How it began
The Persian astronomer and mathematician Nasir al-Din al-Tusi (1201 to 1274) first proposed the device in 1247 in his Tahrir al-Majisti, a commentary on Ptolemy's Almagest, and set it out again in his Tadhkira, a memoir on astronomy. Al-Tusi was the founder of the observatory at Maragheh, in northwestern Iran, built with the Mongol ruler Hulagu Khan. Astronomers there kept their observations and calculations in a large library. The name "Tusi couple" was given to the device many centuries later.
Before it existed
Astronomers had worked for more than a thousand years with the models of Ptolemy's Almagest, which used a device called the equant to match the uneven speeds of the planets. The equant broke the principle of uniform circular motion. Al-Tusi proposed his couple first for the motion in latitude of the inferior planets, and it was then used widely as a substitute for the equant.
How it works
A small circle rotates inside a larger circle of twice its diameter. As both turn, a point on the rim of the small circle moves back and forth in a straight line along a diameter of the large one. In mathematical terms the path is a hypocycloid with two cusps. The construction shows that a straight-line oscillation can be produced from uniform circular motions alone.
How it spread
Ibn al-Shatir used the couple in his own planetary models, and Nicolaus Copernicus used it in De revolutionibus in 1543. In 1973 Willy Hartner showed that Copernicus's diagram matches al-Tusi's letter for letter, with A where al-Tusi has alif and B where he has ba. How the idea reached Europe is debated. Proposed routes include Greek translations by Gregory Chioniades and a Hebrew treatise by Abner of Burgos, who died about 1340.
What lasted
Historians of astronomy regard the couple as a key element of the Copernican system, although neither al-Tusi nor Ibn al-Shatir placed the Sun at the center. Engineers later built straight-line mechanisms on the same geometry, including the Cardan gear and Matthew Murray's hypocycloidal steam engine of the 19th century.
