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The Tusi couple is 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.
Some modern commentators also call the Tusi couple a "rolling device" and describe it as a small circle rolling inside a large fixed circle. However, Tusi himself described it differently:
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,^{[3]} and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy's Almagest.^{[4]}^{[5]}
Nasir al-Din al-Tusi, born in the town of Tus in 1201, is acknowledged throughout the Islamic world as one of the 'Great Wisdoms'. Tusi was the first astronomer to attempt a solution which would provide for latitudinal motion without introducing a longitudinal component. To do so, he proposed in a work called Tahrlr al-Majisti, which was completed in 1247, that the oscillatory motion be produced by the combined uniform circular motions of two identical circles, one riding on the circumference of the other. At that point, Tusi simply states that if one of these circles were to move at a uniform speed equal to twice the speed of the other, and in a direction to it, then any point on the circumference of the first circle would oscillate in a straight line along one of the diameters of the second circle ^{[6]}
The term "Tusi couple" is a modern one, coined by Edward Kennedy in 1966.^{[7]} It is one of several late Islamic astronomical devices bearing a striking similarity to models in Nicolaus Copernicus's De revolutionibus, including his Mercury model and his theory of trepidation. Historians suspect that Copernicus or another European author had access to an Arabic astronomical text, but an exact chain of transmission has not yet been identified,^{[8]} although the 16th century scientist and traveler Guillaume Postel has been suggested.^{[9]}^{[10]}
Since the Tusi-couple was used by Copernicus in his reformulation of mathematical astronomy, there is a growing consensus that he became aware of this idea in some way. It has been suggested^{[11]}^{[12]} that the idea of the Tusi couple may have arrived in Europe leaving few manuscript traces, since it could have occurred without the translation of any Arabic text into Latin. One possible route of transmission may have been through Byzantine science, which translated some of al-Tusi's works from Arabic into Byzantine Greek. Several Byzantine Greek manuscripts containing the Tusi-couple are still extant in Italy.^{[13]}
There are other sources for this mathematical model for converting circular motions to reciprocating linear motion. It is found in Proclus's Commentary on the First Book of Euclid^{[14]} and the concept was known in Paris by the middle of the 14th Century. In his questiones on the Sphere (written before 1362), Nicole Oresme described how to combine circular motions to produce a reciprocating linear motion of a planet along the radius of its epicycle. Oresme's description is unclear and it is not certain whether this represents an independent invention or an attempt to come to grips with a poorly understood Arabic text.^{[15]}
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