Nasir al-Din al-Tusi (nonfiction)

Nasir al-Dīn Tusi at the observatory in Maragha, Persia. British Library. Date: 1562.

Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (Persian: محمد بن محمد بن حسن طوسی‎‎ 18 February 1201 – 26 June 1274), better known as Nasir al-Din Tusi (Persian: نصیر الدین طوسی‎; or simply Tusi /ˈtuːsi/ in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian. He is often considered the creator of trigonometry as a mathematical discipline in its own right. He was a Twelver Muslim. The Muslim scholar Ibn Khaldun (1332–1406) considered Tusi to be the greatest of the later Persian scholars.

Biography

Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan (northeastern Iran) 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.

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 traveled far and wide to attend the lectures of renowned scholars and acquire the knowledge, an exercise highly encouraged in his Islamic faith. 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, the legendary Sufi master who was later killed by the Mongols, and he attended the lectures of Qutb al-Din al-Misri.

In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus (d. AH 639 / AD 1242), a pupil of Sharaf al-Dīn al-Ṭūsī. Later on he corresponded with Sadr al-Din al-Qunawi, the son-in-law of Ibn 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 employed by the Nizari Ismaili state and made his most important contributions in science during this time when he was moving from one stronghold to another. He was captured after the invasion of Alamut Castle by the Mongol forces.

Works

Tusi has about 150 works, of which 25 are in Persian and the remaining are in Arabic,[23] and there is one treatise in Persian, Arabic and Turkish.

His major works include:

• 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 Nasiri – A work on ethics.

al-Risalah al-Asturlabiyah – A Treatise on the 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.
• Talkhis al-Muhassal (summary of summaries).

An example from one of his poems:

Anyone who knows, and knows that he knows,

makes the steed of intelligence leap over the vault of heaven. Anyone who does not know but knows that he does not know, can bring his lame little donkey to the destination nonetheless. Anyone who does not know, and does not know that he does not know, is stuck forever in double ignorance.

Achievements

Tusi couple

During his stay in Nishapur, Tusi established a reputation as an exceptional scholar. Tusi’s prose writing, which numbers 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"). His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, and Theodosius of Bithynia.

Astronomy

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, south of the river Aras, and to the 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[who?] to be one of the most eminent astronomers of his time. His famous student Shams ad-Din Al-Bukhari was the teacher of Byzantine scholar Gregory Choniades, who had in turn trained astronomer Manuel Bryennios about 1300 in Constantinople.

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 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. The Tusi couple was later employed in Ibn al-Shatir's geocentric model and Nicolaus Copernicus' heliocentric Copernican model. 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, only by physical principles found in natural philosophy. Tusi's criticisms of Ptolemy were similar to the arguments later used by Copernicus in 1543 to defend the Earth's rotation.

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 likened to milk in color." [33] 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.

Logic

Nasir al-Din Tusi was a supporter of Avicennian logic, and wrote the following commentary on Avicenna's theory of absolute propositions:

"What spurred him to this was that in the assertoric syllogistic Aristotle and others sometimes used contradictories of absolute propositions on the assumption that they are absolute; and that was why so many decided that absolutes did contradict absolutes. When Avicenna had shown this to be wrong, he wanted to develop a method of construing those examples from Aristotle."[35]

Mathematics

Al-Tusi was the first to write a work on trigonometry independently of astronomy.[36] Al-Tusi, in his Treatise on the Quadrilateral, gave an extensive exposition of spherical trigonometry, distinct from astronomy.[37] 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.[38]

He was the first to list the six distinct cases of a right triangle in spherical trigonometry.

This followed earlier work by Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica, and the earlier Muslim mathematicians Abū al-Wafā' al-Būzjānī and Al-Jayyani.

In his On the Sector Figure, appears the famous law of sines for plane triangles.

{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}}{\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} }

He also stated the law of sines for spherical triangles,[42][43] discovered the law of tangents for spherical triangles, and provided proofs for these laws.

Biology

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. He described "grasses which grow without sowing or cultivation, by the mere mingling of elements,"[44] as closest to minerals. Among plants, he considered the date-palm as the most highly developed, since "it only lacks one thing further to reach (the stage of) an animal: to tear itself loose from the soil and to move away in the quest for nourishment."

The lowest animals "are adjacent to the region of plants: such are those animals which propagate like grass, being incapable of mating [...], e.g. earthworms, and certain insects".[45] The animals "which reach the stage of perfection [...] are distinguished by fully developed weapons", such as antlers, horns, teeth, and claws. Tusi described these organs as adaptations to each species's lifestyle, in a way anticipating natural theology. He continued:

The noblest of the species is that one whose sagacity and perception is such that it accepts discipline and instruction: thus there accrues to it the perfection not originally created in it. Such are the schooled horse and the trained falcon. The greater this faculty grows in it, the more surpassing its rank, until a point is reached where the (mere) observation of action suffices as instruction: thus, when they see a thing, they perform the like of it by mimicry, without training [...]. This is the utmost of the animal degrees, and the first of the degrees of Man in contiguous therewith.

Thus, in this paragraph, Tusi described different types of learning, recognizing observational learning as the most advanced form, and correctly attributing it to certain animals.

Tusi seems to have perceived man as belonging to the animals, since he stated that "the Animal Soul [comprising the faculties of perception and movement ...] is restricted to individuals of the animal species", and that, by possessing a "Human Soul, [...] mankind is distinguished and particularized among other animals."

Some scholars have interpreted Tusi's biological writings as suggesting that he adhered to some kind of evolutionary theory. However, Tusi did not state explicitly that he believed species to change over time.

Chemistry

Tusi contributed to the field of chemistry, stating an early law of conservation of mass.

Influence and legacy

A 60-km diameter lunar crater located on the southern hemisphere of the moon is named after him as "Nasireddin". A minor planet 10269 Tusi discovered by Soviet astronomer Nikolai Stepanovich Chernykh in 1979 is named after him.[51][52] The K. N. Toosi University of Technology in Iran and Observatory of Shamakhy in the Republic of Azerbaijan are also named after him. In February 2013, Google celebrated his 812th birthday with a doodle, which was accessible in its websites with Arabic language calling him al-farsi (the Persian).

See also

• Shen Kuo

References

• Sharaf al-Din al-Muzaffar al-Tusi biography - MacTutor History of Mathematics
• Nasir al-Din al-Tusi at the Mathematics Genealogy Project
• "Tusi". Random House Webster's Unabridged Dictionary.
• Bennison, Amira K. (2009). The great caliphs : the golden age of the 'Abbasid Empire. New Haven: Yale University Press. p. 204. ISBN 978-0-300-15227-2. Hulegu killed the last ‘Abbasid caliph but also patronized the foundation of a new observatory at Maragha in Azerbayjan at the instigation of the Persian Shi‘i polymath Nasir al-Din Tusi.
• Goldschmidt, Arthur; Boum, Aomar (2015). A Concise History of the Middle East. Avalon Publishing. ISBN 978-0-8133-4963-3. Hulegu, contrite at the damage he had wrought, patronized the great Persian scholar, Nasiruddin Tusi (died 1274), who saved the lives of many other scientists and artists, accumulated a library of 400000 volumes, and built an astronomical ...
• Bar Hebraeus; Joosse, Nanne Pieter George (2004). A Syriac Encyclopaedia of Aristotelian Philosophy: Barhebraeus (13th C.), Butyrum Sapientiae, Books of Ethics, Economy, and Politics : a Critical Edition, with Introduction, Translation, Commentary, and Glossaries. Brill. p. 11. ISBN 978-90-04-14133-9. the Persian scholar Naṣīr al-Dīn al-Ṭūsī
• Seyyed Hossein Nasr (2006). Islamic Philosophy from Its Origin to the Present: Philosophy in the Land of Prophecy. State University of New York Press. p. 167. ISBN 978-0-7914-6800-5. In fact it was common among Persian Islamic philosophers to write few quatrains on the side often in the spirit of some of the poems of Khayyam singing about the impermanence of the world and its transience and similar themes. One needs to only recall the names of Ibn Sina, Suhrawardi, Nasir al-Din Tusi and Mulla Sadra, who wrote poems along with extensive prose works.
• Rodney Collomb, "The rise and fall of the Arab Empire and the founding of Western pre-eminence", Published by Spellmount, 2006. pg 127: "Khawaja Nasr ed-Din Tusi, the Persian, Khorasani, former chief scholar and scientist of"
• Seyyed Hossein Nasr, Islamic Philosophy from Its Origin to the Present: Philosophy in the Land of Prophecy, SUNY Press, 2006, ISBN 0-7914-6799-6. page 199
• Seyyed H. Badakhchani. Contemplation and Action: The Spiritual Autobiography of a Muslim Scholar: Nasir al-Din Tusi (In Association With the Institute of Ismaili Studies. I. B. Tauris (December 3, 1999). ISBN 1-86064-523-2. page.1: ""Nasir al-Din Abu Ja`far Muhammad b. Muhammad b. Hasan Tusi:, the renowned Persian astronomer, philosopher and theologian"
• Glick, Thomas F.; Livesey, Steven John; Wallis, Faith (2005). Medieval Science, Technology, and Medicine: An Encyclopedia. Psychology Press. ISBN 978-0-415-96930-7. drawn by the Persian cosmographer al-Tusi.
• Laet, Sigfried J. de (1994). History of Humanity: From the seventh to the sixteenth century. UNESCO. p. 908. ISBN 978-92-3-102813-7. the Persian astronomer and philosopher Nasir al-Din Tusi.
• Mirchandani, Vinnie (2010). The New Polymath: Profiles in Compound-Technology Innovations. John Wiley & Sons. p. 300. ISBN 978-0-470-76845-7. Nasir. al-Din. al-Tusi: Stay. Humble. Nasir al-Din al-Tusi, the Persian polymath, talked about humility: “Anyone who does not know and does not know that he does not know is stuck forever in double ...
• "Al-Tusi_Nasir biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2018-08-05. One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
• "the cambridge history of science".
• electricpulp.com. "ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica". www.iranicaonline.org. Retrieved 2018-08-05. His major contribution in mathematics (Nasr, 1996, pp. 208-14) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
• Ṭūsī, Naṣīr al-Dīn Muḥammad ibn Muḥammad; Badakchani, S. J. (2005), Paradise of Submission: A Medieval Treatise on Ismaili Thought, Ismaili Texts and Translations, 5, London: I.B. Tauris in association with Institute of Ismaili Studies, pp. 2–3, ISBN 1-86064-436-8
• James Winston Morris, "An Arab Machiavelli? Rhetoric, Philosophy and Politics in Ibn Khaldun’s Critique of Sufism", Harvard Middle Eastern and Islamic Review 8 (2009), pp 242–291. [1] excerpt from page 286 (footnote 39): "Ibn Khaldun’s own personal opinion is no doubt summarized in his pointed remark (Q 3: 274) that Tusi was better than any other later Iranian scholar". Original Arabic: Muqaddimat Ibn Khaldūn : dirāsah usūlīyah tārīkhīyah / li-Aḥmad Ṣubḥī Manṣūr-al-Qāhirah : Markaz Ibn Khaldūn : Dār al-Amīn, 1998. ISBN 977-19-6070-9. Excerpt from Ibn Khaldun is found in the section: الفصل الثالث و الأربعون: في أن حملة العلم في الإسلام أكثرهم العجم (On how the majority who carried knowledge forward in Islam were Persians) In this section, see the sentence where he mentions Tusi as more knowledgeable than other later Persian ('Ajam) scholars: . و أما غيره من العجم فلم نر لهم من بعد الإمام ابن الخطيب و نصير الدين الطوسي كلاما يعول على نهايته في الإصابة. فاعتير ذلك و تأمله تر عجبا في أحوال الخليقة. و الله يخلق ما بشاء لا شريك له الملك و له الحمد و هو على كل شيء قدير و حسبنا الله و نعم الوكيل و الحمد لله.
• Dabashi, Hamid. "Khwajah Nasir al-Din Tusi: The philosopher/vizier and the intellectual climate of his times". Routledge History of World Philosophies. Vol I. History of Islamic Philosophy. Seyyed Hossein Nasr and Oliver Leaman (eds.) London: Routledge. 1996. p. 529
• Siddiqi, Bakhtyar Husain. "Nasir al-Din Tusi". A History of Islamic Philosophy. Vol 1. M. M. Sharif (ed.). Wiesbaden:: Otto Harrossowitz. 1963. p. 565
• Peter Willey, The Eagle's Nest: Ismaili Castles in Iran and Syria, (I.B. Tauris, 2005), 172.
• Michael Axworthy, A History of Iran: Empire of the Mind, (Basic Books, 2008), 104.
• H. Daiber, F.J. Ragep, "Tusi" in Encyclopaedia of Islam. Edited by: P. Bearman, Th. Bianquis, C.E. Bosworth, E. van Donzel and W.P. Heinrichs. Brill, 2007. Brill Online. Quote: "Tusi's prose writings, 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 dealt with both religious ("Islamic") topics and non-religious or secular subjects ("the ancient sciences")."
• Seyyed Hossein Nasr. The Islamic Intellectual Tradition in Persia. Curson Press, 1996. See p. 208: "Nearly 150 treatises and letters by Nasir al-Din Tusi are known, of which 25 are in Persian and the rest in Arabic. There is even a treatise on geomancy which Tusi wrote in Arabic, Persian, and Turkish, demonstrating his mastery of all three languages."
• Morris Rossabi (28 November 2014). From Yuan to Modern China and Mongolia: The Writings of Morris Rossabi. BRILL. pp. 281–. ISBN 978-90-04-28529-3.
• Nasir al-Din al-Tusi at the Mathematics Genealogy Project
• Nasir al-Din al-Tusi at the Mathematics Genealogy Project
• Craig G. Fraser, 'The cosmos: a historical perspective', Greenwood Publishing Group, 2006 p.39
• George Saliba, 'Al-Qushji's Reform of the Ptolemaic Model for Mercury', Arabic Sciences and Philosophy, v.3 1993, pp.161-203
• George Saliba, 'Revisiting the Astronomical Contacts Between the World of Islam and Renaissance Europe: The Byzantine Connection', 'The occult sciences in Byzantium', 2006, p.368
• Ragep, F. Jamil (2001), "Freeing Astronomy from Philosophy: An Aspect of Islamic Influence on Science", Osiris, 16, 2nd ser.: 49–64, Bibcode:2001Osir...16...49R, doi:10.1086/649338, JSTOR 301979, at p. 60.
• F. Jamil Ragep (2001), "Tusi and Copernicus: The Earth's Motion in Context", Science in Context 14 (1-2), p. 145–163. Cambridge University Press.
• Ragep, Jamil, Nasir al-Din Tusi’s Memoir on Astronomy (al-Tadhkira fi `ilm al-hay’ a) Edition, Translation, Commentary, and Introduction. 2 vols. Sources in the History of Mathematics and Physical Sciences. New York: Springer-Verlag, 1993. pp. 129
• O'Connor, J. J.; Robertson, E. F. (November 2002). "Galileo Galilei". University of St Andrews. Retrieved 2007-01-08.
• Tony Street (July 23, 2008). "Arabic and Islamic Philosophy of Language and Logic". Stanford Encyclopedia of Philosophy. Retrieved 2008-12-05.
• "trigonometry". Encyclopædia Britannica. Retrieved 2011-04-25.
• Katz, Victor J. (1993). A History of Mathematics: An Introduction, p259. Addison Wesley. ISBN 0-673-38039-4.
• Bosworth, Clifford E.; Asimov (2003). History of civilizations of Central Asia. 4. Motilal Banarsidass. p. 190. ISBN 81-208-1596-3.
• Hayes, John R.; Badeau, John S. (1983). The genius of Arab civilization : source of Renaissance (2nd ed.). Taylor & Francis. p. 156. ISBN 0-262-08136-9.
• http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir.html,"One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth"/
• Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
• Also the 'sine law' (of geometry and trigonometry, applicable to spherical trigonometry) is attributed, among others, to Alkhujandi. (The three others are Abul Wafa Bozjani, Nasiruddin Tusi, and Abu Nasr Mansur). Razvi, Syed Abbas Hasan (1991) A history of science, technology, and culture in Central Asia, Volume 1 University of Peshawar, Peshawar, Pakistan, page 358, OCLC 26317600
• Bijli suggests that three mathematicians are in contention for the honor, Alkhujandi, Abdul-Wafa and Mansur, leaving out Nasiruddin Tusi. Bijli, Shah Muhammad and Delli, Idarah-i Adabiyāt-i (2004) Early Muslims and their contribution to science: ninth to fourteenth century Idarah-i Adabiyat-i Delli, Delhi, India, page 44, OCLC 66527483
• Nasir ad-Din Tusi (1964) The Nasirean Ethics (translator: G.M. Wickens). London: Allen & Unwin, p. 44.
• Nasir ad-Din Tusi (1964) The Nasirean Ethics (translator: G.M. Wickens). London: Allen & Unwin, p. 45.
• Nasir ad-Din Tusi (1964) The Nasirean Ethics (translator: G.M. Wickens). London: Allen & Unwin, p. 45f.
• Nasir ad-Din Tusi (1964) The Nasirean Ethics (translator: G.M. Wickens). London: Allen & Unwin, p. 42 (emphasis added).
• Alakbarli, Farid (Summer 2001). "A 13th-Century Darwin? Tusi's Views on Evolution". Azerbaijan International. 9 (2): 48–49.
• Shoja, M.M.; Tubbs, R.S. (2007). "The history of anatomy in Persia". Journal of Anatomy. 210: 359–378. doi:10.1111/j.1469-7580.2007.00711.x. PMC 2100290. PMID 17428200.
• Alakbarli, Farid (2001). "A 13th-Century Darwin? Tusi's Views on Evolution". Azerbaijan International. 9 (2 (Summer 2001)): 48–49. Retrieved 27 January 2018. While this reasoning may seem backward to today's Western mind, some of Tusi's theories did have merit. For instance, Tusi believed that a body of matter is able to change, but is not able to entirely disappear. He wrote: 'A body of matter cannot disappear completely. It only changes its form, condition, composition, color and other properties and turns into a different complex or elementary matter'.
• "2003ASPC..289..157B Page 157". Adsabs.harvard.edu. Retrieved 2013-02-27.
• 10269 tusi - Mano biblioteka - Google knygos. Books.google.com. Retrieved 2013-02-27.
• "Nasir al-Din al-Tusi's 812th Birthday". Google. Retrieved 19 February 2013.
• "In Persian نگاه عربی به خواجه نصیرالدین طوسی در گوگل". 19 February 2013. Retrieved 19 February 2013.

Further reading

• "Ṭūsī, Muḥammad Ibn Muḥammad Ibn al-Ḥasan". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. 1970–1980. ISBN 978-0-684-10114-9.
• O'Connor, John J.; Robertson, Edmund F., "Nasir al-Din Tusi", MacTutor History of Mathematics archive, University of St Andrews.
• Encyclopædia Iranica, "AḴLĀQ-E NĀṢERĪ", G.M. Wickens
• Encyclopædia Iranica, "AWṢĀF AL-AŠRĀF", G.M. Wickens
• Encyclopædia Iranica, "Nasir al-Din al-Tusi" George Saliba

External links

• Ragep, F. Jamil (2007). "Ṭūsī: Abū Jaʿfar Muḥammad ibn Muḥammad ibn al‐Ḥasan Naṣīr al‐Dīn al‐Ṭūsī". In Thomas Hockey; et al. (eds.). * The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 1153–5. ISBN 978-0-387-31022-0. (PDF version)
• Nasr, Seyyed Hossein (2008) [1970-80]. "Al-Ṭūsī, Muḥammad Ibn Muḥammad Ibn Al-Ḥasan Usually Known as Naṣir Al-Dīn". Complete Dictionary of Scientific Biography. Encyclopedia.com.
• Biography by Islamic Insights
• Biography by Islamic Philosophy Online
• Biography by The Internet Encyclopedia of Philosophy
• Kerry Magruder, History of Science Online: Islamic and Early Medieval Science, University of Oklahoma
• Islam Online.
• http://www.famousmuslims.com/NASIR%20AL-DIN%20AL-TUSI.htm
• "Nasir al-Din al-Tusi (Persian scholar) -- Encyclopædia Britannica". britannica.com. Retrieved 16 January 2014.
• The Rekhaganita. An 18th century Sanskrit translation of Nasir al-Din al-Tusi's recension of Euclid's Elements.
• Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics

Nonfiction cross-reference

• Mathematician (nonfiction)
• Mathematics (nonfiction)
• Tusi couple (nonfiction) - a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a 2-cusped hypocycloid. The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 Tahrir al-Majisti ("Commentary on the Almagest") as a solution for the latitudinal motion of the inferior planets, and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy's Almagest (nonfiction)Almagest.
• Twelver (nonfiction)

External links

• Nasir al-Din al-Tusi @ Wikipedia