# Egyptian and Babylonian Mathematics

Cumulatively, the history of a discipline can provide depth and dimension to an otherwise flat and uninspiring history of civilization. Political, economic, artistic, and scientific achievements give new dimensions to history, and us a greater appreciation for otherwise dry facts and their relative importance. In mathematics, the achievements by the ancient Egyptians and Babylonians remained a well-kept secret until fairly recently. This was part of a larger problem. Mankind had forgotten how to read what it itself had written in the past using hieroglyphics and cuneiform. This loss was a frustration, to put it mildly. A yoke of mystery endured through the Dark Ages and the Renaissance, our own history stared us in the face, undecipherable and begging to be studied. By the 19th and 20th centuries, a few scholars, through strokes of genius, hard work, and information gathered by others, unlocked the riddle of how to read the ancient writing. This breakthrough paved the way for others to embark on the arduous task of reading the large volume of writings that survived, and to this day, the task is far from complete. Still, much has been done, and the story of how we recovered our forgotten past, and some of the mathematics that has been restored to man's historical repertoire as a result of this effort, are worth sharing.

It is necessary first to distinguish the philological breakthroughs that made it possible to decipher cuneiform and hieroglyphics in general, from specific decipherment and interpretation of texts in particular. For example, the deciphering of cuneiform began with the efforts of Georg Friedrich Grotefend and reached its culmination through the efforts of Henry Creswicke Rawlinson (Burton, 20-21), yet, these men were not expressly attempting to translate, say, Plimpton 322, a tablet now known to contain mathematical content. Rather, these men were searching for a general system of decipherment, one which could pave the way for other scholars to focus on actually deciphering the thousands and thousands of tablets that have come down to us. As Otto Neugebauer, the man who deciphered Plimpton 322, put it,

The task of excavating the source material in museums is of much greater urgency than the accumulation of new uncounted thousands of texts on top of the never investigated previous thousands. ...the result of every such dig [actual excavation] is frequently many more tablets than can be handled by one scholar in his lifetime. (Neugebauer, 62)

With this distinction in mind, it will be necessary to take different approaches to our analysis of historical Egyptian and Babylonian mathematics. In the case of Egyptian mathematics, we will focus on the methods used in deciphering hieroglyphics in general, since these methods, evidently, are our primary concern in accepting the validity of translations of particular ancient mathematical texts such as the Rhind Papyrus. In the case of Babylonian mathematics, we will focus on the methods used by Neuegebauer in interpreting particular Babylonian mathematical texts, merely accepting the deciphering principles he used for cuneiform that were pioneered by Rawlinson and others. The reason for a difference in focus lies in the nature of cuneiform syntax and grammar, which necessitates, once a literal translation has been achieved, a greater emphasis on interpretation (at least in the case of mathematical texts) than Egyptian does. As David Burton put it,

Because the Babylonian mode of writing on clay tablets discouraged the compilation of long treatises, there is nothing among the Babylonian records comparable with the Rhind Papyrus. (Burton, 60)

Furthermore, the results of Neugebauer’s interpretative method suffice in some degree in confirming the validity of the underlying system of decipherment, since those results, in all their complexity, are unlikely to be achieved if the underlying system is invalid. As Neugebauer put it,

It is clear that the determination of the meaning of a text is generally the easier the more complicated the mathematical context is because this leaves fewer possibilities for the interpretation of the procedure. (Neuegebauer, 66)With all of these factors in mind, let’s begin our analysis of Babylon and proceed afterwards to Egypt.

By many accounts it seems that Otto Neugebauer was the chief scholar to decipher Babylonian cuneiform texts containing mathematical content, and he is, evidently, the main one responsible for most of what we know today of Babylonian mathematics (Burton, 60, Gillings, 1). In his book The Exact Sciences in Antiquity (63-64), Neuegebauer gives an excellent example of the technique he used to interpret the mathematical texts, and it is this example we will examine.

For the demonstration, Neuegebauer deciphers and interprets a tablet named "YBC 4712 Rev". The tablet is divided into square sections, and by context it is assumed that each square is to be treated individually and consists of math problems without solutions. He begins with the square containing lines 12-17, transliterating the cuneiform using English letters. This yields many words that can then be translated immediately. Among these are "a-rá" ("multiply"), "gar" ("add"), "uš" ("length"), "sag" ("width"), and "-ma" ("equals"). The lines in the section end up reading as follows (note that "," parses lines, "[ ]" indicates a destroyed section of the tablet, "x" and "y" are used for "length" and "width", "+" and "*" are the usual addition and multiplication operators): \[ ++x\text{[= ]}5,\ *2,\ +x=1,\ +y=35,\ *2,\ +( )=50. \] With this done, it is clear that the problem is too short to make sense. To remedy, Neuegbauer assumes a new unknown, $f,$ and tries the interpretation: \[ f+x\text{[= ]}5,\ 2f,\ 2f+x=1,\ f+y=35,\ 2f,\ 2f+(y)=50 \] Plugging y into the parentheses in the last equation is plausible since that would make it a continuation of the one before it and would also avoid introducing a fourth unknown. Under this assumption, he finds $f=15$ and $y=20$ from the last two equations. To test these results, Neuegebauer uses $f=15$ in the first two equations involving $x.$ Since the second equation $2f+x=1$ is impossible if $f+x$ is at least 5 by the first equation, Neuegebauer assumes that 1 in the second equation is in the 60s place, i.e. $1,0;=60.$ Thus, using $2f+x=60$ where $f=15,$ he obtains $x=30,$ and using these values in f+x he gets $f+x=45$ which is in perfect agreement with $f+x\text{[= ]}5.$ Looking for confirmation elsewhere in the tablet, he finds a condition given at the beginning that $x*y=10,0=600,$ and as he already knows that $x=30$ and $y=20,$ the decipherment is complete.

According to Neugebauer, this was the general method used by which the math tablets of Babylon were deciphered, and it certainly seems acceptable. As he put it, “What I have described here is, of course, a simplified story of what actually happened when texts of this type became known, but the essential steps were precisely the same.” (Neuegebauer, 66)

Now that we have discussed in some brief generality how our knowledge of Babylonian mathematics came about, let’s turn to Ancient Egypt. Many books exist giving the general story of how the secret of hieroglyphics was unlocked, and of the special accomplishments of Jean François Champollion (1790-1832) which led to the penetration of the hieroglyphic mystery (Burton 33-35). Still, certain details are worth elucidating to give greater confidence in the validity of the methods. In particular, it is necessary to discuss the discoveries of the Englishman Thomas Young (1773-1830), who pioneered the decoding of hieroglyphics, and was the first to publish the meaning of those hieroglyphics that corresponded to the numerals (Budge, 208-209).

The main key to unlocking the mystery was the Rosetta Stone. It consists of a text that is duplicated in hieroglyphics, demotic script and Greek, and as David Burton points out, “The way to read Greek had never been lost; the way to read hieroglyphics and demotic had never been found.” (Burton, 33) To understand how this stone served the purpose, it is necessary to realize certain facts concerning the Ancient Egyptian language. First of these is that Ancient Egyptian was generally written in four different scripts. The first of these, of course, was hieroglyphics, the second was hieratic, the third was demotic, and the fourth was Coptic. All of these forms represented the same general language of Ancient Egyptian, (disregarding such necessary distinctions as “Middle” or “Late” Egyptian, for varieties that occurred over a period of three or four thousand years) (Gillings, 3). Many texts describe the first three of these written forms of Ancient Egyptian, but the fourth, Coptic, is not so often mentioned. As we shall see, because the way to read Coptic had never been lost either, it provided an important piece to the puzzle of deciphering the other Egyptian scripts. With the exception of some variations in syntax and words borrowed from Greek, Coptic is essentially just Ancient Egyptian written with 24 Greek letters, plus seven additional letters derived from demotic, which represent sounds for which the Greek alphabet contains no equivalents (Budge, 247). Contrary to the common belief that the Rosetta Stone is a trilingual transcription of one text (Burton, 33), it is actually a bilingual transcription, because the demotic and hieroglyphic portions are merely written forms of the same language, i.e. Egyptian (Budge, 40). Only the Greek portion represents a different language. (N.b. the Greek portion is actually Greek, i.e. not to be confused with Coptic). Young’s method was essentially as follows:

He first began by grouping oft-repeated words in the Greek script with what he assumed to be their counterparts (by reason of their position and repetition) in the demotic script, and in this manner accumulated a vocabulary of some 86 Greek-demotic words (Gardiner, 10). At this stage he had no phonetic values assigned to individual demotic characters, but only the semantic values for strings of demotic characters that he believed corresponded to certain Greek words. This done, the next task was to find correspondences between the demotic words he had semantically identified and hieroglyphics. To assist him, he assumed, as others before him had done, that the sets of hieroglyphs contained within the oval “cartouches” were royal names. As one of the words in his accumulated Greek-demotic vocabulary was Ptolemy, he assumed the name in the likely corresponding cartouches of the hieroglyphic portion of the Rosetta Stone to be that of Ptolemy, with variances between cartouches owing to differing honorifics. He then proceeded to assume phonetic values for individual hieroglyphs within the cartouches corresponding to the phonetics of “Ptolemy”. Using his knowledge of Coptic and his assumed phonetic values for certain hieroglyphs, he determined the phonetic (or semantic, if the former wasn’t possible) value of the hieroglyphs comprising Ptolemy’s honorifics (Budge, 210). For instance, “everliving” and “beloved” were among the honorifics given Ptolemy in the Greek script. Thus, he translated these into Coptic to find their Egyptian sounds, then, using the hieroglyphs for which he had assumed phonetic values from Ptolemy’s name and which occurred in the honorific, deduced the phonetic (or semantic) values of the remaining unknown hieroglyphs (Budge, 210). Assumptions about phonetic values of hieroglyphs would get discharged once they were found to agree in places where the same hieroglyphs occurred elsewhere but needed to have the assumed phonetic values. With this general method established by Young, Champollion was then able to proceed and decipher several more names, thereby establishing a considerably larger alphabet of hieroglyphs, and ultimately deriving a complete grammar and translation of several texts including the Rosetta Stone (Budge, 221-225).

With a valid method finally achieved for the decipherment of hieroglyphs came the possibility for the translation of the Rhind Papyrus and other principal texts like the Moscow Papyrus that have yielded us our present knowledge of Egyptian mathematics. One such translation is of the Rhind Papyrus by Arnold Buffum Chace (1845-1932), published under the title The Rhind Mathematical Papyrus, and there are others. These translations generally include both literal and interpretative translations, and, given the methods by which they were achieved as described above, seem both valid and revealing.

In conclusion, it is clear that for their contributions in unlocking ancient history in general and mathematics in particular, those scholars mentioned merit all the honors we can confer upon them. Their contributions are real and not likely to be exhausted for many centuries to come.

## Works Cited

Budge, Ernest Alfred Thomas Wallis, Sir. The Rosetta Stone in The British Museum. New York: AMS Press Inc. 1976

Burton, David M. The History of Mathematics, An Introduction. USA: McGraw-Hill. 1997.

Chase, Arnold Buffum. The Rhind Mathematical Papyrus. USA: Mathematical Association of America. 1979.

Gardiner, Alan, Sir. Egyptian Grammar. 3rd ed. London: Oxford University Press. 1969.

Gillings, Richard J. Mathematics in the Time of the Pharoahs. USA: MIT Press

Neuegebauer, Otto. The Exact Sciences in Antiquity. Denmark: Brown University Press. 1957