What does John O'Sullivan mean when he says national birth?

He means that our national birth was the beginning of a new history, the formation and progress of a political system and not tried that it separates us from the past and it only connects us with the future; and till now as you greet the whole development of the natural rights of man, in the moral life, politics, and national that we can confidently suppose that our country is destined for being the great nation of future