Cuanto mayor sea la distancia que se deba recorrer para la obtención de un bien o servicio mayor será el coste de transporte

suffi-ciently large speed he will be able to do it in a reasonable amount of time.

The calculus has apparently enabled people to show that our com-mon sense ideas about motion do not involve any paradoxes. But wait! Perhaps we have not yet heard the last word on the subject.

Matter was once thought by some people to be infinitely divisible, so that, in principle, a piece of gold, could be divided again and again without end, and still the bits would remain gold. Now, how-ever, we know that gold and other substances are made of atoms, which cannot be subdivided without altering their properties. In other words, if we can figure out a way to continually subdivide a piece of gold, eventually it will cease to be gold. But what has this to do with Zeno’s paradoxes? Just this: perhaps space itself cannot be infinitely subdivided, although we currently have no evidence in this regard. If it should turn out that there is a limitation on the di-visibility of space, then Zeno’s hypothesis that he can subdivide the distance without end would be in error. Zeno’s paradoxes would again be resolved, but the resolution would involve ideas which are quite different from those most of us believe today.

4.2 Aristotle’s views on motion

Aristotle not only had a picture of the universe but also had views on motion. Again, it was Galileo who showed that Aristotle’s ideas were basically wrong. Although not many of Aristotle’s views on science have stood the test of time, it is in the nature of science that

old views are often superseded by new discoveries.

Much of Aristotle’s importance in physics derives from his em-phasis on direct observation of nature and his teaching that theory must follow fact. This was a profound departure from the view of many earlier Greek philosophers (and many later philosophers from various parts of the world) that understanding could be obtained from pure reason.

Today, many scientists believe in a synthesis of the views of Aris-totle and his opponents. They believe that, although observation is a necessary ingredient in science, observation does not by itself lead to the discovery of nature’s laws. According to this view, flights of imagination are also essential. Just as knowledge of the grammar of a language does not by itself make a great writer, knowledge of the

“scientific method,” whatever it may mean, does not make a great scientist. There is an art to being a great scientist.

Now let us turn to Aristotle’s physics. He believed that on earth the natural state of an object was to be at rest. We can easily under-stand how he got that idea. Roll a ball along the floor and it comes to rest. Slide on the ice and you come to rest. Then how did Aristotle explain the fact that motion exists on earth? According to Aristotle, motion on earth is a transitory phenomenon. Objects can be set in motion, but, left to their own devices, sooner or later they will come to rest.

Aristotle also taught that the natural state of motion of objects in the air above the earth was to fall at constant speed. Raindrops obey Aristotle’s law to a good approximation. Aristotle further taught that heavy objects fall faster than light ones. For example, a stone falls

4.2. ARISTOTLE’S VIEWS ON MOTION 49 faster than a feather.

Aristotle must have noticed that some falling objects do not fall at constant speed but accelerate. We have already noted that accel-eration is defined as the rate of change in velocity. Another way of putting this is to say that the acceleration is the change in velocity di-vided by the change in time. Hold an object stationary in your hand, and then let it go. It must accelerate for a time, or it would remain at rest suspended in the air, contrary to the observation that it falls to the ground. Aristotle must have regarded the initial acceleration of a falling object as transitory, as he regarded motion on earth as transi-tory. Sooner or later, Aristotle must have thought, the object would stop accelerating and fall at constant speed, provided it did not hit the ground first.

Although Aristotle was right in many of his observations, he drew the wrong conclusions from them, or rather, more precisely, he failed to understand the essential properties of motion. A simple example will show what Aristotle missed. Take a run and slide on a gravel path. Take a similar run and slide on a patch of smooth ice. It is true that in either case you come to rest. But you slide much further on the ice than on the gravel. Why? Aristotle evidently did not think the answer important. But if we want to go beyond merely noting that objects come to rest on earth, we must face the question of why under different circumstances objects take different times to come to rest.

We here briefly anticipate the answer that Galileo gave, which we shall discuss in more detail in the next section: A person sliding on gravel stops sooner than he does when sliding on ice because the

gravel gives more resistance to the motion than does the ice. Once we have the idea that objects come to rest because of resistance to their motion, we might believe that, contrary to what Aristotle thought, motion tends to persist unless opposed.

Consider another example. Drop a balloon and a stone from the top of a building. The balloon falls slowly, most of the time at con-stant speed, but the stone accelerates all the way to the ground. Why?

Again, Aristotle did not have a satisfactory answer. And again Galileo realized that it was resistance, in this case the resistance of the air, that causes the motion of the ballon and the stone to be so different.

We shall discuss Galileo’s answer further in the next section.