**Diocles** was a contemporary of Apollonius. Practically all that was knew about him until recently was through fragments of his work preserved by Eutocius in his commentary on the famous text by Archimedes *On the sphere and the cylinder.* In this work we are told that Diocles studied the cissoid as part of an attempt to duplicate the cube. It is also recorded that he studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio.

The extracts quoted by Eutocius from Diocles' *On burning mirrors* showed that he was the first to prove the focal property of a parabolic mirror. Although Diocles' text was largely ignored by later Greeks, it had considerable influence on the Arab mathematicians, in particular on al-Haytham. Latin translations from about 1200 of the writings of al-Haytham brought the properties of parabolic mirrors discovered by Diocles to European mathematicians.

Recently, however, some more information about Diocles' life has come to us from the Arabic translation of Diocles' *On burning mirrors* whose discovery is described below. From this work we learn that Zenodorus travelled to Arcadia and entered into discussions with Diocles, so that certainly Diocles was working in Arcadia at the time. This may not seem a very major centre of mathematical importance at the time for such an outstanding scholar as Diocles to be working in, but as Toomer writes in [4]:-

It would be wrong to conclude from this that Archadia was a cultural centre in this period ... : the whole of the introduction confirms the impression we derive from other contemporary sources, that 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.

Let us quote from Diocles' introduction to *On burning mirrors* in the translation by Toomer [4]:-

Pythian the Thasian geometer wrote a letter to Conon in which he asked him how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet the circumference of a circle. And when Zenodorus the astronomer came down to Arcadia and was introduced to us, he asked us how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a point and thus cause burning.

Toomer notes that his translation of *when Zenodorus the astronomer came down to Arcadia and was introduced to us* could, perhaps, be translated *when Zenodorus the astronomer came down to Arcadia and was appointed to a teaching position there.*

It is only recently that an Arabic translation of Diocles *On burning mirrors* has been found in the Shrine Library in Mashhad, Iran. No writing of Diocles was known to Heath in 1921 when he wrote [3], but Toomer translated and published the newly found Arabic translation of the lost treatise *On burning mirrors* by Diocles in 1976.

It has been noticed that *On burning mirrors* is loosely in three parts, for three separate topics are studied. These three topics are burning mirrors, Archimedes' problem to cut a sphere by a plane, and duplicating the cube. Sesiano (see [4]) has suggested that we may have three short works by Diocles combined into one work and this would have a certain attraction since the title *On burning mirrors* fails to reflect properly the contents of the whole. If Sesiano's suggestion is correct then we know that the three were combined early on since by the time of Eutocius they formed a single work.

*On burning mirrors* is a collection of sixteen propositions in geometry mostly proving results on conics. It is thought that three of the propositions are later additions to the text, while the remaining ones give a remarkable insight into the theory of conics in the early second century BC.

The first of these propositions proves what has long been known to have been first established by Diocles, namely the focal property of the parabola. The next two propositions give properties of spherical mirrors and with Propositions 4 and 5 giving the focus directrix construction of the parabola. These constructions are again properties of the parabola that Diocles was the first to give.

The problem of Archimedes to cut a sphere in a given ratio which was known to be in the work through the writing of Eutocius referred to above is studied in Propositions 7 and 8. The duplication of the cube problem, again referred to by Eutocius, is studied by Diocles in Proposition 10. The next two propositions solve the problem of inserting two mean proportions between a pair of magnitudes using the cissoid curve which was invented by Diocles. The final three propositions solve generalisations of the duplication of the cube problem using the cissoid, and another problem of the two mean proportionals type.

There are other fascinating deductions that Toomer makes as editor of [4]. A study of the work lead him to claim, contrary to what has long been believed, that the terms "hyperbola", "parabola", and "ellipse" were introduced into the theory of conics before the time of Apollonius.

In *On burning mirrors* Diocles also studies the problem of finding a mirror such that the envelope of reflected rays is a given caustic curve or of finding a mirror such that the focus traces a given curve as the Sun moves across the sky. The solution of this problem would, of course, have interesting consequences for the construction of a sundial. Neugebauer, in an appendix to [4] (see also [6]), shows that this problem cannot be solved exactly while in [5] Hogendijk shows that, using methods available to Diocles, the problem can be solved approximately. Hogendijk in [5] then considers the interesting possibility that Diocles gave arguments of this type in the original text but that later copiers of the text could not understand this part so omitted it.

**Article by:** *J J O'Connor* and *E F Robertson*

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