How did Greek science influence Arabic science?

Greek science, as well as Greek literature, law, and custom, is generally regarded as the foundation of Western civilization. Yet in the years following the collapse of the Roman Empire, the adoption of Greek values swung eastward as rising Islamic caliphates looked to build their foundation of knowledge by delving into classical Greek texts. In the Abbasid Caliphate, which ruled most of the Middle East from 750 to 1517 (with a brief interruption during the Mongol conquests), scholars and scribes translated ancient Greek texts into Arabic, preserving many scientific works that might otherwise have been lost. The Abbasids also exchanged knowledge with the Byzantines, whose own concepts of science were based on the ancient Greeks.
There are many examples of how Islamic science is based on Greek science. The great Maragheh Observatory in modern-day Iran, built abut 750 years ago, draws upon principles of ancient Greek astronomers and had tens of thousands of books (both ancient and modern) to guide its studies. Arabic medicine was perhaps the most advanced in the medieval world and was built upon the ancient writers of Hippocrates and Galen. Finally, the rapid growth of Islamic cities required advanced mathematics and engineering: the reconstruction of the Dome of the Rock about 1000 years ago required careful precision and knowledge to rebuild the great shrine, reflecting the influence of Pythagoras and Euclid.