FOR BERTRAND PICCARD, the idea to build a solar-powered plane capable of circumnavigating the globe was hatched while running on empty. In March 1999, Piccard was on the final leg of an around-the-world journey by hot air balloon—the first-ever nonstop flight of its kind—when his Breitling Orbiter 3 swept low over the Egyptian desert and skidded to a halt on the corrugated plains. As Piccard stepped out onto the hot sand, he checked the fuel tanks mounted on his gondola and got a shock that became a defining moment. "We had left Switzerland with four tons of propane," he remembers. "We only had 40 kilos...