Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
68
Al-
Khwarizmi’s contribution to making computational-algorithmic
tradition in the era of european renaissance
SHAMSIEVA I.M.
1
МАKHMUTKHODJAEVA L.S.
2
ABDULLAEV U.A.
3
1
PhD, Associate Professor, Tashkent State institute of oriental studies.
2
PhD, Associate Professor, Tashkent Branch of the Plekhanov Russian University of
Economics.
3
PhD, Associate Professor,
Tashkent State institute of oriental studies.
Uzbekistan 100060 Tashkent, Shakhrisabz Street 25
E-mail:
shamsiyevairoda1968@gmail.com
Abstract:
The article tells about the scientific work of the great Uzbek scientist Muhammad
al-Khwarizmi and his historical merits in the world science. Algorithms are one of the hundred
names of al-Khwarizmi. The article details the evolution of this term. The article also shows the role
of algorithmic ideas and the al-Khwarizmi scientific school in becoming of well-known European
mathematicians such as Leonardo Pisansky, Paccioli and many other Renaissance scientists.
Key words:
Al-Khwarizmi, algorithm, Central Asia, mathematics, arithmetic treatise, world
civilization.
Introduction
As is known Uzbekistan is one of the cradles of the world science in the history of
the development of world civilization. It was Uzbek land which had given the world such
prominent thinkers of the Oriental and Western sciences as Beruni, Farabi, Al-Khwarizmi,
Ulugbek and many others.
The achievements of mathematics and computational - algorithmic sciences of
Central Asia during IX-XV cc. have left the lasting trace in the development of the world
scientific thought. In the works of Central Asian mathematicians of paramount importance
was the development of computational techniques. The zeal towards the perfection of
these techniques was revealed in the works of Uzbek classicists of science which served as
the basis for European mathematics and affected greatly the development of the world
computational-algorithmic trend.
The works of prominent mathematicians of Central Asia and Uzbekistan enjoyed an
invariable authority in the Orient in XI-XV cc. and later in Europe.
Literature review
Al-Khwarizmi’s contribution had been studied and commented by the scientists of
subsequent generations being the guidelines for research in various fields of mathematics.
This issues in the Republic of Uzbekistan are considered in the publications of
scientists of Tashkent State University of Economics, M. Ulugbek National University of
Uzbekistan (Uzbekistan), Plekhanov G.V. Russian University of Economics (Russian
Federation), Coordination and Methodological Center of the Uzbekistan’s contemporary
history under the Academy of Sciences of Uzbekistan (Uzbekistan).
Of special interest were scientific works of the outstanding Uzbek scientist
Muhammad ibn Musa al-Khwarizmi who was at the source of contemporary mathematics
and computer science, astronomy, geography; - he made a weighty contribution to the
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
69
treasure-house of the world science: no wonder a noted American scholar of science
George Sarton has named the whole first part of IX с. the epoch of Al-Khwarizmi as “the
most famous mathematician of his time and taking into account all the circumstances –
one
of
the
greatest
of
all
times”[4,p.8].
German scientist K.Vogel named Khwarizmi “the teacher of the West not only in
algebra but also in his introduction of new Indian figures and methods of calculations”[16,
p.89].
K.Vogel continues: … arithmetic’s beginning with words “Dixit Algorizmi” lay down
the foundation of the rapid growth of mathematical knowledge which had occurred in
Western Europe during XVI-XVII cc.[9].
Well-known American scholar of Central Asia Frederick Starr also
argues that,
“In
mathematics, Khwarazmi was the first to elaborate a theory of equations solvable through
radicals, which can be applied to the solution of a variety of arithmetical and geometrical
problems. The result was a book, Algebra, that gave its name to the field; the term
algorithm is a corrupted form of his own name. Khwarazmi advanced the field of spherical
astronomy and did more than anyone else to popularize the decimal system that had been
invented in India“.
The publications of the following authors are devoted to theoretical understanding
of the issues of Khwarazmi contribution to making computational-algorithmic tradition in
the era of european renaissance: G.P.Matvievskaya [14], A.P.Yuschkevich [18],
P.G.Bulgacov [6].
The issues of the need to develop research and educational work, to promote
international cooperation, to improve training in mathematics are considered in the
publications of the Academician of the Academy of Sciences of Uzbekistan K.Kh.
Abdurakhmanov, N.K. Zokirova, R.K. Kabulov, A.F. Fayzullaev, Sh.Nasirov [12].
Research methodology
In fact, the arithmetic treatise of Khwarizmi “Kitab hisab al adad al hindi” (“the book
on counting by Indian figures”) has played a leading role in history not only in arithmetics,
mathematics and its computational – algorithmic trend, but also the world culture at large
as the main source of propagation of positional number notation. It replaced throughout
considerably less elaborated alphabetic system of calculation used by Greeks, cumbersome
Roman number system, complicated Chinese diagrams and the like. Arithmetics based on
positional decimal number notation with the use of zero was first given in it. Since mid XII
c. the principles and techniques of decimal positional arithmetics the first propagandist of
which in Bagdad was al-Khwarizmi gained a wide recognition in Western Europe. At that
time the intense activity of a group of translators of scientific literature from Arabic into
Latin (universally accepted at that time in the learned world) began to develop.
After the conquest of Spain and Portugal by Arabs (early VIII c.) the Arab-Muslim
culture had found its place for long. The Arabic language was an international language of
science for European intellectuals. Therefore, this period in the history of science and
culture is sometimes called “Arabic”. The Spanish towns Madrid, Cordova, Toledo, Seville
were cultural and scientific centres not only in the Pyrenees peninsula, but also in other
states of Western Europe. “The people from neighboring France, Italy and Britain came to
learn sciences from Arabs” [16, p.85].
The period of the highest development of the civilization of the peoples of Central
Asia – the Oriental Renaissance coincided in time with the strongest oppression of Church
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
70
orthodoxy in all fields of European culture and science. As a result, there was a low level of
mathematical knowledge and other natural sciences. Therefore, the propagation of Arab –
Muslim culture in Spain played an important role in the development of the European
culture including mathematics. Of leading value for mathematics here was the propagation
of the achievements of Bagdad school starting from works of Muhammad al-Khwarizmi [2,
p.339].
Western Europe was a successor and bearer of scientific ideas of ancient Rome and
used Roman figures. Therefore, “Indian counting” did not spread here immediately. At first
it was close to deep-rooted Roman number system, -counting board in letter number
system. Being afraid to draw terrible accusation on themselves in heresy the scientists
went to Spain secretly to learn new alien skills for them.
Prior to half XII с. Europeans “learnt Arabic sciences through the language of the
original”. Of paramount importance for the progress of mathematical knowledge in Europe
were as it was already said, the translations from Arabic both original writings and Greek
literature as well which were in the Arabic language. Translations from Arabic were in
great use in XI-XIII cc., however, the European mathematicians continued studying the
Arab manuscripts in XV-XVII cc. as well. Based on the above-mentioned work of Khwarizmi
there came into being algorizmic (algorithmic) tradition in the Latin West in XII-XIII cc.
Analysis and Results
A new arithmetic started to be named al - Khwarizmi in the Latinized form: “algorithm” or
“algorism”, because the Latin translation of the book starts with the words: Algorizmi said
(Dixit Algorizmi).
It is universally known that the Arabic original has not been found so far. The only
copy of the manuscript, which was available in former times in the British abbey, was kept
in the library of Cambridge University. According to the catalogue of the manuscripts,
which was made up in 1858, it was rewritten in small letters (XIII c) [18, p.339].
Each algorizm (algorithm) was intended to popularize a new number system. New
methods of counting with Indian figures suggested by al-Khwarizmi were much superior to
the old, arduous ones. The work of al-Khwarizmi in the Latin translation served as the basis
for the development of “Indian” arithmetics in Europe as well. As G.Sarton put it, this
writing can be the main channel by which a new system became known in Europe [19,
p.18].
The first translations of al-Khwarizmi’s works were performed by: Adelardo from Bat
(1126 y.), Johan Seville (XII c.) Robert from Chester (1145 y.), Gerardo from Cremona (XII
c.), Magister A. (1143 y.), Leonardo of Pisa (early XIII c.) [14, p.45].
The translation work was accompanied by making compilation writings and was of
international nature: those who were involved and accepted Christianity were Arabs,
Spaniards, the families who accepted Islam before, Spanish Jews, Britishers, Italians, Slavs,
Flanderers. There came a great amount of literature on “algorizms” to become one of the
bases of further progress of science in Europe. As M.Simon put it: “Of crucial importance in
accepting in Europe decimal positional number notation and new figures arithmetics,
primarily with arithmetics of al-Khwarizmi [12, p.108]. Among a number of treatizes
writhen in Latin and later in living languages between XII-XV cc. of paramount interest are
and play a vital role the following writings, because they are the most ancient of all known
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
71
and it is these that call al-Khwarizmi the source of inspiration… In particular, we are dealing
with one of the most ancient algorizms written in XII с. and more or less dating back to the
lost Arabic arithmetics of Khwarizmi. One of these treatizes is available in the form of the
only manuscript in Cambridge and was published in transcription in 1857 by Italian
historian B.Boncompagny [3].
The author of translation is not exactly known: probably it was Herardo from
Cremona. Some of the scholars assume that its author is Adelard from Bat, a British scholar
and a thinker who played an important role as a pioneer of propagation of Arab science
and philosophy in the first part of XII с. (J.Ruska, A.P.Yuschkevich, O.Fayzullaev,
G.P.Matvievskaya, A.Akhmedov etc.) According to Andre Allard, a scholar from Belgium
who studied the first translation of the text algorism it is impossible to identify its author
[5, p.55].
However, the manuscript does not represent an exact translation of al-Khwarizmi’s
work at all. The translation is probably a copyist of an earlier period, tailored the work of
the Oriental scientist to the needs of the European reader. He substituted Arabic numerals
and figures used by al-Khwarizmi and which were in use in Western Europe of that time for
Roman figures.
The arithmetic treatise – algorism of Khwarizmi was revised in mid XIIс. by Spanish
scholar Johann Seville. His work was called “Liber Algorismi de pratica arismetrice” (the
book of Algorism on practice of arithmetics) [10].
The Latin revision is kept in Paris national library [6, p.43].
The manuscripts of
J.Seville served as source for propagation of decimal positional arithmetics in Europe.
Another writing often called “the Book on introduction of Alhoarism in actronomy art
compiled by magister A”. (“liber ysagogarum Alchorismi in artem astronomicans a magistro
A”.) [5, p.56] is known in two records, one of which has a date 1143. The authorship of the
book is as cribed to magister A, to be considered by many scientists as Adelard from Bat,
although there is no true evidence in favour of this hypothesis, according to A.Allard is not
available so far. The copies of the manuscript are kept in Viena national library, in Bavaria
state library in Munich, in Parisian national library and in Ambrosan library.
The development of arithmetics and mathematics at large in medieval Europe
down to XVI с. was under way under the strong influence of al-Khwarizmi. It testifies that
the name al-Khwarizmi in Latin transcription (algorithm) came into being in modern
mathematics and the sphere of information technologies in the form of the name for
calculation system performed by certain rules. A prominent scientist S.Gondz wrote: “the
works of Khwarizmi – the start of the European science…” [12, p.110]. The number of
works on algorism grew rapidly, they were available in various countries being first in Latin
(the Latin language was the means of instruction in higher schools in Europe until XVIII с.)
and later in living folk languages.
The arithmetic treatise “Algorismus demonstratus” (“The interpreted algorism”) the
author of which is magister Gernard (Magister Gernardus XIII c.) [13, p.403] arouse a great
interest among the researchers. Also, of great publicity was “Common algorism”
(Algorismus vulgaris) or “a Treatise on art of counting” (circa 1250 y.) (“Tractatus de arte
numerandi”) by an Englishman John Halifax [18, p.343]. The Latin nickname of this scientist
– Johann Sacrobosco (Johannes de Sacrobosco, circa 1200-1256). During XIII-XIV cc.
hundreds of students studied at European Universities. The professors explained the
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
72
manual on “algorism” Sacrobosco [7, p.98] in Latin – the international language of science.
For centuries, this book had become the main textbook on calculations in decimal
positional system of calculus in many European Universities.
Almost at the same time with algorism of Sacrobosco there came “the
Interpretation of algorism” (Demonstratio Jordani de algorismo”) by Jordan Nemorary
mentioned in numerous lists [7, p.344]. Jordan Nemorary, the professor of Paris University
and the General of one of the monks Orders propagated algorism in the Church monk
circles.
French mathematician Alexander de Villa Dei from Normandy (died c.1240 y) set
forth algorism of integer number in poetry – in 284 hexameters. His “Song on algorism
(Carmen de algorizmo”) was translated into French, English and Irish languages [7, p.345].
An anonymous French algorithmic manuscript belongs to the second half of XIII с.
The ancient English algorism “The art of Counting” (“Crafte of Nonbrynge” c.1300 y) as well
as the first famous book on algorism in the Italian language (“Tractatus algorismi” was
drawn up by a Jacobo from Florence in 1307 [7, p.345].
An essay “A book on abacus” by Italian merchant Leonardo of Pisa or Fibonacci
(1202) [18, p.342] played an important role in propagation in Europe of the positional
number system. Much information taken from the mathematical works in the Arabic
language was classified in it. Fibonacci is a fervent supporter of Indo – Arabic arithmetics.
Since the essay was mainly intended for commercial people, it was also common among
the population at large. Although the work bears the name “A book on abacus”, it has
nothing to do at all with calculations using the counting board. The headline seems to be
for Leonardo a term abacus as a synonym for arithmetics – in this sense it was used by
Italians later.
The introduction of positional system into school education, common use of
builders, engineers, calculators, calendars, traders, book-keepers, clerks and others gave
rise to spread of calculations in social life, science (astronomy) and technology. The growth
of algorithmic literature was largely due to invention of book-printing, some treatises were
reprinted more than once. In XIV-XV сc. Indian arithmetics was widely in use among
merchants and calculators (mainly in Italy and later in Germany and France).
In Germany due to growth of towns, expansion of trade and handicrafts there came
a demand of business circles for learning methods of counting. It was in XIII с. that there
appeared specialized schools in which would be calculators were trained. They were called
rechenmeisters (German trade towns.) [7, p.99]. Calculators played a great role in
introduction of Indo-Arabic arithmetics. The instruction was in German and in this manner
the German mathematical terminology was drawn up. Soon alongside Latin there came
into being numerous German textbooks on “algorism”.
Among the first and the most significant writings of such kind published in XV-XVI
cc. one should name the textbooks written by Ulrich Wagner and Johann Widman.
Wagner’s book entitled “Bamberger rechenbuch” (publ. in 1482) was the first printed
German writing on “algorism”. In the book by Johann Widman (1489 y) “Behende und
Rechung auf allen Rauffmanscaff” only one section out of three was devoted to practical
arithmetics [13, p.408].
Over three centuries the European algorithmicans had made a lot
of changes and improvements in the Khwarizmi’s counting system. Particularly, in the
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
73
“Treatise on algorism” by P.Beldomandi (died in 1428) which was published in 1483, the
arrangement of operations takes an up-to-date form [17, p.126].
Muhammad ibn Musa al - Khwarizmi’s Nisba
i
[Nisba – part of a name]in its Latin
forms – more often algorithmus or algorismus turned into the name of new arithmetics,
algorith or algorism
ii
[Orientalist J.Reino in 1849 discovered what was behind the word
Nisba of Al-Khwarizmi]. Later the term algorithm was to be meant any regular
computational process. Approximately, at the same time mention was made on “algorists”,
(for instance, the word “algorists” is to be found in the “A book on algorism from Salem
monastery i.e. adherents to algoristhetic arithmetics, the opposite to “abacusists”) [18,
p.343]. This name comes from the special device – abacus which resembles present-day
counting boards with the help of which all practical calculations were performed.
It should be noted that the spread of algorithmetic arithmetiics was not so fast
(although in the middle of XIIс. new arithmetic was known both in Austria and Bavaria),
and it was not due to the quality of the book and its translation but far more common
reasons: the resistance of the old, strong school, i.e. abacusists, the very limitation of a
group of mathematicians and dissociation, the very small number of schools, the lack of
book-printing etc. Algorithmic arithmetics as from XII с started gaining its positions in a
fierce struggle with abacusists who used Roman numbering and duodecimal Roman
fractions and too cumbersome ways of multiplication and division.
Thus, we see that Latinized name al-Khwarizmi was entered in the title of the book
and nowadays, there is no doubt that the word “algorithm” came into the European
languages because of this writing. It can be said with certain, that the Uzbek scientist
Muhammad al-Khwarizmi has taught Europe the art of counting and his writing on Indian
arithmetics by its worthiness deserves the honour to be a handbook of the first
algorithmicians of the East and West.
Conclusions and Suggestions
Al-Khwarizmi’s works, as it was already said, gave rise to a new mathematical object
and algorithm being widely-used in the field of informational technologies. At the outset
“algorithm” meant the name of the scientist, later numbering in the positional system of
counting, and now any system of calculations made according to stringent certain rules [8,
p.86]. Algorithm - a regular successive process to be used for solving problems of a given
type [12, p.52].
Thanks to al-Khwarizmi the European and world scientific thoughts have got
acquainted with the notion algorithm, one of the basics not only in the sphere of
mathematics and computer science but also in the field of informational technologies. In
the XXI c. algorithm has acquired a special significance. Mathematical logic and computer
science have also raised their claim to it. The present-day Internet and informational
technologies cannot do without the theory of algorithms. Nowadays we know web-
algorithms, algorithmic languages (Alpha, Fortran, Basic, HTML, Java etc, all in all more
than 500 algorithmic languages operating in various countries worldwide), algorithms of
information retrieval in the Internet. The name al-Khwarizmi has been immortalized in the
very name of the scientific notion. The word “al-Khwarizmi” has developed and changed
over twelve centuries and again has taken on the form “al-Khwarizmi”, but along with the
i
Nisba – part of a name.
ii
Orientalist J.Reino in 1849 discovered what was behind the word Nisba of Al-Khwarizmi.
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
74
notion “algorithm’, al-Khwarizmi, Algorizmi, Algorismus, Alchwarism, Hwarism, Algorithm
al-Khwarizmi algorithm.
The first – scientist’s name, the second - notion. It is difficult to overestimate the
great service of al-Khwarizmi for world science [1]. He was a forefather of the strong
cohort of scientists of the Medieval Orient who built the bridge from ancient science
towards the epoch of European Renaissance. The writing of al-Khwarizmi (Khoresmi –
author’s note), particularly his arithmetics served the basis both for the Oriental and for
the European mathematical literature as well during the first years of the era of
Renaissance. If Khwarizmi’s arithmetics was an inspiring source of any chisabu hindi in the
Orient, in the West i.e. in Europe over the last period of Medieval stagnation of scientific
thought and in the first years of Renaissance it performed the same role, it was Oriental
school of Muhammad ibn Musa al-Khwarizmi that trained prominent mathematicians like
Leonardo de Pisa nicknamed Fibonacci, Lukas du Burgo nicknamed Paccioli and others [20,
p.104].
One can say with certainty that al-Khwarizmi and Central-Asian scientists in general
have made their contribution to making Renaissance in Western Europe. Academician
N.I.Conrad [15, p.104] was right when he said that Central-Asian culture at that time as in
many fields of technology and material culture, especially in art and in the field of
legislature, political doctrines, philosophy, historiography, science and fiction had
developed earlier and had been richer in content than all these spheres of culture in the
West.
Acknowledgment
The authors would like to express their appreciation to the editors of Historia
Mathematica, who read and made suggestions on earlier versions of this paper.
References
1.
Abdurakhmanov K.K., Khakimov N.K., Makhmutkhodjaeva L.S. Higher education
as an important factor of sustainable development of the country, «Revista ESPACIOS»,
Venesuela, Vol. 40 (Nº 9), Year 2019.
2.
Akhmedov A. Scientific heritage of al-Khwarizmi and its place in the history of
science and culture. Doctorate thesis. (T., 1986, p.339).
3.
Algoritmi de numero indorum. – in: Trattati d’Arithmetike publicati de
Baldassare Boncom pagni, Roma, 1857.
4.
Al-Khwarizmi in scientific foreign literature. (T., 1984, p. 8).
5.
Allar A. Al-Khwarizmi and the origin of Latin algorism.// In: Mukammad ibn Musa
al-Khwarizmi. For his 1200 anniversary. M., 1983, p.55.
6.
Bulgacov P.G. Et al Muhammed al-Khwarizmi. M., 1983, p.43.
7.
Depman I.J. History of Arithmetics. M., 1959, p.99.
8.
Fayzullaev A.F. Al-Khwarizmi and genesis of algorithmic method of cognition //
Philosophical sciences, 1983 N5, p.96.
9.
Frederick
Starr
S.
Rediscovering
Central
Asia.
http://wilsonquarterly.com/quarterly/summer-2009-thrift-the-double-edged-
virtue/rediscovering-central-asia/
Economics and Innovative Technologies. Vol. 2019, No. 5, september-october
5/2019
(№
00043)
75
10.
Ioanni Hispalensis liber algorismi de pratica arismetrce. Published by
Boncompagni (manuscript kept in Paris) together with “Arithmetics of al-Khwarizmi”.
Roma, 1857.
11.
Jushckevich A.P. History of mathematics in Middle Ages. M., 1961. p.343.
12.
Kabulov R.K., A.F.Fayzullaev, SH.A.Nazirov. Al-Khwarizmi, algorithm and
algorithmization. (T., 2006, p.108).
13.
Matvievscaya G.P. Theory on number in Middle. Ages. Doctorate thesis in
physics and mathematics. T., I.T., 1967, p.403.
14.
Matvievskaya G.P. On history of mathematics in Central Asia. (T., 1962, p.45).
15.
Stein V.M. Participation of the Oriental countries in preparation of European
Renaissance. // China. Japan. History and philology. M., 1961, p.104.
16.
Vogel K. How Al-Khwarizmi became known in Germany || A prominent scientist
of Medieval period Al-Khwarizmi, (T., 1985, p.89).
17.
Yuschkevich A.P. Arithmetic treatise of Muhammad Ben Musa al-Khwarizmi //
Transactions of the Institute of history, natural science and technology. M., 1954, T.I.,
p.126.
18.
Yuschkevich A.P. History of mathematics in Middle ages. (M., 1961, p.339).
19.
Yuschkevich A.P. On the work in arithmetics of Muhammad ibn Musa al-
Khwarizmi // In the book: Muhammad ibn Musa al-Khwarizmi. For his 1200 anniversary.
(M., 1983, p.18).
20.
Yusupov N., Essays in the history of the development of arithmetics in the Near-
East Kazan., 1933, p.27.
