About the Fayum Tablets

In the middle of the past century in Egypt, the collectors found ancient copper tablets on which archaic Greek letters were engraved. Tablets of the New York collector Martin Schoyen measured 215 × 135 mm and 212 × 137 mm, and had a thickness of approximately 1.3 mm (Fig. 1). The third tablet of approximately the same size is housed in the Martin-von-Wagner-Museum of the Julius-Maximilians-Universitat Würzburg, Germany. It is not excluded that more tablets may exist.

Fig. 1. Copper tablets from Fayum, an ancient Egyptian city, now a remote suburb of Cairo, collection of Martin Schoyen (New York).

In some places on the tablets, letters are visible, and the rest of the space is covered with patina. Patina is a product of copper interaction with carbon dioxide of the air, the composition of patina is similar to that of malachite. As it turned out, the tablets were made in the late ninth century BC, and during the past time patina has covered a significant part of the text. An American scientist Roger Woodard headed the work of studying these tablets. In his book [1], he tells a literally detective story about how he has managed to learn about these tablets. Metallographic analysis showed that all the three tablets – both from New York and from Würzburg – are made from one piece of copper.

Since the tablets were the property of collectors, it was impossible to clean the tablet’s patina in order to read the inscriptions. They found the way out using X-ray radiography, but here there were problems, too, because the letters on both sides were simultaneously seen. R. Woodard managed to solve this problem and succeeded in separating letters written on one side from those written on the other. He made trace-drawings of letters engraved on one side according to the radiographs of plates (these trace-drawings are published in the file of the computer application to the book 9781107028111anno_p1-6.pdf). Reading these sketches was made difficult by the background created by the X-rays pictures themselves. Since, according to R. Woodard himself, he did not have any trace-drawings separated from the background of the radiographs, the author together with an engineer Irina Gelberger performed computer processing of trace-drawings (using the AutoCad 2013 program) to remove the X-ray background. One of the results of this treatment is shown in Fig. 2.

Fig. 2. The supposed look of side 1 of a copper tablet from the museum collection in Würzburg at the end of the ninth century BC (trace-drawing on X-ray radiograph by R. Woodard, computer processing by M. Tsayger and I. Gelberger).

The most interesting thing was discovered when a complicated and very expensive work on the study of tablets by modern physical methods was completed. It has turned out that the tables depicted on these tablets contain a repeating alphabet. R.Woodard very clearly compares archaic Greek letters with the well-known Phoenician letters of that time, their amount being twenty-two, just as in the Phoenician (and later – in Hebrew) alphabet . Another peculiarity – in other, later Greek inscriptions, writing and reading are directed from left to right, and on these tables the letters are written in the opposite direction, from right to left. And the text is somewhat incomprehensible: the letters go in the alphabetical order, the alphabet is over and begins again. And this repeats many times, and on all the plates, and nothing else. Nevertheless, we can express appreciation to R.Woodard for a detailed analysis of archaic Greek letters.

Looking from the positions of ancient Greek arithmetic, the purpose of these tablets becomes obvious. In my book [2], which was published in Israel in 2015² I put forward the idea that the Phoenician alphabet was invented by an unknown Phoenician sage specifically for brief writing of numbers, which is very important for trade.

Before the creator of the Phoenician writing, there were examples of recording numbers using seven signs (the hieroglyphic system of the Egyptians), but this recording was very cumbersome due to the fact that many characters had to be repeated several times. The creator of the writing left the numerical system (units, tens, hundreds, thousands) the same (this is not surprising, since the nature created a man with five fingers), but he decided to add special signs to designate each representative in the numerical category – nine signs for units, for tens and for hundreds. It was a magnificent invention, similar to a talented move of a chess player who wins the game sacrificing the queen. The inventor of the alphabet used twenty-seven-sign alphabet for writing numbers, but the record of the number became surprisingly compact. Any number in the range from 1 to 999 became possible to record with three signs only, and in some cases even shorter. And to avoid confusion in the use of letters, it was necessary to introduce a strict sequence of letters one after another.

And what does a shorter number record mean? A number of two or three signs recorded on some trading amphora immediately tells the owner how many measures of goods to be sold it contains. It is very convenient for trade. Thanks to the convenience of short recording of numbers, the Phoenician alphabet began to spread rapidly over all the countries of the Mediterranean basin. Later, probably starting from the tenth or ninth century BC, the alphabet started being used to record sounds, and Aramaic, Hebrew, Arabic, Greek and other alphabets appeared with the aim of recording sounds and words. But these alphabets retained a strict order of letters and could also be used to write numbers. In Hebrew, this p,,rinciple is used in gematria.

The system of recording numbers invented by an unknown Phoenician wise allowed performing simple arithmetic operations, such as addition and subtraction, with such numbers. This could be done using a special table, which was first plotted on the ground or on a board with fine sand (abacus). Later, the Greeks learned to make reusable tables in the form of an ivory plate on which a layer of wax (Marsiliana tablet) was applied.

The peculiarity of this table was that in the header of the table the letters of the alphabet in REVERSE ORDER were written. It is with such recording that the received numerical word read in the accepted direction of writing (for the Greeks – from left to right) contains a correct order of components adopted in Babylon, starting from a larger component and further – in decreasing order (for example, hundreds, tens, units). Only with such order of numerical components according to the Babylonian rule, these components were added up. If a smaller component was followed by a larger component, the number to be written was considered as a result of SUBTRACTION of the smaller component from the larger one. We can see this on the example of Roman numbers, where the record IX (one-ten) is read as the number nine (10-1), and the record XI (ten-one) is read as the number eleven (10 + 1).

Addition of numbers recorded in the Phoenician system was performed according to digit places (units with units, tens with tens, etc.), and the technique and features of this operation are described in my book [2]. It turned out possible to subtract a smaller number from a larger, as well as to perform reduced multiplication of one number by another (so-called Greek multiplication), but we will not dwell on these details here.

When mastering the techniques of ancient Greek arithmetic, much attention is paid to memorizing the ancient Greek alphabet. Without a solid knowledge of the alphabet, one cannot calculate correctly, and in trade it is the most important thing. If a merchant made a mistake not in his favor, he loses as much as his mistake is. And if he made a mistake in his favor, he also loses, because people are not fools, they quickly notice mistakes of a trader, and he gets a bad reputation. That is, it is unprofitable for a trader to make mistakes in his favor, too. In order not to make any mistakes, an ancient merchant had to know very well his alphabet of that time. The Fayum tables were used by ancient teachers of arithmetic to check the knowledge of students, future merchants.

Teachers of arithmetic forced students to memorize the reverse abecedarium of that time. Apparently, in those days the abacus as a special reusable device had not yet been invented, and the calculator each time re-painted the abacus on the ground or on a board with sand. And he had to know (remember) unmistakably the reverse alphabetic sequence of letters (reverse abecedarium). And in the system of notions of that time, the reverse alphabetic sequence was accompanied by writing letters in the “reverse pose”. This is what we see on the drawing of the letters of copper plates (Figure 2), when the letters gamma (Γ) or kappa (Κ) (and all other letters) look to the left, and not to the right.

To check the knowledge of students, the compiler of tables engraved on copper plates specially introduced errors into the alphabetic sequence of letters. Often, one or two letters of a similar shape located in different places of the alphabetical sequence, were interchanged. This is clearly seen in the example of the W-1 plate, whose text is shown in Fig. 2. The interpretation of the letters of this text performed by R. Woodard is shown in Fig. 3. For the ease of reading, the archaic ancient Greek letters in the table are given in the inscription adopted later, at the end of the fifth century BC and preserved to this day. Places of violation of the alphabetical order of letters are shown by a dark background. In parentheses, the omission of a letter is shown (in line 14, a dot in parentheses shows the omission of four letters). In line 18, the letters lambda and mu must exchange places, etc. At the same time, a large number of rows do not have errors in the alphabetical order.

Fig. 3. Interpretation of the letters shown in Fig. 2 by R.Woodard. Letters are given in a later form. Places of violation of the alphabetical order are marked by a dashed background.

All this indirectly indicates that the Fayum tablets were used as a teaching material for the students to learn the ancient Greek alphabetic arithmetic. For example, a teacher could ask a student if everything was right on one or another row of a table, and if the student did not detect the originally introduced violations of the alphabetic sequence, this was an indicator of his knowledge. Tables could be used for a long time by ancient teachers (possibly, by several generations). Therefore, they were made on a strong copper material. And this explanation of the purpose of the Fayum tables serves as an indirect confirmation of the fact that the Phoenician alphabet was used for arithmetic calculations.

On the other hand, the Fayum tablets shed new light on the emergence and change of the alphabetic sequence of ancient Greek letters. Figures 5a and 5b show a comparative table of Phoenician and ancient Greek alphabets.

The letters of the ancient Greek alphabets changed their form at different times. What the archaic Greek letters of the Fayum alphabet looked like is shown in Fig. 2. In those days there was still a practice of changing the letters’ poses at the direction of writing from left to right and from right to left (in Fig. 2 all letters have poses directed from right to left). Letters of the Marsiliana tablet made a century later are shown in Fig. 4 (they are also shown in poses from right to left). In 403 BC, an Athenian archon Euclid legislatively approved Ionian alphabet prepared by the rhetorician Archinch as the state one. This alphabet spread throughout Greece, and became widely used. In this alphabet there were no different directions of the letter and all the letters are shown in poses from left to right.

Fig. 4. Inscription on the upper board of the Marsiliana tablet.

Since for our comparison, it is not the shape of the letters that is important, but the order of their sequencing one after another, in the table in Fig. 5 (a, b) all Greek alphabets are shown by letters of the Athenian alphabet of 403 BC. This makes it easier to compare the positions occupied by different letters and the numerical values assigned to the letters depending on the position occupied.

Fig. 5a. Comparison of the Phoenician and Greek alphabets (beginning).

Fig. 5b. Comparison of the Phoenician and Greek alphabets (e,,nding).

We can see that the archaic Fayum alphabet (late 9th century BC) fully corresponds to the Phoenician alphabet. Four letters: Υ (ipsilon), Χ (chi), Φ (phi) and Ψ (psi) – were added to the Greek alphabet of the Marsiliana tablet (end of the 8th century BC) in comparison with the Fayum alphabet. In the Greek alphabet of the Athenian decree of 403 BC. the letter Ϻ (san) disappeared, and as a result the letters behind it moved forward by one position and changed their numerical value: the letter Ϙ (koppa) began to denote 90, not 100, as in the Fayum and Marsiliana abcedaria, the letter Φ (fi) moved forward by two (!) positions and began to denote 500, and the letter Χ (chi) remained in its previous position and still denoted 600. In the Athenian alphabet, the letters Ω (omega) with the value of 800 and the letter Ϡ (sampi) with the value of 900 were added. The use of Greek letters as numeric signs had a centuries-old practice, and it was almost impossible to change the position of letters in the alphabet on a whim of one or another ruler because of the accepted trade and arithmetic traditions, and the fact that the positions of letters with high numerical values had undergone changes, said that these values were apparently rarely used at that time in trade practice. It is possible that these changes were of a political nature. It must be mentioned that the Athenian Ionic alphabet had existed as a state one without further changes for almost two thousand years until the fall of Constantinople in 1453.

Thus, information about the Fayum tablets becomes understandable from the standpoint of considering the Phoenician alphabet as a numerical system. It is possible that among the Mediterranean peoples it was customary to use the Phoenician alphabet solely for arithmetic purposes, which prompted these peoples to create their local alphabets to record words and concepts, while retaining the ability of the alphabet to write numbers. And for this it was required to establish a strict order of letters following one another.

¹ Mark A. Tsayger, PhD, Beer-Sheva, Israel. Email m_tsayger@hotmail.com

² The book [2] is available at the National Library of Israel in Jerusalem and in the Russian State Library in Moscow (formerly Lenin’s Library). In bookselling networks the book did not appear (Russian bookselling networks, as a rule, do not accept publications sold for private funds).

Bibliography

1. Roger D. Woodard, The Textualization of the Greek Alphabet, Cambridge, 2014
2. М. А. Цайгер, Основы древнегреческой арифметики, Израиль, 2015 (M. A. Tsayger, The Fundamentals of the Ancient Greek Arithmetics. Israel, 2015, (Russian))

Mark Tsayger