Information about the word Algorithm, Al'Khowârizmî and the Translation School of Toledo

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Stamp issued by the Soviet Union on Sept. 6, 1983, to mark the 1200th birth anniversary of Al'Khowârizmî. (Image courtesy of Jeff Miller)

The origin of the word 'Algorithm'

The word "algorithm" itself is quite interesting; at first glance it may look as though someone intended to write "logarithm" but jumbled up the first four letters. The word did not appear in Webster's New World Dictionary as late as 1957; we find only the older form "algorism" with its ancient meaning, i.e., the process of doing arithmetic using Arabic numerals. In the middle ages, abacists computed on the abacus and algorists computed by algorism. Following the middle ages, the origin of this word was in doubt, and early linguists attempted to guess at its derivation by making combinations like algiros [painful] + arithmos [number]; others said no, the word comes from "King Algor of Castile." Finally, historians of mathematics found the true origin of the word algorism: it comes from the name of a famous Persian textbook author, Abu Ja'far Mohammed ibn Mûsâ al'Khowârizmî (c. 825)-literally, "Father of Ja'far, Mohhamed, son of Moses, native of Khowârizm." Khowârizm is today the small Soviet city of Khiva. Al'Khowârizmî wrote the celebrated book Kitab al jabr w'al'muqabala ("Rules of restoration and reduction"); another word, "algebra," stems from the title of his book, although the book wasn't really very algebraic.

Gradually the form and meaning of "algorism" became corrupted; as explained by the Oxford English Dictionary, the word was "erroneously refashioned" by "learned confusion" with the word arithmetic. The change from "algorism" to "algorithm" is not hard to understand in view of the fact that people had forgotten the original derivation of the word. An early German mathematical dictionary, Vollstandiges Mathematisches Lexicon (Leipzig, 1747), gives the following definition for the word Algorithmus: "Under this designation are combined the notions of the four types of arithmetic calculations, namely addition, multiplication, subtranction, and division." The latin phrase algorithmus infinitesimalis was at that time used to denote "ways of calculation with infinitely small quantities, as invented by Leibnitz."


Abu Abdullah Mohammad Ibn Musa al'Khowârizmî was a mathematician, astronomer and geographer. He was perhaps one of the greatest mathematicians who ever lived, as, in fact, he was the founder of several branches and basic concepts of mathematics. He influenced mathematical thought to a greater extent than any other medieval writer. His work on algebra was outstanding, as he not only initiated the subject in a systematic form but he also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. The very name Algebra has been derived from his famous book Al'Jabr wa'al'Muqabilah. His arithmetic synthesised Greek and Hindu knowledge and also contained his own contribution of fundamental importance to mathematics and science.

He treated (probably for the first time) the handling of the elementary arithmetic operations with respect to the Hindu-Arabic decimal numerals. About 750 A.D., the Hindu principles of decimal arithmetic were brought to Persia, as several important works were translated into Arabic. Not long after this, al'Khowârizmî wrote his Arabic textbook on the subject. His work was a strong influence on Leonardo Pisano (Fibonacci), whose book on arithmetic (1202 A.D.) played a major role in the spreading of Hindu-Arabic numerals into Europe.

The importance of Khowârizmî's works was recognized in the twelfth century by the West, when Adelard de Bath (who has been called the first english scientist, 1080 - 1160), Gerard of Cremona (Italy, 1117 - 1187) and others translated his works into Latin.

Both, Adelard and Gerard were scholars in the Translation School of Toledo.

Translation School of Toledo

Through the intermediacy of Arab scholars, the more comprehensive and fundamental achievements of Greek mathematics became available to the Latin scientists of the Middle Ages. This happened in the twelfth and thirteenth centuries, first in Spain, at the western limit of the Islamic world. The Translation School of Toledo was the main place where a great number of books were translated from Arabic into Latin. These included both, original Arab works and those previously translated into Arabic from Hellenistic Greek literature. Among the classics, which had been translated to Arab and edited several times and from different versions, appear the works of Euclid. Of prime importance in the development of mathematics and its teaching was his main work, the Elements, a systematic, axiomatically deductive presentation of 'Euclidean' elementary geometry and elementary arithmetic.

Interestingly enough, by 1950, the word algorithm was most frequently associated with Euclid's algorithm, a process for finding the greatest common divisor of two numbers which appears in Euclid's Elements (Book 7, Propositions 1 and 2.)

Both, Gerard of Cremona and Adelard of Bath, translators of the works of Al'Khowâritzmî and scholars of the Translation School of Toledo, did also translations into Latin of the Arab versions of the Euclidean Elements. An annotated version of the translation made by Adelard of Bath remained the standard Latin translation, and in 1482 it became one of the very first printed mathematics textbooks.

Although numerous translators from Arabic into Latin worked alone, the usual modus operandi was for two scholars to work in tandem, a practice which lent characteristic social coloring to the process. The basic procedure was for one scholar to translate aloud from the Arabic text into the vernacular and for the second to translate from the vernacular, producing a Latin draft. Thus John of Seville characterized the translation of the De Anima of ibn Snna: ``The book... was translated from Arabic, myself speaking the vernacular word by word, and the archdeacon Dominic converting each into Latin.'' Frequently, the translator from Arabic into Castilian (or Catalan) was a Jew (or a converted Jew, as is probable in the case of John of Seville) in which case the other member of the team would be as Christian, typically a cleric. Gerard of Cremona worked with a Mozarab named Galippus, who may merit greater recognition than he has generally received.

The place of Jews in this scheme is obvious: many were trilingual, knowing Hebrew, Arabic, and a romance language. Jews had indeed been accustomed to translate from Arabic into Hebrew, not a difficult task, given the linguistic and semantic similarities between the two languages, or to write in Judeo-Arabic (Arabic written in Hebrew characters). In the latter case, they were able to create a flexible medium for scientific and philosophical expression. This fitted them ideally for the work of translation, which involved the creation in the vernacular and in Latin of virtually an entire new scientific language (particularly in astronomy, where Arab scientists had vastly enlarged the range of observational data, and mathematics, particularly algebra, where the translations were conduits of methodologies unknown in the Latin tradition).

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