Right: Stamp from San Marino, which pays homage to Pythagoras. 

Hippocrates had already shown that duplication of the cube was possible when curves were used which satisfied the following equations a/x=x/y=y/2a

Archytas found the solution through the intersection of a cone, a cylinder and a torus.

The window shows Menaechmus' solution (IVth century B.C.). He discovered the conic sections (the ellipse, the hyperbola and the parabola) which can be used to prove Hippocrates' equation: the x-coordinate of the point of intersection of the two parabolas tells us the segment we're looking for.

Hippasus, a contemporary of Philolaus, was expelled from the Pythagorean society for revealing some of their secrets. Some believe that they were to do with incommensurability, which demonstrated that when a longer segment was compared to a shorter one the result was neither a whole number or one of its ratios (which are now called irrational numbers). For example when comparing the diagonal of a square, a cube or a regular pentagon with its side. A similar thing happens when the diagonals of a regular pentagon are joined together to produce another smaller pentagon inside, a process which can be repeated indefinitely. However, the Pythagorean belief that "numbers constituted everything" came up against a serious problem. As the ratio of the diagonal to the longer segment was neither a whole nor a rational number it couldn't therefore be conceived as a number at all.

We have drawn the diagonal lines inside the pentagon. Each diagonal cuts another at a certain point giving us 5 new points which are then joined together to form another pentagon.

Each one of these points has a special property: it divides the diagonal into two segments (one longer than the other) and the ratio (quotient) of the diagonal to the longer side is equal to the ratio of the longer side to the shorter side. This ratio is known as the GOLDEN NUMBER, which is an irrational number which can be found by solving a quadratic equation. Its value is  . In this diagram we can repeat the process in the inner pentagon and obtain another smaller pentagon inside this one. This process can be repeated indefinitely. 

Zeno attacked Pythagorean beliefs with his paradoxes based on the dialectical method of reasoning. Using his opponents premises as a starting point for his arguments he reduced them to an absurdity (reductio ad absurdum). As a result, he was able to conclude that motion did not exist. His most well-known paradoxes are: the Dichotomy, the Achilles, the Arrow and the Stade.

The Stade paradox: Achilles is racing against a tortoise which is a complete stade ahead of him and is running at a speed ten times faster than the animal. However, when Achilles has run the length of the stade the tortoise has run a tenth of the stade. Then, when Achilles has run 1/10 of the stade the tortoise, which is 1/10 of the stade ahead has run a further 1/100. When Achilles has run this 1/100 of the stade the tortoise, which is now 1/100 of the stade ahead has run a further 1/1000 etc. Therefore, Achilles never catches up with the tortoise.

The fact that this was impossible caused Zeno's reasoning to be doubted. However, his contemporaries were unable to refute his arguments. Nevertheless, the idea of using infinitesimal infinities to prove theories was systematically rejected until the following century.

As well as being known as a materialistic philosopher for his atomic doctrine, Democritus of Abdera was also a reputed geometer. He travelled extensively and wrote a number of mathematical works. According to Cicero, his works were clearly written and even more poetical than the poets. Although non of his actual writings still exist today we do know some of their titles: On Numbers, On Geometry, On Tangencies, On Mappings and On Irrationals. His greatest contribution to mathematics is known as the infinitesimal method. He used this method to prove that the volume  of a pyramid is equal to a third of the area of the base multiplied by the height, something that the Egyptians has already been aware of but not proven. It appears that Democritus divided a prism into three pyramids of equal base area and height and sliced an infinite number of cross sections which were parallel to the base. He then matched these sections up and found that the volumes were equal. This gave an immediate conclusion but it can also be argued that the shape could have been a step pyramid instead of a smooth-surfaced one. Another problem he faced was the fact that he had solved the problem using a new method based on infinitesimal calculations.