Point of Beginning

Web Exclusive: Cubits and Cords

December 1, 2008


The ancient Egyptians had an advanced civilization, and the practice of surveying played an integral part. With simple tools, they managed to create some of the most spectacular monuments mankind has ever produced. But to understand the surveying methods that made the creation of pyramids even possible, it is first necessary to have an understanding of the ways in which the Egyptians manipulated numbers. The basic mathematical processes that relate to surveying include their basic numbering system, simple geometry and their land measurement system.

The knowledge we have today of Egyptian mathematics was obtained from a small number of papyri that were written as textbooks for the scribes, who were the wealthy and educated bureaucrats. In establishing the extent of mathematical knowledge of the Egyptians, two papyri that specifically focus on mathematical problems, the Rhind Mathematical Papyrus and the Moscow Papyrus, are particularly useful.

Figure 1: The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (Figure 1), was found at Thebes and dates to approximately 1650 B.C. The papyrus consists of a collection of mathematical problems from arithmetic, geometry and the measurement of area and length. The problems relate to everyday situations, not theoretical mathematics but rather only the practical-for example, the exchanging of loaves of bread or reckoning the herd of cattle. The second papyrus, the Moscow Papyrus, dates to around 1850 B.C. It contains 25 mathematical problems, similar to the Rhind Papyrus, and also agrees with the rules of the Rhind Papyrus. For example, one problem determines the volume of a truncated pyramid.

Figure 2: Units of Measurement

Measurement

Length

The Egyptians were observant people. They recognized the averaged consistency and proportionality of the parts of the body and, in turn, derived their units of measurement from them. The basic unit was the cubit, which is equivalent to 45 centimeters. This unit was then divided into six palms, each of which could be divided into four thumb widths.

The length of measurement used by surveyors in ancient Egypt was the royal cubit, which was employed for building purposes. The royal cubit consisted of seven palms, rather than six, and was described as the distance between a man’s elbow and the tip of his extended middle finger. It was equivalent to 52.4 centimeters. One hundred royal cubits was called a khet, a common length to be used when calculating and defining parcels of land.

The devices used by surveyors to measure length were all based on the royal cubit. The measuring rods were one royal cubit in length, and the measuring ropes were 100 royal cubits in length. For longer distances, the river measure was used, presumably to describe the distance traveled up the Nile in a day. This measure, the iteru (meaning river) or skhoinos in Greek, was equal to 20,000 cubits, which is around 10.5 kilometers (see Figure 2 for a summary table).

Area

The measure of land area in the Old Kingdom (2780-2100 B.C.) of ancient Egypt was the setjat. In New Kingdom sources, however, the same-sized area is referred to as the aroura, the Greek word. Due to the vast difference in surviving evidence from the New Kingdom as compared with the Old Kingdom, it has come about that the word aroura is far more widely used and associated with the measurement of land in ancient Egypt. The aroura was equal to a square of 1 khet by 1 khet, or 10,000 cubits squared. Like our acre, it was solely for measuring land. The aroura was approximately equal to 0.25 hectare.

Eight problems from the Rhind Mathematical Papyrus deal with area, and six of these refer directly to land or field measurement. One such example is problem number 51 in Figure 3:

Figure 3: Problem number 51 from the Rhind Mathematical Papyrus

Problem number 51:
Example of a triangle of land. Suppose it is said to thee, What is the area of a triangle of side 10 khet and of base 4 khet?

Do it thus: 
1         400
½        200
1        1000
2        2000
Its area is 20 setjat.
Take ½ of 4, in order to get its rectangle. Multiply 10 times 2; this is its area.


This uses the ½ bh rule for the area of a triangle taking half of the base (b=4 khet, or 400 cubits) to multiply by the height (h=10 khet, or 1,000 cubits). The answer is 200,000 cubits squared, or 20 setjat. This is one of the simpler problems. It illustrates the use of the multiplication, which will be explained later. Others explore removing an amount of land from a given number of land parcels such that it equals a given area. All the problems illustrate the use of simple formula, as we would use today, to calculate the areas of simple geometrical shapes.

Basic Arithmetic

Integers

The Egyptians had a series of integers beginning at 1 and increasing. For this series, they had an understanding of the four mathematical operations of addition, subtraction, multiplication and division. Addition and subtraction were straightforward and are carried out as they are today. Multiplication was done through a process of either doubling or multiplying by 10. For example, in problem 79 of the Rhind mathematical papyrus, we have 2,801 multiplied by 7. This would be written:

1/        2801
2/        5602
4/        11204

In each line, the previous line has been doubled. In order to find the answer to 2801 by 7, we can equally say that 2801 by 1, plus 2801 by 2, plus 2801 by 4 is equal to 2801 by 7. By using the first column and selecting the appropriate rows from the list of doubled numbers–in this case all three rows–which add to give the multiplier, we then can add the second column to give the answer. In this case, the second column adds to give 19,607.

Division was similar to multiplication in that the divisor was multiplied until the dividend was found. So, if we were to use the same numbers as above, rather than saying 19,607 divided into 2,801, the Egyptian would look at the question as: How many times must 2,801 be added to give 19,607? The working would therefore be as the multiplication.

Fractions

Apart from integers, the Egyptians also had a series of descending numbers. With the exception of 2/3, all were unit fractions with a numerator of 1. They would therefore not express 5 ÷ 7 as 5/7. Instead, multiplication was used such that 7 as multiplied by the unit fractions needed to obtain 5.

           1/7 × 7 = 1
           (1/4 + 1/28) × 7 = 2
           (1/2 + 1/14) × 7 = 4

Each line is double the previous one. The 1/7 had to be split into ¼ and 1/28 in order to keep to the unit fractions. This conversion was done with the use of tables, which show numerators of 2 and denominators as odd numbers. These are then broken into the equivalent unit fractions. This can be found, for instance, on the Rhind Mathematical Papyrus. In order to solve the problem given above, the final step is to choose those unit fractions that add to give 5. Clearly, the first and third lines do this, and the result is 1/2 + 1/7 + 1/14.

Geometry

Herodotus, a historian who visited Egypt around 450 B.C., attributed the beginnings of geometry in Egypt to the need for understanding and managing the flooding of the Nile. The need for a geometrical proof of land ownership once floodwaters subsided came about because the taxes were paid according to the area and yield of the land. Knowledge of geometrical shapes, such as rectangles, triangles, circles, and their properties are seen by the geometrical problems given in the papyri.

Circles and a value for pi

The area of a circle was given by the rule A = (8/9 d)2 with d denoting the diameter. This was probably because it was found that a cylinder of 9 units diameter filled with water would almost exactly fill a square prism of the same height but whose side was eight units.  Indirectly, this value or 256/81 is used as an approximation of pi in three problems from the Rhind Mathematical Papyrus. The same value is also used on the Moscow Papyrus. The approximation of pi is about 3.1605, which is slightly higher than the value accepted today, 3.1416.

Triangles

In the Middle Kingdom, there is evidence of the use of the formula for the triangle area as we have today. For example, problem number 17 in the Moscow Mathematical Papyrus deals with a given piece of land in the shape of a triangle, and using a set ratio of the base to the height, the values are calculated. The working of the problem is far more complex than we might undertake today. However, it illustrates that the methods used by the Egyptians were empirically derived. Despite what seems like a complex manipulation of numbers, it is in fact the opposite, and each method uses similar and simple calculations.

Some have claimed that the Egyptians knew the Pythagorean theorem, involving the ratio of sides of a right-angle triangle. Despite a similarity in some of their triangle calculations and the theorem we use today, there was no evidence that the Egyptians had knowledge of, for example, the 3,4,5 triangle.



Figure 4: Quadrilateral

Quadrilaterals

The Egyptians calculated rectangles from the length and breadth as we do today, and examples of calculating areas of land or the dimensions of land given a certain area can be found in the mathematical papyri. The formula for quadrilaterals comes much later with an inscription at the temple of Edfu, which dates to around 100 B.C., with a formula that gives erroneous results except in the case of rectangles (see Figure 4):

½ (a+c) × ½ (b+d)

Figure 5: Diagram showing the shadows created by the gnomon and bay to define direction.

Volume Calculation

Furthering their knowledge of areas, the Egyptians were able to calculate the volume of a number of 3D shapes, including such solids as cylinders, parallelepipeds and pyramids. Some historians have stated that there is no evidence to say that the Egyptians knew how to calculate the volume of a pyramid. However, they concede that considering the importance of the pyramid to the Egyptians, it is likely that they did. Questions do remain, however, as to the exact method and formula that the Egyptians used for such a calculation.

Surveying Instruments of Ancient Egypt

The instruments used by the Egyptian surveyors were very primitive compared to those used today. For leveling, a simple level with a plumb bob was used. An instrument comprised of a plumb line and sighter, the Merkhet, was used for positioning and alignment. The measuring cord was used for measuring lengths. To identify true north, a gnomon, or staff, was used to measure shadow lengths and obtain a meridian. Irrigation and building works required a leveling device, and the plumb line was required for erecting vertical walls and orientation.

Gnomon

One of the simplest devices used by surveyors was the gnomon, which was a pole or staff placed vertically on the ground in order to cast a shadow (Figure 5). Essentially, the gnomon gave the Egyptians the ability to mark out the path of the sun thereby ascertaining the position of true north by establishing the meridian.

Figure 6a- Drawing of Amenhophis II holding a Gnomon with graduations

Close to the equator, the high angle of the sun results in a short shadow. At the summer solstice in particular, the shadow can quite easily be obscured by the hand holding the pole. The Egyptians overcame this problem through the use of the bent gnomon, which meant that the shadow could be extended without extending the length of the gnomon to an inconvenient height.

The gnomon would also sometimes have graduations on it (Figure 5). Once the shadow length was marked on the ground, the gnomon could be placed flat in order for the shadow length to be measured.

Figure 6- A variety of ancient surveying instruments. This photo shows a level–the A frame with the plumb bob in the center. The other instruments are a square used for right angles and a plumb device, which was used to make the gnomon perpendicular. They are from the tomb of Sennutem at Thebes dating from the 20th dynasty. The mallet was not found with these instruments.

Plumb Bob and Level

For the purposes of leveling, the Egyptians had a simple plumb bob and level, which was used to determine the horizontal plane and determine the height difference between two points. The leveling instrument was an A-framed instrument that could be calibrated in order to show the true level surface, even if the legs of the instrument themselves were uneven. The plumb bob alone was also used to hold the gnomon vertical so that it could be used for accurate sun observations. An example of such a plumb bob is shown in Figure 6.

Figure 7: Scene from the Tomb of Menna showing surveying of the land for the calculation of taxes.

Measuring Cord

The measuring cord or rope used by the Egyptians has a striking similarity to the surveyor’s Gunter’s Chain (Figure 7). Rather than 100 links, the rope was divided into 100 cubits which were marked by the use of knots. In a striking comparison, just as surveyors assistants today are referred to as chainmen, the assistants in ancient Egypt were called rope stretchers, or harpedonaptae in Greek.

The cord was of great significance to the Egyptians as signified by the “stretching of the cord ceremony.” The ceremony was a ritual performed at the laying of the foundations, and its purpose was to accurately extend a direction.

Figure 8: Merkert: the plumb line with the sighter (sometimes called a bay), which has the small “V” cut in the top.

Merkhet

The Merkhet was a plumb line and sighter used for the purposes of alignment (Figure 8). The sighter was often the central rib of a palm leaf. It was aligned by looking through the slit in the rib of the leaf. Sometimes called a bay, it is also possible that the leaf was subsequently replaced by a wooden stick with a small “V” cut out of it. Due to differing opinions about its use, however, it is possible that the wooden stick (bay) and leaf versions were used for different purposes such as in conjunction with the gnomon to determine direction. The inscription on the merkhet records that it was used as an “indicator for determining the commencement of a festival and for placing all men in their hours.”

Leveling tool

The Egyptians made use of a leveling device that was based on the principal of gravity. It was a square level in the shape of an A with a plumb line hanging from the top (Figure 7). A level surface was achieved when the plumb line was in line with a mark that was on the center of the cross bar. In the event that the legs of the level were not exactly equal, calibration of the instrument could take place. The level was placed on a surface, and the position of the plumb line marked on the crossbar. Then the instrument was reversed and placed in the same position, and the plumb line position marked again. If the surface is level, the plumb line will fall in exactly the same position on the crossbar. If there was any variation in the points on the cross bar, then the surface was not level, and the difference between the two points was marked to be the center point. Then the process could be repeated, using the center point to define the level.

Clearly, there are many similarities in the practices of ancient Egyptian surveyors and surveyors of today. The duties of a surveyor-marking boundaries and setting out of buildings-are remarkably unchanged. Obviously, the difference in the instruments used is significant, yet despite the advanced technology available today, the results produced by ancient Egyptian surveyors are impressive.


Editor’s Note: Look for more of Chapman’s insights into the practice of surveying in ancient Egypt in our March Web exclusive.