Id take a stab at Ladyjeans question; re what went wrong with China and the Middle east.

you could say year 1000 rather than the 15th century,
because at that time both were far more advanced than Europe
by the 15th century things were starting to change. (with Gutenberg and the fall of Constantinople especially, since a lot of refugees and books (greek classics) arrived in Italy at the same time the printing press was

China: up until 1420 or so China was sending out explorers with huge ships, (far bigger than anything Spain or Portugal could muster) but there was a shift in power away from the eunuch philosopher/bureaucrats, further exploration was abandoned and the ships ordered destroyed. China was also a unified kingdom, compared to Europe which was a number of competing states (thereby honing their military skills as well as competing for resources and colonies)

The scientific tradition of logic and reasoning, which started in Ionian Greece, gave the Europeans an advantage as well, since you could make models and test with experiments, the Chinese believed that everything had a living force and thus you could not make models of the world. The chinese were also pigeonholed into their positions in life, one could not improve ones status, so theres no incentive to try.

As far as the ARabs, Cordoba Spain, did in fact have a scientific renaissance of sorts, in the 10-11th century - a lot was learned from translating the greek texts, and algebra, distillation of alcohol, as well as the latine sail, all came from the Arabs, - Id say that ultimately the pervasive religion and status of women hindered development.