Before anything can be said about how Europe learned to settle arguments by measurement, it is worth being clear about what it was settling them by beforehand, and about how good the system was that got replaced.
The usual story starts with a stupid idea, the earth at the centre of everything, and a brave man who noticed it was wrong. That story makes the change impossible to explain. If the old view had been obviously silly, the argument would have lasted a decade rather than a century and a half, and it would not have taken Newton to end it. The earth-centred cosmos was defended by people who could calculate, who knew what evidence was, and who had answers to every objection raised against them. This lesson is about how good those answers were, because the size of the achievement that follows is measured by the quality of what it had to overturn. It assumes no astronomy, and introduces its terms as they arrive.
Two physics, one for each half of the world
The framework was Aristotle's, written in the fourth century BC, recovered in Latin translation in the twelfth and thirteenth centuries AD, and taught in every European university by 1500. Its central claim is that the world is divided in two, and that the division is physical rather than decorative.
Below the moon, everything is made of four elements: earth, water, air and fire. Each has a natural place, and each moves in a straight line towards it when nothing prevents it. Earth and water fall, air and fire rise. Motion in a straight line has a beginning and an end, so this half of the world is a place of coming to be and passing away, where things are generated, change and die. That is a claim about the world you can check by walking outside, and it is broadly what you see.
Above the moon, everything is made of a fifth element with no tendency to fall, no capacity to change, and one natural motion: uniform rotation in a circle. A circle has no beginning and no end, which is why the heavens can go on forever without wearing out. From this it follows that the heavens must be unchanging and that every celestial motion must be circular and uniform, and both consequences were confirmed by every observation anyone had. The stars had kept the same patterns since the Babylonian records, nothing in the sky had been seen to be born or to decay, and the sun, moon and stars came round on time.
The two halves are joined by the earth sitting still at the centre, because that is where earthy matter has fallen to. The cosmos does not have a centre because the earth is there; the earth is there because heavy things fall towards the centre. Gravity, in this scheme, is not a force but a tendency, an object's preference for its own place, and it explains the position of the earth rather than being explained by it.
Three good reasons the earth does not move
Anyone who proposes that the earth moves has to answer three objections. All three were stated in antiquity, all three are empirical, and none of them could be answered in 1543.
The first is that we would feel it, and worse, that things would be left behind. Drop a stone from a tower and it lands at the foot of the tower. If the earth were spinning, the tower would be carried east while the stone fell, and the stone should land some distance west of the base. At the latitude of Rome the surface of a rotating earth would be moving east at about 346 metres per second. A stone falling for three seconds would be left more than a kilometre behind. Nothing of the kind is observed. The objection is not stupid; it is a prediction, and it fails.
The second is that a spinning earth should fling everything off it, in the same way that a stone leaves a sling. This too is a real physical intuition, and it will turn out to be quantitatively answerable rather than wrong. The answer is not available to anyone in 1543.
The third is the sharpest, and it is astronomical. If the earth travels round the sun, then in January we view the stars from one side of that path and in July from the other, from two places separated by twice the radius of the orbit. Nearby stars should shift against the far ones, and shift back six months later. This is stellar parallax, and it is exactly what a surveyor uses to find a distance. No parallax was seen. Either the earth does not move, or the stars are very much further away than anyone had reason to believe.
Example. Take an astronomer who can fix a star's position to about 10 arcminutes, which is roughly a naked-eye limit and one third of the moon's apparent diameter. If the earth's orbit has a radius of one astronomical unit and no parallax is detected, how far away must the stars be, and how does that compare with the size of the cosmos as Ptolemy gave it?
A star at distance shifts by an angle with , working in astronomical units, so a shift too small to see means
Ptolemy put the sphere of the fixed stars at about earth radii, which is km, or AU. So accepting a moving earth means accepting that the stars sit at least times further out than the entire known universe, with nothing whatever in between. That is not a small revision. It is throwing away the scale of the cosmos to save a geometrical convenience, and refusing to do it was a defensible piece of scientific conservatism.
Now you. Suppose an instrument is built that measures positions to 1 arcminute and still sees no parallax. Does that make the earth's motion more or less believable?
Answer
Less, on the face of it, and this is the trap the whole argument sits in. Ten times the precision pushes the minimum distance to AU, which makes the empty gap ten times more absurd. Every improvement in measurement that fails to find parallax strengthens the objection rather than weakening it, and the objection can only ever be dissolved, never answered, until the day someone actually detects the shift. That did not happen for another three centuries. Any account of how this argument was won has to explain how it was won without the one measurement that would have decided it.
Saving the appearances
Aristotle supplied the physics and almost none of the arithmetic. The calculations came from Ptolemy, working in Alexandria around 150 AD, whose Almagest remained the working manual of astronomy for fourteen hundred years, which is by some distance the longest run any scientific text has had.
Ptolemy's problem was that planets do not move uniformly on circles centred on the earth. They speed up and slow down, they vary in brightness, and each of them periodically stops, reverses for weeks or months, stops again and resumes. That reversal is retrograde motion, and it is the central fact any astronomy has to account for. Mars does it for about seventy days every twenty-six months, and while it does it is at its brightest.
The machinery Ptolemy built to reproduce this has three parts. The deferent is a large circle around the earth. The epicycle is a small circle whose centre rides the deferent, with the planet on its rim, so that adding the two motions can carry the planet backwards for part of the cycle. The eccentric offsets the deferent's centre from the earth, so the planet appears to move faster on one side of the sky than the other. Two circles and an offset already reproduce the gross behaviour of a planet, and they do it while keeping every individual motion circular and uniform, as the physics demanded.
The third device is the one that caused trouble. To fit the observations, Ptolemy had to let the epicycle's centre move uniformly not about the deferent's centre, nor about the earth, but about a third point, the equant, placed on the far side of the centre from the earth. The construction works: it is a good approximation to what an ellipse actually does. Its cost is that the epicycle's centre now moves along the deferent at a rate that is not constant. The uniform circular motion required by the physics has been quietly abandoned to make the numbers come out. That was widely felt as a scandal, and it was the specific complaint that Copernicus opened his book with.
How accurate was it, really
Astronomy is judged by tables: given a date, where is the planet. The Alfonsine Tables, computed at Toledo in the 1270s and printed in 1483, were what a European astronomer actually used, and they were Ptolemaic.
They were good but not good enough. Errors in planetary longitude of one to two degrees were routine, and for Mars they could reach five, which is ten times the width of the full moon. In 1563 Tycho Brahe, then sixteen years old, watched a conjunction of Jupiter and Saturn and compared the date with the tables. The Alfonsine prediction was a month out. The Prutenic Tables, computed on Copernicus's own system, were several days out. That comparison is often given as the moment he decided the discipline needed better data, and it is worth noticing what it does not say: the new system did not fix the errors. It merely failed differently.
There was also a failure of a different kind, one that no adjustment of parameters could repair. Ptolemy's model of the moon reproduces its position in the sky well, and does so by swinging the moon towards and away from the earth over the course of a month. In that model the moon's distance runs from about to about earth radii, a ratio of . Since apparent size goes inversely with distance, the moon should appear nearly twice as wide at some times as at others.
Example. The moon's apparent diameter at its most distant is about arcminutes. What does Ptolemy's lunar model predict for its closest, and what is actually observed?
Apparent diameter is inversely proportional to distance, so the model predicts arcminutes, close to a full degree, and the moon at its largest would be visibly almost twice the width of the moon at its smallest. What is observed is a maximum of about arcminutes, a ratio of . The model is wrong by a factor of nearly two on a quantity anyone with a simple sighting instrument can measure. Nobody in the middle ages was unaware of this; it was discussed, and the usual response was that the model was a device for computing positions and should not be read as a claim about distances.
Now you. What does that response cost, if you accept it?
Answer
It costs the connection between the mathematics and the physics, which is most of what a scientific theory is for. If the circles are only a calculating device with no claim to describe where the moon actually is, then astronomy is a technique for predicting positions and says nothing about the world, while physics is a separate discipline that says what the world is like and cannot calculate. The two can then never contradict each other, because they are no longer about the same thing, and no measurement can ever settle a question of physics. That is a comfortable position and a sterile one. The whole story that follows is the story of that separation collapsing, and the reason it mattered so much whether Copernicus was describing the world or merely computing in it.
Settling a disagreement in 1500
Suppose two university men disagree about whether the heavens can change. How is that resolved?
Not by measurement, and not by anyone going outside to look. It is resolved by the quaestio disputata, a formal procedure taught and practised in every faculty. The question is posed; the arguments on one side are set out; the arguments on the other are set out, each supported by a citation from a recognised authority, which in natural philosophy means Aristotle, his commentators, and where relevant scripture; then the master determines the question by showing how the winning position is consistent with the authorities and how the losing citations can be read compatibly with it. A conflict between authorities is real work, and resolving it well took skill. A conflict between an authority and an observation was a much rarer kind of event, and there was no established procedure for it.
This was not obscurantism, and it is worth seeing why it was reasonable. Nobody had a way of establishing that an observation reported by one person on one night was reliable. There were no agreed instruments, no error estimates, no journals, no repetition by other observers, and no mechanism at all for distinguishing a careful report from a careless or dishonest one. A text by Aristotle, by contrast, was stable, publicly available, checkable by anyone who could read, and had been examined by generations of intelligent people. If you have to build knowledge that outlasts the people making it, the text is by far the more defensible foundation. Everything that follows in this subject is, from one angle, the slow construction of the machinery that made a measurement as trustworthy as a text.
The picture of a period incapable of quantitative work is also false. In the 1330s and 1340s the Merton College calculators in Oxford, and Nicole Oresme in Paris, worked out and proved the mean speed theorem: a body under uniform acceleration covers the same distance as a body moving steadily at the speed reached halfway through the interval. Oresme proved it by drawing the speed against time and comparing areas, which is the argument that appears in Galileo's Two New Sciences three hundred years later. What the calculators did not do was measure anything, or suppose that their result described any actual falling body. It was an exercise in the logic of change, done on paper.
The cracks that were visible from inside
Two problems were being complained about by insiders long before anyone proposed moving the earth.
The first was the equant, described above: a mathematical fix that broke the physical principle it was supposed to serve. This is the complaint that opens Copernicus's preface, and it is not an appeal to observation at all but to consistency.
The second was the calendar, and it is the clearest case in the period of a measurement forcing an institution to act. The Julian calendar assumes a year of exactly days. The tropical year, the interval from one spring equinox to the next, is about days.
Example. How fast does the Julian calendar drift, and how much error had accumulated by the 1500s?
The excess is days per year, which is minutes. That accumulates to a whole day in years. The Council of Nicaea in AD 325 had fixed the rules for Easter with the equinox on 21 March, so by the 1580s the drift since then was days, and the equinox was falling on 11 March. Easter, the central date in the calendar, was visibly detaching from the season it was defined by. When Pope Gregory XIII acted in 1582, ten days were removed: 4 October was followed by 15 October.
Now you. The Gregorian reform keeps the leap year but drops it in years divisible by 100 unless also divisible by 400. What average year length does that give, and how good is it?
Answer
In 400 years the Julian rule gives 100 leap days; the Gregorian rule removes three of them, at 1700, 1800 and 1900, keeping 2000. So there are 97 leap days in 400 years and the average year is days. Against the tropical year of that is an excess of days, or 26 seconds, which accumulates to one day in about 3300 years. The reform improved the error by a factor of roughly 26. Notice what kind of achievement this is: it is an entirely conventional fix, driven by a measured quantity, and requiring no view whatever about what moves around what. Copernicus was consulted about the calendar in the 1510s and declined on the grounds that the length of the year was not yet known well enough, which tells you something about the standards he was working to.
What would have to change
The system just described has a physics that explains why things fall and why the heavens are eternal, an astronomy that predicts positions to a degree or two, a procedure for settling disputes that is stable and public, and a set of answers to every objection. It also has an internal inconsistency it cannot repair, a lunar distance that is wrong by a factor of two, tables that are a month out on a conjunction, and a calendar drifting at eleven minutes a year.
That mixture is what a mature theory in difficulty actually looks like, and it is worth remembering how ordinary the difficulties are. None of them, on their own, requires moving the earth. All of them could plausibly have been absorbed by better parameters, and most educated Europeans expected that they would be.
What Copernicus offered was not a solution to any of them. His system was not more accurate, and by one way of counting it was not simpler. What it had was a different kind of virtue, and understanding what that virtue was, and why it was not the sort of thing the existing way of settling arguments knew how to weigh, is where the next lesson starts.