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Nasir al-Din Tusi: The Creator of Trigonometry

posted on: Apr 2, 2021

Nasir al-Din Tusi
“The only laws of matter are those that our minds must fabricate and the only laws of mind are fabricated for it by matter.” James Clerk Maxwell

By: Ahmed Abu Sultan/Arab America Contributing Writer        Muhammad ibn Muhammad ibn al-Hasan al-Tūsī, better known as Nasir al-Din Tusi was a polymath, architect, philosopher, physician, scientist, and theologian. One of the greatest scientists of Islam, he is often considered the creator of trigonometry as a mathematical discipline in its own right. He created a new world of mathematical analysis and discovery. The Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later scholars. Nasir’s work was immortalized due to its relevance and importance nowadays.


Nasir al-Din Tusi was born in the city of Tus in Khorasan in the year 1201 and began his studies at an early age. In Hamadan and Tus he studied the Quran, hadith, Ja’fari jurisprudence, logic, philosophy, mathematics, medicine, and astronomy. Fulfilling the wish of his father, the young Muhammad took learning and scholarship very seriously and traveled far and wide to attend the lectures of renowned scholars and acquire the knowledge. Islam reveres scholars comparing them to Jihadists, due to the struggle caused by traveling, working, and many other medieval dangers. At a young age, he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Attar of Nishapur, and he attended the lectures of Qutb al-Din al-Misri.

In Mosul, he studied mathematics and astronomy with Kamal al-Din Yunus. Later on, he corresponded with Sadr al-Din al-Qunawi and it seems that mysticism, as propagated by Sufi masters of his time, was not appealing to his mind and once the occasion was suitable, he composed his manual of philosophical Sufism in the form of a small booklet entitled Awsaf al-Ashraf. The Mongol invasion and the turmoil it caused in the eastern Islamic territories hardly left the life of any of its citizens untouched. The collapse of Ismaili political power and the massacre of the Ismaili population, who were considered to be a serious threat to the Mongols, left no choice for Tusi except the exhibition of some sort of affiliation to Twelver Shi‘ism, and he denounced his Ismaili allegiances.

Nasir al-Din Tusi
“I was trying to unravel the complicated trigonometry of the radical thought that silence could make up the greatest lie ever told.” Pat Conroy

In Science

During his stay in Nishapur, Tusi established a reputation as an exceptional scholar. Tusi’s prose writing, which numbers over 150 works, represents one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with both religious topics and non-religious or secular subjects. His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, and Theodosius of Bithynia. Through his work, he created a legacy that allowed the growth in many other fields of sciences. His most effective skill was adaptability. When the Mongols invaded under Hulagu Khan, he joined his court at the price of his belief. This decision was only motivated to continue his journey in scientific discovery. At the time, it was considered to be the ultimate form of sacrifice.

Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was constructed in Azarbaijan. Based on the observations in this for the time being most advanced observatory, Tusi made very accurate tables of planetary movements as depicted in his book Ilkhanic Tables. This book contains astronomical tables for calculating the positions of the planets and the names of the stars. His model for the planetary system is believed to be the most advanced of his time and was used extensively until the development of the heliocentric model in the time of Nicolaus Copernicus. Tusi is considered to be the most eminent astronomers of his time.


Tusi valued the importance of logic and its benefits to the evolution of civilization. He was a supporter of Avicennian logic as well. The usage of logic opened the path for the creation of trigonometry. Al-Tusi was the first to write a work on trigonometry independently of astronomy. Al-Tusi, in his Treatise on the Quadrilateral, gave an extensive exposition of spherical trigonometry, distinct from astronomy. It was in the works of Al-Tusi that trigonometry achieved the status of an independent branch of pure mathematics distinct from astronomy, to which it had been linked for so long.

Nasir al-Din Tusi
“Can you imagine young people nowadays making a study of trigonometry for the fun of it? Well I did.” Clyde Tombaugh

Innovation Legacy

Tusi was one of the few innovators of his time to consider the biological status of humans. In his Akhlaq-i Nasiri, Tusi wrote about several biological topics. He defended a version of Aristotle’s scala naturae, in which he placed man above animals, plants, minerals, and the elements. Tusi seems to have perceived man as belonging to the animals since he stated that we have the same instincts of perception and dexterity. However, the difference between us and animals is consciousness. It is the dividing factor that defines us as a species. In addition, scholars translated some of his work that indicated some interest in evolution. Furthermore, he eventually structured the principles that allowed for the discovery of the Law of Mass Conservation.

The ensemble of Tusi’s writings amounts to approximately 165 titles on a wide variety of subjects. Some of them are simply a page or even half a page, but the majority with few exceptions, are well prepared scholarly works on astronomy, ethics, history, jurisprudence, logic, mathematics, medicine, philosophy, theology, poetry, and the popular sciences. Tusi’s fame in his lifetime guaranteed the survival of almost all of his scholarly output. The adverse effect of his fame is also the attribution of several works that neither match his style nor have the quality of his writings. The only remaining memorials are a minor planet named Tusi and a lunar crater on the southern hemisphere of the moon. His life was empirical to much of the progress occurring today, and his work will forever be recognized.

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