# Diocles’s Contribution in Mathematics

Diocles, often called Diocles of  Carystus was a Greek mathematician and geometer. He was born in 240 BC in Carystus, a town on the Greek island Euboea. He was a contemporary of Apollonius. All that is known about his life was through fragments preserved by Eutocius in his commentary, on the famous work of Archimedes, ‘On the sphere and cylinder’. His name is associated with a geometric curve known as Cissoid of Diocles, which was used by him to solve the problem of doubling the cube.  There is no authentic information about his personal life, but recently some information was gathered about him from the Arabic translation of Diocles’s ‘ On burning mirrors.’ which was found in the Shrine Library in Mashhad, Iran, and translated by Toomer in 1976. In this work, it is mentioned that Zenodorus travelled to Arcadia, and had a discussion with Diocles. This shows that Diocles was present in Arcadia at that time. Surprisingly, Arcadia was not a major centre of mathematical sciences that could justify the presence of a Great mathematician at that place. To support the presence of Diocles in Arcadia, Toomer wrote,

It would be wrong to conclude from this that Archadia was a cultural centre in this period, as mathematics during the Hellenistic period was pursued, not in schools established in cultural centres, but by individuals all over the Greek world, who were in lively communication with each other both by correspondence and in their travels.’

He died in 180 BC but the reason for his death is not known.

## On Burning Mirrors

Diocles’s, On Burning Sphere, is a collection of sixteen propositions of geometry, divided into three parts. It is believed that three propositions were added later, but the rest of them were related to the theory of conics, of the 2nd century BC. This book, majorly influenced Arabic mathematicians, particularly al-Haytham, the 11th-century polymath of Cairo, known as “Alhazen” in Europe.

• The first proposition explained what is known to Diocles about the focal property of the parabola.
• Propositions 2nd and 3rd gave the properties of spherical mirrors.
• The 4th and 5th propositions explained the focus directrix construction of the parabola, which was first given by Diocles.
• In propositions 7th and 8th, he discussed the problem given by Archimedes to cut a sphere in a given ratio.
• In Proposition 10th, Diocles studied the problem of doubling the cube.
• Propositions 11th and 12th explained, how to solve the problem of inserting two mean proportions between a pair of magnitudes using the cissoid curve invented by Diocles.
• The remaining propositions gave an insight into the solution of a generalisation, of duplicating the cube problem, using cissoid, and the solution of two mean proportion problems.

In the book ‘On Burning Mirrors,’ Diocles also discussed the problem of finding a mirror, such that the focal point of the mirror traces a curve of the sun rays, as the Sun moves across the sky. The construction of such mirrors helped in the construction of the sundial. After the demonstration of the parabolic mirror, he further added that it was possible to construct a lens with the same properties. Otto Neugebauer, an Austrian mathematician claimed, that this problem cannot be exactly solved, whereas Hogendijk, a Dutch mathematician considered, that arguments related to this problem were given by Diocles in the original copy but the later copiers were unable to understand that text, hence omitted it. He also mentioned that this problem could be approximately solved by the Diocles method. However, Toomer, after a lot of research deducted that the terms ‘hyperbola’, ‘parabola’, and ‘ellipse’ were already known before the era of Apollonius.