By August 1968, the American space program was in danger of failing in its two most important objectives: to land a man on the Moon by President Kennedys end-of-decade deadline, and to triumph over the Soviets in space. With its back against the wall, NASA made an almost unimaginable leap: It would scrap its usual methodical approach and risk everything on a sudden launch, sending the first men in history to the Moon-in just four months. And it would...