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.
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.
Original description 
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:
- if two coplanar circles, the diameter of one of which is equal to half the diameter of the other, are taken to be internally tangent at a point, and if a point is taken on the smaller circle—and let it be at the point of tangency—and if the two circles move with simple motions in opposite direction in such a way that the motion of the smaller [circle] is twice that of the larger so the smaller completes two rotations for each rotation of the larger, then that point will be seen to move on the diameter of the larger circle that initially passes through the point of tangency, oscillating between the endpoints. 
Other sources 
The term "Tusi couple" is a modern one, coined by Edward Kennedy in 1966. It is one of several late Islamic astronomical devices bearing a striking similarity to models in De revolutionibus orbium coelestium, including his Mercury model and his theory of trepidation. Historians suspect that Copernicus or another European author had access to an Arab astronomical text, but an exact chain of transmission has not yet been identified, although the 16th century scientist and traveler Guillaume Postel has been suggested.
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 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. 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.
Since the Tusi-couple was used by Nicolaus 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 both by a historian of Medieval European astronomy and by a historian of Arabic astronomy 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.
- George Saliba (1995), 'A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam', pp.152-155
- "Late Medieval Planetary Theory", E. S. Kennedy, Isis 57, #3 (Autumn 1966), 365-378, JSTOR 228366.
- Craig G. Fraser, 'The cosmos: a historical perspective', Greenwood Publishing Group, 2006 p.39
- Vatican Library, Vat. ar. 319 fol. 28 verso math19 NS.15, fourteenth-century copy of a manuscript from Tusi
- Translated in F. J. Ragep, Memoir on Astronomy II.11 , pp. 194, 196.
- E. S. Kennedy, "Late Medieval Planetary Theory," p. 370.
- E. S. Kennedy, "Late Medieval Planetary Theory," p. 377.
- Saliba, George (1996), "Writing the History of Arabic Astronomy: Problems and Differing Perspectives", Journal of the American Oriental Society 116 (4): 709–18, JSTOR 605441, pp. 716-17.
- Whose Science is Arabic Science in Renaissance Europe? by George Saliba, Columbia University
- I. N. Veselovsky, "Copernicus and Nasir al-Din al-Tusi", Journal for the History of Astronomy, 4 (1973): 128-30
- Claudia Kren, "The Rolling Device," pp. 490-2.
- Claudia Kren, "The Rolling Device," p. 497.
- George Saliba, "Whose Science is Arabic Science in Renaissance Europe?" 
- George Saliba (April 27, 2006). "Islamic Science and the Making of Renaissance Europe". Retrieved 2008-03-01.
- Di Bono, Mario. "Copernicus, Amico, Fracastoro and Tusi's Device: Observations on the Use and Transmission of a Model." Journal for the History of Astronomy 26 (1995):133-154.
- Kennedy, E. S. "Late Medieval Planetary Theory." Isis 57 (1966):365-378.
- Kren, Claudia. "The Rolling Device of Naṣir al-Dīn al-Ṭūsī in the De spera of Nicole Oresme." Isis 62 (1971): 490-498.
- Ragep, F. J. "The Two Versions of the Tusi Couple," in From Deferent to Equant: A Volume of Studies in the History of Science in Ancient and Medieval Near East in Honor of E. S. Kennedy, ed. David King and George Saliba, Annals of the New York Academy of Sciences, 500. New York Academy of Sciences, 1987. ISBN 0-89766-396-9 (pbk.)
- Ragep, F. J. Nasir al-Din al-Tusi's "Memoir on Astronomy," Sources in the History of Mathematics and Physical Sciences,12. 2 vols. Berlin/New York: Springer, 1993. ISBN 3-540-94051-0 / ISBN 0-387-94051-0.
- Dennis W. Duke, Ancient Planetary Model Animations includes two links of interest:
- George Saliba, "Whose Science is Arabic Science in Renaissance Europe?" Discusses the model of Nasir al-Din al-Tusi and the interactions of Arabic, Greek, and Latin astronomers.