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Oh, but they did. Archimedes constructed great cranes that could lift enemy ships out of the water, and we still don't know exactly how he did it. Hero of Alexandria (roughly contemporary with Jesus) wrote many texts about engines and machines of various kinds of the previous three hundred years, many of which show that prototypes may have been made. His coin-operated machines were not too different from those that could be found on any city street in 1930s London or New York, and would probably have been more reliable when it came to disgorging the chocolate, if the Greeks had known about chocolate. The Greeks had elevators, too.

The problem here is that information about the technical aspects of Greek society has been transmitted to us through a bunch of theologians. They liked Hero's steam engine, and indeed many of them have a little glass one on their desk, a sort of Theologians' Toy that they could spin with a candle flame. But the mechanical ideas behind such toys just passed them by. And, just as Greek engineering has not been transmitted to us by theologians, the spiritual attitude of the Renaissance has not come down to us through our 'rational' school teachers. Much of the attempted spirituality within the alchemical position was basically a religious stance, marvelling at the Works of the Lord as they were exposed by the marvels of changes of state an form, when materials were subjected to heat, to 'percussion', and to solution and crystallisation.

This stance has been taken over by today's innocents of rigorous thinking, the New Agers, who find spiritual inspiration in crystals and anodised metals, spherical spark-machines and Newton's pendulums, but do not ask the deeper questions that lie behind these toys. We find the very real awe inspired by science's quest for understanding to be considerably more spiritual than New Age attitudes.

Today there are mystic massage-therapists, aromatherapists, iridologists, people who believe that you can 'holistically' tell what's wrong with someone by examining their irises or the balls of their feet -only -and who root their beliefs in the writings of Renaissance eccentrics like Paracelsus and Dee. But those men would have been horrified to be cited as authorities, especially by such closed-minded descendants.

Prominent among those who refer back to Paracelsus for authority are homeopathists. A basic belief of homeopathy is that medicines become more powerful the more they are diluted. This stance lets them promote their medicine as being totally harmless (it's just water) but also extraordinarily effective (as water isn't). They notice no contradiction here. And homeopathic headache tablets say 'Take one if mild, three if painful'. Shouldn't it be the other way round?

Such people see no need to think about what they are doing, because they base their beliefs on authority. If a question is not raised by that authority, then it's not a question they want to ask.

So, in support of their theories, homeopaths quote Paracelsus: 'That which makes disease is also the cure.' But Paracelsus built his entire career on not respecting authority. Moreover, he never said that a disease is always its own cure.

Contrast this modern spectrum of silliness with the robust, critical attitude of most Renaissance scholars to the idea that arcane practices can lay bare the bones of the world. People such as Dee, indeed Isaac Newton, took that critical position very seriously. To a great extent, so did Paracelsus: for example he repudiated the idea that the stars and planets control various parts of the human body. The Renaissance view was that God's creation has mysterious elements, but those elements are hidden,21 implicit in the nature of the universe, rather than arcane.

This view is very close to Antonie van Leeuwenhoek's marvelling at the animalcules in dirty water, or semen: the astonishing discovery that the Wonders of Creation extended down into the microscopic realm. Nature, God's Creation, was much more subtle. It provided hidden wonders to marvel at as well as the overt artistic vision. Newton was taken with the implicit mathematics of the planets in just this way: there was more to God's invention than was apparent to the unaided eye, and that resonated with his Hermetic beliefs (a philosophy derived from the ideas of Hermes Trismegistos). The crisis of atomism at time was the crisis of pre-formation: if Eve had within her all daughters, each having within her her daughters like a set of Russian dolls, then matter must be infinitely divisible. Or, if not, we could work out the future date of Judgement Day by discovering how many generations there were until we got to the last, empty daughter.

A characteristic of Renaissance thinking, then, was a degree of humility. It was critical about its own explanations. This attitude contrasts favourably with such modern religions as homeopathy, Scientology, creeds that arrogantly claim to offer a 'complete' explanation of the Universe in human terms.

Some scientists are equally arrogant, but good scientists are always aware that science has limitations, and are willing to explain what they are. 'I don't know' is one of the great, though admittedly under-utilised scientific principles. Admitting ignorance clears away so much pointless nonsense. It lets us cope with stage magicians performing the beautiful, and very convincing, illusions -convincing, that is, while we keep our brains out of gear. We know they have to be tricks, and admitting ignorance lets us avoid the trap of believing the illusion to be real merely because we don't know how the trick works. Why should we? We're not members of the Magic Circle. Admitting ignorance similarly protects us against mystic credulity when we encounter natural events that have not yet caught the eye of a competent scientist (and his grant- awarding body), and that still seem to be ... magic. We say 'The magic of nature' ... more the Wonder of Nature, or the Miracle of Life.

This is a stance that nearly all of us share, but it's important to understand the historical tradition it is grounded in. It isn't simply a case of admiring the complexity of God's works. It implies the attitudes of Newton, van Leeuwenhoek and earlier; indeed, right back to Dee. And, doubtless, to some Greek, or several. It involves the Renaissance belief that if we investigate the wonder, the marvel, the miracle, then we'll find even more wonders, marvels and miracles: gravity, say, or spermatozoa.

So what do we, and what did they, mean by 'magic? Dee spoke of the arcane arts, and Newton was committed to many explanations that were 'magickal', especially his commitment to action at a distance, 'gravity', which derived from the mystical attraction/repulsion basics of his Hermetic philosophy.

So 'magic' means three things, all apparently quite different. Meaning one is: 'something to be wondered at', and this ranges from card tricks to amoebas to the rings of Saturn. Meaning two is turning a verbal instruction, a spell, into material action, by occult or arcane means ... turning a person into a frog, or vice versa, or a djinn building a castle for his master. The third meaning is the one we use: the technical magic of turning a light switch on, and getting light, without even having to say 'fiat lux'.

Granny Weatherwax's recalcitrant broomstick is type two magic, but her 'headology' is largely a very, very good grasp of psychology (type three magic carefully disguised as type two). It brings to mind Arthur C. Clarke's phrase 'Any sufficiently advanced technology is indistinguishable from magic', which we quoted and discussed in The Science of Discworld. Discworld exemplifies magic by spells, and indeed is maintained as an unlikely creation by being immersed in a strong magical field (type two). Adults of Earthly cultures, like Roundworld, pretend to have lost intellectual belief in magic of the Discworld kind, while their culture is turning more and more of their technology into magic (third kind). And the development of Hex throughout the books is turning Sir Arthur on his head: Discworld's sufficiently advanced magic is now practically indistinguishable from technology.