Mathematics in medieval Islam in the context of "Islamic science"

The history of Islam is believed, by most historians, to have originated with Muhammad's mission in Mecca and Medina at the start of the 7th century CE, although Muslims regard this time as a return to the original faith passed down by the Abrahamic prophets, such as Adam, Noah, Abraham, Moses, David, Solomon, and Jesus, with the submission (Islām) to the will of God. According to the traditional account, the Islamic prophet Muhammad began receiving what Muslims consider to be divine revelations in 610 CE, calling for submission to the one God, preparation for the imminent Last Judgement, and charity for the poor and needy. As Muhammad's message began to attract followers (the ṣaḥāba) he also met with increasing hostility and persecution from Meccan elites. In 622 CE Muhammad migrated to the city of Yathrib (now known as Medina), where he began to unify the tribes of Arabia under Islam, returning to Mecca to take control in 630 and order the destruction of all pagan idols. By the time Muhammad died c. 11 AH (632 CE), almost all the tribes of the Arabian Peninsula had converted to Islam, but disagreement broke out over who would succeed him as leader of the Muslim community during the Rashidun Caliphate.

The early Muslim conquests were responsible for the spread of Islam. By the 8th century CE, the Umayyad Caliphate extended from al-Andalus in the west to the Indus River in the east. Polities such as those ruled by the Umayyad and Abbasid caliphates (in the Middle East and later in Spain and Southern Italy), the Fatimids, Seljuks, Ayyubids, and Mamluks were among the most influential powers in the world. Highly Persianized empires built by the Samanids, Ghaznavids, and Ghurids significantly contributed to technological and administrative developments. The Islamic Golden Age gave rise to many centers of culture and science and produced notable polymaths, astronomers, mathematicians, physicians, and philosophers during the Middle Ages. By the early 13th century, the Delhi Sultanate conquered the northern Indian subcontinent, while Turkic dynasties like the Sultanate of Rum and Artuqids conquered much of Anatolia from the Byzantine Empire throughout the 11th and 12th centuries. In the 13th and 14th centuries, destructive Mongol invasions, along with the loss of population due to the Black Death, greatly weakened the traditional centers of the Muslim world, stretching from Persia to Egypt, but saw the emergence of the Timurid Renaissance and major economic powers such as the Mali Empire in West Africa and the Bengal Sultanate in South Asia. Following the deportation and enslavement of the Muslim Moors from the Emirate of Sicily and elsewhere in southern Italy, the Islamic Iberia was gradually conquered by Christian forces during the Reconquista. Nonetheless, in the early modern period, the gunpowder empires—the Ottomans, Timurids, Mughals, and Safavids—emerged as world powers.

Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Abbasid Caliphate of Baghdad, the Umayyads of Córdoba, the Abbadids of Seville, the Samanids, the Ziyarids and the Buyids in Persia and beyond, spanning the period roughly between 786 and 1258. Islamic scientific achievements encompassed a wide range of subject areas, especially astronomy, mathematics, and medicine. Other subjects of scientific inquiry included alchemy and chemistry, botany and agronomy, geography and cartography, ophthalmology, pharmacology, physics, and zoology.

Medieval Islamic science had practical purposes as well as the goal of understanding. For example, astronomy was useful for determining the Qibla, the direction in which to pray, botany had practical application in agriculture, as in the works of Ibn Bassal and Ibn al-'Awwam, and geography enabled Abu Zayd al-Balkhi to make accurate maps. Islamic mathematicians such as Al-Khwarizmi, Avicenna and Jamshīd al-Kāshī made advances in algebra, trigonometry, geometry and Arabic numerals. Islamic doctors described diseases like smallpox and measles, and challenged classical Greek medical theory. Al-Biruni, Avicenna and others described the preparation of hundreds of drugs made from medicinal plants and chemical compounds. Islamic physicists such as Ibn Al-Haytham, Al-Bīrūnī and others studied optics and mechanics as well as astronomy, and criticised Aristotle's view of motion.

Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (Persian: غیاث‌الدین جمشید کاشانی Ghiyās-ud-dīn Jamshīd Kāshānī; c. 1380 – 22 June 1429) was a Persian astronomer and mathematician during the reign of Tamerlane.

Much of al-Kāshī's work was not brought to Europe and still, even the extant work, remains unpublished in any form.

Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040) was a mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq. Referred to as "the father of modern optics", he made significant contributions to the principles of optics and visual perception in particular. His most influential work is titled Kitāb al-Manāẓir (Arabic: كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in a Latin edition. The works of Alhazen were frequently cited during the scientific revolution by Isaac Newton, Johannes Kepler, Christiaan Huygens, and Galileo Galilei.

Ibn al-Haytham was the first to correctly explain vision as intromissive rather than extramissive, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience. He also stated the principle of least time for refraction which would later become Fermat's principle. He made major contributions to catoptrics and dioptrics by studying reflection, refraction and nature of images formed by light rays. Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical reasoning – an early pioneer in the scientific method five centuries before Renaissance scientists, he is sometimes described as the world's "first true scientist". He was also a polymath, writing on philosophy, theology and medicine.

Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Isaac Todhunter's textbook Spherical trigonometry for the use of colleges and Schools.Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.