Arrested Development?

History sometimes gives the impression that we are viewing ancient Egypt through minimalist eyes. But what we do have may be that of an archivist who long ago tried to collect what remained from former times in order to preserve it. The Rhind Papyrus is not a single document but appears to have been an ancient text with other material attached to it. So it may be just remnants pasted together and not complete texts.

In ancient times the oral transmission of information was the primary mode of instruction. There was no publishing industry and if someone wanted a text of their own they would have to make a copy of some existing text. And limits of the length of a papyrus roll and time available may have resulted in some judicious editing. Cramming may have been practiced even then.

There may not have been a organized effort to preserve and pass on knowledge in early times. What was needed tended to be passed on. What was not most likely was forgotten. The mathematical texts that we do have come from the Hyksos period when foreigners ruled Egypt. And there appears to have been an effort to recover some of the past.

Egypt may have been the victim of its own success. Its relative stability and isolation resulted in little change over long periods. The rate of change may have been too slow. Time might have passed it by and the illiterate "barbarians" were the ones who were motivated to change and ultimately ended up in control.

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.