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M4005 Geometry

January 23, 2006

Topics discussed in class

  1. Structure of Euclid's system. Definitions. Every definition tells the meaning of a new word in terms of known words. Therefore, not every term can be defined; there must be some first terms whose meaning is determined in some other way than by definition. These terms are called "primitive". (Their meaning is determined by the axioms.). Knowledge, as conceived by Euclid, consists of an organized collection of truths. More complex truths are derived from simpler truths by means of demonstrations or proofs. But not all truths can be found by demonstration. There must be some first truths that are known by some other means than by demonstration. These are called axioms or postulates. Truths derived from axioms by means of demonstration are called propositions.
  2. The meaning of right angle, and the content of Euclid's 4th postulate.
  3. Constructions. Euclid's system also includes constructions. Constructions are procedures for making figures with specified properties. A construction also requires a demonstration; it must be shown that the steps do indeed produce a figure with the properties desired. During class, we viewed Euclid's construction of an equilateral triangle on a given segment. Students attempted to find constructions for the perpendicular bisector of a segment and the bisector of an angle. We compared students' proposals with Euclid's constructions.

What we did

  1. Quiz on reading (5 minutes). View quiz.
  2. Discussion of structure of Euclid (see above)..
  3. Students worked on constructions: a) perpendicular bisector of a segment, b) angle bisector. We noted that Euclid demonstrates (b) first and then uses that to perform (a). View work sheets.

What to think about for next class

    Thales Theorem (see Stillwell) and similar triangles.


  1. Reread Chapter 1 of Stillwell.
  2. Read through the Propositions (but not necessarily the proofs) in Euclid Book I.
  3. Hand in Wednesday written solutions to the following problems from the January 18 handout 1.1, 1.2, 1.3, 1.4, 1.5, 2.1, 2.2.1--3, 2.3, 3.1, 3.2. (January 18 handout available for download here). Written work should be in complete, succinct English sentences; papers should be neat and clearly organized.