Economic Growth
In 636 AD, a great Muslim army stood toe to toe with the forces of the East Roman empire. Neither army saw an advantage and so for several months the armies simply encamped opposite each other across the Yarmuk River. The stalemate was broken by a massive sandstorm which swept up over the camps. Sensing the moment, the Muslims forded the river en masse and charged the Christian army which could only flee or try to fight with heavy winds driving sand into their eyes. The Muslims dispersed the Christian force and claimed Palestine and Syria; neither would permanently leave the Muslim sphere of influence again. Then in 642 AD, at the Battle of Nehavend, the Arabs broke the power of the weakened Sassanian Empire and seized their territory as well.
Muhammad had died in 632 AD, but his followers chose a series of successors (called Caliphs) to unify the movement as spiritual and political leaders. Culturally, the most important was probably Uthman who in the 650s, had the Qur'an compiled into an authoritative written form for the first time. On his death in 656 AD, he was succeeded by the only relative of Muhammad ever to rule, Ali. However, Ali's reign was one of the most fractious in Muslim history, and in 661 AD he was assassinated.
This led to a Muslim civil war which could have broken the power of Islam almost before it had begun. However, the Umayyad merchant clan of Mecca succeeded in destroying their opposition and founded the Umayyad dynasty. This civil war however, was not without consequences. The two factions never made peace, but both survived and have sparred off and on ever since. The Shia Ali, party of Ali, known as the Shiites, are a dissenting minority clan that believes only relatives of Muhammad had the authority to rule as Caliph. They were opposed to the majority Sunni who hold that any Caliph may rule so long as they obey the strictures of Islam. This fracture has never been repaired to this day and Sunnis and Shiites continue to disagree on many points of doctrine.
The Sunni Umayyads established their capital in Damascus and unified their new empire by sending military expeditions in all directions. The Umayyads seized Egypt and stormed across North Africa. They crossed the Straits of Gibraltar and conquered most of Spain. Finally, a massive Muslim force raided as far as central Gaul before being resoundingly defeated by a Christian coalition at the Battle of Poitier in 732 AD. It was a signal that the Muslim tide was really beginning to turn. In 716 AD, the Muslims had marched before the very walls of Constantinople and laid seige to the city for two years. The Byzantines held out despite the determination and dedication of their foes, and in 718 AD they broke the seige of the city. It was the first major victory for the East Romans over the Umayyads. Spain and Constantinople would remain the rough borders between Christian and Muslim spheres of influence for the rest of the Medieval period.
Muslim navies however, would continue to expand until they largely dominated the Mediterranean. While the Romans were the only empire to conquer all the lands surrounding the sea, Muslim ships seized the Mediterranean just as completely using their triangular sailed ships. These were more maneuverable than square sailed Christian ships, and Muslims could sail into the wind, a feat the Christians couldn't duplicate and refused to adopt. As this balance of power was forming, the Umayyads settled down to ostentatious living, something which did not sit well with their devout subjects. In 750 AD, a coup overthrew the decreasingly dynamic empire and Abu al-Abbas founded the Abbasid Dynasty.
The Abbasids recognized the relative importance of their more sophisticated eastern provinces in comparison to the possibility of making advances into barbarian Europe. It was one of the most timely decisions in history as in 751 AD, Abbasid armies were in position to face a major new expansion of the Chinese Tang Dynasty into the Middle East. At the Battle of the Talas River the two armies collided and the Abbasids soundly routed the Chinese force in one of the great turning points in history. With the Tang armies sent back into China, the Abbasids were able to lay claim to the entire Middle East. This massive land empire from India to Spain was the largest empire ever assembled and it had only taken a hundred years to do it. No other people in history conquered so much, so quickly before settling into a peaceful, strong, stable empire. While the Abbasids would settle into their new empire without expanding it significantly, for almost 200 years they would maintain their empire in the face of all challengers. Between the Umayyad and Abbasid dynasties, the Arab Muslims thus hold the best argument for ruling the largest stable empire in world history.
Excepting only the Tang Dynasty which was now on the verge of collapse, the Arab Muslims were also the most cultured empire of the Medieval era. Like all great empires, the Abbasids developed a sophisticated economy fueled by international trade. The Abbasids moved their court from Damascus to Mesopotamia where they founded a beautiful and powerful new city as their capital, Baghdad. Muslim culture was devoted to art, poetry, architecture, gardens, and the greatest scientists and mathematicians of the age. It also created great libraries that preserved and extended Christian learning at a time when European abbeys were storing a few hundred books, and scraping off most of the old writing in order to reuse the parchment for new books about the lives of saints.
This was a key ingredient for the Arab's success. They combined the best of two Eras, the Ancient power of learning with the Medieval power of religion; virtually every other Medieval culture - including the heirs of Arab civilization - witnessed religion's erosion of learning. Religion's emphasis on faith without understanding sometimes gradually, sometimes suddenly sapped the commitment of society to secular education. In contrast, Muhammad had spoken of Allah's command not only to expand the faith but to expand knowledge, and the Abbasids encouraged some of the most dynamic and brilliant scholars in all of history. Christians, Jews, and Muslims all worked and shared their knowledge, but the Muslims had an unusually shrewd practical streak. Where the Greeks loved to take a premise and start working out its implications philosophically, Arabs preferred to take lots of little facts and build them up into one idea. Thus, many of the great precursors of the scientific movement got their start among the Arabs who worked best not in the realms of Greek abstraction but in experimentation and testing. This is how Arab scholars developed detailed astronomical tables derived from practical observations which were used across Eurasia due to their accuracy.
The greatest medical text of the Medieval era was compiled by Ibn Sina of Bukhara - known to Christians as Avicenna, the Arabic Aristotle - Ibn Sina wrote cogently on virtually every topic imaginable and was the greatest scholar of the Medieval era. He was also one of the first Arab scholars to understand the importance of Indian numerals, and was the scholar most responsible for achieving their adoption throughout the Muslim world. The decimal system was so powerful, that Europeans would eventually grasp it and adopt it, calling them Arabic numerals thinking the Arabs had developed them independently.
Also of profound importance was Muhammad ibn Musa al-Kwarazmi (of Kwarazm); al-Kwarazmi created an entirely new field of mathematics never imagined in Greek geometry. Rather than sketching geometric proofs visually, he wrote down equations of letters and numbers to indicate relationships, ratios, values, and other mathematical concepts or elements. He then developed the discipline of al-muqabala "comparing" which demonstrated that quantities (even if unknown or undetermined) could be subtracted from each side of an equation as long as they could be shown to be equal. He developed this into the more powerful discipline of al-jabr "restoring" in which an unknown quantity could be subtracted or removed from one side of an equation if it's opposite was added to the other side. The name al-jabr would become westernized in Europe to algebra, and al-Kwarazmi's equations would eventually replace geometry as a superior tool in most mathematical disciplines.