Abu-Mahmud Khojandi

Abu Mahmud Hamid ibn Khidr Khojandi (known as Abu Mahmood Khojandi, Alkhujandi or al-Khujandi,  was a Central Asian astronomer and mathematician with Mongol orijin who lived in the late 10th century and helped build an observatory, near the city of Ray (near today’s Tehran), in Iran. He was born in Khujand; a bronze bust of the astronomer is present in a park in modern-day Khujand, now part of Tajikistan. The few facts about Khujandi’s life that are known come from his surviving writings as well as from comments made by Nassereddin Tusi. From Tusi’s comments it is fairly certain that Khujandi was one of the rulers of the Mongol tribe in the Khudzhand region, and thus must have come from the nobility


In Islamic astronomy, Khujandi worked under the patronage of the Buwayhid Amirs at the observatory near Ray, Iran, where he is known to have constructed the first huge mural sextant in 994 AD, intended to determine the Earth’s axial tilt (“obliquity of the ecliptic”) to high precision. He determined the axial tilt to be 23°32’19” for the year 994 AD. He noted that measurements by earlier astronomers had found higher values (Indians: 24°; Ptolemy 23° 51′) and thus discovered that the axial tilt is not constant but is in fact (currently) decreasing. His measurement of the axial tilt was however about 2 minutes too small, probably due to his heavy instrument settling over the course of the observations.[2][3]


In Islamic mathematics, he stated a special case of Fermat’s last theorem for n = 3, but his attempted proof of the theorem was incorrect. The spherical law of sines may have also been discovered by Khujandi, but it is uncertain whether he discovered it first, or whether Abu Nasr Mansur, Abul Wafa or Nasir al-Din al-Tusi discovered it first.[4][5].[1]
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Jamshīd al-Kāshī

Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī)[1] (Persian: غیاث‌الدین جمشید کاشانیGhiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician


Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as Timur. Al-Kashi lived in poverty during his childhood and the beginning years of his adulthood. The situation changed for the better when Timur died in 1405, and his son, Shah Rokh, ascended into power. Shah Rokh and his wife, Goharshad, a Persian princess, were very interested in the sciences, and they encouraged their court to study the various fields in great depth. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world’s greatest mathematicians.
Eight years after he came into power in 1409, their son, Ulugh Beg, founded an institute in Samarkand which soon became a prominent university. Students from all over the Middle East, and beyond, flocked to this academy in the capital city of Ulugh Beg’s empire. Consequently, Ulugh Beg harvested many great mathematicians and scientists of the Muslim world. In 1414, al-Kashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg, and it is said that he was the king’s favourite student. Al-Kashi was still working on his book, called “Risala al-watar wa’l-jaib” meaning “The Treatise on the Chord and Sine”, when he died in 1429. Some scholars believe that Ulugh Beg may have ordered his murder, while others say he died a natural death. The details are unclear.


Khaqani Zij

Al-Kashi produced a Zij entitled the Khaqani Zij, which was based on Nasir al-Din al-Tusi‘s earlier Zij-i Ilkhani. In his Khaqani Zij, al-Kashi thanks the Timurid sultan and mathematician-astronomer Ulugh Beg, who invited al-Kashi to work at his observatory (see Islamic astronomy) and his university (see Madrasah) which taught Islamic theology as well as Islamic science. Al-Kashi produced sine tables to four sexagesimal digits (equivalent to eight decimal places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations between coordinate systems on the celestial sphere, such as the transformation from the ecliptic coordinate system to the equatorial coordinate system.[2]

Astronomical Treatise on the size and distance of heavenly bodies

He wrote the book Sullam al-Sama on the resolution of difficulties met by predecessors in the determination of distances and sizes of heavenly bodies such as the Earth, the Moon, the Sun and the Stars.

Treatise on Astronomical Observational Instruments

In 1416, al-Kashi wrote the Treatise on Astronomical Observational Instruments, which described a variety of different instruments, including the triquetrum and armillary sphere, the equinoctial armillary and solsticial armillary of Mo’ayyeduddin Urdi, the sine and versine instrument of Urdi, the sextant of al-Khujandi, the Fakhri sextant at the Samarqand observatory, a double quadrant Azimuthaltitude instrument he invented, and a small armillary sphere incorporating an alhidade which he invented.[3]

Plate of Conjunctions

Al-Kashi invented the Plate of Conjunctions, an analog computing instrument used to determine the time of day at which planetary conjunctions will occur,[4] and for performing linear interpolation.[5]

Planetary computer

Al-Kashi also invented a mechanical planetary computer which he called the Plate of Zones, which could graphically solve a number of planetary problems, including the prediction of the true positions in longitude of the Sun and Moon,[5] and the planets in terms of elliptical orbits;[6] the latitudes of the Sun, Moon, and planets; and the ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.[7]


Law of cosines

In French, the law of cosines is named Théorème d’Al-Kashi (Theorem of Al-Kashi), as al-Kashi was the first to provide an explicit statement of the law of cosines in a form suitable for triangulation.

The Treatise on the Chord and Sine

In The Treatise on the Chord and Sine, al-Kashi computed sin 1° to nearly as much accuracy as his value for π, which was the most accurate approximation of sin 1° in his time and was not surpassed until Taqi al-Din in the 16th century. In algebra and numerical analysis, he developed an iterative method for solving cubic equations, which was not discovered in Europe until centuries later.[2]A method algebraically equivalent to Newton’s method was known to his predecessor Sharaf al-Dīn al-Tūsī. Al-Kāshī improved on this by using a form of Newton’s method to solve x^P - N = 0 to find roots of N. In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.[8]In order to determine sin 1°, al-Kashi discovered the following formula often attributed to François Viète in the 16th century:[9]
\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,\!

The Key to Arithmetic

Computation of 2π

In his numerical approximation, he correctly computed 2π (or \tau) to 9 sexagesimal digits[10] in 1424,[2] and he converted this approximation of 2π to 17 decimal places of accuracy.[11] This was far more accurate than the estimates earlier given in Greek mathematics (3 decimal places by Archimedes), Chinese mathematics (7 decimal places by Zu Chongzhi) or Indian mathematics (11 decimal places by Madhava of Sangamagrama). The accuracy of al-Kashi’s estimate was not surpassed until Ludolph van Ceulen computed 20 decimal places of π nearly 200 years later.[2] It should be noted that al-Kashi’s goal was not to compute the circle constant with as many digits as possible but to compute it so precisely that the circumference of the largest possible circle (ecliptica) could be computed with highest desirable precision (the diameter of a hair).

Decimal fractions

In discussing decimal fractions, Struik states that (p. 7):[12]
“The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).[13]

Khayyam’s triangle

In considering Pascal’s triangle, known in Persia as “Khayyam’s triangle” (named after Omar Khayyám), Struik notes that (p. 21):[12]
“The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written by Yang Hui, one of the mathematicians of the Sung dynasty in China.[14] The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in his Key to arithmetic of c. 1425.[15] Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of the Renaissance mathematicians, and we see Pascal‘s triangle on the title page of Peter Apian‘s German arithmetic of 1527. After this we find the triangle and the properties of binomial coefficients in several other authors.[16]

Biographical film

In 2009 IRIB produced and broadcast (through Channel 1 of IRIB) a biographical-historical film series on the life and times of Jamshid Al-Kāshi, with the title The Ladder of the Sky [17][18] (Nardebām-e Āsmān [19]). The series, which consists of 15 parts of each 45 minutes duration, is directed by Mohammad-Hossein Latifi and produced by Mohsen Ali-Akbari. In this production, the role of the adult Jamshid Al-Kāshi is played by Vahid Jalilvand.[20][21][22]
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Shams al-Dīn al-Samarqandī

Shams al-Dīn Muḥammad ibn Ashraf al-Ḥusaynī al-Samarqandī (c. 1250 – c. 1310) was a 13th-century astronomer and mathematician from Samarkand. Nothing is known of al-Samarqandi’s life except that he composed his most important works around 1276. He wrote works on theology, logic, philosophy, mathematics and astronomy which have proved important in their own right and also in giving information about the works of other scientists of his period.
Al-Samarqandi wrote a work Risala fi adab al-bahth which discussed the method of intellectual investigation of reasoning using dialectic. Such methods of enquiry were much used by the ancient Greeks. He also wrote Synopsis of astronomy and produced a star catalogue for the year 1276-77. In mathematics al-Samarqandi is famous for a short work of only 20 pages which discusses 35 of Euclid’s propositions. Although a short work, al-Samarqandi consulted widely the works of other Muslim mathematicians before writing it. For example he refers to writings by Ibn al-Haytham, Omar Khayyam, al-Jawhari, Nasir al-Din al-Tusi, and Athīr al-Dīn al-Abharī.
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Qutb al-Din al-Shirazi

Qutb al-Din al-Shirazi (1236 – 1311) was a 13th-century Persian polymath[1] and poet who made contributions to astronomy, mathematics, medicine, physics, music theory, philosophy and Sufism.


He was born in Kazerun in October 1236 to a family with a tradition of Sufism. His father, Zia’ al-Din Mas’ud Kazeruni was a physician by profession and also a leading Sufi of the Kazeruni order. Zia’ Al-Din received his Kherqa (Sufi robe) from Shahab al-Din Omar Suhrawardi[citation needed]. Qutb al-Din was garbed by the Kherqa (Sufi robe) as blessing by his father at age of ten[citation needed]. Later on, he also received his own robe from the hands of Najib al-Din Bozgush Shirazni, a famous Sufi of the time[citation needed]. Quṭb al-Din began studying medicine under his father. His father practiced and taught medicine at the Mozaffari hospital in Shiraz. After the passing away of his father (when Qutb al-Din was 14), his uncle and other masters of the period trained him in medicine. He also studied the Qanun (the Canon) of the famous Persian scholar Avicenna and its commentaries. In particular he read the commentary of Fakhr al-Din Razi on the Canon of Medicine and Qutb al-Din raised many issues of his own. This led to his own decision to write his own commentary, where he resolved many of the issues in the company of Nasir al-Din al-Tusi.
Qutb al-Din lost his father at the age of fourteen and replaced him as the ophthalmologist at the Mozaffari hospital in Shiraz. At the same time, he pursued his education under his uncle Kamal al-Din Abu’l Khayr and then Sharaf al-Din Zaki Bushkani, and Shams al-Din Mohammad Kishi. All three were expert teachers of the Canon of Avicenna. He quit his medical profession ten years later and began to devote his time to further education under the guidance of Nasir al-Din al-Tusi. When Nasir al-Din al-Tusi, the renowned scholar-vizier of the Mongol Holagu Khan established the observatory of Maragha, Qutb al-Din Shirazi became attracted to the city. He left Shiraz sometime after 1260 and was in Maragha about 1262. In Maragha, Qutb al-din resumed his education under Nasir al-Din al-Tusi, with whom he studied the al-Esharat wa’l-Tanbihat of Avicenna. He discussed the difficulties he had with Nasir al-Din al-Tusi on understanding the first book of the Canon of Avicenna. While working in the new observatory, studied astronomy under him. One of the important scientific projects was the completion of the new astronomical table (zij). In his testament (Wasiya), Nasir al-Din al-Tusi advises his son ṣil-a-Din to work with Qutb al-Din in the completion of the Zij.
Qutb-al-Din’s stay in Maragha was short.
Subsequently, he traveled to Khorasan in the company of Nasir al-Din al-Tusi where he stayed to study under Najm al-Din Katebi Qazvini in the town of Jovayn and become his assistant. Some time after 1268, he journeyed to Qazvin, Isfahan, Baghdad and later Konya in Anatolia. This was a time when the Persian poet Jalal al-Din Muhammad Balkhi (Rumi) was gaining fame there and it is reported that Qutb al-Din also met him. In Konya, he studied the Jam’e al-Osul of Ibn Al-Athir with Sadr al-Din Qunawi. The governor of Konya, Mo’in al-Din Parvana appointed him as the judge of Sivas and Malatya. It was during this time that he compiled the books the Meftāḥ al-meftāh, Ekhtiārāt al-moẓaffariya, and his commentary on Sakkāki. In the year 1282, he was envoy on behalf of the Ilkhanid Ahmad Takudar to Sayf al-Din Qalawun, the Mamluk ruler of Egypt. In his letter to Qalawun, the Ilkhanid ruler mentions Qutb al-Din as the chief judge. Qutb al-Din during this time collected various critiques and commentaries on Avicenna‘s Canon and used them on his commentary on the Kolliyāt. The last part of Qutb al-Din’s active career was teaching the Canon of Avicenna and the Shefa of Avicenna in Syria. He soon left for Tabriz after that and died shortly after. He was buried in the Čarandāb cemetery of the city.
Shirazi identified observations by the scholar Avicenna in the 11th century and Ibn Bajjah in the 12th century as transits of Venus and Mercury.[2] However, Ibn Bajjah cannot have observed a transit of Venus, as none occurred in his lifetime.[3]Qutb al-Din had an insatiable desire[1] for learning, which is evidenced by the twenty-four years he spent studying with masters of the time in order to write his commentary on the Kolliyāt. He was also distinguished by his extensive breadth of knowledge, a clever sense of humor and indiscriminate generosity.[1] He was also a master chess player and played the musical instrument known as the Rabab, a favorite instrument of the Persian poet Rumi.



  • Tarjoma-ye Taḥrir-e Oqlides a work on geometry in Persian in fifteen chapters containing mainly the translation of the work Nasir al-Din Tusi, completed in November 1282 and dedicated to Moʿin-al-Din Solaymān Parvāna.
  • Risala fi Harkat al-Daraja” a work on Mathematics

Astronomy and Geography

  • Eḵtiārāt-e moẓaffari It is a treatise on astronomy in Persian in four chapters and extracted from his other work Nehāyat al-edrāk. The work was dedicated to Mozaffar-al-Din Bulaq Arsalan.
  • Fi ḥarakāt al-dahraja wa’l-nesba bayn al-mostawi wa’l-monḥani a written as an appendix to Nehāyat al-edrāk
  • Nehāyat al-edrākThe Limit of Accomplishment concerning Knowledge of the Heavens (Nehāyat al-edrāk fi dirayat al-aflak) completed in 1281, and The Royal Present (Al-Tuhfat al-Shahiya) completed in 1284. Both presented his models for planetary motion, improving on Ptolemy‘s principles.[4] In his The Limit of Accomplishment concerning Knowledge of the Heavens, he also discussed the possibility of heliocentrism.[5]
  • Ketāb faʿalta wa lā talom fi’l-hayʾa, an Arabic work on astronomy, written for Aṣil-al-Din, son of Nasir al-Din Tusi
  • Šarḥ Taḏkera naṣiriya on astronomy.
  • Al-Tuḥfa al-šāhiya fi’l-hayʾa, an Arabic book on astronomy, having four chapters, written for Moḥammad b. Ṣadr-al-Saʿid, known as Tāj-al-Eslām Amiršāh
  • *Ḥall moškelāt al-Majesṭi a book on astronomy, titled Ḥall moškelāt al-Majesṭi


  • Dorrat al-tāj fi ḡorrat al-dabbāj Qutb al-Din al-Shirazi’s most famous work is the Pearly Crown (Durrat al-taj li-ghurratt al-Dubaj), written in Persian around AD 1306 (705 AH). It is an Encyclopedic work on philosophy written for Rostam Dabbaj, the ruler of the Iranian land of Gilan. It includes philosophical outlook on natural sciences, theology, logic, public affairs, ethnics, mystiicsm, astronomy, mathematics, arithmetics and music.
  • Šarḥ Ḥekmat al-ešrāq Šayḵ Šehāb-al-Din Sohravardi, on philosophy and mysticism of Shahab al-Din Suhrawardi and his philosophy of illumination in Arabic.


  • Al-Tuḥfat al-saʿdiyah also called Nuzhat al-ḥukamāʾ wa rawżat al-aṭibbāʾ, on medicine, a comprehensive commentary in five volumes on the Kolliyāt of the Canon of Avicenna written in Arabic.
  • Risāla fi’l-baraṣ, a medical treatise on leprosy in Arabic
  • Risāla fi bayān al-ḥājat ila’l-ṭibb wa ādāb al-aṭibbāʾ wa waṣāyā-hum

Religion, Sufism, Theology, Law, Linguistics and Rhetoric and others

  • Al-Enteṣāf a gloss in Arabic on Zamakhshari‘s Qurʾan commentary, al-Kaššāf.
  • Fatḥ al-mannān fi tafsir al-Qorʾān a comprehensive commentary on the Qurʾan in forty volumes, written in Arabic and also known by the title Tafsir ʿallāmi
  • Ḥāšia bar Ḥekmat al-ʿayn on theology; it is a commentary of Ḥekmat al-ʿayn of Najm-al-Din ʿAli Dabirān Kātebi
  • Moškelāt al-eʿrāb on Arabic syntax
  • Moškelāt al-tafāsir or Moškelāt al-Qorʾān, on rhetoric
  • Meftāḥ al-meftāhá, a commentary on the third section of the Meftāḥ al-ʿolum, a book on Arabic grammar and rhetoric by Abu Yaʿqub Seraj-al-Din Yusof Skkaki Khwarizmi
  • Šarḥ Moḵtaṣar al-oṣul Ebn Ḥājeb, a commentary on Ebn Ḥājeb’s Montaha’l-soʾāl wa’l-ʿamal fi ʿelmay al-oṣul wa’l-jadwal, a book on the sources of law according to the Malikite school of thought
  • Sazāvār-e Efteḵā, Moḥammad-ʿAli Modarres attributes a book by this title to Quṭb-al-Din, without providing any information about its content
  • Tāj al-ʿolum A book attributed to him by Zerekli
  • al-Tabṣera A book attributed to him by Zerekli
  • A book on ethnics and poetry, Quṭb-al-Din is also credited with the authorship of a book on ethics in Persian, written for Malek ʿEzz-al-Din, the ruler of Shiraz. He also wrote poetry but apparently did not leave a divan (a book of poems)
Qutb al-Din was also a Sufi from a family of Sufis in Shiraz. He is famous for the commentary on Hikmat al-ishraq of Suhrawardi, the most influential work of Islamic Illuminist philosophy.
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Muhammad ibn Hasan Tūsī

Khawaja Muhammad ibn Muhammad ibn Hasan Tūsī  (born 18 February 1201 in Ṭūs, Khorasan – died on 26 June 1274 in al-Kāżimiyyah district of metropolitan Baghdad), better known as Nasīr al-Dīn Tūsī (Persian: نصیر الدین طوسی‎; or simply Tusi in the West), was a Persian[1][2][3][4] polymath and prolific writer: an architect, astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed.[citation needed] He was of the Ismaili-, and subsequently Twelver Shī‘ah Islamic belief.[5] The Muslim scholar Ibn Khaldun (1332–1406) considered Tusi to be the greatest of the later Persian scholars


Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan (in north-eastern Iran) in the year 1201 and began his studies at an early age. In Hamadan and Tus he studied the Qur’an, Hadith, Shi’a jurisprudence, logic, philosophy, mathematics, medicine and astronomy.[7]
He was apparently born into a Shī‘ah family and lost his father at a young age. Fulfilling the wish of his father, the young Muhammad took learning and scholarship very seriously and travelled far and wide to attend the lectures of renowned scholars and acquire the knowledge which guides people to the happiness of the next world. At a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib.[8] He met also Farid al-Din ‘Attar, the legendary Sufi master who was later killed by Mongol invaders and attended the lectures of Qutb al-Din al-Misri.
In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus (d. 639/1242). Later on he corresponded with Sadr al-Din al-Qunawi, the son-in-law of Ibn al-‘Arabi, 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 own manual of philosophical Sufism in the form of a small booklet entitled Awsaf al-Ashraf “The Attributes of the Illustrious”.
As the armies of Genghis Khan swept his homeland, he was captured by the Ismailis and made his most important contributions in science during this time when he was moving from one stronghold to another. He finally joined Hulagu Khan‘s ranks, after the invasion of the Alamut castle by the Mongol forces.


Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry.
  • Al-Tadhkirah fi’ilm al-hay’ah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah (A Commentary on al-Tadhkirah) – Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi and by Nazzam Nishapuri.
  • Akhlaq-i-Nasri – A work on ethics.
  • al-Risalah al-Asturlabiyah – A Treatise on astrolabe.
  • Zij-i ilkhani (Ilkhanic Tables) – A major astronomical treatise, completed in 1272.
  • sharh al-isharat (Commentary on Avicenna’s Isharat)
  • Awsaf al-Ashraf a short mystical-ethical work in Persian
  • Tajrīd al-iʿtiqād (Summation of Belief) – A commentary on Shia doctrines.


During his stay in Nishapur, Tusi established a reputation as an exceptional scholar. “Tusi’s prose writing, which number over 150 works, represent one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with both religious (“Islamic”) topics and non-religious or secular subjects (“the ancient sciences”).[9] His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, and Theodosius of Bithynia.[9]


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, west of Maragheh, the capital of the Ilkhanate Empire.
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 Zij-i ilkhani (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. Between Ptolemy and Copernicus, he is considered by many to be one of the most eminent astronomers of his time.
For his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions. He used this technique to replace Ptolemy‘s problematic equant[10] for many planets, but was unable to find a solution to Mercury, which was solved later by Ibn al-Shatir as well as Ali Qushji.[11] The Tusi couple was later employed in Ibn al-Shatir‘s geocentric model and Nicolaus Copernicusheliocentric Copernican model.[12] He also calculated the value for the annual precession of the equinoxes and contributed to the construction and usage of some astronomical instruments including the astrolabe.
Ṭūsī criticized Ptolemy’s use of observational evidence to show that the Earth was at rest, noting that such proofs were not decisive. Although it doesn’t mean that he was a supporter of mobility of the earth, as he and his 16th-century commentator al-Bīrjandī, maintained that the earth’s immobility could be demonstrated, but only by physical principles found in natural philosophy.[13] Tusi’s criticisms of Ptolemy were similar to the arguments later used by Copernicus in 1543 to defend the Earth’s rotation.[14]
About the real essence of the Milky Way, Ṭūsī in his Tadhkira writes: “The Milky Way, i.e. the galaxy, is made up of a very large number of small, tightly-clustered stars, which, on account of their concentration and smallness, seem to be cloudy patches. because of this, it was likend to milk in color.” [15] Three centuries later the proof of the Milky Way consisting of many stars came in 1610 when Galileo Galilei used a telescope to study the Milky Way and discovered that it is really composed of a huge number of faint stars.[16]


In his Akhlaq-i-Nasri, Tusi put forward a basic theory for the evolution of species. He begins his theory of evolution with the universe once consisting of equal and similar elements. According to Tusi, internal contradictions began appearing, and as a result, some substances began developing faster and differently from other substances. He then explains how the elements evolved into minerals, then plants, then animals, and then humans. Tusi then goes on to explain how hereditary variability was an important factor for biological evolution of living things:[17]
“The organisms that can gain the new features faster are more variable. As a result, they gain advantages over other creatures. […] The bodies are changing as a result of the internal and external interactions.”
Tusi discusses how organisms are able to adapt to their environments:[17]
“Look at the world of animals and birds. They have all that is necessary for defense, protection and daily life, including strengths, courage and appropriate tools [organs] […] Some of these organs are real weapons, […] For example, horns-spear, teeth and claws-knife and needle, feet and hoofs-cudgel. The thorns and needles of some animals are similar to arrows. […] Animals that have no other means of defense (as the gazelle and fox) protect themselves with the help of flight and cunning. […] Some of them, for example, bees, ants and some bird species, have united in communities in order to protect themselves and help each other.”
Tusi recognized three types of living things: plants, animals, and humans. He wrote:[17]
“Animals are higher than plants, because they are able to move consciously, go after food, find and eat useful things. […] There are many differences between the animal and plant species, […] First of all, the animal kingdom is more complicated. Besides, reason is the most beneficial feature of animals. Owing to reason, they can learn new things and adopt new, non-inherent abilities. For example, the trained horse or hunting falcon is at a higher point of development in the animal world. The first steps of human perfection begin from here.”
Tusi then explains how humans evolved from advanced animals:[17]
“Such humans [probably anthropoid apes] live in the Western Sudan and other distant corners of the world. They are close to animals by their habits, deeds and behavior. […] The human has features that distinguish him from other creatures, but he has other features that unite him with the animal world, vegetable kingdom or even with the inanimate bodies. […] Before [the creation of humans], all differences between organisms were of the natural origin. The next step will be associated with spiritual perfection, will, observation and knowledge. […] All these facts prove that the human being is placed on the middle step of the evolutionary stairway. According to his inherent nature, the human is related to the lower beings, and only with the help of his will can he reach the higher development level.”

Chemistry and Physics

In chemistry and physics, Tusi stated a version of the law of conservation of mass. He wrote that a body of matter is able to change, but is not able to disappear:[17]
“A body of matter cannot disappear completely. It only changes its form, condition, composition, colour and other properties and turns into a different complex or elementary matter.”.[6]
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Mu’ayyad al-Din al-‘Urdi

Mu’ayyad al-Din al-’Urdi (died 1266) was an Muslim astronomer, mathematician, architect and engineer working at the Maragheh observatory. He was born in Aleppo,todays Syria, and later moved to Maragheh, Azarbaijan, to work at the Maragha observatory under the guidance of Nasir al-Din Tusi.[1]He is known for being the first of the Maragha astronomers to develop a non-Ptolemaic model of planetary motion.[2] In particular, the Urdi lemma he developed was later used in the geocentric model of Ibn al-Shatir in the 14th century and in the heliocentric Copernican model of Nicolaus Copernicus in the 16th century. As an architect and engineer, he was responsible for constructing the water supply installations of Damascus, Syria, in his time

The Urdi Lemma

“Urdi’s lemma” was an extension of Apollonius’ theorem that allowed an equant in an astronomic model to be replaced with an equivalent epicycle that moved around a deferent centered at half the distance to the equant point. Anythony Grafton’s demonstration of Maestlin’s proof to Kepler may help to visualize. You can also drag the al-Shatir and al-Tusi sliders to zero in Dennis Duke’s animation to see al-‘Urdi’s equant-less model for Mars in operation.


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Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (died 1213/4)[1] was a Persian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).


Tusi was born in Tus, Iran. He taught various mathematical topics including the science of numbers, astronomical tables and astrology, in Aleppo and Mosul. His best pupil was Kamal al-Din ibn Yunus. In turn Kamal al-Din ibn Yunus went on to teach Nasir al-Din al-Tusi, one of the most famous of all the Islamic scholars of the period. By this time Tusi seems to have acquired an outstanding reputation as a teacher of mathematics, for some travelled long distances hoping to become his students.
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