Let it be tried whether any middle ground can be taken which will at once support a due supremacy of the national authority. what does it mean?

This means that let the suggestion/method be tried in order to come to a point of agreement that will be adopted to support and provide the expected superiority to the national authority.

Explanation:

The statement was made by James Madison to Edmund Randolph in 1787. Madison wanted a strong central government to provide order and stability. He wanted the "middle ground" that will help provide a due supremacy to the national authority. He was right to have supposed that it will not be well suited for the national government to oversee the entire task of a vast country.  

James Madison was an American statesman and Founding Father. He served as the fourth president of the United States from 1809 to 1817.