Newton was definitely a scientist, by any reasonable meaning of the word. But not all the time.

The mystical passage that we've quoted, complete with alchemical symbols59 and obscure terminology, is one that he wrote in the 1690s after more than twenty years of alchemical experimentation. He was then aged about 50. His best work, on mechanics, optics, gravity, and calculus, was done between the ages of 23 and 25, though much of it was not published for decades.

Many elderly scientists go through what is sometimes called a 'philosopause'. They stop doing science and take up not very good philosophy instead. Newton really did investigate alchemy, with some thoroughness. He didn't get anywhere because, frankly, there was nowhere to go. We can't help thinking, though, that if there had been somewhere, he would have found the way.

We often think of Newton as the first of the great rational thinkers, but that's just one aspect of his remarkable mind. He straddled the boundary between old mysticism and new rationality. His writings on alchemy are littered with cabbalistic diagrams, often copied from early, mystical sources. He was, as John Maynard Keynes said in 1942, 'the last of the Magicians ... the last wonder-child to whom the Magi could do sincere and appropriate homage'. What confuses the wizards is an accident of timing -well, we must confess that it is actually a case of narrative imperative. Having homed in on Newton as the epitome of scientific thinking, the wizards happen to catch him in post-philosopausal mode. Hex is having a bad day, or perhaps is trying to tell them something.

If Archimedes wasn't a scientist and Newton was only one sometimes, just what is science?

Philosophers of science have isolated and defined something called the 'scientific method', which is a formal summary of what the scientific pioneers often did intuitively. Newton followed the scientific method in his early work, but his alchemy was bad science even by the standards of his day, when chemists had already moved on. Archimedes doesn't seem to have followed the scientific method, possibly because he was clever enough not to need it. The textbook scientific method combines two types of activity. One is experiment (or observation - you can't experiment on the Big Bang but you can hope to observe traces that it left). These provide the reality-check that is needed to stop human beings believing something because they want it to be true, or because some overriding authority tells them that it's true. However, there is no point in having a reality-check if it's bound to work, so it can't just be the same observations that you started from.

Instead, you need some kind of story in your mind.

That story is usually dignified by the word 'hypothesis', but less formally it is the theory that you are trying to test. And you need a way to test it without cheating. The most effective protection against cheating is to say in advance what results you expect to get when you do a new experiment or make a new observation. This is 'prediction', but it may be about something that has already happened but not yet been observed. 'If you look at red giant stars in this new way then you will find that a billion years ago they used to ...' is a prediction in this sense.

The most naive description of the scientific method is that you start with a theory and test it by experiment. This presents the method as a single-step process, but nothing could be further from the truth. The real scientific method is a recursive interaction between theory and experiment, a complicity in which each modifies the other many times, depending on what the reality-checks indicate along the way.

A scientific investigation probably starts with some chance observation. The scientist thinks about this and asks herself 'why did that happen?' Or it may be a nagging feeling that the conventional wisdom has holes in it. Either way, she then formulates a theory. Then she (or more likely, a specialist colleague) tests that theory by finding some other circumstance in which it might apply, and working out what behaviour it predicts. In other words, the scientist designs an experiment to test the theory.

You might imagine that what she should be trying to do here is to design an experiment that will prove her theory is correct.60 However, that's not good science. Good science consists of designing an experiment that will demonstrate that a theory is wrong -if it is. So a large part of the scientist's job is not 'establishing truths', it is trying to shoot down the scientist's own ideas.

And those of other scientists. This is what we meant when we said that science tries to protect us against believing what we want to be true, or what authority tells us is true. It doesn't always succeed, but that at least is the aim.

This is the main feature that distinguishes science from ideologies, religions and other belief systems. Religious people often get upset when scientists criticise some aspect of their beliefs.

What they fail to appreciate is that scientists are equally critical about their own ideas and those of other scientists. Religions, in contrast, nearly always criticise everything except themselves.

Buddhism is a notable exception: it emphasises the need to question everything. But that may be going too far to be helpful.

Of course, no real scientist actually follows the textbook scientific method unerringly. Scientists are human beings, and their actions are driven to some extent by their own prejudices. The scientific method is the best one that humanity has yet devised for attempting to overcome those prejudices. That doesn't mean that it always succeeds. People, after all, are people.

The closest that Hex manages to come to genuine science is Phocian the Touched's lengthy and meticulous investigation of Antigonus's theory of the trotting horse. We hope that you have heard of neither of these gentlemen, since, to the best of our knowledge, they never existed. But then, neither did the Crab Civilisation -which didn't stop the crabs making their Great Leap Sideways. Our story here is modelled on real events, but we've simplified various otherwise distracting issues. With which we shall now distract you.

The prototype for Antigonus is the Greek philosopher Aristotle, a very great man who was even less of a scientist than Archimedes, whatever anyone has told you. In his De Incessu Animalium

{On the Gait of Animals) Aristotle says that a horse cannot bound. The bound is a four-legged gait in which both front legs move together, then both back legs move together. He's right, horses don't bound. But that is the least interesting thing here. Aristotle explains why a horse can't bound: If they moved the fore legs at the same time and first, their progression would be interrupted or they would even stumble forward ... For this reason, then, animals do not move separately with their front and back legs.

Forget the horse: many quadrupeds do bound, so his reasoning, such as it is, must be wrong. And a gallop is very close to a bound, except that the left and right legs move at very slightly different times. If the bound were impossible, then by the same token so should the gallop be. But horses gallop.

Oops.

You can see that all this is a bit too messy to make a good story, so in the interests of narrativium we have replaced Aristotle by Antigonus, and credited him with a very similar theory about a long-standing historical conundrum: does a trotting horse always have at least one hoof on the ground? (In a trot, diagonally opposite legs move together, and the pairs hit the ground alternately.) This is the kind of question that must have been discussed in ale-houses and public baths since well before the time of Aristotle, because it's just out of reach of the unaided human eye. The first definitive answer came in 1874 when Eadweard Muybridge (born Edward Muggeridge) used high-speed photography to show that sometimes a trotting horse has all four feet off the ground at once. The proportion of times this occurs depends on the speed of the horse, and can be more than Phocian's 20 per cent. It can also be zero, in a slow trot, which further complicates the science. Allegedly, Muybridge's photographs won Leland Stanford Jr, a former Governor of California, the tidy sum of $25,000 in a bet with Frederick MacCrellish.