Between A.D. 800 and 1450, the most important centers for the study of what we now call "the exact sciences" -- including the mathematical sciences of arithmetic, geometry, and trigonometry and their applications in such fields as astronomy, astrology, geography, cartography, and optics -- were not in Europe but in the vast, multinational Islamic world. Research from the last few decades has profoundly changed our understanding of the Islamic scientific tradition. We now know that it was richer and more profound and had more complex relations to other cultures than we had previously...