The claim that the earth goes round the sun was not new in 1543, was not more accurate than what it replaced, and did not simplify the calculations, so the interesting question is what it did do.
The previous lesson left the inherited system with real internal problems and no obvious reason to abandon it. Nicolaus Copernicus, a canon of Frombork cathedral in Polish Prussia with a doctorate in canon law and a working practice in medicine, spent thirty years on an alternative and published it in the year he died. Understanding what he offered means being unsentimental about what he did not.
The book and the delay
Copernicus circulated a short handwritten sketch, the Commentariolus, to a few friends around 1510. It lists seven assumptions, of which the important ones are that the earth is not the centre of the universe, that it turns daily on its own axis, and that it revolves annually about the sun. He then did nothing publicly for a quarter of a century.
What broke the silence was a visitor. Georg Joachim Rheticus, a young Lutheran mathematician from Wittenberg, arrived in 1539, stayed two years, and published a summary of the system, the Narratio Prima, in 1540 to test the reaction. The reaction was survivable, and De revolutionibus orbium coelestium was printed at Nuremberg in 1543, with a copy reportedly reaching the author on the day he died.
It arrived with a lie in the front of it. Andreas Osiander, the Lutheran theologian who saw the book through the press, inserted an unsigned preface stating that the hypotheses in it need not be true, nor even probable, and that it is enough if they yield a calculation agreeing with the observations. That is precisely the retreat described in the previous lesson, astronomy as a computing technique with no claim about the world, and Copernicus's own text refuses it flatly. Readers took the preface for the author's until Kepler exposed it in 1609, which shaped sixty years of reception: the book could be used by anyone as a source of tables while its central claim was treated as a manner of speaking.
What did not change
Copernicus was a conservative in almost every respect that matters to physics. Motion in the heavens is still circular, still uniform, and the circles are still real spheres carrying the planets round. There is no suggestion of a force, no account of why the earth should move, and no reply to the falling-stone objection beyond the assertion that air and earth share the same motion.
He also kept the epicycles. This is the point on which the usual account is simply wrong. Copernicus removes the epicycles that produce retrogression, which is the great gain, but he refuses the equant, and refusing it costs him: to reproduce the non-uniform speed the equant was invented for, he adds small epicycles back in. Counting circles is a slippery exercise, and different reckonings of both systems give different totals, but the honest summary is that the two systems use comparable numbers of circles, in the mid thirties, and that Copernicus's is no simpler as a calculating machine.
Nor is it more accurate. The Prutenic Tables of 1551, computed on his system by Erasmus Reinhold, are a modest improvement on the Alfonsine ones, and that improvement comes from fresh parameters rather than from the arrangement. Both were several days out on the conjunction of 1563. If accuracy had been the test, the argument would have been over quickly and in the wrong direction.
Retrogression stops being a device
Here is the first thing the new arrangement actually buys, and it is a matter of explanation rather than of fit.
In Ptolemy's system, every superior planet, meaning Mars, Jupiter and Saturn, needs an epicycle to make it turn back. Each of those epicycles has a period of exactly one year. Not approximately: exactly. And the line from the epicycle's centre to the planet stays parallel to the line from the earth to the sun at all times, for all three planets, forever. For Mercury and Venus, the inner pair, the roles swap: it is the deferent that has a period of exactly one year.
Ptolemy knew this. It is built into his models as a stipulation, because that is what the observations require. What his system cannot do is say why the sun's year should appear inside the machinery of five other bodies that have nothing to do with it.
On a moving earth, every one of those coincidences is the same fact stated five times. Retrogression happens when the earth, on a faster inner track, overtakes an outer planet, and while it does the outer planet appears to slide backwards against the stars, exactly as a slower car appears to move backwards when you pass it. The period of the effect is a year because the effect is the earth's own orbit reflected in the sky. Mars is brightest at exactly the moment it retrogresses because that is when the earth is nearest to it. Venus and Mercury never stray far from the sun because their orbits are inside ours.
This is a different kind of argument from anything in the first lesson, and it is worth naming. Copernicus is not claiming a better fit to the data. He is claiming that his arrangement derives what the other has to assume, and that a theory forced to stipulate the same unexplained coincidence five separate times is telling you something about itself. Nothing in the quaestio disputata knows how to weigh that.
The order of the planets is no longer a choice
The second gain is sharper still, because it converts a matter of taste into a matter of arithmetic.
Ptolemy has no way to determine the order of the planets. The models fix each planet's angular behaviour and say nothing about distance, so the sequence outwards from the earth is chosen on grounds of plausibility: faster-moving bodies are put nearer. That works for the moon, Mars, Jupiter and Saturn, whose motions round the zodiac take a month, two years, twelve years and thirty. It fails completely for Mercury, Venus and the sun, which all go round in exactly one year on average and therefore cannot be ordered by that rule at all. Some astronomers put Mercury and Venus below the sun, some above, and the disagreement had run for a thousand years with no way to settle it.
On a heliocentric arrangement the question answers itself. The time a planet takes to complete one circuit of the sun, its sidereal period, is not what we observe; what we observe is the synodic period, the time between successive alignments with the sun as seen from a moving earth. The two are related by simple arithmetic. For a planet outside the earth's orbit, the earth gains one full lap on it in each synodic period, so
where days is the earth's own period, the planet's sidereal period and the observed synodic period. For a planet inside the earth's orbit the planet is the one gaining laps, and the two terms on the right swap places.
Example. Mars returns to opposition every days. What is its sidereal period?
Rearranging, . The two reciprocals are and per day, so and days, or one year and 322 days. That is the number Copernicus gives, and it is right to better than a tenth of a per cent. Note what has happened: a quantity nobody can observe directly, because we are riding on a moving platform, has been extracted from one that anybody can time with a calendar.
Now you. Jupiter comes to opposition every days. Find its sidereal period in years.
Answer
per day, so days. Dividing by gives years, or 11 years and 315 days. The ordering that follows is forced: 88 days for Mercury, 225 for Venus, 365 for the earth, 687 for Mars, 4333 for Jupiter and about for Saturn. The sequence outwards from the sun is not a convention any more, and the thousand-year argument about where to put Mercury and Venus is over.
Distances become measurable
The third gain is the largest, and it is the one that makes the solar system into an object rather than a set of angles.
Take Venus. Seen from the earth it swings out to one side of the sun, comes back, crosses, swings out to the other side. Its greatest angular separation from the sun, its maximum elongation, is about 46 degrees. If Venus travels a circle around the sun inside our own, then at that moment our line of sight grazes its orbit, which means the angle at Venus between the sun and the earth is a right angle. The triangle is then fixed by one angle and the earth-sun distance.
Example. Venus reaches a maximum elongation of . How far is it from the sun, in units of the earth's distance?
In the right-angled triangle, the side opposite the elongation angle is the Venus-sun distance and the hypotenuse is the earth-sun distance, so
The modern value is , so this is low by per cent. What matters is not the accuracy but that the question has an answer at all. Ptolemy cannot ask it: his models are indifferent to scale, and doubling every distance in them changes no prediction.
Now you. Mercury's maximum elongation is about . Find its distance from the sun, and say why the answer is less trustworthy than the one for Venus.
Answer
, against a modern semi-major axis of , so about one per cent high. It is less trustworthy because Mercury's maximum elongation is not a single number: it ranges from about to about depending on where in its orbit the elongation happens, and while . The spread in the answer is over fifty per cent of its value. The method assumes a circular orbit centred on the sun, and Mercury has the most eccentric orbit of the planets, so it is the case where that assumption is worst. This is the first appearance of a problem that will not be solved for another seventy years, and it is worth noticing that the data were shouting about it from the beginning.
For a planet outside the earth's orbit the trick is different but the principle is the same. Watch Mars at opposition, when the sun, earth and Mars are in line. Then wait until Mars is at quadrature, meaning it appears exactly from the sun, at which point the angle at the earth is the right angle. Measure the elapsed time, work out how far each body has gone in that interval, and the geometry closes.
Example. Mars takes about 106 days to go from opposition to quadrature. Find its distance from the sun.
In 106 days the earth covers of its orbit and Mars, with its period of days, covers . The angle at the sun between the two bodies has therefore opened to . In the right-angled triangle with its right angle at the earth, the earth-sun distance is the side adjacent to that angle and the Mars-sun distance is the hypotenuse, so
against a modern value of . Everything on the right-hand side is a timing, and timings are the one thing pre-telescopic astronomy could do superbly.
Now you. Ptolemy had the same observations of oppositions and quadratures. Why could he not run this calculation?
Answer
Because the calculation is about a triangle whose vertices are the sun, the earth and Mars, and in Ptolemy's system the sun is not a vertex of anything. The sun is one more body going round the earth, with no special relation to Mars, so the moment of quadrature is a fact about angles in the sky and not about a right angle in a physical triangle. Ptolemy can note that Mars retrogresses at opposition, and he does, but he has no reason to treat the earth-sun line as one side of a figure containing Mars. The general point is that a measurement is only possible inside a theory that says what is being measured. The observations were public property for fourteen centuries; what was missing was an arrangement that made them mean a distance.
Doing this for every planet gives the system its scale, in units of the earth's distance from the sun. Copernicus's values against the modern ones:
| Planet | Copernicus | Modern | Error |
|---|---|---|---|
| Mercury | 0.376 | 0.387 | 2.8% |
| Venus | 0.719 | 0.723 | 0.6% |
| Mars | 1.520 | 1.524 | 0.3% |
| Jupiter | 5.219 | 5.204 | 0.3% |
| Saturn | 9.174 | 9.539 | 3.8% |
Three of the five are within half a per cent, obtained with naked-eye sightings and a calendar. The two worst are the innermost, where the elongation method is defeated by eccentricity, and the outermost, where a slow planet makes the timings hardest. Nobody before had any number in that table at all.
What it cost
Set against those gains is a bill that no contemporary could pay.
The stars had to be moved to an unimaginable distance, for the reasons given in the previous lesson, leaving a void whose size had no purpose. The physics of the first lesson had to be abandoned, since a moving earth is no longer the place towards which heavy things fall, and nothing was offered in its place. Scripture had to be reinterpreted, though this was the least of it at first, and both Luther and Melanchthon dismissed the idea in passing in the 1540s while the Catholic Church did not act on it for seventy years. Against all that stood an argument about explanatory economy and a table of distances nobody had asked for.
It is also worth correcting the legend that the book was ignored. Owen Gingerich spent thirty years locating and examining surviving copies of the first two editions, some six hundred of them, and published the census in 2002. They are heavily annotated, by identifiable astronomers, and the annotations concentrate on the technical models rather than on the cosmology. The book was read closely by exactly the people qualified to read it, and most of them treated it the way Osiander's preface invited: an ingenious set of models, to be mined for parameters, without committing to the earth actually moving.
That is where the argument sat for a generation. Two geometries, each reproducing the observed positions to a degree or two, one with better explanations and worse physics. Nothing available could choose between them, because the observations were not good enough to expose the difference between two theories that both fitted them badly. The next step was not an idea at all. It was a Danish nobleman spending roughly one per cent of his country's annual revenue on the instruments to measure the sky an order of magnitude better than anyone ever had.