That's a tough question to answer, since I think most historians would challenge the assumption that the Islamic world—sometimes referred to as the "Islamicate" following the work of MGS Hodgson—ever did stagnate. Certainly, Eurocentric narratives tend to sideline the Islamic world, but perhaps it's worth briefly considering European history from an Islamic-centered perspective.

The Rise of Islam is sometimes seen as a freak historical event, as religious fanatics roared out of Saudi Arabia, conquered much of Afro-Eurasia, and then promptly fragmented into a variety of lesser caliphates, sultanates, and emirates. That's not at all the case. Robert Hoyland has recently argued that Arab mercenaries became influential through the constant fighting between the Byzantine and Persian empires. The rise of Islam was in fact facilitated in large part by Byzantine war-hawkishness and religious intolerance, which could include persecuting any sect of Christianity not sanctioned by the Emperor. Peoples throughout the Mediterranean and even into Spain and southern France turned to Islam as an advantageous alternative to Byzantine leadership and Christian monarchs throughout southern and western Europe.

The Carolingian Period of Charlemagne (r.768-814) is often seen as one of European formation and renaissance. Since the Mediterranean sailing economy had been largely replaced by the desert caravan economies stretching through Saharan Africa and across the Middle East, the Carolingians began to foster their own modest economy in the North Sea. Nevertheless, the Islamic economies were so flush that some wealth began to overflow into Charlemagne's Europe, both across the Mediterranean and also the long way around via Scandinavia, where 9th-century hoards are primarily comprised of Islamic dirhams rather Frankish and Anglo-Saxon coins. Politically speaking, Charlemagne's renaissance struck the few visitors from Harun al-Rashid (who inspired the 1001 Nights) as outlandish and barbaric. Charlemagne's biographer Notker, in Book 2, ch. 8, describes how envoys arrived with an impressive elephant and quantities of silk. Although Notker tried to make Charlemagne seem impressive in this interaction, it's clear that the Arab envoys had set out not even knowing where the Franks lived, couldn't help but laugh when confronted by Charlemagne's court in their Easter best, were horrified that at Charlemagne's pride in his hunting, and when they were drunk, admitted that the court was even poorer in its hospitality than rumored in the east.

The High and Late Middle Ages which are sometimes seen as Europe's return from the Dark Ages, are marked by such illuminated personalities as Thomas Aquinas and Bernard of Clairvaux. Their religious and intellectual activities mimicked in modest forms the bolder florescence of Islamic cultures ranging from Cordoba to Kabul. This was the Golden Age of Islam, when Greek works handed down in Islamic libraries gradually began to be translated into Latin, sometimes via Arabic translations and synopses. The great Muslim philosopher Avicenna was celebrated alongside the likes of Aristotle and Anselm, and the translation of Arabic medical works spurred the emergence of European science. Muslim scholars were already engaging in close studies of manuscript traditions that wouldn't be paralleled in Europe until the German philological schools of the 1800s. And Ibn Khaldun wrote a landmark historical treatise that parallels many of the ideas of later European thinkers like Marx and Durkheim.

The Colonial Period is often seen as a time when European might came to the fore, but this misses the point that Muslims factored into and influenced all these developments. The Portuguese began to sail around Africa to reach the predominantly Muslim markets of the Indian Ocean while cutting out Islamic middlemen in the Ottoman Empire. The Spanish bumped into America hoping to do the same. Iberian brutality in the American colonies arose in part from worldviews honed during the Reconquista, as Iberians learned to view themselves as ferocious dogs cornered on the edge of the Islamic world. Many of the slaves they eventually transported to the Americas were Muslims. Initially, European slave traders benefitted by syphoning off a small portion of what was a thriving Muslim slave trade. But as the demands of the brutal European-American economy turned Atlantic slavery into an unprecedented horror, Muslims in Africa rose up to fight a jihad against the European slave system. Meanwhile, Muslim peoples continued to flourish across North Africa, the Middle East, Central Asia, and out into Indonesia.

The 1800s is seen as the period of High Colonialism. Europe lost its dominance over the Americas during the Age of Revolutions, but Britain reinvented itself as an empire in India. This development occurred as an ongoing competition between Britain and the Muslim-led Mughal Empire, and it came to a head when the British leadership faced a joint Muslim-Hindu revolt in 1857. This is, perhaps, the period which might best be referred to as the period of Islamic stagnation, but that ignores major shifts in the Islamic world that would have enduring effects. To name just two, this was a period when Wahhabism became a significant force in the Ottoman Empire and when Muslim identity began emerge as the basis for a modern nation-state in Pakistan.

The 1900s is also easily portrayed as a period of European dominance, with Europeans and their allies emerging as victors in two world wars. The flipside, of course, is that Europeans and their allies also suffered two devastating defeats in two world wars. Colonial troops were vital to both endeavors, and colonies were central to both fights. The Islamic world was no mere pawn—Muslims were influential actors and Islamic regions were key battlegrounds. The Ottoman Empire, for example, was a key player in the First World War, remembered best for inflicting a punishing defeat on Australians at Gallipoli. And during WWII, the US came into the war initially fighting on fronts where Muslim were prominent—opposing German troops in North Africa and cutting off Japanese troops in Indonesia. During the subsequent Cold War, Muslims took center stage as leaders of decolonization movements and the promotion of the Third World and the Non-Aligned Movement, which countered the growing rivalry between the US and the USSR. And as the Cold War came to an end, the Muslim world is again asserting itself as a leading player in global politics, both unconventionally through organizations like Al-Qaeda and institutionally through political and economic organizations like OPEC.

This is, of course, a picture of history biased toward an Islamic perspective, but that's the point: perspective matters. To say the Islamic world stagnated is to ignore both the successes of the Islamic world and to neglect the many failures of the West (or whatever 'world' is seen as the model of comparison and progress). So even if you don't agree with this perspective on history, it's important to recognize and interrogate the danger of the single story.