« He influenced mathematical thinking more than any other medieval writer».

Philip Hitty, famous American scientist, professor at Princeton University (1886-1978)

In 832, the Abbasid caliph Al-Mamun founded a research and development center in the capital of his state, Baghdad. The educational center modeled on the ancient Museum in Alexandria, which became known as the House of Wisdom. In essence, the House of Wisdom was the Academy of Sciences. Many scientists from various regions of Central Asia and the Arab East worked there; they had at their disposal a rich library of ancient manuscripts, as well as a large specially built astronomical observatory. The House of Wisdom became a center for the study of mathematics, astronomy, medicine and chemistry. For a significant period of his life he headed the library of the “House of Wisdom” Abu Abdullah Muhammad ibn Musa Al-Khwarizmi, nicknamed the "father of algebra". It was thanks to Al-Khorezmi that Europe learned what decimal counting and numbers are. It was he who first proposed dividing Earth to meridians and parallels. Thus, every point on the earth had clear coordinates, latitude and longitude. It was Al-Khorezmi who created the first reliable model of the earth, the prototype of the modern globe, and this was 700 years before the famous Italian scientist Galileo Galilei.

It is largely thanks to this Muslim scientist that the world is what it is today.

Al-Khwarizmi's contributions to mathematics

The modern word “algorithm” comes from the name of Al-Khwarizmi, and it is associated with the title of his book “ Al-kitab al-mukhtasar fi hisab al-jabr wa-al-mukabala " This book was translated into English twice in the 12th century. Latin language and played an extremely important role in the development of mathematics in Europe. Translated, the title means “A short book on completion and opposition.” The word “algebra” comes from the title of this book. Al-jabr means the operation of transferring negative terms from one side of an equation to another to obtain positive terms on both sides. " Al-muqabala" means "contrast", that is, bringing similar terms in both sides of the equation. This work Al-Khwarizmi became the first written work on algebra.

Thanks to Latin translations, he gained fame in Europe and had a huge influence on the development of Western science. His book on algebra introduced Europeans to a hitherto unknown discipline and served as a classic mathematical text for European university students for several centuries. Al-Khorezmi first introduced algebra as an independent science of general methods solutions to linear and quadratic equations, gave a classification of these equations. Also Al-Khwarizmi in 834 he separated algebra from geometry.

Al-Khwarizmi's contributions to astronomy

Astronomy occupied a leading place among the exact sciences in the medieval East as one of the most necessary sciences in practice, without which it was impossible to do either in irrigated agriculture or in sea and land trade. By the 9th century. include the first independent works on astronomy in Arabic, a special place among them was occupied by “ziji” - collections of astronomical and trigonometric tables (at that time trigonometry was part of astronomy). Using these tables, the position of the luminaries on the celestial sphere, the time of solar and lunar eclipses. They also served to measure time. Among the first zijs is the zij of Al-Khorezmi, which began with a section on chronology and calendar. This was very important for practical astronomy, since different peoples At different times, different calendars were used, and when making observations, universal dating is important. His works on astronomy were based on the works of Indian astronomers. He made detailed calculations of the positions of the sun, moon and planets, solar eclipses. Al-Khwarizmi's astronomical tables were translated into European languages, and later into Chinese. Trigonometric and astronomical tables (Zij al-Khwarizmi) formed the basis of medieval research in the field of astronomy in both the East and Western Europe.

Al-Khorezmi’s book on the astrolabe, the main instrument of astronomical measurements of that time, is also important. “The Book on the Construction of the Astrolabe” has not survived and is known only from mentions in other sources. From the astronomical works of Al-Khorezmi, the “Book of sundial" and "Book about action with the help of an astrolabe" (incompletely included in the work of Al-Fergani). In sections 41-42 of this book, a special compass was described for determining the time of prayer.

Al-Khwarizmi's contribution to geography

Al-Khorezmi’s works on geography were also associated with works on mathematics and astronomy. He is considered the author of the first work on mathematical geography. Al-Khorezmi was the first to describe in Arabic the inhabited part of the Earth known at that time, gave a map with the coordinates of the most important settlements, with seas, oceans, mountains, rivers.

In many ways, he relied on Greek works (Ptolemy’s Manual of Geography), but his “Book of the Picture of the Earth” is not just a translation of the works of his predecessors, but an original work containing a lot of new data. The book included a description of the world, a map and a list of coordinates of the most important places. Despite the fact that Al-Khwarizmi's map was more accurate than the map of the ancient Greek astronomer, his works did not replace the Ptolemaic geography used in Europe.

Using his own discoveries, Al-Khwarizmi corrected Ptolemy's studies of geography, astronomy and astrology. To create a map " known world» Al-Khwarizmi studied the works of 70 geographers. He also organized scientific expeditions to Byzantium, Khazaria, and Afghanistan, under his leadership the length of one degree of the earth's meridian was calculated (very accurately for those times).

Conclusion

It cannot be said that there was no algebra before Al-Khwarizmi. In ancient times, people solved simple algebraic problems; There were techniques for solving individual specific problems, but Al-Khorezmi was the first to introduce algebra as a science of general methods for solving numerical linear and quadratic equations, and gave a classification of these equations.

Muhammad ibn Musa Al-Khwarizmi ranks important place among the scientists of Central Asia, whose names went down in the history of exact natural science. In the 9th century. - at the dawn of medieval Eastern science - the scientist made a great contribution to the development of arithmetic and algebra. Al-Khwarizmi's algebraic treatise was among the first works on mathematics translated in Europe from Arabic into Latin.

The modern name of algebra, as we noted above, comes from the word “al-jabr”, and the word “algorithm” comes from the name Al-Khorezmi. The works of Al-Khwarizmi had a great influence on scientists of the East and West for several centuries and served as a model for writing mathematics textbooks for a long time.

Historians of science highly appreciate both the scientific and popularization activities of Al-Khorezmi. Renowned historian of science George Sarton called him "the greatest mathematician of his time, and, all things considered, one of the greatest of all time."

And most importantly, Al-Khwarizmi emphasized that he wrote his famous book on algebra (Al-kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala) to serve the practical needs of the people in matters relating to inheritance, partition property, litigation and commerce. He first of all considered his work as worship of the Almighty and only then as helping people.

May Allah grant us to be pleasing to Allah and useful to society. Amine!

Makhach Gitinovasov

MINISTRY OF EDUCATION AND SCIENCE OF THE RB

Bashkir State Pedagogical University

"Al Khwarizmi -

outstanding mathematician and astronomer"

Ufa - 2004
Content

Introduction........................................................ ............................................... 3

Motherland al Khwarizmi.......................................................................... 4

Works of al Khwarizmi................................................... .................... 6

Algebra by al-Khwarizmi.................................................... ....................... 8

Conclusion................................................. ...................................... eleven

Literature................................................. ...................................... 12

Al Khwarizmi's full name is Abu Adallah (or Abu Jafar) Muhammad ibn Musa al Khwarizmi. Translated from Arabic, this means: father of Abdallah (or father of Jafar), Muhammad, son of Musa from Khorezm. Sometimes, in accordance with the Arabic spelling, it is called al Khuwarizmi.

History has preserved almost no biographical information about al-Khorezmi. Even the exact dates of his birth and death have not reached us. It is only known that he was born at the end of the eighth century, and died in the second half of the ninth, more precisely after 847. Now it is conventionally accepted to consider the year of his birth to be 783, and the year of death to be 850.

In some historical sources, al-Khorezmi is called “al Majusi,” i.e., a magician. From this they conclude that his ancestors were magicians - priests of the Zoroastrian religion, widespread in Central Asia.

Al Khwarizmi's homeland

The scientist’s homeland was Khorezm, a vast region of Central Asia, which corresponds to the modern Khorezm region of Uzbekistan, Tashauz region of Turkmenistan. There is no mention in historical sources of the specific place of birth of al-Khorezmi, but some indirect considerations allow us to assume that he came from ancient Khiva.

In Khorezm by the beginning of the 9th century. traditions of an ancient and original culture have developed. We find evidence of this in the works of medieval Eastern historians. More details about ancient history of this region were obtained thanks to archaeological excavations that began to be carried out here during Soviet times. Valuable finds of archaeologists, complementing the reports of medieval writers, made it possible to get an idea of ​​​​the highly developed civilization of ancient Khorezm.

Remains of a grandiose monument were discovered on the territory of Khorezm. irrigation system. It was created long before the beginning of our chronology - in the 2nd millennium BC. e. The developed irrigation system of Khorezm determined high level the entire economy of the area. In ancient books there are reports of large, well-fortified cities of Khorezm. For example, Fir Castle, built on the banks of the Amu Darya at the beginning of the 4th century, was surrounded by three rows of high walls and was visible at a distance of about twenty kilometers.

During the excavations, magnificent works of Khorezmian artists and sculptors were found. Khorezm merchants conducted brisk trade with India and China, the Middle East, the Caucasus and Eastern Europe. They exported furs, livestock, and fish.

Already in very distant times, the Khorezmians owned writing. Monuments of this writing were discovered during archaeological excavations and deciphered by scientists. Already in ancient times, the foundations of exact sciences were formed in Khorezm. The achievements of the Khorezmians in the field of economic life would have been impossible without certain knowledge in mathematics, geodesy, astronomy, etc.

For example, the construction of canals, fortresses, and multi-story palaces required not only practical skills, but also the ability to accurately level the terrain and perform complex calculations and measurements. Traveling to distant countries through deserts would be impossible without the ability to navigate by the stars, that is, without mastering the rudiments of astronomy.

Founded in the 60s. VIII century the city of Baghdad became new capital Arab Caliphate. Baghdad quickly became an important center of trade, science and culture. The city, where people came from various regions of the caliphate, was crowded and lively, famous for its bazaars.

A large scientific school arose in Baghdad, which attracted outstanding scientists from different countries. A library was created, replenished with valuable scientific works. The “House of Wisdom” was founded - an institution that performed the functions of an academy of sciences. At the “House of Wisdom” there was a rich library of ancient manuscripts and an astronomical observatory. Al Khorezmi was also recruited to work at the House of Wisdom.

Works of al Khwarizmi

Diverse scientific interests al Khorezmi concerned mathematics, theoretical and practical astronomy, geography and history. Not all of the works he wrote have survived. Some of them, mentioned by medieval writers, were subsequently lost.

Information about the works of al-Khorezmi reported by Eastern historians does not always coincide. It has now been established that al Khwarizmi was the author of the following works:

1. “The Book of Indian Accounting”;

2. “A short book on the calculus of al-jabr and al-muqabala”;

3. “Astronomical tables”;

4. “Book of the Earth’s Picture”;

5. “Book on building an astrolabe”;

6. “Book about actions using an astrolabe”;

7. “The Book of the Sundial”;

8. “Treatise on the definition of the era of the Jews and their holidays”;

9. “Book of History.”

Of these works, only seven have reached us - in texts belonging either to al-Khwarizmi himself or to his medieval commentators.

The geographical treatise “The Book of the Picture of the Earth” is the first known work on geography in Arabic. He had a strong influence on the further development of this science in the countries of the East.

Al Khorezmi paid great attention to astronomy. His main task in this area is the compilation of zij, i.e., astronomical and trigonometric tables necessary for solving problems of theoretical and practical astronomy. In this work, for the first time in literature in Arabic, a table of sines was given and a tangent was introduced. Zij al Khorezmi was very popular not only in the East, but also in Europe. It was not him that the largest eastern astronomers referred to. At the beginning of the 12th century. it was translated into Latin and then became available to European scientists. In addition to zij, al-Khorezmi described the calendar systems of different peoples.

Al Khorezmi has important achievements in the development of practical astronomy. He wrote a treatise on the design and use of the astrolabe, the main instrument used in the Middle Ages for observing the starry sky.

The “Book of History” or “Book of Remembrance” is mentioned in several medieval works. Therefore, al-Khorezmi is considered one of the earliest historians who wrote in Arabic.

Al Khorezmi's mathematical works brought him the greatest fame in the history of science.

Algebra by al-Khwarizmi

Algebraic treatise of al-Khorezmi is known under the title: “A short book of completion and opposition” (in Arabic: “Kitab mukhtasar al-jabr wal-mukabala”). The treatise consists of two parts – theoretical and practical. The first of them sets out the theory of linear and quadratic equations, and also touches on some issues of geometry. In the second part, algebraic methods are applied to the solution of specific household, commercial and legal problems.

In the introduction, al-Khwarizmi talks about what prompted him to write the essay: “I composed short book about the calculus of algebra and almukabala, which includes simple and difficult questions arithmetic, because people need it when dividing inheritance, drawing up wills, dividing property and court cases, in trade and all kinds of transactions, as well as in measuring land, drawing canals, geometry and other types of similar matters.” Thus, it is emphasized that with the help of algebraic methods it is possible to solve various applied problems.

Next, al Khorezmi shows which numbers are used in algebra. If arithmetic operates with ordinary numbers that are “made up of units,” then algebra involves numbers special type– unknown quantity, its square and free term of the equation.

Al Khorezmi calls an unknown quantity the term “root” (jizr) and gives the following definition: “A root is any thing that is multiplied by itself, whether it is a number equal to or greater than one, or a fraction less than it.” This definition is due to the fact that when solving equations, we always looked for not only x, but also x2. Therefore, the unknown was considered as the root of the square of the unknown. The definition also emphasizes that the unknown can take on both integer and fractional values. The term “root” used by al-Khwarizmi is, in all likelihood, a translation of the Sanskrit word “mula” (“plant root”), which was used by Indian mathematicians to denote the unknown in an equation. Later in Arabic literature the term “thing” (“shai”) was used for the same purpose.

The square of the unknown is called the word “property” (“small”) and is defined as “that which is obtained from the root when multiplied by itself.”

Al Khorezmi calls the free member of the equation – a “prime number” – a “dirham”, i.e. a monetary unit.

Next he moves on to classify linear and quadratic equations. Currently, it seems completely unnecessary, since all special cases are combined using the notation ax 2 +bx+c=0, where the coefficients a, b and c can take positive, negative and zero values. But during the time of al-Khwarizmi, the situation was different: not only did there not exist letter designation, but also the concept of a negative number. Therefore, the equation only made sense if all its coefficients were positive.

Al Khorezmi identifies the following six types of equations:

1. “squares are equal to roots”, which in modern notation means ax 2 = bx;

2. “squares are equal to numbers,” i.e. ax 2 =c;

3. “the roots are equal to the number,” i.e. ax=c;

4. “squares and roots are equal to the number”, i.e. ax 2 +bx=c;

5. “squares and numbers are equal to roots,” i.e. ax 2 +c=bx;

6. “roots and numbers are equal to the square”, i.e. bx+c=ax 2.

Examples are given for each of these types.

In order to bring this equation to one of the indicated types, al Khorezmi introduces two special actions. The first is al-jabr, which means replenishment. It consists of moving the negative term from one side of the equation to another. From this term arose modern word"algebra".

The second action is al-muqabala, which means opposition. It consists of reducing equal terms on both sides of the equation.

In addition, it was required that the coefficient of the leading term be equal to one. Later, in some works of Eastern scientists, even special algebraic operations appeared - “additions” (al-takmil) and “reduction” (ar-rad). The first of them consisted of multiplying all terms of the equation by the inverse of the coefficient a in the equation ax 2 + bx + c = d, if a> 1. The second meant a similar operation if a<1. Встречался также специальный термин (аль-хатт), обозначающий действие деления коэффициентов уравнения на общий множитель.

Al Khorezmi considers various problems regarding the division of inheritance. For example: “A man died, leaving two sons, and bequeathed a third of his property to another person. He left 10 dirhams in cash and a loan equal to the share of one of them.”

Following al-Khwarizmi's reasoning, let's denote the debt by x. Then the entire property is equal to 10+x. since three heirs receive equal shares, then (10+x)/3=x, whence x=5.

Algebraic methods of al-Khwarizmi were also used in the chapter on geometry.

Conclusion

Muhammad ibn Musa al Khorezmi occupies an important place among the scientists of Central Asia, whose names have entered the history of exact natural science. In the 9th century. - at the dawn of medieval Eastern science - the scientist made a great contribution to the development of arithmetic and algebra. Algebraic treatise of al-Khwarizmi was among the first works on mathematics translated in Europe from Arabic into Latin. In Europe until the 16th century. algebra was called “the art of algebra and almukabala.” The modern name algebra comes from the word al-jabr. And the word algorithm comes from the name of al-Khorezmi.

Al Khwarizmi gives rules for calculating the area of ​​a square, triangle and rhombus. Gives rules for calculating volume, including that of a truncated square pyramid. He compiled calendars and wrote about chronology. His merits in astronomy are great, although, like his astronomers of his contemporaries, he proceeded from the geocentric system of the world. He made a great contribution to mathematical geography. Al Khorezmi, for the first time in Arabic, described in detail the inhabited part of the Earth known at that time, gave a map of it indicating the coordinates of the most important settlements, depicting seas, islands, mountains, rivers, etc.

The works of al-Khorezmi had a strong influence on scientists of the East and West for several centuries and for a long time served as a model for writing mathematics textbooks.

Literature

1. S. Kh. Sirazhetdinov, G. P. Matvievskaya. Al Khorezmi is an outstanding mathematician and astronomer of the Middle Ages. M.: Education, 1983.

2. Yushkevich A.P. History of mathematics in the Middle Ages. M.: Fizmatgiz, 1961.

Abu Ab-dal-lah Mu-ham-mad ibn Mu-sa al-Ho-rez-mi /783 - 850/ - one of the largest scientists of the Middle Ages. His birthplace is Kho-rezm. Your knowledge al-Ho-res-mi so-vershenst-vo-val in the “House of wisdom-ros-ti” in Bagh-da-de. This institution was of its own kind, Aka-de-mi-ey na-uk, in which many Arab scientists worked -skogo Vos-ka. “The House of Mud-ros-ti” is famous for its bo-ga-that bib-lio-te-coy of the ancient ru-ko-pi-seys and as-ro-no-mi-ches -koy ob-ser-va-to-ri-ey.

Is-trace-to-va-te-li-but-vi-li that al-Ho-res-mi was the author of 9 co-chi-ne-nies:

Book about Indian arith-me-ti-ke;

A short book about the is-number of al-geb-ry and al-mu-ka-ba-ly;

Ast-ro-no-mi-ches-tables (zij);

Books of maps of the Earth;

Books about the construction of as-ro-la-bia;

Books about actions with the help of ast-ro-la-bia;

Books about solar hours;

Treatise on the definition of the era of the Jews and their holidays;

Books of history.

So-chi-ne-nie al-Ho-rez-mi about arif-me-ti-ke syg-ra-lo the most important role in the history of ma-te-ma-ti-ki and ho Its original Arabic text has been lost, based on the Latin translation of the 12th century. In this co-chi-ne-nii for the first time yes-but sis-te-ma-ti-ches-some of the arif-me-ti-ki, os-no-van-noy on de -tich-tich-no-zi-tsi-on-noy calculation system.

Al-geb-ra-i-ches-kaya book-ga al-Ho-rez-mi (Ki-tab mukh-ta-sab al-jabr and wa-l-mu-ka-ba-la) with-sto- it from two parts - te-o-re-ti-ches-koy (te-o-ria of solving linear and square equations, not-ko- the same questions of geo-metry) and practical (use of al-geb-ra-i-ches-methods in the solution -institutions of economic-vein-but-to-vy, trade-go-go-vyh and legal-di-ches-for-dachas – de-laying on-the-consequences, making-up for -material, division of property, various personal transactions, land acquisition, construction of sewers). Thankfully, the Arabic word “al-jabr” has come to be known as “al-geb-ra”. Unas-le-do-van-noe from the eastern ma-te-ma-ti-kov the doctrine of linear and square equations has become os -new development of al-geb-ry in Europe. La-ti-ni-zi-ro-van-the name of the scientist entered into science under the term “al-go-rhythm”.

The geo-metric part of the track is dedicated to the area and volume of geo-met-ri-ches figures (triangle-nick, square-rat, rhombus, par-ral-le-lo-gram, circle, circle segment, four-corner-nickel with different hundred-ro -na-mi and ug-la-mi, pa-ral-le-le-pi-ped, round qi-lin-dr, prism-ma, con-nus).

Og-ro-men’s contribution to the scientist and to as-ro-no-miya, which was not-about-ho-di-ma for oro-sha-e-mo-go-earth le-de-lia, sea and land trade. Zij (collection of as-ro-no-mi-ches and three-go-no-met-ri-ches-tables) al-Ho-res-mi is sacred to the chron-no- lo-gies and ka-len-da-ryu (important scientific direction, since different peoples used different -mi sis-te-ma-mi time-me-no-th account). His book about ast-ro-la-bia was of great importance for ast-ro-no-mia at that time.

Al-Kho-rez-mi also made a significant contribution to geography. He is considered the author of the first so-chi-no-niya according to ma-te-ma-ti-chesh geography. For the first time in Arabic, he described a part of the Earth that was known at that time, and gave a map with co-ordi -on-the-most-important village-points, with seas and oceans, mountains and rivers. What does it have to do with him Books of the Earth– not just a re-translation of the co-chi-ne-pre-six-ven-ni-kov, but an original work containing many new ones data. He or-ga-ni-zo-val on-scientific ex-pe-di-tions in Vi-zan-tiya, Kha-za-riya, Af-ga-nis-tan. Under his leadership, you were numbered for the length of one city on earth.

Despite his wide-ranging circle of scientific in- te-re-s, the main science of his life is ma-te-ma-ti-ka . al-Ho-res-mi first presented al-geb-ru as a science about general methods for solving numerical linear and square equations, gave a class-si-fi-ka-tion of these equations.

The well-known is-to-rik in science J. Sar-ton called him “the great ma-te-ma-ti-com of his time and , if you take into account everything about the story, one of the greatest of all times.”

According to ma-te-ri-a-lam: Si-razh-di-nov S., Mat-vi-ev-skaya G. Al-Ho-rez-mi - you-da-yu-shchi-sya ma-te-ma-tik and as-ro-nom environment -not-ve-kovya. M., “Pro-lighting”, 1983

In 1983, the world scientific community will celebrate the 1200th anniversary of al-Khorezmi, the great Central Asian scientist, whose works left a deep mark on the history of science. In the 9th century, at the dawn of the heyday of medieval Eastern mathematics, he made important contributions to the development of arithmetic and algebra. His achievements in astronomy and mathematical geography are great. For several centuries, the works of al-Khorezmi had a strong influence on scientists of the East and West and for a long time served as a model for writing mathematics textbooks.

There is very little biographical information about al-Khorezmi. His full name is Abu Abdallah (or Abu Ja'far) Muhammad ibn Musa al-Khwarizmi. Sometimes - in accordance with the Arabic spelling - he is called al-Khuwarizmi. Even the exact dates of his birth and death have not reached us. It is known that he was born at the end of the 8th century, and died in the second half of the 11th century, more precisely after 847. Now it is conventionally accepted to consider the year of his birth to be 783, and the year of death to be 850.

In some medieval sources he is called “al-Majusi”, i.e. magician. From this they conclude that his ancestors were magicians - priests of the Zoroastrian religion, widespread in the Central Asian states before the advent of Islam.

The scientist’s homeland is Khorezm, a vast region of Central Asia, which corresponds to the modern Khorezm region of the Uzbek SSR (center - the city of Urgench), part of the Karakalpak Autonomous Soviet Socialist Republic and the Tashauz region of the Turkmen SSR. Historical sources do not mention the place of birth of al-Khorezmi, but some indirect considerations allow us to assume that he came from ancient Khiva.

In Khorezm by the 8th century. An ancient and distinctive culture emerged. We find evidence of this in the works of medieval Eastern historians. More detailed information about the ancient history of this region has been obtained in recent decades thanks to archaeological excavations. By supplementing the reports of medieval scientists, they made it possible to get an idea of ​​the highly developed civilization of ancient Khorezm.

We have little reliable information about the sciences known to the ancient Khorezmians and about the level of development of these sciences. But since the history of scientific thought is inextricably linked with the economic, social and cultural history of society, there is no doubt that already in ancient times the foundations of the exact sciences were formed in Khorezm. All these achievements of the Khorezmians in the field of economic life, of course, were impossible without certain knowledge in mathematics, geodesy, astronomy, etc. The construction of canals, fortresses, and multi-story palaces required not only practical skills, but also the ability to level the terrain and perform complex calculations and measurements. Traveling to distant countries through deserts would be impossible without the ability to navigate by the stars, that is, without mastering the principles of astronomy. The development of astronomy was also stimulated by the needs of irrigated agriculture. When planning agricultural work that depended on seasonal changes in nature, in particular on floods, a calendar was needed, the creation of which requires a thorough acquaintance with the patterns of the apparent movement of celestial bodies. The Khorezmians developed their own calendar system, described in detail by Biruni in his work “Monuments of Past Generations”.

At the beginning of the 8th century. Central Asia, including Khorezm, was captured by Arab troops. The war brought with it a lot of destruction and casualties. The conquerors, trying to introduce a new religion - Islam, eradicated everything that was associated with religions common in Central Asia in the pre-Muslim period. Cultural and scientific monuments were also destroyed.

Biruni, talking about this difficult period in the history of his homeland - Khorezm, wrote that the Arab governor in Central Asia, Kuteiba ibn Muslim, persecuted “people who knew the Khorezm written language well, knew their traditions and taught them the sciences that existed among the Khorezmians, and subjected them to all sorts of torments ". However, the cultural traditions that developed in Khorezm many centuries ago were not destroyed. The wounds inflicted by the wars gradually healed, and by the 9th century. conditions began to emerge for a new upsurge in the spiritual life of the peoples of Central Asia.

This period was marked by important achievements in the field of exact sciences. Among those Khorezmians who glorified their homeland with their labors, Muhammad ibn Musa al-Khorezmi should be named first. The ancient traditions of Khorezm science, which found a remarkable continuation in his work, undoubtedly played a major role in his formation as a scientist.

Al-Khorezmi belonged to the numerous Central Asian scientists brought to work in the capital of the Arab Caliphate, Baghdad. Among al-Khorezmi's contemporaries who lived in Baghdad, one can name, for example, the famous astronomers Abu l-Abbas Ahmad al-Ferghani and Ahmad ibn Abdallah al-Marwazi, known as Habash al-Khasib. The first of them came from Fergana, the other from Merv.

Baghdad was founded in the 60s of the 8th century. Caliph al-Mansur from the Abbasid dynasty, who ruled in 754 - 775. The new capital of the caliphate, which at that time occupied a vast territory, quickly became an important center of trade, science and culture. The rulers of the caliphate understood that their economic and military plans could not be realized unless the knowledge possessed by the conquered peoples was mastered. Therefore, they contributed in every possible way to the development of science. A large scientific school arose in Baghdad, which attracted outstanding scientists from different countries. A library was created, replenished with valuable books.

Particular attention at this time was paid to the study of ancient Greek and Hellenistic science. The works of the classics of antiquity were collected and translated into Arabic. Special expeditions were sent to purchase scientific manuscripts. Of particular interest were the exact sciences - mathematics, astronomy, geodesy, mathematical geography. Euclid's Elements, Ptolemy's Almagest, Menelaus' Spherics, etc. were translated. Indian astronomical works were also studied. However, Baghdad scientists of the 8th - 9th centuries. were not only translators and commentators. They also engaged in independent research and achieved remarkable results in various fields of knowledge.

The successors of Caliph al-Mansur continued to patronize science. His grandson Harun ar-Rashid, who reigned from 786 to 809, is known (albeit in a very idealized form) from the tales of One Thousand and One Nights. Science in Baghdad reached its greatest flourishing under the son of Harun ar-Rashid, Caliph al-Ma’mun, who ruled from 313 to 833. Under him, the “House of Wisdom” (Bayt al-Hikmah) was founded, an institution that performed the functions of the Academy of Sciences. At the “House of Wisdom” there was a rich library of ancient manuscripts and an astronomical observatory.

Al-Khorezmi worked in Baghdad, among other scientists, for many years. Until 813, al-Ma'mun was the governor of the eastern provinces and lived in Merv. It is possible that here he met with al-Khorezmi, and subsequently invited him to Baghdad.

In one of his writings, al-Khwarizmi praised al-Ma'mun. It is unknown how active al-Ma’mun’s personal participation in scientific work actually was, but there is no doubt that the scientists who worked at the “House of Wisdom” made enormous contributions to mathematics, astronomy and other sciences. No detailed data has been preserved about the Baghdad period of al-Khorezmi’s life either.

There is information that he made two trips (one to the country of the Khazars, and the other to Byzantium), but it is difficult to say that this information is reliable.

The latest date associated with the name of al-Khorezmi is 847. Caliph al-Wasiq died this year, and al-Khwarizmi is mentioned among those present at his death.

Al-Khorezmi's diverse scientific interests concerned mathematics, theoretical and practical astronomy, geography and history. His writings played an important role in the development of these sciences.

Not all of the works written by al-Khwarizmi have survived. Some of them, mentioned by medieval writers, were subsequently lost. But the surviving works also allow us to evaluate the scientific heritage of the great scientist. Finally, there is information about al-Khorezmi’s work on history. It was titled. "Book of History" [Excerpts from recently discovered surviving fragments from this book are published in this issue of the journal.] and was mentioned in several medieval writings. Therefore, al-Khorezmi is considered one of the earliest historians who wrote in Arabic.

Al-Khorezmi was ahead of many of his contemporaries in developing new scientific issues and at the same time did a lot to promote and popularize the achievements of his predecessors. Grateful descendants appreciated his services. [D. Sarton called al-Khwarizmi "the greatest mathematician of his time and, all things considered, one of the greatest of all time."]

Arithmetic treatise of al-Khwarizmi. Al-Khwarizmi's work on arithmetic played a vital role in the history of mathematical science. It was the first systematic presentation of arithmetic based on the decimal positional number system using zero. It originated in India, and therefore al-Khwarizmi, and after him other medieval mathematicians, called it “Indian.” Thanks to al-Khorezmi’s book, “Indian” arithmetic became widespread in the countries of the Near and Middle East, and then in Europe.

Before al-Khwarizmi, various ways of designating numbers and methods of calculation were common in the East. In business transactions, the so-called “manual” or “finger” counting, which had ancient origins, was widely used. Fingers, joints, various finger bends, and hand gestures were given certain numerical values, and people knew how to perform the necessary arithmetic operations with their help. This type of account was widely used by merchants - representatives of different nations. His techniques were also presented in European textbooks until the 16th-17th centuries. To denote numbers, mathematicians of the Near and Middle East also used letters of the Arabic alphabet (“abjad” or “jumal”). However, the Old Arabic computational methods came into use most firmly. Numbers and calculations were written down here not with the help of signs, but with words. They especially differed from the Indians in the study of fractions.

The original Arabic text of al-Khwarizmi's arithmetic treatise has been lost. However, its content is well known to us from a Latin translation made in the 12th century. Gerardo of Cremona. At this time, a group of scientists was actively working in Spain, translating the most famous works in the East from Arabic into Latin. They sought to introduce Europe to the best achievements of Eastern science. Among the first mathematical works, al-Khwarizmi's treatise on Indian arithmetic was translated.

This Latin translation has survived to this day in a single manuscript, which is kept in Cambridge. It suffers from many shortcomings: it contains typos, gaps, and the end of the text is missing. To get a more accurate idea of ​​al-Khwarizmi’s treatise, researchers studied the manuscripts of two more Latin works, also written in the 12th century: “The Book of Introduction of Alkhorism to Astronomical Art, compiled by Master A.” and "The Book of Algorism on the Practice of Arithmetic." The first of these is attributed to the famous translator Adelard of Bath, who worked around 1120-1130. The second belongs to the prominent scientist from Toledo, John of Spain, who worked in the second half of the 12th century.

These works provide a detailed presentation of the work of al-Khwarizmi, and the rules formulated by him are explained with numerous examples. Comparing all these. manuscripts, modern scientists were able to completely restore the content of al-Khwarizmi’s treatise and find out its role in the history of mathematics.

The treatise essentially sets out the rules of our modern arithmetic, Al-Khwarizmi teaches how to write any number using nine signs that take on a value depending on the digit in which they are located. He pays special attention to the sign introduced to designate an empty digit, i.e., zero, which is represented by a “small circle.” The following explains how to perform basic arithmetic operations on numbers written using “Indian” signs.

After the appearance of al-Khwarizmi's treatise on Indian arithmetic, the methods outlined in it began to quickly spread among mathematicians and astronomers in the countries of the Near and Middle East, gradually displacing old computational techniques. The introduction of the decimal positional number system ensured the rapid development of computational mathematics. In this area, scientists from the Near and Middle East who worked in the 9th - 15th centuries have made many important achievements. They developed methods for extracting roots of any degree, applied the rule, now called Newton's binomial, to any natural exponent, discovered decimal fractions, etc. A remarkable proof of their success could be the calculation of numbers? with seventeen decimal places.

Role al-Khwarizmi V development of algebra. Al-Khwarizmi's algebraic treatise has come down to us in the original: an Arabic manuscript is kept in Oxford (Bodleian Library, Ms Hunt, 214), rewritten in 1342. Its text was published in 1831 by F. Rosen along with an English translation. Since then, this text has been reprinted several times and translated into European languages, including Russian.

In addition to the Arabic original, there are two medieval Latin translations, which were completed in the 12th century. The first of them, dated 1145, belonged to the great scientist and translator Robert of Chester, the second to Gherardo of Cremona. These translations immediately gained popularity among European mathematicians.

Al-Khwarizmi's algebraic work, like his arithmetic treatise, has been studied by many researchers, and the results of their work are reflected in an extensive literature. It has been established that al-Khwarizmi played no less a role in the history of algebra than in the history of arithmetic. In al-Khwarizmi's book, algebra was first presented as a science of general methods for solving numerical linear and quadratic equations. Although the form of presentation of these methods is now unusual (at the time of al-Khwarizmi, alphabetic symbolism had not yet been introduced, and therefore all the rules are given in verbal expression), in essence, his reasoning is quite understandable and close to modern mathematicians.

Algebraic treatise of al-Khwarizmi is entitled “A short book on the calculus of completion and opposition” (kitab mukhtasarf-l-hisab al-jabr wa-l-mukabala) and consists of two parts - theoretical and practical. The first of them sets out the theory of linear and quadratic equations, and also touches on some issues of geometry. In the second part, algebraic methods are applied to solving specific legal, household, and trade problems.

In the introduction, al-Khorezmi talks about the reasons that prompted him to start writing the essay: “I compiled a short book on the calculation of algebra and almukabala, which contains simple and complex questions of arithmetic, because this is necessary for people when dividing inheritance, drawing up wills, dividing property and court cases, in trade and all kinds of transactions, as well as in measuring lands, drawing canals, geometry and other types of similar matters." Thus, the need to solve applied problems directly prompted the study of theoretical issues.

Al-Khorezmi explains which numbers are used in algebra. If arithmetic operates with ordinary numbers that are “composed of units,” then algebra involves numbers of a special type - an unknown quantity, its square (in modern notation x and x 2) and a free term equations The square of the unknown is called the word “property” (small) and is defined as “that which is obtained from the root when multiplied by itself.” Al-Khorezmi calls the free member of the equation - a “prime number” - “dirhams,” i.e., monetary units. He also gives a classification of linear and quadratic equations. Currently, it seems unnecessary, since all special cases are combined using the notation ax 2 +bx+c=0, where the coefficients a, b And c can take positive, negative and zero values. But at the time of al-Khorezmi, not only did not the letter designation exist, but also the concept of a negative number, so al-Khorezmi had to distinguish six types of linear and quadratic equations, which became canonical for a long time.

In order to bring this equation to one of the indicated types, al-Khwarizmi introduces two special actions, the names of which appear in the title of the book. The first of these is al-jab, “replenishment.” It consists of moving the negative term from one side of the equation to another. (It is from this term that the modern word “algebra” arose.) The second action - almukabala, “opposition” - consists of bringing similar terms on both sides of the equation.

The tasks for the division of property constitute the content of the entire second part of al-Khorezmi’s work, which is called “The Book of Wills.” At one time, it served as a practical guide for lawyers involved in the division of inheritance. According to Muslim law, each family member inherits a strictly defined share of the property left behind. In specific cases, for example, if some part of the inheritance was bequeathed to a stranger under certain conditions, the task became more complicated. Algebra helped to get out of the difficulty: the question boiled down to solving a linear equation.

Geometry of al-Khwarizmi. Mathematicians of the Near and Middle East in the Middle Ages paid great attention to geometry. Of particular interest was Euclid's Elements, translated into Arabic at the end of the 8th - beginning of the 9th century. But along with questions of theory, Eastern scientists were also occupied with problems of practical geometry, which land surveyors, artisans, and builders constantly had to solve. Therefore, mathematicians wrote special essays that served as guides for practitioners. They usually did not contain proofs, but only provided definitions of basic geometric concepts and rules for measuring figures and bodies. The rules were explained with numerous specific examples.

For the first time in literature in Arabic, such a collection of information necessary in the practical activities of man was given by al-Khorezmi. The geometric section of his “Algebra”, called “Chapter on Measurement,” is devoted to these issues. It focuses on the measurement of figures and bodies. After al-Khwarizmi, questions of practical geometry were considered in the works of many outstanding mathematicians, on whom he had a strong influence.

Astronomical works of al-Khwarizmi. Astronomy occupied the leading place among the exact sciences in the medieval East as a science essential for practice. Therefore, already in the 8th century. Intensive astronomical research began in Baghdad and other cities of the caliphate. Ancient Greek works on astronomy were translated into Arabic. Among the first to be translated and carefully studied was Ptolemy’s Almagest (2nd century). This work, which summarized the achievements of ancient scientists, was the foundation of the entire medieval astronomical theory.

Indian science had a strong influence on the development of astronomy in the countries of the Near and Middle East. Medieval historians say that in 773 a man well-versed in astronomical teachings arrived in Baghdad from India. Through him, Baghdad scientists became acquainted with the Siddhantas - Indian works that provided information on mathematics and astronomy.

Already in the 8th century. Astronomical observations were made at an observatory built in Damascus. They developed especially widely in Baghdad after a large observatory was built here at the “House of Wisdom” in 829. Scientists compiled astronomical tables and strived for them to be more accurate than the tables of their predecessors. This required more advanced astronomical instruments. Eastern masters achieved high art in the design of astrolabes, quadrants, and sundials.

In the 9th century. The first independent works on astronomy appeared in Arabic. Among them, a significant place was occupied by zijs - collections of astronomical and trigonometric tables needed to solve many problems of practical astronomy. With the help of these tables, time was measured, the positions of the luminaries on the celestial sphere were calculated, the moments of the beginning of solar and lunar eclipses were determined, etc. In the zijs, the tables were supplemented with detailed theoretical explanations.

Among the first zijs is zij al-Khorezmi. [A fragment from this essay is published in this issue of the journal.] This work brought him fame during his lifetime and was highly valued by astronomers of later times. Medieval historians write that it existed in two versions, but when they were compiled is unknown.

The zij of al-Khwarizmi has not been preserved in the Arabic original. We are familiar with this work from the Latin translation of 1126, owned by Adelard of Bath. Unfortunately, it was made not from the work of al-Khorezmi itself, but from its adaptation, which was compiled at the end of the 10th - beginning of the 11th century. Maslama ibn Ahmad al-Majriti was an Arab scholar who worked in Spain. Al-Majriti tried to follow the original exactly, but changed some of the astronomical values ​​that al-Khwarizmi gave for the latitude of Baghdad; in al-Majriti's text they are recalculated for Cordoba. The Latin translation of these tables became widespread in Europe and served as the basis for astronomical research here. They have now been published, translated into English and carefully studied by historians of science.

Zij al-Khwarizmi has been commented on by many scholars from the Near and Middle East. Among them was the outstanding Central Asian astronomer Abu-l-Abbas Ahmad al-Fergani, the author of the famous work “The Beginnings of Astronomy,” which also became famous in Europe in the 12th century. Al-Ferghani was a contemporary of al-Khorezmi and also worked in Baghdad. His commentary on zij al-Khwarizmi has not reached us.

Three voluminous works by Abu Rayhan Biruni dedicated to this zij also turned out to be lost. Their content can be judged by the statements of Biruni himself in other works. He discussed and substantiated the tables given by al-Khwarizmi, and defended them from the unfair criticism of some astronomers. At the same time, he sought to clarify al-Khorezmi’s data and correct those that seemed erroneous to him. Biruni's attention to al-Khorezmi's zij once again proves the high authority this work enjoyed among the most prominent eastern astronomers a century and a half after it was written.

Nowadays there is a known commentary on the zij of al-Khwarizmi, which was compiled by a scientist of the 10th century. Ahmad ibn Musanna. It helps researchers more accurately reconstruct the content of zij.

Al-Khorezmi's work is of great interest to those who study the history of eastern astronomy. IX century scientists working in Baghdad sought to combine the theories of ancient astronomers with Indian astronomical theories and teachings common in pre-Islamic Iran. Al-Khwarizmi in his zij explained in detail the methods that were developed in India. They complemented the theory of Ptolemy, which became the basis of astronomy in the Near and Middle East.

In the zij of al-Khwarizmi, for the first time in literature in Arabic, a table of sines was given and a tangent was introduced.

In addition to zij, al-Khwarizmi also wrote other works on astronomy. He devoted three treatises to the astrolabe, a portable astronomical instrument widely used in the Near and Middle East. The treatises set out the rules for using this complex instrument, described the types of astrolabes known in the 9th century, and provided examples of solving problems of practical astronomy with their help. Al-Khwarizmi gave the first known description of another astronomical instrument, the sine quadrant. Al-Khorezmi also wrote an essay on the sundial and the calendar.

Geography. Works on geography were closely connected with the mathematical and astronomical works of Eastern scientists. Particular attention to issues of geographical science during this period was due to the urgent needs of practice, since long journeys across the territory of the caliphate associated with trade, administrative needs, etc., required clarification of the world map.

Creating a geographical map is associated with considerable difficulties: after all, the convex surface of the spherical Earth must be depicted on a plane, that is, a complex mathematical problem must be solved - projecting a sphere onto a plane. The astronomical tasks that arose in this regard were also not easy: in particular, it was necessary to accurately determine the geographical latitude and longitude of places, which required great knowledge of astronomy.

The first successful attempt to solve these problems is associated with the name of the great ancient Greek mathematician and astronomer Hipparchus (2nd century BC), but exact information about the map he compiled and the methods he used has not been preserved. His ideas were developed by later scientists. All geographical knowledge of the ancients was generalized in the 2nd century. in the works of Marinus of Tire and Ptolemy.

Medieval scientists of the Near and Middle East relied on the foundations of mathematical geography, which were developed during the period of antiquity. The author of the first geographical work, which laid the foundation for their activity in this field of science, was al-Khorezmi. His essay entitled "The Book of the Picture of the Earth" [An excerpt from this book and a map compiled by al-Khwarizmi are published in this issue of the magazine.](“Kitab surat al-ard”), preserved in a single Arabic manuscript, which is in the library of Strasbourg. This work, discovered only at the end of the 19th century, aroused great interest among researchers (K. Nallino, H. Mzhik, E. Honigman, V.V. Bartold, I.Yu. Krachkovsky, etc.), who showed how important the role in he played a role in the development of geography. According to V.V. Bartold, “The Book of the Picture of the Earth” marked the beginning of Arab geographical science.

Al-Khorezmi was the first to describe in detail in Arabic the inhabited part of the Earth known at that time and gave a map of it indicating the coordinates of the most important settlements, depicting seas, islands, mountains, rivers, etc. He relied on Greek writings. However, “The Book of the Picture of the Earth” is not a simple translation of the works of its predecessors, but an original work containing a lot of completely new data. Academician I. Yu. Krachkovsky noted that in it al-Khorezmi showed himself to be no less an independent scientist than in his mathematical works.

The work was written, apparently, in connection with the work in the field of geodesy and geography that was carried out in Baghdad under Caliph al-Ma'mun. The purpose of the work was to clarify the size of the Earth, previously calculated by Greek scientists. To do this, the length of one degree of the earth's meridian was directly measured, which was quite close to the true one (approximately 111 km). The measurements were carried out on flat terrain in the desert by a group of leading Baghdad astronomers using specially made instruments. Apparently, al-Khwarizmi also took an active part in this important work.

“The Book of the Picture of the Earth” was completed around 840, as it mentions the city of Samarra, located not far from Baghdad, where, under the heirs of al-Mu’mun, the capital of the caliphate was temporarily moved. The construction of Samarra began only in 836.

According to ancient tradition, al-Khorezmi divided the part of the earth then considered inhabited (ecumene) into seven “climates”. “Climates” are latitudinal zones that differ from each other in the length of the summer day (summer solstice) by half an hour. In al-Khorezmi they are limited by geographical parallels 16°27°, 24°, 30°22°,36°,41°, 45°, 48°. Here he showed originality in comparison with his Greek predecessors, who gave slightly different boundaries for “climates”.

For each “climate,” al-Khorezmi gave tables of coordinates of cities and gave a description of mountains, seas, islands and rivers. He indicated the latitudes and longitudes of 489 settlements. Some data are borrowed from Ptolemy, others are given in an updated form. For example, he corrected the coordinates of the border points of the Mediterranean Sea given by Ptolemy. The most significant additions To Ptolemy's world map made by al-Khwarizmi concerns Central Asia. He provided new information about the cities of this region, changed the description of the rivers, etc.

The four geographical maps available in the surviving manuscript of the “Book of Pictures of the Earth” are, according to I. Yu. Krachkovsky, “the oldest monuments of Arab cartography that have come down to us.” Citing the ancient names of places, al-Khorezmi also indicates the names that were used in his time.

Al-Khwarizmi’s work served as the basis for subsequent work by scholars of the medieval Near and Middle East in the fields of geography, geodesy and cartography.

Thus, a brief overview of the works of al-Khwarizmi shows that they had a huge influence on the development of science in both the East and the West.

In al-Khorezmi’s homeland in Khorezm, as well as throughout Central Asia, after the establishment of Soviet power, an unprecedented rise in culture and science began. The peoples who are the descendants of the ancient Khorezmians have given the world many prominent scientists, widely known for their works far beyond the borders of our country. Mathematics has reached a high level of development in Central Asia.

In 1979, the homeland of al-Khorezmi hosted participants in a conference on the modern theory of algorithms (this modern mathematical term goes back to the Latin form of his name) and its applications, which was organized in Urgench by the USSR Academy of Sciences and the Uzbek SSR Academy of Sciences on the basis of the Institute of Cybernetics of the Uzbek SSR Academy of Sciences . The guests, among whom were many prominent specialists in mathematical logic and the theory of algorithms, paid tribute to the memory of Muhammad ibn Musa al-Khwarizmi. In a report dedicated to his work, the Austrian professor Zemanek said: “We can express only one wish: that in a thousand years those who discover any of us would look at what we have created with the same respect with which we look today at al-Khorezmi and his colleagues in the “House of Wisdom.”

The text is reproduced from the publication: Muhammad ibn Musa al-Khorezmi and his contribution to the history of science // Questions of the history of natural science and technology, No. 1. 1983

Abu Abdullah (or Abu Jafar) Muhammad ibn Musa al-Khorezmi was born in 783 Khiva, Khorezm - and died in 850 Baghdad - one of the largest medieval Persian scientists of the 9th century, mathematician, astronomer, geographer and historian.

Very little information about the scientist’s life has been preserved. Presumably born in Khiva in 783. In some sources, al-Khorezmi is called “al-Majusi,” that is, a magician, from which it is concluded that he came from a family of Zoroastrian priests who later converted to Islam. Al-Khorezmi's homeland is Khorezm, which included the territory of modern Uzbekistan and part of Turkmenistan.

Al-Khorezmi was born in an era of great cultural and scientific upsurge. He received his primary education from outstanding scientists of Movarounnahr and Khorezm. At home, he became acquainted with Indian and Greek science, and he arrived in Baghdad as a fully established scientist.

In 819, al-Khwarizmi moved to the Baghdad suburb of Kattrabbula. He spent a significant period of his life in Baghdad, heading the “House of Wisdom” (Arabic: Bayt al-Hikmah) under Caliph al-Mamun (813-833). Before becoming caliph, al-Mamun was the governor of the eastern provinces of the Caliphate, and it is possible that from 809 al-Khwarizmi was one of al-Mamun’s court scholars.

In one of his writings, al-Khwarizmi praised al-Mamun, noting his “love of science and desire to bring scientists closer to himself, extending the wing of his patronage over them and helping them in clarifying what is unclear to them and in facilitating what is difficult for them."

The “House of Wisdom” was a kind of Academy of Sciences, where scientists from Syria, Egypt, Persia, Khorasan and Movarounnahr worked. It housed a library with a large number of ancient manuscripts and an astronomical observatory. Here many Greek philosophical and scientific works were translated into Arabic. At the same time, Khabbash al-Khasib, al-Fargani, Ibn Turk, al-Kindi and other outstanding scientists worked there.

Commissioned by Caliph al-Mamun, al-Khwarizmi worked on creating instruments for measuring the volume and circumference of the earth. In 827, in the Sinjar desert, al-Khwarizmi took part in measuring the length of the degree of arc of the earth's meridian in order to clarify the circumference of the Earth, found in antiquity. Measurements made in the Sinjar Desert remained unsurpassed in accuracy for 700 years.

Around 830, Muhammad ibn Musa al-Khwarizmi produced the first known Arabic treatise on algebra. Al-Khwarizmi dedicated two of his works to Caliph al-Mamun, who provided patronage to the scientists of Baghdad.

Under Caliph al-Wasiq (842-847), al-Khwarizmi led an expedition to the Khazars. The last mention of it dates back to 847.

Contribution to world science

Al-Khorezmi was the first to introduce algebra as an independent science about general methods for solving linear and quadratic equations, and gave a classification of these equations.

Historians of science highly appreciate both the scientific and popularization activities of al-Khorezmi. The famous historian of science J. Sarton called him "the greatest mathematician of his time and, all things considered, one of the greatest of all time."

Al-Khwarizmi's works were translated from Arabic into Latin, and then into new European languages. Various mathematics textbooks were created on their basis. The works of al-Khorezmi played an important role in the development of science during the Renaissance and had a fruitful influence on the development of medieval scientific thought in the countries of the East and West.

Mathematics

Al-Khwarizmi developed detailed trigonometric tables containing sine functions. In the 12th and 13th centuries, based on the books of al-Khwarizmi, the works Carmen de Algorismo and Algorismus vulgaris were written in Latin, which remained relevant for many centuries. Until the 16th century, translations of his books on arithmetic were used in European universities as the main textbooks on mathematics. In 1857, Prince Baldassare Boncompagna included a translation of the "book of Indian arithmetic" as the first part of a book entitled Treatises on Arithmetic.

Astronomy

Al-Khorezmi is the author of serious works on astronomy. In them, he talks about calendars, calculations of the true positions of the planets, calculations of parallax and eclipses, compiling astrological tables (zij), determining the visibility of the moon, etc. His works on astronomy were based on the works of Indian astronomers. He carried out thorough calculations of the positions of the sun, moon and planets, and solar eclipses. Al-Khwarizmi's astronomical tables were translated into European and later Chinese languages.

Geography

In the field of geography, al-Khwarizmi wrote the book “The Book of the Picture of the Earth” (Kitab surat al-ard), in which he clarified some of Ptolemy’s views. The book included a description of the world, a map and a list of coordinates of the most important places. Despite the fact that al-Khwarizmi’s map was more accurate than the map of the ancient Greek astronomer, his works did not replace the Ptolemaic geography used in Europe. Using his own discoveries, al-Khwarizmi corrected Ptolemy's studies of geography, astronomy and astrology. To compile a map of the “known world,” al-Khwarizmi studied the works of 70 geographers.

Works of al-Khwarizmi

Book on Indian Accounting (Arithmetic Treatise, Book on Addition and Subtraction);

A short book on the calculus of algebra and al-muqabala (“Kitab mukhtasab al-jabr and wa-l-muqabala”);

A book about actions with the help of an astrolabe (“Kitab al-amal bi-l-asturlabat”) is included in an incomplete form in the work of al-Fargani; in sections 41-42 of this book a special compass for determining the time of prayer was described.

Book about the sundial (“Kitab ar-ruhama”);

Book of the Picture of the Earth (Book of Geography, “Kitab surat al-ard”);

Treatise on the definition of the era of the Jews and their holidays (“Risala fi istihraj tarikh al-yahud wa ayadihim”);

The book about the construction of the astrolabe has not survived and is known only from mentions in other sources.

Astronomical tables (“Zij”);

History book - contained horoscopes of famous people.

Of these 9 books, only 7 have reached us. They have been preserved in the form of texts either by Al-Khwarizmi himself, or in translations into Latin, or by his Arab commentators.

Main article: Kitab al-jabr wa-l-muqabala

Al-Khwarizmi is best known for his “Book of Complementation and Opposition” (“Al-kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala”), which played a vital role in the history of mathematics. The word “algebra” comes from the title of this book. The original Arabic text is lost, but the contents are known from a Latin translation of 1140 by the English mathematician Robert of Chester. The manuscript, which Robert of Chester entitled "The Book of Algebra and Al-Mukabal" is kept in Cambridge. Another translation of the book was made by the Spanish Jew John of Seville. Conceived as an initial guide to practical mathematics, “Kitab al-jabr ...” in its first (theoretical) part begins with a consideration of equations of the first and second degrees, and in the final two sections moves on to the practical application of algebra in matters of measurement and inheritance. The word al-jabr (“replenishment”) meant the transfer of a negative term from one side of the equation to another, and al-muqabala (“opposition”) meant the reduction of equal terms in both sides of the equation.

The first page of the book Kitab al-jabr wa-l-mukabala.

Theoretical part

In the theoretical part of his treatise, al-Khorezmi gives a classification of equations of the 1st and 2nd degrees and identifies six types of quadratic equations (\displaystyle ax^(2)+bx+c=0) ax^(2)+bx+c= 0:

“square” is equal to “root” (\displaystyle ax^(2)=bx) ax^(2)=bx (example (\displaystyle 5x^(2)=10x) 5x^(2)=10x);

“square” is equal to the free term (\displaystyle ax^(2)=c) ax^(2)=c (example (\displaystyle 5x^(2)=80) 5x^(2)=80);

the “root” is equal to the free term (\displaystyle bx=c) bx=c (example (\displaystyle 4x=20) 4x=20);

“square” and “root” are equal to the free term (\displaystyle ax^(2)+bx=c) (\displaystyle ax^(2)+bx=c) (example (\displaystyle x^(2)+10x=39 ) x^(2)+10x=39);

the “square” and the free term are equal to the “root” (\displaystyle ax^(2)+c=bx) ax^(2)+c=bx (example (\displaystyle x^(2)+21=10x) x^( 2)+21=10x);

The “root” and the free term are equal to the “square” (\displaystyle bx+c=ax^(2)) bx+c=ax^(2) (example (\displaystyle 3x+4=x^(2)) 3x+4 =x^(2)).

This classification is explained by the requirement that both sides of the equation have positive terms.

Having characterized each type of equations and showing with examples the rules for solving them, al-Khorezmi gives a geometric proof of these rules for the last three types, when the solution is not reduced to simply extracting the root.

To reduce quadratic canonical species, al-Khwarizmi introduces two actions. The first of them, al-jabr, consists of transferring a negative term from one part to another to obtain positive terms in both parts. The second action - al-muqabala - is to bring similar terms in both sides of the equation. In addition, al-Khwarizmi introduces the rule for multiplying polynomials. He shows the application of all these actions and the rules introduced above using the example of 40 problems.

The geometric part is mainly devoted to measuring the areas and volumes of geometric figures.

Practical part

In the practical part, the author gives examples of the use of algebraic methods in solving household problems, measuring land, building canals, etc. The “chapter on transactions” discusses the rule for finding an unknown member of a proportion from three known terms, and in the “chapter on measurement” - rules for calculating the area of ​​various polygons, an approximate formula for the area of ​​a circle, and a formula for the volume of a truncated pyramid). It is also accompanied by the “Book of Wills,” dedicated to mathematical problems arising during the division of inheritance in accordance with Muslim canon law.

Algebra by al-Khorezmi, which laid the foundation for the development of a new independent scientific discipline, was later commented on and improved by many eastern mathematicians (Ibn Turk, Abu Kamil, al-Karaji, etc.). This book was translated twice into Latin in the 12th century and played an extremely important role in the development of mathematics in Europe. Such an outstanding European mathematician of the 13th century as Leonardo of Pisa was directly influenced by this work.

Algorithm

The Latin translation of the book begins with the words "Dixit Algorizmi" (said Algorizmi). Since the essay on arithmetic was very popular in Europe, the Latinized name of the author (Algorizmi or Algorizmus) became a common noun, and medieval mathematicians called arithmetic based on the decimal positional number system. Later, European mathematicians began to call this any calculation according to strictly defined rules. Currently, the term “algorithm” means a set of instructions that describe the order of actions of the performer to achieve the result of solving a problem in a finite number of actions.

Astronomical tables (zij)

Astronomy occupied a leading place among the exact sciences in the medieval East. It was impossible to do without it either in irrigated agriculture or in sea and land trade. By the 9th century. The first independent works on astronomy appeared in Arabic, among which collections of astronomical and trigonometric tables (ziji) occupied a special place. Zijs served to measure time and with their help the positions of luminaries on the celestial sphere, solar and lunar eclipses were calculated.

Among the first zijs is “Zij al-Khwarizmi,” which served as the basis for medieval research in this area in both the East and Western Europe. Although "Zij al-Khwarizmi" is mainly an adaptation of Brahmagupta's "Brahmaguphuta-siddhanta", much of the data in it is given at the beginning of the Persian era of Yazdegerd and, along with the Arabic names of the planets, their Persian names are given in the tables of equations of the planets of this zij. The Treatise on the Calculation of the Era of the Jews is also adjacent to this zij. The “Book of Chronicles” of al-Khwarizmi, mentioned in various sources, has not survived.

The book began with a section on chronology and calendar, which was very important for practical astronomy, since due to the difference in calendars it was difficult to determine the exact dating. The existing lunar, solar and lunisolar calendars and different beginnings of chronology led to many different eras and the same event was dated differently among different peoples. Al-Khwarizmi described the Islamic Julian calendar (calendar of “rums”). He also compared different eras, among them the ancient era of India (began in 3101 BC) and the “age of Alexander” (began in October 1, 312 BC).

According to al-Khwarizmi's calculations, the beginning of the Islamic era corresponds to July 16, 622. Al-Khwarizmi accepted the meridian passing through a place called Arin as the prime meridian from which time was measured; I. Yu. Krachkovsky identified Arin with the city of Ujjain in India. The Zij speaks of the "Dome of Arina" because it was believed that the meridian of Ujjain coincided with the meridian of the island of Sri Lanka, supposedly lying on the equator; According to the ideas of Indian geographers, in the “middle place” of the Earth, the point of intersection of the prime meridian and the equator, there is a certain “dome”, or “Dome of Ujjain”. In Arabic spelling, the words Ujjain and Arin differ little, so the "Dome of Ujjain" became the "Dome of Arin", or simply Arin.

Book on Indian Accounting

Al-Khwarizmi wrote “The Book of Indian Counting,” which contributed to the popularization of Arabic numerals and the decimal positional system for recording numbers throughout the Caliphate, right up to Muslim Spain. The Arabic text has been lost, but a 12th-century Latin translation, Algoritmi de numero Indorum, survives. The Latin translation did not preserve many details of al-Khwarizmi's original text. The book had a huge influence on later leadership not only of the East, but also of the West.

The book describes how to find a decimal number consisting of nine Arabic digits and a zero. It is possible that al-Khwarizmi became the first mathematician to use zero in notation of numbers. The original Book of Indian Calculus described a method for finding the square root, but it is not found in the Latin translation.

Two hundred years after the writing of On Indian Accounting, the Indian system spread throughout the Islamic world. In Europe, “Arabic” numerals were first mentioned around 1200. Arabic numerals were originally used only in universities. In 1299, a law was passed in Florence, Italy, prohibiting the use of Arabic numerals. But since Arabic numerals began to be widely used by Italian merchants, by the 16th century all of Europe switched to them. Until the beginning of the 18th century, Russia used the Cyrillic number system, after which it was replaced by a number system based on Arabic numerals.

Earth Picture Book

His works on geography were also associated with his works on mathematics and astronomy. Al-Khwarizmi’s “Book of the Picture of the Earth,” the first geographical work in Arabic and the first work on mathematical geography, had a strong influence on the development of this science.

For the first time in Arabic, he described the inhabited part of the Earth known at that time, gave a map with 2402 settlements and coordinates of the most important settlements. In many ways, he relied on Greek works (Ptolemy's Geography), but his Book of Pictures of the Earth is not just a translation of the works of his predecessors, but an original work containing a lot of new data. He organized scientific expeditions to Byzantium, Khazaria, Afghanistan, under his leadership the length of one degree of the earth's meridian was calculated (very accurately for those times), but his main scientific achievements were related to mathematics. The “Book of the Picture of the Earth” gave a definition of latitude and longitude.

In essence, the House of Wisdom was the Academy of Sciences. Many scientists from various regions of Central Asia and the Arab East worked there; they had at their disposal a rich library of ancient manuscripts, as well as a large specially built astronomical observatory. The House of Wisdom became a center for the study of mathematics, astronomy, medicine and chemistry.

For a significant period of his life, the library of the “House of Wisdom” was headed by Abu Abdullah Muhammad ibn Musa Al-Khorezmi, nicknamed the “father of algebra.” It was thanks to Al-Khorezmi that Europe learned what decimal counting and numbers are. It was he who first proposed dividing the globe into meridians and parallels. Thus, every point on the earth had clear coordinates, latitude and longitude. It was Al-Khorezmi who created the first reliable model of the earth, the prototype of the modern globe, and this was 700 years before the famous Italian scientist Galileo Galilei.

It is largely thanks to this Muslim scientist that the world is what it is today. Al-Khwarizmi's contribution to mathematics The modern word “algorithm” comes from the name of Al-Khwarizmi, and it is associated with the title of his book “Al-Kitab al-mukhtasar fi hisab al-jabr wa-al-muqabala”. This book was translated into Latin twice in the 12th century and played an extremely important role in the development of mathematics in Europe.

Translated, the title means “A short book on completion and opposition.” The word “algebra” comes from the title of this book. Al-jabr means the operation of transferring negative terms from one side of an equation to another to obtain positive terms on both sides. “Al-muqabala” means “opposition”, that is, bringing similar terms in both sides of the equation. This work of Al-Khwarizmi became the first written work on algebra. Thanks to Latin translations, he gained fame in Europe and had a huge influence on the development of Western science.

His book on algebra introduced Europeans to a hitherto unknown discipline and served as a classic mathematical text for European university students for several centuries. Al-Khorezmi was the first to introduce algebra as an independent science about general methods for solving linear and quadratic equations, and gave a classification of these equations. Al-Khwarizmi also separated algebra from geometry in 834.

Al-Khorezmi's contribution to astronomy Astronomy occupied a leading place among the exact sciences in the medieval East as one of the most necessary sciences in practice, without which it was impossible to do either in irrigated agriculture or in sea and land trade. By the 9th century. include the first independent works on astronomy in Arabic, a special place among them was occupied by “zijis” - collections of astronomical and trigonometric tables (at that time trigonometry was part of astronomy). Using these tables, the position of the luminaries on the celestial sphere and the time of solar and lunar eclipses were calculated. They also served to measure time.

Among the first zijs is the zij of Al-Khorezmi, which began with a section on chronology and calendar. This was very important for practical astronomy, since different peoples used different calendars at different times, and universal dating was important for observations. His works on astronomy were based on the works of Indian astronomers.

He carried out thorough calculations of the positions of the sun, moon and planets, and solar eclipses. Al-Khwarizmi's astronomical tables were translated into European languages, and later into Chinese. Trigonometric and astronomical tables (Zij al-Khwarizmi) formed the basis of medieval research in the field of astronomy in both the East and Western Europe. Al-Khorezmi’s book on the astrolabe, the main instrument of astronomical measurements of that time, is also important. “The Book on the Construction of the Astrolabe” has not survived and is known only from mentions in other sources.

From the astronomical works of Al-Khorezmi, the “Book of the Sundial” and the “Book of Action with the Help of the Astrolabe” (incompletely included in the work of Al-Fergani) are also known. In sections 41-42 of this book, a special compass was described for determining the time of prayer. Al-Khorezmi's contribution to geography Al-Khorezmi's works on geography were also associated with his works on mathematics and astronomy.

He is considered the author of the first work on mathematical geography. Al-Khorezmi was the first to describe in Arabic the inhabited part of the Earth, known at that time, and gave a map with the coordinates of the most important settlements, with seas, oceans, mountains, and rivers. In many ways, he relied on Greek works (Ptolemy’s Manual of Geography), but his “Book of the Picture of the Earth” is not just a translation of the works of his predecessors, but an original work containing a lot of new data.

The book included a description of the world, a map and a list of coordinates of the most important places. Despite the fact that Al-Khwarizmi's map was more accurate than the map of the ancient Greek astronomer, his works did not replace the Ptolemaic geography used in Europe. Using his own discoveries, Al-Khwarizmi corrected Ptolemy's studies of geography, astronomy and astrology. To compile a map of the “known world,” Al-Khwarizmi studied the works of 70 geographers.

He also organized scientific expeditions to Byzantium, Khazaria, and Afghanistan, under his leadership the length of one degree of the earth's meridian was calculated (very accurately for those times). Conclusion It cannot be said that there was no algebra before Al-Khwarizmi. In ancient times, people solved simple algebraic problems; There were techniques for solving individual specific problems, but Al-Khorezmi was the first to introduce algebra as a science of general methods for solving numerical linear and quadratic equations, and gave a classification of these equations.

Muhammad ibn Musa Al-Khorezmi occupies an important place among the scientists of Central Asia, whose names have entered the history of precise natural science. In the 9th century. - at the dawn of medieval Eastern science - the scientist made a great contribution to the development of arithmetic and algebra.

Al-Khwarizmi's algebraic treatise was among the first works on mathematics translated in Europe from Arabic into Latin. The modern name of algebra, as we noted above, comes from the word “al-jabr”, and the word “algorithm” comes from the name Al-Khorezmi. The works of Al-Khwarizmi had a great influence on scientists of the East and West for several centuries and served as a model for writing mathematics textbooks for a long time.

Historians of science highly appreciate both the scientific and popularization activities of Al-Khorezmi. Renowned historian of science George Sarton called him "the greatest mathematician of his time, and, all things considered, one of the greatest of all time."