How we cite our quotes: (Paragraph.Sentence)
Quote #1
Let us all understand that the question before us is not whether some Americans are for peace and some Americans are against peace. (19.1)
Think about this in terms of delicious pizza. Everybody loves pizza—this much we can agree on. Nixon is just saying that those protesters who don't support his policies in Vietnam are the pizza weirdos: the ones who put anchovies, pineapple, and probably grape salad all over their pizzas.
Quote #2
Well, one of the strengths of our free society is that any American has a right to reach that conclusion and to advocate that point of view. But as President of the United States, I would be untrue to my oath of office if I allowed the policy of this Nation to be dictated by the minority who hold that point of view and who try to impose it on the Nation by mounting demonstrations in the street. (116.1-116.2)
Here's the meat and potatoes of Nixon's whole argument. Nixon really didn't want all those anti-war protestors to change the events in Vietnam. So, he's saying that they're just a small fraction of Americans, and therefore shouldn't really have that strong a voice in politics, especially in regard to the war in Vietnam.
Quote #3
If a vocal minority, however fervent its cause, prevails over reason and the will of the majority, this nation has no future as a free society. (117.2)
If the previous statement represented Nixon's meat and potatoes, this one is the side salad with a healthy dollop of contempt for dressing. Nixon really didn't want those anti-war protestors to change the events in Vietnam, so here he's basically saying that if the anti-war protestors get their way, the United States would crumble, and war would probably break out anyway.