By the time of the Merton Calculators, Greek philosophy had been fully assimilated into Christian theology. In the twelfth century, western scholars had complained about ‘the poverty of the Latins’.1 They had lacked the best mathematics and natural philosophy – and they knew it. By 1300, though, riches had replaced dearth. The most advanced ancient thought had been discovered and translated. Arabic writers contributed arithmetic and algebra, adding new branches of mathematics to the geometry and number theory known in the ancient world. Thomas Bradwardine and his colleagues started to use mathematics as a tool to generate discoveries in physics and begin the process of combining numerical analysis and natural philosophy. Although Oxford ceased to produce original thinkers to match this golden generation, their ideas spread to France where they were taken up by the most accomplished natural philosophers of the Middle Ages. It was in Paris that medieval science reached its peak.
The Rector of Paris
John Buridan (c.1300–c.1361) was the most remarkable philosopher of the fourteenth century. He was born in Arras, northern France and, as far as we know, never left his home country. He spent his entire career at the university of Paris where he received his Master of Arts degree in about 1320 and was elected rector twice.2 Unlike most of the individuals we have encountered so far, he did not train as a theologian. This was unusual at the time because theology was such an important subject. People who had mastered the requisite philosophy wanted to go on and study the science of the divine. Buridan also declined to join either the Dominicans or Franciscans, although he did enter the priesthood. Staying out of the religious orders meant that he was something of a free agent, but it also left him without the support of a powerful organisation. Luckily, he was perfectly capable of making his way in the world without any outside help.
Buridan’s life as a scholar was not very eventful and so, as usual, the mythmakers have stepped in to liven up his biography. One story tells of how the king of France ordered him thrown into the River Seine for having an affair with the Queen.3 Another rumour claims that the university expelled him for his support of William of Ockham. As we saw in the last chapter, there was a half-hearted attempt to ban William’s teaching at Paris in 1339, which is probably where this tale derives from. What is clear is the high esteem in which he was held. He was called a ‘very distinguished man’ and a ‘celebrated philosopher’ by his contemporaries.4
Buridan’s first love was logic, where he was firmly in the ‘nominalist’ camp that followed William of Ockham in rejecting the reality of universals. Buridan’s nominalism meant that his natural philosophy had an empirical bent. The principles of science, he wrote, are accepted because we frequently observe them to be true and never come across a counter-example.5 He believed that the job of physics was to explain things based on how they normally appeared, ‘assuming the ordinary course of nature’. It did not matter to him that God, by his absolute power, could bring about miracles. As Buridan explains, ‘it is evident to us that every fire is hot and that the heavens are moved, even though the contrary is possible by God’s power. And it is evidence of this sort that suffices for the principles and conclusions of natural philosophy.’6
This logic led him to reject Aristotelian ideas about violent motion because they did not correspond to what he saw in the real world. Instead, Buridan formulated an alternative theory around the concept of ‘impetus’ that had had its genesis in the work of John Philoponus in the sixth century.7 Buridan was familiar with this (albeit possibly only at second hand) and combined it with the insights of William of Ockham.
Contrary to Aristotle, it was clear to Buridan that nothing, especially not the air, pushes a thrown ball after it leaves the hand that threw it. It was equally clear that the ball moves as a direct result of the movement of the hand. Buridan suggested that the hand gives the ball a quality, which he called impetus. To hurl a heavy rock, you would have to give it more impetus than to lob a pebble the same distance. Likewise, faster-moving objects have more impetus than slow ones. Thus, Buridan decided that impetus must be a quality whose magnitude was proportional to both weight and speed.8 This makes it very similar to, but not quite the same as, the concept of momentum in modern physics.
According to Buridan, an object falling due to gravity will gain impetus as it speeds up. Conversely, once a thrown ball has left the hand, it expends impetus to overcome air resistance. When it has used up all of this impetus, the ball stops moving forward and falls to the ground.
He realised that this led to a radical implication of his theory: ‘Impetus’, he said, ‘would last forever if it were not diminished and corrupted by an opposing resistance or a tendency to contrary motion.’9 Therefore, if there is no air resistance, such as in a vacuum, then an object will continue moving forever. Looking to the heavens, Buridan suggested that this might be the case for the planets orbiting the earth. He did not believe that they moved in a vacuum. Rather, he agreed with Aristotle that the heavens were a perfect world of ‘quintessence’ without decay and without friction. This meant that the planets would not meet any resistance and should, therefore, keep moving forever. Buridan had solved a problem that had exercised the minds of both pagan Greek and Christian thinkers: what keeps the planets moving in their orbits?
Recall that when we met Adelard of Bath in chapter 4, he was discussing with his nephew whether or not the stars needed to eat. He thought that they may be living creatures in some sense because they moved of their own accord. In Timaeus, Plato had suggested that the planets might possess some sort of soul, which caused them to move.10 It was a short step for medieval thinkers to replace these pagan souls with Christian angels.11 This, together with some fanciful iconography, is the source of an unfair caricature of Christian thinking. Medieval theologians did not have an image of angels, complete with wings and haloes, pushing the planets around the sky. Rather, they thought that angels were immaterial spiritual beings who certainly did not need wings to get around. All the theologians were trying to do was stick to Aristotle’s law that everything that moves has to be moved by something else. It was obvious that the planets were in motion, so something had to be pushing them.
Now, with his theory of impetus, Buridan had shown that the planets did not need angels or anything else to get around. Bradwardine had already specifically compared the universe to a clock.12 Buridan may well have been inspired by this analogy, and imagined that God had wound up the ‘world machine’ at the beginning of time. As he explained:
In the celestial motions, there is no opposing resistance. Therefore, when God, at the creation, moved each sphere of the heavens with just the velocity he wished, he then ceased to move them himself and since then those motions have lasted forever due to the impetus impressed on the spheres.13
God, he was suggesting, had endowed each celestial sphere, and the planet embedded in its rim, with a certain amount of impetus at the creation. Because there was no resistance in the heavens, this impetus would be sufficient to keep the sphere rotating and the planet moving on its circular course until doomsday. This comes quite close to the modern principle of inertia. However, it is not quite the same. No one is suggesting that Buridan had mechanics done and dusted 300 years before Newton. And while Buridan came close to describing inertia when he realised that the planets would keep moving if there was nothing to stop them, he never formulated it in the right way. Aristotle said that if a moving object is left undisturbed, it will stop. The modern principle of inertia states that a moving object will keep going at the same speed in a straight line until it is subjected to another force. Buridan thought the planets would keep moving in circles but he never extended his principle to movement in a straight line. Nevertheless, he had successfully challenged Aristotle’s natural philosophy and laid the foundations for the new science of mechanics.
John Buridan’s other great achievement was to ignite discussion of the subject that is emblematic of the beginning of modern science – the motion of the earth. Although he never thought that the earth orbited the sun, he did give serious consideration to the possibility that the earth might be turning.
According to almost all Greek cosmologists, the earth did not rotate each day. The entire heavens turned full circle every 24 hours while the earth remained stationary at the centre. The problem that Buridan had with this was that it seemed rather ugly. The heavens were very large and causing them to turn had to be less efficient than rotating the earth, which was, relatively speaking, minute. Like many medieval Christians, Buridan expected God to have arranged things in an elegant way, always allowing that he could do as he pleased. However, although there was a presumption towards elegance, you still had to check the empirical facts to see if God really had operated in that way. Buridan knew that the night sky appears to turn around, but realised that this could just be an illusion caused by the motion of the earth. To explain why we cannot directly observe the earth turning, he compared the situation to a boat sailing down a river. Imagine, he said, that you are on board the boat. If someone else is watching you from another boat moored to the riverbank, they can see that you are moving relative to them. But without also observing the surrounding landscape, they cannot tell whether they are the ones in motion or standing still. We are all in the same situation with the earth’s rotation. Unless we have some exterior viewpoint, it is impossible to tell whether the earth is rotating together with everything on it, or if it is stationary.14 Buridan needed another way to determine whether the earth itself was turning.
Ptolemy had tried to settle the argument by stating that there would be a great rushing wind as the surface of the earth moved while the atmosphere stayed put.15 Not so, said Buridan. The atmosphere must be rotating together with the earth and so there is no reason for us to feel the air being left behind as the world turns under it.
At that point, unfortunately, he made a mistake by following Aristotle too closely. Surely, Buridan said, if you fire an arrow straight up into the air, it will eventually fall back on top of you. But if the earth really was rotating, you should have moved with it and thus have avoided the falling arrow. Of course, today we know that the earth does turn and we do not see this happening. As we will see, it was Buridan’s most brilliant student, Nicole Oresme (c.1325–82), who first correctly analysed this problem.
Nicole Oresme: The Bishop Philosopher
Unlike his master, Oresme qualified as a doctor of theology and forged an impressive career outside the university. He was tutor to the future Charles V of France (1338–80) for whom he later translated a number of scientific treatises into French. As a reward, the king elevated him to the bishopric of Lisieux in 1377.16 Despite this busy professional life, he wrote several important treatises on mechanics and mathematics that built on the work of Buridan and the Merton Calculators. Unusually for his time, he was also an implacable opponent of astrology.17
Oresme is most celebrated for his contribution to the argument about the rotation of the earth. Taking forward Buridan’s analysis, he explained that when we fire an arrow into the air, it already shares the earth’s rotational motion. As the atmosphere is also moving with the earth, there is no reason that the arrow should not fly straight up and down relative to the archer. In proposing this, Oresme was actually challenging Aristotle’s law that different kinds of motion are contrary. In his scheme, the arrow must simultaneously move naturally with the rotation of the earth and violently from the bow that fired it. He concluded that we cannot use either reason or observation to conclusively determine if the earth moves. This left Oresme with an unanswered question and he tried to find the solution in the one book he considered completely reliable – the Bible.
Christians already realised that the Bible cannot always be taken literally. They believed that the Holy Spirit had inspired the biblical authors to write in the everyday language of the man in the street, and that it was not a scientific text. In that case, Oresme said, it was hardly surprising that the Bible assumed that the earth is stationary, because that is our everyday perception of the matter. He could treat a passage that says the earth does not move as figurative ‘by saying that this passage conforms to the normal use of popular speech just as it does in many other places … which are not to be taken literally.’18 Only an extra-terrestrial view of the earth could determine the truth about its motion. God, of course, has such a viewpoint and Oresme examined each of the relevant biblical passages to see if they provided evidence either way. Eventually, he came across Psalm 93:1, which reads: ‘The world is established that it cannot be moved.’ This, he decided, was good evidence that the earth was not rotating. However, he could easily have employed his reasoning to deal with this verse too if he had had any good reasons for thinking that the earth was in fact moving. We cannot blame him for eventually supporting the common-sense position of Aristotle and all the other ancient and contemporary authorities. What Oresme had done was prepare the groundwork. He refuted most of the objections to a moving earth two centuries before Copernicus had suggested it might actually be in motion.
Oresme’s other major achievements were in mathematics. The Merton Calculators had derived the mean speed theorem but Oresme found an elegant way to prove it. One of the best ways to represent motion is on a graph. If you plot a graph with speed on the vertical axis and time on the horizontal, you can see relationships that you previously had to imagine. For example, the slope of a line that plots the change of speed over time represents acceleration. A constant slope means that acceleration is constant. Oresme also realised that the area underneath the line must represent the distance travelled. Simply by cutting and pasting, he could show that the mean speed theorem was true. The area under a horizontal line at the average speed was the same as the area under a line with a constant slope.19
9. An excerpt from a fifteenth-century manuscript showing Nicole Oresme’s proof of the mean speed theorem
Using graphs to illustrate motion is an extremely powerful technique. The next step would be to describe the general mathematical relationship between the area under the graph, the line itself and its gradient. Oresme lacked the mathematical tools, now called calculus, to achieve this but his establishment of the meaning of the components of a graph plotting speed against distance was an essential first step.
Nicole Oresme’s proof of the mean speed theorem in graphical form
The work done by the Merton Calculators and the natural philosophers of Paris became widely diffused across western Europe. The free movement of scholars and common knowledge of Latin meant that a pan-Catholic republic of letters existed in the late Middle Ages. For example, at the same time as Buridan and Oresme were working at Paris, a German by the name of Albert of Saxony (c.1316–90) arrived to teach philosophy. He began his academic career rather late and did not receive his Master of Arts degree until he was in his mid-thirties. He probably studied under Buridan himself and followed him as rector of the university in 1353. Nine years later the Pope called him to Avignon. Albert took advantage of his closeness to the Pope to get permission to found a new university in Vienna before he was appointed a bishop in 1366.20 Thereafter his career was more political than academic, but he carried the work of his master John Buridan far beyond the confines of Paris.
Albert’s own books are less advanced than either Buridan’s or Oresme’s, which arguably made them more popular. In one small way, though, he did strike a blow against Aristotelian physics. Recall from chapter 9 how Aristotle implied that a cannon ball would drop out of the sky like a cartoon character running off a cliff when it ran out of speed. Albert could see that this was not so. Instead, he drew a diagram showing a cannon ball shooting out of the barrel of the gun in a straight line but then curving gently to earth when it reached the limit of its range.21 This is not mathematically accurate, but it is the earliest picture we have of a curved trajectory.
Another travelling philosopher, Paul of Venice (1369–1429), carried the new ideas to Italy. He had spent three years of his early academic career at Oxford but, in 1393, returned to the Veneto to take up a post at the university of Padua. Venice ruled the Mediterranean Sea and made its money by trading with the Byzantine Empire and the Turks. The city also controlled a large chunk of northern Italy including, after 1404, the city of Padua, home to the famous university. For the Venetians, having their own university was a matter of considerable pride. They supported it by providing fat salaries for the professors and protection for the students. After the Reformation, Protestants still continued to attend the university of Padua even though Catholic Italy officially considered them heretics.
Paul of Venice brought the mathematical work of the Merton Calculators back with him from Oxford.22 His colleagues realised that it had a wide range of applications, especially in the field of medicine for which Padua was most famous. They began to consider how they could best calculate the effects of combining drugs with different properties using the techniques developed by Thomas Bradwardine and Richard Swineshead.23 By 1400, tracts by all the Merton Calculators could be found in Padua’s university library.24The new science spread to Germany too where, in 1425, a document from Cologne refers to the era as the ‘age of Buridan’.25
Bread, Wine and Atoms
The Church looked benignly upon all this speculation. Almost all the practitioners were members of the clergy and many, like Nicole Oresme and Albert of Saxony, were in senior positions. Mechanics and mathematics did not cause any concern. Even though Aristotle’s natural philosophy was almost the official position of the Church in many areas, thinkers could still challenge and reform it. In one field, though, matters were considerably more delicate. Back in the eleventh century, Lanfranc had used Aristotle’s theory of substance and accident to explain transubstantiation. This linked natural philosophy closely to the Eucharistic miracle when bread and wine turned into the body and blood of Christ. In 1215, transubstantiation had become the official dogma of the Catholic Church. From then on, an attack on Aristotle’s theory of matter looked like an attack on a key Christian doctrine. Anti-Aristotelian reformers had to tread carefully. One idea that was completely incompatible with the theory of substance and accident was atomism. Medieval philosophers did not know very much about this alternative ancient Greek theory of matter beyond Aristotle’s criticisms of it. His predecessor, Democritus (c.460–c.370BC), had suggested that all matter consisted of tiny indivisible particles swirling around in a void. Aristotle had no time for this idea because he rejected the void and believed that everything was infinitely divisible. Most of his medieval readers concurred with his opinion. Thomas Bradwardine even went to the trouble of writing a refutation of atomism called On the Continuum.26
A few scholars did think that atomism was at least worthy of consideration. Nicholas of Autrecourt (c.1300–69) went further. After gaining his Master of Arts degree, he joined the Paris theology faculty and began to study for his doctorate. Along the way, he produced a commentary on the Sentences of Peter Lombard, the standard theological textbook of the time. Again, as was usual, the faculty subjected this to a test of Christian orthodoxy. It failed spectacularly. The case ended up before the Pope and did not reach a final settlement until 1347. The Pope found Nicholas guilty of holding heretical opinions and ordered him to recant. The offending commentary was consigned to the flames. His academic career was over but he received a lucrative appointment as dean of Metz Cathedral in compensation and lived out his days in peace.27
Nicholas was the ultimate anti-Aristotelian. He thought that the world would be a better place if people stopped reading Aristotle and rejected almost all of his philosophy. It is possible that he was espousing atomism simply because Aristotle did not agree with it. The trouble was that, for medieval theologians, atomism left transubstantiation without any philosophical foundation. Lanfranc had explained how the underlying substance of the bread and wine could change while the accidents such as taste and appearance stayed the same. But the concepts of ‘substance’ and ‘accident’ came from Aristotle. According to atomism, they were meaningless. Instead, the properties of matter were due to the shapes of the individual atoms. For example, fire atoms were sharp and prickly which is why they burn. Water atoms were round and smooth so they flow across each other easily. With the atomic model, there did not seem to be any way to allow the bread and wine actually to be the body and blood of Christ. Either the host contained ‘bread’ atoms or it was made of ‘flesh’ atoms. There was no room in atomic thinking for the underlying substance to be different from the outward appearance.
Modern readers might find it hard to understand what all the fuss was about. After all, transubstantiation is accounted a miracle and trying to explain it scientifically is a fool’s errand. This misses the point of just how rationalistic medieval theology was. One consequence of this was the near-universal belief among theologians that logic limited God’s absolute power. He could not bring about a logical contradiction. The answer to the old canard ‘Can God make a weight so heavy that he cannot lift it?’ was a straightforward ‘no’. More interesting was the idea that God cannot make a human being sin. Theologians thought that this was a logical contradiction because doing God’s will can never be a sin. So if God makes you sin, you are not actually sinning.28Likewise, not even God could make the host into bread and flesh at the same time. Aristotle’s substance and accident concepts had allowed Lanfranc to perform some mental gymnastics to separate the idea of flesh from the idea of bread. By assigning these two ideas to different concepts, he gave the impression that no contradiction existed. Atomism did not provide any such way out, so it seemed to make transubstantiation logically impossible. If that was the case, not even God could perform it. Thus Nicholas was required to specifically abjure his ideas about the Eucharist.29
Admittedly, the philosophical arguments for atoms were not very strong. There was no evidence that they existed and they explained nothing that Aristotle could not handle just as well. In fact, atoms were exactly the kind of superfluous concept that Ockham’s razor was supposed to dispose of. With hindsight, we now know that atomism turned out to be an extremely fruitful theory of matter. Perhaps it would have been better if the Church had not stamped so hard on the idea. Certainly, this was a clear-cut case of theological orthodoxy curtailing philosophical enquiry. But this happened so rarely that we cannot maintain that the Church held back science in general. In Nicholas of Autrecourt’s case, he specifically linked his ideas about matter to the all-important doctrine of the Eucharist and so the authorities felt compelled to act. Supporting atomism without pursuing the theological implications would not have provoked such a hostile reaction.
The popular image of the medieval Church as a monolithic institution opposing any sort of scientific speculation is clearly inaccurate. Natural philosophy had proven itself useful and worth supporting. It is hard to imagine how any philosophy at all would have taken place if the Church-sponsored universities had not provided a home for it. But the price of having a rich sponsor is having to bend to their interests and avoid subjects that they find controversial. Modern scientific researchers competing for funding from big companies have exactly the same problem. The Church allowed natural philosophers a much wider dispensation than many corporate interests allow their researchers today. They were free to speculate as much as they pleased as long as they avoided religious controversy. Even atomism would make a triumphant comeback in the seventeenth century.
Nicholas of Autrecourt disagreed with John Buridan as well. Their dispute was over the empirical status of natural science. Buridan, as we have seen, thought that if nature always behaves in one way and never in another, that is quite enough evidence for the truth of a natural law. Nicholas disagreed. He demanded incontrovertible proof before he would believe any scientific proposition. This placed the burden of proof so high that science could never have advanced based on his method.30
Buridan took a more cautious line with the theologians of Paris. In one treatise, while he was discussing whether a vacuum can exist, he suddenly states that his colleagues in the theology faculty have accused him of straying into forbidden territory. Buridan responds by asserting that he is allowed to consider the question as long as he settles it in an orthodox fashion.31 He agreed with Aristotle that a vacuum was naturally impossible, but admitted that God could create one if he wanted to. This was an example of the 1277 condemnations in action. They stated that it was heretical to limit God to what Aristotle said was possible. So, in following the condemnations to the letter, Buridan was not saying anything controversial. Rather, the oath he had made when he received his Master of Arts degree compelled him to settle the dispute on the side of orthodoxy. In chapter 6 we saw how, following the arguments surrounding Siger of Brabant, philosophers at Paris had promised not to encroach on the theologians’ territory, or at least to ensure that they did not make any heterodox claims. The deal still held and so immediately God came into the question, Buridan thought it best to make clear he was sticking to the rules.
The Decline of Medieval Science
As it turned out, the ban on natural philosophers covering theological questions was not much of a handicap to science in the Middle Ages. Most of the writers on natural philosophy were also theologians and so did not suffer from any restriction. Thomas Aquinas, Nicole Oresme and Albert of Saxony, among others, could all discuss the interaction between God and nature to their hearts’ content. And they did. In fact, there was so much natural philosophy in medieval theological books that some clerics started to complain about it.32 Questions about the nature of infinity, what lies outside the universe and how God created the world all appear frequently. Theologians used their training in Aristotle’s ideas together with their knowledge of Christian doctrine to try to come up with some answers.
Buridan and Oresme represent the high water mark of medieval natural philosophy. Some interesting work was done through the fifteenth century at the university of Padua where Paul of Venice had brought the discoveries of the Merton Calculators. And we will shortly meet two more innovative thinkers who combined fruitful speculation in natural philosophy with their job of cardinals in the Catholic Church. Furthermore, as we will also see in the next chapter, the invention of spectacles, the compass and the mechanical clock were followed by even more impressive developments in the fifteenth century. But given how close Oresme and Buridan had come to some of the key concepts of modern science, it seems a disappointment that their immediate successors did not take their insights forward. What went wrong?
The deadly incursion that stopped all of Europe in its tracks entered Italy through Venice in 1348 and swept through the continent, leaving between a third and a half of the population dead. This first wave of the Black Death took Thomas Bradwardine with it. The university towns, crowded with students from far and wide, were especially badly hit. As the students fled from the plague, they took it with them back to their homes and monasteries. One noted theologian and astrologer, Richard Holcott (d.1349), had used his art to confidently predict a peaceful death for himself. Maybe, lying in his pallet as the Black Death ravaged his body, he wondered where his calculations had gone wrong.33
The initial assault of the plague had burnt itself out by 1350 but it returned in 1360 – when John Buridan was probably among its victims – in 1369, in 1375 and intermittently for the following three centuries. Even those who survived one epidemic could never be sure that they would live through the next. The effects of the Black Death went beyond the enormous death toll, which included many of Europe’s finest natural philosophers. It mocked the ability of man to control his destiny and made fools of the doctors. Perhaps the scholars of Europe lost their nerve. In finding it again, they would discard almost the entire legacy of medieval philosophy.