A quarter of a century later, it is a shock to realise that Neil Armstrong was not addressing the whole world when he became the first human to walk on the surface of the Moon. He was talking to the people in a building that looks no further along the technological continuum than my comprehensive school - though with a little less graffiti.
Mission Control, which has talked astronauts up and down from the triumph of Apollo XI to the catastrophe of the Challenger, is a modest room in a squat, plain building, the Johnson Space Center, in a suburb of Houston. It looks nothing as grand as the flight deck of the Starship Enterprise.
Yet using equipment which today looks suspiciously like a Blue Peter project that has got out of hand, they managed to put men on the Moon, and bring them back. To give some idea of the primitive tools with which Nasa's scientists worked, this story is being written on a pounds 500 laptop with more computing power than that available for the Apollo missions.
The first word spoken on the Moon was 'Houston' - as in 'Houston. Tranquillity Base here. The Eagle has landed.' And it was addressed to the modest room.
The man whose surname won the accolade, Sam Houston, would have made a good public relations operator. He persuaded investors to lay the foundations for a city that had no rational grounds for existence. Houston (the place, not the man) was a patch of swamp inland from the Gulf of Mexico, where only mosquitoes thrived. The settlement lurched from one epidemic to the next, sustained first by the arrival of the railway, next by the discovery of oil in 1903.
If Houston had a good excuse for being, it might have accumulated some civic history. Instead, the city centre is a grid of fast, wide streets creating blocks on which skyscrapers have been plonked - or not. Whether a particular lot supports a high-rise office complex in reflective green glass or (the only alternative) a car park seems to be based on a random process.
Ground control in the city is carried out by police wearing sunglasses tinted the same sinister reflecting shade as most of the buildings in the city. In July, the tourist office advises the visitor to stick to the five miles of underground tunnels, which constitute a refrigerated alternative to being baked by the Texas sun.
Through the heat haze, Houston appears a graceless collation of high- rises and low-hopers. America's fourth-largest city (New York, Los Angeles and Chicago have more people and allure) is never going to win a beauty contest, but oil money can do more than build big, ugly cities.
When Houstonians hear the word 'culture', they evidently reach for their wallets. The city's array of theatres, museums and art galleries is commensurate with an overflow of disposable income, and nowhere is this more elegantly manifested than in the Menil Collection. The exterior looks like Mission Control should: a vast, sheer, gleaming white slab. One small step inside transports you to a miraculous universe where exquisite Byzantine relics are displayed 10 paces away from some of Warhol's soup cans. Space, time and art have become hopelessly and wonderfully confused.
What Houston lacks in body and soul, it makes up for in phallic symbols. Head into the south-eastern suburbs and you suddenly spot a phalanx of them. Is that a Saturn V in their parking lot, you wonder, or are they just pleased to see the tourists?
The Johnson Space Center was on the other end of Neil Armstrong's mobile phone. It looks like a Sixties industrial park, with only the scattering of cylindrical skyrockets to indicate the center's extra-terrestrial activities. For two decades, it opened its doors only coyly to visitors. It was educational, in the grimmest sense of the word. Visitors were shown around not by slightly manic scientists in white coats and thick spectacles, but by insufferably earnest National Park Rangers in silly cowboy hats.
Two years ago, Nasa's fun consultants got on the case, changed the name and cut out all the technical tosh. So now visitors get trundled around the heart of the space effort in ridiculous trolleys. (They missed a trick here: why not imitation lunar landing vehicles?) Guides in space uniforms lead you from the flotation chamber (a huge swimming pool where astronauts train for weightlessness) to Earth's link with the Moon, Mission Control.
You file in to sit and gawk in the same seats where astronauts' relatives sit and fret. When no mission is taking place, tourists are allowed in to look through thick glass at the communications hub; during a mission the hottest seat, the one labelled 'Flight Director', is occupied by an astronaut who translates the requirements of the dozens of ground staff into instructions digestible in orbit.
It all makes you want to take to the stars. Happily, the Johnson Space Center now also has the best little theme park in Texas. After a hike around the rockets in the parking lot, where you discover just how ludicrously large the Saturn V actually is, you are free to indulge in some final- frontier fun. You can see the flight deck of a space shuttle, try on an astronaut's helmet and walk through the Skylab trainer - a padded-cell version of the rotation spaceship.
The next time men or women land on the Moon, my father reckons, the first word they are likely to say is 'Darmstadt' (the German city where European Space Agency missions are controlled). It hasn't got the same electric crackle as Houston, and I bet it hasn't got a built-in theme park either.
British Airways and Continental Airlines fly between Gatwick and Houston. The cheapest official return fare on BA is pounds 501. Major Travel (071-485 7017) has a fare of pounds 424 on Continental.
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