# Ancient Persian Mathematics

### A Note on the History of Ancient Persian Mathematics

In memory of "Maryam Mirzakhani"

Introduction. In my opinion, nothing captures the importance of history better than the following quote by Edmund Burke:

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

Furthermore, the history of mathematics gives us a better understanding of why mathematicians worked on their ideas over the centuries. In this direction, I use historical points in my teaching to make my classes and lessons more useful and meaningful. Here is my note on the history of ancient Persian mathematics:

### Persian or Arabic Mathematics?

In sum, before I criticize the great works of two Scottish mathematicians who have worked on the history of mathematics, let me congratulate them on their magnificent works. I also express my desire to seek reality and to avoid nationalism in my criticisms.

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

Different ethnic groups live in Iran, and some of them are Arabs who mostly live in the Khuzestan province, one of the Iranian southern provinces. Furthermore, we should remember that in the past Iran (the Persian Empire) was much larger and, during history and after various divisions, Iran's political borders have changed numerous times.

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

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

The title of this fascinating article is "Arabic mathematics: forgotten brilliance?". They start their article with this passage that "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?

And this passage becomes more interesting when they add that "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."

And at the end when the authors want to describe the period that 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 [B] 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 [R1] and [R2]. 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 was centred 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."

And when one refers to the biographies of the considered mathematicians, one finds that most of them have been from Persia (now Iran), so why those mathematicians must be considered as Arabs. And since Persians have had a very great and glorious culture and civilization in pre-Islamic ages, then why there is no mention to 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, I consider their works very useful for the history of mathematics, since thanks to the significant efforts of these specialists in the history of mathematics, we now know that the works of those historical mathematicians was not just copying Greek works!

### Persian mathematicians

Appendix. In the following, I list some world-famous Persian mathematicians (in Persian ریاضیدانان ایرانی):

Marwazi (766 – d. after 869): Marwazi مروزی was the first to describe the trigonometric ratios sine, cosine, tangent, and cotangent.

Khwarizmi (c. 780 - c. 850): Khwarizmi خوارزمی presented the first systematic solution to linear and quadratic equations. He is considered the father of algebra. Khwarizmi (in Arabic Al-Khwarizmi الخوارزمی earlier transliterated as Algoritmi) is the eponym of algorithm.

Banu Musa (9th century): Banu Musa بنو موسی (meaning sons of Moses) were three brothers from Khorasan. They were the sons of Musa ibn Shakir موسی ابن شاکر who was most probably an astronomer. The Banu Musa brothers collaborated on different projects in such a way that their contributions are often hard to distinguish. However, we know that the eldest brother Mohammad (c. 800 - after 873) was excellent in astronomy and geometry, Ahmad (c. 805 - after 873) worked mainly on mechanics, and Hasan (c. 810 - after 873) was a specialist of geometry.

Mahani (c. 820 - c. 880): Mahani ماهانی was a Persian mathematician. Mahani's works on mathematics covered the topics of geometry, arithmetic, and algebra. He was the eponym of the Mahani's equation in Persian mathematics.

Abu Hanifa Dinawari (c. 828 - c. 896): Dinawari دينوری was a Persian astronomer, agriculturist, botanist, metallurgist, geographer, mathematician, and historian.

Nayrizi (c. 865 - 922): Nayrizi نیریزی was an astronomer and a mathematician. He gave a proof of the Pythagorean theorem using the Pythagorean tiling.

Khazin (900-971): Abu Jafar Khazin ابو جعفر خازن worked on astronomy and number theory.

Buzjani (940-998): Buzjani بوزجانی made significant innovations in spherical trigonometry, and his work on arithmetic for businessmen contains the first instance of using negative numbers in a medieval text.

Abu-Mahmud Khojandi (c. 940 - c. 1000): Khojandi خجندی stated a special case of Fermat's Last Theorem for n = 3, though his attempted proof of the theorem was incorrect.

Abu Sahl Bijan al-Quhi (c. 940 - c. 1000): al-Quhi also known as Kuhi کوهی is considered one of the greatest geometers.

Sijzi (c. 945 - c. 1020): Sijzi سجزی studied intersections of conic sections and circles and proposed that the Earth rotates around its axis in the 10th century.

Karaji (953-1029): Karaji کرجی used mathematical induction to prove the binomial theorem. According to some historians, Karaji was the first to use mathematical induction to prove a statement in mathematics.

Kushyar Gilani (c. 971 - c. 1029): Gilani گیلانی was a Persian mathematician, geographer, and astronomer who contributed to trigonometry. The mathematician Ali ibn Ahmad al-Nasawi علی بن احمد نسوی (c. 1011 - c. 1075) was his student.

Biruni (973-1048): Biruni بیرونی was a Persian mathematician. He has been called variously the "founder of Indology", "father of comparative religion", "father of modern geodesy", and the "first anthropologist".

Avicenna (980-1037): Ibn Sina ابن سینا (commonly known in the West as Avicenna) is regarded as one of the most significant physicians. He was also an astronomer, a logician, philosopher, and mathematician.

Omar Khayyam (1048-1131): Known for his contributions to mathematics, astronomy, philosophy, and Persian poetry, Khayyam خیام is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics. Using Khayyam-Saccheri quadrilateral, Khayyam also contributed to the understanding of the parallel axiom.

Sharaf al-Din al-Tusi (c. 1135 - c. 1213): Sharaf al-Din Tusi شرف الدین توسی used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in geometry and the mathematical sciences, having no equal in his time".

Afzal al-Din Kashani (? - c. 1214): Afzal al-Din Kashani افضلالدین کاشانی (also known as Baba Afzal بابا افضل) was a logician, philosopher, and poet.

Nasir al-Din al-Tusi (1201-1274): Tusi توسی was a Persian polymath who had about 150 works and vastly contributed to architecture, astronomy, logic, mathematics, color theory, biology, chemistry, and philosophy. For example, the Tusi couple was first proposed by Tusi in his 1247 Tahrir al-Majisti (Commentary on the Almagest) as a solution for the latitudinal motion of the inferior planets.

Qutb al-Din al-Shirazi (1236-1311): Shirazi شیرازی made contributions to astronomy, mathematics, medicine, physics, music theory, philosophy, and Sufism.

Farisi (1267-1319): Kamal al-Din Farisi کمال الدین فارسی worked on number theory (amicable numbers) and contributed to optics extensively.

Jamshid al-Kashi (c. 1380 - 1429): Kashi کاشی was an astronomer and a mathematician. Among his works, one can mention the law of cosines (in France, still referred to as the Théorème d'Al-Kashi), the estimation of 2π to 16 decimal places of accuracy, and the most accurate approximation of sin 1° in his time.

Ulugh Beg (1394-1449): Ulugh Beg اُلُغ بیگ worked on trigonometry and spherical geometry.

Muhammad Baqir Yazdi: Yazdi یزدی lived in the 16th century. He gave the pair of amicable numbers 9,363,584 and 9,437,056 many years before Euler's contribution to amicable numbers.

Mohsen Hachtroudi (1908-1976): A student of Élie Joseph Cartan (1869-1951), Hachtroudi هشترودی contributed to differential geometry and was the inventor of the Hachtroudi curvature.

Maryam Mirzakhani (1977-2017): Maryam Mirzakhani مریم میرزاخانی was an Iranian mathematician. Mirzakhani's research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics. She became the first woman and the first Iranian to be honored with the award.

References.

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

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

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

[G] A Ghorbani, Riyazidanan-e Irani (Tehran, 1971).

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

[R1] R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).

[R2] R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).

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

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

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.