This is hardly "medieval" as most people think of it: overlapping (say) both Shakespeare and Plymouth Rock.
What if Muslims had captured the heart of Europe in the 1500's or so?
Good points. Also, remember when the fanatics overran the Byzantine (East Roman) Empire when they captured and pillaged Constantinople in 1453. That's why Constantinople is now called "Istanbul". By the way, what few elements of civilization the Arabs preserved was largely the product of what they discovered in Byzantium. In fact, even the Koran contains elements (much misunderstood by Muhammed) of Judaism and Christianity that Islam (mis)appropriated.