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Islamic astronomers take the equant apart

Al-Battani re-measured every planetary motion, Ibn al-Haytham argued the equant was physically impossible, and Ibn al-Shatir built the models Copernicus would mirror.

An open manuscript on cream paper. The left page is filled with closely written black Arabic naskh, several lines overruled in red. The right page carries two compass-drawn figures with a few lines of text between them: above, a set of nested circles cut by a vertical diameter with a small circle riding on it at the top, annotated in red Arabic; below, a red circle crossed by curved arcs that converge on a point at the top and a point at the bottom, with black annotations fanned around the outside.
A page from 'Ali al-Jurjani's commentary on al-Tusi's Tadhkira, an Arabic manuscript copied in Timurid Persia in 813 AH/1410 CE. The upper diagram sets out the Ptolemaic planetary orbs, labelled in red with the centre of the world, the centre of the deferent, and da'irat mu'addil al-masir, the equant circle. · Image: Sotheby's

From 877 to 918 al-Battani observed at ar-Raqqa on the Euphrates with a gnomon, sundials, a triquetrum, parallactic rulers, a new kind of armillary sphere and a mural quadrant. His Zij al-Sabi runs to fifty-seven chapters and a catalogue of 489 stars, every planetary mean motion redetermined, not copied from the Almagest. He confirmed al-Ma'mun's astronomers' finding that the Sun's apogee moves roughly a degree in 66 Julian years; Ptolemy had held it fixed. His solar eccentricity is reckoned better than Copernicus's and Tycho Brahe's, six and seven centuries later.

Between 1025 and 1028 in Cairo, Ibn al-Haytham wrote al-Shukuk 'ala Batlamyus, Doubts Concerning Ptolemy. The equant predicted well; his objection was that it requires a solid sphere to turn uniformly about a point that is not its centre, which no sphere can do. It offered no replacement.

In 1259 Nasir al-Din al-Tusi founded an observatory at Maragha in Azerbaijan under the Ilkhanid ruler Hulagu: library, instrument workshops, a staff including Mu'ayyad al-Din al-Urdi. It brought together the Urdi lemma and the Tusi couple, a small circle rolling inside one twice its diameter that turns two uniform circular motions into a straight-line oscillation — enough to reproduce the equant's effect with uniform rotations about genuine centres alone.

Ibn al-Shatir (Damascus, about 1305 to about 1375), head muwaqqit, or timekeeper, at the Umayyad Mosque and designer of the marble sundial on its northern minaret, removed the equant and the eccentric from the planetary models in Nihayat al-sul fi tashih al-usul, "The Final Quest Concerning the Rectification of Principles", substituting additional epicycles. His lunar and Mercury models are mathematically identical to Copernicus's of 1543; for the other planets, including Mars, Copernicus uses the same devices at the same points, Sun and Earth exchanged.

No document carrying this to Poland has been found, and Copernicus never names him. George Saliba argued for transmission through Byzantine Greek manuscripts and Jewish intermediaries; Viktor Blåsjö and others for independent rediscovery. The route is not settled.