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There are two separate articles: How do we know about Greek mathematics? and "How do we know about Greek mathematicians?". Before reading this second article on how we can find out about the lives of the ancient Greek mathematicians, it will help if the reader first looks at the previous article on how the works of these mathematicians have reached us.

Perhaps the most important fact about the lives of the mathematicians, if we are to have a proper appreciation of their work, is a knowledge of the period during which they lived. Some mathematicians added a date to their work and this has been preserved during the copying process described in the article How do we know about Greek mathematics?. Some are referred to by other authors and at least an approximate date can be given. Otherwise much more indirect evidence needs to be used.

The following type of argument is typical of the type used. What works does the mathematician refer to? Clearly the mathematician must have lived after these works were written. What works refer to the mathematician? One has to be particularly careful about using data of this type since during the copying of the texts additional references may have been added which the original author could never have known about since they are from a later date. This method may leave a span of more than 200 years during which the mathematician may have lived. Particularly useful are cases where the mathematician made astronomical observations. Often these can be dated with great accuracy and, as we shall see in the example below, even having some mathematicians dates known accurately will help to date others.

As an example we examine how the dates of Diocles given in this archive have been determined. First let us see what facts Heath knew when he wrote his famous book [1], *A history of Greek mathematics*, which he began in 1913. In the 5^{th} century AD, Proclus wrote his *Commentary on Euclid* which is our principal source of knowledge about the early history of Greek geometry. This commentary has survived and in the work Proclus wrote that Geminus used the term "cissoid" for a curve he describes. Now Heath knew, from other sources which we will describe in a moment, that the "cissoid" was used by Diocles to solve the classical problem of doubling the cube (and almost certainly invented by him for this purpose). Hence, Heath can deduce that [1]:-

The other bound for Diocles dates which was deduced by Heath comes from Eutocius who wrote commentaries on three works of Archimedes in the first quarter of the 6^{th} century AD. Eutocius's commentary on Archimedes *On the Sphere and Cylinder II* includes a quotation from Diocles solving the following problem of Archimedes (see for example [1]):-

To cut a given sphere by a plane in such a way that the volumes of the segments are to one another in a given ratio.

Although Archimedes promises a solution later in his text, it does not appear. Eutocius quotes from a solution by Diocles of this problem. In the quote from Diocles, reference is made to Apollonius. Eutocius states that the quote he gives is from Diocles' *On burning mirrors* but at the time Heath wrote his book no version of Diocles's text had been found, either in Greek or Arabic. Heath deduces from the quotes in Eutocius that Diocles [1]:-

... was later than Archimedes and Apollonius. He may therefore have flourished towards the end of the second century or at the beginning of the first century BC.

Heath also writes [1]:-

Diocles ... writing a century or more after Apollonius ...

Now the dates deduced by Heath, say 130 BC - 70 BC, are based on Heath's dating of Apollonius and of Geminus both of which were inaccurate. The range of dates (after Apollonius and before Geminus) is convincing but, as Toomer writes in [2]:-

There are no grounds for Heath's further refinements.

In the paper *La cissoid et Diocles* written by Paul Tannery, which was certainly known to Heath, it was noted that the reference to Apollonius in Eutocius's commentary on Archimedes' *On the Sphere and Cylinder II* must be an addition by Eutocius, but Heath clearly must have felt that Paul Tannery was wrong. However, time would show that Paul Tannery was correct and Heath was wrong.

Before continuing with the description of what has been discovered since Heath wrote his *A history of Greek mathematics* it is reasonable to ask: Does it matter when Diocles lived? The answer is certainly "yes" in deciding how fine a mathematician Diocles was. If indeed Diocles lived a century or more after Apollonius, as Heath "proves", then he would be very familiar with Apollonius's *Conics* and so his own work on conic sections would have to be evaluated in that light. However, if Diocles did not know of Apollonius's *Conics* then his own work clearly must be considered far more innovative than is the case otherwise.

We now come to the work [2] by Toomer. It presents an Arabic translation of Diocles *On burning mirrors*, from a manuscript copied in 1462, together with an English translation and commentary. The publication of this work in 1976 was an event of major importance in the history of mathematics adding a previously lost piece to the jigsaw. How does it help us to know the dates when Diocles lived? To answer this we quote from Diocles' introduction to *On burning mirrors* in the translation by Toomer [2]:-

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'.

This certainly allows us to give quite accurate dates for Diocles since the dates of Zenodorus are known accurately (although to explain how we know this is another story!). Looking at the text of *On burning mirrors * shows that Paul Tannery was correct and Heath was wrong: Eutocius did insert the reference to Apollonius. In fact Eutocius did quite a lot more for, although claiming to give a direct quote, Eutocius rewrote Diocles' proof in a later style so he converts Diocles' text into what for him would be a modern form.

How then do the dates of Apollonius and Diocles fit? Did Diocles know of Apollonius's *Conics* ? We can now say with certainty that Apollonius and Diocles were contemporaries, with Diocles perhaps slightly the elder. Does the text of On burning mirrors show whether Diocles knew Apollonius's work? Here the evidence is somewhat contradictory. The whole work is written in a language used for conics which predated Apollonius's contributions except for one theorem, Proposition 8, where Apollonius's terminology is used. Of course some later copyist could have inserted this later terminology into Proposition 8 but, as Toomer writes, that someone should make the change to just this one proposition and leave the old terminology everywhere else is unbelievable. We are left with the possible explanation, offered by Toomer in [2], that Diocles learnt of Apollonius's work (perhaps before its publication) and made the change in his nearly completed work. This, however, leaves the problem of why Diocles only made this change in one place. A number of different solutions have been proposed but this is leading us away from the question of dating which we are discussing in this article.

When we asked how the dates of Apollonius and Diocles fitted we were thinking of those of Apollonius as being known while those of Diocles being less well known. We shall consider later in this article how we have learnt about Apollonius but before going on to this discussion it is worth thinking about how certain we are that our information about Diocles is now correct. Are we sure that the dates given for Diocles above are at least close to being correct? Unfortunately the answer must be no. We cannot be certain. Although the argument is based on the best knowledge available at present, we certainly cannot rule out the possibility that further evidence may be discovered some time in the future which could point to a different answer. It is tempting to think that this could not happen given the "facts" which are now known. However, to give just one possible source for doubt, Toomer might have incorrectly identified Zenodorus for he writes [2]:-

Although the Arabic text is slightly corrupt at both places where this person's name is mentioned, that is the only plausible way to read the name.

The other point worth mentioning is that there must be other mathematicians whose dates have been worked out using as strong a logical argument as Heath used for Diocles, but nevertheless they too are incorrect.

Let us now turn to other ways to gain information about the ancient Greek mathematicians. Another useful piece of information which can often be found in mathematical works is a dedication to a patron who had supported the mathematician. This dedication can provide dating information if the patron and his dates are known. It can also provide other information since it will almost certainly allow the place where the mathematician carried out the work to be identified. Other works are written in the form of a letter sent to a colleague and may contain information concerning why the work was written. As an example let us examine the problem mentioned above of finding out about Apollonius. For this his famous work *Conics* is an extremely valuable source, in fact it is almost our only source.

The first book of *Conics* was sent by Apollonius to Eudemus of Pergamum. Let us give a little background about the city and state of Pergamum. It was a Greek city some 25 km from the Aegean Sea; today the town of Bergama, Izmir, Turkey stands on the site. The ruler Eumenes I had declared Pergamum an independent state in 263 BC and ruled it until he was succeeded by his nephew Attalus I (269 BC - 197 BC) in 241 BC. After defeating the Galatians in 230 BC, Attalus declared himself King of Pergamum. There followed a period of political treaties (particularly with Rome), and of battles against the neighbouring states with frequent wars.

Book I of the *Conics* of Apollonius begins:-

Apollonius to Eudemus, greetings.

If you are in good health and things are in other respects as you wish, it is well; with me too things are moderately well. During the time I spent with you at Pergamum I observed your eagerness to become acquainted with my work in conics; I am therefore sending you the first book, which I have corrected, and I will forward the remaining books when I have finished them to my satisfaction. I dare say you have not forgotten my telling you that I undertook the investigation of this subject at the request of Naucrates the geometer, at the time when he came to Alexandria and stayed with me, and, when I had worked it out in eight books, I gave them to him at once, too hurriedly, because he was on the point of sailing...

This is an extremely useful introduction. It not only tells us that Apollonius was living in Alexandria but, of course, that he visited Pergamum. It also gives a feeling for the way that the mathematicians of this time travelled around, visiting each other and stimulating each other with questions and ideas.

The second book of the *Conics* was also sent by Apollonius to Eudemus of Pergamum:-

Apollonius to Eudemus, greetings.

I have sent you my son Apollonius bringing you the second book of the conics as arranged by us. Go through it then carefully and acquaint those with it worthy of sharing in such things. Philonides the geometer, who I introduced to you in Ephesus, if ever he comes to Pergamum, acquaint him with it too. Take care of yourself, and be well.

Now we learn that, by the time he sent this book, Apollonius was old enough to have a son (also called Apollonius) who could travel to Pergamum to deliver Book II. Also note that Philonides, Eudemus of Pergamum and Apollonius had all met at Ephesus, again an indication of the way that these mathematicians travelled around.

Although Book III of the Conics has no preface in the form it has reached us, it seems from the Preface to Book IV, however, that it almost certainly had one at the time it was sent to Pergamum. Again Book III was sent to Eudemus of Pergamum but, before Apollonius had completed Book IV, Eudemus had died. Apollonius therefore sent Book IV to Attalus I. As we mentioned above he was the ruler of Pergamum from 241 to 197 BC. This now provides us with a firm point on which to base an estimate of the years through which Apollonius lived.

Apollonius to Attalus, greetings.

Some time ago I expounded and sent to Eudemus of Pergamum the first three books of my conics which I have complied in eight books, but as he has passed away, I have resolved to dedicate the remaining books to you because of your desire to possess my works.

Now in addition to having a reputation for military and diplomatic skill, Attalus also had a reputation as a generous patron of the arts. It is likely that indeed he had given Apollonius more support that just the desire to possess his works.

Of course references may be found in other literature concerning the mathematician in whose life we are interested. There are difficulties with this information - are we sure that it refers to the right person (for example Euclid was a very common name); taking various pieces of such information together may lead to inconsistencies which need to be resolved; and at ever stage we need to make an evaluation of the likely accuracy of the information. We must be careful here not to make the mistake of assuming that if the same information is given by a number of sources then it must be correct. The sources may have all taken the information from the same place.

To give an example of references which might help put events into perspective we gave an example still related to the events described above. Relevant information was found in a papyrus discovered at Herculaneum. When Vesuvius erupted in 79 AD Herculaneum, together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 m deep which preserved the city until excavations began in the 18^{th} century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular papyri which give us important information. One papyrus states [1]:-

Philonides was a pupil, first of Eudemus, and afterwards of Dionysodorus, the son of Dionysodorus the Caunian.

Now Eudemus referred to in this quote is Eudemus of Pergamum and Philonides is the geometer who Apollonius met at Ephesus. So we have one more small piece of the jigsaw, the knowledge that Philonides was a pupil of Eudemus of Pergamum.

**References (2 books/articles)**

**Other Web sites:**

Astroseti (A Spanish translation of this article)

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

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