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## M4005 Geometry

**January 23, 2006**

**Topics discussed in class**

**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.
- The meaning of
**right angle**, and the content of Euclid's 4th postulate.
**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

- Quiz on reading (5 minutes). View quiz.
- Discussion of structure of Euclid (see above)..
- 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.

### Homework

- Reread Chapter 1 of Stillwell.
- Read through the Propositions (but not necessarily the proofs) in Euclid
Book I.
- 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.