That’s what transfixed my family and hundreds of millions of others. We were watching human beings overcome what we had always assumed was a basic limitation of our species to break a barrier that was physically or psychologically insurmountable. We were watching a triumph of the human spirit — and that, of course, is exactly why so many people are getting ready to enjoy this week’s Paralympic Games.

Not since the Paralympic movement began in 1948 has there been such a rush for tickets. Of the total of 2.4 million, there are only 200,000 still to be sold, and they are being snapped up as fast as Olympic tickets were — as fast as Locog can put them on to the website. We are getting ready for live sites in Trafalgar Square and Potters Fields, and Channel 4 is going to be running 150 hours of prime-time television. The International Paralympic Committee say they have never seen such interest and enthusiasm from any public. Without counting our chickens, we are confident that London will complete the success of the summer of 2012, and put on a wonderful Paralympic Games.

There will be prodigious displays of sporting excellence (for my money, wheelchair basketball is far better and more murderous than the other type). Indeed, there are some purists who say that we shouldn’t really be talking about the “back stories” of these amazing performers. They argue that we should simply see them as athletes, and blot out the difficulties they overcome. I hope I will be forgiven if I say I can’t see it like that. All athletes have a back story. Every athlete has had a barrier to surmount.

Is it relevant to our appreciation of Laura Trott that she was born with a collapsed lung? Too right it is. There isn’t a sportsman or woman alive who hasn’t at some time or other been called upon to show inner steel, and the Paralympians have had to overcome adversity in a way that is mesmerising, heartbreaking and inspiring. There are men and women who have been told by their doctors that they are beyond hope, who have been left for dead, who have come back from terrible accidents or injuries. These are people who have so comprehensively vanquished disability that they are defined by what they can do rather than what they can’t do. When Oscar Pistorius was a baby, his mother wrote him a letter that she intended for him to read as an adult. “The real loser is never the person who crosses the finishing line last,” she told him. “The real loser is the person who sits on the side, and does not even try to compete.”

Oscar Pistorius, who has no legs below the knee, is now one of the fastest men in the world. These people have The Right Stuff by the bucketload, exploding from every pore. When we watch them, we see the same phenomenon that drew the world round the TV set in 1969. It is the human urge to try, to risk, to dare, to master our fear and dive in; and then to keep doing the same until we succeed.

We admire that spirit in some deep and probably primeval way. It is, after all, the same competitive spark that produced just about every example of social, cultural, material and economic progress in the past 10,000 years. It was the spirit that drove Mo Farah and Jessica Ennis in those amazing and unforgettable Olympics of earlier this month. It was what got Neil Armstrong on to that Gemini programme, into the lunar module and on to the surface of the Moon.

It’s The Right Stuff, and the Paralympians have more of it than just about anyone else in the world.