The Muslim mind has always been attracted to the mathematical sciences in accordance with the "abstract" character of the doctrine of Oneness which lies at the heart of Islam. The mathematical sciences have traditionally included astronomy, mathematics itself and much of what is called physics today. In astronomy the Muslims integrated the astronomical traditions of the Indians, Persians, the ancient Near East and especially the Greeks into a synthesis which began to chart a new chapter in the history of astronomy from the 8th century onward. The Almagest of Ptolemy, whose very name in English reveals the Arabic origin of its Latin translation, was thoroughly studied and its planetary theory criticized by several astronomers of both the eastern and western lands of Islam leading to the major critique of the theory by Nasir al-Din al-Tusi and his students, especially Qutb al-Din al-Shirazi, in the 13th century.
The Muslims also observed the heavens carefully and discovered many new stars. The book on stars of 'Abd al-Rahman al-Sufi was in fact translated into Spanish by Alfonso X el Sabio and had a deep influence upon stellar toponymy in European languages. Many star names in English such as Aldabran still recall their Arabic origin. The Muslims carried out many fresh observations which were contained in astronomical tables called Zij. One of the acutest of these observers was al-Battani whose work was followed by numerous others. The Zij of al-Ma'mun observed in Baghdad, the Hakimite Zij of Cairo, the Toledan Tables of al-Zarqali and his associated, the II-Khanid Zij of Nasir al-Din al-Tusi observed in Maraghah, and the Zij of Ulugh-Beg from Samarqand are among the most famous Islamic astronomical tables. They wielded a great deal of influence upon Western astronomy up to the time of Tycho Brahe. The Muslims were in fact the first to create an astronomical observatory as a scientific institution, this being the observatory of Maraghah in Persia established by al-Tusi. This was indirectly the model for the later European observatories. Many astronomical instruments were developed by Muslims to carry out observation, the most famous being the astrolabe. There existed even mechanical astrolabes perfected by Ibn Samh which must be considered as the ancestor of the mechanical clock.
Astronomical observations also had practical applications including not only finding the direction of Makkah for prayers, but also devising almanacs (the word itself being of Arabic origin). The Muslims also applied their astronomical knowledge to questions of time-keeping and the calendar. The most exact solar calendar existing to this day is the Jalali calendar devised under the direction of 'Umar Khayyam in the 12th century and still in use in Persia and Afghanistan.
As for mathematics proper, like astronomy, it received its direct impetus from the Quran not only because of the mathematical structure related to the text of the Sacred Book, but also because the laws of inheritance delineated in the Quran require rather complicated mathematical solutions. Here again Muslims began by integrating Greek and Indian mathematics. The first great Muslim mathematician, al-Khwarazmi, who lived in the 9th century, wrote a treatise on arithmetic whose Latin translation brought what is known as Arabic numerals to the West. To this day guarismo, derived from his name, means figure or digit in Spanish while algorithm is still used in English. Al-Khwarzmi is also the author of the first book on algebra. This science was developed by Muslims on the basis of earlier Greek and Indian works of a rudimentary nature. The very name algebra comes from the first part of the name of the book of al-Khwarazmi, entitled Kitab al-jabr wa'l-muqabalah. Abu Kamil al-Shuja' discussed algebraic equations with five unknowns. The science was further developed by such figures as al-Karaji until it reached its peak with Khayyam who classified by kind and class algebraic equations up to the third degree.
The Muslims also excelled in geometry as reflected in their art. The brothers Banu Musa who lived in the 9th century may be said to be the first outstanding Muslim geometers while their contemporary Thabit ibn Qurrah used the method of exhaustion, giving a glimpse of what was to become integral calculus. Many Muslim mathematicians such as Khayyam and al-Tusi also dealt with the fifth postulate of Euclid and the problems which follow if one tries to prove this postulate within the confines of Eucledian geometry.
Another branch of mathematics developed by Muslims is trigonometry which was established as a distinct branch of mathematics by al-Biruni. The Muslim mathematicians, especially al-Battani, Abu'l-Wafa', Ibn Yunus and Ibn al-Haytham, also developed spherical astronomy and applied it to the solution of astronomy and applied it to the solution of astronomical problems.
The love for the study of magic squares and amicable numbers led Muslims to develop the theory of numbers. Al-Khujandi discovered a particular case of Fermat's theorem that "the sum of two cubes cannot be another cube", while al-Karaji analyzed arithmetic and geometric progressions such as: 13+23+33+...+n3=(1+2+3+...+n)2. Al-Biruni also dealt with progressions while Ghiyath al-Din Jamshid al-Kashani brought the study of number theory among Muslims to its peak.
In the field of physics the Muslims made contributions in especially three domains. The first was the measurement of specific weights of objects and the study of the balance following upon the work of Archimedes. In this domain the writings of al-Biruni and al-Khazini stand out. Secondly they criticized the Aristotelian theory of projectile motion and tried to quantify this type of motion. The critique of Ibn Sina, Abu'l-Barakat al-Baghdadi, Ibn Bajjah and others led to the development of the idea of impetus and momentum and played an important role in the criticism of Aristotelian physics in the West up to the early writings of Galileo. Thirdly there is the field of optics in which the Islamic sciences produced in Ibn al-Haytham (the Latin Alhzen) who lived in the 11th century, the greatest student of optics between Ptolemy and Witelo. Ibn al-Haytham's main work on optics, the Kitab al-manazir, was also well known in the West as Thesaurus opticus. Ibn al-Haytham solved many optical problems, one of which is named after him, studied the property of lenses, discovered the Camera Obscura, explained correctly the process of vision, studied the structure of the eye, and explained for the first time why the sun and the moon appear larger on the horizon. His interest in optics was carried out two centuries later by Qutb al-Din al-Shirazi and Kamal al-Din al-Farisi. It was Qutb al-Din who gave the first correct explanation of the formation of the rainbow.
It is important to recall that in physics as in many other fields of science the Muslims observed, measured and carried out experiments. They must be credited with having developed what came to be known later as the experimental method.