# How do we know about Greek mathematics?

There are two separate articles in this archive: "How do we know about Greek mathematics?" and How do we know about Greek mathematicians?. There is a common belief that the question posed in this article, about Greek mathematics rather than Greek mathematicians, is easy to answer. Perhaps all we need to do to answer it is to read the mathematical treatises which the Greek mathematicians wrote. We might think, very naively, that although some of the origainal texts have been lost there should be plenty left for us to be able to gain an excellent picture of Greek mathematics.

The truth, however, is not nearly so simple and we will illustrate the way that Greek mathematical texts have come down to us by looking first at perhaps the most famous example, namely Euclid's Elements. When we read Heath's The Thirteen Books of Euclid's Elements are we reading an English translation of the words which Euclid wrote in 300 BC? In order to answer this question we need to examine the way the Elements has reached us, and, more generally, how the writings of the ancient Greek mathematicians have been preserved.

Rather surprisingly, from the earlier era of Babylonian mathematics original texts survive. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. The Greeks, however, began to use papyrus rolls on which to write their works.

Papyrus comes from a grass-like plant grown in the Nile delta region in Egypt which had been used as a writing material as far back as 3000 BC. It was not used by the Greeks, however, until around 450 BC for earlier they had only an oral tradition of passing knowledge on through their students. As written records developed, they also used wooden writing boards and wax tablets for work which was not intended to be permanent. Sometimes writing from this period has survived on ostraca which are inscribed pottery fragments. One assumes that the first copy of the Elements would have been written on a papyrus roll, which, if it were typical of such rolls, would have been about 10 metres long. These rolls were rather fragile and easily torn, so they tended to become damaged if much used. Even if left untouched they rotted fairly quickly except under particularly dry climatic conditions such as exist in Egypt. The only way that such works could be preserved was by having new copies made fairly frequently and, since this was clearly a major undertaking, it would only be done for texts which were considered of major importance.

It is easy to see, therefore, why no complete Greek mathematics text older than Euclid's Elements has survived. The Elements was considered such a fine piece of work that it made the older mathematical texts obsolete, and nobody was going to continue to copy these older texts onto new rolls of papyrus just to preserve them for historical purposes. The Elements was continually copied, but there are two distinct problems which occur when works such as this are copied. Firstly they might have been copied by someone who had no technical knowledge of the material being copied. In this case many errors would be made in the copying process (although as we shall see below this can be used by historians to their advantage). On the other hand the copy might have been made by someone of considerable learning who knew of later developments in the topics being described and so might have added material which was not in the original text.

From 300 BC until the codex form of book was developed, the Elements must have been copied many times. The codex consisted of flat sheets of material, folded and stitched to produce something much more recognisable as a book. Early codices were made of papyrus but later developments replaced this by vellum. Codices began to appear around the 2nd century AD but they were not the main vehicle for works until the 4th century.

There were not only developments in the material on which the works were written but also in the script that was used in writing. On the original papyrus rolls the writing was all in capital letters with no spaces between the words. It required much material to write relatively little and it was also hard to read. Minuscule script, which developed around 800 AD, consisted of lower case letters and was much more compact and easy to read. A process of turning the old unspaced capital scripts into minuscule began and much of the mathematical writing which have survived have done so because they were copied into this new format.

We have now reached the oldest surviving complete copy of the Elements written in minuscule in 888 AD. Arethas, bishop of Caesarea Cappadociae (now in central Turkey), built up a library of religious and mathematical works and one of the eight works from this library to survive is the Elements copied by the scribe Stephanus for Arethas. The cost to Arethas was 14 gold pieces, about one fifth of what a scribe would expect to receive in a year. Let us call this manuscript of Euclid's Elements E888 to make reference to it easier. There are a number of points that we should now make:
i) The year 888 AD seems a long time ago but it is only 1100 years ago while it is 1200 years after the Elements was written. This oldest surviving text of the Elements was written closer to the present day than to the time when the original was written.

ii) Although E888 is the oldest surviving complete text of the Elements there are older fragments (see for example [8] and [9]). Six particularly old fragments (dating from around 225 BC) of what may be parts of the text were found on Elephantine Island in 1906. Experts argue whether these were written by someone studying the Elements or examining a book which Euclid incorporated into the Elements.

iii) Some surviving texts which were written later than E888 nevertheless are based on an earlier version of the Elements than E888.

iv) The manuscript E888, as is typical of such manuscripts, contains annotations which were made on the earlier copy which the scribe Stephanus used and copied by him onto E888. It also contains annotations made by later readers.

v) Most of the manuscripts of the Elements which have survived are based on a version with commentary and additions, produced by Theon of Alexandria (with perhaps the help of his daughter Hypatia) in the 4th century AD. E888 is indeed based on Theon's work.

vi) The first versions of the Elements to appear in Europe in the Middle Ages were not translations of any of any of these Greek texts into Latin. At this time no Greek texts of the Elements were known and the only versions of the Elements were those which had been translated into Arabic.

vii) It is worth recording that reason often given that no earlier copies of texts survived is because the Arabs burnt the library in Alexandria in 642 AD. It appears, however, that there is no truth in the story that the Arabs burnt this library, see for example [1].
In fact the first Arabic translation was made by al-Hajjaj early in the 9th century. Another translation by Hunayn was revised by Thabit ibn Qurra also in the 9th century. Gherard of Cremona translated the Thabit version into Latin in the 12th century. An earlier Latin translation from Arabic by Adelard of Bath around 1120 survives. These translations from Arabic are all of versions which trace back through the edition by Theon of Alexandria.

The relations between the different versions of a large number of Greek mathematical manuscripts was brilliantly worked out by the Danish scholar J L Heiberg towards the end of the 19th century. It would be impossible to do justice to the scholarly work that was involved in such a task, but we at least indicate the way that it is tackled. If we compare two manuscripts $A$ and $B$ say, and find that the errors present in $A$ are also present in $B$ but there are some errors in $B$ which do not appear in $A$, then it is reasonable to deduce that $B$ was copied from $A$ or that it was copied from a copy of $A$. If we find that $A$ and $B$ have common errors but also each has distinct errors of its own, then it is likely that both $A$ and $B$ were copied from $C$. If no manuscript has survived which fits the role of $C$, then $C$ can be reconstructed from $A$ and $B$ with some degree of certainty.

Using methods of this type Heiberg showed that all except one of the surviving manuscripts of the Elements derived from the edition by Theon of Alexandria. The one exception was based on an earlier version of the text than Theon's edition but this earlier version was itself later than the version on which Theon must have based his edition. Between 1883 and 1888, Heiberg published an edition of the Elements which was as close to the original as he was able to produce (see [5]). Heath's edition of 1908 ([4] is a later edition of this work) was based on Heiberg's edition and contains a description of the different manuscripts which have survived.

We have only given a brief indication of the way that the Elements have come down to us. We refer to [4] and [5] for a detailed description. Let us now turn to the works of perhaps the greatest of the Greek mathematicians, namely Archimedes.

William of Moerbeke (1215-1286) was archbishop of Corinth and a classical scholar whose Latin translations of Greek works played an important role in the transmission of Greek knowledge to medieval Europe. He had two Greek manuscripts of the works of Archimedes and he made his Latin translations from these manuscripts. The first of the two Greek manuscripts has not been seen since 1311 when presumably it was destroyed. The second manuscript survived longer and was certainly around until the 16th century after which it too vanished. In the years between the time when William of Moerbeke made his Latin translation and its disappearance this second manuscript was copied several times and some of these copies survive. Up until 1899 Heiberg had found no sources of Archimedes' works which were not based on the Latin translations by William of Moerbeke or on the copies of the second Greek manuscript which he used in his translation.

In 1899 an exceptionally important event occurred in our understanding of the works of Archimedes. An Archimedes palimpsest was listed in a catalogue of 890 works in the library of the Metochion of the Holy Sepulchre in Istanbul. In 1906 Heiberg was able to start examining the Archimedes palimpsest in Istanbul. What exactly was Heiberg examining? A palimpsest is a text which has been washed so that another text can be written on top. The underlying text, in this case works of Archimedes, is said to be "in palimpsest". The two main reasons to do this were either cost, it was cheaper to reuse an old parchment rather than purchase a new one, or often Greek texts were deliberately destroyed for it was considered by some Christians to be a holy act to destroy a pagan text and replace it by a Christian one.

The Archimedes palimpsest had been copied in the 10th century by a monk in a Greek Orthodox monastery Constantinople. Then in the 12th century the parchment had been washed and religious texts written on top. Originally the pages were about 30 cm by 20 cm but when they were reused the pages were folded in half to make a book 20 cm by 15 cm with 174 pages. Of course this involved writing the new texts at right angles to the Archimedes text and, since it was bound as a book, part of the Archimedes text was in the spine of the "new" 12th century book. To make Heiberg's task even harder, the pages of the Archimedes text had been used in an arbitrary order in making the new book. However, Heiberg had all the skills necessary to deal with these problems.

What did Heiberg find? The palimpsest contained four of Archimedes' works which were already known, but the versions on the palimpsest were independent of the two lost manuscripts used by William of Moerbeke in his Latin translations. This was an exciting find for scholars wanting to gain more insight into the original contents of Archimedes' work. Better still the palimpsest also contained a text of On floating bodies which up until that time was only known through Latin translations. Best of all however, was the fact that a work of Archimedes was found on the palimpsest for which no copy in any language was known prior to Heiberg studying the palimpsest. It was the extremely important Method of mechanical theorems which we describe in Archimedes' biography.

Heiberg published his reconstruction of the works of Archimedes found in the palimpsest while the palimpsest itself remained in the monastery in Istanbul. However before publication of Heiberg's new edition [6] of Archimedes' works incorporating these remarkable new discoveries was complete, the region was plunged into war along with the rest of Europe. During World War I, the allies planned to partition the Ottoman empire but Mustafa Kemal, later known as Atatürk, had different ideas. Atatürk faced local uprisings, official Ottoman forces opposed to him, and Greek armed forces. However, Turkey was declared a sovereign nation in January 1921 but, later that year, the Greek armies made major advances almost reaching Ankara. The survival of the library of the Metochion of the Holy Sepulchre in Istanbul could not be guaranteed amid the fighting, and head of the Greek Orthodox Church requested that the books from the library be sent to the National Library of Greece to ensure their safety. Of the 890 works in the library only 823 reached the National Library of Greece and the Archimedes palimpsest was not among them.

Exactly what happened to the Archimedes palimpsest is unclear. It was, it appears, in the hands of an unknown French collector from the 1920s although the palimpsest remained officially lost and most people assumed that it had been destroyed. The French collector may have sold it quite recently, but all we know for certain is that the palimpsest appeared at auction in Christie's in New York in 1998 sold on behalf of an anonymous seller. It was put on display with the spine broken open to reveal all the original text which had been in the spine when it had been examined by Heiberg. It was sold to an anonymous buyer for 2 million dollars on 29 October 1998 but the new owner has agreed to make it available for scholarly research.
The Archimedes palimpsest

Click on the picture above to see a bigger version.
A number of mysteries remain regarding the palimpsest in addition to who the present owner is:
Was the palimpsest sold or stolen in 1922?

Who owned the palimpsest during the years 1922 to 1998?

The palimpsest was seen to have a number of icons on it when displayed by Christie's in New York in 1998 but Heiberg had not mentioned any icons on the work. Were the icons added by one of its owners to try to increase its value?

The diagrams seen in the version of On floating bodies in the palimpsest are different from those in the translation by William of Moerbeke. More strangely they are different from those appearing in Heiberg version of On floating bodies in [6] which has the text from the palimpsest. Where did the diagrams as produced by Heiberg come from if not the palimpsest?

### References (show)

1. A J Butler, The Arab conquest of Egypt and the last thirty years of the Roman Dominion (Oxford, 1902 reprinted 1978).
2. D H Fowler, The mathematics of Plato's Academy : A new reconstruction (New York, 1990).
3. T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
4. T L Heath, The Thirteen Books of Euclid's Elements (3 Volumes) (New York, 1956).
5. J L Heiberg, H Menge and M Curtze (eds.), Euclid Opera Omnia (9 Vols.) (Leipzig, 1883-1916).
6. J L Heiberg (ed.), Archimedes Opera omnia cum commentariis Eutocii (Leipzig, 1910-15, reprinted 1972).
7. J Gray, Sale of the century?, The Mathematical Intelligencer 21 (3) (1999), 12-15.
8. J L Heiberg, Quelques papyrus traitant de mathématique, Proc. Danish Acad. Sci. 2 (1900), 147-171.
9. J L Heiberg, Paralipomena zu Euklid, Hermes 39 (1930), 46-74; 161-201; 321-356.