An instrument that shows things nobody has seen before poses a problem that the things themselves do not: nobody knows yet whether to believe it.
The previous lesson left astronomy with exact laws, better tables and no way of deciding what actually moves. What changed the situation was not a better argument but a tube with two pieces of ground glass in it, and the two years from 1610 to 1613 produced more new astronomical facts than the previous fourteen centuries. This lesson is about what those facts established, what they did not, and why refusing to be convinced by them was not simply stupidity.
The instrument
The telescope was not invented by Galileo Galilei. In October 1608 Hans Lipperhey, a spectacle maker in Middelburg in the Netherlands, applied for a patent on a device for seeing distant things as if nearby; the States General refused the patent on the grounds that the idea was already too widely known. Within months the instruments were being sold as novelties across Europe.
What Galileo did, in the summer of 1609, was make a much better one. He worked out the arrangement from a description, ground his own lenses, and reached about eight times magnification by August and about twenty by that November, when everyone else was selling three. The improvement came from grinding the weak convex objective more accurately and from stopping the objective down with a ring of card to use only its central portion, which cuts the aberration of a simple lens at the cost of light. His field of view was tiny, about 15 arcminutes, or half the moon's width, and the image quality at the edge was poor. It was, for about eighteen months, the best telescope in the world.
He published in March 1610, in a short book called Sidereus Nuncius, the starry messenger. The print run of 550 sold out immediately.
The moon is a place
Turned on the moon, the telescope shows that the line dividing the lit half from the dark, the terminator, is not smooth. It is ragged. Bright points appear in the dark side, separated from the lit region, and grow and join it as the sun rises over them. On the lit side, dark shapes stretch away from features and shorten as the sun climbs.
Anyone who has watched a sunrise in mountains knows what that is: peaks catching the light before the valleys. Galileo said so, and then did something more useful than saying it. He measured one.
Example. A peak is lit while the surrounding surface is still dark, and it lies about one twentieth of the moon's diameter from the terminator. Taking the moon's radius as km, how high is the peak?
The terminator is where the sun's rays graze the surface. A peak of height at a distance from that line, measured across the face, is lit if it pokes up into the sunlight, and the geometry is a right-angled triangle with legs and and hypotenuse :
Here is one twentieth of the diameter, which is one tenth of the radius, so km, and
Galileo's own figure was four miles, about km. Either way the answer is that lunar mountains are higher than the Alps, whose greatest peak reaches km. The point is not the number but the kind of statement it is: a quantity, in miles, about a body in the heavens, obtained from a shadow. The moon is not a polished sphere of the fifth element. It is a place, with terrain, and its terrain can be surveyed from here.
Now you. A different peak is lit at one fortieth of the moon's diameter from the terminator. How high is it, and what does the relationship between the two answers tell you about the method's sensitivity?
Answer
Now km, and km. Halving the distance from the terminator quarters the height, because for small the expression is very close to . That square is a problem: an error in judging is doubled in the height, so an estimate of the distance good to twenty per cent gives a height good to only forty. It is also why the method works at all, since a mountain a few kilometres high on a body km across produces a lit patch you can see. Galileo's four miles is in the right range, and modern values for the highest lunar terrain are around 8 to 10 km above the mean radius, so if anything he was conservative.
Four objects that go round something else
On the night of 7 January 1610 Galileo saw three small bright points in a line beside Jupiter. On the 8th they had moved, and not in the way fixed stars would. By the 13th there were four. Over the following weeks he tracked them, and by the time the book went to press he could state that they are four bodies revolving about Jupiter, with the closest going round in under two days and the furthest in about sixteen.
He named them the Medicean stars, after Cosimo II de' Medici, Grand Duke of Tuscany, and was appointed court mathematician and philosopher at Florence within months. Everyone else has called them the Galilean moons ever since.
The astronomical argument is narrow but sharp. One standard objection to a moving earth was that the earth is the centre of all revolution, so that a second centre is impossible; a related objection was that if the earth moved it would leave its moon behind. Jupiter's moons refute both at once. Here is a body that is certainly not the centre of the universe, with four satellites, and it moves across the sky for years without shedding any of them.
The system also turns out to be a scale model of the one the previous lesson described. Measured in units of the innermost moon's distance and period, the four give
| Moon | Distance | Period | |
|---|---|---|---|
| Io | 1.000 | 1.000 | 1.000 |
| Europa | 1.591 | 2.007 | 1.000 |
| Ganymede | 2.538 | 4.045 | 1.000 |
| Callisto | 4.465 | 9.434 | 1.000 |
Example. Callisto orbits at times Io's distance. If Kepler's third law holds here as it does for the planets, what should its period be, given Io's days?
The law says , so in Io units: Io periods, and days. The measured period is days. The agreement is to about one part in ten thousand, in a system a thousand times smaller than the solar one, involving a central body Galileo could not have weighed.
Now you. Galileo could not have run that check. Why not, and what does its success later tell you?
Answer
The third law was not published until 1619, nine years after the moons were found, and even then Kepler stated it for planets going round the sun with no suggestion that it was general. The check needs accurate distances of the moons from Jupiter in units of Jupiter's diameter, which took decades of careful micrometer work to obtain. What its eventual success shows is that the law is not a fact about the sun in particular. The same relation, with a different constant, governs any set of bodies circling a common centre, which means the constant is telling you something about the central body rather than about the arrangement. That is the thread Newton eventually pulls, and it is already lying on the table in 1610 without anyone being able to see it.
Venus kills Ptolemy
The decisive observation came late in 1610, and Galileo first announced it as an anagram to establish priority without revealing the result, a practice the later lesson on publication returns to. Unscrambled, it read: the mother of loves imitates the figures of Cynthia. Venus shows phases like the moon.
To see why that matters, consider what each arrangement predicts. In Ptolemy's system Venus rides an epicycle whose centre is always on the line from the earth to the sun, and always between the two. Venus is therefore always closer to the earth than the sun is, and always lit from behind or the side, so it should show crescent and new phases only, and never a full or nearly full disc.
If Venus goes round the sun, then when it is on the far side of the sun it is fully lit and far away, and when it is on the near side it is a thin crescent and close. Phases run through the whole cycle, and, crucially, the apparent size varies inversely with the distance, so the full phase is small and the crescent is large.
Example. Venus is AU away when full and AU away when a thin crescent. Its diameter is km and one astronomical unit is km. What angular diameters does the heliocentric arrangement predict?
At AU the distance is km, so the angular diameter is radians. Converting, one radian is arcseconds, giving arcseconds. At AU the distance is km and the same calculation gives arcseconds. So Venus should be about six times wider when a crescent than when full, in the same ratio as the distances, . That is exactly what the telescope shows, and it is a quantitative prediction, not merely a shape.
Now you. Which of the three systems on the table does this observation eliminate, and which does it leave standing?
Answer
It eliminates Ptolemy's, and only his. A full Venus at small angular size is flatly impossible if Venus's epicycle is always between us and the sun. But the observation shows that Venus goes round the sun, and both surviving systems already say that: the Copernican, where the earth does too, and the Tychonic, where the sun and its retinue of planets go round a stationary earth. The two are geometrically identical for every angle and every distance in the sky, as the earlier lesson set out, so no telescopic observation of phases, sizes or positions can separate them. The telescope killed the system that had been dying for a century and left the live rival untouched. It is a good illustration of a general point: a decisive observation usually decides between two of the options rather than all of them, and the one it leaves standing is usually the one that was constructed with it in mind.
Spots on the sun
Between 1611 and 1613 Galileo, the Jesuit Christoph Scheiner and others turned telescopes on the sun, safely by projecting its image onto paper. Dark spots appear, move steadily across the disc, and take about a fortnight to cross it.
Scheiner's first interpretation was that they are small bodies orbiting the sun, passing in front of it, since a blemish on the sun itself was hard to reconcile with the incorruptible heavens. Galileo's argument that they lie on the surface is geometrical and good. A spot near the centre of the disc appears round; the same spot near the edge appears squeezed into a narrow ellipse, exactly as a mark on a rotating sphere would when foreshortened. Spots also change shape, break up and dissolve over days, and they all move at the rate corresponding to a sun turning once in about 27 days.
The sun, in other words, is a changeable body that rotates. The Letters on Sunspots of 1613 is where Galileo first put in print, over his own name, that the Copernican arrangement is the true one.
Why refusing to look was not merely foolish
The best-known anecdote of this period is that the Aristotelian philosopher Cesare Cremonini declined to look through the telescope at all. Reported more carefully, several people did look and did not see, including at a demonstration in Bologna in April 1610 arranged by Giovanni Antonio Magini, where a room full of astronomers failed to confirm Jupiter's moons.
It is worth taking that seriously rather than laughing at it. The instruments were bad. A simple lens shows coloured fringes, ghost images and flare; the field was minute and the mounting was a hand; observers unused to it saw double stars that were not there and missed objects that were. Nobody had a theory of how the instrument formed its image, since Kepler's optical account appeared in 1611 and was not widely absorbed for years. So the question of the day was entirely reasonable: how do you know the instrument is not manufacturing the appearances rather than revealing them?
The answer that was eventually given, and it took a decade, was not an argument but a practice. Telescopes were shown to work reliably on terrestrial targets whose truth could be checked independently, a distant inscription read through the tube and then walked to and read directly. Different instruments in different hands showed the same moons of Jupiter, at the same times, and their positions could be predicted in advance. Kepler received one of Galileo's instruments and confirmed the observations in print in September 1610, which mattered because Kepler had every professional reason to be a rival. What makes an instrument trustworthy is that its results are reproducible, predictive and consistent with things you can check by other means, and that case has to be built rather than asserted. Every instrument since has had to earn its credibility the same way.
By 1613 the situation was this: the Ptolemaic system was dead, the moon and sun were ordinary changeable bodies, the earth was not the only centre of motion, and the choice between a moving earth and Tycho's compromise was exactly where it had been. Galileo, now famous and confident, pressed the Copernican case in public. What that provoked is the subject of the next lesson.