COUNT LIKE AN EGYPTIAN
A HANDS-ON INTRODUCTION TO ANCIENT MATHEMATICS
By DAVID REIMERPRINCETON UNIVERSITY PRESS
Copyright © 2014 Princeton University Press
All rights reserved.
ISBN: 978-0-691-16012-2
Contents
PREFACE, VII,
INTRODUCTION, IX,
COMPUTATION TABLES, XI,
1 NUMBERS, 1,
2 FRACTIONS, 13,
3 OPERATIONS, 22,
4 SIMPLIFICATION, 55,
5 TECHNIQUES AND STRATEGIES, 80,
6 MISCELLANY, 121,
7 BASE-BASED MATHEMATICS, 144,
8 JUDGMENT DAY, 182,
PRACTICE SOLUTIONS, 209,
INDEX, 235,
CHAPTER 1NUMBERS
THE WORDS OF THE GODS
Hieroglyphic Numbers
In the primal waters at the dawn of time, the Egyptian god Ptah brought himself into being. This bearded god had skin blue as the night sky, and he carried a scepter whose form combined the Egyptian symbols of stability, dominion, and life. In his heart, Ptah conceived of the world, and his tongue turned his thoughts into words. At the sound of his voice, the universe changed. The amorphous eight gods of the Ogdoad, including the primeval waters, darkness, chaos, and the invisible power, came together. There they formed the primeval mound, the first piece of the earth. The act drained the power of the Ogdoad and the mound became their tomb, but their sacrifice created the birthplace of the sun, the father of the Egyptian pantheon.
This mound was the center of the earth, which the Egyptians believed resided right in the middle of their nation. The Egyptians called the central part of the world the Mansion of the Life Force of Ptah, which the ancient Greeks translated as Aigyptos, the origin of our word "Egypt."
The magic of Ptah's words created the world, and words in ancient Egypt had real power. This was especially true for hieroglyphics, which the Egyptians called the words of the gods. These artistic writings, along with other magical diagrams, cover the walls of their tombs and temples. But hieroglyphs are more than mere writing. When Egyptians wanted just to write, they used the hieratic script, a simplified form of hieroglyphs. They used the hieroglyphs only when their words needed a small portion of the same power that Ptah had used to create the world. They used the magic of words to protect themselves from the evil that was in the spirit world.
Such spells usually took the form of either monologues or stories. In monologues, Egyptians spoke directly to the gods, and they would plead with a god for his or her assistance. However, the words contained so much power that these spells contained threats directed at the gods. The magic in the monologues' words was apparently strong enough to prevent divine retribution for their harsh words. Similarly, words infused stories, the second form of a magical spell, with divine power. Hence telling a tale of a god healing another god had the power to heal.
The diagrams that accompanied the hieroglyphs were also magic. Spells granted them the ability to come to life to serve the dead or protect the living. One such spell, the opening of the mouth, allowed the spirits both of the dead and the divine to enter or leave a mummy, statue, or drawing. Ptah's name literally translates into the words "the opener," interpreted precisely in this sense. Ptah, in fact, was the patron god of the craftsmen who built and decorated the tombs and temples of ancient Egypt.
These craftsmen had to create the images according to precise specifications because of their mystical nature. Important objects needed to have more magic and hence needed to be drawn bigger. They also had to be drawn with attention to mathematical proportion so they wouldn't come to life misshapen and malformed. These "magical blueprints" required that all the parts were carefully detailed. Hence the figures took on odd poses to clearly depict each essential body part. Many of the poses also possessed symbolic value and in turn conveyed different occult powers. Egyptians were quite capable of accurately drawing figures in natural postures, but these images were not art but, rather, detailed specifications for their afterlife.
Words had so much power that they were often dangerous, even to their users. The bad parts of a magical story could harm someone as easily as the good parts could help. So when a tale included an evil event such as a murder, it often skipped these parts or made a vague reference to such events. Even the symbols used to make up words presented some danger. Imagine the frustration you'd feel if your soul woke up shortly after your funeral only to be chased around your tomb by the spirit of a venomous snake. This would have happened because some craftsman didn't take the proper precautions when writing a word containing the j sound, whose symbol takes the form of a cobra. A better-trained craftsman would have drawn the snake sliced up or impaled for the safety of the deceased.
There is no mathematics written in hieroglyphs, but numbers are used for the occasional date or quantity. They use a straight vertical line, A, to represent the number one. This is no surprise since virtually every culture uses a similar symbol to represent 1 just as we do. This practice is tens of thousands of years old and far predates writing, which is a mere five thousand years old. It seems to have been started by hunter-gatherers who used notched bones or sticks to record quantities. While it's easy to cut a straight line with a knife across a piece of wood, a curved shape, like our 2, would be needlessly difficult. So, when a denizen of the ice age needed to remember the number 5, he or she would make five straight cuts into a stick. The Egyptians carried on this practice in their writing. Hence, the Egyptian 3 appears as AAA, just like three notches on a bone.
Unlike their contemporaries, such as the Mesopotamians, the Egyptians didn't group their 1s in specific patterns. For example, 4 could be written in one line as [??], or in two rows of two as [??]. This is consistent with their other hieroglyphs, since the layout was concerned more with the aesthetic look of the word than with a systematic layout. For example, the word "day" could be written [??]. These three symbols represented a hut, a mouth, and a quail chick and made the sounds of h, r, and w respectively. Because the first two symbols were short compared to the picture of the quail, it was often written as below, filling up the space on the temple wall more uniformly.
The numbers 1 and 3 had special use in Egyptian hieroglyphs. As we've seen, the symbol [??] can represent the sound made by the letter w, but it could also represent an actual quail. In order to help the reader distinguish between the two, the Egyptians wrote a symbol identical to the number 1 below the drawing when they wished to identify the object and not the sound. Similarly they could pluralize the object by writing the number 3 below it. For example, the following depicts both the singular and plural of fish.
The system of writing numbers as a bunch of 1s has a serious flaw. Look at the number [??]. It's far from obvious that this is the number 21. Too many lines blur together making them difficult to count. The Egyptians, like most ancient cultures, used symbols to represent groups larger than 1. For example, they used [??], a picture of a cattle hobble, to represent the number 10. Using the [??] and the A symbols, they represented numbers up to 99. For example, the number 21 could be written [??].
For larger numbers they used the symbols [??],[??],[??], and [??] to represent 100; 1,000; 10,000; and 100,000 respectively. The pictures represent a coil of rope, a lotus flower, a finger, and a tadpole. So we can write the number 251,342 as follows:
[ILLUSTRATION OMITTED]
The pictures used for the numbers may give clues to how the words were pronounced. Words in ancient Egypt were usually spelled without vowels. If we used a similar system, we could use the symbol, [??], to represent the word "bell." However, it could also be used to write "ball" and "bull." The symbol for cattle hobble, [??], is composed of three consonants: m, j and w. So the Egyptian word for 10 could sound something like "mojaw," "mijow," and so on. We need to remember that Egyptian is a dead language and no one is really sure how any of the symbols sounded. Egyptologists have made intelligent guesses based on their knowledge of the ancient language Coptic, which evolved from Egyptian. But this problem is compounded when we realize that even if we knew all the written sounds, we still don't know what vowels go in between. So when we see the coiled rope symbol, [??], we believe the consonants are s, h, and t and can only guess the vowel. While most of us can think of a few interesting words using these letters, each adjacent pair has an unknown vowel sound between them suggesting 100 is pronounced something like "sehet" or "sahot."
The number for a million is depicted by a man holding his hands in the air, [??]. Egyptians used this word to represent extremely large numbers in exactly the same way we do when we say we've done something "a million times." It's difficult to say what the pose means. Some have suggested it's the arms of man outstretched, overwhelmed when confronted with the concept of infinity. It's also reminiscent of the pose the air god Shu takes while holding up the sky, restraining her from embracing her lover, the earth, and crushing all things between heaven and earth. The symbol, [??], also stands for each of the Heh gods. These are the spirits of the Ogdoad, who died to form the earth and coincidentally help Shu hold up the sky.
The number 1 million was used repeatedly in the Egyptian mythos. Perhaps the most important example is the barque of millions. A barque was a boat a god used to sail across the sky, which according to the Egyptians, was made of water. The barque of millions was the sun god Ra, which was navigated by the god Thoth and his wife Ma'at across the sky each day. The "millions" refers to all the souls who had achieved salvation and manned the barque as it descended into the nether world each night. Together with Seth, the god of strength and violence, they defended the boat on its journey to the dawn, when the sun god would be reborn anew in order to shine another day.
THOTH, SCRIBES, AND BUREAUCRACY
Hieratic Numbers and Addition
On the day an ancient Egyptian was born, Thoth, the god of scribes and wisdom, would change into his ibis form. He needed this form so he could fly from his barque, the moon, down to the earth, where he would carry out the will of the gods. Unseen by human eyes, Thoth would find one of the bricks of the house in which the baby was born and write down the day the child was fated to die. When that day would finally arrive, the soul of the mortal would once again encounter Thoth in the Hall of Osiris. Here, in the land of the setting sun, which formed the barrier between heaven and hell, the soul would be given final judgment. Regardless of the outcome, Thoth would dutifully record the result.
In order to fully appreciate the importance of Thoth and the scribal class in Egypt, we need to understand the central role of bureaucracy in Egypt. Contemporary movies about the ancient world seem to invariably include scenes of a vast marketplace where the characters are offered a wide array of treats and forbidden goods. This imagery is based on a modern misconception. The economies of ancient civilizations were, by and large, controlled by a central government. The state provided everything its ruling class thought the citizens needed, and the former took what they considered to be their share. When not working for the government, individuals would occasionally exchange goods and services with each other for a few items the government wouldn't provide. Relatively speaking, it was a small part of the economy, which otherwise was dominated by Egypt's ruling class.
The pharaoh, governors, and high priests who controlled the government needed an army of bureaucrats to manage the economy. The scribes of Egypt performed this function, documenting every aspect of ancient life. Just like Thoth, they were there from a person's birth to their death, recording all. Scribes were on the farms, in the storehouses, and on the factory floors. They were even on the battlefields, recording the details of the fight and tabulating the casualties by counting thousands of hands severed from the dead.
The constant need to keep records on every aspect of Egyptian life was a huge drain on the time of the scribes. Hieroglyphics consist of detailed pictures that take a long time to properly write. Apparently, the ancient scribes didn't have the time or patience to make their records with hieroglyphics, so they invented hieratic. This is essentially a cursive form of hieroglyphics, but it is different enough to be considered its own script.
Some of the hieratic symbols are recognizable from their hieroglyphic roots. For example, the hieratic number 2 evolved from the two straight lines of the hieroglyphics. When an ancient scribe painted the first line, he wouldn't lift his brush all the way before painting the second line, making the motion a little faster. This would have the effect of connecting the two vertical lines near the bottom.
Our number 2 was created in much the same way long ago in India. The only significant difference is that they started with two horizontal lines. The curve of the 2 is nothing more than the backstroke to begin the second line. The number 3 evolved in much the same way.
As the figures grew more complicated, the Egyptian scribes reduced parts of the hieroglyphic symbols to simple strokes. Consider the hieroglyphic 7. Normally it could be written in four vertical strokes on top of three more. The impatient scribe would paint all four as one horizontal brush stroke and zigzag back for the next row. He could not make a stroke for the second row because it would be unclear how many ones it contained. So he would jiggle his brush representing two and follow it with a slash down representing the third one. While the figure hardly looks like the original seven lines, all that really matters is that the scribes recognized this symbol and were able to easily distinguish it from similar figures.
The ancient scribes of Egypt, like any other accountants, eventually needed to find the total of the values of the objects they inventoried. This is perhaps the most difficult subject for me to write about. I can easily explain how they calculated the volume of an unfinished pyramid, multiplied mixed numbers, and created fractional identities, but I can't be sure how they added 15 and 12. Both of the ancient math scrolls that have been found regard addition as being too simple to detail. Hence the work of the solutions is not shown. Only the answers are given. With no written record, we can do little but guess.
On the surface, the problem doesn't seem that bad. Consider the following addition of 82 and 54. The Egyptians would have written the numbers in hieratic, but I'm using hieroglyphics just because they're more recognizable. The above sum is easily added in one's head. We can first combine the ones, [??], adding 2 and 4 to get 6. Then we can compute the tens, [??], adding 8 and 5 to get 13. We can interpret this as ten 10s and three 10s. Since ten 10s is 100, or D, the answer is [??] ??] ??].
We can't automatically assume that the ancient Egyptians used this method. Although there are hints in ancient texts, there is no direct evidence as to how they added. We also don't see their work when they added lists of fractions often having different denominators. I can't imagine doing such problems in my head. If we know they didn't do all additions in their head, how can we be sure they did any in their head?
We've fallen into the trap of mathematical familiarity. Subconsciously we think to ourselves, "They must do their math the way we do, because our way is the right way." Of course it's foolish to think that anything we do is the so-called right way. It's often simply one choice of many.
In Mesopotamia at that time, there is some evidence that the people used tokens to perform calculations. In order to explain this in modern terms imagine that you have 82 cents in one hand and 54 in the other. In the first hand you have 8 dimes and 2 pennies. Note the relationship between the 8 dimes and the eight [??] in the scribe's number. Similarly in the second hand you have 5 dimes and 4 pennies. All you need to do to add these numbers is to pour the coins from one hand into the other. So you now have 13 dimes and 6 pennies in one hand. You now decide you have too many coins, so you replace 10 dimes with a one-dollar bill. The addition is now complete. You now write [??] representing the dollar, [??] representing the three dimes, and [??] the six pennies. Note that we never actually added any digits. We simply pushed the piles together and made a currency exchange.
I'm not advocating that the Egyptians used tokens. You could rightly argue that there is no physical evidence of tokens being used in ancient Egypt. I could counter by saying that there is no need for physical tokens. They might have done their mathematics on a dust board. This is roughly the equivalent of doing math on a dirty car window. They could have "placed tokens" by making marks with their fingers and "picked up tokens" by smearing dirt over the marks they wished to erase. I'm simply pointing out that there are other ways, beside the modern methods, to solve problems. In the absence of evidence, speculation is fine, but we have to understand it for what it is. Ignorance has never stopped me in the past, so let's add some Egyptian numbers.
(Continues...)Excerpted from COUNT LIKE AN EGYPTIAN by DAVID REIMER. Copyright © 2014 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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