Tusi couple

In its familiar circular form, a smaller circle rolls inside a larger circle whose radius is twice as great. When the circles rotate in opposite directions at the appropriate relative rates, a point on the smaller circle moves back and forth along a straight diameter of the larger circle. The theorem therefore shows how a rectilinear oscillation can arise from combined circular motions.

Nasir al-Din al-Tusi introduced a rudimentary version of the result in his 1247 redaction of Ptolemy's Almagest. He developed the construction further in later astronomical writing, including the Taḏkera, where he stated and proved it. Encyclopaedia Iranica describes its original astronomical role as helping address problems in planetary latitude and lunar motion within mathematical models based on uniform circular motion.

The construction is an important example of mathematical innovation in the astronomy of the Maragha tradition. Historians have discussed its relationship to similar constructions in Copernicus's work. Encyclopaedia Iranica reports that the theorem appears in Copernicus, while the MacTutor biography notes that historians disagree on how the result reached him. The connection is therefore a subject of historical analysis, not proof by itself of a simple, established chain of transmission.