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Natural Science - Year I

Unit 7: The Greeks and Early Mathematics

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History Weblecture for Unit 7


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History Lecture for Unit 7: Greek Mathematics

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Outline/Summary

Mathematics in Ancient Greece

This week, we depend heavily on the web for our history reading as we take a look at the beginning of Greek mathematics. The Greeks, through scholars like Pythagoras, were well aware of the work that the Egyptians and Babylonians had done in mathematics, but they were able to take it one step further and abstract the concept of pure number from the concept of counting. To understand the difference, think of how we normally teach a child to count. We hold up fingers, or spoons, or blocks, and we count things. Our first idea of number is always associated with the how-many-ness of real objects.

The Greek philosophers in Ionia on the coast of Asia Minor appear to be the earliest philosophers in the Western tradition to look at number as separate from the objects. Instead of adding 2 cookies + 2 cookies to get 4 cookies, they realized that 2 of anything plus 2 of anything made 4 of that thing, and then that 2 + 2 = 4, without regard to what things were being counted and summed. Once the Greeks had made this step, they could look for patterns and associations between numbers by themselves.

Pythagoras

The first Greek mathematician of note is Pythagoras. Rather than have you read through a brief biography here, I'm going to send you to another site, one which has been developing an extensive site on the history of mathematics for the past decade.

Read through the short biography of Pythagoras at website for the School of Mathematics at the University of St. Andrew in Scotland.

As you read, keep in mind the following questions:

  • Why was Pythagoras in Egypt? How did he wind up in Metapontium?
  • What would have been some of the mathematical ideas he would have learned in Babylon and Egypt?
  • Which of Pythagoras' ideas about numbers seem odd or unusual to you?
  • Which of Pythagoras' ideas about numbers might have applications to the way we use mathematics in science?

We have no direct writings by Pythagora himself, but reports of his teachings were widespread Within two generations, Pythagorean ideas of the soul, life and death, mathematics, and the nature of the universe had became part of the intellectual inheritance of Plato and Aristotle.

Now read the accounts of Plato and Aristotle about Pythagoras at the Hanover Historical Texts Project.

As you read, keep in mind the following questions:

  • What Pythagorean concepts does Plato record? Does he mention Pythagoras by name?
  • What is a "void"? How do Pythagorean beliefs about a void differ from other Greek beliefs about the void?
  • To what other "first prinicples" does Aristotle compare Pythagorean numbers?
  • Why did the Pythagoreans assume a counter-earth existed?
  • What problems does Aristotle find in the Pythagorean account of nature?
Euclid

We jump forward several hundred years to Euclid, or rather to the book Euclid wrote about Geometry. We actually are not sure exactly when Euclid lived, or where, or even if the really was an historical Euclid. Some historians think the book of geometry that bears this name was really written by a team of mathematicians.

Read through the St. Andrews biography about Euclid.
  • Why is The Elements of Geometry important?
  • What is a postulate? A definition? An axiom? (Look up the definition of these terms in a dictionary if you need to.)
  • As you look through the summary of its chapters, see if you can determine which parts of the Elements might have relied on the work of the Pythagoreans.
  • How might the ideas of mathematical proof shaped our ideas of what proof in science should look like?

Euclid uses deduction to show that if certain ideas are true, then we must conclude mathematical figures have very specific relationships. To prove a proposition, Euclid starts with definitions and postulates.

A definition describes an object or explains the meaning of a term, but does not guarantee the existing of the thing it describes. For example, "A point is that which has no part" (i.e., no dimensions) is a definition. Euclid's geometry relies on twenty-three defintions that describe points, lines, straight lines, surfaces, plane surfaces, plane angles, right angles, obtuse and acute angles, circles, the center, radius, and diameter of a circle, three-sided and four-sided figures (triangles and rectangles), and parallel lines.

A postulate or axiom is a statement that must be accepted without proof. Three of Euclid's postulates are constructions, or instructions on how to draw straight lines and circles. The fourth postulate claims that all right angles are equal to one another, and the fifth that two non-parallel lines lying in the same plane will meet.

Euclid also relies on some "common sense" notions, for example, that two things that are equal to a third thing are equal to each other, and that in mathematics, the whole of something is greater than any fraction of it.

By combining definitions and postulates, Euclid shows a number of relationships between angles and lines lying in the same plane (plane geometry). Proposition I in Book I contains a proof that if you start with a short straight line, you can construct and equilateral triangle and know that all three sides are the same length.

You can verify this for yourself and explore some of Euclid's other propositions by looking at the Interactive Elements at Clark University. This requires Java, so it may not work on your computer.
  • What postulates does Euclid use in this proof?
  • What definitions does Euclid use in this proof?
  • What common notions does Euclid use in this proof?

This proposition ends with the abbreviation "QEF", which means "quod erat faciendum", that which was to be done, since the objective here is to draw a certain figure. Most other proofs end with "QED", or "quod erat demonstrandum", which means "that which was to be demonstrated".

Apollonius

Our third Greek mathematician is Apollonius of Perga.

Read through the St. Andrews biography of Apollonius, the man whom the Greeks called "The Great Geometer".
  • Who are some of the ancient Greek philosophers who might have influenced Apollonius?
  • How do we know what Apollonius wrote? Do we have all of his works?
  • What are conics? What are conic sections?

Apollonius did a lot of work on what we now call conic sections -- slices and lines drawn on the surface of a cone. Using the same methods as Euclid, and starting with definitions and postulates, he proved that the different figures such as a circle, an ellipse, a parabola, and a hyperbola can be defined from specific intersections of a flat surface, or plane, and a cone. You can see how this works by looking at how mycroft's very short movie on Conics, which has four sections:

  • A cone by itself, tilting so that you can see inside.
  • The cone with a plane (flat surface) that intersects it parallel to the plane of the base. This kind of intersection creates a circle.
  • The cone with the plane intersecting it at an angle greater than the angle of the side. This kind of intersection creates an ellipse if the plane cuts all the way across the cone.
  • The cone with the plane intersecting it parallel to the axis of the cone. This kind of intersection creates a parabola.
  • Not shown is the case where the plane intesects the cone in an angle equal to the side of the cone. This kind of intersection is a hyperbola.

Understanding these figures will be of great importance to Kepler and Newton in determining how the planets move around the sun. Later on, Einstein would realize that the "well" of force caused by gravitational mass maps to a funnel-like surface similar to a cone, which is why orbits of planets and comets take the shape of ellipses and parabolas.

Study/Discussion Questions:

Further Study/On Your Own

Optional Websites to check

We actually know very little of Pythagoras directly, since he worked in community, and we have none of his own writings. We have to look at the collection of fragments in which he is mentioned to piece together his life. These fragments have been collected and translated as part of the Hanover Historical Texts Project.