# Math Tutorials and Moreby George

## Math Tutorials and Moreby George

### Ptolemy (c.85–165 A.D.)

Aristotle's view of the solar system was greatly expanded by Claudius Ptolemy. Not much is known about Ptolemy's life except that he was a mathematician and astronomer that lived in Alexandria, Egypt during the last part of the first century A.D. into the second century A.D. At that time Egypt was a Roman province. His famous work was the Almagest in which he layed out a model for the motion of the planets that was based on the ideas of Aristotle. Each planet was modeled separately. To handle the retrograde movement of the planets he introduced epicycles. An epicycle is a smaller circle whose center moves in a circular orbit around the earth. Like Aristotle, Ptolemy took the earth as the center of the universe. As such, the earth was the center of the sphere containing the fixed stars. However, the earth was not taken as the center of a planet's orbit. A planet moved at a constant angular velocity around an epicycle whose center moved at a constant angular velocity around a larger circle whose center was displaced somewhat from the earth. Epicycles were introduced primarily to handle retrograde motion. It was observed that planets moved faster in some parts of their orbit than in others. To handle this nonuniform motion, Ptolemy introduced another point called the `equant.' It was assumed that the angular velocity relative to this point was constant. This produced a nonuniform angular velocity relative to the center. Figure 2 shows the basic parts of Ptolemy's model of planetary motion.

Since Mercury and Venus always appeared close to the sun, Ptolemy assumed that the centers of their epicycles lay on the line connecting the earth and the sun and that they rotated around the earth at the same rate as the sun does. This is illustrated in Figure 3.

The retrograde motion of the outer planets (Mars, Jupiter, and Saturn) always occurs in the night sky and the planet is always brightest during this retrograde motion. To accomplish this Ptolemy required that the arrow from the center of a planet's epicycle to the planet always be parallel to the arrow from the earth to the sun. This configuration is shown in Figure 4.

Using models such as this, Ptolemy was able to predict the motion of the planets fairly accurately. Sometimes he needed to introduce epicycles riding on other epicycles to get the desired accuracy. Ptolemy's planetary model was the generally accepted model for planetary motion up into the 16th century. It is not known how Ptolemy viewed his planetary model. Did he think that his model represented reality, or was it merely a computational tool for producing results that agree with observation. Even today Scientists often disagree on whether certain theories really represent reality or not. At the time of Galileo the church didn't object to a sun-centered universe as a computational tool, but it did object to claims that it represented reality.

There were two aspects of Ptolemy's models that persisted in the sun-centered models of Copernicus. These were the requirements that all motions be circular and at a uniform speed. These requirements were not dropped until Kepler developed his laws of planetary motion.