Al-Khwarizmi Persian Mathematician, Astronomer and Geographer
December 20, 2016 by Editor
۲nd & last part – Nasser Kanani, (Prof. Dr.Ing. Dr. Habil.) Berlin
Measurement of the circumference of the Earth
To accomplish a desire of Caliph al-Ma′mun, al-Khwarizmi joined a team of Muslim scholars consisting of geographers, astronomers and mathematicians, who were commissioned to measure the circumference of the Earth, which had long been known to be spherical, by determining the length of a degree of a meridian.
The men of science travelled a north-south road in the plain of Sinjār in Iraq until they observed a change of one degree in the meridian. They measured the distance travelled to the amount of 115,350 kilometers. By calculating 115,350×۳۶۰° they succeeded in determining the Earth’s circumference to be 41,526 kilometers, which is amazingly accurate in comparison to the modern value of 40,075 or 40,007 kilometers (equatorial or meridional, respectively).
In addition to his works in astronomy, al-Khwarizmi made important strides in mathematics. In fact, his everlasting fame is based on his achievements in this field of science. He is considered the greatest among the Muslim mathematicians and the founder of some of the most important concepts of mathematics. The eminent Belgian-American science historian, George Sarton (1884-1956), has described him as: “… the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time….”۱
Al-Khwarizmi is in particular well-known for his books on algebra and arithmetic.
Al-Khwarizmi’s most famous and exceedingly influential work in the realm of mathematics was his Magnus Opus entitled “The Compendious Book on Calculation by Restoring and Balancing”, which was written around 830 CE under the patronage of Caliph al-Ma‘mun. He was the first mathematician to use the two expressions “al-jabr” (restoring) and “al-muqābala” (balancing) in the title of his book to designate algebra as a separate discipline.2 In the preface, he described the purpose of his book as follows:
“That fondness for science, by which God has distinguished the Imam al-Ma‘mun, the Commander of the Faithful…that affability and condescension which he shows to the learned,… has encouraged me to compose a short work on calculating by al-jabr and al-muqābala, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealing with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.”۳
The Arabic version of “The Compendious Book on Calculation by Restoring and Balancing” has survived and consists of four parts. In the first part Al-Khwarizmi explains the fundamentals of algebra and introduces his systematic approach to solving linear and quadratic equations, which he classifies as six standard types of equations as follows (in all cases a, b and c are positive integers):
۱٫ bx = c
۲٫ ax2 = bx
۳٫ ax2 = c
۴٫ ax2 + bx = c
۵٫ ax2 + c = bx
۶٫ bx + c = ax2
Given that the coefficients a, b and c are to be positive, it is seen that al-Khwarizmi’s systemization covers all possible forms of linear and quadratic equations.
As one can see, the above equations are expressed in accordance with the language of modern mathematics. It goes without saying that at the time of al-Khwarizmi this type of mathematical notation had not yet been invented so that he had to use ordinary language to define the equations. He expressed them entirely in words calling x “shay”, meaning thing, and x2 “māl”, signifying wealth, and the coefficients a, b, c “dirham”, meaning coin.
Al-Khwarizmi goes on to show how to solve the six standard types of equations by both algebraic and geometric methods of solution. He then explains the basic algebraic reckoning with expressions involving an unknown or square roots and discusses six examples of problems each ending with one of the six basic equations. He also demonstrates how to multiply out expressions such as (a + bx) × (c + dx) and also discusses methods of extracting square roots.
The second part of the book contains a few considerations on the application of the Rule of Three5 to commercial transactions. Part three covers surfaces and volumes of elementary plane and solid figures such as the circle, sphere, cone, and pyramid, mostly without any use of algebra. It also looks at rules for measuring of lands, the digging of canals. Part four, which is the final and the longest chapter of the book, is devoted to the application of algebraic methods to a wide range of problems in trade, surveying, lawsuits, partition, legacies and inheritance. In Particular, it deals with the complicated Islamic rules for inheritance and resolves problems regarding the division of money and real estate according to the complex requirements of the Islamic inheritance laws.
To sum up, “The Compendious Book on Calculation by Restoring and Balancing” is a practical book replete with worked examples and applications. Solomon Gandz (1883-1954), a historian of science from Austria, gives this opinion of al-Khwarizmi’s algebra: “Al-Khwarizmi’s algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called “the father of algebra” because he is the first to teach algebra in an elementary form and for its own sake.”۶
Roshdi Rashed (1936-), an Egyptian-French scholar, characterizes al-Khwarizmi’s concept of algebra as follows:
“Al-Khwarizmi’s concept of algebra can now be grasped with greater precision; it concerns the theory of linear and quadratic equations with a single unknown. If al-Khwarizmi was confined to the second degree at best, it was simply through the notation of solution and proof in the new discipline. The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. In fact, only a solution by means of the proof answered to al-Khwarizmi’s requirements.”۷
Around 825 CE, al-Khwarizmi wrote a treatise on arithmetic entitled “The Book of Addition and Subtraction according to the Hindu Calculation”, or for short, “Hindu Art of Reckoning.” In this book, in which al-Khwarizmi explicitly confirmed the Indian influence on his arithmetic, he introduced the modified Hindu numerals (1, 2, 3, 4, 5, 6, 7, 8, 9, and 0) and described the place-value decimal system of numerals. In particular, he stressed the dominant role of zero as a numeral in its own right and of fundamental importance.
The following passage from al-Khwarizmi’s book on Algebra gives an idea of his depth of abstraction and understanding of numbers: “When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred; then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; the thousand can be thus repeated at any complex number; so forth to the utmost limit of numeration.”۸ Al-Khwarizmi had fully grasped the revolutionary concept of zero, which was to change the future of mathematics and many other fields of science for that matter. He had realized how important it was to use zero as a place-holder9 in positional base notation of numerals. In addition, he provided methods for arithmetical calculations, and it is believed that he also included a procedure for finding square roots.
Al-Khwarizmi’s book on “Hindu Art of Reckoning” was principally responsible for spreading the Indian system of numeration in the Islamic world. The use of these numbers spread throughout the Muslim world over the next two centuries, leading to the speedy development and progress of science.
The original Arabic version of “Hindu Art of Reckoning” was lost but a Latin translation of it entitled “Algoritmi10 de numero Indorum” survived and it was this Latin version that introduced the Hindu numerals to Europe, now generally known as Arabic numerals. The introduction of Hindu place-value system of numerals and zero as a place-holder in positional base notation is probably one of Al-Khwarizmi’s most important contributions.
“Of all the great thinkers who have enriched the diverse branches of knowledge during the era of early Islam, Muhammad bin Musa Khwarizmi occupies an outstanding place. Being one of the greatest scientists of all times and the greatest of his age, Khwarizmi was a versatile genius, who made lasting contributions to the field of Mathematics, Astronomy, Music, Geography and History.”۱۱ Al-Khwarizmi left among others a Golden Rule for the world to create science with. He himself used it to solve many major problems of algebra, geometry, astronomy, and in other fields of science. This Golden Rule henceforth referred to as algorithm states that all complex problems can be solved by means of the following five simple steps:12
۱٫ First, break down each problem into a number of elemental steps. An elemental step is one, which cannot be simplified any further. 2. Second, arrange all the elemental steps in an order or sequence, such that each step can be taken up and solved one at a time, without affecting other parts of the problem. 3. Third, find a way to solve each of the elemental steps separately. Because each elemental step has been simplified to the maximum, it is very likely that the solution of an elemental step will itself be extremely simple making it available with relative ease. 4. Forth, proceed to solve each elemental step, either one at a time or several at a time, but in the correct order. 5. Fifth, when all elemental steps are solved, the original problem itself has also been solved.
The following citation taken from al-Khwarizmi’s book on algebra shows how he used to define an algebraic equation in words, and demonstrates the logical order he followed to solve it.
“One māl and ten shay are equal to thirty nine dirham.”
“The question in this type of equation is as follows: what is the square, which combined with ten of its roots, will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this, which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents the root of this square, which itself, of course is 9. Nine therefore gives the square.” In modern notation, the above equation may be written as x2 + 10x = 39. Following the al-Khwarizmi’s proposed method the solution is:
(x+5)2 = 39 + 25 = 64 x + 5 = √۶۴ = ± ۸
According to modern mathematics the quadratic equation involved has two solutions (+8 and -8). Since al-Khwarizmi was unable to accept the existence of negative numbers, he did not take into consideration -8 as a root for this equation and he accepted only +8. Consequently, he came to the conclusion that x = 8 – 5 = 3. After al-Khwarizmi, other Muslim mathematicians built on his concrete ideas of algebra and arithmetic.
Latin Translations of al-Khwarizmi’s works
Al-Andalus was the geographic term Muslim Arabs used to denote those areas of Spain that came under their control in the Middle Ages for 781 years (from 711 to 1492 CE). Cordoba, the capital of Muslim Spain, became soon the center for intellectual enlightenment and learning. Scholars from various parts of Europe were attracted to Cordoba to study Arabic and Islamic sciences. Al-Khwarizmi’s books were also brought to al-Andalus and were translated into Latin in the early 12th century. These Latin translations exerted enormous influence on the development of science in Europe and had a profound impact on the advance of mathematics, astronomy and geography. Al-Khwarizmi has been held in high esteem throughout the centuries since then.
In the tenth century the Spanish astronomer Maslamah ibn Ahmad al-Majriti (c. 1000) made a critical revision of the original Arabic treatise of al-Khwarizmi’s Zij al-Sind-hind, which later was lost. It is assumed that al-Khwarizmi’s detailed trigonometric tables containing the sine functions were probably extrapolated to tangent functions by al-Majriti.13 The al-Majriti’s revision was translated into Latin presumably by Adelard of Bath (1080-1152) in 1126.14 This Latin translation made al-Khwarizmi’s Zij into the western canon of astronomical and mathematical studies.
Al-Khwarizmi’s work on algebra15 was translated into Latin by the English scholar Robert of Chester in 1145 CE, and another Latin version was done by the Italian translator, Gherardo da Cremona (1117-1187) in 1150 CE.16 The Spanish scholar, Juan Hispalense or John of Seville (?-1180), produced another Latin translation. These Latin translations were used as textbooks in various European universities till as late as the 16th century. It was due to these Latin versions that the term algebra found its way into the European languages.17
In the course of time European mathematicians discovered the outstanding significance of al-Khwarizmi’s work and began to apprehend that he not only had initiated the subject in a systematic form, but had also demonstrated how to solve linear and quadratic equations. During the time of Renaissance, al-Khwarizmi was classified as being the “original inventor and founder of algebra.”
Al-Khwarizmi’s treatise on “Hindu Art of Reckoning” survived only in its Latin translation most probably done by Adelard of Bath in the 12th century. This Latin translation, henceforth referred to as “Algoritmi de numero Indorum”۱۸, had a twofold effect: On the one hand it introduced to Europe the numbers that are called “Arabic numerals,” on the other hand it gave rise to the word algorithm deriving from al-Khwarizmi’s name rendered in Latin as Algoritmi.19 The term “Arabic numerals” comes from the fact that Europe learned of them from the al-Khwarizmi’s Arabic text. It should be reminded however that al-Khwarizmi himself had called them “Hindu numerals.” Today these numbers are correctly described as Hindu-Arabic numerals.
The term algorithm came to mean in medieval Europe the whole system of decimal numbers and arithmetical operations such as addition, subtraction, multiplication and division as well as extracting of square root.20 Later on, it came to denote any regular process of solving problems by following a finite number of logical steps.21 The Arabic numerals introduced by al-Khwarizmi, like much of his new mathematics, were not welcomed in Europe wholeheartedly. In 1299 there was a law in the commercial center of Florence (Italy) forbidding the use of the Arabic numerals.22 The idea of zero was met with great skepticism. Many mathematicians even as late as Renaissance Europe believed zero to be a “worthless nothing.”۲۳ The Arabic numerals along with zero were finally accepted during the 16th century.
Al-Khwarizmi’s influence in the realm of arithmetic was tremendous. Two important books on arithmetic, “Carmen de Algorismo” written in the twelfth century by the French teacher and poet, Alexandre de Villedieu (1175-1240), and “Algorismus vulgaris” written in the thirteenth century by the English monk and astronomer, Johannis de Sacrobosco (1195-1256), owe a lot to al-Khwarizmi’s book and were used for several hundred years in Europe.
۱٫ George Sarton: “Introduction to the History of Science,” Volume 1, “The Time of al-Khwarizmi,” Carnegie Institution, Washington, 1927, pp. 563-564
۲٫ Restoring and balancing refer to the transposition of terms in a given algebraic equation. Restoring is the process of removing negative quantities from the equation by adding the same quantities to each side. For example, the algebraic equation x – ۲ = ۱۲ becomes x – ۲ + ۲ = ۱۲ + ۲ through “restoring,” and as a consequence one obtains x = 14. Balancing is the process of cancelling out quantities of the same type on both side of a given equation. For instance, the algebraic equation x + 3 = 7 + 3 turns into x = 7 by “balancing” the two equal numbers 3. By the help of these two operations it is possible to convert any algebraic equation into its simplest form.
۳٫ As quoted in “The Teaching of Mathematics” by Victor J. Katz, see
۴٫ “Al-Khwarizmi’s Six Types of Rhetorical Algebraic Equations” translated and annotated by Louis C. Karpinski, in “A Source Book in Medieval Science” edited by Edward Grant, Harvard University Press, Cambridge (USA), 1974, pp. 106-110
۵٫ The Rule of Three is a shorthand version for a particular form of cross-multiplication. For an equation of the form a/b = c/x, the Rule of Three states that x = bc/a.
۶٫ Solomon Gandz: “The Sources of al-Khwarizmi’s Algebra,” Osiris, i (1936), pp. 263-277
۷٫Roshdi Rashed: “al-Khwarizmi’s Concept of Algebra”, in “Arab Civilization – Challenges and Responses”, Studies in Honor of Constantine K. Zurayk, edited by George N. Atiyeh and Ibrahim M. Oweiss, published by State University of New York Press, Albany, 1988, p. 104
۸٫ “The Algebra of Mohammed Ben Musa,” edited and translated by Frederic Rosen, Kessinger Publishing LLC, London, 1831, p. 3
۹٫ As an example, the number 508 contains a zero as a place-holder, since it means that there are 5 hundreds, 0 tens, and 8 ones.
۱۰٫ It is to be noted at this stage that in the course of time, Algoritmi, the Latin corruption of al-Khwarizmi’s name, metamorphosed into algorithm.
۱۱٫ Ahmad Kh. Jamil: “Hundred Great Muslims,” Library of Islam, Des Plaines, IL U.S.A., 1987, p. 140
۱۳٫ George Sarton: “Introduction to the History of Science,” Volume 1, “The Time of al-Khwarizmi,” Carnegie Institution, Washington, 1927, pp. 563-564
۱۴٫ Four manuscripts of this Latin translation have survived and are kept at the Bibliothèque Publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).
۱۵٫ A unique Arabic copy of Al-Khwarizmi’s algebra that was kept at The Bodleian Library of the University of Oxford was translated in 1831 by Frederic Rosen into English.
۱۶٫ Both, Adelard of Bath and Gerard of Cremona were scholars in the Translation School of Toledo in Spain where a great number of books were translated from Arabic into Latin.
۱۷٫ European scholars and monks who studied the Latin translation of Al-Khwarizmi’s algebra often would simply shorten the long Arabic title to the words al-jabir. Using the characters of the Latin alphabet to transliterate al-jabir, it became algebra.
۱۸٫ The Latin translation was originally untitled and commonly referred to by the first two words with which it started: Dixit algoritmi (“So said al-Khwarizmi”). Using these two words and taking into consideration the content of the original Arabic text, Baldassarre Boncompagni (1821-1894), an Italian science historian who printed the Latin manuscript in 1857 after its discovery, suggested “Algoritmi de numero Indorum” (“Al-Khwarizmi on the Hindu Art of Reckoning”) as the title for the Latin translation.
۱۹٫ Al-Khwarizmi’s name is the origin of guarismo in Spanish and algarismo in Portuguese, both meaning digit.
۲۰٫ An early German mathematical dictionary “Volltandiges Mathematisches Lexicon” (Leipzig, 1747), gives the following definition for the German word Algorithmus: “Under this designation are combined the notions of the four types of arithmetic calculations, namely addition, multiplication, subtraction, and division.”
۲۱٫ The Latin phrase algorithmus infinitesimalis invented by the German polymath Gottfried Wilhelm von Leibniz (1646-1716) was to denote “ways of calculation with infinitely small quantities.” Nowadays, any algorithm refers to a set of instructions and numerical calculations that produces various kinds of results when carried out. Algorithms are critical to computers, programming, engineering, and software design.
۲۲٫ Dirk Jan Strujk: “The Prohibition of the Use of Arabic Numerals in Florence,” Archives internationales d’histoire des sciences, 84-85, 1968, pp. 291-295
۲۳٫ After its introduction by al-Khwarizmi, zero was used in the Islamic world for about 250 years before the Western world ever knew of it.