Early Arabic scholars were much more free to pursue their studies than their Christian counterparts. However, this compromise with Islam perhaps meant there was not ultimately the same level of commitment on the part of scientists to explore fully objective realities. The desire to accord with the Qur'an always remained strong.
Many early Islamic scientists drew inspiration from the Greek tradition of Aristotle and also from the new lands in India, which had come under Islamic influence. The learning of these civilizations was greatly respected within Islam as indeed were all forms of knowledge. It is worth recalling that the oldest university in the world is that of Al-Azhar in Cairo. Throughout the Islamic world the Madrassas or schools continued the tradition of disciplined education in both the Qur'an and also secular matters.
Mathematics
There were a number of important contributions made to mathematic by Islamic scholars. Perhaps the two most important and yet very basic ones were:
i) The Number System
Western mathematics had previously depended on Roman numerals which were long and cumbersome to use. Arabic scholars adapted certain Indian symbols and gradually transformed them into the numerals we are familiar with today. They were adopted by Fibonacci in the 14th century for use by Italian merchants and gradually became indispensable to Western mathematicians.
ii) The Numeral 0
Previous to its introduction by Arabic scholars there had been no way of indicating an empty place in the decimal system. Trying to do calculations with very large numbers like 1,000,000 or indeed any number with a zero in it was therefore very time consuming and complex.
Other contributions were:
The introduction of Algebra - for example in the work of Al-Kwarizhi.
The use of decimal fractions - for example in the 9th century treatise 'Key to Arithmetic' by al-Kashi.
Chemistry
One of the greatest was the alchemist Jabir whose experiments led to use of compounds such as saltpetre and alum. Distillation was also another skill Arabic alchemists developed to a high level.
Astronomy
Islamic astronomers were heavily influenced by Ptolemy in their observations of the sky and hence somewhat limited in the progress they were able make.
In addition, by the time the telescope became widely available the golden period of Islamic scholarship had passed. There were however many sophisticated observatories built which did not need a telescope. Instead, they used large structures of various shapes to determine, with great accuracy, the position of heavenly bodies. One good example is the one built by Sultan Ulagh Beg in 15th century Sarmakand.