The observation that would have satisfied Bellarmine was made in 1728, ninety-five years after Galileo's trial, and by then there was nobody left to convince.
The previous lesson left the Principia published, its predictions coming true one after another, and its central concept regarded as an absurdity by the best physicists in Europe. This lesson closes three threads: the missing evidence for the earth's motion, which finally arrives; the question of what actually ended the argument, which is not what the evidence would suggest; and an honest accounting of what the method turned out to be and how much of the usual story about it is wrong.
The proof arrives by accident
James Bradley, later Astronomer Royal, set out in 1725 to detect stellar parallax, the shift described in the first lesson that had defeated everyone since Aristarchus. He chose the star gamma Draconis for a good instrumental reason: it passes almost exactly overhead at London, so it can be observed through a fixed vertical telescope with no correction for atmospheric refraction, the largest error in the game.
He found a shift. Over a year the star moved back and forth by about 20 arcseconds, which is a large effect by the standards of the search. It was also the wrong shift. Parallax should put a star at its extreme displacement when the earth is at the extreme of its orbit in the corresponding direction. Bradley's displacement was at its maximum a quarter of a year away from that, exactly when the earth's velocity, rather than its position, pointed the right way.
The explanation, which he is said to have arrived at watching a boat's wind vane change direction as the boat changed tack on the Thames, is that light has a finite speed and the observer is moving. A telescope must be tilted slightly into the direction of the earth's motion, in the way an umbrella is tilted forward when you walk through falling rain. The tilt depends on the ratio of the observer's speed to the speed of light, and it is at its greatest when the two motions are perpendicular, which is precisely the phase Bradley saw.
Example. The earth's orbital speed is km/s and light travels at km/s. What tilt does that predict?
One radian is arcseconds, so arcseconds. The modern value of the constant of aberration is arcseconds, and Bradley's own determination was about . This is a real, direct, quantitative demonstration that the earth is moving, and it is the first one in the whole story: not an argument from explanatory economy, not a removal of an objection, but a measured angle that has no value at all unless the observer is in motion.
Now you. Bradley did not know independently to any accuracy. What could he get out of the measurement instead, and how does aberration differ from parallax in what it tells you?
Answer
He could get the time light takes to travel from the sun to the earth, since the aberration angle is the earth's orbital speed divided by , and the orbital speed is AU per year, so the angle immediately gives the AU divided by in units of time. Bradley published a light travel time of about 8 minutes 12 seconds, against a modern seconds, or 8 minutes 19 seconds, a confirmation of Ole Rømer's much rougher result of 1676 from the eclipses of Jupiter's moons. The difference from parallax is instructive. Parallax depends on how far away the star is and so differs from star to star, which is what makes it useful as a distance measure and hard to detect. Aberration depends only on the observer's velocity, so it is the same for every star in a given direction, which is why it is far larger and why it turned up first. Bradley found the more spectacular effect while hunting the more useful one, which is a common shape for a discovery.
The distance to a star
Parallax itself took another 110 years, and it fell in 1838 to Friedrich Bessel at Königsberg, using a heliometer, an instrument with a split objective lens whose halves can be slid against each other to measure small angular separations precisely.
The choice of target was the clever part. Bessel picked 61 Cygni, an undistinguished star chosen because Giuseppe Piazzi had found it moving across the sky at arcseconds a year, faster than any star then known. Large apparent motion suggests proximity, and proximity is what a parallax hunter needs. He measured its position against two faint neighbours for eighteen months and reported an annual parallax of arcseconds.
Example. How far away is that, and what does it do to Tycho's objection?
Distance in astronomical units is divided by the parallax in arcseconds:
which is light years; the modern parallax of arcseconds gives . Compare the lower bound Tycho's precision imposed, from the lesson on his instruments: no parallax above 1 arcminute means the stars are beyond AU. The true distance is times that bound, so Tycho's empty gap was not merely large but a hundred and ninety times larger than the version he found unbelievable. His objection was quantitatively correct and it was answered by the universe simply being much bigger than anyone was prepared to imagine, together with the discovery that the stellar discs he had measured were an artefact of the eye.
Now you. Both Copernicus and Tycho could have been convinced by a parallax measurement. Why does the fact that it arrived in 1838 matter for how this subject should be read?
Answer
Because by 1838 there had been no serious dispute for a century and a half. Every working astronomer in Europe had been a Copernican since around 1700, the Principia was 151 years old, and the Catholic Church had dropped the general prohibition on heliocentric books in 1758 and the specific ones in 1835. So the measurement that would have decided the argument arrived long after the argument was over, which means it did not decide it. Something else did, and identifying that something is the point of the next section. It also sets a warning about how scientific disputes are usually narrated: the decisive experiment is a convenient story, and the more common pattern is a theory winning by a long accumulation of quantitative successes in areas nobody was arguing about, with the crucial evidence arriving afterwards as a formality.
The last piece, in a Paris basement
The earth's daily rotation, separate from its annual motion, got its own demonstration in 1851, when Léon Foucault hung a 28 kg brass bob on a 67 metre wire from the dome of the Panthéon in Paris. A pendulum's swing plane stays fixed while the earth turns underneath it, so the plane appears to rotate, at a rate of degrees per hour, where is the latitude.
Example. What does that give at the latitude of Paris, ?
degrees per hour, so the plane comes back to its starting orientation after hours. Over a few minutes of watching, the swing visibly walks around the marker, and there is nothing to see except the earth turning.
Now you. What does the formula give at the equator and at the pole, and why is the pendulum made so long?
Answer
At the pole, , so the plane turns a full 360 degrees in 24 hours, which is the easiest case to picture: the earth simply rotates beneath a fixed swing. At the equator, and the effect vanishes entirely, so the experiment fails there. The length has two purposes. It makes the period long, seconds, so the pendulum swings for hours before friction stops it and the slow rotation has time to accumulate into a visible angle. And a long wire with a heavy bob is less disturbed by the small asymmetries in the suspension that would otherwise make the plane precess for reasons having nothing to do with the earth. It is worth noticing what the demonstration owes to the earlier lessons: it is Galileo's pendulum, understood through Huygens's formula, used to detect an effect that only Newton's mechanics predicts.
So what did settle it
By 1700 essentially every competent astronomer in Europe worked in a heliocentric frame, with no direct evidence of the earth's motion whatever. The evidence that closed the case is not any single observation but a network of quantitative agreements, and it is worth setting them out together because the pattern is the answer to the question this subject asks.
Kepler's laws made tables an order of magnitude better than any before them, and navigators and calendar makers used them regardless of what they believed about the sky. The moon test tied the fall of a stone to the orbit of the moon within about one per cent. Jupiter's mass, calculated from Io's orbit, agreed with the perturbations Jupiter produced in other bodies. The earth's shape, predicted from its rotation, was confirmed by two expeditions to opposite ends of the world. A comet returned within a month of a computed date, having been delayed by a computed amount by two planets whose masses came from a different measurement entirely.
None of those is a proof that the earth moves. Every one of them is a case where a quantity derived one way agreed with a quantity measured another way, in a situation nobody had rigged. That is what accumulated until the alternative became untenable, and Bellarmine's condition, described in the lesson on the trial, was in the end not met so much as bypassed: what arrived was not a demonstration in his sense but a convergence of independent measurements, a category he had no place for.
What the method actually was
There is no single method that these people followed, and it is worth saying so bluntly, because the textbook version is a tidy sequence of hypothesis, experiment and conclusion that nobody in this subject used. Bacon, Descartes and Newton each published a method, the three are mutually inconsistent, and none of them describes what its author did.
What was actually built, over roughly a century and a half, is a set of practices. Instruments that state their errors, so that a discrepancy can be told from noise. Situations contrived so that one factor can be varied while others are held still, and controls run alongside so that the obvious alternative explanation can be excluded. Predictions of numbers rather than of tendencies, so that a theory can fail by an amount. Phenomena manufactured on purpose, so that nature does not have to supply them. Publication with dates, witnesses and enough detail to rebuild the apparatus. And a willingness to accept a mathematical law with confirmed consequences even when nobody can say what produces it, which was the largest concession of all and the one that the ablest people of the time refused.
Not one of those was obvious, and several were resisted by people who had good reasons. Believing an instrument over a text requires the instrument to have earned its credibility, which took the telescope a decade. Accepting an unexplained law means giving up the older and more satisfying idea of what an explanation is. Treating an artificial case as evidence about nature needs an argument that the artifice does not distort what it isolates. The method looks obvious now because we inherited the finished version, and every part of it was somebody's contested proposal.
The honest accounting
Three qualifications, all of which a reader will meet elsewhere and should have in advance.
The term is retrospective. Nobody in the seventeenth century thought they were living through the Scientific Revolution; the phrase was popularised by Alexandre Koyré and Herbert Butterfield in the middle of the twentieth century, and Steven Shapin's history of 1996 opens by saying that there was no such thing as the Scientific Revolution, and that his book is about it. The joke has a serious point: the episode is a construction imposed on a messy period, and the tidiness is ours.
The continuities are real and are usually suppressed. The mean speed theorem is fourteenth century, as the first lesson noted. The mathematical devices Copernicus used to eliminate the equant appear in the work of the Maragha astronomers of thirteenth and fourteenth century Persia and in Ibn al-Shatir's models, which are geometrically identical to some of Copernicus's, and how the transmission happened is still argued about. Optics ran continuously from Ibn al-Haytham in the eleventh century through the medieval Latin tradition into Kepler.
And the people are not modern scientists, which is the qualification most worth sitting with. Kepler earned part of his living casting horoscopes and regarded the harmony of the spheres as a literal musical fact. Newton wrote more words on alchemy and biblical chronology than on mathematics and physics combined, and when John Maynard Keynes bought a trunk of those papers at auction in 1936 he called their author not the first of the age of reason but the last of the magicians. Boyle believed in transmutation and successfully lobbied for the repeal of the English statute against multiplying gold. They were not doing modern science with the mysticism as a regrettable hobby; the two were parts of one enterprise, and the sorting of them into respectable and disreputable is something later generations did.
What can be said, without exaggeration, is this. In 1543 a disagreement about nature was settled by weighing authorities, and the best available answer to a factual question was the one most consistent with the texts. By 1759 a disagreement about nature was settled by computing a number in advance and going out to see whether it was right, and a comet arrived to schedule. The people who built that machinery did not know they were building it, disagreed with each other about how it worked, and held beliefs that would embarrass any of their successors. It is still the most consequential thing Europe did, and none of it was obvious at the time.