Khwarizm – The City of the Birth of Algebra:

Khwarizm is the city of the birth of algebra, where Al-Biruni corrected and refined the sciences of the past and thought of the earth spinning on its axis many centuries before Copernicus.

The Muslims entered Khwarizm in 712 CE, led by Kutayba ibn Muslim al-Bahili, the lieutenant in Khorasan of the governor Al-Hadjadj b.Yusuf. Islam gradually spread throughout the region in Khwarizm, just as it did in Khorasan and Transoxenia. Khwarizm in the earlier Middle Ages had two capitals, one on the Persian side of the Oxus called Jurjaniyah, or Urganj, the other on the eastern Turkish side called Kath, which in the tenth century was held to be the capital in chief of the province. Geographers of the 10th century give detailed accounts of the topography and climate of Khwarizm, with much emphasis on the very cold winters and on the flourishing economy and commerce. The great prosperity of the region is attested by the fact, that Ibn Fadlan on his mission to the King of the Bulghars left Gurgandj in a caravan of 3000 camels and 5000 people. Khwarizm was celebrated for its fruit, carpets, brocades of mixed cotton, silk and numerous canals.

Besides the economic prosperity, Khwarizm witnessed a sharp rise in the numbers of traditionalists (i.e Hadith scholars) , lawyers and theologians at an early stage. Khwarizm was to produce one amongst the great scholars Al-Khwarizmi (d.847); then in the following century, we have such outstanding figures as the poet and prose stylist Abu Bakr Muhammad b. Al-Abbas al-Khwarizmi (d. 993); and the Samanid secretary and official,Abu Abd Allah Muhammad B.Ahmad al-Khwarizmi, author of the pioneering encyclopaedia of the sciences, the mafatih al-ulum (The Keys to Sciences). The Islamic literary and philological sciences flourished there as is shown by the section on the fudala khwarizm in Tha’alibi’s anthology, the Yatimat al-Dhar; and by the figure of al-Biruni.

The focus here is on two of the greatest figures of all: Al-Khwarizmi, and Al-Biruni.


Muhammad ibn Musa Khwarizmi (780-850 CE), from Khwarizm (now Khiva), south of the ‘Arab Lake,’ was the greatest scientist of his time, working in mathematics, geography, and astronomy. Al-Khwarizmi wrote in about 825 CE a treatise on Algebra entitled ‘Kitab al Mukhtassar fi’l hisab al jabr wa’l muqabalah‘ (Compendious Book of Calculation by Completion and Balancing). Translated in Latin simply as ‘algebra’. In addition to writing the first arithmetic using numerals, this work became ‘the prototype’ for all Islamic and medieval Latin works on Algebra. Al-Khwarizmi’s work is undoubtedly the beginning of algebraic calculus and decimal arithmetic; the first work in which that word appears in the mathematical sense, ‘Algebra’ meaning in Arabic ‘restoration’, that is the transposing of negative terms of an equation. It set forth solution paradigms for six types of problems expressed in modern symbolism by the equations (1) ax2 = bx, (2) ax2 = b, (3) ax = b (4) ax2 + bx = c (5) ax2 + c = bx and (6) bx + c = ax2 , where a, b and c are all positive rational numbers.

Al-Khwarizmi may be regarded as the inaugurator, or at any rate the populariser, of algebra as a subject independent of geometry. It was not exactly algebra in the modern sense of the term, but more of an introduction to applied calculus, based on numerous detailed examples. Khwarizmi gave the value of pi thus: ‘If a circle has a diametre of 7, it has a circumference of: 22’ This is an approximation accurate to about 1 in a thousand.

Al-Khwarizmi’s Kitab al-mukhtasar (Compendious book on calculation by completion and balancing) translated in Latin simply as Liber algebre. The term “algebra” was first applied only to the first part translated by Robert of Chester in 1145. Al-Khwarizmi’s work was studied until the sixteenth century as the principal textbook of European universities, introducing into the West the science of algebra. But Robert was not alone. Muslim Algebra was also transmitted to the West through the Latin translations of Adelard of Bath, John of Seville and indirectly through Fibonacci’s writings. It was also transmitted through the Hebrew treatise of Abraham Bar Hiyya, itself translated into Latin by Plato of Tivoli. Plato’s translation appeared in the same year, 1145, as Robert of Chester’s translation of Al-Khwarizmi’s Algebra. The year 1145 might well be counted as the birth year of European Algebra; but if we mean to reserve that title for the publication of the first original treatise (as opposed to translations), then the birth year was 1202 when the Liber Abaci appeared. By the middle of the twelfth century thanks to Plato of Tivoli and Robert of Chester, Latin mathematicians could become acquainted, if they were sufficiently eager to do so, with the main results of the Arabic algebra of the ninth and tenth centuries.  It amounts to this: that they could find the positive roots of quadratics and could solve imperfectly special cubics. Some sixty and eighty years-that is, two and three generations-later the first Christian algebraist added a few contributions of his own, first in the Liber Abaci (1202), then in the Flos (1225).

  1. C. Karpinski’s: edition of the Algebra (1915) contains a general summary of al-Khwarizmi’s life and works. Of its original three parts, only two were transmitted to Europe, and those two arrived separately. Hence, it took Latin scholars some time to realise that the two parts compliment each other. Algebra, however, made little progress in Europe. It only appeared in university curriculum when it was taught at Leipzig in the late fifteenth century.

From a variant of the name of the author “Al-Khwarizimi”, we obtain the word “algorism,” and logarithm. The word ‘algorithm’ for a long time signified arithmetic, or at any rate any process involving repeated calculation. His book on arithmetic, the Arabic text of which has been lost but which survives in a Latin version, was instrumental in introducing the numerical system. With this system also spread the use of the zero, which derives from the Arabic sifr, ‘void’; the earliest use of the Arabic zero was in a number indicating the year 260 AH (CE 873) in a deed written on parchment, less than twenty-five years after Khwarizmi’s death.  But there is no doubt that, as far as the West was concerned, the zero was a direct importation from Arabic.  A Latin translation of an arithmetic text was discovered in 1857 at the University of Cambridge Library. Entitled Algoritimi de Numero Indorum, the work opens with the words: “Spoken has Algoritimi. Let us give deserved praise to God, our Leader and Defender.” It is believed that this is a copy of Al-Khwarizmi’s arithmetic text which was translated into Latin in the twelfth century by Adelard of Bath. This work deals with arithmetic, geometry, music, and astronomy; it is possibly a summary of al-Khwarizmi’s teachings rather than an original work.

Al-Khwarizmi’s astronomical and trigonometric tables, which were revised by Maslama al-Majriti (q. v., second half of tenth century), were translated into Latin as early as 1126 by Adelard of Bath. They were the first Muslim tables and contained not simply the sine function but also the tangent (Maslama’s interpolation?). On top of Al Khwarizmi’s achievements referred to already, and amongst other things, he introduced a method similar to long division to extract the square root (jithr) of a number. He was the first to introduce the concept of mal (power) for the squared unknown variable. He also gives geometrical solutions of quadratic equations, for e.g., x2 + 10x = 39, an equation often repeated by later writers. He also perfected the geometric representations of quadratic equations having two variables, e.g. the circle, ellipse, parabola and hyperbola (conic sections) etc.

With regard to al-Khwarizmi’s geographical work, his maps which were made at the command of the caliph Ma’mun and seen by Masudi in the tenth century, have not survived. It consists of a representation of Ptolemy’s material in tabular form, with the interpolation of further data which was available in the Muslim period, these tables were arranged on the system of seven climates, six climates grouped around a central one.

In gratitude to God, for his scholarly achievements, Al-Khwarizmi, in the preface of his book Kitab al-Jabr wal muqabala, says:

‘I have been guided by good intentions, and I hope this work will meet amongst people of good letters, who have been gifted by God’s higher skills, a favourable welcome. May God guide me in the right path, either in this enterprise or in others; I trust in God, Master of the Highest Realm; May God’s blessings be on his Prophets and Messengers.’


Al-Biruni, Muhammad Ibn Ahmad Abu’l- Rayhan (973-1050 CE). Al-Biruni studied mathematics, history, and medicine. His interest extended to nearly all the sciences. His production exceeds 146 titles in more than twenty different disciplines, such as mathematics, mathematical geography, chronology, mechanics, pharmacology, mineralogy, history, literature, religion, and philosophy. He composed an encyclopaedia of astronomy, a treatise on geography, and an epitome of astronomy. But the bulk of his work lies in mathematics and related disciplines (ninety-six titles). He contributed to geometry the solution of theorems that thereafter bore his name. He wrote histories of Mahmud’s reign, of Subukrigin, and of Khwarizm. He explained the workings of natural springs and artesian wells by the hydrostatic principle of communicating vessels. Only twenty-two works of his have survived the ravages of time and only thirteen of these works have been published. His work on the demarcation of the coordinates of cities (Tahdid) was written so as to determine the direction to Mecca (Qibla); Al-Biruni determined the local meridian and the coordinates of any locality. In one of his works, he even outlines a psychological theory of error, listing six causes liable to cause men to lie. The fruit of his endeavours was an excellent and original book on the chronology of the nations, which he dedicated to prince Kabus, a work, which brought him into contact with Ibn Sina, with whom he engaged in unremitting polemics. In his work Biruni surveys the calendars of the various peoples: Persians, Greeks, Egyptians, Jews, Melkite and Nestorian Christians, Sabaeans, and the ancient Arabs. He was one of a group of scholars compulsorily assembled by the Ghaznavid sultan Mahmud, and went then to India, the outcome of the journey was the publication of his masterly work on India. He initiates his readers into the science of the Indians, and leads them on to an understanding of their thought, as well as propounding views on the geology of the Indus valley which were well in advance of his time. He speculated on the possibility that the Indus valley had been once the bottom of a sea. In astronomy, Al-Biruni wrote treatises on the astrolabe, the planisphere, the armillary sphere; and formulated astronomical tables for Sultan Mas’ud. Indeed, Al-Biruni’s astronomical findings are well elaborated in his al-Qanun al-Mas’udi (dedicated to the ruler Mas’ud), a work which is a most extensive astronomical encyclopaedia, slightly short of 1,500 pages, in which he determines the motion of the solar apogee, corrects Ptolemy’s findings and is able to state for the first time that the motion is not identical to that of precession, but comes very close to it. Al-Biruni had his doubts about Ptolemy’s view that distance of the sun from the earth is 286 times the latter’s circumference, al-Biruni’s argument being that Ptolemy based his claim on total eclipses but disregarded annular eclipses which implied larger distances. Al-Biruni was unable to observe a total eclipse. Therefore, he could not verify the findings of Ptolemy, a fact which he frankly admits, that measurement of the moon’s distance from the earth was possible, but he found the sun’s distance immeasurable by the instruments of that age and its distance remained an object for conjecture. Al-Biruni employs mathematical techniques unknown to his predecessors that involve analysis of instantaneous motion and acceleration, described in terminology that can best be understood if we assume that he had “mathematical functions” in mind. In the eighth maqalah of the Qanun, al-Biruni presented a masterly exposition of both the solar as well as the lunar eclipses, especially the section dealing with al-kusufin (the images of the eclipses) which pass on the faces of the sun and the moon without affecting their body; he referred in his letter to a book on the two united and equal axes. Six hundred years before Galileo, Al-Biruni discussed the theory of the earth rotating about is own axis. The following in relation to this subject is worthy of record as an indication of Al-Biruni’s independence of mind:

He took it for granted that the earth is round, and noted “the attraction of all things towards the centre of the earth,” and remarked that astronomic data can be explained as well by supposing that the earth turns daily on its axis and annually around the sun, as by the reverse hypothesis:

‘Rotation of the earth would in no way invalidate astronomical calculations, for all the astronomical data are as explicable in terms of the one theory as of the other. The problem is thus difficult of solution.’

Al-Biruni wrote about the astrolabe, the planisphere and the armillary sphere, besides inventing an astrolabe which he called cylindrical, but which is now referred to as orthographical. Using the astrolabe and the presence of a mountain near a sea or flat plain, he calculated the earth’s circumference by solving a highly complex geodesic equation. With the aid of mathematics, he also enabled the direction of the Qibla to be determined from anywhere in the world. Max Meyerhof observed earlier this century that most of al-Biruni’s mathematical works and many other writings have not been published yet.

Al-Biruni was also very active in physics. He composed an extensive lapidary, describing a great number of stones and metals from the natural, commercial, and medical points of view. He determined the specific gravity of eighteen precious stones, and laid down the principle that the specific gravity of an object corresponds to the volume of water it displaces. To determine specific gravity, he used a ‘conical vessel’ (which al-Khazini calls the conical instrument of abu’l-Rayhan (al-Biruni), to find the ratio of the weight of water displaced to the weight of a substance in air. Al-Khazini in his Kitab Mizan al-hikma (the Book of the balance of Wisdom) makes a detailed description of such an instrument and its uses (including by himself). In his work on precious stones, he gave an account of the correspondence to be found between these and the metals, and determined their specific weights. Mieli explains how Al-Biruni weighted meticulously the substance he wanted to study, then dipped it into his conical instrument that was filled with water. He weighed the water which had been displaced by the immersed substance and which was escaping the instrument through a hole conveniently placed. The ratio between the weight of the body and that of the same water volume, gave the specific weight sought. Many of the specific aspects of such an instrument owe to the fact that al-Biruni was extremely careful in ensuring his results were as accurate as possible. Al-Biruni carried out a series of measurements of specific weights, and summarised his findings in a number of tables. His results are very close to modern data, some of the deviations are explained by the impurity of the specimen and by temperature differences in his experiments. He also determined the specific weight of some liquids, and established the differences in the specific weights of hot and cold water along with fresh and salt water. He was the first in history to introduce checking tests in the practice of experiments.

Al-Biruni also wrote on descriptions of several different forms of chess, and elaborates on the moves of the different pieces. The most popular form of chess was played with dice, which allowed chance to predominate over the skill of the player. It should be mentioned, too, that Biruni initiated and studied the problem of grains of corn placed on squares of the chess-board, repeatedly doubling the number of grains on each square, a procedure related to geometric progression. He found, indeed, a method of calculating, without laborious additions, the result of the repeated doubling of a number, as in the Hindu story of the chessboard squares and the grains of sand.

But the best conclusion on this genius is left to Sarton:

‘Oriental historians call him “the Sheik”—as if to mean “the master of those who know.” His multifarious production in the same generation with Ibn Sina, Ibn al-Haytham, and Firdausi, marks the turn of the tenth century into the eleventh as the zenith of Islamic culture, and the climax of medieval thought’

Concluding words

In 1221, like in many other places, the Mongols inflicted great damage on Khwarizm. Yaqut was at Jurjaniyah, or Gurganj as he calls it, shortly before the place was devastated by the Mongols under Genkhis Khan; and he writes that he had never seen a mightier city, nor one more wealthy or more beautiful. In 1220 all this was changed to ruin. The great canal dykes having been broken down, the waters of the Oxus flowed off by a new course, and the whole city was laid under water. The Mongol hordes when they marched away left nothing, according to Yaqut, but corpses and the ruined walls of houses to mark the place of the great city.

However, a century afterwards it began to recover and Ibn Battuta could find and praise the madrassa, the hospitals, the charitable institutions, the fine markets. It recovered up until in 1388 when Timur fell on Khwarizm. Timur razed the capital of Khwarizm Urgene to the ground, and inflicted so massive a devastation, that Khwarizm never recovered and lost its cultural and economic splendour forever.

Before such onslaughts, Khwarizm like the eastern most parts of Islam produced a great number of scholars. Setting aside those considered at great length already, such as al-Razi, Al-Farghani, Al-Tabari, Al-Khazini, etc, also can be cited lesser known ones such as Umar al-Nasafi who came from the vicinity of Samarkand; the grammarian al-Zamarkashshari, from Khwarizm; the mathematician al-Kharaqi who lived in Merv; the grammarian al-Maidani who came from Nishapur; the mathematician Muzaffar al-Asfusari, and the historian of religions al-Shahrastani, from other parts of Khorasan. A fine group of men, says Sarton. Khwarizm, however, produced two of the greatest scholars of Islam: Al-Khwarizmi and al-Biruni. They gave civilisation and science some of its best, on Al-Khwarizmi, Arndt reflecting, ‘works of a learned man, from a nation that has ceased to exist and in a time that passed away more than a millennium ago, are still benefiting us today.’

(This article was written at and modified by AMYN Staff)