The Tusi couple is a mathematical device consisting of a small circle rotating inside a larger circle of exactly twice its diameter, such that a point on the circumference of the smaller circle traces a straight line - specifically, a diameter of the larger circle. The construction was introduced by the Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 work Tadhkira fi 'Ilm al-Hay'a (Memoir on Astronomy) as a means of generating linear motion from circular motion without recourse to the equant, a theoretical point used in Ptolemaic astronomy that al-Tusi and others found geometrically objectionable. The device has since been studied in the history of mathematics, the history of astronomy, and the historiography of the Scientific Revolution.

Ptolemaic astronomy, as codified in the Almagest (c. 150 CE), accounted for the observed non-uniform motions of the planets partly through the equant - a point offset from the center of a deferent circle, from which a planet's motion appeared uniform. Many medieval astronomers, both in the Islamic world and later in Europe, regarded the equant as a violation of the principle that celestial motion must be composed of uniform circular motions. Al-Tusi's couple offered a geometrically rigorous alternative: by combining two uniform circular motions, it produced the oscillating linear motion needed to model planetary latitude variations and other irregularities without the equant device. Al-Tusi deployed it within a broader reform of Ptolemaic models, a tradition of mathematical astronomy centered at the Maragha observatory in northwestern Iran known as the Maragha school, active during the 13th and 14th centuries.

The proof underlying the couple is straightforward: if a circle of radius r rolls inside a circle of radius 2r, every point on the rolling circle's rim moves along a diameter of the fixed circle. This is a special case of a hypocycloid, a family of curves generated by a point on a circle rolling inside another circle. When the radius ratio is exactly 2:1, the hypocycloid degenerates to a line segment.

A contested question in the history of science is whether Nicolaus Copernicus, writing in the early 16th century, arrived at an identical mathematical device independently or derived it - directly or indirectly - from Islamic sources. Copernicus employed a construction equivalent to the Tusi couple in his De revolutionibus orbium coelestium (1543) to solve similar problems in planetary theory, and used a closely related device, sometimes called the “Tusi couple variant” or “Copernicus theorem,” to generate uniform circular motion along a straight line. The formal similarity is exact, and no Latin or Greek intermediary source for these constructions has been identified that predates known Arabic transmission routes. Some historians argue the correspondence is too precise to be coincidental and implies transmission through as-yet-unidentified channels; others maintain that independent discovery of an elementary geometric result cannot be ruled out. See Tusi Couple - Transmission Debate for a full treatment of the competing arguments.

Beyond its original astronomical context, the Tusi couple is recognized as an early example of a four-bar linkage - a class of mechanisms that convert rotational motion into other motion types, foundational in mechanical engineering. The Scottish engineer James Watt arrived at a related concept, the Watt's linkage, in the late 18th century in the context of steam engine design, independently and in a different mechanical tradition. The couple also appears in discussions of epicyclic gear trains and, in a more abstract setting, in the mathematics of Chebyshev polynomials, which describe the same parametric curve family.

There is broad consensus among historians of mathematics and astronomy that al-Tusi invented the device described above, that Copernicus employed a formally equivalent construction, and that the Maragha school produced a significant body of mathematical astronomy that anticipated several structural features of Copernican models. The question of transmission - whether and how these ideas reached Copernicus - is not settled. See Tusi Couple - History of Science Consensus for a summary of areas of agreement and ongoing dispute.

* Independent Discovery - The view that Copernicus, working from Ptolemaic sources available in Latin Europe, independently reconstructed the same device; the geometric result is simple enough that parallel invention is plausible. * Transmission from Islamic Sources - The view that the structural identity between al-Tusi's and Copernicus's constructions, combined with the absence of a Latin intermediary, implies transmission through Byzantine, Jewish, or other intermediary scholarly networks. * The "Maragha Revolution" Framing - The view, associated particularly with historian Noel Swerdlow and others, that Copernican mathematical astronomy is best understood as a continuation of, rather than a break from, Islamic reform of Ptolemy; critics of this framing argue it overstates structural continuity and understates Copernicus's heliocentric innovation.

* Tusi Couple - History * Tusi Couple - Transmission Debate * Tusi Couple - History of Science Consensus * Maragha School * Nasir al-Din al-Tusi * Nicolaus Copernicus * Ptolemaic Astronomy * Hypocycloid * Scientific Revolution * History of Astronomy

1. Nasir al-Din al-Tusi, Tadhkira fi 'Ilm al-Hay'a, 1247. Critical edition and translation: F. J. Ragep, Nasir al-Din al-Tusi's Memoir on Astronomy, 2 vols. (New York: Springer, 1993). 2. Noel M. Swerdlow and Otto Neugebauer, Mathematical Astronomy in Copernicus's De Revolutionibus, 2 vols. (New York: Springer, 1984). This work first drew systematic scholarly attention to the parallel between al-Tusi and Copernicus in modern historiography. 3. George Saliba, Islamic Science and the Making of the European Renaissance (Cambridge, MA: MIT Press, 2007). Argues for transmission and situates the Tusi couple within a broader account of Islamic contributions to Copernican astronomy. 4. Edward S. Kennedy, “Late Medieval Planetary Theory,” Isis 57, no. 3 (1966): 365-378. An early survey of Maragha-school planetary models. 5. F. J. Ragep, “Copernicus and His Islamic Predecessors: Some Historical Remarks,” History of Science 45 (2007): 65-81. Reviews the transmission question and the state of evidence.