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 thought. This book offers an overview of this newly energized field of historical investigation. The areas discussed include cross-cultural transmission; transformations of Greek optics; the philosophy and practice of mathematics; numbers, geometry, and architecture; the transmission of astronomy; and science and medicine in the Maghrib. The emphasis throughout the book is on the transmission of scientific knowledge, either from one culture to another or within the medieval Islamic world. The book also presents many unsolved historical problems, such as the question of when and where the Hindu-Arabic number symbols evolved from the Eastern Islamic forms to the Western Islamic forms, which are virtually identical to the modern forms 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.