Two systems that both predict planetary positions to within a degree or two cannot be told apart by observations that are themselves good to a degree or two, so somebody had to make measurement itself an order of magnitude better.
That is what Tycho Brahe did, and the previous lesson ended where he began: with the conjunction of Jupiter and Saturn in 1563 that the Alfonsine Tables missed by a month and the Copernican tables by several days. He was sixteen. The conclusion he drew was not that one system was right but that the discipline was working from data unfit for the question, and that fixing that came before choosing sides.
A new star
On 11 November 1572 a star appeared in Cassiopeia that had not been there. It was brighter than Venus for a fortnight, visible in daylight, and it faded over eighteen months until it disappeared in March 1574. We now know it was a supernova, the explosion of a white dwarf some 8000 light years away, and its remnant is still detectable in radio and X-rays.
For Aristotle's physics, described two lessons ago, a new object in the heavens is not surprising so much as forbidden. Everything above the moon is made of an unchangeable fifth element. So the standard response was to place the object below the moon, in the region of change, and call it a vapour or a comet in the upper air. That is not evasion; it is the reasonable interpretation, and comets had been treated that way since antiquity.
The response is also testable, and Tycho tested it. A nearby object is seen from a slightly different direction as the earth's rotation carries the observer across the diameter of the earth, so it should shift against the background stars over the course of a night. This is diurnal parallax, and it is large for the moon: the moon's position shifts by up to about 57 arcminutes, nearly a degree, between rising and setting. Tycho measured the new star's angular distance from nine reference stars in Cassiopeia, repeatedly, through the night and over months, with a sextant he had built himself.
Example. Tycho found no shift larger than his measurement error, which was about 4 arcminutes. How far away is the new star, at least?
Parallax falls off as the inverse of distance, so if the moon at distance shows 57 arcminutes and the star shows less than 4, then
The new star is at least fourteen times as far as the moon, and therefore firmly in the region where nothing is supposed to change. Note the shape of the argument: the conclusion comes from a failure to detect something, and it is only worth anything because Tycho could state how small a shift he would have detected. An observation with no error estimate attached could not have supported it.
Now you. In 1577 a great comet appeared, and Tycho bounded its parallax at under 15 arcminutes. How far away is it, and what does that do to the crystalline spheres?
Answer
lunar distances, or about 230 earth radii, which is roughly what Tycho reported. That places the comet not merely above the moon but out among the planets, and the comet's path across the sky over several months carried it clean through the region assigned to Mercury, Venus and the sun. If the planets ride on nested solid spheres, a comet cannot pass through them, so either the comet is impossible or the spheres are not there. Tycho drew the second conclusion and said so in print: the heavens are fluid, and the machinery that had carried the planets since Aristotle does not exist. That is a serious loss for everyone, including Tycho, because the spheres were the only account anybody had of what keeps a planet moving.
What an arcminute costs
The King of Denmark, Frederick II, gave Tycho the island of Hven in 1576 along with the income to run it. He built Uraniborg, a house that was also an observatory, and then in 1584 Stjerneborg beside it, with the instruments sunk into vaults below ground level so that the wind could not move them. The whole enterprise consumed something like one per cent of the Danish crown's annual revenue for twenty years. It is worth asking what all that money was buying, because the answer is not obvious.
Part of it was simply size, and the reason is geometrical.
Example. You want to read an angle to 1 arcminute off a graduated arc, and the smallest division a careful eye can distinguish and a craftsman can engrave is about 0.5 mm. How large must the instrument be?
An angle in radians on an arc of radius subtends a length . One arcminute is of a degree, or radians, so
The instrument has to be nearly two metres in radius to have arcminute divisions at all, which is exactly the scale of Tycho's great mural quadrant. Precision in a pre-telescopic instrument is bought by the metre, and the mass that comes with it is why the instruments ended up in vaults, and why they could not be carried anywhere.
Now you. How large would the same instrument have to be to read to 1 arcsecond, and what does that tell you about where naked-eye astronomy was heading?
Answer
One arcsecond is radians, so m. An instrument the length of a football pitch, engraved to half a millimetre and rigid enough not to sag, is not merely expensive but impossible, and it would be pointless anyway: the unaided eye cannot resolve two points closer than about an arcminute, so there would be nothing to see through the sights. Naked-eye astronomy was within a factor of a few of a hard ceiling, and Tycho was already close to it. Getting past that ceiling needs a different principle, not a bigger budget, and the principle arrives with the telescope.
Knowing your own error
Size was not the whole of it. Tycho added transversal scales, rows of oblique dots that let a reader interpolate within a division, so that an instrument divided to an arcminute could be read to a fraction of one. He built several instruments of different designs and measured the same stars with each, which is how you find out that an instrument is wrong rather than assuming it is right. He observed each object many times rather than once. He compiled the first systematic table of atmospheric refraction, correcting for the fact that light from a low star is bent so that the star appears higher than it is, by as much as 34 arcminutes at the horizon, which is more than the sun's own diameter. And he wrote down what he thought his errors were.
That last habit is the real invention, and it is worth separating from the money. An observation with no error attached is a claim; an observation with an error attached is a constraint, because it tells you in advance what would count as a discrepancy. Every argument in this lesson depends on it. The new star is beyond the moon because the shift was smaller than four arcminutes, a number that means nothing unless Tycho can say how he knows he would have seen four. The comet passes through the spheres for the same reason. Nothing about a null result is informative until the size of the null is stated, and Tycho is the first astronomer to do this systematically.
The catalogue he produced gives positions for 777 stars, the best of them good to about an arcminute, against Ptolemy's catalogue of 1022 stars good to perhaps fifteen or twenty. He also broke with the practice of observing an object only when it was best placed, at opposition or at a station, and instead followed the planets continuously round their circuits, which is what makes it possible to test a model where it is weakest instead of where it was fitted. An improvement of that size in the accuracy and coverage of data is rarer in the history of science than any single idea, and it is what everything in the next lesson runs on.
The parallax that was not there, and the stars that were too big
Tycho pointed his instruments at the question that mattered and looked for annual stellar parallax: the shift of a nearby star against distant ones as the earth, if it moves, carries the observer from one side of its orbit to the other. He found none, at his precision of about an arcminute.
The inference is the one from the first lesson, sharpened by a factor of ten. No parallax above 1 arcminute means the stars are more than astronomical units away. Copernicans were untroubled and said the stars are simply very far off. Tycho then made the argument that, on the evidence available to him, is unanswerable.
Example. A bright star seen with the naked eye appears as a disc of roughly 1 arcminute across, and every careful observer including Tycho measured something of that order. If such a star is at the minimum Copernican distance of 3438 AU, how big is it?
Its diameter is the distance times the angle in radians:
The star would be one astronomical unit across, which is to say as wide as the earth's entire orbit. The sun's diameter is km, or AU, so every one of these stars would be about 107 times the diameter of the sun, and that is for the nearest ones. Tycho's conclusion follows: the Copernican arrangement requires a universe in which the sun is a dwarf among a swarm of monsters, with a vast empty gap between them and us, and no reason for any of it. He rejected it, and he rejected it on a measurement.
Now you. The nearest bright star turns out to be about AU away, and stars really are comparable in size to the sun. What has gone wrong with the argument above?
Answer
The measured disc is not there. At AU a sun-sized star subtends radians, which is arcseconds, twenty thousand times smaller than the disc everyone was measuring. What Tycho, and everyone else with a naked eye or an early telescope, was measuring was an artefact: the optics of the eye and the diffraction of light spread a point source into a small blob whose size depends on brightness rather than on the star. Nobody could have known this, since diffraction was not described until Grimaldi in 1665 and not understood until the nineteenth century. The lesson is uncomfortable and worth sitting with. Tycho's reasoning was valid, his instruments were the best in the world, his number was carefully measured, and his conclusion was wrong, because the quantity he measured was not a property of the thing he thought he was measuring. Precision is not accuracy, and no amount of care inside a measurement protects against the artefact you have not thought of.
A third system
Having rejected Copernicus and being unable to accept Ptolemy, whose equant offended him as much as it had offended Copernicus and whose lunar distances were wrong, Tycho published his own arrangement in 1588.
In the Tychonic system the earth stands still. The moon and the sun go round the earth. All five planets go round the sun, and are carried along with it as it circles the earth. This looks like a fudge and is nothing of the kind. Geometrically it is exactly equivalent to the Copernican system for every angle in the sky: take the heliocentric arrangement, hold the earth fixed instead of the sun, and every observed direction is unchanged. Every explanatory gain of the previous lesson survives intact. Retrogression happens because the planet's motion round the sun is combined with the sun's annual circuit, so the one-year period is explained rather than stipulated; the planetary order is fixed by the sidereal periods; the distances come out the same.
And it keeps the physics. The earth does not move, so falling stones land at the foot of the tower, nothing is flung off, and there is no parallax to explain away, which means the stars can sit just beyond Saturn at a sane distance and be sane sizes. It fails only against the crystalline spheres, since the Mars orbit intersects the sun's, and Tycho had already abolished those with his comet.
This is the crux of the whole subject and it is worth stating baldly. After the most accurate measurements ever made, there were three systems on the table, and observation of positions could not choose between two of them, because they are the same geometry seen from different fixed points. The Tychonic system was not a rearguard action by a reactionary; it was the option a careful empiricist should have preferred in 1590, and it remained live for another century. Measurement alone was never going to end this argument. What was needed was a physics that could say which body is really moving, and there was none.
The data change hands
Frederick II died in 1588. His successor Christian IV had no interest in subsidising an astronomer who quarrelled with his tenants, and by 1597 the funding was gone. Tycho left Denmark with his instruments and his records, and in 1599 became Imperial Mathematician to Rudolf II in Prague.
In 1600 he hired an assistant: a poor, short-sighted, argumentative Lutheran schoolteacher from Graz named Johannes Kepler, who had published a book in 1596 proposing that the spacing of the planetary orbits was set by the five regular solids nested inside one another. Tycho thought the book was nonsense, which it is, and thought its author could calculate, which he could. He gave Kepler the orbit of Mars, the hardest of the planets and the one whose eccentricity had defeated every model, and kept the rest of the data to himself.
Tycho died on 24 October 1601, eighteen months later, of a bladder complaint after a banquet. The observations passed, not entirely legally, to Kepler. Two people had what nobody else had: a set of positions good to an arcminute, and a man willing to believe them.