To Explain the World: The Discovery of Modern Science (20 page)

Rather than supposing that air keeps projectiles moving, Buridan supposed that this is an effect of something called “impetus,” which the hand gives the projectile. As we have seen, a somewhat similar idea had been proposed by John of Philoponus, and Buridan’s impetus was in turn a foreshadowing of what Newton was to call “quantity of motion,” or in modern terms momentum, though it is not precisely the same. Buridan shared with Aristotle the assumption that something has to keep moving things in motion, and he conceived of impetus as playing this role, rather than as being only a property of motion, like momentum. He never identified the impetus carried by a body as its mass times its velocity, which is how momentum is defined in Newtonian physics. Nevertheless, he was onto something. The amount of force that is required to stop a moving body in a given time is proportional to its momentum, and in this sense momentum plays the same role as Buridan’s impetus.

Buridan extended the idea of impetus to circular motion, supposing that planets are kept moving by their impetus, an impetus given to them by God. In this way, Buridan was seeking a compromise between science and religion of a sort that became popular centuries later: God sets the machinery of the cosmos
in motion, after which what happens is governed by the laws of nature. But although the conservation of momentum does keep the planets moving, by itself it could not keep them moving on curved orbits as Buridan thought was done by impetus; that requires an additional force, eventually recognized as the force of gravitation.

Buridan also toyed with an idea due originally to Heraclides, that the Earth rotates once a day from west to east. He recognized that this would give the same appearance as if the heavens rotated around a stationary Earth once a day from east to west. He also acknowledged that this is a more natural theory, since the Earth is so much smaller that the firmament of Sun, Moon, planets, and stars. But he rejected the rotation of the Earth, reasoning that if the Earth rotated, then an arrow shot straight upward would fall to the west of the archer, since the Earth would have moved under the arrow while it was in flight. It is ironic that Buridan might have been saved from this error if he had realized that the Earth’s rotation would give the arrow an impetus that would carry it to the east along with the rotating Earth. Instead, he was misled by the notion of impetus; he considered only the vertical impetus given to the arrow by the bow, not the horizontal impetus it takes from the rotation of the Earth.

Buridan’s notion of impetus remained influential for centuries. It was being taught at the University of Padua when Copernicus studied medicine there in the early 1500s. Later in that century Galileo learned about it as a student at the University of Pisa.

Buridan sided with Aristotle on another issue, the impossibility of a vacuum. But he characteristically based his conclusion on observations: when air is sucked out of a drinking straw, a vacuum is prevented by liquid being pulled up into the straw; and when the handles of a bellows are pulled apart, a vacuum is prevented by air rushing into the bellows. It was natural to conclude that nature abhors a vacuum. As we will see in
Chapter 12
, the correct explanation for these phenomena in terms of air pressure was not understood until the 1600s.

Buridan’s work was carried further by two of his students:
Albert of Saxony and Nicole Oresme. Albert’s writings on philosophy became widely circulated, but it was Oresme who made the greater contribution to science.

Oresme was born in 1325 in Normandy, and came to Paris to study with Buridan in the 1340s. He was a vigorous opponent of looking into the future by means of “astrology, geomancy, necromancy, or any such arts, if they can be called arts.” In 1377 Oresme was appointed bishop of the city of Lisieux, in Normandy, where he died in 1382.

Oresme’s book
On the Heavens and the Earth
¹⁰
(written in the vernacular for the convenience of the king of France) has the form of an extended commentary on Aristotle, in which again and again he takes issue with The Philosopher. In this book Oresme reconsidered the idea that the heavens do not rotate about the Earth from east to west but, rather, the Earth rotates on its axis from west to east. Both Buridan and Oresme recognized that we observe only relative motion, so seeing the heavens move leaves open the possibility that it is instead the Earth that is moving. Oresme went through various objections to the idea, and picked them apart. Ptolemy in the
Almagest
had argued that if the Earth rotated, then clouds and thrown objects would be left behind; and as we have seen, Buridan had argued against the Earth’s rotation by reasoning that if the Earth rotated from west to east, then an arrow shot straight upward would be left behind by the Earth’s rotation, contrary to the observation that the arrow seems to fall straight down to the same spot on the Earth’s surface from which it was shot vertically upward. Oresme replied that the Earth’s rotation carries the arrow with it, along with the archer and the air and everything else on the Earth’s surface, thus applying Buridan’s theory of impetus in a way that its author had not understood.

Oresme answered another objection to the rotation of the Earth—an objection of a very different sort, that there are passages in Holy Scripture (such as in the Book of Joshua) that refer to the Sun going daily around the Earth. Oresme replied that this was just a concession to the customs of popular speech, as
where it is written that God became angry or repented—things that could not be taken literally. In this, Oresme was following the lead of Thomas Aquinas, who had wrestled with the passage in Genesis where God is supposed to proclaim, “Let there be a firmament above the waters, and let it divide the waters from the waters.” Aquinas had explained that Moses was adjusting his speech to the capacity of his audience, and should not be taken literally. Biblical literalism could have been a drag on the progress of science, if there had not been many inside the church like Aquinas and Oresme who took a more enlightened view.

Despite all his arguments, Oresme finally surrendered to the common idea of a stationary Earth, as follows:

Afterward, it was demonstrated how it cannot be proved conclusively by argument that the heavens move. . . . However, everyone maintains, and I think myself, that the heavens do move and not the Earth: For God has established the world which shall not be moved, in spite of contrary reasons because they are clearly not conclusive persuasions. However, after considering all that has been said, one could then believe that the Earth moves and not the heavens, for the opposite is not self-evident. However, at first sight, this seems as much against natural reason than all or many of the articles of our faith. What I have said by way of diversion or intellectual exercise can in this manner serve as a valuable means of refuting and checking those who would like to impugn our faith by argument.
¹¹

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.