THE GREEK HEROIC AGE 
History
 
1.- THE HEROIC AGE  (Vth century B.C.)                                                                                
 One of the most important personalities of this century is Pericles. Athens attracted intellectuals from all parts of the Greek world wanting to satisfy their thirst for knowledge. Rather than coming up with necessary solutions to practical problems at that time, the scholars were more interested in developing their own personal intellect. This desire for wisdom lead them to focus their learning on theoretical issues. During this period the three famous (or classical) problems were dealt with and two methods of reasoning were put into use. 

The table below lists the mathematicians who lived during this period and the problems that formed the focus of their study.

Anaxagoras of Clazomenae (Athens)

Hippocrates of Chios (Athens) 

 squaring the circle or how to draw a square whose area is the same as that of a circle using a ruler and compass.

 

Hippias de Elis (Attic peninsular) the trisection of the angle or how to construct an angle equal to a third of another given angle 
Philolaus of Tarentum (Southern Italy)

Archytas of Tarentum

 the duplication of the cube or how to construct another cube whose volume is double that of the given cube 
Hippasus of Metapontum (Southern Italy) Incommensurability or  line segments which are not in rational proportion to one another (THE GOLDEN SECTION)
Zeno of Elea (Athens) The paradoxes on motion or  the reductio ad absurdum method
Democritus of Abdera(Thrace) Infinitesimal techniques or the method of breaking a problem down into infinite steps 
2.- ANAXAGORAS OF CLAZOMENAE
Anaxagoras of Clazomenae ( +428 B.C.): went to prison for asserting that the sun was not a deity, but a red hot stone and that the moon was an uninhabited earth which received its light from the sun. He spent his life studying living matter which he claimed was made up of infinite individual indivisible elements. According to Plutarch, while he was imprisoned he kept himself busy with his study of squaring the circle. His conclusions are unknown. However, those of another mathematician have had more success.
3.- HIPPOCRATES OF CHIOS  (430 B.C.)
  He spent his life studying geometry after becoming a victim of fraud in Byzantium and losing all his money. According to Proclus, he wrote about "Elements of Geometry" which was later lost like all other texts written during this century. In one of his works Simplicius (VIth century A.D.) states that he had copied extracts from Eudemus' "History of Mathematics" which is also now lost. In his work Simplicius describes part of the work of Hippocrates about the quadrature of lunes (a lune is a curved shape bounded by two circular arcs whose radii are different) along with the following theorem:
"Similar segments of circles are in the same ratio as the squares on their bases"

Hippocrates used this theorem to square the circle.
In this window Pythagoras' theorem is explained in geometrical terms together with a visual demonstration of the theorem:
C1+C2=C3
The area of the semicircle constructed on each side of the triangle is proportional to the area of the corresponding square on each side.
 
In this window you can see  three semicircles S1, S2 and S3 which satisfy the following equation:
S1+S2=S3
This is an example of how Pythagoras' theorem can be applied to semicircles which are similar to the squares constructed on each side of the triangle.
From here on the conclusion is straightforward
The right-angled triangle must not be an isosceles triangle.

 

The sum of the area of the small semicircles S1 and S2 is equal to the area of the big semicircle.
S1+S2=S3
If you first take away the two white sections from the small semicircles and then take them away from the big semicircle you will be left with the two half-moons (lunes) in the first case or the triangle in the second case.

In this window you can see how the first half-moon is constructed.

The three circular sections in this diagram are similar and satisfy Pythagoras' theorem. S1+S2=S3

If you take away the two smaller sections from the semicircle in the first example and the big section away in the other example then we can prove that the area of the half-moon is equal to that of the triangle.

The first squaring of a half-moon

    As the area of a triangle is equal to that of a square whose side is half the length of the base of the triangle, then we can use this to obtain the first example of squaring a curvilinear shape.
       
           
  Rosa Jiménez Iraundegui
 
Spanish Ministry of Education. Year 2001