We do not know if Oresme really was unwilling to take the final step toward acknowledging that the Earth rotates, or whether he was merely paying lip service to religious orthodoxy.
Oresme also anticipated one aspect of Newton’s theory of gravitation. He argued that heavy things do not necessarily tend to fall toward the center of our Earth, if they are near some other world. The idea that there may be other worlds, more or less like the Earth, was theologically daring. Did God create humans on those other worlds? Did Christ come to those other worlds to redeem those humans? The questions are endless, and subversive.
Unlike Buridan, Oresme was a mathematician. His major mathematical contribution led to an improvement on work done earlier at Oxford, so we must now shift our scene from France to England, and back a little in time, though we will return soon to Oresme.
By the twelfth century Oxford had become a prosperous market town on the upper reaches of the Thames, and began to attract students and teachers. The informal cluster of schools at Oxford became recognized as a university in the early 1200s. Oxford conventionally lists its line of chancellors starting in 1224 with Robert Grosseteste, later bishop of Lincoln, who began the concern of medieval Oxford with natural philosophy. Grosseteste read Aristotle in Greek, and he wrote on optics and the calendar as well as on Aristotle. He was frequently cited by the scholars who succeeded him at Oxford.
In Robert Grosseteste and the Origins of Experimental Science,12 A. C. Crombie went further, giving Grosseteste a pivotal role in the development of experimental methods leading to the advent of modern physics. This seems rather an exaggeration of Grosseteste’s importance. As is clear from Crombie’s account, “experiment” for Grosseteste was the passive observation of nature, not very different from the method of Aristotle. Neither Grosseteste nor any of his medieval successors sought to learn general principles by experiment in the modern sense, the aggressive manipulation of natural phenomena. Grosseteste’s theorizing has also been praised,13 but there is nothing in his work that bears comparison with the development of quantitatively successful theories of light by Hero, Ptolemy, and al-Haitam, or of planetary motion by Hipparchus, Ptolemy, and al-Biruni, among others.
Grosseteste had a great influence on Roger Bacon, who in his intellectual energy and scientific innocence was a true representative of the spirit of his times. After studying at Oxford, Bacon lectured on Aristotle in Paris in the 1240s, went back and forth between Paris and Oxford, and became a Franciscan friar around 1257. Like Plato, he was enthusiastic about mathematics but made little use of it. He wrote extensively on optics and geography, but added nothing important to the earlier work of Greeks and Arabs. To an extent that was remarkable for the time, Bacon was also an optimist about technology:
Also cars can be made so that without animals they will move with unbelievable rapidity. . . . Also flying machines can be constructed so that a man sits in the midst of the machine revolving some engine by which artificial wings are made to beat the air like a flying bird.14
Appropriately, Bacon became known as “Doctor Mirabilis.”
In 1264 the first residential college was founded at Oxford by Walter de Merton, at one time the chancellor of England and later bishop of Rochester. It was at Merton College that serious mathematical work at Oxford began in the fourteenth century. The key figures were four fellows of the college: Thomas Bradwardine (c. 1295–1349), William Heytesbury (fl. 1335), Richard Swineshead (fl. 1340–1355), and John of Dumbleton (fl. 1338–1348). Their most notable achievement, known as the Merton College mean speed theorem, for the first time gives a mathematical description of nonuniform motion—that is, motion at a speed that does not remain constant.
The earliest surviving statement of this theorem is by William of Heytesbury (chancellor of the University of Oxford in 1371), in Regulae solvendi sophismata. He defined the velocity at any instant in nonuniform motion as the ratio of the distance traveled to the time that would have elapsed if the motion had been uniform at that velocity. As it stands, this definition is circular, and hence useless. A more modern definition, possibly what Heytesbury meant to say, is that the velocity at any instant in nonuniform motion is the ratio of the distance traveled to the time elapsed if the velocity were the same as it is in a very short interval of time around that instant, so short that during this interval the change in velocity is negligible. Heytesbury then defined uniform acceleration as nonuniform motion in which the velocity increases by the same increment in each equal time. He then went on to state the theorem:15
When any mobile body is uniformly accelerated from rest to some given degree [of velocity], it will in that time traverse one-half the distance that it would traverse if, in that same time, it were moved uniformly at the degree of velocity terminating that increment of velocity. For that motion, as a whole, will correspond to the mean degree of that increment of velocity, which is precisely one-half that degree of velocity which is its terminal velocity.
That is, the distance traveled during an interval of time when a body is uniformly accelerated is the distance it would have traveled in uniform motion if its velocity in that interval equaled the average of the actual velocity. If something is uniformly accelerated from rest to some final velocity, then its average velocity during that interval is half the final velocity, so the distance traveled is half the final velocity times the time elapsed.
Various proofs of this theorem were offered by Heytesbury, by John of Dumbleton, and then by Nicole Oresme. Oresme’s proof is the most interesting, because he introduced a technique of representing algebraic relations by graphs. In this way, he was able to reduce the problem of calculating the distance traveled when a body is uniformly accelerated from rest to some final velocity to the problem of calculating the area of a right triangle, whose sides that meet at the right angle have lengths equal respectively to the time elapsed and to the final velocity. (See Technical Note 17.) The mean speed theorem then follows immediately from an elementary fact of geometry, that the area of a right triangle is half the product of the two sides that meet at the right angle.
Neither any don of Merton College nor Nicole Oresme seems to have applied the mean speed theorem to the most important case where it is relevant, the motion of freely falling bodies. For the dons and Oresme the theorem was an intellectual exercise, undertaken to show that they were capable of dealing mathematically with nonuniform motion. If the mean speed theorem is evidence of an increasing ability to use mathematics, it also shows how uneasy the fit between mathematics and natural science still was.
It must be acknowledged that although it is obvious (as Strato had demonstrated) that falling bodies accelerate, it is not obvious that the speed of a falling body increases in proportion to the time, the characteristic of uniform acceleration, rather than to the distance fallen. If the rate of change of the distance fallen (that is, the speed) were proportional to the distance fallen, then the distance fallen once the body starts to fall would increase exponentially with time,* just as a bank account that receives interest proportional to the amount in the account increases exponentially with time (though if the interest rate is low it takes a long time to see this). The first person to guess that the increase in the speed of a falling body is proportional to the time elapsed seems to have been the sixteenth-century Dominican friar Domingo de Soto,16 about two centuries after Oresme.
From the mid-fourteenth to the mid-fifteenth century Europe was harried by catastrophe. The Hundred Years’ War between England and France drained England and devastated France. The church underwent a schism, with a pope in Rome and another in Avignon. The Black Death destroyed a large fraction of the population everywhere.
Perhaps as a result of the Hundred Years’ War, the center of scientific work shifted eastward in this period, from France and England to Germany and Italy. The two regions were spanned in the career of Nicholas of Cusa. Born around 1401 in the town of Kues on the Moselle in Germany, he died in 1464 in the Umbrian province of Italy. Nicholas w
as educated at both Heidelberg and Padua, becoming a canon lawyer, a diplomat, and after 1448 a cardinal. His writing shows the continuing medieval difficulty of separating natural science from theology and philosophy. Nicholas wrote in vague terms about a moving Earth and a world without limits, but with no use of mathematics. Though he was later cited by Kepler and Descartes, it is hard to see how they could have learned anything from him.
The late Middle Ages also show a continuation of the Arab separation of professional mathematical astronomers, who used the Ptolemaic system, and physician-philosophers, followers of Aristotle. Among the fifteenth-century astronomers, mostly in Germany, were Georg von Peurbach and his pupil Johann Müller of Königsberg (Regiomontanus), who together continued and extended the Ptolemaic theory of epicycles.* Copernicus later made much use of the Epitome of the Almagest of Regiomontanus. The physicians included Alessandro Achillini (1463–1512) of Bologna and Girolamo Fracastoro of Verona (1478–1553), both educated at Padua, at the time a stronghold of Aristotelianism.
Fracastoro gave an interestingly biased account of the conflict:17
You are well aware that those who make profession of astronomy have always found it extremely difficult to account for the appearances presented by the planets. For there are two ways of accounting for them: the one proceeds by means of those spheres called homocentric, the other by means of so-called eccentric spheres [epicycles]. Each of these methods has its dangers, each its stumbling blocks. Those who employ homocentric spheres never manage to arrive at an explanation of phenomena. Those who use eccentric spheres do, it is true, seem to explain the phenomena more adequately, but their conception of these divine bodies is erroneous, one might almost say impious, for they ascribe positions and shapes to them that are not fit for the heavens. We know that, among the ancients, Eudoxus and Callippus were misled many times by these difficulties. Hipparchus was among the first who chose rather to admit eccentric spheres than to be found wanting by the phenomena. Ptolemy followed him, and soon practically all astronomers were won over by Ptolemy. But against these astronomers, or at least, against the hypothesis of eccentrics, the whole of philosophy has raised continuing protest. What am I saying? Philosophy? Nature and the celestial spheres themselves protest unceasingly. Until now, no philosopher has ever been found who would allow that these monstrous spheres exist among the divine and perfect bodies.
To be fair, observations were not all on the side of Ptolemy against Aristotle. One of the failings of the Aristotelian system of homocentric spheres, which as we have seen had been noted around AD 200 by Sosigenes, is that it puts the planets always at the same distance from the Earth, in contradiction with the fact that the brightness of planets increases and decreases as they appear to go around the Earth. But Ptolemy’s theory seemed to go too far in the other direction. For instance, in Ptolemy’s theory the maximum distance of Venus from the Earth is 6.5 times its minimum distance, so if Venus shines by its own light, then (since apparent brightness goes as the inverse square of the distance) its maximum brightness should be 6.52 = 42 times greater than its minimum brightness, which is certainly not the case. Ptolemy’s theory had been criticized on this ground at the University of Vienna by Henry of Hesse (1325–1397). The resolution of the problem is of course that planets shine not by their own light, but by the reflected light of the Sun, so their apparent brightness depends not only on their distance from the Earth but also, like the Moon’s brightness, on their phase. When Venus is farthest from the Earth it is on the side of the Sun away from the Earth, so its face is fully illuminated; when it is closest to the Earth it is more or less between the Earth and the Sun and we mostly see its dark side. For Venus the effects of phase and distance therefore partly cancel, moderating its variations in brightness. None of this was understood until Galileo discovered the phases of Venus.
Soon the controversy between Ptolemaic and Aristotelian astronomy was swept away in a deeper conflict, between those who followed either Ptolemy or Aristotle, all of them accepting that the heavens revolve around a stationary Earth; and a new revival of the idea of Aristarchus, that it is the Sun that is at rest.
PART IV
THE SCIENTIFIC REVOLUTION
Historians used to take it for granted that physics and astronomy underwent revolutionary changes in the sixteenth and seventeenth centuries, after which these sciences took something like their modern form, providing a paradigm for the future development of all science. The importance of this revolution seemed obvious. Thus the historian Herbert Butterfield* declared that the scientific revolution “outshines everything since the rise of Christianity and reduces the Renaissance and Reformation to the rank of mere episodes, mere internal displacements, within the system of medieval Christendom.”1
There is something about this sort of consensus that always attracts the skeptical attention of the next generation of historians. In the past few decades some historians have expressed doubt about the importance or even the existence of the scientific revolution.2 For instance, Steven Shapin famously began a book with the sentence “There was no such thing as the scientific revolution, and this is a book about it.”3
Criticisms of the notion of a scientific revolution take two opposing forms. On one hand, some historians argue that the discoveries of the sixteenth and seventeenth centuries were no more than a natural continuation of scientific progress that had already been made in Europe or in the lands of Islam (or both) during the Middle Ages. This in particular was the view of Pierre Duhem.4 Other historians point to the vestiges of prescientific thinking that continued into the supposed scientific revolution—for instance, that Copernicus and Kepler in places sound like Plato, that Galileo cast horoscopes even when no one was paying for them, and that Newton treated both the solar system and the Bible as clues to the mind of God.
There are elements of truth in both criticisms. Nevertheless, I am convinced that the scientific revolution marked a real discontinuity in intellectual history. I judge this from the perspective of a contemporary working scientist. With a few bright Greek exceptions, science before the sixteenth century seems to me very different from what I experience in my own work, or what I see in the work of my colleagues. Before the scientific revolution science was suffused with religion and what we now call philosophy, and had not yet worked out its relation to mathematics. In physics and astronomy after the seventeenth century I feel at home. I recognize something very like the science of my own times: the search for mathematically expressed impersonal laws that allow precise predictions of a wide range of phenomena, laws validated by the comparison of these predictions with observation and experiment. There was a scientific revolution, and the rest of this book is about it.
11
The Solar System Solved
Whatever the scientific revolution was or was not, it began with Copernicus. Nicolaus Copernicus was born in 1473 in Poland of a family that in an earlier generation had emigrated from Silesia. Nicolaus lost his father at the age of ten, but was fortunate to be supported by his uncle, who had become rich in the service of the church and a few years later became bishop of Varmia (or Ermeland) in northeast Poland. After an education at the University of Cracow, probably including courses in astronomy, in 1496 Copernicus enrolled as a student of canon law at the University of Bologna and began astronomical observations as an assistant to the astronomer Domenico Maria Novara, who had been a student of Regiomontanus. While at Bologna Copernicus learned that, with the help of his uncle’s patronage, he had been confirmed as one of the 16 canons of the cathedral chapter of Frombork (or Frauenburg), in Varmia, from which for the rest of his life he derived a good income with little in the way of ecclesiastical duties. Copernicus never became a priest. After studying medicine briefly at the University of Padua, in 1503 he picked up a degree of doctor of laws at the University of Ferrara and soon afterward returned to Poland. In 1510 he settled in Frombork, where he constructed a small observatory, and where he remained until his death in 1543.
Soon after he came to Frombork, Copernicus wrote a short anonymous work, later titled De hypothesibus motuum coelestium a se constitutis commentariolus, and generally known as the Commentariolus, or Little Commentary.1 The Commentariolus was not published until long after its author’s death, and so was not as influential as his later writings, but it gives a good account of the ideas that guided his work.
After a brief criticism of earlier theories of the planets, Copernicus in the Commentariolus states seven principles of his new theory. Here is a paraphrase, with some added comments:
1. There is no one center of the orbits of the celestial bodies. (There is disagreement among historians whether Copernicus thought that these bodies are carried on material spheres,2 as supposed by Aristotle.)
2. The center of the Earth is not the center of the universe, but only the center of the Moon’s orbit, and the center of gravity toward which bodies on Earth are attracted.
3. All the heavenly bodies except the Moon revolve about the Sun, which is therefore the center of the universe. (But as discussed below, Copernicus took the center of the orbits of the Earth and other planets to be, not the Sun, but rather a point near the Sun.)
4. The distance between the Earth and the Sun is negligible compared with the distance of the fixed stars. (Presumably Copernicus made this assumption to explain why we do not see annual parallax, the apparent annual motion of the stars caused by the Earth’s motion around the Sun. But the problem of parallax is nowhere mentioned in the Commentariolus.)
To Explain the World: The Discovery of Modern Science Page 15
