THE IONIAN AGE 
History  
 

1. PYTHAGORAS OF SAMOS

 

 

 

Pythagoras of Samos (580-500 B.C.): Pythagoras, who lived some 50 years after Thales, was a more mysterious character who also travelled in his youth to Egypt, Babylon and possibly India. It was in these countries where he acquired and developed his knowledge of mathematics and philosophy. He was a contemporary of Buddha, Confucius and Lao-Tze which perhaps explains his interest in religious mysticism. He is said to have been vegetarian which was common amongst believers in the transmigration of souls. He settled at Croton, in southeast Italy, which was part of Greece at that time, where he set up a secret sect called the Pythagoreans, which helped the development and extension of mathematical knowledge within the Hellenic world. Proclus (410-485 A.D.) also provides the first written reference of Pythagoras in his "Commentary on the First Book of Euclid's Elements". Immediately following on from his writings on Thales he says of Pythagoras "(He) transformed this science into a liberal form of education, examining its principles from the beginning and probing the theorems in an immaterial and intellectual manner. He discovered the theory of proportionals and the construction of the cosmic figures". The Pythagoreans were the first scholars who were more inspired by their love of wisdom and beauty than by practical questions. It is difficult to separate history and legend when talking about Pythagoras the man but there is not doubt that the Pythagorean school played an important role in the development of ancient Greek mathematics. Let us now have a closer look at some of his most important discoveries:

2. PYTHAGORAS' THEOREM 

The expressions were used to obtain Pythagorean triads, although they had already been discovered by the Babylonians.

They were believed to be used to prove Pythagoras' theorem, which the Babylonians were also conscious of but for which no proof had been demonstrated up until this point in time. In this window you can see a demonstration which makes use of similar triangles and segments which are in proportion.

The altitude (perpendicular height) is the mean proportional of the two segments that the hypotenuse is divided into

1.- When the perpendicular height is drawn on the diagram two similar triangles are produced. We are going to call them mbh and nhc. Note that they have equal angles and therefore their corresponding sides are also in proportion.

            

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.


In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides

2.- The triangle with sides a, b and c is a right-angled triangle. There are actually three right-angled triangles in the diagram as a is the circle diameter, and therefore the angle formed by sides b and c is also a right angle. We shall call these three triangles mbh, nhc and abc.

a/b=b/m; a·m=bas abc and mbh are similar  (1)

 a/c=c/n ;a·n=c2 as abc and nhc are similar    (2)

b2+c2=a·m+a·n= a·(m+n)=a·a=a2 according to (1) and (2). Therefore:

       a2 = b2 + c2

If you click on the right-hand button on the mouse a menu will appear. This menu allows you to change the length of m


3. THE FIVE-POINTED STAR

Although Proclus ascribes him to the construction of the five regular polyhedrons it is far more likely that Pythagoras only knew three of them: the cube, the octahedron and the dodecahedron. A dodecahedron-shaped stone with pentagon-shaped faces dating back to the Etruscan period (around 500 B.C.) was discovered in Padua. The five-pointed star, which was constructed from a regular pentagon, was believed to be the symbol used between the members of the Pythagorean sect. By studying the properties of the segments in this star we are able to discover what is known as the golden section.

The demonstration shows how the five-pointed star is constructed from the diagonals in the pentagon.

Use the zoom or move the OX and OY axes if necessary.


4. THE GOLDEN SECTION

Find point H on line BC which divides it into two segments BH and HC in such a way that BH is the mean proportional of BC and HC. The ratio of similitude is referred to as phi and is an irrational number.

 

We have to divide segment BC into two segments which are in golden proportion. In order to do this we need to find point H which satisfies the following equation:

    

Select the segment length and adjust the axes and zoom so that you can see the construction clearly. You can choose the number of decimal places for phi, which is the GOLDEN NUMBER

Click on the diagram on the right-hand button of the mouse, then animate the demonstration to see how to locate point H.


       
           
  Rosa Jiménez Iraundegui
 
Spanish Ministry of Education. Year 2001