Euclid and Proofs

The concept of a mathematical proof was formulated by the ancient Greeks, and first written in an organized way by Euclid. His set of books is called The Elements. It starts by describing basic geometric objects and rules, then combines them using logical reasoning to come up with facts about the geometric objects.

Euclid's starting points:

• Definitions. Short names for useful combinations of ideas. Definition 15 defines the term circle:

  A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

  Modern mathematicians would call Euclid's early definitions, such as

  Definition 1: A point is that which has no part.

  as undefined terms: his definition only gives you a vague idea what he's talking about, and you can't look up what he means by "that which has no part". Try looking of the meaning of some word in the dictionary, then look up the definitions of the words used in its definition. Generally you will find that you eventually get back to your original word. That is, you have to start with something undefined when making definitions.
• Postulates (also called axioms). These are basic rules that are supposed to be more or less obvious, such as the axiom that two points determine a line (Postulate 1).
  
  At around the turn of the century (that is, 1900) mathematicians started examining the foundations of mathematics. They found that Euclid was missing some postulates--some of his theorems used facts that were neither postulates nor proved anwhere else. So several people developed alternative systems of postulates for Euclidean geometry; they all have more than 5 postulates.
• Common notions. These are axioms or facts from other parts of mathematics that aren't particularly geometric; for example, Things which equal the same thing also equal one another (Common notion 1).
• Propositions (also called theorems. A lemma is a term for a minor theorem whose main use is to help prove another theorem.) Theorems are facts about the objects in geometry that have been shown to be true by logical reasoning using axioms, definitions, and previously proved theorems. The reasoning used is called a proof of the theorem.

Euclid's work is organized into 13 books of related theorems and definitions. All the common notions and axioms occur in Book I.

A mathematician at Clark University has been putting Euclid's Elements on the Web; so far he is up to Book VI. Each proposition comes with an interactive diagram* illustrating the proposition, a proof of the proposition, and notes of the other propositions, axioms, common notions, and definitions used in the proof.

Individual problems

These questions are based on the Euclid site on the Web. Or you could even use a traditional book of the works of Euclid.

1. Read all the definitions for Book I. In Def. 22, what is our term for Euclid's oblong? What is our term for Euclid's rhomboid? Note that trapezium (singular of trapezia) does not mean trapezoid.
2. Read the five postulates for Book 1 and the Guides you get by clicking on the Postulate number. (You don't need to read the part about "neusis" for Postulate 2). Restate each of the postulates in your own words (not the Guide's words).
3. Read all the common notions for Book I.
4. Read Book I, Proposition 1 and its proof. Note that the first part of the "proof" is actually instructions for the construction, and the second part is the justification.
   • Compare the proof to your group's proof of the same proposition. Do you think that you and Euclid said essentially the thing?
   • What definitions, common notions, postulates, and other propositions does the proof use?
   • You may be amused (or horrified) to read, at the end of the proof, about all the holes in Euclid's proof of this proposition people have found over the millenia.
5. Draw a flow chart or tree diagram that shows how the proof of Proposition 6 depends on other propositions, definitions, postulates, and common notions. Connect two items with an arrow from A to B if A is directly used in the proof of B. A flow chart has only one node for each proposition, postulate, etc. (the tree diagram below repeats nodes) but the diagram may look more complicated because the arrows have to be drawn crossing each other.
   For example, Prop. I.5 uses Def. 20, Posts. 1 and 2, C.N. 3, and Props. I.3 and I.4. Prop. I.4 uses C.N.s 3 and 4. Prop. I.3 uses Def. 15, Post. 3, C.N. 1, and Prop. I.2. Prop. I.2 uses Def. 15, Posts. 1, 2, and 3, C.N.s 1 and 3, and Prop. I.1. Finally, Prop. I.1 uses Defs. 15 and 20, Posts. 1 and 3, and C.N. 1. So all the reasons can be traced back to Definitions, Postulates, and Common Notions.
   
   There is a table of dependencies among the Propositions at the end of the Book I page.
   

* The interactive diagrams use a programming language java, which can be run by recent versions of Web browsers, such as Netscape v. 3.0. If you are viewing this page from a computer whose browser doesn't have java capability, you won't get an interactive diagram.

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