It was in Greece that mathematics were invented. « Invented »? This shortcut seems to neglect the Babylonian scholars, Indian and Egyptian who had preceded them. « Invented » because it was our Greek friends who first had the aim of the demonstration. Their ancestors’ constataient « properties, the Greeks showed them.

# The three enigmas of Greek mathematicians

Descartes had not yet invented the coordinates that still bear his name; also demonstrations Were geometric.

Pythagoras, for example, was a puzzle expert: it was this talent that allowed him to demonstrate his theorem. A cutting force, he found indeed that the mosaic he had built on the side of the hypotenuse well understood :

- the red square, he had drawn on the side of the small side;
- other pieces gray, blue, green and yellow, he had drawn on the side of the large side.

QED and no math!

**The ruler and compass**

The Greeks were passionate geometry. Two instruments were intended to build everything: the rule (non-graduate) and compass. The construction of the circle and the square was of course obvious! So were the two fundamental figures, given to men by the gods. With persistence, imagination suffering too, Euclid, Archimedes, raised all challenges. Yet three figures resisted all attempts.

# History of Mathematics

### The three enigmas of Greek mathematicians

## **The duplication of the cube**

**Hippocrates of Chios (430 BC)**

We were in the sixth century before Christ. The priests were gathered around the Oracle. The latter promised them to free Greece from the brown plague. But that prophecy come true, it was necessary that the volume of the altar of Apollo is doubled. This altar was a large cube of side and therefore d³ volume: the problem was simple: a volume of 2 cubic d³. We went for the greatest mathematicians; they doubled the length of one side and thus obtained a cube of side 2d and therefore volume 8 d³ and non-2d³ not. The plague devastated the country. It was perhaps take the problem the other way. Find a volume of cube ³√ (2d³) or simplifying d³√2. It took until 2500 years to discover that the cube root of 2 was (and still is) not constructible with a ruler and compass.

# History of Mathematics

### The three enigmas of Greek mathematicians

## **The angle trisection**

**Hippias of Elis (425 BC)**

It was easy to cut an angle into two. It was located on the intersection of two segments that form the corner and drew a circle with a compass.

This circle cut the segments in a point. two new circles is traced from these two new centers. These two circles intersected in two points. The straight line passing through these two points was that bisects the angle cut in half.

But how to cut the angle in three? Impossible. The greatest mathematicians broke teeth, the tip of their compass and their rules.

2500 years were still necessary so that it was impossible.

# History of Mathematics

### The three enigmas of Greek mathematicians

**Squaring the circle**

**Artemon of Clazomenae (435 BC)**

As we saw the square was magical. It was the standard of measurement. Any surface must be reduced to a square surface. For a rectangle, it was pretty simple: its surface was (and still is) equal to d × D. The square of the same area must have a side of √d × D.

We became interested in the circle. He had a radius R. How to build a square of the same area? The Egyptians already knew an approximate value of Π . The Greeks had refined. The surface of the circle was to Π R². The square had to have one side a = √ (ΠR²) or a = R√Π, nothing more simple. He had to wait the nineteenth century to discover that Π is transcendent; that is to say, it is not the solution of an equation of the type a² = Π. We must demonstrate to *Carl Louis Ferdinand von Lindemann.*