The Scientific Revolution

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 $d$ shifts by an angle $\theta$ with $\tan \theta =1/d$, working in astronomical units, so a shift too small to see means

Ptolemy put the sphere of the fixed stars at about $19865$ earth radii, which is $19865\times 6371$ km, or $0.85$ AU. So accepting a moving earth means accepting that the stars sit at least $344/0.85=406$ 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?

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 $33.5$ to about $64.2$ earth radii, a ratio of $64.2/33.5=1.91$. 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 $29.4$ 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 $29.4\times 1.91=56.2$ 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 $33.5$ arcminutes, a ratio of $33.5/29.4=1.14$. 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?

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 $365.25$ days. The tropical year, the interval from one spring equinox to the next, is about $365.2422$ days.

Example. How fast does the Julian calendar drift, and how much error had accumulated by the 1500s?

The excess is $365.25-365.2422=0.0078$ days per year, which is $0.0078\times 24\times 60=11.2$ minutes. That accumulates to a whole day in $1/0.0078=128$ 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 $(1582-325)\times 0.0078=9.8$ 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?

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.

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 $E=365.256$ days is the earth's own period, $P$ the planet's sidereal period and $S$ 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 $779.94$ days. What is its sidereal period?

Rearranging, $1/P=1/E-1/S=1/365.256-1/779.94$. The two reciprocals are $0.00273779$ and $0.00128215$ per day, so $1/P=0.00145564$ and $P=686.98$ 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 $398.88$ days. Find its sidereal period in years.

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 $46°$. 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 $0.7233$, so this is low by $0.6$ 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 $23°$. Find its distance from the sun, and say why the answer is less trustworthy than the one for Venus.

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 $90°$ 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 $106\times 360/365.256=104.5°$ of its orbit and Mars, with its period of $686.98$ days, covers $106\times 360/686.98=55.6°$. The angle at the sun between the two bodies has therefore opened to $104.5-55.6=48.9°$. 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 $1.5237$. 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?

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:

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.

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 $d_m$ 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?

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 $\theta$ in radians on an arc of radius $R$ subtends a length $R\theta$. One arcminute is $1/60$ of a degree, or $2.909\times {10}^{-4}$ 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?

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 $1/\tan (1^{\prime })=3438$ 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 $1.393\times {10}^6$ km, or $0.0093$ 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 $657000$ AU away, and stars really are comparable in size to the sun. What has gone wrong with the argument above?

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.

A model of the orbit of Mars that predicted every observed position to within eight arcminutes would have been the best astronomy ever done, and the reason it counts as a failure is the whole content of this lesson.

Johannes Kepler arrived in Prague in 1600 to work for Tycho Brahe, whose observations, described in the previous lesson, were good to about one arcminute. He was given Mars because Mars was the problem: the most eccentric orbit of the planets then known apart from Mercury, close enough to be observed well, and the body on which every existing model failed worst. He expected to finish in eight days. It took him five years, and the book that came out of it, Astronomia Nova of 1609, is the first work in which a physical cause is offered for a planetary motion and the first in which a theory is discarded because of a residual too small to see.

Kepler wanted a cause

Kepler is a strange figure to have this role, and it is worth being clear about what he actually believed, because it explains what he did.

His first book, the Mysterium Cosmographicum of 1596, argues that there are exactly six planets because there are exactly five regular solids, and that the spacing of their orbits is set by nesting a cube, tetrahedron, dodecahedron, icosahedron and octahedron between six spheres. This is not a metaphor. He calculated it, compared it with Copernicus's distances, found agreement of a few per cent, and regarded it as the discovery of his life. He was still defending it twenty years later.

Alongside that runs the idea that made him useful. In the same book he noticed that the further a planet is from the sun, the more slowly it moves, and not just in the sense that a longer orbit takes longer: the actual speed along the path is lower. Mercury takes 88 days, Saturn takes over 29 years, and the ratio is far greater than the ratio of the orbit lengths. To Kepler this meant that something emanating from the sun drives the planets and weakens with distance. That single thought sets him apart from everyone before him. Ptolemy, Copernicus and Tycho all wanted a geometry that reproduced the positions. Kepler wanted a physical cause, and he gave Astronomia Nova the subtitle Physica Coelestis, a physics of the heavens, which had previously been a contradiction in terms.

The consequence is methodological. If your model is a calculating device, a residual of eight arcminutes is a triumph. If your model is a claim about a real body being pushed by a real cause, a residual of eight arcminutes is a fact about the world that you do not understand.

The vicarious hypothesis

Kepler's first attack used the classical machinery: a circular orbit, with the sun displaced from the centre by some distance, and an equant point on the far side about which the motion is uniform. Ptolemy had always placed the equant exactly as far from the centre as the earth was, splitting the offset evenly. Kepler let the two distances be free parameters and fitted them to four accurately observed oppositions of Mars.

The result was excellent. The fitted model, which he called the vicarious hypothesis because he did not believe it was true, reproduced the longitudes of Mars, its positions round the zodiac, to within about 2 arcminutes across the whole set of oppositions. That is the accuracy of the data themselves. By every standard then in use the problem was solved.

He kept testing. The model that fitted the longitudes gave a splitting of the offset that disagreed with the value obtained from Mars's latitudes, its distances north and south of the ecliptic, and the disagreement was not tiny. Forcing the two into agreement, and then predicting positions near the middle of the orbit rather than at opposition, left errors of up to 8 arcminutes.

Example. How big is 8 arcminutes, and why is it decisive here when it would have been invisible fifty years earlier?

The full moon is about 31 arcminutes across, so 8 arcminutes is roughly a quarter of the moon's width: a gap you could not see between two stars without an instrument. The question is not whether it is small in absolute terms but how it compares with the error of the data. Tycho's positions are good to about 1 to 2 arcminutes, so an 8 arcminute residual is four to eight times the uncertainty. It is not noise. Had Kepler been working with the older observations, good to perhaps 10 arcminutes, the residual would have been comfortably inside the error bars and the vicarious hypothesis would have been declared correct. Tycho's money is what makes the discrepancy exist as a fact rather than as a possibility.

Now you. Kepler wrote that these eight minutes alone pointed the way to a complete reformation of astronomy. What would the alternative response have been, and could anyone have defended it?

Fixing the earth first

Before the orbit of Mars could be found, Kepler had to solve a problem nobody had taken seriously: every observation of Mars is made from a moving platform whose own motion was only approximately known. Copernicus had given the earth a uniform circular orbit with an offset sun, and no equant, on the grounds that the earth was special.

Kepler's method for testing this is the most elegant thing in the book. Mars returns to the same point in its orbit every sidereal period, $686.98$ days. So take observations of Mars separated by exactly that interval, and however many times you repeat it, Mars is in the same place in space. The earth is not: it has moved to a different point on its own orbit each time. Each observation then gives the direction of Mars and the direction of the sun as seen from the earth, and with Mars pinned down as a fixed reference the earth's position can be triangulated. Kepler used Mars as a surveying benchmark to map the earth's orbit from outside.

The answer came back that the earth does have an equant, and that its offset splits evenly, exactly like the other planets. The earth's speed varies in its orbit in the same way and for the same reason. That is a small technical result with a large consequence: the earth is not a special case with special rules, but a planet, behaving like the rest. Whatever drives Mars drives us.

The area law, then the ellipse

With the earth's orbit known, Kepler could convert Tycho's angles into actual distances from the sun to Mars at many points, and the shape that emerged was not a circle with an equant.

He got the speed law first, which is the reverse of the order in which the laws are usually listed. Working from his conviction that the sun drives the planets, he tried the rule that a planet's speed is inversely proportional to its distance from the sun. That is not exactly right, but summing the distances round the orbit to get the time was hopelessly laborious, so he replaced the sum with the area swept out by the line from the sun to the planet, which he could compute geometrically. The result is the area law: the line joining a planet to the sun sweeps equal areas in equal times. He regarded it as an approximate computing device standing in for the real distance law. It is exact, and the distance law is the approximation, which is one of the luckier mistakes in the history of science.

The shape took another three years. He tried an oval, could not compute with it, and eventually noticed a number. The width of the orbit he needed was less than the width of the circumscribed circle by a factor of $1.00429$, and the maximum angular offset between the two ways of computing the planet's position was $5°{18}^{\prime }$. He then observed that $1/\cos (5°{18}^{\prime })=1.00429$, the same number. He wrote that he awoke as from sleep. The curve that has that property is an ellipse with the sun at one focus.

Example. Mars has an eccentricity of $e=0.0934$ and a semi-major axis of $1.5237$ AU. How different is its orbit from a circle, and how far is the sun from the centre?

The semi-minor axis is $b=a\sqrt{1-e^2}=1.5237\times \sqrt{1-0.00872}=1.5237\times 0.99563$, so $b/a=0.9956$. The orbit is narrower than a circle by $0.44$ per cent, which no measurement of that era could have detected as a shape. The sun sits at a focus, a distance $ae=1.5237\times 0.0934=0.1423$ AU from the centre, which is $9.3$ per cent of the semi-major axis. So what Kepler detected was not that the orbit is an oval; it is very nearly circular. What he detected is that the sun is well off centre and that the planet's speed varies accordingly. Every model since Ptolemy had known that and had handled it with an equant. The ellipse is the correct way of handling it, and the difference between correct and nearly correct is those 8 arcminutes.

Now you. Find the perihelion and aphelion distances of Mars, and the ratio of its speeds at those two points.

The third law

The two laws of Astronomia Nova describe a single orbit. Neither says anything about how one orbit relates to another, and Kepler had been hunting for that relation since 1596, because it is the question the polyhedra were supposed to answer.

He found it on 15 May 1618 and published it in Harmonices Mundi the following year, buried in a book largely concerned with the musical intervals sounded by the planets. The third law states that the square of a planet's period is proportional to the cube of its semi-major axis, so that with the period in years and the distance in astronomical units, $T^2=a^3$ exactly.

Example. Jupiter's sidereal period is $11.862$ years. Where should it be?

$a=\sqrt[3]{T^2}=\sqrt[3]{{11.862}^2}=\sqrt[3]{140.71}=5.201$ AU. The modern semi-major axis is $5.2038$ AU, so the law is right to within $0.05$ per cent for a planet five times as far out as the earth.

Now you. Saturn's period is $29.457$ years. Predict its distance, and check it against the value Copernicus obtained.

What it left unexplained, and who believed it

The three laws are exact, they are simple, and they are entirely without foundation. Nothing in them says why the orbit should be an ellipse rather than an oval, why the areas rather than the arcs should be equal, or why the exponent in the third law should be $3/2$ rather than anything else. Kepler supplied a physical cause and it was wrong: he imagined the rotating sun sweeping the planets round with an immaterial emanation that thins with distance like light, plus a magnetic action pulling each planet alternately in and out to make the orbit oval. He knew it was unsatisfactory.

Nor was the reception rapid. Galileo, who corresponded with Kepler, never mentioned the ellipse in the Dialogue of 1632 and went on using circles to the end. The book is dense, argumentative, and written as a narrative of the author's own failures over five years, which is admirable and unreadable. Astronomers adopted the laws slowly and mostly for a practical reason.

That reason was the Rudolphine Tables, published in 1627, the last thing Tycho's data were used for. Where earlier tables were wrong by degrees, these were wrong by arcminutes, an improvement of one to two orders of magnitude. Kepler used them to predict a transit of Mercury across the face of the sun on 7 November 1631, which Pierre Gassendi observed in Paris, finding Mercury within about a quarter of a degree of the predicted place, while predictions on the older tables were out by degrees. Kepler had died the year before. He also predicted a transit of Venus for 1631 that was not visible from Europe, and missed the one in 1639, which the English curate Jeremiah Horrocks caught by correcting Kepler's own elements.

That is the mechanism by which the new astronomy actually spread: not by convincing anyone of the earth's motion, but by making better tables, which navigators, calendar makers and astrologers wanted regardless of what moved. Meanwhile, in Padua, a mathematics professor had pointed a new instrument at the sky and found things nobody had known were there.

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 $1737$ km, how high is the peak?

The terminator is where the sun's rays graze the surface. A peak of height $h$ at a distance $d$ 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 $R$ and $d$ and hypotenuse $R+h$:

Here $d$ is one twentieth of the diameter, which is one tenth of the radius, so $d=173.7$ km, and

Galileo's own figure was four miles, about $6.4$ km. Either way the answer is that lunar mountains are higher than the Alps, whose greatest peak reaches $4.8$ 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?

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

Example. Callisto orbits at $4.465$ 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 $1.769$ days?

The law says $T^2\propto a^3$, so $T=a^{3/2}$ in Io units: ${4.465}^{1.5}=9.434$ Io periods, and $9.434\times 1.769=16.69$ days. The measured period is $16.689$ 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?

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 $1.72$ AU away when full and $0.28$ AU away when a thin crescent. Its diameter is $12104$ km and one astronomical unit is $1.496\times {10}^8$ km. What angular diameters does the heliocentric arrangement predict?

At $1.72$ AU the distance is $2.573\times {10}^8$ km, so the angular diameter is $12104/2.573\times {10}^8=4.70\times {10}^{-5}$ radians. Converting, one radian is $206265$ arcseconds, giving $9.7$ arcseconds. At $0.28$ AU the distance is $4.19\times {10}^7$ km and the same calculation gives $59.6$ arcseconds. So Venus should be about six times wider when a crescent than when full, in the same ratio as the distances, $1.72/0.28=6.1$. 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?

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.

The condemnation of Galileo in 1633 is usually told as a collision between science and religion, and that framing hides the only thing about it that helps explain how measurement eventually won.

The previous lesson left the argument in a specific state. Ptolemy's system was dead, killed by the phases of Venus. Two systems remained, the Copernican and the Tychonic, which are geometrically identical and cannot be separated by any observation of position. Nobody had a physics in which a moving earth was possible. In that situation Galileo Galilei, the most famous natural philosopher in Europe, campaigned publicly for one of the two, and this lesson is about what happened and what it shows about the standing of evidence in 1633.

The first act, 1613 to 1616

Trouble began over dinner. In December 1613 Galileo's pupil Benedetto Castelli was questioned at the Medici court about whether the earth's motion contradicted scripture, in particular the passage in Joshua where the sun is commanded to stand still. Galileo wrote Castelli a long letter setting out how he thought the two should be related, and copies circulated.

In December 1614 a Dominican, Tommaso Caccini, preached against the Galileists in Florence, and in February 1615 another Dominican submitted a distorted copy of the Castelli letter to the Roman Inquisition. Galileo responded by expanding his argument into the Letter to the Grand Duchess Christina of 1615, which is one of the sharpest statements ever written of the claim that natural philosophy should be left to settle its own questions. Its core move is that scripture accommodates itself to ordinary speech and is not a textbook of astronomy, and it quotes Cardinal Baronio's line that the Bible teaches how one goes to heaven, not how the heavens go.

The Inquisition acted in 1616. On 24 February a panel of eleven consultants judged the proposition that the sun is the immobile centre of the world to be foolish and absurd in philosophy and formally heretical, and the proposition that the earth moves to be at least erroneous in faith. On 5 March the Congregation of the Index suspended De revolutionibus until corrected, the corrections appearing in 1620 and amounting to about ten passages altered so that the earth's motion reads as a hypothesis. Foscarini's book defending the system was banned outright. Galileo himself was not condemned. He was called in by Cardinal Robert Bellarmine and told not to hold or defend the opinion, and a disputed minute in the file, unsigned, records a stricter injunction not to teach it in any way whatever. That minute is what he was tried on seventeen years later.

Bellarmine's condition

The most important document of the whole affair is not a verdict but a letter Bellarmine wrote to Foscarini on 12 April 1615, before any of the machinery moved. It says three things, and they need separating.

First, there is no difficulty in treating the earth's motion as a mathematical supposition, a way of computing that saves the appearances. That is Osiander's position from two lessons ago, and it was uncontroversial.

Second, if there were a true demonstration that the sun really stands at the centre and the earth revolves, then one would have to proceed with great care in explaining the scriptures that appear contrary, and would have to say that we do not understand them rather than that what is demonstrated is false.

Third, he does not believe there is such a demonstration and has not been shown one, and until he is, the received reading of the texts stands.

Example. State exactly what Bellarmine's letter makes the question turn on, and whether Galileo could meet the condition in 1615.

It makes it turn on the burden of proof, and it concedes the principle that a demonstration would win. Bellarmine is not saying that evidence about nature is irrelevant to reading scripture; he says the opposite, that a demonstrated truth about nature obliges a rereading. What he denies is that anything on offer amounts to a demonstration, and by the standards of the day he was right. Galileo had the phases of Venus, which as the previous lesson showed refute Ptolemy and leave Tycho untouched; the moons of Jupiter, which remove an objection without establishing anything positive; and mountains on the moon, which are irrelevant to the question. Nothing in that list distinguishes the two live systems. The three observations that would eventually do it, stellar aberration, stellar parallax and the swing of a long pendulum, lay 113, 223 and 236 years in the future.

Now you. A common reading is that Bellarmine's position is simply that authority beats evidence. Is that fair?

The wrong proof

Galileo understood the requirement and spent years looking for a physical demonstration. He believed he had found it in the tides, and the fourth and final day of the Dialogue is devoted to it.

The argument is genuinely ingenious. Combine the earth's daily rotation with its annual motion around the sun. At midnight a point on the surface is moving in the same direction as the earth's orbital motion, and at noon it is moving backwards against it, so the actual speed of that point through space varies over the day. The seas, not rigidly attached, cannot follow the changing speed of their basins, and slosh. Tides follow from the earth's double motion, and if the earth stood still there would be none.

Example. Work out the effect quantitatively, and compare the prediction with what the tides actually do.

The earth's orbital speed is $29.79$ km/s and its equatorial surface speed is $0.4646$ km/s, so the modulation is $0.4646/29.79=1.56$ per cent of the orbital speed, once per day. That is a real effect and a small one. The fatal problem is not its size but its period: the mechanism gives one acceleration and one deceleration per rotation, so it predicts one high tide a day, at a fixed hour. The observed tide is semidiurnal, two highs in each period of 24 hours 50 minutes, which is $12$ hours $25$ minutes apart, and the time of high water shifts by about 50 minutes each day. That 50 minutes is the giveaway: it is exactly the daily lag of the moon.

Now you. Galileo knew the tides track the moon and rejected it. Why, and what did the rejection cost him?

The second act, 1632 to 1633

In 1623 Maffeo Barberini, a Florentine who had admired Galileo for years, became Pope Urban VIII. Galileo went to Rome, had six long audiences, and came away believing he had permission to write a book treating both systems provided it reached no conclusion. Urban asked that a particular argument be included: since God is omnipotent, any set of appearances could have been produced by means beyond our conception, so no physical account of them can be held to be necessarily true.

The Dialogue Concerning the Two Chief World Systems was licensed and printed in Florence in February 1632. It is a masterpiece of scientific writing and a political catastrophe. It is not balanced: the Copernican speaks with all the arguments and the Aristotelian, named Simplicio, is given the losing side and often made to look foolish. Urban's argument appears, but in Simplicio's mouth, in the closing pages. And the two chief world systems of the title are Ptolemy's and Copernicus's, which means the book stages a contest with a corpse and omits the Tychonic system entirely, the one arrangement that fitted every telescopic observation as well as Copernicus did.

Sales were stopped in August. The trial ran from April to June 1633, and its legal question was narrow: not whether the earth moves, but whether Galileo had violated the 1616 injunction and obtained his licence by concealing it. On 22 June 1633 he was sentenced as vehemently suspect of heresy, made to abjure kneeling, and placed under house arrest for the remaining nine years of his life. Seven of the ten cardinals of the Holy Office signed the sentence.

What the episode actually shows

Three corrections to the usual story, each of which matters for the argument of this subject.

The first is that the opposition was not uniformly clerical or anti-astronomical. The Jesuit mathematicians of the Collegio Romano, the best-equipped astronomers in Italy, confirmed Galileo's telescopic discoveries in 1611 and held a ceremony in his honour. Christoph Clavius had reformed the calendar; Christoph Grienberger and others did serious observational work. Most of them adopted the Tychonic system, and they did so with a defensible reason: it fitted every observation, and it did not require a physics nobody had. The people who knew the most were not persuaded, and their not being persuaded was rational.

The second is that the real question was jurisdictional. The Council of Trent had reserved the interpretation of scripture in matters of faith and morals to the Church, and Galileo, a layman, was publishing rules for reading scripture. Beneath the astronomy sits a fight about who is entitled to settle a dispute when a text and an observation appear to conflict, which is a question about authority rather than about the sky.

The third is that the ruling did not stop the work. Descartes, in the Netherlands, suppressed his own cosmological book Le Monde on hearing the news, and it was not published until after his death, which is a real cost. Italian astronomy did suffer. But Galileo, under house arrest and forbidden to publish, wrote the best book of his life, had it smuggled out to the Elzevirs in Leiden, and published it in 1638.

Example. The condemnation is often used to show that authority can suppress a scientific truth. What does the sequence of events actually establish about the standing of measurement in 1633?

That measurement had real but insufficient standing, and that both sides agreed on the criterion. Bellarmine conceded that a demonstration would compel a reinterpretation of scripture; Urban conceded that hypotheses which save the appearances are legitimate to compute with; Galileo accepted that he needed a demonstration and went looking for one. Nobody in the dispute held that evidence about nature was irrelevant. What the episode establishes is that in 1633 an observation could not yet outrank a text, that the standard for a demonstration was set at a height no astronomical evidence of the period could clear, and that the Copernicans could not clear it because they genuinely did not have the goods. The change that this subject is about is not that people started caring about evidence. It is that within two generations a body of evidence accumulated that met even a demanding standard, and that a method emerged for building such bodies deliberately.

Now you. Why does omitting the Tychonic system from the Dialogue count as a scientific failing and not just a tactical one?

What was missing

Strip out the personalities and the jurisdiction and one gap remains. The objection of the very first lesson was never answered: on a moving earth, why does a dropped stone land at the foot of the tower, and why is nothing flung off?

Galileo had the answer, or most of it, and it had nothing to do with astronomy. It came from rolling balls down inclined planes and timing them, from watching pendulums, and from thinking hard about what happens to a ball on a ship. Under house arrest at Arcetri, forbidden the subject of the earth's motion, he wrote it down. That book, and the science of motion in it, is where the next lesson goes.

The objection that a dropped stone would be left behind by a moving earth is a question about motion, not about the sky, and answering it meant measuring how things move, which nobody had ever done.

The previous lesson ended with Galileo under house arrest at Arcetri, forbidden to discuss the earth's motion. What he wrote there, and had smuggled to the Elzevir press in Leiden in 1638, is Discourses and Mathematical Demonstrations Concerning Two New Sciences. It never mentions the Copernican question. It is nonetheless the book that removes the objection, and it does so by a move that matters more than any of its results: instead of observing motion where it occurs naturally, it builds an artificial situation designed to make one quantity measurable.

Why fall cannot simply be timed

Take the standard demonstration, a weight dropped from a tower. The tower of Pisa is about 56 metres tall, and a body falling that far takes

Three and a half seconds, over which the body covers the last twenty metres in under half a second. To learn anything about how the speed changes you would need to mark positions to a fraction of a second, and in 1600 there was no instrument that could do it. Mechanical clocks of the period kept time to a quarter of an hour a day at best. There was no second hand anywhere in Europe, because there was nothing that could usefully drive one.

This is the real obstacle, and it explains why two thousand years of watching things fall produced nothing quantitative. The phenomenon runs faster than the available clocks. Aristotle's account, that heavier bodies fall proportionally faster and that speed is proportional to weight and inversely to the resistance of the medium, was not tested because it was not testable with anything to hand.

Galileo's solution was to change the phenomenon. A ball rolling down a gentle slope is doing the same thing as a falling body, if fall is what happens when a body is free to move towards the earth and a slope is a fall restrained. Tilt the plane less and the motion is slower, in a known proportion, and the same law can be examined at a speed a crude clock can handle. The acceleration along a plane inclined at angle $\theta$ is $g\sin \theta$, so a shallow angle divides it down as far as you like.

Example. Galileo describes a board about 12 braccia long, which is close to 7 metres, with a groove lined with parchment, raised at one end by 1 braccio in 12. What is the acceleration along it, and how long does a ball take to run its length?

The elevation of 1 in 12 makes $\sin \theta =1/12=0.0833$, an angle of $4.8°$. So

in metres per second squared, and from $s=\frac{1}{2}a{t}^2$,

Four seconds instead of the one second the same drop in height would take in free fall, and, more importantly, four seconds spread over seven metres of track that can be marked and subdivided. A real ball rolling rather than sliding is slower still, by a factor of $5/7$ in the acceleration, which puts the descent nearer $4.9$ s; that changes the timescale and not the law being tested. Galileo timed it by weighing the water that ran from a hole in a raised vessel during the descent, since the weight of water is proportional to the time, and reported that repeated trials agreed to within a tenth of a pulse beat.

Now you. Raise the same board by 2 braccia in 12 instead. What happens to the acceleration and to the time?

The law that comes out

The result is that the distance covered from rest goes as the square of the time. Divide the descent into equal intervals and the distances covered in successive intervals are in the ratio 1, 3, 5, 7, the odd numbers, since the cumulative distances go 1, 4, 9, 16 and the differences between consecutive squares are the odd numbers.

The derivation, given the assumption, is short. Suppose the speed grows in proportion to the time elapsed, $v=at$. Then over an interval starting at rest the average speed is half the final speed, $at/2$, and the distance is the average speed times the time:

The step in the middle, replacing a uniformly changing speed by its value at the midpoint, is the mean speed theorem, proved at Merton College and by Nicole Oresme in the fourteenth century, as the first lesson noted. Galileo did not discover it. What he did was assert that it describes actual bodies actually falling, and then go and check.

He also had to choose the assumption. Speed could grow in proportion to the time, or in proportion to the distance fallen, and both sound equally plausible; Galileo entertained the second before showing it leads to absurdity. Nothing decides between them except measurement, and it is the inclined plane that decides.

Example. A body falls from rest for one second and covers $4.9$ m. How far does it fall during the fourth second alone?

Cumulative distances go as $t^2$, so after three seconds it has fallen $9\times 4.9=44.1$ m and after four seconds $16\times 4.9=78.5$ m. The fourth second alone contributes $78.5-44.1=34.3$ m, which is $7\times 4.9$, the fourth odd number times the first distance.

Now you. A ball takes $4.1$ s to run the whole 7 m board. Where is it after half that time, and after a quarter of it?

Did he actually do it

The question is worth asking, because Galileo reports his results with suspicious tidiness and because a substantial school of historians said he did not.

Alexandre Koyré argued in the middle of the twentieth century that the experiments in the Discorsi are largely rhetorical: that a water clock could not deliver the precision claimed, that a real ball on a real board is spoiled by friction and by the energy taken up in spinning, and that Galileo reached his law by mathematical reasoning and then dressed it in apparatus for the reader's benefit. The charge is serious, and it has a general form worth noticing: a reported experiment that comes out exactly right is evidence of something, and not always of what the author intends.

Two things have answered it. In 1961 Thomas Settle built the apparatus as described, with a grooved board and a vessel of water, and got the times-squared law to about the accuracy Galileo claimed, which establishes that the reported precision was achievable with the reported equipment. And Galileo's working papers, examined closely by Stillman Drake in the 1970s, contain sheets of raw numbers, including one recording distances for a ball rolling off the end of a table, with arithmetic in the margin and values that do not fit the published account cleanly. Those are the traces of somebody measuring rather than illustrating.

The honest verdict is mixed and more interesting than either extreme. Galileo did experiments, reported them selectively, rounded in his own favour, and presented as a demonstration what had been a long and messy business of adjustment. The rolling ball does in fact lose about two sevenths of its acceleration to rotation, so his measured accelerations were never $g\sin \theta$, and none of that affects the proportionality he was testing, which is why the result survived the sloppiness.

The tower objection dissolved

None of this yet answers the tower. For that Galileo needed a second principle, and he states it in the Dialogue as an argument rather than an experiment.

Shut yourself below decks in a large ship with some flies, a bowl of water, and a friend throwing a ball. While the ship is at rest, observe how everything behaves. Now let the ship move steadily, without pitching. Nothing changes. The flies do not pile up at the stern, the ball takes the same effort to throw forwards as backwards, and drops of water fall straight into the vessel below. From no experiment inside the cabin can you tell whether the ship is moving, provided the motion is steady.

The reason is that everything in the cabin shares the ship's motion and keeps it. A body in motion continues in that motion unless something acts to change it, so the stone released from the tower keeps the eastward motion it had while held, and travels east while it falls at exactly the rate the tower does. It lands at the foot. The prediction of the first lesson was based on a hidden assumption, that a released body loses whatever motion it had, and once that assumption is denied the objection evaporates.

Pierre Gassendi settled the maritime version experimentally in 1640, dropping stones from the mast of a galley rowed at speed across Marseille harbour, and found them landing at the foot of the mast every time.

Be precise about the limit here. Galileo's conserved motion is not Newton's. He believed the natural persisting motion was along a circle concentric with the earth, on the grounds that a body moving in a straight line would eventually leave the earth's surface and rise, so his principle is a horizontal or circular persistence rather than a rectilinear one. Descartes states the straight-line version in 1644, and it becomes Newton's first law in 1687. Galileo's version does the job in the ship and on the tower, and it is wrong in the general case.

Two motions at once

The other consequence of persistence is what happens when a body has motion in two directions. Fire a ball horizontally off a cliff. Its horizontal motion continues unchanged, so horizontal distance grows in proportion to time. Its vertical motion is fall from rest, so vertical distance grows as the square of time. Eliminating the time between the two gives a vertical drop proportional to the square of the horizontal distance, which is a parabola.

This was not an academic result. Gunners had used range tables since the sixteenth century and Niccolò Tartaglia had published on the problem in 1537, without any correct account of the trajectory. Galileo's composition of motions gives one, along with the result that the maximum range comes at an elevation of $45°$ and that elevations equally above and below $45°$ give the same range.

Example. A cannon fires at $200$ m/s at $45°$. What range does the parabolic theory predict, and what actually happens?

The range of a parabolic trajectory over level ground is

Just over four kilometres. Seventeenth century guns of that muzzle velocity did not achieve anything like it, and the shortfall is air resistance, which for a heavy fast ball is not a small correction but comparable to the whole effect. Galileo knew this perfectly well and said so: his theorems hold exactly in a medium with no resistance, and the treatment of resistance he offers is frank guesswork. Naming the idealisation and its cost, rather than hiding it, is what makes the result usable.

Now you. The same gun fires at $30°$. Compare the predicted range with the one at $45°$, and say what other elevation gives the same range as $30°$.

What this science does not contain

It is worth listing what is missing, because the gap is exactly the size of Newton.

There is no concept of force. Acceleration is described, and its cause is not named. There is no idea of mass, no relation between force and acceleration, and no reason why bodies fall at all beyond the old talk of heaviness. Galileo does not state a value for $g$; he works entirely in proportions, and the first careful measurement of the actual distance fallen in a second was made by Giovanni Battista Riccioli in Bologna in the 1640s, timing drops from the city towers against a pendulum he had calibrated by counting swings through a whole night against the stars. Riccioli was an opponent of Copernicus and set out to test Galileo's law; he confirmed it, and published the confirmation alongside his arguments against the earth's motion, which is a good example of how a result can travel further than the person who found it.

Nor does any of this prove the earth moves. It removes an objection, which is a different and lesser thing. After 1638 a Tychonic astronomer could accept the whole of the new science of motion without changing his mind about the sky, and many did.

What has been established is something else, and it is the reason this lesson sits where it does. A question that seemed to be about the world as we find it, do falling bodies keep the motion they had, was answered by building a situation that does not occur in nature, a parchment-lined groove on a tilted board with water dripping into a bucket, on the argument that the artificial case exhibits the natural one with the confusions removed. Nothing in the older way of settling arguments licenses that. The next lesson follows the same move as it spreads to magnets, to blood and to a programme for knowledge in general.

A ball rolling down a parchment groove is not a falling body, and a stone sphere on a workbench is not the earth, so anyone arguing from one to the other owes an account of why the artificial case tells you about the natural one.

The previous lesson watched Galileo build a situation that does not occur in nature in order to measure something that does. That move is more important than any of his results, and it was being made at the same time by people with no interest in astronomy. This lesson takes three of them: a physician who built a model of the earth, a physician who did arithmetic on the heart, and a Lord Chancellor who tried to write down the rules. Between them they establish what an experiment has to be like before its result can be believed about anything outside the room it was done in.

A little earth on the bench

William Gilbert was president of the College of Physicians and physician to Elizabeth I, and he spent seventeen years and much of his own money on magnets. De Magnete, published in 1600, is the first book of its kind: a long, sober report of experiments, most of them his own, with the failures included.

The puzzle he attacked was the compass. A magnetised needle points roughly north, which had been known and used for four centuries and explained by an attraction to the pole star, or to a magnetic mountain in the far north, or to a point in the heavens. In 1576 the London instrument maker Robert Norman had added a second fact: a needle balanced so that it can turn in a vertical plane does not lie flat but dips downwards, in London by about $71°{50}^{\prime }$, nearly three quarters of the way to vertical. No attraction from a distant point in the sky explains that.

Gilbert's answer was to build a model. He had a lodestone turned into a sphere, called it a terrella, a little earth, and mapped the behaviour of a small pivoted needle over its surface. The needle points along the meridian, towards the pole of the sphere. Placed at the equator of the terrella it lies flat; nearer a pole it dips, and at the pole it stands upright. Break the sphere in two and each piece is a complete magnet with two poles. Heat it past a certain point and it loses its virtue.

Every one of those behaviours matches what a compass does on the earth. Gilbert's conclusion is the flat sentence that the earth itself is a great magnet, that its directive power comes from its own body rather than from the sky, and that the compass is responding to the ground under the sailor's feet.

Example. For a uniformly magnetised sphere, the dip angle $\delta$ at latitude $\lambda$ satisfies $\tan \delta =2\tan \lambda$. What dip does that predict for London, and how does it compare with Norman's measurement?

London is at latitude $51.5°$, so $\tan \lambda =1.257$ and $\tan \delta =2.515$, giving

Norman measured $71.8°$, so the model is $3.5$ degrees low. That is a substantial error by the standards of Tycho's astronomy and a small one for a first physical theory of a planetary property. Turned round, the measured dip corresponds to a latitude of $56.7°$ on a uniform sphere, five degrees north of where London actually is.

Now you. What should be concluded from the $3.5°$ discrepancy?

Where the model stops working

Gilbert did not stop where the evidence did. He argued that because a magnetic sphere has an axis it must rotate, treating the earth's daily rotation as a magnetic consequence, and he suggested that magnetic virtue holds the parts of the cosmos in their relations. Kepler picked that up and built his wrong physics of the planets on it. The same method, the same man, the same book: one conclusion that stands and one that does not, and nothing internal to the method separates them. What separates them is that the first was checked against measurements made by other people in other places, and the second was not checkable at all.

Half an hour of arithmetic

The received account of the blood, from Galen in the second century, was that blood is made in the liver out of digested food, is carried outwards through the veins to the parts of the body, and is there consumed. A small quantity seeps through invisible pores in the wall between the two sides of the heart, mixes with air from the lungs, and is distributed by the arteries. There is no circuit. Blood is manufactured, delivered and used up.

William Harvey, physician to James I and then Charles I, had studied at Padua under Fabricius, who had described the valves in the veins in 1603 without working out what they were for. In De Motu Cordis, published in 1628, Harvey killed the received account with a calculation that anybody could have done at any point in the preceding fourteen centuries.

Example. Take the left ventricle to hold about 2 ounces, suppose that only an eighth of its contents is expelled at each beat, and take a slow pulse of 1000 beats in half an hour. How much blood leaves the heart in that half hour, and what follows?

An eighth of 2 ounces is $0.25$ ounces, so the output is $0.25\times 1000=250$ ounces, which at 16 ounces to the pound is

A man of 150 lb contains something like 11 lb of blood in total. So on deliberately pessimistic assumptions, the heart pushes out more than the body's entire content of blood in half an hour, and several times a man's weight in a day. It cannot all be manufactured from food, since nobody eats that; and it cannot all be consumed at the periphery, since there is nothing there to consume it. The only possibility left is that the same blood goes round again.

Now you. Redo the argument with modern figures: a stroke volume of 70 mL, a pulse of 72 a minute, and a blood volume of 5 litres.

The route the blood takes

The arithmetic says blood must return; it does not say by what route. Harvey supplied the route with experiments on the arm that anyone can repeat. Bind a limb tightly and the arteries are compressed, the limb below goes pale, and the veins do not fill. Slacken the ligature so that the deeper arteries are open but the surface veins are still pressed, and the limb swells and the veins below the binding stand out, because blood is arriving through the arteries and cannot get back. Press a finger on a raised vein and slide a second finger along it towards the heart: the vein empties and stays empty, and it refills only from the far end, never from the heart's side. The valves Fabricius had drawn all face the same way, towards the heart. Blood goes out by arteries, back by veins, and only in that direction.

The honest limit is that Harvey never saw the join. Nothing in his account explains how blood crosses from the smallest artery to the smallest vein, and he was reduced to speaking of porosities in the flesh. Marcello Malpighi saw the capillaries in a frog's lung with a microscope in 1661, four years after Harvey's death. The circulation was accepted for thirty years on the strength of an argument whose last step was missing, because the alternative had been arithmetically destroyed.

Bacon's programme, and the ledger on it

Francis Bacon, Lord Chancellor of England until his conviction for taking bribes in 1621, wrote no experimental treatise of any consequence and discovered nothing. His importance is that he tried to write down what the new practice should be before there was much of it.

The Novum Organum of 1620 sets out to replace Aristotle's logic. Its diagnosis is that the mind is not a clean mirror: Bacon's four idols name the systematic errors of human beings in general, of each person's own temperament and training, of the words we inherit, and of the philosophical systems we are taught. Its remedy is to gather instances in an organised way. For any property under investigation, compile a table of cases where it is present, a table of the nearest comparable cases where it is absent, and a table of cases where it varies in degree, and then work upwards to what is always present when the property is and always absent when it is not. Where two explanations survive, look for an instantia crucis, an instance at a fork in the road, a case where the two accounts predict opposite things.

Set against that, what Bacon actually got wrong is a long list. He rejected the Copernican system. He rejected Gilbert's magnetic earth. He had no use for mathematics, which was already the most productive tool in the field, and he thought the labour of collecting instances could be distributed among ordinary men of no special ability. The method as literally specified does not work: there is no theory-free collection of instances, since deciding which cases are the comparable ones is already a theoretical act, and no amount of listing would have produced the law of fall or the ellipse.

What survives is worth more than the method. The idea that knowledge of nature is a collective enterprise rather than a personal accomplishment, that it accumulates, that it requires organisation and funding and a division of labour, that negative instances must be sought rather than avoided, and that the point of it is use: all of that is Bacon, and the next lesson shows an entire institution built on his prospectus by men who had read him carefully and quietly dropped his logic.

What makes a contrived case count

Pull the three together and the conditions under which an artificial situation licenses a claim about nature start to be visible.

The contrivance has to isolate the factor of interest while leaving it recognisably itself. Galileo's incline slows fall without changing what fall is; Gilbert's terrella is a sphere of magnetic material like the one he claims the earth to be. The step is defensible exactly to the extent that the resemblance is argued rather than assumed, which is where Gilbert's rotation argument fails: nothing about the terrella spins.

The result has to be checkable against something you did not arrange. Harvey's arithmetic predicts consequences in the intact living arm, and those consequences can be produced in front of witnesses. Gilbert's dip law predicts a number a navigator can measure at sea.

And the phenomenon has to survive other hands. This is the condition none of the three could yet guarantee, because there was no reliable way for a report to travel, no agreed way to describe an apparatus so that someone else could build it, and no forum in which a contested result could be tried.

Example. Harvey's ligature demonstration and Gilbert's terrella both persuade, but not in the same way. What is the difference?

Harvey's ligature is a demonstration on the very thing in question: the arm is a real arm, the blood is real blood, and the audience watches the actual system misbehave when interfered with. No inference from model to original is needed, and the only question is whether the interference has been correctly described. Gilbert's terrella is an analogy, and it carries an unavoidable extra premise: that the earth is relevantly like this stone. That premise is supported by the number of independent behaviours the model reproduces, direction, dip, the two poles, the effect of breaking the stone, but it is a premise, and it is exactly where Gilbert went wrong when he pushed on to rotation. The general lesson is that a model earns credibility by the count of independent things it gets right, and loses it the moment it is used for a property the resemblance was never tested on.

Now you. Bacon's crucial instance is meant to decide between two surviving explanations at a stroke. Why is a single such instance rarely as decisive in practice as it sounds?

That vacuum experiment is where this goes next. The question of whether nature abhors a vacuum had been argued from texts for four hundred years, and it was closed in four, by a glass tube full of mercury and a walk up a mountain.

Every pump maker in Europe knew that a suction pump cannot lift water higher than about thirty-four feet, and no natural philosopher had ever asked them why.

The previous lesson set out three attempts to make a contrived situation count as evidence, and ended with the condition none of them could meet: a result has to survive other hands. The vacuum experiments of the 1640s and 1650s are where that condition is first met properly, and the whole sequence, from a workshop complaint to a quantitative law, takes about twenty years. It is the best worked example in the century of an old question being settled by measurement.

The pump makers' limit

The question itself was ancient. Aristotle had argued that a void is impossible, since motion through it would be instantaneous and therefore absurd, and the scholastic tradition turned this into the principle that nature abhors a vacuum. Everything from a syringe to a siphon to a suction pump was explained by it: nature will not tolerate an empty space, so water rises to fill one.

The trouble is that the abhorrence has a limit. A lift pump raises water by taking away the air above it; when the pipe is longer than about $34$ feet, or $10.4$ metres, the water simply stops following the piston up, and a space is left at the top. Galileo discusses this in Two New Sciences in 1638, and his response is characteristic of a theory in trouble: he keeps the abhorrence and gives it a finite strength, as though the water column were a rope that snaps under its own weight at a certain length.

Evangelista Torricelli, who had been Galileo's assistant in his last months, took the other road in 1644. Suppose the water is not being pulled up by anything. Suppose it is being pushed up, by the weight of the atmosphere pressing on the surface of the well, and that the column rises until its own weight balances that push. Then the height at which it stops is not a property of the water's cohesion at all. It is a measure of how heavily the air presses.

Example. Water stops at $10.33$ m. Mercury is $13.6$ times as dense as water. If the air's weight is what holds the column up, how high a column of mercury should the same air support?

The pressure at the bottom of a column of liquid is $\rho gh$, so balancing the same air pressure with a denser liquid needs a proportionally shorter column:

Seventy-six centimetres, a length that fits in a room. Torricelli filled a glass tube about a metre long with mercury, stoppered it, inverted it into a dish, and released it. The mercury fell until it stood about $76$ cm above the dish, leaving an empty space at the top of the tube. He wrote to Michelangelo Ricci in June 1644 that we live submerged at the bottom of an ocean of the element air, and that the experiment was intended to make an instrument that shows the changes of the air, thicker and heavier or thinner and lighter.

Now you. Pascal repeated the experiment at Rouen in 1646 with tubes over 12 metres long, using water and also wine, in public. What height should the wine stand at, taking its density as $0.99$ that of water, and why was wine the interesting choice?

Why the tube alone does not settle it

It is worth being careful here, because the mercury tube is often presented as decisive and it is not.

A defender of the old view can accept every measurement. Let nature's resistance to a void have a fixed strength per unit area of the surface being torn open. Then the length of column it can support is inversely proportional to the liquid's density, exactly as the weight-of-air account predicts, and the mercury height of $76$ cm follows just as well. Both theories fit the numbers, because both are saying that a fixed force per unit area balances the weight of a column.

There was also a third position, defended by the English Jesuit Franciscus Linus, that an invisible membrane of rarefied matter, a funiculus, stretches from the top of the tube and holds the mercury up by tension. That sounds desperate now and was not obviously so then, since it explains why you feel a pull when you put a finger over the open end.

To separate the accounts you need a case where they disagree, and Blaise Pascal saw what it was. If the column is held up by the weight of the air above it, then carrying the tube up a mountain, so that less air is above it, must shorten the column. If it is held up by nature's abhorrence, or by a membrane in the tube, altitude is irrelevant: the tube does not know how high it is.

The Puy de Dôme

Pascal could not make the climb himself, so in November 1647 he wrote to his brother-in-law Florin Périer at Clermont, who lived at the foot of the Puy de Dôme, and set out the design. It took Périer until 19 September 1648 to get a clear day.

The design is what matters. Périer assembled witnesses, including clerics and laymen of standing, and prepared two identical tubes with mercury from the same vessel. He set up the first at the monastery of the Minims in Clermont, at the foot of the mountain, and left one of the fathers to observe it at intervals through the whole day and record whether it changed. He carried the second up the mountain, measuring at the summit and at points on the way down.

That first barometer is the point of the exercise, and it is the earliest well-documented control in the modern sense. Without it, a shorter column at the summit could be dismissed as an effect of the weather changing during the day, or of the tube having been disturbed on the walk. With it, the two possibilities are separated: the base instrument did not move all day, while the other fell as it climbed and returned to its old height on coming back down.

Example. At the Minims the column stood at 26 pouces 3.5 lignes; at the summit it stood at 23 pouces 2 lignes. Taking the Paris pouce as $27.07$ mm and the ligne as one twelfth of that, $2.256$ mm, what is the difference, and what altitude does it correspond to?

The base reading is $26\times 27.07+3.5\times 2.256=703.8+7.9=711.7$ mm and the summit reading is $23\times 27.07+2\times 2.256=622.6+4.5=627.1$ mm, a fall of $84.6$ mm, or 3 pouces 1.5 lignes, which is what Périer reported. Air pressure falls roughly exponentially with height, $p=p_0{e}^{-h/H}$ with a scale height $H$ of about $8400$ m in the lower atmosphere, so

The summit of the Puy de Dôme stands about $1065$ m above Clermont. Périer had no such formula and drew no altitude from his numbers; he simply reported that the column fell by three inches and more. But the reading was already an altimeter, and within a few years that is what it was used as.

Now you. Pascal repeated the test on the tower of Saint-Jacques in Paris, about 50 m high. What fall should he have seen, in millimetres and in lignes?

Making a vacuum to order

Otto von Guericke, mayor of Magdeburg, took the other route: rather than letting a vacuum appear at the top of a tube, he built a pump to make one. By 1650 he had an air pump that could evacuate a sealed copper vessel, and in 1654 he demonstrated it before the Emperor at Regensburg with the experiment everyone remembers, repeating it at Magdeburg a few years later. Two copper hemispheres, greased and fitted together with no fastening, were evacuated, and teams of horses harnessed to each side failed to pull them apart.

The size of the effect is easy to work out. The hemispheres were about 20 inches across, or $0.51$ m, and the atmosphere presses on the outside with about $101325$ Pa. The effective area is the circle the rim encloses rather than the curved surface, because only the component of the pressure along the axis contributes, so the force is $101325\times \pi \times {0.254}^2=20500$ N, or about $2.1$ tonnes. Eight horses to a side were not enough.

The arrangement also hides a physical joke. Two teams of eight pulling in opposite directions exert exactly the same separating force as one team of eight pulling against a wall, because the wall supplies the equal and opposite pull for nothing. Doubling the horses doubled the spectacle and not the force, and the whole thing was designed as spectacle, for an audience of princes whose support von Guericke wanted. That is worth noticing without sneering. The vacuum was a contested and expensive research programme, and a demonstration in front of the Emperor did more to establish that the air has real mechanical force than any quantity of careful reporting would have.

Boyle's law, and what Boyle actually claimed

Robert Boyle, in Oxford, read about von Guericke's pump and had a better one built in 1659 by his assistant Robert Hooke, with a glass receiver so that experiments could be watched inside it. New Experiments Physico-Mechanical, Touching the Spring of the Air appeared in 1660, and it is the first book to report a long programme of experiments on a single manufactured phenomenon.

The results inside the receiver are worth listing because each closes an escape route. A ringing bell falls silent as the air is withdrawn, so sound needs air. A candle goes out and a bird faints, so burning and breathing need air. A feather and a coin fall at the same rate, which is Galileo's claim freed from the medium at last. And a Torricellian tube placed inside the receiver behaves as it should: as the air around it is pumped out, the mercury column falls, and when air is let back in it rises. That last one is decisive against both rivals. If the mercury were held up by an abhorrence of the empty space at the top of the tube, or by a membrane inside it, nothing done to the air outside could matter.

Boyle then answered Linus with a measurement. Take a J-shaped tube, sealed at the short end, and pour mercury into the long open end to compress the trapped air. The trapped volume is read on the short arm; the pressure on it is the height difference of the mercury plus the atmosphere's own $29\frac{2}{16}$ inches. His published figures, in his own units, are these.

Example. Volume falls by a factor of four from 48 to 12 while pressure rises from $29.13$ to $117.56$ inches. How well does the product hold?

$48\times 29.13=1398$ and $12\times 117.56=1411$, a difference of $0.9$ per cent across a fourfold compression. Taking the product to be constant, $pV=1400$ or so, and predicting the pressure at a volume of 16 gives $1400/16=87.5$ inches against a measured $87.88$, an error of $0.4$ per cent.

Now you. Boyle called the relation a hypothesis of Mr Towneley's and did not claim it held universally. Was that caution justified?

What was actually settled

In 1640 the existence of a vacuum was a question about the interpretation of Aristotle, argued for four hundred years without resolution. By 1660 it was a piece of equipment, with a pump, a gauge, a price and a set of reproducible effects, and the residual argument was about details of the apparatus.

Three things made the difference. The prediction was quantitative, so a theory could fail by a number rather than by a debating point. The design isolated the disagreement, since the trip up the mountain is the one circumstance on which the accounts differ, and it was run with a control that closed off the obvious alternative explanation. And the phenomenon was manufactured to order, so that anyone with a pump could produce it at will, which is not a small point: nature does not offer vacuums, and this fact had to be made before it could be studied.

That last feature is also the vulnerable one. A fact that exists only inside an expensive machine, witnessed by whoever the owner invites, is a strange thing to build knowledge on, and Thomas Hobbes said so at length. Answering him meant deciding what a public fact is and who gets to certify one, which is where the argument goes after this, once the mechanical picture of the world that all these experiments assumed has been set out.

Experiments only tell you something if you already have a view about what sort of thing the world is, and by the 1640s a new one was on offer: matter, motion, and nothing else whatever.

The preceding lessons have been full of people measuring: a slope, a magnet, a heart, a column of mercury. None of that yet says what a good explanation looks like when the measuring is done. The answer that spread across Europe in the middle of the century is called the mechanical philosophy, and its central rule is that everything happens by contact, by one piece of matter pushing another. This lesson sets out what that bought and what it forbade, because the prohibition is what makes Newton's achievement so unwelcome to the people best equipped to understand it.

Matter is extension

René Descartes served in two armies, settled in the Dutch Republic, and suppressed his own cosmology, Le Monde, on hearing of Galileo's condemnation in 1633, as the earlier lesson noted. The physics appeared instead inside the Discourse on Method of 1637 and, in full, in the Principles of Philosophy of 1644.

Its starting point is a definition that does enormous work. The essence of body is extension: to be a material thing just is to occupy space, with length, breadth and depth, and nothing else belongs to matter at all. Colour, taste, smell, heat and heaviness are not in bodies; they are effects produced in us by the size, shape, arrangement and motion of the parts of bodies.

Three consequences follow immediately, and they are not modest.

There is no vacuum. If matter is extension, then a region with extension is a region with matter in it, and an empty space is a contradiction in terms. The universe is a plenum, full everywhere, and the space at the top of Torricelli's tube is filled with a subtle matter fine enough to pass through glass. Descartes said so explicitly, and the previous lesson's experiments were, from his point of view, about the mechanical force of the air and not about the void at all.

There is no action at a distance. A body can only affect another by touching it, because there is nothing to a body but extension and nothing it can do but occupy space and move. Any explanation appealing to attraction, sympathy or influence across a gap is not an explanation but a confession, and this is why the whole generation reacted to such talk as a return to the occult qualities of the schools.

And the world is one kind of stuff. The distinction of the very first lesson, a corruptible sublunary region and an unchanging heaven of a fifth element, is not so much refuted as made unstateable. The moon and a millstone are the same sort of thing, differing in the arrangement of their parts, so a physics that works here has to work there.

What it could compute

The programme would be worthless if it were only a metaphysics, and its most impressive product is a number.

Descartes published the sine law of refraction in 1637: light crossing between two media obeys $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, with a constant for each pair of media. Willebrord Snel had the same relation unpublished in 1621 and Ibn Sahl had it in Baghdad in 984, so the priority is tangled, but Descartes is the one who put it into print and used it.

Example. Light strikes water at $45°$ from the normal. Water has $n=1.333$ and air effectively $1$. Where does the ray go, and at what angle does light inside the water fail to get out at all?

Refraction into the water gives $\sin {\theta }_2=\sin 45°/1.333=0.7071/1.333=0.5305$, so ${\theta }_2=32.0°$. Going the other way, light inside the water meeting the surface at angle $\theta$ escapes only if $1.333\sin \theta \le 1$, so the critical angle is $\arcsin (1/1.333)=48.6°$; beyond that it is totally reflected back into the water.

Now you. Why is the second answer needed for the first proper theory of the rainbow?

The rainbow, computed

Descartes took a ray entering a spherical drop at height $b$ above the axis, traced it through with the sine law, and worked out its total deviation. He did this numerically, for ten thousand rays, by hand, and found that as the entry point moves across the drop the deviation reaches an extreme and turns back. Near that extreme, many rays leave in almost the same direction, so light piles up there and that direction is bright.

The algebra behind his arithmetic is short. With incidence angle $i$ and refraction angle $r$, the total deviation of a ray that refracts, reflects once and refracts again is

and this is stationary when

Example. Compute the rainbow angle for water with $n=1.333$.

$(n^2-1)/3=(1.7769-1)/3=0.2590$, so $\cos i=0.5089$ and $i=59.4°$. Then $\sin r=\sin (59.4°)/1.333=0.8607/1.333=0.6457$, giving $r=40.2°$. The deviation is

A deviation of $137.9°$ means the light comes back towards the observer at $180-137.9=42.1°$ from the direction pointing directly away from the sun. The bow is a circle of radius about $42°$ centred on the shadow of your own head, which is exactly where it is, and anyone can check it with an outstretched hand. This is the first time a common natural appearance was derived, quantitatively and correctly, from a law of matter and a piece of geometry.

Now you. Refractive index varies with colour: red light in water has $n=1.3318$ and violet $n=1.3435$. What does that do to the bow, and what could Descartes not explain?

Vortices

Having banned empty space and action at a distance, Descartes had to explain the solar system with pushes, and his answer was the boldest thing in the book.

The plenum is in motion, and motion in a full space has to be circulatory, since matter can only move by displacing other matter round in a closed loop. The universe is therefore divided into vast whirlpools, each with a star at its centre. Our sun sits at the middle of one, and the planets are carried round in the swirl like straws in an eddy, each settling at the radius where its bulk is in balance with the surrounding matter. Gravity is the same machinery: the fine matter of the vortex, whirling fastest, presses outwards most strongly and so forces coarse bodies such as stones towards the centre.

Give it its due. It explains, effortlessly, three facts that had never been explained: why all the planets go round the same way, why they lie in nearly the same plane, and why gravity acts towards a centre without anything reaching out to grab. It requires no attraction whatever, and it makes the earth's motion inoffensive to the Church, since the earth is at rest relative to the matter immediately around it and merely carried along, an argument Descartes made explicitly after 1633.

Its defect is that it computes nothing. There is no way to derive Kepler's ellipse from a vortex, no way to get the area law, and no way to obtain the three-halves power in the third law. Newton devoted the whole second book of the Principia to showing that fluid vortices cannot produce the observed motions, and that a vortex obeying Kepler's third law would have to have properties no fluid has. That did not settle it quickly: Cartesian physics remained the standard teaching in France until about 1740, and the argument between vortices and attraction ran for two generations.

Mechanism done properly

The strongest work in this tradition was Christiaan Huygens's, and it shows what the programme could achieve when the mathematics was taken as seriously as the metaphysics.

Descartes had stated seven rules of impact, and six of them are wrong, because he took the conserved quantity to be size times speed without regard to direction. Huygens, using the relativity principle from the lesson on motion, worked out the correct rules and showed that what is conserved in an elastic collision is the vector quantity and also the sum of the products of mass with the square of speed. In 1668 the Royal Society asked for solutions to the collision problem, and Huygens, John Wallis and Christopher Wren independently supplied compatible answers.

The pendulum clock is the other achievement, and it is the one that changed daily life. Galileo had noticed that a pendulum's period is nearly independent of the size of the swing. Huygens turned that into a timekeeper in 1656, and analysed it properly in the Horologium Oscillatorium of 1673, where he proved that the exactly isochronous path is a cycloid rather than a circle, derived the relation

and published the formula for the outward tendency of a body moving in a circle, proportional to $v^2/r$. That last result is the one Newton needed and did not have to invent.

Example. How long must a pendulum be to beat seconds, meaning a period of two seconds with one swing each way per second?

Rearranging, $L=g{T}^2/4{\pi }^2=9.81\times 4/39.48=0.994$ m. Just under a metre, which is why longcase clocks are the height they are. A pendulum of exactly one metre has a period of $2\pi \sqrt{1/9.81}=2.006$ s.

Now you. Mechanical clocks before 1656 kept time to about 15 minutes a day; a good pendulum clock kept it to about 10 seconds. What does that factor buy, and what does the formula above make possible besides timekeeping?

The prohibition

Step back and the mechanical philosophy is doing two things at once. It is a research programme, extremely productive, which produced the rainbow, the laws of impact, the pendulum clock and the whole idea that qualities reduce to the arrangement of parts. It is also a rule about what counts as an explanation, and that rule is a prohibition: no attraction, no influence across a gap, no property of a body that is not the size, shape or motion of its parts.

Everyone signed it. Galileo invoked it to dismiss the moon's effect on the tides, as the lesson on the trial described. Boyle called himself a corpuscularian and wrote a book against the notion of nature as an agent. Huygens and Leibniz held to it all their lives.

That is the situation into which the Principia arrives, with a force that acts instantaneously between bodies separated by empty space, in inverse proportion to the square of their distance, with no mechanism whatever offered for how it does so. Before that, though, there is a question this lesson has taken for granted: how any of these results reached anyone else at all. Descartes computed the rainbow in a house in the Dutch countryside, Huygens in The Hague, Boyle in Oxford. What made those private activities into a shared body of knowledge is the subject of the next lesson.

Boyle's pump stood in a house in Oxford, Descartes computed the rainbow alone in the Dutch countryside, and Galileo saw Jupiter's moons through the only good telescope in the world, so the obvious question is how any of that became something the rest of Europe could use.

The previous lessons have produced a stock of results and one unmet condition, stated two lessons ago: an experimental fact has to survive other hands. Meeting it took a set of institutions and conventions built deliberately between roughly 1640 and 1670, and they are as much a part of the change this subject describes as any telescope. What follows is about how a private observation was turned into a public fact, and about the best objection anyone made to the whole arrangement.

The switchboard

Before there were societies there was correspondence. The central node was a Minim friar in Paris, Marin Mersenne, who from the 1620s until his death in 1648 wrote to and received letters from something like 140 correspondents across Europe, including Descartes, Galileo, Torricelli, Hobbes, Fermat, Pascal and Huygens.

What Mersenne did was more than pass letters on. He posed problems to several people at once, circulated an answer to the person best placed to attack it, and arranged for experiments to be repeated in another city. When Pascal's brother-in-law climbed the Puy de Dôme, the design and the result travelled through channels of this kind. A network of that sort is fragile in one specific way: it depends on a person. Mersenne died and it dissolved.

Alongside the letters ran the older system of patronage. Galileo named Jupiter's moons after the Medici and got a court appointment for it, and disputes at court were adjudicated by the patron's favour rather than by the evidence, since a client's credibility was a function of his standing. That arrangement can produce excellent work and it cannot produce a stable public fact, because the verdict changes when the patron does.

Why publish at all

There is a prior question that a modern reader skips: why tell anyone. The craft tradition, from which most instrument makers came, kept its knowledge secret because secrecy was its living. Mathematicians in sixteenth century Italy fought public problem-solving contests for university chairs and concealed their methods for the same reason. A discovery was an asset, and giving it away was giving away the asset.

The transitional device is the anagram, and it is a wonderfully exact expression of the dilemma. Galileo announced the phases of Venus in 1610 as a scrambled Latin sentence, which established that he had something on a date without saying what. Huygens did the same in 1656 for the ring of Saturn, releasing the solution three years later when he was ready. Hooke published ceiiinosssttuv in 1676 and revealed it in 1678 as ut tensio sic vis, as the extension so the force, which is the law of the spring.

Example. What is the anagram trading, and what replaced it?

It trades disclosure for a dated claim. The author gets a public, verifiable record that he possessed the result at a certain moment, without letting a rival use it, which is exactly what a secretive practitioner wants. What replaced it is better in every respect for everyone but the author: a journal that prints the result in full, with a date on the issue. Publication buys the same priority and pays for it by handing the content over, and it only becomes attractive when the currency changes, when a reputation built on published results is worth more than the private use of an unpublished one. That change is what the institutions of the 1660s brought about, and once it had happened the anagram disappeared within a generation.

Now you. Newton had his method of fluxions by 1666 and first let anything of it appear in print in 1693. Leibniz reached his version of the calculus in 1675 and published in the Acta Eruditorum in 1684. What does that pair of dates say about what a publication system rewards?

An institution with rules

On 28 November 1660, after a lecture by Christopher Wren at Gresham College, a group of twelve agreed to meet weekly to promote experimental learning. Charles II granted a charter in 1662 and a second in 1663, which lists 131 original fellows. The society took as its motto nullius in verba, on no one's word, a phrase from Horace that reads as a direct repudiation of the method of settling arguments described in the first lesson.

The practices matter more than the charter. Experiments were performed at the meeting, in front of everyone, and Robert Hooke was appointed Curator of Experiments in 1662 and required to bring three or four to each weekly meeting, which is a punishing schedule and a revealing one: the society's business was to make things happen in the room rather than to discuss reports of them. Results were entered in a register book with a date, so a claim of priority could be checked against a record the claimant did not control. Attendance was recorded, which turns the audience into named witnesses.

Two models

The Académie Royale des Sciences, founded in Paris in 1666, made the opposite choices on almost every point. Its members were few, salaried by the crown, chosen by the state, and organised in ranks; it worked on problems the state wanted solved, and it published collectively. The English body was larger, unpaid, self-financed, open to anyone respectable who would pay a shilling a week, and correspondingly amateur. Both models still exist, and the difference between them is not a national quirk. A salaried academy can direct effort at a problem the state wants solved and can sustain a programme over decades, which is how French astronomy came to dominate the measurement of the earth. A subscription society cannot direct anything, and depends on whoever turns up, but it is much harder to silence, since no one can be dismissed from it for reaching an unwelcome result. Every arrangement for funding enquiry since has sat somewhere between the two, and has inherited the same trade.

The literary technology

The most easily missed invention is a way of writing.

Boyle's experimental reports are extraordinarily long-winded by any earlier standard, and the prolixity is the point. He describes the apparatus in detail sufficient to rebuild it, names the people present, gives the date, reports the trials that failed as well as those that worked, and hedges his conclusions with qualifications about what he did not establish. His books carry detailed engravings of the pump, which is not decoration but specification.

The historians Steven Shapin and Simon Schaffer named the effect virtual witnessing: the reader is put in the position of someone who was in the room, and a fact witnessed by a hundred readers is more solid than one witnessed by six fellows. Reporting failures is central to it. An account in which everything worked reads as a claim about the author's skill; an account including the trials that leaked and the seals that failed reads as a description of an object that behaves in a certain way, which is what an experimental fact is meant to be.

Around this sat a convention of civil dispute. Disagreement was to be about the fact and not the person, hypotheses about causes were to be kept separate from reports of what happened, and a fellow who could not reproduce a result was to say so plainly and without imputing dishonesty. Those conventions look like manners. They are the mechanism by which a contested claim can be argued to a conclusion instead of becoming a quarrel between gentlemen, which in that period could be settled with a sword.

Example. Boyle reports the trials in which his pump leaked and the experiment failed. Why does including them strengthen rather than weaken the report?

Because the reader's problem is not whether Boyle is clever but whether the phenomenon is real, and a report of unmixed success is compatible with a great many things other than a real phenomenon, including selective reporting and an author skilled at making an apparatus behave. Failures give the reader the information needed to distinguish those cases: they show the conditions under which the effect does not appear, which is how you learn what the effect depends on, and they show that the author is describing the machine's behaviour rather than curating an impression. There is a practical benefit too, which is that the next person to build a pump knows where the seals fail and does not conclude from a leak that Boyle was lying.

Now you. Hobbes objected that a fact produced by an expensive machine in a closed room, witnessed by a self-selected club, is no foundation for knowledge at all. Where is he right?

The journal

The final piece arrived on 6 March 1665, when Henry Oldenburg, the society's secretary, brought out the first issue of Philosophical Transactions, a private venture he edited and largely paid for and from which he hoped, unsuccessfully, to make a living. The Journal des Sçavans had appeared in Paris two months earlier with a broader remit.

Oldenburg's innovation is the combination of four things in one object. It appears on a date, so priority is fixed by a public record. It appears at intervals, so the reader has a reason to keep looking. It is a compilation, so the news of the whole network reaches someone who belongs to none of it. And its contents were vetted before printing, informally by Oldenburg and by fellows he consulted, which is the ancestor of refereeing, made a formal requirement when the society took the journal over in 1752.

The volume of work behind it is worth stating. Oldenburg conducted the society's foreign correspondence himself, in several languages, at a rate that reached tens of letters a week, and in 1667 he was imprisoned in the Tower of London for two months on suspicion of intelligence with the enemy during the Dutch war, because a man writing constantly to foreigners looked exactly like a spy. Philosophical Transactions has appeared continuously ever since, which makes it the longest-running scientific journal in the world.

Example. List what a fact had to acquire, by 1670, before it could outrank a text of Aristotle.

It had to be produced by an apparatus described in enough detail for someone else to build it. It had to be witnessed, by named people, on a stated date, and entered in a record the author did not control. It had to be reproduced, by other people using other instruments, with the failures reported too. It had to be published in a dated periodical that put it in front of readers who had no stake in the outcome. And it had to survive the objections of anyone who cared to make them, argued under conventions that made disagreement about the claim rather than about the man. Every one of those is a social arrangement rather than a discovery, and together they answer the question the first lesson posed. A text was preferred to an observation in 1500 because a text was stable, public and checkable while an observation was none of those things. By 1670 an observation could be all three.

Now you. Why does none of this machinery guarantee that the facts it certifies are true?

By 1670 there was an apparatus for producing and certifying knowledge about nature: instruments with known errors, a technique of contrived experiment, a mathematics adequate to describe motion, a mechanical picture of what explanations must look like, and institutions that made a result public and durable. What there was not was a physics. Nothing yet explained why Kepler's ellipses have the shape they do, and nothing connected the fall of a stone to the motion of the moon. The next lesson is about the calculation that did.

By 1670 there were two separate bodies of exact knowledge about motion, one for the heavens and one for the ground, and nothing whatever connecting them.

Kepler's three laws, from the earlier lesson, describe how planets move and offer no cause. Galileo's law of fall describes how bodies drop and offers no cause either. Both were solid, both were quantitative, and each was silent about the other. Isaac Newton's achievement is to show that they are two consequences of one rule, and the argument that persuaded him is a single arithmetical comparison that this lesson works through in full.

Where the inverse square comes from

Start with the one thing everyone agreed on by 1670: Kepler's third law, that the square of a planet's period goes as the cube of its distance, $T^2=k{r}^3$ for some constant $k$ shared by all the planets.

Then take the result from the lesson on the mechanical philosophy. A body moving in a circle of radius $r$ at constant speed is continually changing direction, and Huygens had published in 1673 the size of the acceleration required to keep it there:

The speed of a body going once round a circle in time $T$ is $v=2\pi r/T$, so

Example. Substitute Kepler's third law into that expression and see what the acceleration towards the sun depends on.

Putting $T^2=k{r}^3$ into the denominator:

The $r$ in the numerator cancels two of the three in the denominator and an inverse square law falls out, with a constant that is the same for every planet because $k$ is. So if planets are held in their orbits by an acceleration directed at the sun, that acceleration must weaken as the square of the distance. Nothing in this derivation is difficult, which is why several people had it: Newton by about 1666, Robert Hooke by 1670, and Edmond Halley and Christopher Wren independently. In January 1684 the three Londoners discussed it over coffee and none of them could do the hard part, which is to show that an inverse square force produces an ellipse rather than merely being consistent with a circle. In August, Halley went to Cambridge and asked Newton, who replied that it would be an ellipse and that he had calculated it.

Now you. Suppose observation had given $T^2\propto r^2$ instead. What force law would follow, and what does that tell you about the status of the inverse square?

Why the circle is the easy case

Everything above assumes a circular orbit, and no planetary orbit is circular. That restriction is not a detail, and it is the reason four capable men could all have the inverse square law and still have nothing.

The derivation runs in the wrong direction. It starts from an observed regularity across the planets, Kepler's third law, and extracts the exponent of a force. What is needed is the reverse: assume a force falling off as the inverse square of the distance from a fixed point, let a body move under it with whatever speed and direction it happens to have, and prove that the path traced out is a conic section with that point at a focus. Only then are Kepler's first two laws consequences of the force rather than facts sitting alongside it, and only then does the theory apply to a body whose distance is changing, which is every real planet and every comet.

That problem needs a mathematics of continuously changing quantities, and it is genuinely hard. Hooke wrote to Newton in 1679 with the physical idea, that orbital motion is straight-line motion continuously bent by a central attraction, and Hooke could not do the mathematics; he said as much, and later claimed the result anyway. Halley could not do it. Wren offered a prize of a book worth forty shillings to whoever could produce a proof within two months, and nobody collected it.

Newton could, and the nine-page tract he sent Halley in November 1684 is the proof. It also settles the converse, that an inverse square attraction admits ellipses, parabolas and hyperbolas depending on the speed at a given distance, which is what makes a comet on an enormous elongated orbit the same kind of object as a planet. Kepler's first two laws stop being descriptions of what planets happen to do and become theorems.

Comparing an apple with the moon

The bold step is to suppose that the acceleration holding the moon in orbit is the very same one that makes a stone fall in an orchard, weakened by distance according to the law just derived. That is testable with numbers available in the 1660s.

The moon's distance had been known reasonably well since antiquity, from the size of the earth's shadow in a lunar eclipse and from parallax, at about 60 earth radii. The modern figures are an earth radius of $6371$ km and a mean lunar distance of $384400$ km, so

The moon's sidereal period is $27.3217$ days, which is $2.3606\times {10}^6$ seconds.

Example. Compute the moon's acceleration towards the earth, and compare it with $g$ divided by the square of $60.3$.

The centripetal acceleration is

which is $2.723\times {10}^{-3}$ metres per second squared. The prediction from an inverse square law anchored at the earth's surface is

in the same units. The two agree to $1.1$ per cent. The moon is falling, continuously, at the rate a stone would fall if a stone could be carried out to sixty earth radii, and the two numbers were obtained from completely independent measurements: one from an orchard and a pendulum, the other from an eclipse and a calendar.

Now you. Newton's early attempt used the common estimate that a degree of latitude is 60 miles, giving an earth radius of $3438$ miles or $5530$ km. Redo the comparison with that figure and see what happens.

The obstacle that actually mattered

The comparison above quietly assumes something enormous. It treats the distance from the earth to the moon as the distance between two points, and the distance from the earth to the apple as the earth's radius. But the earth is not a point. It is a ball of matter, and every part of it is a different distance from the apple, with the near parts a few metres away and the far parts twelve thousand kilometres away, each pulling with a different strength and in a different direction.

For the moon, sixty radii out, treating the earth as a point is obviously a fair approximation. For an apple three metres above the ground it is not obviously anything, and until you know what the total pull of a whole sphere amounts to, the number $g$ cannot be compared with anything.

The result Newton needed is the shell theorem: a uniform spherical shell attracts an external body exactly as though its entire mass were concentrated at its centre, provided the attraction of each piece falls off as the inverse square. It is a special property of that exponent and it fails for others. He proved it in the spring of 1685, twenty years after the first version of the moon test, and it is Proposition LXXI of the Principia. Only with that theorem in hand does the apple's distance from the earth become the earth's radius, and only then does the calculation above mean anything.

There was a second obstacle, conceptual rather than mathematical. Newton's early work, following Huygens and Descartes, was framed in terms of a body's outward endeavour in a circle, balanced by something. The modern framing, in which a body would travel in a straight line and is continuously deflected inwards by a force, arrived clearly in a letter Hooke wrote him in November 1679, proposing exactly that decomposition. Newton found the suggestion useful and spent the next thirty years refusing to admit it, which is one reason the two men detested each other.

Predicting rather than checking

A comparison of two numbers is one thing; a prediction is stronger. Turn the relation round and use the surface value of $g$ to predict how long the moon should take to go round.

Setting the required acceleration equal to the inverse square law gives $4{\pi }^{2}r/T^2=g{R}^2/r^2$, so

Example. Evaluate that with $r=3.844\times {10}^8$ m, $R=6.371\times {10}^6$ m and $g=9.81$.

The numerator inside the root is $r^3=5.680\times {10}^{25}$ and the denominator is $9.81\times 4.059\times {10}^{13}=3.982\times {10}^{14}$, so the ratio is $1.4265\times {10}^{11}$ and its square root is $3.777\times {10}^5$. Multiplying by $2\pi$ gives $2.373\times {10}^6$ seconds, which is $27.47$ days. The observed sidereal period is $27.32$ days, so the prediction is high by $0.53$ per cent. One measurement made in a laboratory, plus two distances, predicts the length of the month.

Now you. Where does that remaining half a per cent come from?

What the test does and does not establish

Be careful about the size of the conclusion. The moon test establishes that one law of attraction towards the earth, weakening as the inverse square, fits both a falling stone and the moon's orbit. That is already a repudiation of the division that opened this subject, since a single rule now spans the sublunary and the celestial.

It does not establish that gravitation is universal. It says nothing about whether the moon pulls back on the earth, whether Jupiter pulls on Saturn, or whether two stones on a bench attract each other. Those claims need the third law of motion, that action and reaction are equal and opposite, and a great deal more work, and they bring consequences the moon test does not: the tides, the precession of the equinoxes, the perturbations of one planet by another, and the shape of the earth.

Nor does it explain anything. The whole apparatus says how strongly bodies are attracted, and nothing whatever about why or by what means, across a quarter of a million miles of empty space, with no contact of any kind. Everything the mechanical philosophy stands for forbids it. That is the bill for the Principia, and the next lesson is about what it delivered and what it cost.

The book that ends this story does something no book had done before: it predicts, from one rule, a set of quantities that had never been connected, and it declines to say why the rule holds.

The previous lesson got as far as a comparison of two numbers, the fall of a stone and the fall of the moon, agreeing to about one per cent under an inverse square law. That is a good result about the earth and its satellite. This lesson is about what happens when the same rule is applied to every pair of bodies in the universe, what it bought, and the bill that came with it.

How it got written

In August 1684 Edmond Halley travelled to Cambridge and asked Isaac Newton what path a planet would follow under an inverse square attraction. Newton said an ellipse, and that he had calculated it, and then could not find the paper. He reworked it and sent Halley a nine-page tract, De motu corporum in gyrum, that November. Halley saw what it was and pushed for more.

Two and a half years later the Philosophiae Naturalis Principia Mathematica was printed, in July 1687. The Royal Society had agreed to publish it and then discovered it had no money, having spent its budget on Francis Willughby's History of Fishes, so Halley paid for the printing himself out of a modest income. Something like three or four hundred copies were made. Halley also wrote the review, and the Latin ode at the front.

The structure is worth knowing. Definitions and axioms come first, including the three laws of motion, of which the third, that action and reaction are equal and opposite, does the work that turns attraction into a mutual relation. Book I develops the dynamics of bodies moving without resistance, and contains the proof that an inverse square force gives conic sections and the shell theorem of the previous lesson. Book II treats motion through resisting fluids, and ends by demonstrating that the vortices of the mechanical philosophy cannot reproduce Kepler's third law. Book III, the System of the World, applies all of it to the solar system.

Weighing a planet

The central claim of Book III is that every body attracts every other with a force proportional to the product of their masses and inversely as the square of the distance between them. Newton never wrote it as an equation with a constant in front, and the constant itself was not measured until Henry Cavendish weighed the earth with a torsion balance in 1798, obtaining a value equivalent to about $6.67\times {10}^{-11}$ in modern units. Everything in the Principia is done in ratios, which is why it can determine the relative masses of the sun and planets without knowing any of them absolutely.

The method is the third law applied twice. For a small body circling a large one, the same substitution as in the previous lesson gives $T^2=4{\pi }^2{a}^3/GM$, so the combination $a^3/T^2$ measures the mass of the central body. Compare the value for a moon of Jupiter with the value for a planet of the sun, and the masses of Jupiter and the sun come out in ratio.

Example. Io orbits Jupiter at $421700$ km with a period of $1.769$ days. What is Jupiter's mass as a fraction of the sun's?

Convert into astronomical units and years, so that the earth's own orbit gives $a^3/T^2=1$ for the sun. Io's distance is $421700/1.496\times {10}^8=2.819\times {10}^{-3}$ AU and its period is $1.769/365.25=4.843\times {10}^{-3}$ years. Then

which is one part in $1047$. The modern value is one part in $1047$. Newton's own figure in the third edition was one in $1067$, limited by the accuracy of Io's measured distance from Jupiter. Nobody had ever weighed anything that was not on a balance.

Now you. Titan orbits Saturn at $1221870$ km in $15.945$ days. Find Saturn's mass in the same way.

Three predictions nobody else could make

The case for the Principia is not that it explains gravity, since it does not, but that one law delivers a list of unrelated things that had never been connected to anything.

The tides get their first correct account. The moon pulls harder on the near side of the earth than on the centre, and harder on the centre than on the far side, so the water is drawn up in two bulges on opposite sides. Two high tides per day follow, and the timing follows the moon, which is exactly the fact Galileo threw away, as the lesson on the trial described. The sun does the same thing more weakly, in a ratio of about $2.17$ to $1$ in the moon's favour, so tides are large when the two act together at new and full moon and small when they act at right angles. Every part of that is a consequence of the inverse square law and none of it was available before.

The precession of the equinoxes gets its first explanation. Hipparchus had discovered around 130 BC that the equinox drifts slowly round the zodiac, taking about $26000$ years, a rate of some $50$ arcseconds a year. Nobody had ever proposed a cause. Newton showed that the earth's equatorial bulge gives the sun and moon something to take hold of, and that the resulting torque makes the axis wheel round like a slow top.

And the earth turns out not to be round.

Example. A rotating fluid body bulges at its equator. Newton calculated a flattening of about 1 part in 230, and the true value is 1 in 298. What does the difference amount to on the ground, and how was the question settled?

A flattening of $1/298$ on a radius of $6371$ km means the equatorial radius exceeds the polar one by about $21$ km; Newton's $1/230$ would make it $28$ km. Either way the earth is an oblate spheroid, wider than it is tall. The observable consequence is that the length of one degree of latitude, measured along the surface, is longer near the poles, where the surface curves less sharply: $111.7$ km at the pole against $110.6$ at the equator, a difference of $1.1$ km per degree. This mattered because the Cassini family, from measurements within France, had concluded the opposite, that the earth is a lemon rather than an orange. The Académie settled it by sending two expeditions: Maupertuis to Lapland in 1736, and Godin, Bouguer and La Condamine to Peru from 1735, an eight-year ordeal at high altitude. The Lapland degree came back longer than the French one. Newton was right, Cassini was wrong, and Voltaire congratulated Maupertuis on having flattened both the earth and the Cassinis.

Now you. Why is Newton's $1/230$ too large?

The comet

The most public vindication came from Halley. In his Synopsis of the Astronomy of Comets of 1705 he computed orbits for 24 comets using Newton's methods, and noticed that those of 1531, 1607 and 1682 had nearly identical elements and were spaced about 76 and 75 years apart. He proposed that they were one object on a long ellipse, and predicted its return around 1758.

That prediction is unlike anything before it. It is specific, it concerns a class of object regarded for millennia as portents outside the order of nature, it could not be hedged, and it would be tested long after its author was dead. Halley died in 1742.

Example. As 1758 approached, Alexis Clairaut, working with Joseph Lalande and Nicole-Reine Lepaute, spent six months computing how much Jupiter and Saturn would delay the comet, and announced perihelion around mid-April 1759, with a stated uncertainty of about a month. The comet passed perihelion on 13 March 1759. How good is that?

The prediction was about 32 days early relative to the event, inside the uncertainty Clairaut had quoted. Measured against the orbital period of roughly 76 years, or $27760$ days, the error is $32/27760=0.12$ per cent. What deserves emphasis is not the accuracy but the structure of the exercise: a law inferred from planetary motions was used to calculate a perturbation of one body by two others, applied to an object last seen in the previous century, and checked against an event nobody could influence. There is no way to fake that and no way to explain it away if it fails.

Now you. Why does the perturbation calculation matter more than the bare prediction of a return?

The price

Set against all this is one thing, and every competent mechanical philosopher in Europe saw it at once. Gravitation acts between bodies separated by empty space, instantaneously, in proportion to a quantity of matter, with no intervening medium and no contact whatever. The lesson on the mechanical philosophy set out the prohibition that everyone in the field had signed: no action at a distance, no influence across a gap, no property of a body that is not the size, shape or motion of its parts. The Principia breaks it on the first page of Book III.

Huygens, who admired the mathematics without reservation, wrote that the principle of attraction seemed to him absurd. Leibniz called it a return to the occult qualities of the schoolmen and, later, a perpetual miracle. The charge was not obscurantism: it was that Newton had stopped doing physics and started doing mathematics, describing the phenomena in equations without giving any account of how the world produces them.

Newton agreed with them about the problem. In a letter to Richard Bentley in 1693 he wrote that for gravity to be innate and essential to matter, so that one body could act upon another at a distance through a vacuum without anything to carry the action, was so great an absurdity that no one competent in philosophical matters could fall into it. He spent decades looking for a medium that would carry the force, and published none of it.

What he did instead is in the General Scholium added to the second edition in 1713. He has not been able to discover the cause of gravity from the phenomena, and he frames no hypotheses, because whatever is not deduced from the phenomena is to be called a hypothesis and hypotheses have no place in experimental philosophy. It is enough, he says, that gravity really exists and acts according to the laws set out, and suffices to account for all the motions of the heavenly bodies and of the sea.

That is the price, and it should be stated plainly, because it is the largest single change in this whole subject. Before the Principia, an explanation meant an account of how something is brought about. After it, a mathematically exact law with a wide range of confirmed predictions counts as knowledge whether or not anyone can say what produces it. Understanding is demoted; prediction is promoted. Everything that has come since has been done on those terms, and the people who objected were not fools defending superstition. They were defending the older and more ambitious idea of what it means to explain something, and they lost.

How long it took

Britain accepted the Principia quickly, helped by the fact that Newton was its author and by a run of textbooks and public lectures. On the Continent it took roughly fifty years. French universities taught Cartesian vortices into the 1740s, and the shift came from a small number of advocates and two decisive results: Voltaire's Lettres philosophiques of 1734 and his Elements of Newton's Philosophy of 1738, Émilie du Châtelet's French translation with her own commentary, finished shortly before her death in 1749 and published in full ten years later, the Lapland expedition's confirmation of the earth's shape in 1737, and the comet in 1759.

Notice what did the persuading. Not the elegance of the argument, and not the metaphysics, on which Newton was and remained vulnerable. What settled it was that the theory kept producing quantities that could be checked, and the checks kept coming out right, in cases the theory had not been built to handle. That is a claim about method as much as about gravity, and drawing out what the method turned out to be, and when it was actually adopted, is the business of the last lesson.

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 $29.79$ km/s and light travels at $299792$ km/s. What tilt does that predict?

One radian is $206265$ arcseconds, so $\theta =20.5$ arcseconds. The modern value of the constant of aberration is $20.49$ arcseconds, and Bradley's own determination was about $20.2$. 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 $c$ 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?

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 $5.2$ 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 $0.314$ arcseconds.

Example. How far away is that, and what does it do to Tycho's objection?

Distance in astronomical units is $206265$ divided by the parallax in arcseconds:

which is $10.4$ light years; the modern parallax of $0.286$ arcseconds gives $11.4$. Compare the lower bound Tycho's precision imposed, from the lesson on his instruments: no parallax above 1 arcminute means the stars are beyond $3438$ AU. The true distance is $191$ 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?

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 $15\sin \lambda$ degrees per hour, where $\lambda$ is the latitude.

Example. What does that give at the latitude of Paris, $48.87°$?

$15\times \sin (48.87°)=15\times 0.7532=11.3$ degrees per hour, so the plane comes back to its starting orientation after $360/11.3=31.9$ 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?

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.