- ...
- Theodorus proved the incommensurability of , , , ...,.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
- Archytas solved the duplication of the cube problem at the intersection of a cone, a torus, and a cylinder.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...histories
- Here the most remarkable fact must be that knowledge at that time must have been sufficiently broad and extensive to warrant histories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Anaximander
- Anaximander further developed the air, water, fire theory as the original and primary form of the body, arguing that it was unnecessary to fix upon any one of them. He preferred the
*boundless*as the source and destiny of all things.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Anaximenes
- Anaximenes was actually a student of Anaximander. He regarded air as the origin and used the term 'air' as god
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...proofs.
- It is doubtful that proofs provided by Thales match the rigor of logic based on the principles set out by Aristotle found in later periods.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...incommensurables.
- The discovery of incommensurables brought to a fore one of the principle difficulties in all of mathematics - the nature of infinity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...discovered
- as attested by Archimedes. However, he did not rigorously prove these results. Recall that the formula for the volume pyramid was know to the Egyptians and the Babylonians.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Persians.
- This was the time of Pericles. Athens became a rich trading center with a true democratic tradition. All citizens met annually to discuss the current affairs of state and to vote for leaders. Ionians and Pythagorean s were attracted to Athens. This was also the time of the conquest of Athens by Sparta.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Elis
- a city in the Peloponnesus. According to Plato, Hippias was a 'vain and boastful man'.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
**Menaechmus** - Menaechmus invented the conic sections. Only one branch of the hyperbola was recognized at this time.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
**Dinostratus** - Dinostratus showed how to square the circle using the trisectrix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
**Theaetetus** - Theaetetus proved that there are only five regular solids: the tetrahedron (4 sides, triangles), cube (6 sides, squares, octahedron (8 sides, triangles), dodecahedron (12 sides, pentagons), and icosahedron (20 sides, hexagons). Theaetetus was a student of Theodorus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...analysis
- where what is to be proved is regarded as known and the consequences deduced until a known truth or a contradiction is reached. A contradiction renders the proposition to be false.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...absurdum.
- where what is to be proved is taken and false and consequences are deduced until a contradiction is produced, thus proving the proposition. This, the indirect method, is also attributed to Hippocrates.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...diameters.
- At this time there is still no apparent concept of a formula such as .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...world,
- This is still an issue of debate and contention
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...substance
- Morris Kline,
*Mathematical Thought From Ancient to Modern Times*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
**definition** - and hence also undefined terms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...infinity
- This is still a problem today.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fri Jan 31 13:12:00 CST 1997