Al-Tusi introduces in the Taḥrīr al-Majisṭī (Commentary on the Almagest, 1247) the 'Tusi couple': a geometric mechanism generating linear motion from two rotating circles, eliminating the Ptolemaic equant without violating the principle of uniform circular motion. It is the first treatment of spherical trigonometry as an autonomous mathematical discipline, separate from astronomy. His Treatise on the Quadrilateral (c. 1260) sets out the six cases of the right spherical triangle. Copernicus uses the Tusi couple almost unmodified in De Revolutionibus (1543), though he does not cite him. He founds the Maragha observatory (1259), the first institutionalized observatory with a program of collective research.