National Air and Space Museum audio guide,
five stops, free.

Take a moment before anything else. Wood, pale muslin, two long skids where you would expect wheels, and not a drop of paint anywhere; the fabric is left in its natural finish. It looks like something a strong gust could destroy, and the honest answer is that one did. This is the actual machine that flew at Kitty Hawk, North Carolina, on the seventeenth of December 1903, and the whole gallery around you is built around it.

Wilbur and Orville Wright ran a bicycle shop in Dayton, Ohio. In May 1899 Wilbur wrote a letter to the Smithsonian, the same institution whose building you are standing in, asking for whatever it had published about flight. Four years of patient work followed: a kite in 1899, full-size gliders in 1900, 1901 and 1902, and, when those gliders lifted far less than their calculations predicted, a small wind tunnel of their own to test wing shapes. By 1902 they could glide under control. What they still needed was power.

So look behind the wings. Two propellers, pushing rather than pulling, turned by chains from a twelve-horsepower engine they built with their shop mechanic, Charles Taylor. The propellers were the cleverest idea of all. The brothers treated each blade as a wing turned on its side, making lift forwards instead of upwards. Now look to the front. That horizontal surface out ahead of everything is the elevator; there is no tailplane at all. To bank, the pilot twisted the wingtips in opposite directions, a trick called wing warping.

On the fourteenth of December Wilbur won a coin toss, took the first try, and damaged the machine. Three days later it was Orville's turn. At 10:35 in the morning he flew for twelve seconds and thirty-six metres. They made four flights that day, and on the last one Wilbur stayed up for fifty-nine seconds and covered 255 metres. Then, while they stood talking about it, a gust flipped the Flyer over and sent it tumbling across the sand. It never flew again.

Its afterlife is almost stranger. It sat crated in a shed in Dayton, spent eleven days under floodwater and mud in 1913, and then became the prize in a feud. The Smithsonian's secretary, Charles Walcott, refused to credit the Wrights and honoured the failed Aerodrome of his predecessor Samuel Langley instead. In 1928, with the quarrel still unsettled, Orville lent the Flyer to the Science Museum in London. It spent the Second World War in an underground store near Corsham, not in the London Underground, whatever you may have heard. Only after the Smithsonian printed a retraction in 1942 did it come home, and it was handed over on the seventeenth of December 1948, forty-five years to the day after Kitty Hawk.

The fabric you see dates from 1985, cut using a large piece of the original 1903 covering as its pattern. Some of that original went much further. In July 1969 a scrap of cloth from the upper left wing and a sliver of wood from the left propeller were inside the lunar module Eagle when it landed on the Moon. Sixty-six years separate the sand at Kitty Hawk from the Sea of Tranquility.

Gold foil, spindly legs, and a grey cabin on top that looks as if it was folded rather than built. If it reminds you of the photographs of Eagle on the Moon, that is exactly the intention. But the label beside it is honest about the twist: this lunar module has never been to space.

Grumman built it at Bethpage, on Long Island. By the end of 1966 it and its twin, LM-1, were almost finished, meant for uncrewed test flights in Earth orbit. Then came the tragic Apollo 1 fire in January 1967, and the safety rules for any spacecraft that would carry astronauts were rewritten. Rather than rebuild two nearly completed landers, NASA kept them for uncrewed missions only. LM-1 flew on Apollo 5 in January 1968 and did well enough that a second test was judged unnecessary. LM-2's delivery to the launch site was delayed, then cancelled.

So it went to work on the ground. In April 1968 the uncrewed Apollo 6 launch showed that the Saturn V rocket could shudder violently up and down, an effect engineers called pogo, and in May LM-2 became part of the test system built on the ground to study it. Then it was dropped, again and again, to prove its electronics and mechanics would survive a jarring touchdown. The fifth and final drop test was on the seventh of May 1969, less than three months before Neil Armstrong and Buzz Aldrin walked on the Moon.

Next came a detour to Japan. At Expo 70 in Osaka the American pavilion wanted a real lunar module, and the real Eagle was not available: its lower half is still sitting on the Sea of Tranquility. So Grumman paired LM-2's cabin with a test landing stage and dressed it to look as if it had just touched down. Afterwards it came to the Smithsonian, and from July 1976 until late 2015 it stood on an imitation lunar surface at the far east end of this building.

Now look at the two halves. Up top is the ascent stage, the pressurised cabin with its own rocket engine, the only part that would have left the Moon. Below it is the descent stage with the legs, the landing engine and the surface experiments, wrapped in gold and silver blankets of aluminized Mylar and Kapton. When the museum moved it into this hall, the team took it apart and set out to make it match Eagle in virtually every detail, and every scrap of material they removed was photographed, documented and kept. Even the ladder on the front leg is there, as on Eagle.

Between 1969 and 1972, six lunar modules identical to this one set twelve American astronauts down on the Moon. This one never flew at all, and that is precisely why it is here for you to stand beside.

Look at the front of it and try to find where the pilot looked out. You will not find it, and neither could he. There is no windscreen. The fuel tanks sit ahead of the cockpit, and Charles Lindbergh crossed the Atlantic seeing forward only through a small periscope on the left side, or by swinging the plane round to peer out of a side window.

In 1919 a man named Raymond Orteig offered twenty-five thousand dollars to the first pilot to fly nonstop between New York City and Paris. Lindbergh was a former barnstormer who had become an airmail pilot on the run between Saint Louis and Chicago, and in February 1927, with money from a group of backers in Saint Louis, he ordered an aeroplane from Ryan Airlines in San Diego. They built it in two months. It started as an ordinary Ryan M-2 and was stretched to carry fuel: ten more feet of wing, two more feet of fuselage, the engine pushed forward and the cockpit pushed back, so that the tank sat at the centre of gravity and the pilot sat behind it, where he would be safer in a crash.

He flew it east to New York first, with a single stop in Saint Louis, and set a transcontinental record on the way, twenty-one hours and forty minutes. Then, on the morning of the twentieth of May 1927, he took off alone for Paris. Thirty-three and a half hours and 3,610 miles later he came down at Le Bourget, and a crowd of a hundred thousand people was waiting for him. There are pencil tallies inside the cockpit showing how he did it: he kept count of the fuel he drew from each tank, switching between them to keep the plane in balance.

The crowd in Paris wanted souvenirs, and took them. Look at the rudder. On the left side there is a winged letter R. On the right side there is only a blank patch, because somebody cut the fabric away.

Now look at the nose. Those flags are the countries of the tour he flew afterwards, from Washington to Mexico City and on through Central America, Colombia, Venezuela and the Caribbean. The Puerto Rican one is wrong, by the way: it should have five stripes. The two tubes under the fuselage were added for that tour, to drop flares so he could see to land at night. And the cowling is covered in signatures, pencilled or scratched with penknives by the guards and mechanics who watched over the plane wherever it went.

President Calvin Coolidge called Lindbergh a messenger of peace and goodwill who had broken down another barrier of time and space, and the museum reckons that nothing in aviation stirred the whole world as much again until Apollo 11, whose capsule you have just left. On the thirtieth of April 1928 Lindbergh flew the Spirit one last time, from Saint Louis to Washington, and gave it to the Smithsonian. Its door stands open, and it has stayed that way because he asked for it.