Esîruddin Abheri's parallel postulate proof

Euclid, one of the prominent philosophers of the Hellenistic Age, taught mathematics at the Museum of Alexandria in the 3rd century BCE. Euclid compiled the works of all mathematicians and the systematic theories of philosophers who preceded him, and authored his work Elements, consisting of 13 books and 456 propositions. In Elements, Euclid introduced the Fifth Postulate, also known as the Parallel Postulate, which not only altered the course of mathematics and science but also changed its methodology, opening new fields in the world of science. Euclid's parallel postulate, with its implication or expression of infinity, has attracted the attention of numerous mathematicians, both from the Western and Eastern worlds, due to its complexity. It is well known that the Islamic world reached its peak in translation activities during the 8th to 12th centuries. During this period, when translation efforts were at their peek, scholars in the Islamic world also believed that the parallel postulate needed to be proven and made various attempts to do so. For this reason, Euclid's Elements held a significant place in the scientific life of the Ottoman Empire. In the Ottoman scientific world, Euclid's Elements became well-known under the name Usul al-Handasa wa'l-Hisab (The Principles of Geometry and Arithmetic). Scholars in the Islamic world who were interested in geometry did not remain indifferent to Euclid's Parallel Postulate. The dilemma of the parallel postulate was addressed by many geometers with great attention and care, who made efforts to prove it. To enhance understanding, various commentaries and annotations were written in places where the meaning was unclear. In the 13th century, Esîrüddin Ebherî, a prominent scholar of the Islamic world during the Middle Ages, did not remain indifferent to Euclid's Parallel Postulate. He approached Euclid's work, Elements, with great care, attempting to explain the proof of the parallel postulate in his treatise titled Islahu Kitabi'l-Ustukussât fi'l-Hendese li-Öklidîs. Ebherî's work attracted the attention of many scholars, including Muhammad b. Ashraf Samarkandî, Kamal al-Din al-Farisî, Qadi Zadeh al-Rumi, Ali Qushji, and Abu Ishaq Abdullah al-Kirmanî, who wrote various commentaries and annotations on it. These commentaries and annotations were based on Ebherî's Islahu Kitabi'l-Ustukussât fi'l-Hendese li-Öklidîs, which underscores the significance of Esîrüddin Ebherî in both the Islamic world and later among Ottoman mathematicians. In this study, we will examine how Esîrüddin Ebherî approached the proof of Euclid's Parallel Postulate in his work Islahu Kitabi'l-Ustukussât fi'l-Hendese li-Öklidîs. The thesis will also analyze the proofs related to the Parallel Postulate found in the works of Muhammad b. Ashraf Samarkandî, Kamal al-Din al-Farisî, Qadi Zadeh al-Rumi, Kamal al-Din al-Farisî, and Ali Qushji. These proofs, grounded in Ebherî's work, will be examined by identifying and explaining the commentaries, annotations, or critiques.