THE IONIANS
History
 

1. THE ANCIENT GREEKS

Between 800 B.C. and 800 A.D. the Mediterranean coast became a hub of intellectual activity. The alphabet seems to have originated from the Sinai peninsula in Egypt and found its way into Greek, Roman and Carthaginian colonies through the commercial activity in the region. In the VIIIth century B.C. the first Olympic games were held and the first Greek literature appeared in the works of writers such as Homer and Hesiod. However, there was no evidence of mathematical activity until 600 B.C. when the emergence of two men, Thales of Miletus and Pythagoras of Samos, made the first mathematical discoveries of this period. Despite the lack of documented evidence, tradition has nevertheless given a lot of importance to early Greek mathematics. During the VIth century B.C. the Ionians, who formed Greek settlements along the Mediterranean coast, developed their commercial activity with the Egyptian and Mesopotamian colonies. This gave the Greeks the opportunity to come into contact with Egyptian and Babylonian culture. It is highly probable that both Thales and Pythagoras acquired their knowledge of ancient mathematics from their travels to these regions, which they later developed back in their homeland.

2. THALES OF MILETUS

 

 

 

 'Capitolino' Museum (Rome)

 The earliest written document about the life and work of Thales is found in the book "Commentary on the First Book of Euclid's Elements" written by the philosopher Proclus (410-485 A.D.) approximately 1,000 years after his death. Proclus writes that Thales "...first went to Egypt and thence introduced this study into Greece. He discovered many propositions himself, and instructed his successor in the principles underlying many other, his method of attack being in some cases more general, in others more empirical". Thales is considered to be the first known mathematician and the creator of demonstrative geometry. Tradition has lead us to associate him with the following five theorems, which were possibly learned on his travels whilst trading in Babylon, as well as finding a proof for the last theorem.                                   

                              

3.- THE FIRST THEOREM.
A circle is bisected by its diameter
If you click on the right-hand button on the mouse a menu will appear. This menu allows you to change the size and position of the diagram with the zoom and centre controls.

 

 

 

 

The Init button restores the initial vales. The Animate button activates the demonstration. The Pause button stops the demonstration.

4.- THE SECOND THEOREM.
  The base angles of an isosceles triangle are equal           

If you click on the right-hand button on the mouse a menu will appear. This menu allows you to change the size and position of the diagram with the zoom and centre controls.

Change the length of the sides by using the arrows or writing in the new values and pressing the enter key.
The Init button restores the initial vales. 

5.- THE THIRD THEOREM.
The opposite angles at a vertex formed by two intersecting lines are equal      
 

If you click on the right-hand button on the mouse a menu will appear. This menu allows you to change the size and position of the diagram with the zoom and centre controls.

Move the straight lines by changing the slope of the line and look at the size of the angles.
The Init button restores the initial vales. 

 


6. - THE FOURTH THEOREM. 
If two triangles are such that two angles and a side of one are equal respectively to those of the other, then the triangles are congruent (equal)
Select the angles and side length and note how the second triangle always remains equal to the first.
Use the zoom or move the OX and OY axes if necessary.

7. - THE FIFTH THEOREM
An angle inscribed in a semicircle is always a right angle

 

Click on the pink point and drag it with the mouse to change the inscribed angle and note that it is always a right angle.

       
           
  Rosa Jiménez Iraundegui
 
Spanish Ministry of Education. Year 2001