# History of Persian Mathematics

### A Note on the History of Persian Mathematics

In memory of "Maryam Mirzakhani"

Abstract. The history of mathematics in Iran (formerly known as Persia) can be categorized into three distinct periods: ancient, medieval, and modern. This note aims to explore and discuss the developments and contributions in each of these periods.

Introduction

The author believes that nothing captures the importance of history better than the following quote by George Santayana [S1]:

"Those who don't know history are doomed to repeat it."

Furthermore, the history of mathematics provides us with a deeper understanding of why mathematicians pursued their ideas over the centuries. In this context, the author incorporates historical points into his teaching to make his classes and lessons more engaging and meaningful.

The history of mathematics in Iran (Persia) can be divided into three main periods:

Ancient Period in the History of Mathematics in Persia (Iran)

In the remnants of buildings, water facilities, road construction, canal systems, tools, and machines of Persia , one can see technical and engineering mathematics in a tangible and visible form. The precise alignment of bricks in ancient structures, the sophisticated design of aqueducts and irrigation systems, the strategic planning of road networks, and the intricate mechanisms of early machinery all reflect a deep understanding of mathematical principles. These artifacts not only showcase the ingenuity of Persian engineers but also highlight the advanced level of technical knowledge that was applied in various aspects of daily life and infrastructure development.

This period encompasses the early contributions during the Elamites period (3200-539 BC), the Achaemenid Empire (550-330 BC), the Parthian dynasty (247 BC - 224 AD), and the Sasanian period (224-651 AD).

The Elamites (3200 BCE - 539 BCE) played a significant role in the politics and culture of the ancient Near East. Their influence extended through various periods, sometimes as a collection of city-states and at other times as a unified empire.

The Elamites had knowledge of astronomy and developed their own calendar systems. They were also skilled in construction and engineering, as evidenced by the impressive structures in their capital, Susa [BA].

Recent research by Heydari and Muroi highlights the advanced mathematical knowledge of the Elamites. Their studies, particularly on the Susa Mathematical Texts, reveal that the Elamites had a sophisticated understanding of geometry. For example, the Susa Mathematical Text (SMT) No. 2, dating from 1894 BCE to 1595 BCE, provides a formula for computing the approximate area of a regular heptagon [HM5]. The careful study by the same researchers reveals that the Elamites were not only aware of the Pythagorean theorem but also knew how to apply it to solve geometric problems [HM2]. On the other hand, the Elamite scribes, like their Babylonian counterparts, were familiar with the basics of solid geometry and knew how to compute the volume of three-dimensional figures such as cubes, prisms, and truncated pyramids [HM6].

The investigation of SMT No. 5 and SMT No. 6 shows that the Susa scribes had good skills in working with quadratic equations. The fact that only the statements of problems are provided suggests that these texts were likely prepared for educational purposes. The investigation also suggests that the scribes of Susa were employing methods that likely laid the foundation for "geometric algebra" in Greek mathematics [HM3]. Finally, it is important to mention that Susa scribes were familiar with complicated systems of polynomial equations and knew how to solve them [HM4]. These series of studies highlight the Elamites' significant contributions to mathematics, demonstrating their ability to solve complex problems related to construction and engineering.

Evert M. Bruins, based on his research, provides enough documents to show that the Susa scribes performed some of their calculations from a purely mathematical point of view [B3]. This observation suggests that in ancient Susa, mathematical skills were most likely taught in an educational setting which might reasonably be named "The Susa School of Mathematics" [HM1]. Since the Elamites were eventually absorbed into the Persian Achaemenid Empire around 539 BCE, we now turn to discuss the mathematics of the Achaemenid Empire.

During the Achaemenid Empire, mathematics was extensively used in administration, finance, and architecture. The discovery of clay tablets in Persepolis provides evidence of arithmetic and accounting systems used during this time [V2]. Achaemenid architects used precise geometric proportions in their designs. For example, the bell-shaped column bases and bull-shaped capitals at Persepolis were crafted with specific geometric ratios [VNKN]. Another astonishing achievement is the construction of Xerxes Canal, led by two Persian engineers, Artachaees and Bubares [F1].

The Parthian period is less documented in terms of mathematical advancements compared to other periods in Persian history. However, the Parthians were known for their contributions to astronomy and engineering, which often involved mathematical principles. During this period, the Parthians made significant strides in architecture and construction, requiring practical applications of geometry and arithmetic.

The status of science, including mathematics, during the Sasanian period is crucial. This era was significant for the development of science and mathematics in Persia. One of the most notable institutions was the Academy of Gondishapur, founded in the 3rd century AD. It became a major center of learning, attracting scholars from various fields, including mathematics, astronomy, medicine, and philosophy. The academy played a crucial role in preserving and advancing different branches of science and engineering, including mathematical knowledge. The Sasanians made significant contributions to astronomy and astrology, which often involved complex mathematical calculations. The "Royal Astronomical Tables" (in Persian, Zig e Shahriyaran زیگِ شهریاران) are an example of the advanced mathematical work carried out during this period. The Sasanians were also known for their impressive architectural achievements, such as the construction of the Taq Kasra. These projects required sophisticated knowledge of geometry and arithmetic. Above all, the Sasanian period served as a bridge between ancient and medieval scientific traditions. The knowledge preserved and developed during this time was later transmitted to the Islamic world, where it flourished during the "Islamic Golden Age" [F3].

Medieval Period in the History of Mathematics in Persia (Iran)

In the period often referred to as the "Islamic Golden Age" (8th to 16th centuries), scholars from various backgrounds made groundbreaking contributions to the fields of science and engineering. The translation movement, particularly during the Abbasid Caliphate, played a crucial role in the preservation and expansion of scientific knowledge. Scholars, especially Persians, translated Greek, Middle Persian (Pahlavi), Sanskrit, Syriac, and other scientific texts into Arabic, and later into Persian, thereby enriching the scientific literature of the time.

While a comprehensive exploration of Indo-Persian interactions in mathematics would require a lengthy paper, we can briefly highlight their significant influences. Indian mathematics profoundly impacted Persian mathematics, especially during the medieval period. The Indian decimal place-value system, including the concept of zero, was adopted by Persian mathematicians. This system was crucial for simplifying calculations and was later transmitted to the Islamic world and Europe.

The Persian mathematician Khwarizmi, often referred to as the "father of algebra", was heavily influenced by Indian mathematics. His works on arithmetic were based on Indian sources, playing a key role in introducing these concepts to the Islamic world.

Indian advancements in trigonometry, particularly the use of sine and cosine functions, were incorporated into Persian astronomical works. These functions were essential for astronomical calculations and were further developed by Persian scholars.

The works of Indian mathematicians like Brahmagupta and Aryabhata were translated into Arabic and Persian, influencing scholars such as Biruni and Omar Khayyam. These translations helped integrate Indian mathematical concepts into Persian scholarship [R2].

During the medieval era, particularly under the Mughal Empire, there was a significant influx of Persian immigrants. Numerous Persian scholars, artists, and craftsmen moved to India, greatly enhancing the cultural and intellectual fabric of the region. These immigrants seamlessly integrated into Indian society, making notable contributions in literature, art, governance, and science. The Persian language was adopted as the court language of the Mughal Empire, and many Persian cultural practices became widely embraced. For instance, Mohammad Sa'id Ashraf Mazandarani was a distinguished Persian scholar, poet, and calligrapher. He excelled in various fields, including astrology, calligraphy, jurisprudence, mathematics, medicine, and poetry, during his studies in Isfahan. In 1658/59, he moved to India, leveraging his exceptional talents and the connections of his mentor, Saib Tabrizi. His calligraphy teacher, Abd al-Rashid Daylami, who had previously served at the Timurid-Mughal court of Shah Jahan, also facilitated his integration into the Indian scholarly community. Upon his arrival at the Mughal court, Ashraf quickly gained recognition. Within a year, Emperor Aurangzeb personally requested him to teach his daughter, Zeb al-Nisa Makhfi.

Many original scientific works were also written in Persian and Arabic during the medieval period. Our understanding of the history of science in medieval Persia is derived from these treatises as well as archaeological evidence. However, many manuscripts and archaeological sites remain unanalyzed and unexplored. Consequently, while we have a reasonably good understanding of the history of mathematics during this period, our knowledge is still incomplete and continues to evolve as new discoveries are made.

Celebrated Persian figures from this era include Jabir ibn Hayyan, Marwazi, Mohammad Khwarizmi, Mahani, Zakariya-ye Razi, Nayrizi, Farabi, Sufi (Azophi), Buzjani, Khujandi, Quhi, Ibn Haytham, Sijzi, Karaji, Kushyar Gilani, Biruni, Avicenna, Khayyam, Sharaf al-Din Tusi, Nasir al-Din Tusi, Bi Bi Monajemeh, Qutb al-Din Shirazi, Kamal al-Din Farsi, Maraghi, Jamshid Kashi, Mohammad Baqir Yazdi, and many others. See Persian mathematicians.

However, there are misunderstandings in the history of Persian mathematics that must be resolved carefully. Firstly, some historians have incorrectly labeled Persian mathematicians as Arabs, which is not true. Recent historians have started to address and correct this misconception, ensuring that the contributions of Persian scholars are accurately recognized. Secondly, there is a misconception that the science during this period was merely a reproduction of Greek teachings. In reality, Persian scholars not only preserved Greek knowledge but also expanded upon it, making original contributions and advancements in various scientific fields. To clarify, the author wishes to illustrate these misunderstandings with specific examples.

In sum, before the author critiques the great works of two Scottish mathematicians who have worked on the history of mathematics, congratulates them on their magnificent contributions. The author also expresses his desire to seek the truth and avoid nationalism in his criticisms.

The question is why most historical Iranian (Persian) scholars are considered Arabs. For example, while Khayyam is considered a Persian poet, he is introduced as an Arab mathematician!

Different ethnic groups have lived in Persia (and modern Iran). In modern Iran, some of them are Arabs who mostly reside in the Khuzestan province, one of Iran's southern provinces. Furthermore, we should remember that in the past, Iran (the Persian Empire) was much larger, and throughout history, Iran's political borders have changed numerous times.

It is true that many of these scholars wrote their works in Arabic (the international scientific language of the time), but this should not lead us to think that they were Arab. Today, most scholars write in English. Should this cause us to consider them American, for instance?

It is wonderful that the two esteemed mathematicians have been publishing fascinating articles about the contributions of Persian mathematicians to mathematics on the Internet. However, it is surprisingly odd that they categorize the mathematics nurtured and flourished by these accomplished mathematicians as Arabic, many of whom were not even Arab!

The title of this fascinating article is "Arabic mathematics: forgotten brilliance?". They start their article with the passage: "Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly, many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth, and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks."

Again, the question arises: why Arabic/Islamic contribution?

This passage becomes more interesting when they add: "There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in [D]: ... Arabic science only reproduced the teachings received from Greek science."

At the end, when the authors describe the period they want to discuss, they write: "Before we proceed, it is worth trying to define the period that this article covers and give an overall description to cover the mathematicians who contributed. The period we cover is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century. Giving a description to cover the mathematicians who contributed, however, is much harder. The works [K] and [B2] are on "Islamic mathematics," similar to [A] which uses the title "the Muslim contribution to mathematics." Other authors try the description "Arabic mathematics," see for example [R3] and [R4]. However, certainly not all the mathematicians we wish to include were Muslims; some were Jews, some Christians, some of other faiths. Nor were all these mathematicians Arabs, but for convenience, we will call our topic "Arab mathematics."

The inconvenience shows itself when they vividly express that "the regions from which the "Arab mathematicians" came were centered on Iran/Iraq but varied with military conquest during the period. At its greatest extent, it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China."

When one refers to the biographies of the considered mathematicians, one finds that most of them were from Persia (now Iran), so why must these mathematicians be considered Arabs? Since Persians had a very great and glorious culture and civilization in pre-Islamic ages, why is there no mention of the probable influence of pre-Islamic Persian mathematics on "Persian/Arabic mathematics from the end of the eighth century to about the middle of the fifteenth century"?

Finally, despite this criticism, the author considers their works very useful for the history of mathematics. Thanks to the significant efforts of these specialists in the history of mathematics, we now know that the works of those historical mathematicians were not just copying Greek works!

Modern Period in the History of Mathematics in Persia (Iran)

As Iranian historian A Ghorbani explains in [G3], perhaps the last Persian mathematician of the medieval period with significant innovations was Mohammad Baqir Yazdi of 16th century, who worked on amicable numbers and identified the pair 9,363,584 and 9,437,056 well before Euler's contributions. After that, apparently, nothing important happened in the advancement of science in Iran until the mid-19th century.

Following a severe defeat in the Russo-Persian War (1804-1813), Abbas Mirza (1789-1833), in consultation with his minister Mirza Agha Bozorg (1754 - 1822 or 1823), initiated several reform programs in Iran. One of his key actions was dispatching Iranian students to Europe for a Western education. Abbas Mirza also established a modern school for engineering called Mohandeskhaneh in Tabriz. However, this school was not successful and terminated due to the early demise of Abbas Mirza. Later, the modernization of science and engineering was furthered by Amir Kabir, who played a pivotal role in the establishment of Dar al-Fonun [G1].

From the 19th century to the present, the establishment of modern educational institutions and societies has played a pivotal role in advancing mathematical research and education in Iran. The founding of Dar al-Fonun, a polytechnic college, in 1851 marked the beginning of modern education in Iran. This institution was crucial in advancing scientific and mathematical knowledge, laying the groundwork for future educational establishments.

For instance, the University of Tehran, founded in 1934, became a central hub for mathematical research and education. It has produced numerous prominent mathematicians and has been instrumental in fostering a culture of scientific inquiry and innovation.

Another key institution is the Sharif University of Technology, established in 1966. It has become one of Iran's leading institutions for science and engineering. The Department of Mathematical Sciences at Sharif University has been particularly influential, producing top-tier research and fostering a strong mathematical community. The university consistently ranks highly in national and international mathematics competitions, highlighting its role in nurturing mathematical talent. Additionally, Sharif University collaborates with various international research institutions, significantly contributing to global mathematical research.

Additionally, the Amirkabir University of Technology, founded in 1958, has made significant contributions to the field of mathematics and engineering. Its Department of Mathematics and Computer Science is known for its rigorous academic programs and research output.

The Institute of Mathematical Research, founded by Gholam Hossein Mossaheb in 1966, is another key institution. It has been instrumental in advancing mathematical research and education in Iran, training numerous mathematics teachers and researchers.

Shahid Beheshti University, established in 1968, has also contributed to advancing statistics, computer science, and mathematics education and research in Iran.

The Institute for Research in Fundamental Sciences (IPM), established in 1989, is another notable institution. It focuses on advanced research in various fields, including mathematics, and has become a leading center for scientific research in Iran.

The Institute for Advanced Studies in Basic Sciences (IASBS) in Zanjan, established in 1991, has also played a pivotal role in advancing modern mathematics in Iran. Its Department of Mathematics offers advanced education programs and conducts significant research, contributing to the global mathematical community.

Other Iranian universities such as Khaje Nasir Toosi University of Technology, Iran University of Science and Technology, Shahid Chamran University of Ahvaz, Shahid Bahonar University of Kerman, University of Guilan, University of Kashan, Tabriz University, Isfahan University, Ferdowsi University of Mashhad, Isfahan University of Technology, Shiraz University, Yazd University, and Babol Noshirvani University of Technology have also made significant contributions to the advancement of mathematics, computer science, and statistics in Iran.

References.

[A] A A al'Daffa, The Muslim contribution to mathematics (London, 1978).

[BA] G P Basello and E Ascalone. Cuneiform culture and science, calendars, and metrology in Elam. In: The Elamite World, pp. 697-728. London, 2018.

[B2] J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.

[B3] E M Bruins, Quelques textes mathématiques de la mission de Suse. Proceedings of the Amsterdam Academy 53, pp. 1025-1033, 1950.

[C] H Corbin, History of Islamic Philosophy, (London, 2006).

[D] P Duhem, Le système du monde (Paris, 1965).

[F1] M Farshad, History of Engineering in Iran (Tehran, 2011).

[F2] M Farshad, History of Science in Iran (Tehran, 1985).

[F3] R N Frye, The Cambridge History of Iran Vol. 4 (Cambridge, 1975).

[G1] F Ghassemlou, Rahyafti be Tarikh-e Riaziat dar Iran-e Mo'aser (Tehran, 2017).

[G3] A Ghorbani, Zendeginameh-ye Riyazidanan-e Doreh-ye Eslami [Biographie des mathématiciens de l'époque islamique] (Tenran, 1996).

[HM1] N Heydari and K Muroi. Circular Figures in Elamite Mathematics. arXiv preprint arXiv:2212.12423 (2022).

[HM2] N Heydari and K Muroi. Pythagorean Theorem in Elamite Mathematics. arXiv preprint arXiv:2305.17753 (2023).

[HM3] N Heydari and K Muroi. Quadratic Equations in Elamite Mathematics. arXiv preprint arXiv:2310.06101 (2023).

[HM4] N Heydari and K Muroi. Systems of Equations in Elamite Mathematics. arXiv preprint arXiv:2310.06994 (2023).

[HM5] N Heydari and K Muroi. The Elamite Formula for The Area of a Regular Heptagon. arXiv preprint arXiv:2209.14289 (2022).

[HM6] N Heydari and K Muroi. Volumes of Solid Objects in Elamite Mathematics. arXiv preprint arXiv:2303.13230 (2022).

[J] F Justi, Iranisches Namenbuch (Marburg, 1895).

[K] E S Kennedy et al., Studies in the Islamic Exact Sciences (1983).

[R2] K Ramasubramanian. Indian Influence on Early Arabic and Persian Writers of Mathematical Sciences. In: K Ramasubramanian (eds) Gaṇitānanda. Springer, Singapore, 2019.

[R3] R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).

[R4] R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).

[S1] G Santayana, The Life of Reason: The Phases of Human Progress (New York, 1905).

[S2] G Sarton, Introduction to the History of Science (Baltimore, 1927).

[V1] BL van der Waerden, A History of Algebra: From al-Khwarizmi to Emmy Noether (Berlin, 1985).

[V2] G Vollmers, Accounting and control in the Persepolis fortification tablets, Accounting Historians Journal: Vol. 36: Iss. 2, Article 7, 2009.

[VNKN] M Veisi, A H Nobari, S M Koohpar, and J Neyestani, An Investigation of the Geometric Proportions of Bell-Shaped Column Bases and Bull Capitals at Persepolis and in Caucasian Achaemenid Sites. Ancient Civilizations from Scythia to Siberia 20, 2, 195-211, 2014.

Keywords. Persian, Persia, Iran, Iranian, Mathematics, Mathematician.

Also, see my notes on the history of algebra, Persian language, Persian mathematicians, and Iranian modern mathematicians.

The Persian Mathematician and Father of Algebra Al-Khwarizmi

The Persian Mathematician, Astronomer, Philosopher, and Poet Omar Khayyam

Note. The picture on the top of the current page, taken from Wikipedia, is a page from the book "The Compendious Book on Calculation by Completion and Balancing" written by Al-Khwarizmi.