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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:

"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."

Nasir al-Din al-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 give a way of construing those examples from Aristotle."

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.

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.

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

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

In 1265, Tusi wrote a manuscript regarding the calculation for nth roots of an integer. Moreover, he revealed the coefficients of the expansion of a binomial to any power giving the binomial formula and the Pascal triangle relations between binomial coefficients. He also wrote a famous work on theory of color, based on mixtures of black and white, and included sections on jewels and perfumes.

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. The K.N. Toosi University of Technology in Iran and Observatory of Shamakhy in the Republic of Azerbaijan are also named after him.

Ala Al-Din Abu'l-Hasan Ali Ibn Ibrahim Ibn al-Shatir (1304 – 1375) (Arabic: ابن الشاطر‎) was an Arab Muslim astronomer, mathematician, engineer and inventor who worked as muwaqqit (موقت, religious timekeeper) at the Umayyad Mosque in Damascus, Syria.

His most important astronomical treatise was the Kitab nihayat al-sul fi tashih al-usul (The Final Quest Concerning the Rectification of Principles), in which he drastically reformed the Ptolemaic models of the Sun, Moon, and planets, by his introducing his own non - Ptolemaic models which eliminates the epicycle in the solar model, which eliminate the eccentrics and equant by introducing extra epicycles in the planetary models via the Tusi - couple, and which eliminates all eccentrics, epicycles and equant in the lunar model.

While previous Maragha school models were just as accurate as the Ptolemaic model, Ibn al-Shatir's geometrical model was the first that was actually superior to the Ptolemaic model in terms of its better agreement with empirical observations. Another achievement of Ibn al-Shatir was the rejection of the Ptolemaic model on empirical rather than philosophical grounds. Unlike previous astronomers before him, Ibn al-Shatir was not concerned with adhering to the theoretical principles of cosmology or natural philosophy (or Aristotelian physics), but rather to produce a model that was more consistent with empirical observations. His model was thus in better agreement with empirical observations than any previous models produced before him. His work thus marked a turning point in astronomy, which may be considered a "Scientific Revolution before the Renaissance".

Unlike previous astronomers, Ibn al-Shatir generally had no philosophical objections against Ptolemaic astronomy, but was only concerned with how well it matched his own empirical observations. He employed careful observations of the diameters of the Sun and Moon to test the Ptolemaic models on empirical grounds, testing "the Ptolemaic value for the apparent size of the solar disk by using lunar eclipse observations." His work on his experiments and observations, however, has not survived, but there are references to this work in his The Final Quest Concerning the Rectification of Principles.

As a consequence of his observations, he formulated his own modification of the Ptolemaic model. Ibn al-Shatir's concern with the observed solar diameter led him to replace Ptolemy's epicycle and equant solar model with a model using three spheres, a large sphere centered on the Earth which he called the parecliptic, a smaller sphere carried by the pareclptic, which he called the deferent, and an even smaller sphere carried by the deferent, which he called the director. The Sun was then carried by the director.

Although his system was firmly geocentric, he had eliminated the Ptolemaic equant and eccentrics, and the mathematical details of his system were identical to those in Nicolaus Copernicus' De revolutionibus. His lunar model was also no different from the lunar model used by Copernicus. This suggests that Ibn al-Shatir's model may have influenced Copernicus while constructing the heliocentric model. Though it remains uncertain how this may have happened, it is known that Byzantine Greek manuscripts containing the Tusi - couple which Ibn al-Shatir employed had reached Italy in the 15th century. It is also known that Copernicus' diagrams for his heliocentric model, including the markings of points, was nearly identical to the diagrams and markings used by Ibn al-Shatir for his geocentric model, making it very likely that Copernicus may have been aware of Ibn al-Shatir's work.

Y. M. Faruqi writes:

"Ibn al-Shatir’s theory of lunar motion was very similar to that attributed to Copernicus some 150 years later".

"Whereas Ibn al-Shatir’s concept of planetary motion was conceived in order to play an important role in an earth - centered planetary model, Copernicus used the same concept of motion to present his sun - centered planetary model. Thus the development of alternative models took place that permitted an empirical testing of the models."

Ibn al-Shatir constructed a magnificent sundial for the minaret of the Umayyad Mosque in Damascus which gave both seasonal and equinoctial hours. The fragments of this sundial in a Damascus museum make this the oldest polar - axis sundial still in existence.He devised a timekeeping device incorporating both a universal sundial and a magnetic compass.The compendium, a multi - purpose astronomical instrument, was first constructed by Ibn al-Shatir. His compendium featured an alhidade and polar sundial among other things. These compendia later became popular in Renaissance Europe.Ibn al-Shatir described another astronomical instrument which he called the "universal instrument" in his Rays of light on operations with the universal instrument (Al-Ashi'a al-lāmi'a fī 'l-'amal bi-'l-āla al jāmi'a). A commentary on this work entitled Book of Ripe Fruits from Clusters of Universal Instrument (Kitab al-thimār al-yāni'a ‘an qutāf al-āla al-jāmi'a) was later written by the Ottoman astronomer and engineer Taqi al-Din, who employed the instrument at the Istanbul observatory of Taqi al-Din from 1577 - 1580.