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Mathematics, Astronomy, and Chronology

Not much information is available on the status of these sciences in the pre-Islamic Iran and whatever knowledge we possess is based on the archeological discoveries and the written works that have found their way into the Islamic sources from the Pahlavi language. Nevertheless, the information available about the engineering, naval, astrological, and astronomical activities as well as the complicated tax system used in ancient Iran reflect on the mastery of the Iranians in the field of mathematics and other related sciences, a significant part of which wass transferred to the Islamic era.

Some the most important achievements of the Iranians during the Islamic era include:

Writing the first mathematical works of the Islamic period in the fields of algebra, arithmetic, geometry, and the first independent works on trigonometry;

Acquainting the Muslims and thereafter the Europeans with the Indian numeric system and its application in making calculations;

Classification of quadratic equations and their geometrical as well numeric solutions;

Engagement in some classic mathematical problems like the quadrature of a circle and the construction of heptagons and nonagons. While solving the first problem was redeemed impossible, the construction of heptagons and nonagons could not be done with the use of a scales and compasses. Such problems could only be solved after a series of discussions and correspondence among some Iranian mathematicians with the help of other methods like geometry and demonstrative geometry and with the employment of conical sections;

Conducting researches on Euclid’s fifth principle and finding ways to prove this principle. Although such attempts had begun in ancient Greece and continued until the last part of the 19th century, they did not produce any direct results even if they contributed towards the emergence of non-Euclidean geometry. Almost all the Muslim mathematicians who worked on the Euclidean principles were Iranians;

Preparing the various earliest trigonometric tables as well as the application of what is today known as the tangent as an independent trigonometric function;

Creating, proving and applying the sinusoidal theorem in place of Menelaus’s theorem in trigonometry as well as creating, proving, and applying the tangent theorem;

Providing solutions toequations up to 9th degree;

Research on other theories on numbers including the Fermat’s theorem by a scholar called Māhāni;

The science of astronomy of the early Islamic period was founded on the basis of the astronomical traditional of Iran, India, and Greece and almost all the pioneer astronomers of the Abbasid court were either Iranians or were deeply influenced by the Iranian astronomy. The proof of this fact is the existence of such words as zij (astronomical table), hilāj, kadkhodā, jān bakhtān, jozhar, and even hendeseh (geometry) – taken from the Pahlavi term “hendāzag” meaning “measure” – as well as a number of other words of Pahlavi origin that are found in the various mathematical and astronomical sources. Besides transferring their own astronomical experiences to the world of Islam, the Iranians also played a significant role in the translation of Indian and Greek works and putting them at the disposal of other Islamic scholars. From among the ancient Iranian works in astronomy to which references have been made in the sources of the Islamic period mention can be made of the Zij-e Shah/Shahriyār or the Zij Shatr regarding which works, information is available indicating that they were written during the reigns of Anushirvān and Yazdgerd III, even though some research scholars have said that these zijs date back to earlier periods. For instance, Abu Mash’ar has written about a very ancient zij that was the source of the above mentioned Zij-e Shahriyār. Ibn Rasteh’s report very clearly shows the level of the acceptability of the Zij-e Shahriyār during the Islamic period, to which Muslim astronomers referred as an authentic source. Not only was this zij used in the Islamic territories but it was also used in the Western world, particularly in Andalusia, and it continued to be considered as an authentic source in the Sind province of India even after the spread of Ptolemy’s “Almagest”.

It is important to note that since trigonometry was regarded as a premise of astronomy prior to being considered as a branch of mathematics, all the innovations introduced by the Iranians in the field of trigonometry could therefore be treated as part of their venture into astronomy. As a matter of fact, many works of astronomy written by the Iranians, particularly the zijs, were of great importance from the viewpoint of their trigonometric tables. Nonetheless, some of the achievements of the Iranians in the field of astronomy included:

1. The first astronomical observations, most of which were carried out during the early Islamic period; 2. Two out of three of the most significant observations of the Islamic period; 3. Efforts towards the rectification of Ptolemy’s astronomy, which along with the efforts of the Andalusian scholars, paved the path for the emergence of the Copernican theory that the sun is at rest near the center of the universe, and that the earth, spinning on its axis once daily, revolves annually around the sun; and 4. Invention of a number of observational instruments used in astronomy from among which mention can be made of Fakhr-e Rāzi’s sextant, Ibn Sin’s (Avicenna) observational instruments famous for their precision, and some other instruments like Tusi’s linear astrolabe, famous for its convenient usage.

The Iranians also played a significant role in the field of, the impact of which is evident till date. In ancient Iran two systems of chronology were used over an area extending from Soghd to Armenia and Asia Minor. The first type divided the year into 12 months of thirty days each with an additional five days that were referred to as “Andargāh” or “Khamseh Mostarqeh” and since the actual length of the year is 365.24 days the beginning of the year (or Noruz) was not fixed. A second system of chronology was used by the mobeds (Zoroastrian priests) and the government offices, particularly the tax bureaus. In this system of chronology, instead of adding 5 days to the 360 days, a different method was used according to which after every 120 years (according to some sources every 116 years) there would be a leap year, which consisted of 13 months. Interestingly, with the coronation of every new king there would be a new beginning to the Iranian Calendar, which would start at the Noruz of the same year. However, since the computation of the leap year had been abandoned long before the advent of Islam the time of Noruz was subject to change. For instance, the year in which (11 AH/632 AD) Yazdgerd III held his coronation, Noruz fell on June 16 i.e. 91 days after the beginning of the spring and this day became the beginning of the Iranian calendar. Even after the conquest of Iran by the Muslims, the old system of Iranian chronology continued to be used by the astronomers, Iranian Zoroastrians, and government departments (without calculating the leap year). Interestingly, this calendar continues to be in use by the Zoroastrians of India. Despite the fact that the omission of the leap year caused a lot of problems in the computation of taxes, the caliphs, who considered such a practice as being against the tenets of Islam - and thus unlawful - refrained from restoring the addition of the leap year into the solar calendar. It was only during the reign of the Abbasid caliph, Motewakkel, and his successor, Mo’tazed, that the omitted leap years were included into the calendar and fixed calendar years were adopted. Finally, during the rule of the Saljuq king, Malek Shah, the Jalāli Calendar, which was the most accurate calendar of the world, was instituted and adopted. In the year 1924 this calendar was renamed as the Hejri Shamsi (lit. “the Solar Hejri”) calendar, with some minor changes, and came to be adopted as the official calendar of Iran. The Hejri Shamsi calendar is the only calendar, the commencement of which has been instituted on the basis of an astronomical event (i.e. the beginning of spring), and unlike other calendars its beginning remains fixed and unchanged.

On the subject of prominent Iranian scholars in this field, besides those already mentioned, mention must be made of Nobakht (d. about 160 AH) and Fazāri or perhaps more accurately “Firuzān”, the father of Yahyā bin Abi Mansur, who were the first astronomers of the Islamic period. Nobakht was the first outstanding astronomer from the famous Iranian family of Āl-e Nobakht and he was the first scholar to determine a suitable date and time for the beginning of the construction of Baghdad (145 AH/ 762 AD). Nobakht’s son, Abu Sahl, and his two grandchildren, Hasan and Abdullah, too, served as astronomers in the Abbasid court. As recorded by Ibn Nadim, Abu Sahl and most of the prominent personalities of Āl-e Nobakht were great translators who translated Pahlavi books into Arabic.

Fazāri and Ya’qub bin Tāreq, who were also probably of Iranian origin, were the first Muslim scholars to initiate the writing of trigonometric tables on the basis of Indian and Iranian works. It has also been recorded that the Indian (Gooya) numbers were first spread through the zijs of Fazāri or the Khwārazmi zijs and the first Ethiopian zijs. ‘Amr bin Farkhān Tabari (d. About 200 AH), too, was an Iranian scholar who, besides writing commentaries on Ptolemy’s works, wrote a book on births/birth signs and translated a number of other works from ancient Persian into Arabic. According to Al-Biruni, Fazāri and Māshāllāh, who was in all probability a Jewish Iranian settled in Basra, were the links between Abu Mash’ar and the astronomical works of the Sassanian period.

Ahmad Nahāvandi, a contemporary of Yahyā bin Khāled Barmaki (160 AH), was the first Islamic scholar to carry out some astronomical observations in Jondi Shāpur which are recorded in a zij called the Zij-e Moshtamal.

Another Iranian astronomer of repute was Bezist Firuzān, who had adopted the name “Yahyā bin Abi Mansur” after embracing Islam, and whose ancestors, too, were mostly astronomers and were known as the Bani Monajjem (lit. the sons of the astronomer). Apparently Yahyā was in charge of the observatories that had been built in Shamāsiyah of Baghdad on the instructions of Ma’mun, the Abbasid caliph, from 213 AH onwards. His grandson, Hārun bin Ali (d. 288 AH, Baghdad), too, was famous for writing an outstanding zij. Similarly, Sahl bin Boshr Ahkāmi (d. after 236 AH), a Jewish astronomer from Khorāsān, had become very famous in medieval Europe following the translation of many of his works into Latin. Abu Sa’id Zarir Jorjāni was yet another scholar in the field of astronomy who presented a theory for determining the degree of longitude on the same pattern on which the book “Analemma” had been written.

Ahmad bin Mohammad bin Kathir Forghāni was yet another great astronomer who not only influenced the science of astronomy during the Islamic era through his book “Jawami’ ‘Elm al-Nojum wa Harakāt al-Samāwiyah”, but who had also greatly impacted the astronomy of medieval Europe through the translation of his books into Latin by Jonnes Hispalensis. These books were later translated from Latin into Hebrew by a person called Jacob Anatoli.

Mohammad bin Musā Khwārazmi was one of the greatest mathematicians of all times who wrote the earliest works on algebra and arithmetic in the Islamic era. He is also considered to have influenced the mathematical thought more than any other mathematician of the middle ages. His book “Al-Mokhtasar Hesāb al-Jabr wa al-Moqābelah” contains analytical solutions to equations of the first and second degrees and he can, thus, be considered as one of the founders of non-geometrical analysis in the field of mathematics. Indian numerals found their way into Europe through the Latin translation of his book “Al-Jam’ wa al-Tafriq”. The zij written by him is one of the earliest works that have been written on various astronomical and trigonometric tables of functions. Many of his works have been translated into Latin, Hebrew, English, French and German. Furthermore, the two terms commonly used in the field of mathematics, viz. algebra and algorithm (as well as the similar terms used in other European languages) have been derived from the name of his book “Al-Jabr” and his own name “Al-Khwārazmi” respectively.

Ahmad bin ‘Abdullah Marvzi popularly known as Habash Hāseb was one of the greatest astronomers of Ma’mun and Motewakkel, the Abbasid caliphs who was engaged in astronomical observations from 214 to 250 AH. He also wrote three different zijs, two of which were written on the basis of the methods used by Indians, while the third one was written as a completion to his observations. He was also the first scientist to have calculated the time by measuring the height of a cosmic object (on the basis of the solar eclipse of the year 829 AD). He further formulated the oldest table of invert cotangents and applied it as an independent trigonometric line.

Mention was earlier made of the Banu Musā family. The most outstanding contribution of this family to mathematics was the drawing of the oval shape by the method popularly known as Bāghbāni, as well as the writing of explanatory notes and commentaries on the conic sections of Apollonius, the Greek mathematician, who lived during the early 300’s and late 200’s BC. A number of later scholars have mentioned the contributions of Banu Musā towards astronomical observations and tables in their works.

From among other great Iranian mathematicians and astronomers mention can also be made of Māhāni who ventured into solving the fourth theorem from his treatise “On the Sphere and Cylinder” through the equation x3 + a = cx2, which later on came to be known as the Māhāni Equation that was subsequently declared to be an unsolvable theorem.

Abu Ma’shar Balkhi, too, was a prominent Iranian astronomer who had greatly impacted the history of astronomical laws. However, very few of his works on astronomy have survived.

The only information available about Soleimān bin ‘Esmat Samarqandi is that he was engaged in astronomy the year c. 275 AH and had written a zij called “Nayyerin” as well as a commentary on Ptolemy’s “Almagest”.

Fazl bin Hātam Neyrizi (d. about 310 AH/922 AD) had written a commentary on the second version of Hajjāj bin Yusof’s translation of Euclidean principles and had invented an instrument with great precision for measuring the dimensions of non-reachable objects. His treatise on this subject was greatly admired by Al-Biruni. 

Abu Nasr Fārābi, another prominent Iranian scholar of the Islamic period, had written numerous treatises on geometry, the Almagest and the classification of the various branches of science.

Abdullah bin Amājur and his son Ali bin Abdullah as well as Ali’s slave, Mofleh, who was popularly known as Ibn Amājur, too, were among the greatest Muslim astronomers of Shirāz and Baghdad during the period 272-321 AH. Ibn Yunos has praised their precision in astronomical observations and their command over geometry and astronomy and has even written about a zij that Mofleh had written single-handedly.

Owing to their deep interest in the Pythagorean School, the Ekhwān al-Safā gave a lot of importance to what was referred to as the secret behind the alphabets and the theory of numbers and left behind some treatises on mathematics, natural sciences, and laws of astronomy.

Abu Ja’far Khāzan succeeded in solving the Māhāni equation with the help of conic sections and wrote a zij called the “Zij al-Safā” (not currently available), which was greatly praised by Qafti. He also presented a discussion on the solutions of the equation x2 + y2 = z2 in his treatise entitled “Enshā’ al-Mothallathāt al-Qā’emah al-Zāviyah” (A Commentary on Right-Angled Trigonometry), which has been mistakenly attributed to Abu Ja’far Mohammad Hosein (6th Century AH/12th Century AD) by some writers.

‘Abd al-Rahmān Sufi (d. 376 AH/986 AD) made some astronomical observations and recorded them in his book “Sowar al-Kawākeb” in the form of a treatise, which is considered as one of the three greatest works by Muslims in this field of science. He has also written another important book on cosmic planets. Ibn A’lam, too, had made some precise observations with the instrument that had been made by him. He had also written a zij that was in use for nearly two centuries. It is not yet known as to why Sarton has considered him to be non-Iranian.

In all probability it was Sāghāni (d. 379 AH/989 AD) who had made the astronomical instruments for the Sharaf al-Dolah Buyahi’s observatory. He had made some researches on dividing the angle into three parts and dividing the circle into seven parts. He had also adopted a strange method for measuring the surface of the globe. The only information we have about Abu al-Fazl Heravi (d. about 371 AH) is from what has been recorded by Al-Biruni who has praised him for his mastery in mathematics and his precision in astronomical observations.

Abu al-Wafā Buzjāni was a great Iranian mathematician who had greatly contributed to the progress of trigonometry. Some of his contributions included the theorem on tangents, the invention of the formula sinx = 2*sin(x/2)*cos(x/2), and a method for computing the half degree sine up to eight digit decimals.

According to Al-Biruni, Abu Mahmud Hāmed bin Khezr Khojandi was the greatest craftsman in making astrolabes and other instruments used in astronomy. He had made a large sextant called “Fakhri” in the Mt. Tabrak near the Rey city, which was one of the largest and yet the most precise instruments for astronomical observations during the entire Islamic period. It is said that with the help of this instrument he managed to measure the general mile (an astronomical unit) in the year 384 AH. He had also reasoned out that the equation x3 + y3 = z3 could not be solved.

Abu Sahl Bizhan bin Rostam Kuhi (d. about 405 AH), the head of the astronomers of the Sharaf al-Dolah observatory of Baghdad had carried out some outstanding researches, which are considered to be some of the best geometrical works written by Muslim scholars. Abu al-Jud converted the regular nonagon into the equation x3 + 1 = 3x and also converted another geometrical problem into a 4th degree equation and subsequently solved it by simplifying them into parabola and hyperbola. He further managed to solve the mathematical problem that would result in the equation “cx2 = x3 + bx + a” that other scholars of ‘Azad al-Dolah’s court like Buzjāni, Sāghāni, Kuhi, etc. had failed to solve. He was the first person to discover the scientific method for drawing the regular heptagon. However, owing to a small computation mistake on his part, half of the credit went to Abu Sa’d ‘Alā bin Sahl (d. about 303-304 AH) who rectified the mistake made by Abu al-Jud. During the same period, Abu Ali Khiyuqi (from Khiweh) wrote his treatise entitled Al-Esteqsā’ on the computation of inheritance some of whose problems were related to linear or second degree equations. He had applied his own innovative methods to solve these problems.

Abu Nasr ‘Arāq, Al-Biruni’s teacher, too, was a great mathematician. He was the first person to have made a reference to the concept of the polar triangle, long before François Viete (1540-1603 AD) had spoken about it, and had even applied it in drawing a triangle with predefined angles. Although he has made important written contributions to astronomy, he is primarily known for his works in trigonometry.

Abu Sa’id Ahmad bin Mohammad bin ‘Abd al-Jalil Sajzi was one of the greatest Iranian mathematicians and astronomers of the 4th Century AH/10th Century AD. Some writers are of the opinion that by inventing the Persian Astrolabe he had demonstrated his belief in the rotational movement of the earth on its axis. He was the first scholar to have solved the problem of dividing the angle into three parts by resorting to a non-geometrical method and had named it static geometry. Abu Sa’id had also written a commentary on Menelaus’s theorem in trigonometry.

Abu Al-Hasan Qā’eni, also called Ibn Bāmshāz, was the author of one of the oldest works on the Jewish calendar. He had also written a treatise entitled “From Dawn to Sunrise”. Abu Bakr Karaji (d. about 420 AH), was yet another renowned Iranian mathematician and engineer who, in his book, “Al-Badi’” had discussed a number of new issues and who had for the first time used the term “Esteqrā’” (induction) for solving equations and had written a book on it.

Even though Ibn Sina’s (Avicenna) expertise in mathematics was in no way comparable to his proficiency in medicine, he too had made some valuable contributions to the theory of the “Jacob’s Rod” formulation that has mistakenly been attributed to Regio Montanus or Lois Ben Grison.

In his books “Al-Estekhraj al-Autār” and “Al-Qānun al-Mas’udi”, the great Iranian scholar Abu Reihān Biruni, popularly know as Al-Biruni, who had authored some valuable works in mathematics, astronomy, and chronology, managed to solve mathematical problems that cannot be solved merely with the help of a compass and ruler. Later on these problems became famous as Al-Biruni’s problems. His other book, the “Maqālid-e Elm-e Elāhiyah” is the first independent work in trigonometry. Similarly, his book, the “Rāshikāt al-Hend” presents the best explanations on Indian arithmetic during the medieval ages. Al-Biruni’s innovative methods for measuring the globe became very popular among the later mathematicians, one of which was published by G. B. Nicholasi De Paterno in the year 1660. Al-Biruni also wrote a book on the principles of measuring the distance between various cities which was entitled “Nahāyāt al-Amāken”. It was because of these books that Al-Biruni has come be known as one of the greatest geographers of all times.

Abu al-Hasan bin Ahmad Nasvi (393-473 AH or later) - who has been acclaimed by Nasir al-Din Tusi - authored his famous book, the “Al-Moqna’ fi al-Hesāb al-Hendi”, both in the Persian and Arabic languages. Most of Nasvi’s books have been translated into Latin and modern European languages. Yet another great Iranian mathematician was Abu al-Fath Esfahāni who had written elaborately on the “Conic Sections” of Apollonius and it was through this work that the Europeans gained acquaintance with Apollonius’s treatise on this subject.

Omar Khayyām, too, was one of the greatest Iranian mathematicians and astronomers of the medieval ages who had classified the cubic equations (13 different versions) and solved them through geometrical methods in his treatise entitled the “Maqālah fi al-Jabr wa al-Moqābelah”. Newton’s Binomial Series (the table of which is known as the Pascal’s Triangle) in reality was first formulated by Khayyām. In the year 467 AH he became engaged with several other scientists in order to reform the calendar; their efforts resulted in the adoption of a new era, called the Jalāli which is the most accurate calendar of the world.

Ghazāli, too, had written a treatise on the essence and movement of stars as well as a paper on astronomy. Even though there is no consensus among the scholars about Ghazāl’s negative impact on the progress of science among the Muslim scholars, the fact of the matter is that he had categorically expressed his opposition to the study of exact sciences, in general, and mathematics in particular.

Abd al-Rahmān Khāzani (lived in the 6th Century AH/12th Century AD), who, according to some scholars, was the main contributor to the Jalāli calendar, is one of the twenty-two people who had made independent and accurate astronomical observations during the Islamic period. His book, the “Zij al- Mo’tabar al-Sanjari” contains important mathematical calculations. Contrary to what has been mistakenly written by Rāshed and Qorbāni, Sharaf a-Din Mas’udi and Sharaf al-Din Tusi were two different people. As regards Mas’udi, he was a prominent mathematician who had authored two books, viz. the “Jahān-e Dānesh va Āthār-e Alavi” and the “Al-Jabr wa al-Moqābelah” (that has wrongly been attributed to Tusi) in Persian and Arabic. The latter book is one of the mathematical masterpieces and contains the best classifications of the cubic equations of the olden days as well as their numerical solutions. Mas’udi’s methodology can easily be applied to other equations of higher degrees and it is the same methodology that later on came to be known as the Rofini-Horner methodology. Sharaf al-Din Tusi, too, had written a few short treatises in mathematics and invented a simple astrolabe known as the linear astrolabe or Tusi’s stick. Similarly, Abu al-Mahāmed Ghaznavi was yet another Iranian scholar whose most important work on astronomy was written in Persian and was called the “Kefāyah al-Ta’lim fi Sanā’ah al-Tanjim”.

Athir al-Din Abhari (d. 663 AH) and Nasir al-Din Tusi were the two Iranian scholars who tried to prove Euclid’s fifth principle. Shams al-Din Samarqandi is of the opinion that Athir al-Din had done a better and more scientific job in this regard. On the other hand, Nasir al-Din Tusi was the founder of the Marāgheh observatory and was the principal of the Marāgheh School of astronomy. Tusi and some of the members of this school including Qotb al-Din Shirāzi, Mo’ayyed al-Din ‘Arazi, and Mohi al-Din Maghrebi critically examined the Ptolemaic system and presented their own systems. The outcomes of the studies of these scholars as well as their observations were compiled in a book called the “Zij-e Ilkhāni”. Salibā is of the opinion that the planetary view of Qotb al-Din Shirāzi presented in the book “Nahāyah al-Edrāk fi Derāyah al-Aflāk” is the same as Mo’ayyed al-Din ‘Arazi’s view put forward in his book, the “Elāhiyah”.

Najm al-Din Dabirān Kātebi, too, was a colleague of Nasir al-Din Tusi in Marāghah whose views on whether the earth is static or stationary had engrossed many scholars for a long time. Shams al-Din Samarqandi was another scholar who had written important commentaries on the Euclidian geometry as well Euclid’s fifth principle in the form of a treatise entitled, the “Ashkāl al-Ta’sis”. 

Another prominent Iranian mathematician was Kamāl al-Din Fārsi (665-718 AH) who propounded some novel numerical theorems and proved them in his famous treatise entitled, the “Tazkerah al-Ahbāb fi Bayān al-Tahāb”. He had also pointed out the mistakes made by some mathematicians as regards numbers and had explained the causes of such mistakes.

Ghiyāth al-Din Jamshid Kāshāni (d. 832 AH) was the most proficient Muslim mathematician who was also in charge of the Samarqand observatory and had succeeded in writing the Khāqāni Zij as a complementary to the Ilkhāni Zij. He also invented a new observation instrument called the “tabaq al-manāteq”. In two of his most important treatises entitled, the “Mohitiyah” and the “Wetr wa Jaib” Kāshāni, he calculated the value of π (with such accuracy that it remained unchallenged for 150 years) and the value of 1 degree sine respectively. Similarly, he also reinvented and popularized decimal fractions – that had first been referred to by Euclid – vis-à-vis sexagesimal fractions. His calculative methods were much ahead of his own times and could be compared with the method that was much later introduced by François Witt.

Qāzizadeh Rumi (766-840 AH), who succeeded Kāshāni after his death, and Chaghmini, who was the author of the book “Molakhkhas fi ‘Elm-e Elāhiyah”, were two of Kāshāni’s colleagues in the Samarqand observatory. Qāzizadeh was in turn succeeded by ‘Ali Qushchi(d. 879 AH) as the head of the observatory. The Timurid prince, Ologh Beig, who ruled Samarqand was himself an expert astronomer and had initiated the Samarqand observatory and school. The zij of Ologh Beig is the outcome of his efforts as well as those of his colleagues.

During the 9th Century AH/15th Century AD there emerged a number of mathematicians like Khalil bin Ebrāhim and ‘Abd al-Ali Birjandi who produced important works. One of such scholars was Bahā’ al-Din ‘Āmeli Sheikh Bahā’i (953-1031 AH) whose works, and particularly his “Kholāsah al-Hesāb” and “Tashrih al-Aflāk, were used as reference textbooks for a long time. Mohammad Bāqer Yazdi is the last prominent Iranian mathematician who had come up with many innovative theorems on balanced numerals.

* source: Keramati , yunes " Iran Entry " The Great Islamic Encyclopedia . Ed. Kazem Musavi Bojnourdi.Tehran: The Center of Great Islamic Encyclopaedia , 1989-, V.10 , pp.395 - 396

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