Tusi Pair

The Tusi Couple is a mathematical mechanism developed by the 13th-century astronomer Nasir al-Din al-Tusi (1201–1274). Its primary function is to generate linear motion from the combination of two circular motions. Tusi developed this mechanism to address and correct a fundamental problem in Ptolemaic astronomy, particularly the use of the equant point.

The term Tusi Couple is not limited to a single device; it is also used as a general term encompassing various mathematical tools Tusi developed throughout his career for different purposes.

Development and Versions

Initial Concept and Mathematical Linear Version

Tusi first articulated his search for a solution to the irregularities in Ptolemy’s models of celestial motion—particularly the equant problem—in his 1235 Persian-language work Risale-i Mu‘îniyye. However, the complete mathematical formulation and application of the mechanism were presented in his 1245 supplementary treatise, Hall-i Müşkilât-ı Mu‘îniyye (Resolution of the Difficulties of the Mu‘îniyye) or Zeyl-i Mu‘îniyye.

The “mathematical linear version” of this period operates as follows:

• Two circles are internally tangent; the radius of the larger circle is twice that of the smaller circle.
• The two circles rotate in opposite directions at constant speeds.
• The rotational speed of the smaller circle is exactly twice that of the larger circle.
• A point on the circumference of the smaller circle, initially at the point of tangency, undergoes a linear back-and-forth motion along the diameter of the larger circle as a result of this combined motion.

Adhering to the physical principle that celestial spheres are solid bodies incapable of passing through one another, Tusi specifically emphasized that the smaller circle does not “roll” within the larger circle but is “carried” by it (in Persian: mî bard).

Two Equal Circles Version (So-Called Curvilinear)

In his 1247 work Tahrîru’l-Mecistî (Revision of the Almagest), Tusi presented a modified version of the mechanism. In this version, two circles of equal radius are used to resolve problems in Ptolemy’s latitude models. However, this model produced oscillation along a straight line (chord), rather than the intended curvilinear motion.

Physicalized and Fully Curvilinear Versions

Tusi developed the most advanced forms of the mechanism in his 1261 work et-Tezkire fî İlmi’l-Hey’e (Treatise on the Science of Astronomy).

• Physicalized linear version: He transformed the circles into solid celestial spheres and added a third “enclosing sphere” (muhîta) to prevent the disruption of the epicycle.
• Fully curvilinear version: He constructed a three-sphere curvilinear oscillation model to explain planetary latitudinal motion and the motion of the Moon’s prosneusis point.

In Tezkire, Tusi also revised certain terms used in the lunar model from his earlier Hall treatise and corrected various errors.

Resolution of the Equant Problem

The equant point used in Ptolemy’s planetary models required a celestial body or the center of an epicycle to move uniformly in a circular path around a point distinct from its geometric center. This contradicted the Aristotelian physical principle that celestial bodies must rotate uniformly around their own centers.

The Tusi Couple eliminated this physical inconsistency by providing a mechanism that reproduced observed irregular motions solely through combinations of uniform circular motions. Tusi applied this mechanism in Tezkire to model the motions of the Moon and the superior planets (Mars, Jupiter, Saturn).

Impact on Aristotelian Physics

The Tusi Couple had a significant impact on Aristotelian natural philosophy. By mathematically demonstrating that linear motion (oscillation) could arise from the combination of two uniform circular motions, it challenged the rigid Aristotelian distinction between celestial (circular) and terrestrial (linear) motion.

Tusi’s student at the Maragha Observatory, Kutb al-Din al-Shirazi, discussed the philosophical implications of this mechanism and refuted Aristotle’s claim that ascending and descending motions must include a moment of rest (quies media) by demonstrating the continuous oscillatory motion produced by the Tusi Couple.

Merâga School and Subsequent Influence

The Tusi Couple became one of the fundamental tools of the astronomical tradition known as the Maragha School.

• Kutb al-Din al-Shirazi (1236–1311), as a student of Tusi, employed this mechanism in his model for the planet Mercury.
• Ibn al-Shatir (1304–1375), a Syrian astronomer, completely restructured Ptolemy’s models using Maragha techniques such as the Tusi Couple and the Urdi Lemma. He also applied the Tusi Couple in his Mercury model.

Transmission to Europe and Copernicus

The Tusi Couple reached Europe by various means before 1543. Traces of the mechanism appear in translations into Byzantine Greek (via Gregory Chioniades) and in the works of European scholars such as Nicole Oresme and Giovanni Battista Amico.

One of the most prominent manifestations of this mechanism is found in the work of Nicolaus Copernicus (1473–1543). Copernicus adopted the Tusi Couple for the same reason as the Maragha astronomers: to eliminate Ptolemy’s equant model.

In his work De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres) (Book III, Chapter 4), Copernicus applied the “Two Equal Circles Version” from Tusi’s 1247 Tahrîr. He used this model in the following contexts:

• The inequality of precession,
• The variation of the obliquity of the ecliptic,
• The longitudinal model of Mercury,
• The latitudinal theory of the planets.

Copernicus also utilized the linear (physicalized) version of the Tusi Couple in his earlier work Commentariolus to vary the radius of Mercury’s orbit.