The Treaty of Westphalia in 1648 was a general European settlement following the 30 Years War, and is a generally accepted turning point away from the feudal dynastic sovereignty of the Medieval Period towards the centralized nation state model of the Modern Era (obviously this wasn't an overnight change, but rather a general transition at different paces over different areas of Europe). As Europe's polities morphed from heterogenous polyglots of overlapping political control to a handful of homogenous, more socially and politically stable nations the first Great Powers began to assert themselves. These Powers would spend the next century and a half in what was dubbed the Stately Quadrille of shifting alliances which prevented any single Power or alliance from gaining an hegemony (or even the potential of one) in European politics. The strategic thinking shared by these powers was zero sum in nature: Each Great Power had too much potential militarily and economically to outright annex another without a huge expenditure of treasure and manpower - which would then leave the victor vulnerable to others in turn - so the balance of power between these powers had to be maintained for the benefit of all.

This strategic thinking didn't prevent conflict between the Great Powers (they warred more or less ceaselessly), but did have the effect of involving powers that would otherwise have been neutral in conflicts to get involved in order to maintain the all important balance of power. This can be seen in the many general settlements following many general conflicts: The War of the Spanish Succession, The War of the Austrian Succession, The 7 Years' War, etc. culminating in the Congress of Vienna following the Napoleonic Wars in which the status quo ante bellum was more or less recreated - with France accepted as a Great Power once again despite all the damage she had caused the remainder of the Great Powers (to not accept this as fact would have totally upset the balance of power). Following this settlement the lines of modern Europe were more or less fixed, and the Great Powers went back to their same strategic, zero sum thinking: Gains by a Great Power without gains to the others upset the balance, therefore a power vacuum was abhorrent to the leaders of the Great Powers.

France, Britain, Austria-Hungary, The Prussian Empire and Russia had no strong kindred feelings towards the Ottoman Empire, in fact Russia and Austria-Hungary had been implacable foes for generations. However, Ottoman Turkey controlled the Dardanelles and nominally the Suez as well as the Red Sea and Persian Gulf - areas of obvious, vital strategic importance even to this day. Which power could the remainder agree to hand over control of any of these areas to should the Ottoman Empire collapse? If they couldn't agree would it mean war? If it meant war, who would ally with whom? It is clear to see why the Great Powers thought it better for all parties to maintain the status quo since it didn't generate a causus belli between them.