The Battle of Tours

Hmm... the entire population of Europe at this time, excepting the Byzantines and Iberia, was only around nine or ten million. These were divided into hundreds of little factions. None of these people were particularly wealthy or sophisticated in terms of tactics or equipment. The fifty thousand warriors that the Franks brought to the Battle of Tours was probably their ENTIRE army and if not the entire, certainly the majority of it.

Contrast these with the resources of the Umayyad Empire. The Arabs controlled an empire of perhaps more than fifty million people. The Arabs held several cities that had over a hundred thousand people when Charlemagne's capital at Aachen during the peak of Frankish power never had over ten thousand. By conquering two of the greatest powers of Late Antiquity, they were made grealty sophisticated and wealthy. Furthermore, the Arabs were united. Their large population and wealth allowed them to easily field and equip armies of tens of thousands fighting in several different fronts.

Now, machine, can you honestly tell me that the petty kingdoms of Europe living in their tiny unwalled towns could defeat the Arabs at the peak of their power?