"in a republican nation, whose citizens are to be led by reason and persuasion and not by force, the art of reasoning becomes of first importance.."

If you want to lead a a republic government by opinions you must compromise with everyone and not by forced laws. If it helps understanding a republic government is a government led by people that is not biased.

In a democratic nation, the people are used to being led by reasonable people who have the gift of persuasion to their side of an argument if they have enough evidence to support it. We are not a people who want to be told what to do, when to do it and how to do it. We believe our minds are worth using and to that end, we have the right to research a problem or issue with all the resources available and come to a conclusion. Reasoning is part of coming to that conclusion about the government and the people in office. We have to discuss the issues and decide who is doing the best job of representing us. If we no longer use our reasoning power as citizens we have given in and become nothing more than a society of people with no rights.  (I think it means something like this.)