During the Middle Ages, the best scientists and doctors were not in Europe, but in the Islamic Empire to the south and east. Most of the science and medicine that people were doing in Europe was learning from Islamic scientists and doctors. The Crusades, by sending a lot of Europeans to go live in West Asia for a while, helped to spread Islamic science to Europe. Other scientific knowledge came from Islamic Spain to France. Because people thought Arabs were the best doctors, this Italian jar for holding medicine is decorated in an Islamic style, even though neither the jar nor the medicine inside it came from West Asia.
In medieval Europe, people were very interested in scientific observation. Together with Islamic scientists, they learned a lot about plants, and also astronomy. Europeans learned how to use an astrolabe from Islamic astronomers in Spain. On the other hand, Europeans seem to have learned about the compass about 1200 AD from the Chinese through the Mongol Empire, and then to have passed that knowledge on to the Seljuks. About the same time, Europeans also learned about hot air balloons from the Mongols.
One important kind of medieval science was alchemy (al-KEM-ee), which we call chemistry today. Alchemists tried to find ways to help people live forever, and they were especially interested in turning lead into gold (so they could get rich). They never were able to do that, but along the way they did learn a lot about chemistry. The word "alchemy" comes from the Arabic word for "the chemistry", originally from a Greek word meaning "fluids".
Logic was also an important part of medieval science, mainly through the Catholic Church. Men like Peter Abelard and Thomas Aquinas learned from Islamic Empire scholars like Ibn Rushd and Maimonides to use logic to find relationships between Aristotle's philosophy and Christian faith. Professors taught Aristotle and Socrates' methods of argument in European universities, just getting started in the 1100s AD. In the 1200s AD, universities opened all over Europe, from the University of Paris to the University of Naples.
In mathematics, the Middle Ages saw the introduction of what we call Arabic numbers (0,1,2,3,4,5,6,7,8,9) from India through West Asia to Europe. The new numbers led to a shift in mathematical interests from geometry to algebra. By about 1200, Ibn al-Haytham's work on optics encouraged Europeans to work with glass lenses, and by 1286 some Italians were using reading glasses.
Europeans seem to have been more independent and original when it came to engineering. They invented new tools for farmers. During the early Middle Ages, the horseshoe was invented, and also a new kind of horse harness that worked better than what the Romans had. By about 1000 AD, northern Europeans had figured out how to make glass from local materials. People began to use the harrow to turn over plowed earth, and the pitchfork. Chimneys were also invented in Europe in the Middle Ages. And once they had learned about gunpowder, a Chinese invention which reached Europe through the Mongol Empire, Europeans led the way in the invention of cannons, which seem to have first been used around 1320 AD in the Hundred Years' War.