Copernicus' complex Mercury model in De revolutionibus is virtually identical, geometrically, to Ibn al-Šāṭir's (ca. 1305 – ca. 1375). However, the model in his earlier Commentariolus is different and in many ways unworkable. This has led some to claim that the younger Copernicus did not understand his predecessor's model; others have maintained that Copernicus was working totally independently of Ibn al-Šāṭir. We argue that Copernicus did have Ibn al-Šāṭir's models but needed to modify them to conform to a “quasi-homocentricity” in the Commentariolus. This modification, and the move from a geocentric to heliocentric cosmology, was facilitated by the “heliocentric bias” of Ibn al-Šāṭir's models, in which the Earth was the actual center of mean motion, in contrast to Ptolemy and most Islamicate astronomers. We show that: 1) Ibn al-Šāṭir sought to reproduce Ptolemy's critical elongation at the trines(±120°), but changed the Ptolemaic values at 0, ±90, and 180°; 2) in the Commentariolus, Copernicus does not try to produce viable elongations for Mercury; and 3) by the time of writing De revolutionibus, Copernicus is in full control of the Mercury model and is able to faithfully reproduce Ptolemy's elongations at all critical points. We also argue that claims regarding “natural” solutions undermining transmission are belied by historical evidence.

In this paper, I examine the opposition between direct proof and proof per impossibile, introduced by al-Sijzī (second half of the 10th century) in his treatise entitled The side is not commensurable with the diagonal. From this example he gives to illustrate this opposition, al-Sijzī argues the superiority of the direct proof. In this paper, I also give the editio princeps of his treatise, as well as its first translation.

In this paper, we present the Aristotelian theory of comets, which is well known in the Muslim West through the commentaries of Ibn Rušd and Ibn Bāǧǧa. This aspect is covered three centuries later in an unknown manuscript attributed to the famous mathematician Ibn Ġāzī al-Miknāsī (1437-1513), and this text is not present in the known list of his works. The author devotes a part of his manuscript to the comet astrology following Ptolemy, and introduces a critical position of Iḫwān al-Ṣafā’, characterized by the rejection of the sidereal comets idea. We are also interested in comet classifications given by Ibn Ġāzī and another author ‘Alī al-Antākī. We found that the content of the De cometis of pseudo-Ptolemy is analogous with the text of Ibn Ġāzī, and almost identical to the text of al-Antākī. Then, we are interested in observations of comets from the 9th to the end of the 19th century in North Africa and in Islamic Spain (Andalusia), recorded in some Arabic manuscripts on astronomy and history, which have never been the subject of an extensive study. The studied observations are: the two comets X/975 P1 and X/998 D1 reported by an anonymous author of the 15th century, the X/1381 V1 comet reported by Ibn al-Qāḍī (1553-1616), the passage of Halley's comet in 1456 reported by Muḥammad al-Zarkašī (1434-1525), the C/1743 X1 comet observed by ‘Abd al-Razzāq ibn Ḥamadūš (1695-1785), which confirm its fan structure, the two observations of Ibn ‘Alī al-Šrīf al-Šalāṭī of the D/1770 L1, and especially the C/1769 P1 comet. The graphic representation of this latter comet is unprecedented in the Muslim West.

Avicenna's logic is presented in many treatises, of which the most important is Al-Qiyās (Prior analytics), which is included in his encyclopaedic book Al-Šifā’. But the treatise entitled Manṭiq al-mašriqiyyīn is said to express Avicenna's own logic by some commentators. One can thus ask: is this treatise in conflict with Al-Qiyās and the other logical writings? In this paper, we will answer this question by comparing Avicenna's analyses of the absolute – i. e. non modal – propositions. We will show that there is no radical difference between both treatises in this respect, since most conditions added to the absolute propositions in Al-Qiyās can be found in Manṭiq al-mašriqiyyīn. These conditions are: (1) “as long as S exists”, (2) “as long as it is S”, (3) “as long as it is P”, (4) “at some times”, (5) “at one determined time”, (6) “sometimes but not permanently”. In Manṭiq al-mašriqiyyīn, the ḍarūrīya contains (1), the lāzima contains (2), the muntašira contains (4), the mafrūḍa contains (5), the wujūdiyya contains (6), but (3) is abandoned. The waqtiyya was evoked in Al-Qiyās, although not classified. So the temporal analysis initiated in Al-Qiyās is pursued and developed in Manṭiq al-mašriqiyyīn, even if in the latter it is sometimes confused.

In several places and for different purposes, Ibn al-Haytham has studied the variations of a well-known function. The context of these applications reveals some characteristics of Ibn al-Haytham's mathematical and physical thought. An in-depth study of his mathematical proofs of monotony show how close he was from an elaborate concept of the continuous.