Al Khwarizmi: A Medieval Polymath

A review article| Khalid Bin Umar 

During the time of the Renaissance, Europe labelled a medieval polymath as being the “original inventor of algebra.” The person who wrote the first book on Algebra, & first to describe the systematic solution and quadratic equations. He was the one who wrote pieces on mechanical devices such as the clock, astrolabe, and sundial. He was also accredited for familiarizing the Western World with the decimal positional number system; moreover he was the man accountable for the use of Arabic numerals in Mathematics. People marveled, how he brought about the concept of Algorithm in Mathematics. It explains how the word “algorithm” came from his name. Without Algorithms, we would not be able to read this on the computer. In his book, Introduction to the History of Science, George Sarton refers to the first half of the ninth century as the “ The time of Al Khwarizmi“.

They were talking about Muḥammad ibn Mūsā al-Khwarizmi (780–850) who was a scholar from Golden age of Islam, an eternal student, a practitioner, and a teacher. As a polymath, his work was multidisciplinary that covered mathematics, astrology, astronomy, and geography. His academic writings have withstood centuries. He presented the first systematic solution of linear and quadratic equations along with the decimal system and introduced reduction and balancing (e.g., like and unlike terms). It was these contributions that gave him honor as the “father or founder of algebra.”

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Although little is known about his early life, we know that he lived from 780 to 850 AD. His name tells that he was the son of Musa and father of Jafar.   He lived and worked in Baghdad, but his family came from Khiva in Khwarizm in Central Asia (then part of  Greater Khurasan of Iran). Presently its a part of Xorazm Province in Uzbekistan. Historical records left no clue about his daily activities, friends, and family of Khwarizmi.

In 772, when the seat of the Islamic Caliphate was transferred from Damascus to Baghdad, it became the center of scientific studies and trade at the height of the Islamic Khilafat. Merchants and scientists far from China & India traveled to this city. A more rational point of view emerged, knowledge from all over the world were collected, processed, and synthesized scientifically. In 820, Mamun Al Rasheed founded a scientific society named Bait-ul-Hikma (House of Wisdom) which consisted of a library, & an astronomical observatory where scientists were invited to do research. It made a significant impact on the sciences, especially mathematics and algorithms, along with medicine. Al-Khwārizmī was the lead astronomer and head of the library of the House of Wisdom. He accomplished most of his work here in Baghdad during the period starting from 813 to 833.

He was fortunate that he lived at a time when his works spread and made a mark in the academic history of the world. In 12th century, when his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe.

Algebra

The word Algebra is derived from the title of his seminal book called, al-Kitab al-mukhtasar fi hisab al-jabrwa’l-muqabala (The Compendious Book on Calculation of Integration and Equation) (الكتابالمختصرفيحسابالجبروالمقابلة).

The word “algebra” is derived from the Arabic term “al-jabr” which means “reunion of broken parts.”  Al-Khwarizmi used algebra to solve quadratic equations and the ideas that al-Khwarizmi developed, helped usher in the European Renaissance during the 14th, 15th, and 16th centuries. The book was translated in Latin as Liber algebraeetalmucabala by Robert of Chester hence the word “algebra” came into prevalence. It gives you several hundred simple quadratic equations by analysis as well as by geometrical example. It also has substantial sections on methods of dividing up inheritances and surveying plots of land. It is mostly concerned with methods for solving practical computational problems rather than algebra as the term is now understood.

This book covered linear and quadratic equations, which in turn solved trade imbalances, inheritance questions and problems arising from land surveyance and allocation. In passing, he also introduced into common usage of our present numerical system, which replaced the old, cumbersome Roman one.

The impetus behind the development of algebra was the Islamic law of inheritance, Hisab al-jabrw’al-muqabala devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as linear equations; hence, quadratic equations were not required. This book of Calculation of Integration and Equation changed mathematics forever.

This book is considered as the foundation text of modern algebra. It provided an exhaustive account of solving polynomial equations up to the second degree. It also introduced the fundamental methods of “reduction” and “balancing”. He invented the study of algebra, & was a pioneer to consider algebra as an independent discipline.  He introduced the methods of “reduction” and “balancing” (the transposition of subtracted terms to the other side of an equation, i.e., the cancellation of like terms on opposite sides of the equation] and “balancing” (subtraction of the same quantity from both sides of an equation, and the cancellation of like terms on opposite sides. Al-Khwarizmi might not have invented algebra in the modern sense as Indians and Greeks were reportedly practicing this art but it was al- Khwarizmi who introduced algebra to the West. His style was so clear and authoritative that his volume became the standard mathematical text in Europe for hundreds of years.

In particular, Al-Khwarizmi developed a formula for systematically solving quadratic equations (equations involving unknown numbers to the power of 2, or x2) by using the methods of completion and balancing to reduce any equation to one of six standard forms, which were then solvable. He described the standard forms in terms of “squares” (what would today be “x2″), “roots” (what would today be “x“) and “numbers” (regular constants, like 42), and identified the six types as: 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), and roots and number equal squares (bx + c = ax2).

Having introduced the natural numbers, al-Khwarizmi introduced the main topic of this first section of his book, namely the solution of equations. His equations were linear or quadratic and composed of units, roots, and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. However, although we shall use the familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi’s mathematics is done entirely in words with no symbols being used.

He first reduced an equation (linear or quadratic) to one of six standard forms:

1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39.
5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2.

The reduction is carried out using the two operations of al-jabr and al-muqabala. Here “al-jabr” means “completion” and is the process of removing negative terms from an equation.

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If we use one of al-Khwarizmi’s own examples, “al-jabr” transforms x2 = 40 x – 4 x2 into 5 x2 = 40 x. The term “al-muqabala” means “balancing” and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of “al-muqabala” reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots).

Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. For example to solve the equation x2 + 10 x = 39

The book was translated in Latin as Liber algebraeetalmucabala by Robert of Chester hence the word “algebra came into prevalence. Gerard of Cremona later translated it. He formulated the rules of inheritance as linear equations; hence, quadratic equations were not required. This book of Calculation of Integration and Equation changed mathematics forever.

Arithmetic 

He wrote a treatise on Indo-Arabic numerals Kitāb al-Jamʿwal-tafrīq bi-ḥisāb al-Hind (“The Book of Addition and Subtraction According to the Indian Calculation”) in 825 AD. It was translated into Latin as Algoritmi de numero Indorum. This way he introduced Arabic numerals into the Latin West, based on a place-value decimal system developed in Indian sources. A fundamental role was played by the introduction of zero by means of a special symbol, a small circle or a dot. The decimal positioning system was known for a long time by Mesopotamian in a sense of positional system but the arithmetical treatise by Khwarizmi was the first mathematical work to provide a detailed presentation of the rules for executing four basic operations and for computing with fractions according to such notation. The efficiency of such computing methods contributed in making the name of Khwarizmi famous all over the Europe.

Algorithms

Today, people use algorithms to do addition and long division on the principles that are found in Al-Khwarizmi’s text written about 1200 years ago. The modern meaning of the word relates to a specific practice for solving a particular problem. He is the person who gave his name to one of the hottest ideas in 21st-century business: algorithms he developed the concept of the algorithm in mathematics and binary mathematics used in computer programming is based on the algorithmic modules, hence science of computer would not have developed as we see in our life. The programming language is based on trigonometry and derivative which was developed by him.

Robert of Chester’s translation of al-Khwarzmi’s treatise on algebra opens with the words dixit Algorithmi, “Algorithmi says.”  What happened is that over the period of time al-Khwarizmi’s name became in the West to denote first the new process of reckoning with Hindu-Arabic numerals, algorithmus,

Since his name was translated in latin as Algoritm, this led to the term “algorithm” and “algorism” which are derived from the latinized forms of al-Khwarizmi’s name, Algoritmi and Algorismi. 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.

Traditional systems had used different letters of the alphabet to represent numbers or cumbersome Roman numerals, and the new system was far superior. It allowed people to multiply and divide easily in addition also gave chance to check their work. The merchant Leonardo Fibonacci of Pisa, who had learned about Arabic numerals in Tunis, wrote a treatise rejecting the abacus in favor of the Arab method of reckoning, and as a result, the system of Hindu-Arabic numeration caught on quickly in Central Italy. By the fourteenth century, Italian merchants and bankers had abandoned the abacus and were doing their calculations using pen and paper, in much the same way we do today.the entire step-by-step process of solving mathematical problems.

The original Arabic text of his work is lost but a Latin translation, Algoritmi de numeroIndorumsurvived. The work describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work. Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version.

Trigonometry

Al-Khwārizmī (c. 780-850) compiled a set of tables for the trigonometric functions of sinesevidentl for a circle of radius of 150 degrees and cosine in the Zīj al-Sindhind, alongside the first tables for tangents. He was also an early pioneer in spherical trigonometry, and wrote a treatise on the subject, Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents which encompassed the tables of trigonometric functions of sines and cosines.

Astronomy

Al-Khwārizmī’s Zīj al-Sind hind (Arabic: زيج السند هند “astronomical tables of Sind and Hind“) is a work on calendrical and astronomical calculation to determine the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. The sine quadrant, invented by al-Khwārizmī, was used for astronomical calculations. It was used to determine time (especially the times of prayer) by observations of the sun or stars. 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 prayers. Al-Khwarizmi made several important improvements to the theory and construction of sundials, 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. Like his treatises on algebra and Indo-Arabic numerals, this astronomical work (or an Andalusian revision thereof) was translated into Latin.

His books on sundialswas Kitab al Rukhama,this work included a table of sines, evidently for a circle of radius 150 units. 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 Khwarizmi. 

Geography

Al-Khwārizmī’s third major work is his Kitāb ṣūrat al-Arḍ (Arabic: الأرضصورةكتاب “Book on the appearance of the Earth” or “The image of the Earth” translated as Geography), which was finished in 833.

It was a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map. The book, which is based on Ptolemy‘s Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. The manuscript does include maps which on the whole were more accurate than those of Ptolemy. In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy.

He systematized and corrected Ptolemy’s research in geography, using his own original findings .The text exists in a manuscript; the maps have unfortunately not been preserved, although modern scholars have been able to reconstruct them from al-Khwarizmi’s descriptions. He supervised the work of 70 geographers to create a map of the then “known world”. When his work became known in Europe through Latin translations, his influence made a permanent mark on the development of science in the West.

A single surviving copy of the text (no maps) has been kept at the Strasbourg University Library in France, while a Latin translation is available at the BibliotecaNacional de España, in Madrid. Kharazmi’s world-map has been reconstructed in detail by RadiyaJafri of the Aligarh Muslim University, and published as :kharI.tahjughrAfiyA-ye jahAnazrU-ye ketAb .sUrat al-‘ar.dKhwArazmI, ta.hlIlenteqAdIwa-tafsIrbehqalam-e rA.diyahJafrI,

He assisted a project to determine the circumference of the Earth which had long been known to be spherical and in making a world map for al-Ma’mun, the caliph, overseeing 70 geographers by measuring the length of a degree of a meridian through the plain of Sinjar in Iraq.

Jewish calendar 

(Risāla fi istikhrājtaʾrīkh al-yahūd “Extraction of the Jewish Era”) – Al-Khwārizmī wrote a treatise on the Hebrew calendar. 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 AbūRayḥān al-Bīrūnī and Maimonides.

Al-Khwarizmi wrote a work on history which has not reached to us, but which was quoted by several later historians. A copy of his Kitāb al-Taʾrīkhhad 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

al-jadwal al-`ishrinI (The vicesimal table)This is  the spherical astronomy to determine the direction of Mecca (Qibla). In this book of tables he presented all twenty values of longitude difference x and twenty values of latitude difference y the angle q(x,y) between the qibla and the south in degrees and minutes. This is for a locality whose geographical latitude is the sum of y and the latitude of Mecca, and whose geographical longitude differs x from that of Mecca. The qibla was computed according to an approximate method equivalent to (tan (q(x,y))=sin(y)/sin(x)), as shown by the numbers 45;0 in the diagonal of the table.

Ma‘rifatsa‘at al-mashriqfīkullbalad –on the morning width

Ma‘rifat al-samt min qibal al-irtifā -for the determination of the azimuth from a height ( ‘).

Kitab al-`amalbi’l-asturlab, a book on operating with the astrolabe. Preserved in Berlin

Zara’if min `amal -Witty things on the Astrolabe

Marifat al-samtbil-asturlab the determination of the azimuth by means of the astrolabe

It is very much evident from these works that Al Khwarizimi was a outstanding scholar and a genius of all time.

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’.  

In the episode of Science in a Golden Age, theoretical physicist, Jim al-Khalili explores Al-Khwarizmi’s 9th century treatise that also underpins the science of flight and the engineering behind the fastest car in the world

In the history of Arabs – Philip Hitti wrote – “One of the greatest scientific minds of Islam, he influenced mathematical thought to a greater extent than any other medieval writer“.

While a good deal of controversy lingered on his major contributions — as to whether they were the result of original research or based on Indian and Greek sources — few can deny that beyond his ability to synthesize existing knowledge that the Greeks, Indians and others assembled. Saidan states that it should be understood as arithmetic done “in the Indian way”, with Hindu-Arabic numerals, rather than as simply “Indian arithmetic”. The Arab mathematicians incorporated their own innovations in their texts, Bridget Lim &Corronna Brezina accept that he might have taken some hint from Indian texts but they say Al Khwarizmis’ presentation of Algebra cannot be traced directly to any Hindu work. 

There is no doubt that Al-Khwārizmī’s contribution to mathematics, geography, astronomy, and cartography were phenomenal and this efforts were and holistic since they ranged from one subject to another; he took risk and didn’t hesitate in writing his own findings and exploring new dimensions in fields that intrigued him. Without his use of the zero in the Indian numbers that were introduced to Europe, the discipline may not have accomplished the kind of progress that gave rise to contemporary mathematics. He clearly left his mark in Mathematics, Astronomy and Geography; his achievements within that particular time period have continued their legacy till present times.

In memory of the great mathematician, a crater on the far side of the moon was named after Al Khwarizmi in 1973, which illustrated that he has been held in high esteem by the international scientific community and that his works have stood the test of time. 

References:

Brentjes, Sonja (2007). “Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī” in Thomas Hockey et al.(eds.).

The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 631–633

Goldstein, B.R. (1968). Commentary on the Astronomical Tables of Al-Khwarizmi: By Ibn Al-Muthanna. Yale University Press.

Gandz, Solomon (1938). “The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī”. Osiris. 5 (5): 319–391. doi:10.1086/368492. 

Burnett, Charles (2017), “Arabic Numerals”, in Thomas F. Glick (ed.), Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, 

Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer Science+Business Media

Daffa, Ali Abdullah al- (1977).The Muslim contribution to mathematics. London: Croom Helm. 

Saidan, A. S. (Winter 1966), “The Earliest Extant Arabic Arithmetic: Kitab al-Fusul fi al Hisab al-Hindi of Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi”,

Al-Khwarizmi: Father of Algebra and Trigonometry By Bridget Lim, Corona Brezina

The Algebra of Mohammad ben Musa, edited and translated by Frederic Rosen. London 1831.