St. Pius X on Separation of Church and State

In the previous post I quoted the position Martin Luther held regarding the relation between a ruler and the Church. In his 1905 encyclical Vehementer nos, St. Pius X writes:

That the State must be separated from the Church is a thesis absolutely false, a most pernicious error. Based, as it is, on the principle that the State must not recognize any religious cult, it is in the first place guilty of a great injustice to God; for the Creator of man is also the Founder of Human societies, and preserves their existence as he preserves our own. We owe Him, therefore, not only a private cult, but a public and social worship to honor Him. Besides, this thesis is an obvious negation of the supernatural order. It limits the action of the State to the pursuit of public prosperity during this life only, which is but the proximate object of political societies; and it occupies itself in no fashion (on the pleas that this is foreign to it) with their ultimate object which is man's eternal happiness. The same thesis also upsets the order providentially established by God in the world, which demands a harmonious agreement between the two societies...
It follows necessarily that there are many things belonging to them in common in which both societies must have relations with one another... Finally, this thesis inflicts great injury on society itself, for it cannot either prosper or last long when due place is not left for religion, which is the supreme rule and the sovereign mistress in all questions touching the rights and duties of men.