A new science of motion

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

t = \sqrt{\frac{2 s}{g}} = \sqrt{\frac{2 \times 56}{9.81}} = 3.4 s

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 $θ$ is $g \sin θ$ , 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 $\sin θ = 1 / 12 = 0.0833$ , an angle of $4.8 °$ . So

a = g \sin θ = \frac{9.81}{12} = 0.818

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

t = \sqrt{\frac{2 \times 7.0}{0.818}} = 4.1 s

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?

Answer

The acceleration doubles to $1.64$ m/s², since $\sin θ$ doubles, and the time falls by a factor of $\sqrt{2}$ to $2.9$ seconds, since $t \propto 1 / \sqrt{a}$ . That the time changes as the square root rather than in proportion is itself a check on the law, and it is the check Galileo actually ran: he repeated the whole experiment at many different inclinations and found the same rule each time. Doing it at one angle would show only that a ball reaches the bottom. Doing it at several, and finding that the pattern of distances against times keeps its form while the timescale stretches, is what makes the result about acceleration rather than about a particular board.

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, $v = a t$ . Then over an interval starting at rest the average speed is half the final speed, $a t / 2$ , and the distance is the average speed times the time:

s = \frac{a t}{2} \times t = \frac{1}{2} a t^{2}

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?

Answer

Distance goes as the square of the time, so at half the time it has covered a quarter of the distance, $1.75$ m, and at a quarter of the time a sixteenth, $0.44$ m. Turned around, this is how the experiment was actually done: the marks were laid out at $1 / 16$ , $1 / 4$ , $9 / 16$ and the full length, and the check was whether the ball reached them at equally spaced instants. Laying out distances precisely with a rule is easy; measuring short times precisely is hard. The design converts the hard measurement into an easy one, and that trade is the whole art of the thing.

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 $g \sin θ$ , 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

R = \frac{v^{2} \sin 2 θ}{g} = \frac{200^{2} \times 1}{9.81} = 4078 m

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 $30 °$ .

Answer

$R = 200^{2} \sin 60 ° / 9.81 = 40 000 \times 0.866 / 9.81 = 3531$ m, which is $87$ per cent of the maximum. Since $\sin 2 θ$ has the same value at $2 θ = 60 °$ and $2 θ = 120 °$ , the elevation of $60 °$ gives the same range as $30 °$ , the two being equally spaced about $45 °$ . The practical value of that pairing is real: a gunner who wants to clear an obstacle can choose the high trajectory and reach the same target. It is also a genuine prediction, one that could be and was checked against range tables compiled by people who had never heard of a parabola, which is a good illustration of a theory earning trust by reproducing what practitioners already knew before it is used for anything new.

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.