How were the European and Atlantic area revolutions of the 1700s and 1800s influenced by the religious, philosophical, and technological innovations of the 1500s to1700s? 

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The major religious innovation of the early modern period that created the preconditions for the subsequent revolutions was the rise of Protestantism. The first reason this enabled revolutions was that it broke down the monolithic alliance of "throne and altar" in which the Roman Catholic church and European monarchies (especially the Hapsburgs) provided mutual support. The second way Protestantism led to revolution was its insistence on "the right and duty of private judgment" as opposed to the Roman Catholic notion of a divinely sanctioned church hierarchy. Also, due to the existence of official state churches, religious and political dissent were often connected.

Philosophically, much of the rediscovery of ancient Greek and Roman texts in the Renaissance, and the growth of humanism and critical philology, destabilized the authority of the Roman Catholic church. 

Several technological innovations created the condition for the revolutions of the eighteenth and nineteenth centuries. One of them was the invention of print, allowing for new ideas to be widely and rapidly disseminated. Another important innovation was the agricultural revolution, including such technologies as the seed drill and horse hoe. The improved agricultural productivity and trade of the early modern period led to a rising middle class, often composed of Protestants, who served as an educated counterweight to established regimes.