When you say ‘to the west’ I am going to assume you mean Western/Latin Europe and ‘the east’ the (former) territories of the Byzantine Empire.

Much of the intellectual inheritance in the Islamic World was Greek. Many ancient Greek works are primarily known to Latin/European culture because Arab scholars translated these works from Greek into Arabic, which allowed them to survive and later be translated and ‘re-introduced’ to Europe. This process also allowed knowledge from India and Persia to become known to the Western Europeans. This cross-pollination happened mainly in Iberia and Sicily, territories that were conquered by the Caliphate.

Another way in which one could say the Arab world contributed to the ‘educating’ Latin Europe was through Byzantine scholars. It is a commonly accepted theory that, as the Byzantine Empire declined, many of its intellectuals moved to Italy, taking their knowledge and works with them. This decline came at the hands of both the Muslim invasions and the Fourth Crusade, culminating in the 1453 Ottoman capture of Constantinople.

So it can be stated that both through erudite work of Arab scholars and rapid conquest by the Turks, Western Europe was made aware once more of the knowledge of Antiquity.