How a Scribe Might Have Done It

The guesses for the formulas for triangular and pyramidal numbers used rational fractions. This might have been a little advanced for the scribes of the Pyramid Age who only used unit fractions. The guesses can be rewritten using the four arithmetical operations addition, subtraction, multiplication and whole number division. The notation is a common one used for unit fractions. The Egyptian method for writing numbers was a little more cumbersome that decimal notation. There was a special glyph that represented 2/3rds. The scribes would have had the skills needed to do calculations for following problem. And the need to know the number of blocks required to build a pyramid would have suggested it.

Another Derivation of Triangular and Pyramidal Numbers

The methods previously indicated are not the only way that one could arrive at the formula for the triangular and pyramidal numbers. If one guesses that the triangular numbers are proportional to n then one finds, on dividing the triangular numbers by the corresponding value of n, that the result is linear as seen below and the unknown factor is easily guessed at.

Similarly, one might guess that the pyramidal numbers are proportional to the triangular numbers and divide a pyramidal number by its corresponding triangular number. One again gets a linear sequence of numbers and the factor can be easily determined.

So it is not always obvious how a particular formula was arrived at. The formulas found in the ancient papyri were probably intended for use by scribes functioning as clerks and probably do not comprise a mathematical treatise. The ultimate source of the procedures may have been lost with the passage of time and what we now have may have been copied and recopied over thousands of years.

The procedures given in the mathematical papyri are in the most general form. One finds both the method for finding the area of a truncated triangle (with the upper portion cut off parallel to the bottom) and the volume of a truncated pyramid.

For more information see,

The Rhind Mathematical Papyrus by Gay Robins and Charles Shute

Mathematics and Measurement by O. A. W. Dilke

Mathematics in the Time of the Pharaohs by Richard J. Gillings

Pyramidal Numbers

One can ask how many identical blocks would be needed to create a pyramid. Suppose there is one block at the very top, four block blocks beneath it, nine blocks in the third row down, etc. To get the total number one needs to know the sum of a series of square numbers. If one considers a series of sums, a particular sum is equal to the prior sum plus k².

The sum of n 1s is n. The sum of 1 through n is n(n+1)/2. So it would seem likely that the sum we are looking for is some cubic expression and by substituting k-1 for k we get the prior sum and the difference.
We can equate the coefficients of k on both sides of the second equals sign to obtain equations for the coefficients.
Note that only unit fractions are needed for the sum. Finally we deduce the formula for the total number of blocks in a pyramid of n rows.
If we assume that the blocks are square in shape with width w and height h the volume of the pyramid is

V = n(n+1)(2n+1)w²h/6.

If we ignore the additive constants and set W = nw and H = nh we get the usual formula for the volume of a pyramid,

V = W²H/3.

A method for computing the volume of a truncated pyramid if found in the Moscow Mathematical Papyrus.

It is not necessary that all the scribes knew this rudimentary level of geometry and algebra. It would take only one master of sufficient skill to deduce the methods of calculation. The builder of the first Step Pyramid was known as Imhotep. He was immortalized by the ancient Egyptians.

It could be that the pyramids contained the mathematical knowledge of the ancient Egyptians. It is not unlikely that they would set this knowledge down in stone. At least the pyramids provided an opportunity to do so.

Sum of 1, 2, 3, ... , n

The sum of a linear series is fairly easy to deduce. As seen below one writes down the series from 1 to n on the first row. The second row is the same series in reverse order. Adding each column separately one finds each sum is n+1. So twice the sum is n(n+1). Therefore, the sum is n(n+1)/2 as indicated in the last blog.

The Area of a Triangle

The ancient Egyptians knew how to compute the area of a triangle. There is a simple way that they might have arrived at a method for computing the area. Consider a stack of blocks which decreases by one block for each higher row as in the figure below. As can be seen the total number of blocks in n(n+1)/2. One then has to subtract the number of blocks outside the triangle which is n/2. So the total number of blocks is n²/2. If the width of a block is w and its height is h then the width of the triangle is W = n·w and its height is H = n·h. So the area of the triangle is A = ½·H·W.

(click on the image above to enlarge it)

Is it likely that the ancient Egyptians knew of the formula for the total number of blocks, n(n+1)/2? Well, consider doubling an odd unit fraction 1/n by finding the difference between 2/n and 2/(n+1). The difference is 2/[n(n+1)] so

2/n = 2/(n+1) + 2/[n(n+1)]

and, since n is odd, n+1 is even and divisible by 2. The second term is just the inverse of n(n+1)/2. It is likely that the ancient Egyptians could easily compute the total number of blocks and therefore determine the area of a triangle.

This sequence of "triangular numbers" was known to Pythagoras who studied under the Egyptian priests in Memphis, Egypt. And so it is likely that the sum of a linear series was known much earlier in time.

The Slope of the Sides of a Pyramid

The ancient Egyptians knew how to computed the slope of the sides of a pyramid which we know from the Rhind Mathematical Papyrus and other ancient scrolls. The value they used was known as the seked which was the base of a right triangle whose height was a unit length. It is similar to an Egyptian fraction which were the sums of unit fractions. The fractions were just a series of divisors.

The procedure for computing the seked was to take half the base of the pyramid and divide by the height. This makes computing the slope of the Great Pyramid of Giza especially simple since the ratio of the base to height was 11/7. The seked is just 11/2 or 5½ (palms) since the unit length, the cubit, is 7 palms.

The pyramid designers needed to keep track of the slope of the pyramids since they were working near the limits of the construction material. The Meidum Pyramid may have collapsed due to internal stresses which can result in cracks within the structure. A sand dune similarly has a critical slope. It is believed that the Great Pyramid was reentered for inspection after it had been sealed because there is a tunnel from beneath the pyramid to the Grand Gallery that was concealed afterwards.

It is doubtful that 11/7 had anything to with π but may have just been a convenient slope. Still the scribes were devoted to Thoth and may have shared his attitude on secrecy. Their education consisted of figuring things out for themselves. Problems in the scrolls were worked out in detail for the reader but he had to determine the general method. The problems just illustrated the procedure involved.

The Royal Cubit & Foot of the Fourth Dynasty

A significant event in the history of Ancient Egypt was the unification of Upper and Lower Egypt. This was accomplished by focussing on similarities rather than differences. The subsequent events show what happens if two people work with each other instead of against each other. The central administration of the Old Kingdom resulted in a new class of scribes and a general advance in society over previous times.

But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.

Ignoring the sequence after 1,1,1,2 tells us that the ratio is nearly equal to 8/5 while ignoring that after 1,1,1,2,1 suggests a ratio of 11/7. Comparing these multiples of the royal foot and cubit with the dimensions of the pyramid gives the following information,

which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,

1 royal foot = 299.35 mm

1 royal cubit = 523.86 mm

which is very close to the values for the cubit stick from the 18th Dynasty.

An interesting coincidence is that,

11/7 = 1.5714

π/2 = 1.5708

π/2 is the arc of a circle corresponding to 90°.

Empowerment

There is an ancient Egyptian fable about the Book of Thoth which illustrates the sense of empowerment. It is from the Ptolemaic period when the Greeks ruled Egypt and so is several periods removed from the time of Pyramid building. It is not known how old the story is but was written on papyrus in Demotic and found in the grave of a Coptic monk at Thebes. Thoth is a god of wisdom and is attributed as the source of writing and the arts and sciences. What is interesting about the story is that Thoth goes to Ra to be empowered to seek vengeance for the theft of his book. The moral seems to be that even kings are subject to the will of the gods.

Another aspect of the story is the secrecy associated with the Book of Thoth. It was hidden away and protected by a series of defenses. But no matter how well something is protected the defenses can always be defeated.

Thoth appears in the story to have a dark side. While Ra, the Sun god, is the source of truth and light Thoth, the Moon, is more obscure. To obtain vengeance on the thief he first gets power over him by taking him and his family to the land of the dead. Thoth is jealous of his power and it is not easily taken.

Pharaoh

The title of Pharaoh was not solely a personal honorific. Literally it means "great house" so the power of the Pharaoh extended to those who served him. It is somewhat akin to the "Crown" in England and the "Presidency" in the US. He was the Egyptian Chief of State. Servants were referred to as "arms". Their power came from the Pharaoh whose power ultimately came from the gods.

An Emblem of Authority

The cubit, foot and other units of length can be viewed as a tool to extend the Pharaoh's power throughout Egypt. The sacred foot was an emblem of power as the ideographs below indicate. There is an ideograph similar to that of the tcheser (foot) in which the hand is holding what appears to be a scepter. The royal cubit may therefore be a symbol of excellence.

One of the duties of the scribes was to distribute goods. When precision was required the royal measures were beyond question.

Tcheseru

The royal cubit was considered sacred and used to determine the dimensions of temples. I found the hieroglyphics for a tcheser length in Wallis Budge's Egyptian Hieroglyphic Dictionary which confirms its special status. The name also suggests a connection with the Eye of Horus fractions mentioned earlier. It could be that tcheser conveys the meanings hallowed, exalted and royal. There is a sense of truth and justice associated with it also. Copies were made of the royal cubit for the use of workers and compared with the original monthly. The penalty for tampering with the royal cubit was death. For more information see Thoth: Architect of the Universe by Ralph Ellis.

The Royal Cubit in the 18th Dynasty

On 18th Dynasty Egyptian cubits the 20th finger is labeled a remen. The 16th finger is indicated as being a "t'eser" and the hieroglyphic symbol on the 15th and 16th fingers is similar to the remen but overlaps the border between them. The glyph appears to be that of a forearm with the hand grasping a rod which can be seen more clearly on the cubit at the Museo Egizio in Turin, Italy. There is a picture of this cubit stick in Mathematics and Measurement by O. A. W. Dilke on p. 23.

t'eser (foot) indicated as being 16 fingers in length

Mathematics in the Time of the Pharoahs by R. Gillings p.220

If one does an actual measurement one does get a value very close to a foot. If you want to check this measurement I would recommend an assistant since balancing the tape measure while taking a picture is rather difficult.

measuring the foot

Plausible Deniability

So without evidence to the contrary, i.e., no evidence of ETs or temporal agents being involved, we have plausible deniability on the ancient foot or remen being based on the nanosecond. There is indeed a natural explanation for it. But why so much misdirection in calling it a foot? Was it ancient propaganda? The conspiracy theorist might be harder to convince. Ockham's razor says that a simple explanation is to be preferred over a more elaborate one if they both fit the available evidence.

The Remen?

Just measured the distance from the lower tendon of the biceps of my right arm to the notch between my thumb and the palm of my hand and it turned out to be 300 mm. Could the remen have been the distance from the upper arm to where one grasps something? One could probably check out that measurement on the statues of the period and compare them with values for the remen. The remen glyph shows part of the upper arm and the notch between the thumb and hand. Calling this distance a foot is a misnomer.

Cubits and Palms

The hand's breadth, a 4 fingered palm, may be more fundamental than the Egyptian bd (foot). The cubit was most likely a means of counting palms. The royal cubit was 7 palms but there was a more common cubit of 6 palms. The counting scheme seems to be based on the powers of two idea. 4 fingers = 1 palm. 16 fingers = 4 palms = 1 bd (foot). But the cubit doesn't fit in with this scheme. Neither 6 or 7 palms is a power of 2. There was a cubit rod found a Memphis which had the 7 palm royal cubit on one side and the 6 palm common cubit on the other. So the simplest explanation of the cubit is that it was a measuring stick which allowed the user to measure lengths without counting them one palm at a time. Lengths could just be read off in terms of palms and fingers.

The fingers of the royal cubit were marked as follows: 1finger, 2 fingers, 3 fingers, 4 fingered palm, five fingered palm, fist, and so on up to 8 fingers. The glyph for the remen is placed above the 15th finger on the scale. This appears to be what Wikipedia calls the bw (foot) which was 15 or 16 fingers. The term remen was also used to represent half the diagonal of a cubit square which was about 20 fingers. And a remen square would have a diagonal about equal to a cubit. The remen marked on the cubit seems to be associated with the forearm and not a foot. Perhaps this is an older usage that of the diagonal of a square came later.


It you look at the side of a royal cubit you will see that there are fingers which are subdivided into parts varying from 2 to 16. The Egyptian scribes needed all these divisions since they used unit fractions. But the scale shows that with 16 subdivisions they were capable of millimeter accuracy. One can compare the royal cubit with the modern engineer's scale or architect's scale.

The Egyptian Cubit

I have decided to continue my "researches" here where postings are less restricted.

We can compare various feet which have been used and the nsec,

1 pes (Roman foot) = 296 mm
1 pous (Ionian foot) = 296 mm
1 English foot = 305 mm
1 Egyptian bd = 300 mm

1 nsec = 300 mm

The Egyptian bd is 4 "hand breadths" of 4 fingers and there are 7 hand breadths to a cubit so the bd was 16 fingers and the cubit was 28 fingers. If the bd was the primary unit the cubit would be 7/4 of it. Is there any significance to this?

7 = 1 + 2 + 4 → 7/4 = 1 + 1/2 + 1/4

These are Eye of Horus fractions and the representation of integers as a sum of powers of 2 was used for multiplication.

The Egyptians made the conversion 2/7 → 1/4 +1/28 so 4/7 = 1/2 + 1/14 could also be easily represented and calculations involving 4/7 would not pose any difficulty.

Eye of Horus fractions would have been of greater significance to the Egyptians. 7/4 was more fundamental.

Update on Ancient Units of Measurement vs the Nanosecond

Lately, as some already know, I have been researching ancient units of measurement and their approximation of the nanosecond (nsec) as a measure of length. Some of the ancient "feet" are a little too close and we can speculate on whether or not they knew about the light foot. We will most likely end up with a reductio ad absurdum or an incredible mystery.

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An Impression

Impression by Monet

This painting is eponymous for the Impressionist movement.