You can answer this by reading in the Epilogue of the book. On p. 410 and following in the paperback edition, Diamond discusses why the Fertile Crescent lost its lead and why Europe took over.
Diamond argues that the Fertile Crescent lost its lead due to geographic factors. It had an early lead because it was well-suited to the development of agriculture. However, it was not well-suited to having big civilizations. Once civilizations arose, they destroyed the environment of the Fertile Crescent. The area that was once the Fertile Crescent is now mainly desert. By contrast, Europe had an environment that could support much larger-scale agriculture and could therefore support large civilizations.