Ibn al-Shatir constructed mathematical models of planetary motion from combinations of uniform circular motions. His work in the Nihayat al-Sul treated both the angular progress of planets and their displacement above or below the reference orbital plane.
The Venus model offers a concrete example: the same carefully organized rotations can describe longitude and latitude without adding an independent collection of spheres merely for latitude. A modern mathematical reconstruction and edition of the relevant chapter makes this economy of construction explicit. The achievement concerns the geometry of a predictive astronomical model; it does not require asserting a particular unverified route of transmission to later European work.[ref]
Lunar-model diagram attributed to Ibn al-Shatir
. Lunar-model diagram attributed on Commons to Ibn al-Shatir, dated there to the fourteenth century; reproduction from CCVALG, uploaded by Viaros17. Public domain in its source country and in the United States, according to the linked file page. Source.
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