It was an ammonia nitrate bomb, said Chris Cole, an agent with the FBI in Madison, which investigated the crime.
"It was generally viewed as the largest, most destructive terrorist attack occurring on U.S. soil prior to the Oklahoma City Bombing," Cole said.
"It was something that I'd only ever seen in film," said Soglin, "and in scenes of buildings blowing up during World War II - things that happened in war."
And while the bombing was timed to make certain the building was empty, the reality, Cole said, "is there was someone there, and he was killed."
Armstrong and his "confederates" heard the news in their getaway car.
"Well, hear on the radio that, basically, that the announcer said they were pulling a body out of Army Math," said Armstrong. "And it looked like someone had died. All of us were saying, 'Oh no...' And then I told everybody in the car, I said, 'Maybe with time, we'll feel better about this.'"
"Has that time come?" Greenberg asked.
"Oh, I don't think you can ever feel good about it," Armstrong said. "I felt good about doing the bombing, the bombing per se, but not taking someone's life."
"There had been deaths on American college campuses already," said Greenfield. "It's just that the deaths before that were perpetrated by the forces of authority. This time, it was the dissidents."
In a contemporary report CBS News correspondent Ike Pappas said, "There's no question in the minds of authorities that this was the work of dissident radicals."
"This was off the charts," said Soglin. "This was so different than anything that had happened."
And Paul Soglin said it hurt the cause it was supposed to help. "When school reopened a couple of weeks later, it was as though the life had been sucked out of the anti-war movement," he said.
"I think the Army Math Research Center bombing was the moment when most everybody in the movement had to look into their own souls and minds, and say, 'What are we about?'" said Greenfield.
Armstrong's father urged Karl and his brother Dwight to turn themselves in: "They'll find you eventually anyway," he said.
Karl and Dwight - along with David Fine - were eventually caught and imprisoned. Karl served 8 years of a 23-year sentence.
But the fourth bomber, Leo Burt, disappeared - vanished.
Four decades and counting since that early morning in August, there's still a $150,000 reward for his capture. But the trail seems to have grown cold.
As for Karl Armstrong, he returned to Madison after his prison term.
In fact, until recently, Armstrong operated a juice stand just blocks from the campus building he destroyed.
Karl's brother Dwight also returned to Madison - he died last year.
And David Fine, who went on to law school but was denied admission to the Oregon Bar, has worked as a paralegal.
Robert Fassnacht's widow, Stephanie, remained in Madison, where she raised their three children. She never remarried
She declined to appear in this story. But she did give us a message for Karl Armstrong: "I would like him to know that I harbor no ill will towards him - and I never did."
"I've always felt shame for, you know, it's - but I felt that the original motivation for the bombing was true," said Armstrong.
Still, Armstrong's message for his fugitive confederate may surprise you:
"If he's watching this show, if he's watching you right now, what would you want to say to Leo?" Greenberg asked.
"Good job keeping yourself free, Leo, good job," Armstrong replied.
"No hesitation on that message?" Greenberg asked.
"No. Because, you know, to us there was purity of purpose. And it just went bad."