematically presented algebra in an elementary format for its intrinsic value, whereas Diophantus's focus was predominantly on number theory.

Victor J. Katz further asserts:

The earliest authentic algebra treatise that remains extant is Mohammad ibn Musa al-Khwarizmi's work on al-jabr and al-muqabala, composed in Baghdad approximately in 825 CE.

Within the MacTutor History of Mathematics Archive, John J. O'Connor and Edmund F. Robertson observed:

A pivotal advancement in Arabic mathematics likely originated during this period with al-Khwarizmi's contributions, specifically the genesis of algebra. The profound significance of this novel concept cannot be overstated. It represented a radical departure from the predominantly geometric Greek mathematical paradigm. Algebra emerged as a unifying theoretical framework, enabling the treatment of rational numbers, irrational numbers, and geometric magnitudes as cohesive "algebraic objects." This innovation forged an entirely new trajectory for mathematical development, vastly expanding its conceptual scope beyond previous limitations and establishing a foundation for future disciplinary progress. Furthermore, the integration of algebraic principles facilitated an unprecedented self-application of mathematics.

Roshdi Rashed and Angela Armstrong state:

Al-Khwarizmi's seminal text distinguishes itself not only from ancient Babylonian tablets but also from Diophantus' Arithmetica. Rather than presenting a series of problems for resolution, it offers an exposition commencing with fundamental terms, designed to generate all conceivable prototypes for equations, which are subsequently established as the primary subject of inquiry. Furthermore, the intrinsic concept of an equation emerges from the outset in a generalized manner, not merely as a byproduct of problem-solving, but as a specific construct defining an infinite array of mathematical challenges.

According to Florian Cajori, a distinguished Swiss-American historian of mathematics, Al-Khwarizmi's algebraic methodology diverged from that of Indian mathematicians, who lacked analogous rules such as restoration and reduction. Carl B. Boyer further elaborated on the distinctiveness and significance of Al-Khwarizmi's algebraic contributions compared to those of the Indian mathematician Brahmagupta, stating:

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 scholarly contribution focused on arithmetic, surviving exclusively in Latin translations as the original Arabic texts are no longer extant. His writings encompassed the work titled kitāb al-ḥisāb al-hindī ('Book of Indian computation'), and potentially a more foundational text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic'). These treatises detailed algorithms for decimal numbers (Hindu–Arabic numerals) designed for execution on a dust board. Known as takht in Arabic (Latin: tabula), this board, coated with a thin layer of dust or sand, facilitated calculations by allowing figures to be inscribed with a stylus and subsequently erased or modified with ease. Al-Khwarizmi's algorithms remained in use for nearly three centuries until they were superseded by Al-Uqlidisi's methods, which permitted calculations using pen and paper.

As part of the 12th-century influx of Arabic scientific knowledge into Europe through translations, these texts proved revolutionary. Al-Khwarizmi's Latinized name, Algorismus, subsequently became the etymological root for the term "algorithm," which denotes a method for computation. This new approach progressively supplanted the abacus-based calculation techniques previously prevalent in Europe.

Four Latin texts, which represent adaptations rather than literal translations of Al-Khwarizmi's methodologies, have been preserved:

• Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum)
• Liber Alchoarismi de Practica Arismetice
• Liber Ysagogarum Alchorismi
• Liber Pulveris

The manuscript Dixit Algorizmi ('Thus spake Al-Khwarizmi'), which begins with this phrase, is housed in the University of Cambridge library and is commonly referenced by its 1857 publication title, Algoritmi de Numero Indorum. It is attributed to Adelard of Bath, who also translated astronomical tables in 1126, and is considered potentially the closest surviving work to Al-Khwarizmi's original writings.

Al-Khwarizmi's contributions to arithmetic were instrumental in disseminating Arabic numerals, which originated from the Hindu–Arabic numeral system developed in Indian mathematics, throughout the Western world. The word "algorithm" itself stems from "algorism," a method for performing arithmetic using Hindu-Arabic numerals, a technique pioneered by al-Khwārizmī. Both "algorithm" and "algorism" are etymologically linked to the Latinized versions of al-Khwārizmī's name, specifically Algoritmi and Algorismi.

Astronomy

Al-Khwārizmī's significant astronomical treatise, Zīj as-Sindhind (Arabic: زيج السند هند, meaning "astronomical tables of Siddhanta"), comprises approximately 37 chapters dedicated to calendrical and astronomical computations, alongside 116 tables containing calendrical, astronomical, and astrological information, including a table of sine values. This work represents the earliest of numerous Arabic Zijes that drew upon Indian astronomical methodologies, collectively referred to as the sindhind. The term "Sindhind" itself is a linguistic adaptation of the Sanskrit word Siddhānta, which commonly denotes an astronomical textbook. Notably, the mean motions presented in al-Khwarizmi's tables originate from the "corrected Brahmasiddhanta" (Brahmasphutasiddhanta) by Brahmagupta.

This treatise includes comprehensive tables detailing the movements of the Sun, Moon, and the five planets recognized during that era. Its publication signified a pivotal moment in Islamic astronomy, as prior to this, Muslim astronomers predominantly engaged in a research-oriented approach, focusing on translating existing works and assimilating established knowledge.

Although the original Arabic manuscript, composed around c. 820, is no longer extant, a version by the Spanish astronomer Maslama al-Majriti, dating to approximately c. 1000, has been preserved through a Latin translation, likely undertaken by Adelard of Bath on January 26, 1126. Four extant manuscripts of this Latin translation are currently housed in the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid), and the Bodleian Library (Oxford).

Trigonometry

Al-Khwārizmī's Zīj as-Sindhind incorporated tables for the trigonometric functions of sine and cosine. Furthermore, a treatise on spherical trigonometry is also ascribed to his authorship.

Al-Khwārizmī developed precise tables for sine and cosine values.

Geography

Al-Khwārizmī's third principal scholarly contribution is his Kitāb Ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض, translated as "Book of the Description of the Earth"), alternatively referred to as his Geography, completed in 833. This extensive revision of Ptolemy's second-century Geography presents a comprehensive list of 2402 coordinates for cities and various geographical landmarks, preceded by a general introductory section.

Only a single extant copy of Kitāb Ṣūrat al-Arḍ remains, housed at the Strasbourg University Library, while a Latin translation is preserved at the Biblioteca Nacional de España in Madrid. The treatise commences with an ordered compilation of latitudes and longitudes, organized by "weather zones"—specifically, in blocks of latitudes, with longitudes ordered within each zone. Paul Gallez observes that this systematic arrangement facilitates the inference of numerous latitudes and longitudes even when the sole surviving document is in a severely deteriorated, almost illegible state. Although neither the Arabic original nor the Latin translation contains a world map, Hubert Daunicht successfully reconstructed the absent map using the provided coordinate list. Daunicht meticulously extracted the latitudes and longitudes of coastal points from the manuscript, or inferred them from contextual clues when illegible. He then plotted these points onto graph paper and connected them with straight lines, thereby approximating the coastline as it would have appeared on the original map. A similar methodology was applied to delineate rivers and towns.

Al-Khwārizmī significantly rectified Ptolemy's substantial overestimation of the Mediterranean Sea's length, spanning from the Canary Islands to its eastern coasts. Ptolemy had erroneously calculated this distance as 63 degrees of longitude, whereas al-Khwārizmī provided a nearly accurate estimate of approximately 50 degrees of longitude. Furthermore, he "portrayed the Atlantic and Indian Oceans as expansive, open bodies of water, contrasting with Ptolemy's depiction of them as land-locked seas." Consequently, al-Khwārizmī's Prime Meridian, situated at the Fortunate Isles, was positioned approximately 10° east of the meridian adopted by Marinus and Ptolemy. The majority of medieval Muslim gazetteers subsequently continued to employ al-Khwārizmī's prime meridian.

Jewish calendar

Al-Khwārizmī authored several additional treatises, including one on the Hebrew calendar, specifically titled Risāla fi istikhrāj ta'rīkh al-yahūd (Arabic: رسالة في إستخراج تأريخ اليهود, "Extraction of the Jewish Era"). This work elucidates the Metonic cycle, a 19-year intercalation period, and outlines the principles for ascertaining the weekday upon which the first day of the month Tishrei occurs. Furthermore, it computes the temporal difference between the Anno Mundi (Jewish year) and the Seleucid era, and provides methodologies for determining the mean longitudes of the sun and moon utilizing the Hebrew calendar. Comparable content is also present in the scholarly contributions of Al-Bīrūnī and Maimonides.

Other Scholarly Contributions

Ibn al-Nadim's comprehensive index of Arabic literature, Al-Fihrist, references al-Khwārizmī's Kitāb al-Taʾrīkh (Arabic: كتاب التأريخ), a historical chronicle. Although no original manuscript of this work is extant, a copy was reportedly discovered in Nusaybin during the 11th century by Mar Elias bar Shinaya, the metropolitan bishop. Elias's own chronicle incorporates excerpts from al-Khwārizmī's text, covering events from "the death of the Prophet" up to 169 AH, at which point Elias's narrative becomes incomplete.

Numerous Arabic manuscripts housed in collections in Berlin, Istanbul, Tashkent, Cairo, and Paris contain additional content that is either definitively or very likely attributable to al-Khwārizmī. Notably, the Istanbul manuscript includes a treatise on sundials. The Fihrist specifically attributes the work Kitāb ar-Rukhāma(t) (Arabic: كتاب الرخامة) to al-Khwārizmī. Other scholarly papers, such as one detailing the methodology for determining the Qibla (direction of Mecca), delve into topics within spherical astronomy.

Particular attention is warranted for two specific texts: one concerning the "morning width" (Ma'rifat sa'at al-mashriq fī kull balad) and another addressing the determination of azimuth from an elevated position (Ma'rifat al-samt min qibal al-irtifā'). Furthermore, al-Khwārizmī authored two distinct volumes dedicated to the utilization and fabrication of astrolabes.

Commemorations and Recognitions

• The Al-Khwarizmi crater, located on the lunar far side, is named in his honor.
• Asteroid 13498 Al Chwarizmi, a main-belt asteroid, was discovered on August 6, 1986, by E. W. Elst and V. G. Ivanova at Smolyan.
• Asteroid 11156 Al-Khwarismi, also a main-belt asteroid, was discovered on December 31, 1997, by P. G. Comba at Prescott.

Notes

References

Sources

• The earliest known manuscript of Kitab Surat al-Ard is located in the Strasbourg National Library.
• Earliest Manuscript of Kitab Surat al-Ard in the Strasbourg National Library