The mathematical device from Islamic astronomy that appears in Copernicus's lunar model is the Tusi couple.
Developed by the Persian astronomer Nasir al-Din al-Tusi in the 13th century, the mechanism combines two circular motions to produce a point moving back and forth in a straight line. A small circle rotates inside a larger circle with twice its diameter, creating a two-cusped hypocycloid.
The device helped astronomers model irregular-looking motions without abandoning the ancient ideal of uniform circular movement. Copernicus used the Tusi couple in mathematical models presented in works including De revolutionibus orbium coelestium. Similarities between his models and those of Islamic astronomers such as al-Tusi and Ibn al-Shatir have fueled extensive historical research.
One common oversimplification is that historians have proven a direct, documented handoff from al-Tusi to Copernicus. The mathematical parallels are striking, but the exact route of transmission remains uncertain; some scholars have also proposed independent rediscovery or transmission through Byzantine sources.