Galileo called his defence of the heliocentric hypothesis Dialogue concerning the Two Chief World Systems: the Ptolemaic and the Copernican (1632). Its author intended it for lay people and so he wrote in Italian rather than in Latin. The two world systems of the title exclude Tycho Brahe’s and so, to some extent, Galileo was not putting up the strongest competitor to Copernicus. Likewise, he made no use of Kepler’s discovery that planets move in ellipses rather than circles, nor the extraordinary accuracy of Kepler’sRudulphine Tables. Thus the Dialogue did not represent the cutting edge of science at the time. Rather it was a popularisation of the issues intended to bring the advantages of Copernicus to as wide an audience as possible.
This means that Galileo’s Dialogue concerning the Two Chief World Systems is not a masterwork of science. Instead, it is a first-class piece of rhetoric aimed squarely at non-experts. The modern genre it most resembles is popular science of the sort that tries to convince lay readers that they can understand relativity or string theory while glossing over all the difficult points. There are three characters in the Dialogue, Salviati (who represents Galileo), Simplicio (a naïve Aristotelian) and Sagredo (a not-terribly-neutral chairman) who engage in discussion over four days. The book as a whole is curiously unstructured but always reasonably entertaining. Simplicio’s job is to put up the arguments that Galileo wants to refute, which inevitably makes him appear foolish. Sagredo is supposed to represent the reader who initially has no opinion, but always ends up agreeing with Galileo.
Galileo’s arguments for Copernicus
The first part of the book refutes the Aristotelian doctrine that the region of the heavens is unchanging and perfect whereas the earth is subject to corruption. The contrary view was becoming widely accepted after the novas of 1572 and 1604 as well as occasional comets, most recently in 1618. Galileo seems now to accept the position of the Jesuit Grassi that comets are a heavenly and not an atmospheric event.1 Galileo also brings forth his own telescopic discoveries that the moon is not a perfect sphere and the sun has blemishes.
The second day of discussion in the Dialogue is devoted to the question of the earth’s rotation. Again, there is nothing much new here. Galileo trots out the old objection that the rotation of the earth should give rise to a great wind as the atmosphere is left behind and the old explanation about relative motion on board a ship. This had all been around since the fourteenth century but there is no reason why his lay readers should have been familiar with any of it. After all, Galileo’s discussion is a great deal more detailed and he expands the number of explanatory thought experiments. In one scenario, he suggests dropping a stone from the mast of a ship while it is moving and again when it is stationary. In both cases the stone will land at the base of the mast.2 There is no evidence that Galileo ever carried out any of these experiments, as he admits in the text, but he was correct in his deductions about their result.
In the third part, the conversation finally turns to Copernicus. Galileo runs through the reasons for favouring the sun being the centre of the planets’ orbits. He covers the way that planets’ movements all seem linked to the sun, as well as his discovery of the phases of Venus and sunspots (which show that the sun rotates on its axis).
The final part of the dialogue is devoted to the cause of the tides and represents Galileo’s most serious scientific mistake. He thought that it was possible to prove a scientific theory – to demonstrate it, in the jargon of the time. For him, the tides were proof that Copernicus was right and that the earth was rotating.3 He imagined that as the earth turned and moved through space, the oceans got slightly left behind and bunched up on one side of their basin before overcompensating and sloshing over to the other side.4 This is an extremely bad argument that directly contradicts Galileo’s own explanations about how the atmosphere is carried along as the earth rotates. He seemed to think that whereas air in the atmosphere moves with the earth, water in the seas is slightly retarded. In fact, the tides are caused by the gravitational pull of the moon dragging them around as it orbits the earth. They are perfectly explicable in the Ptolemaic universe and offer no evidence at all that Copernicus is correct. The link between the moon and the tides had been noted in antiquity and we have seen how it had been suspected, if not always admitted, throughout the early and late Middle Ages.
Still, Galileo believed that the tides argument was the demonstration that he needed to prove Copernicus correct. This left him with a problem. He was not allowed to defend the belief that the earth orbited the sun, only to treat it as a hypothesis. If he wrote that he had proved it, he would be stepping outside his remit. His solution was extremely foolish. What he should have done was to explain why the tides argument might be false (which would not have been difficult, because it is). Instead, he put the Pope’s argument about how God could engineer circumstances to produce any end result into the mouth of Simplicio.5 The Pope’s argument was weak and inimical to natural philosophy. By slotting it in at the end, Galileo showed what he thought of it and thereby ensured that His Holiness would take umbrage.
The Trial of Galileo
Before the Dialogue could be printed, Galileo needed to obtain official sanction from the Congregation of the Index. After a certain amount of negotiation, this was forthcoming and the book duly came out in Florence in February 1632. When the Pope read it, he was furious. An eyewitness reported: ‘His Holiness exploded into great anger and suddenly told me that even our Galileo had dared to enter where he should not have, into the most serious and dangerous subjects which could be stirred up at this time.’6 The Pope ordered the book withdrawn and appointed a special commission to examine it. Galileo was summoned to Rome in February 1633 to stand trial for heresy. His efforts to persuade the Church to abandon its opposition to Copernicus had failed. He had only succeeded in reinforcing it.
Scholars still argue about exactly why Galileo’s former friend Pope Urban put him on trial. The reason was probably that he felt betrayed by Galileo mocking his own argument and felt he had to either react or lose credibility. He had enough problems hanging on to his authority in Rome without some jumped-up mathematician making him look silly. The Church is culpable for banning heliocentricism back in 1616, but Galileo himself shares some of the blame for escalating the crisis in 1632, even if he was acting for the purest of motives.
Everyone knew he was guilty of defending and holding Copernicus’s views. The problem for the Inquisition was that the Congregation of the Index had actually permitted the publication of the Dialogue. How could they convict Galileo of heresy when he had their prior approval? The solution presented itself during the run-up to the trial. It turned out that Galileo had not merely been told not to hold or defend the views of Copernicus back in 1616. He had actually been enjoined not to teach them in any way at all. This was a stronger prohibition than Galileo had revealed to the Congregation of the Index. He denied that he remembered exactly those words, and the memorandum on the Inquisition’s files that contained them was not signed. This has led a few scholars to suggest that it was faked in order to secure a conviction. For some historians, its discovery just when it was needed is almost too convenient. That said, there is no evidence of forgery and Galileo would certainly have been convicted even without the suspect memorandum.7
His trial began on 12 April 1633. During the hearing, he was given comfortable quarters but this did not lessen the seriousness of his position. He started off by claiming that he did not agree with Copernicanism and that his book refuted the position. No one believed this but he stuck to his guns. Lying to the Inquisition was an extremely dangerous thing to do. There was adequate proof of his guilt and he could not expect mercy if he refused to own up. One of the inquisitors decided to meet Galileo informally to talk him around. As a result, at his next interrogation Galileo admitted that an uninformed reader of his book would get the impression that he thought Copernicanism was true. He also admitted to arrogance and vainglory in making his arguments appear stronger than they were. On 21 June, the Inquisition gave Galileo a final chance to admit that he did hold to Copernicanism. When he refused, he was reminded that the evidence against him was sufficiently strong for torture to be justified in obtaining a confession.8 Still he refused but stated that he would submit to whatever the Church decided to do with him. The Inquisition decided that this was a sufficient reply and Galileo was not to be tortured.
The next day, a committee of cardinals chaired by Pope Urban met to consider the verdict. The committee found Galileo seriously suspected of heresy and ordered him to admit his errors. Furthermore, he was sentenced to life imprisonment for failing to obey the Inquisition’s order not to hold, defend or teach Copernicanism. This sentence was immediately commuted to house arrest and Galileo returned home. In an unprecedented step, the Pope ordered that copies of Galileo’s sentence should be dispatched throughout the Catholic world.
Galileo must have felt wretched. His plan to help the Church correct its mistake had failed and his friendship with the Pope was ruined. Soon after he returned home, his beloved daughter, now Sister Marie Celeste, died and Galileo’s own health was rapidly declining. A lesser man might have given up the hard labour of natural philosophy to sink into endless melancholy. Not Galileo. He began to write again and produced the book that sealed his reputation as a titan among mankind’s intellectual champions. Because of his disgrace, the new book, Discourses on Two New Sciences, had to be smuggled out and published by a printer in the Netherlands.9 Despite this, it was never banned by the Congregation of the Index and there is no evidence that it provoked further trouble for its author from the authorities. Any action would have been purely vindictive in any case. By the time Galileo received his own copy of the book, he was completely blind. He was no longer a threat. When the English Protestant poet John Milton (1608–74) visited him in the autumn of 1638, no one objected.10 Galileo’s last years were peaceful and as comfortable as extreme old age could be in those days. He died on the night of 8 January 1642. Today, he lies in a grand tomb in the Franciscan Church of Santa Croce in Florence, outside of which Cecco D’Ascoli had burnt three centuries earlier. The machinations of Urban VIII denied Galileo the fitting memorial that we see today for almost a century after his death.
Two New Sciences: The Legacy of Medieval Science
Discourses on Two New Sciences (1638) represents the culmination of four centuries of work by medieval mathematicians and natural philosophers. For this book, Galileo brought back the three characters that he had created in Dialogue concerning Two Chief World Systems – Salviati, Sagredo and Simplico – and again provided a record of their fictitious discussions over four days. Some of the conclusions on local motion and projectiles had been previewed on the second day of dialogue in the earlier book, but now Galileo provides much more detail. According to the conceit of Discourses on Two New Sciences, Salviati has been reading a treatise by Galileo from which he quotes lengthy excerpts.
Galileo admits that not all of the material in the book originates from his own work and has rightly been criticised for not fully acknowledging his sources. Modern scholars have found that a great deal more of the Discourses owe a debt to medieval and sixteenth-century predecessors of Galileo than previously realised. In the book, Salviati admits, ‘some of the conclusions have been reached by others, first of all by Aristotle’,11 but this is something of an understatement. There is more truth in his next remark: that Galileo had demonstrated the conclusions much more rigorously than hitherto.
The first day of the Discourses begins with Salviati musing on the arsenal of Venice which had, in 1571, produced many of the ships that had defeated the Turks in the sea battle of Lepanto. Nicolò Tartaglia’s own work on projectiles, also called New Science and written in 1537, had been dedicated to Venice’s efforts to defeat the Turks. Perhaps this is how Galileo chose to acknowledge his intellectual debts.
The conversation on the first day ranges widely. Galileo states unambiguously that a vacuum can really exist, at least instantaneously as two flat plates are parted.12 There must briefly be a vacuum between the plates before the surrounding air can rush in to fill the gap. He goes on to explain that vacuums ‘suck’ because nature resists their formation and attempts to test the strength of this force. (In fact, Galileo is wrong about this. Vacuums do not ‘suck’, rather air pressure pushes.)
Later in the day, the conversation moves to falling objects. Sagredo points out to Simplicio, who is defending Aristotle’s doctrine that heavy weights fall faster than light ones: ‘I, who have made the test, can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound.’13 This repeats the observation that had been made a millennium earlier by John Philoponus and much more recently by Simon Stevin in Holland. In one of his early manuscripts dated around 1590 Galileo had already noted and criticised Philoponus’s ideas on falling bodies, but appears now to essentially agree with his observation.14
Galileo goes on to provide some intriguing thought experiments. He considers the case where two objects of differing weights are connected together and let fall. If Aristotle were right, when a small stone was attached to a large stone, the former would have to retard the fall of the latter. Hence, the combination would fall more slowly than the heavy stone on its own.15 This is nonsense, and Galileo thereby proves that objects fall at the same rate both experimentally and from first principles. His results may not be original, but his treatment is certainly definitive.
Galileo and Free Fall
The second day of discussion deals with the science of materials. We will leave that to one side and move straight on to day three. Here, Galileo tackles free fall – how objects move under gravity. Again, he is not quite honest about the extent to which he draws on prior work on the subject:
I have discovered by experiment some properties of motion that are worth knowing and which have not hitherto been observed or demonstrated. Some superficial observations have been made, as for instance, that the free motion of a heavy falling body is continuously accelerated. But to just what extent this acceleration occurs has not yet been announced.16
In fact it had been announced, as we have seen, by Domingo de Soto. It is one of the great mysteries of the history of science how much Domingo’s widely circulating textbook influenced Galileo, if only at second hand. We will probably never know. However, Galileo must have been familiar with the mean speed theorem of the Merton Calculators.17 It had appeared in at least seventeen printed books before 1600, not to mention countless manuscripts.18
Galileo also claims that ‘so far as I know, no one has yet pointed out that the distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same relation as the odd numbers beginning with unity.’19 With the aid of a diagram from the Discourses, it is easy to see what he means by this. The graph of time plotted against speed for an object subject to a constant rate of acceleration will be a straight but sloping line. It is straight because the acceleration is constant, and the slope corresponds to the acceleration. The steeper the slope, the higher the acceleration represented.
As Nicole Oresme had realised, the distance the object travels is represented by the area under the graph. Point A represents the object at rest. At time C, it has accelerated to speed B. Thus, the distance it has travelled is the area of the triangle, ABC. Recall
15. A reproduction of the mean speed theorem’s proof in Galileo’s Discourses on Two New Sciences (1638) from the 1914 English translation
from the diagram in chapter 12 that the mean speed theorem of the Merton Calculators implies that the area of triangle ABC, representing motion at a constant acceleration, equals the area of rectangle ADEC, representing motion at the average speed of the accelerating object. Clearly, Galileo is using this result here and demonstrating it in the same way as Nicole Oresme had done.
If we define the distance travelled in time AC as one unit, represented by the triangle ABC, it is clear from the diagram that the distance travelled in the next increment of equal time CI is the area BCIF. This is, as you can see by rearranging the pieces, three times the area of ABC. And the distance travelled in the following increment of time IO is the area FIOP which is five times the area of ABC. This is what Galileo meant by ‘the distances traversed stand to one another in the same relation as the odd numbers beginning with unity’. The distances covered increase in the series 1, 3, 5, 7 and so on.
Compare this to the words of William Heytesbury in the early fourteenth century. He wrote: ‘When the acceleration of a motion takes place uniformly from zero to some amount, the distance it will traverse in the first half of the time will be exactly one third of that which it will traverse in the second half of the time.’20 And the diagram from the 1494 edition of William Heytesbury’s Rules for Solving Logical Puzzles (following the original drafted by Nicole Oresme) really does look uncannily similar to the one used by Galileo to make the same point in the Discourses. William did not quite generalise the distance travelled beyond one unit and three units in the first two time intervals, but shortly afterwards Oresme did state that the distances follow the series 1, 3, 5, 7 et cetera as Galileo’s would.21
16. An illustration of the mean speed theorem from a 1494 printed edition of William Heytesbury’s Rules for Solving Logical Puzzles
There is good evidence that Galileo knew of the very book by William Heytesbury quoted above, because in unpublished notes that he took as a student at Pisa he refers directly to it.22 What is more, the same set of notes also mentions the Calculator Richard Swineshead and an Italian scholar, Gaetano di Thiene (1387–1465), who wrote the commentary printed in the same volume as William’s text. It is true that humanists had obscured the achievements of medieval scholarship, but the relevant books were still easily available and familiar to specialists. Even if the natural philosophers of Galileo’s time were beholden to Aristotle, mathematicians still used the conclusions of the Merton Calculators. One such mathematician, Niccolò Cabeo (1586–1650), was dismissive of Galileo’s claims to priority. ‘Another notable thing in Galileo is an intolerable boasting’, Cabeo says. ‘He wants absolutely everyone to have been in shameful ignorance from the time of Adam to our own era such that they should not know by what proportions the speed of falling weights increase.’23
Galileo’s contribution to the theory of falling objects was to realise that the distance fallen was proportional to the square of the time elapsed.24 This result follows from the diagrams above quite easily but does not appear to have been specifically stated by the Merton Calculators. One axis of the graphs corresponds to time and the other to speed. The distance travelled, that is the area under the graph, is speed multiplied by distance divided by two. But where an object is accelerating at a constant rate, speed equals time multiplied by acceleration. That means that the other axis is proportional to time elapsed as well. So the distance must be acceleration (‘a’) multiplied by time (‘t’) squared, divided by two or ½at2. Anyone who remembers the mechanics they did at school will recognise this formula.
The triumph of Galileo was not his connection between the mean speed theorem and gravity. Nor was it his description of how an accelerated body moves. Rather it was his experimental proof that the relationships derived back in the fourteenth century held true in nature. He realised that in order to carry out accurate experiments, he had to slow the action down. To do this, he decided to replace dropping balls with rolling them down a slope (the slope is called an ‘inclined plane’ in scientific parlance). For timing, he used miniature water clocks – mechanical clocks were not yet sufficiently accurate for precision work. To time a short interval, he let water flow through a narrow spout and weighed the amount that escaped in the moment the spout was open.25
The idea of rolling balls down inclined planes was not new. John Marliani (d.1483), a physician in Pavia, had published a number of books on natural philosophy in the late fifteenth century. In his Question on Proportion, printed in 1482, Marliani discussed experiments he had carried out with pendulums and balls on slopes. Unlike Galileo, he was not testing the right hypothesis and consequently did not manage to make any breakthroughs.26
Before Galileo could rely on his experiments he had to deal with a complication: could he be sure that rolling a ball down an inclined plane was exactly equivalent to dropping it? For his results to be valid, there had to be a precise parallel between the two situations. The inclined plane was a problem that had exercised mathematicians since antiquity. How was it that a man could push up a slope a weight that he could not lift? Pushing it up the slope involved raising it just as high as picking it up. The ancient Greeks failed to provide a definitive answer to the question, but a medieval scholar, Jordanus de Nemore (c.1225–60), managed to crack it. He, or one of his early anonymous followers, proved that the force exerted by an object on a slope was proportional to the steepness of the slope.27 This means that rolling balls down an inclined plane is analogous to dropping them. The speed of the balls will be slower but directly proportional to their speed when dropped.
In typical sixteenth-century style, Nicolò Tartaglia had reproduced and printed this important medieval result as his own work.28 At the same time, the failed proofs attempted by ancient Greek writers came to light and mathematicians, Jerome Cardan among them, tried to understand where they had gone wrong.29 Galileo, who must have been familiar with Jordanus’s proof, spent a lot of time on the question and eventually managed his own solution based on classical methods.
The fourth and final day of the dialogue in Discourses on Two New Sciences covers projectile motion. Galileo begins with a statement of the law of inertia that finally puts paid to Aristotle’s maxim that no object can move unless another object is moving it. Galileo tells us: ‘Imagine any particle moving along a surface without any friction. Then we know … this particle will move along a plane with a motion that is constant and perpetual, provided the surface has no limits.’30 John Buridan had explained how the planets keep moving forever in a frictionless environment. Galileo now generalises the law and realises that strictly it applies to a flat plane rather than circles in the heavens. He then goes on to show that a projectile, such as a cannon ball, will follow a curved trajectory.
In his own book New Science, Tartaglia had already shown that projectiles follow curved paths, and Jerome Cardan had suggested the curve in question ‘imitates the form of a parabola’.31 However, Galileo rightly pointed out that no one had demonstrated that this was correct. He now did so from first principles. He assumed that a cannon will push the ball in the direction in which it is fired at a constant speed. Under the law of inertia, if there were no other forces acting on the ball, it would keep going in a straight line in that direction forever. But two other forces act upon it – air resistance (which we can ignore for the purposes of this discussion) and gravity. During the previous day of the dialogue, Galileo had shown what happens when an object falls under gravity. The path of the cannon ball, he explained, is the combination of it moving in a straight line and falling under gravity at the same time. Aristotle’s statement that the violent motion from the cannon and the natural motion from gravity are incompatible is false. So, to calculate the path of the cannon ball you just add the two kinds of motion together. If you do so, you will find, as Cardan predicted, that the resulting path is a special kind of curve called a parabola.32
Galileo’s achievement lay in bringing together what had been done before, disposing of the vast amount that was irrelevant or simply wrong, and then proving the remainder with controlled experiments and brilliant arguments. A kind of new science did indeed begin with him, but there is no denying that he built on medieval foundations. Without them, he would never have been able to cover a fraction of the ground that he did, even in the long life he was granted.
When he solved the problems with which medieval natural philosophers had long struggled, Galileo eclipsed his predecessors. Like Kepler on optics, he rendered much of what had gone before obsolete. That did not mean that it was not important. To historians who want to learn where Galileo and Kepler found their ideas, medieval natural philosophy is indispensable. The achievements of their generation, outstanding though they were, should not obscure the breakthroughs made by Thomas Bradwardine, John Buridan, Nicole Oresme and others.
However, the most significant contribution of the natural philosophers of the Middle Ages was to make modern science even conceivable. They made science safe in a Christian context, showed how it could be useful and constructed a worldview where it made sense. Their central belief that nature was created by God and so worthy of their attention was one that Galileo wholeheartedly endorsed. Without that awareness, modern science would simply not have happened.