# Math Tutorials and Moreby George

## Math Tutorials and Moreby George

### Aristotle (384–322 B.C.)

Aristotle was born in a small Greek colony in northern Greece called Stagira. His father was the personal physician to Philip of Macedon, the grandfather of Alexander the Great. Presumably, it was his father who taught him to take an interest in the details of natural life. At the age of eighteen, Aristotle became a student at the Academy of Plato. After Plato's death, Aristotle spent four years as a tutor of Alexander (later to become Alexander the Great). Later on, during his military conquests, Alexander helped to spread Greek culture (including Aristotle's work) to other regions. In 335 B.C. Aristotle returned to Athens and started his own school called the Lyceum. Aristotle was interested in almost everything — government, ethics, philosophy, science, etc. He pretty much invented modern logic. Although Aristotle had great admiration for Plato, he disagreed with his teacher on a number of things. One of these disagreements was on the nature of forms (ideals). Plato was searching for a certain and unchanging basis on which to base knowledge. Thus, he postulated an unchanging world of ideals or forms for which visible objects were merely an imperfect representation. For example, there is an ideal concept of a chair that is universal and applies to all chairs at all times. However, there are many different representations of a chair in the visible world. Central to the concept of forms was the idea of purpose. For example, the purpose of a chair was to provide a place to sit. To Plato these forms could only be approached through reason. Aristotle also believed in universals and ideals, but he didn't believe that they could be separated from the visible world. In fact, he believed that these ideals or forms could only be recognized by studying the real world of concrete objects. Aristotle used the word 'essence' for these ideals, and believed that every substance was a unity of essence and matter.

Aristotle's primary scientific interest was in biology. He observed and cataloged a large number of plants and animals, and divided them into classes. He observed how living things developed. For example, an acorn always developed into an oak tree. Thus, he believed that the acorn had built into it an essence or purpose that was fully realized when it became an oak tree. He transferred this idea of essence to inanimate objects as well. He thought that all objects here on earth were made up of four basic elements — earth, water, air, and fire. He believed that the essence or goal of the earth element was to move toward the center of the universe which he took as the center of the earth. That is why heavy objects containing a lot of the earth element tend to fall toward the ground. The essence of water is also to move toward the center of the universe, but this nature is not as strong as it is for earth. Air and fire have a natural goal to rise above the earth. This natural essence is greater for fire than it is for air. There was one class of objects that didn't fit into this scheme, namely the stars and planets. They seemed always to be moving in a circular path at a uniform speed. Therefore, Aristotle divided the universe into two regions: the region consisting of the earth and everything in between the earth and the moon (sublunar region) and the region beyond the moon (the superlunar region). The sublunar region is composed of the four basic elements. The superlunar region is composed of a fifth weightless element called aether or quintessence. The nature of aether is to move in a circular path with a constant speed. It seems strange to us to assign purposes and goals to inanimate objects, but to Aristotle this was what caused objects to behave as they do.

Instead of avoiding motion and change as Plato did with his world of unchangeable forms, Aristotle embraced change and sought out causes for change and motion. Aristotle considered two types of causes for motion. There is natural motion that is due to an object's built in essence and there is motion that is caused by an outside agent. He called motion produced by an external agent `violent motion.' All motion in the sublunar region eventually stops. Violent motion stops when the external agent stops pushing or pulling. Natural motion stops when the object has reached its natural place or when it is prevented from doing so by another object, e.g. a falling rock stops when it hits the ground. Based largely on Aristotle's ideas a view of the physical universe was developed that was the dominant view from about 300 BC up into the sixteenth century. Here are some of the components of this view

1. The universe is eternal. It had no beginning and no foreseeable end.
2. The earth is spherical.
3. The earth is located at the center of the universe.
4. The earth is stationary, i.e., it doesn't rotate or revolve around any other object.
5. The stars, the moon, the planets, and the sun revolve around the Earth, completing a revolution about every 24 hours.
6. The sublunar region is composed of four basic elements: earth, water, air, and fire. This region is subject to change and decay.
7. The superlunar region is composed of a fifth element called aether whose natural motion is circular at a uniform rate. The superlunar region is perfect and unchangeable.
8. All motion in the sublunar region eventually stops.
9. An object that is stationary will remain stationary unless there is some source of motion.

You may be surprised that it was realized at this early date that the earth is spherical. However, there were several observations that led the ancient Greeks to this conclusion.

• The sun rises and sets at different times as you move in an east-west direction.
• The positions of constellations in the sky change as you move in a north-south direction. New constellations may come into view and others may disappear.
• As a ship approaches land the land appears to rise out of the sea. High points appear above the horizon first and later on the lower regions.
• During a lunar eclipse, the shadow of the earth on the moon appears circular.

In fact Eratosthenes measured the circumference of the earth in about 240 B.C. He Knew that the sun would be directly above the city of Syene at noon on the summer solstice. He knew this by looking at the reflection of the sun at the bottom of a deep well. He placed a vertical pole in the city of Alexandria which was 5000 stadia (about 925 Km) due north of Syene. By looking at the shadow cast by the pole he determined the sun's rays made an angle of 7.2° with the pole. It can be seen in Figure 1 that 7.2° is also the angular separation of the two cities. As the ratio 360°/7.2°=50, it follows that the ratio the circumference to the distance between the cities is also 50. Thus the circumference is 50×5000 stadia or 46,250 Km. Current Satellite measurements give a value of 40,008 Km for the circumference. Thus, he was off by only about 16%.

This experiment by Eratosthenes is further verification of a spherical earth as the pole would have cast no shadow if the earth were flat. The Greek's idea of a spherical earth spread to other regions, and by 800 A.D. it was rare to find any people group that believed in a flat earth.

There are a number of observations that led to the belief that the earth is stationary.

1. It doesn't feel like we are moving.
2. We don't feel the wind-like effects of the air as we move through it.
3. Dropped objects fall straight down. It was believed that if we were moving, then we would have moved ahead a certain distance as the object was falling and the object would land behind us.
4. The earth is very big and heavy, and there is no obvious source capable of moving it.
5. We don't observe stellar parallax (the change of relative positions of stars due to changes in the observer's position).

Although the reasons given for a stationary earth seemed very reasonable at the time, most of us today believe that the earth does move and that it revolves around the sun. However, you might ask yourself why you believe this. For most of us, it is because someone told us or because we read it in a book. It may surprise you that even today we don't have any direct observations that we revolve around the sun. As we will see, the reasons that this is the accepted viewpoint today are very subtle and not at all obvious.

It should be noted that we now believe that the earth is orbiting the sun with a velocity of about 70,000 mph. Due to the earths rotation a point on the equator is moving at a velocity in excess of 1000 mph. Yet we don't feel like we are moving and we don't feel any super strong winds due to these high velocities. At first glance these facts seem to support a stationary earth.

The idea of the earth being the center is perfectly natural if the earth is stationary since all the heavenly bodies appear to revolve around the earth. Belief in the earth as the center was not motivated primarily by the belief that the center was a place of great importance, but was believed primarily on physical grounds. In fact, the superlunar region was considered to be the place of greatest perfection and importance. The earth was characterized by imperfection and decay.

The idea of parallax probably requires some explanation. If you hold up a piece of paper in front of you and hold up a finger in front of the paper, you will notice that your finger appears to move from one side of the paper to the other as you move your head from side to side. This phenomenon is called parallax and involves the apparent change of the relative positions of objects due to changes in position of the observer. If the earth were circling the sun, the relative position of stars should exhibit parallax when viewed from different positions in the orbit. Since none was observed, this was taken as evidence of a stationary earth. It turns out that parallax is real, but because the stars are so far away our instruments were not accurate enough to measure it until the year 1838 A.D. Stellar parallax is probably the best empirical evidence we have today that the earth revolves around the sun.

The reason that objects appear to drop straight down is that the velocity of the dropped object has two components — the downward velocity produced by gravity and the velocity in the direction of motion that it shares with us. The velocity it initially shares with us persists as the object falls, since there is no force to change it (the principal of inertia). Thus, the dropped object not only moves downward, but also moves forward just as we do. A true understanding of relative motion and inertia was not obtained until the 16th and 17th centuries (Galileo and Newton).

Everyone realizes that it takes considerable effort to move a heavy object such as a large rock. The earth was certainly a very large and heavy object, and there didn't appear to be any agent around powerful enough to move it and to keep it moving. The concept of inertia was not understood until the time of Galileo and Newton. Newton's laws imply that once an object is in motion it takes a force sufficient to overcome its inertia in order to bring it to a stop. The earth was set in motion during the creation of the solar system. The major force acting on the earth today is the gravitational attraction of the sun. This force causes the earth to speed up in certain parts of its orbit and to slow down in others, but it never acts in such a manner as to bring it to a halt. In fact it can be shown that the earth-sun system is periodic. As an aside, the idea of gravitational attraction itself is somewhat mysterious. How can an object exert a force on another object when there is no contact? Many thought that Newton was engaging in witchcraft when he first proposed the idea.

Although we now believe that most of Aristotle's beliefs concerning the solar system are false, they seemed to be very reasonable at the time. In fact these views were not seriously challenged until the 16th century.

Today we know that stars consist of hot gases and planets are composed of hard materials much like we have on earth. However, prior to the development of the telescope there was no reason to think that the lights seen in the sky were made of materials like those on the earth. The fact that the stars moved in a regular pattern year after year led to the idea that the superlunar region represented perfection. The motion of the stars was very different from observed motion on the earth which always eventually comes to a halt. Thus, these ideas of Aristotle seemed reasonable prior to the time of Galileo.