Al Tusi's diagram of linear motion from
circular motion later used by Copernicus
Because West Asia was such an economic crossroads in the medieval Islamic period - because of the Silk Road that connected China and India in the east to Europe and Africa in the West - there were always lots of new scientific ideas coming through West Asia too. Educated West Asian scholars were able to make use of these foreign ideas to develop new scientific theories and approaches.
One example from the East is the use of "Arabic" numbers, which really came from India, about 630 AD. The Arabic word for numbers, in fact, is hindsah, which means "from India". Arab scientists, especially the Persian Mohammed Al-Khwarizmi, were able to make use of the new numbers (and possibly the work of Greek mathematicians like Diophantus of Alexandria) to develop algebra around 830 AD (The English word "algorithm" comes from Al-Khwarizmi). (Ordinary people, however, kept on using the Greek system of numbers; only mathematicians used Arabic numbers).
In the 800s AD, the great schools at Cordoba in Spain, under Umayyad rule, inspired many scholars to investigate new scientific ideas. Among them was a man of Berber origin, Ibn Firnas, who designed the first glider, which he successfully used in 875, when he was 65 years old, to fly down from a cliff near Cordoba (though he hurt his back when he landed). This was the first controlled human flight.
A more successful invention also from Islamic Spain was the glass mirror, invented around 1000 AD. Even earlier, in the 900s, Ibn Sahl and others made curved glass mirrors that concentrated sunlight to focus heat.
About 1000 AD, West Asian blacksmiths also learned how to make steel from India, and then they developed the idea further to produce the very high quality Damascus steel that was used in fighting the Crusades.
Another example from the East is the use of paper, which the Arabs learned from the Chinese about 750 AD, fighting in Central Asia. Also from about 750, Islamic scientists invented a better sugar press to make use of sugar cane from India, and figured out how to make white glass pottery glazes to imitate Chinese porcelain. The magnetic compass also came to West Asia from China, about 1100 AD.
From the West, Arabic scholars were able to read the books of the Greek philosophers Plato and Aristotle, and the Roman encyclopedist Pliny the Elder, and they translated these books into Arabic. They were especially interested in Aristotle and Pliny's studies of plants and animals, and produced many new studies like that of their own, often with beautifully detailed and accurate illustrations. This led to the classification and description of many new species of plants and animals, and also to advances in medicine. All through the Middle Ages, everyone knew that the best doctors, men like Ibn Sina or Maimonides, lived in the Islamic kingdoms.