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The telescope with the crimson covering magnified nine times. Galileo made many of them during the course of the autumn. He did not feel constrained by his promise to the Senate to keep the invention secret - the more so since basically it was not his own invention - but he had to admit that the products were of variable quality depending on how well he succeeded with the lenses.

The objective lenses in the first telescopes were small. Galileo discovered that he could cut them more accurately if he made them larger, but this affected the sharpness of the image. Then he began to stop down the objective, to cover most of it so that only a small opening was left. This was an improvement, and so were his experiments at fashioning the tube so that it could be pulled out or pushed in.

Although Galileo also tried to describe the theory of "perspective" and "refraction", it was his practical experiments that gradually improved the telescope. By early autumn he had an example that would magnify twenty times.

As soon as the nights at Padua grew long and dark, he took the next step. He raised his cannocchiale and pointed it to the sky.

Galileo still had not gone public with his views on cosmology. As far as the Copernican system was concerned, he still lived and taught according to a Sarpi maxim:

"Your innermost thoughts should be guided by reason, but you should act and speak only as others do"18.

But he had a definite feeling that the general consensus about astronomical truths was basically just as flawed as he had shown the "truth" about bodies in free fall to be. Indeed, he suspected, though he could not prove it, that the spherically shaped bodies in the sky in some way adhered to the same simple laws as the earthly balls he had carefully timed on hundreds of occasions rolling down his inclined planes.

The telescope now afforded him a golden opportunity to observe heavenly phenomena that no one else had yet seen. But competitors would soon have similar instruments in their hands, so he had to make the best use of his head start.

The Moon was the obvious and easiest object to observe. From the end of November he took regular observations which provided a totally new and startling picture. Instead of the smooth spherical surface that the textbooks described, the Moon's surface was clearly disturbed and rough, with valleys, mountains and craters. No circular, Aristotelian perfection there.

A new world opened up as he lifted the telescope yet higher: the broad band of the Milky Way dissolved into stars, myriads of unknown stars! They had never been seen by Ptolemy or anyone else, were not marked on any celestial chart, had never been accorded any astrological importance. But they existed.

All this was important enough. But his great, ground-breaking discovery began on the evening of 7 January 1610.

Galileo, a professor of mathematics, was no practised astronomer of the Tycho Brahe school accustomed to making painstakingly accurate reckonings. But that evening he tried to locate Jupiter whose aspect was then favourable. As the objective lens had had to be stopped down using a piece of cardboard with a hole in it, the angle of vision was minimal. When the professor finally found the planet, he noticed that it appeared as a tiny flat disc, and not just one point of light. In addition he saw three small, unknown stars which were directly in line with Jupiter and lying close to the planet. Two of them were due east of Jupiter, the third further west.

The sighting was odd, but Galileo had already found that his telescope picked out many new and unknown stars. And so he calmly noted down the new find, pleased that he had made yet another discovery that would astound the astronomical establishment and bring him yet more kudos -and, perhaps, in the long run, an even greater income than the Venetian Senate's thousand scudi.

The very next evening, 8 January, found him ready to make more observations of the moving planet Jupiter, which according to the calculations and tables should now have shifted its position in the night sky a little in relation to the fixed stars.

As Jupiter, seen from the Earth, should have been moving westwards just then, Galileo expected the planet to have passed the third of the newly discovered stars such that all three would now be east of the planet. But when at last he managed to find Jupiter in his small field of vision, there was nothing to be seen on the eastern side. The three small stars all sat neatly on its western side.

Galileo sketched the constellation on a piece of paper and compared it with his notes from the evening before. There was no room for doubt: Jupiter was not behaving the way it was supposed to. He had no idea why. But as most apparent mysteries are eventually found to have simple explanations, he imagined it might have to do with some error in the tables. For most of the year, Jupiter's observed orbit moved eastwards in relation to the fixed stars. If this held good for these January days as well there would be nothing to explain at all.

Even so, he was in some anticipation about what he would see on the next evening. But 9 January was a mild and overcast day in Padua. The sky was a uniform grey, not the slightest glimpse of any star penetrated the cloud cover.

10 January 1610 was, by contrast, a cool clear day and the good weather lasted until nightfall. As soon as it got dark, Galileo set up his telescope. He fixed it to a tripod - without support it was difficult to hold the instrument still enough when one was looking at distant objects. Then he cleaned the lenses and turned the telescope to the point in the sky where Jupiter should now be.

What he saw was enough to convince him that he was on the track of something incomprehensible, something entirely new and unknown.

Only two of the stars were now visible, but this evening both were back where he had seen them three days earlier, in a line east of Jupiter.

Perhaps Galileo Galilei's principle characteristic as a scientist was his uncanny ability to draw fast, almost intuitive conclusions from a limited number of observations. He once wrote that this was the way God himself reasoned: "... immediate conclusions, without transitions, is what characterises God's mind." (But when it was Galileo and not God who was making the mental leaps, the conclusions were not always correct.)

He sketched the three heavenly bodies once again and pondered the phenomenon. The tables of Jupiter's movements had no bearing on the matter. Regardless of any failings there, planets always moved more or less evenly eastwards or westwards, they did not jump backwards and forwards from one evening to the next.

There was obviously a possibility that, from one observation to the next, he had mixed up the small new stars with other stars near by. But no matter how carefully he searched the sky around Jupiter, he found no other stars. And the only reasonable explanation for the apparent absence of one of them, was that for the time being, it was hidden behind Jupiter.

Only one explanation was left: the strange phenomenon had nothing to do with Jupiter's orbit. It must be connected with a motion in the small stars themselves!

The problem was that according to all astronomical wisdom from the Greeks to Tycho Brahe, the fixed stars stood still - that was why they were called fixed stars. Only planets moved, and then only in their fixed orbits and not in this confusing way.

Even that night Galileo had suspicions about what he was looking at. But on the next evening, 11 January, when the small stars were displaying yet another configuration, he was certain. The matter was, he wrote, "as clear as day": the small points of light were neither stars nor planets. They were - no matter how ridiculous it sounded - moons.

In contrast to the vast majority of his contemporaries, Professor Galilei certainly despised astrology. But he could read a portent in the sky when he saw one. He knew very well what these satellites - two evenings later he discovered there were actually four of them - heralded. They presaged not just a revolution in astronomy, but if he acted swiftly and surely, they might also bring about a dramatic change in his own fortunes.

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