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would command. It easily overcame Wilson’s pacifism. Feynman himself visited an army recruitment office and offered to join the Signal Corps. When he was told he would have to start with unspecialized basic training—no promises—he backed down. That spring, in 1941, after three years of frustration, he finally got a job offer from Bell Laboratories in New York, and he wanted to accept. When his friend William Shockley showed him around, he was thrilled by the atmosphere of smart, practical science in action. From their windows the Bell researchers could see the George Washington Bridge going up across the Hudson River, and they had traced the curve of the first cable on the glass. As the bridge was hung from it, they were marking off the slight changes that transformed the curve from a catenary to a parabola. Feynman thought it was just the sort of clever thing he might have done. Still, when a recruiter from the Frankford Arsenal nearby in Philadelphia—an army general—visited Princeton seeking physicists, Feynman did not hesitate to turn down Bell Laboratories and sign up with the army for the summer. It was a chance to serve his country.

In one way or another, by the time the United States entered the war in December, one-fourth of the nation’s seven-thousand-odd physicists had joined a diffuse but rapidly solidifying military-research establishment. A generation brought up with the understanding that science meant progress, the harnessing of knowledge and the empowerment of humanity, now found a broad national purpose. A partnership was already forming between the federal establishment and the leaders of scientific institutions. The government created in the summer of 1941 an Office of Scientific Research and Development, subsuming the National Defense Research Committee, charged with coordinating research in what MIT’s president, Karl Compton, the epitome of the new partnership, called “the field of mechanisms, devices, instrumentalities and materials of warfare.” Not just radar and explosives but calculating machines and battlefield medicines occupied the urgent war effort. An area like artillery was no longer a matter of haphazard trial-and-error lobbing of randomly designed shells. The nuclear physicist Hans Bethe had turned on his own initiative to a nascent theory of armor penetration; he also took on the issue of the supersonic shock waves that would shudder from the edge of a projectile. Less glamorously, Feynman spent his summer at the Frankford Arsenal working on a primitive sort of analog computer, a combination of gears and cams designed to aim artillery pieces. It all seemed mechanical and archaic—later he thought Bell Laboratories would have been a better choice after all.

Still, even in his college workshops, he had never confronted such an urgent blending of mathematics and metal. To aim a gun turret meant converting sines and tangents into steel gears. Suddenly trigonometry had engineering consequences: long before the tangent of a near-vertical turret diverged to infinity, the torque applied to the teeth of the gears would snap them off. Feynman found himself drawn to a mathematical approach he had never considered, the manipulation of functional roots. He divided a sine into five equal subfunctions, so that the function of the function of the function of the function of the function equaled the sine. And the gears could handle the load. Before the summer ended he was given a new problem as well: how to make a similar machine calculate a smooth curve—the path of an airplane, for example—from a sequence of positions coming in at regular intervals of a few seconds. Only later did he learn where this problem had arisen—from radar, the new technology from the MIT Radiation Laboratory.

After the summer he returned to Princeton, nothing remaining in his graduate education except the final task of writing his thesis. He worked slowly, trying out his least-action view of quantum mechanics on a variety of basic, illustrative problems. He considered the case of two particles or particle systems, A and B, which