Egyptian Mathematics, Research Paper Example

Historical Introduction

The written use of mathematics dates back to approximately 3000 B.C. (Gillings, 1982).  Labels were inscribed on tombs and graves throughout Egypt carried what are now thought to be the first remnants of numeric representations.  Areas of the first documented measurements and use of mathematics were found in the civilization known as the Old Kingdom (2690-2180BC)(Gillings, 1982).  This measurement showed the guidelines of slope and distance.  Through the advancement of trade and market the Egyptians developed a foundation for fractions.  The Egyptians use of mathematics include the use of fractions and the corollary use of fraction tables for the use of the fractions of ½, 1/3, 2/3, and 3/4s.  Evidence of the use of mathematics for surveying land are present in historical literary documentation such as the Papyrus A Nastasi I and Papyrus Wilbour.  These are thought to have evolved from what initially started as land surveyors using knotted rope to define certain distanced for building homes and marking pastures. During the timeframe of ancient Egypt mathematics was used as a tool for construction, market and daily tasks in the same sense as a shovel, rope or oxen are used as tools.  Although the Egyptian mathematics were not well suited for calculations since straight forward multiplication and division were impossible so they adapted their system in such a way that addition and subtraction could be manipulated to essentially perform multiplication and division.

Improvements and Adaptations

Through the research of the mathematics in Ancient Egypt and how the pyramids were built there were other important discoveries and adaptations made through necessity.  Initially when numeric systems were needed and determined that it was needed for future reference it was written onto stone in the area in which it would be needed to be used.  Hieroglyphics were carved into the stone surrounding the buildings or burial sites.  Papyrus was used as a writing medium and the tip of a reed was used as a pen to write.  Papyrus is a form of paper that is formed from the papyrus plant which was quite abundant in the Nile Delta in Egypt.  Papyrus surpassed the writing medium of choice over previously used scrolls made of animal hide.

The downside of papyrus is that of its fragility and lack of structural integrity over the course of time especially if it is exposed to any sort of sub-optimal environmental elements.  The development of hieratic writing and numerals developed as a result of the development of a faster and more efficient way to record.  With the advent of a writing medium and ease of use of writing, early mathematicians and builders could use the tools to calculate equations with relative ease.  Although there are very few remaining papyrus material left due to the fragile nature, two major mathematical documentations do survive.  They are the Rhind Papyrus and the Moscow Papyrus.

The Rhind Papyrus

The Rhind papyrus is one of the famous documents from the Egyptian kingdom dating to 1650 B.C.  This document was purchased in 1858 by Henry Rhind and placed in the British Museum in 1864.  This documented historical representation shows that the Egyptians utilized fractions and eighty four problems and solutions.  The papyrus was divided into three books each consisting of separate and integral portions of Egyptian mathematics.

The first book consisted of reference tables ranging from simple fractional expressions to very involved linear equations.  The first collection consists of twenty arithmetic and twenty algebraic equations.  The first forty equations break out into the following:

• 1 through 6–Computations for a certain amount of loaves of bread and how it was divided among ten people
• 7 through 20—Show how to multiply the expressions by different fractions
• 21 through 23—Subtraction problems
• 24 through 34—Linear equations
• 35 through 38—Divisions of hekat
• 39 through 40—Divide loaves of bread by arithmetic progressions

The second book consists of geometric problems also known as mensuration problems (Katz 2007).  They problems in book two are broken down into the following:

• 41 through 46—Show how to find the volume of a cylinder and rectangle using granaries as an example
• 47—This problem is more of a table of fractions and their equivalent conversion
• 48 through 55—Shows the computations for area
• 55 through 60—These calculations are related to the slopes of pyramids. This allowed the pyramids to be built with stability and continuity

Book three of the Rhind Papyrus has a collection of twenty four problems.

• 61—This is a two part equation with the first part containing multiplication of fractions and the second part computes 2/3 of 1/n where n is odd
• 62 through 68—are algebraic equations
• 69 through78—are computations of pefsu involving the strength of bread and beer
• 79—is the summation terms in geometric progression
• 80 through 81—compute fractions of hekats which is then followed by a table
• 82 through 84—compute the amount of feed needed for a certain amount of livestock

The Moscow Papyrus

The second major papyrus relating to mathematics is the Moscow papyrus.  It is approximately the same age as the Rhind papyrus.  The Moscow papyrus is now in the Museum of Fine Arts.  In the Moscow papyrus there are approximately twenty five problems.  The order of the problems is not in any particular order nor are they as complex as those found in the Rhind papyrus.  That does not mean that they are not just as significant as the Rhind as they still depict the usage of mathematics in the Egyptian life.  Problems ten through fourteen calculate the surface area and volume of a frustum, which is portion of a solid such as a square or pentagonal, divided by two parallel lines.  This was a method for hidden surface determination.

Problems two and three calculate the length of a ship’s rudder and the length of a ship’s mast according to the original log of thirty cubits long.  Interesting enough problems 1, 19, and 25 are “Aha” problems in which there are problems solving for unknown quantities.  Eleven and twenty three are Baku problems which calculate output of workers.

Pefsu problems make up ten of the twenty five problems involving the strength or output of certain definable inputs.  A typical example would be the amount of grain input to make loaves and depending on the strength would determine how much the loaves of bread equated to in terms of beer.  Seven of the twenty five problems are geometry problems involving calculating the area of triangles and finding surface area of a hemisphere and volume of a frustrum in pyramid form.

The best known result of ingenious arithmetic computations and the fallen blood, sweat and tears of the laborers is that of the Egyptian pyramids, the Temple of Karnak and the Great Sphinx.  These were built by Egyptians and during the time frame ranging from 2700 B.C. and 1700 B.C. (Imhausen, 2006).  This much is fairly certain.  The real questions are how they were built, why they were built and were the Egyptians scientifically and mathematically more astute than everyone else in world or were their mathematical savvy and efforts focused primarily on their functions of building pyramids that allowed them, through iterative advancements, to build the spectacular wonders of the world that are known to this day.

Through the use of the remaining papyruses it is known that the Egyptians could calculate the amount of work each worker could provide, the amount of area a truncated pyramid would encompass and how much material it would need to build such a colossal structure.  Approximately twenty to thirty thousand and ranging upward of sixty thousand workers were used to build the pyramids.  According to the calculations provided for work output, eight to twelve men were used to pull the 2.5 ton block over land and approximately twenty men were needed to pull the block up the sloped side of the pyramid.

So how did mathematics help build the pyramids?  The pyramids are a conglomeration of multiple mathematic formulas put together which are scaled upward.  Through the mathematical relics of the papyrus the Egyptians could calculate slope, volume, weight and inputs and outputs of various variables such as work output and grain.  Using the same calculations they determined how much stone, work output and corresponding land mass was needed to build a pyramid.  Through the calculations of mass deductions could be made on the amount of raw material and labor would be needed to input into the pyramid.  From the raw material and people calculations were made to show time, team breakdown, skill set, logistics of travel and sustainment, and rudimentary work break down structures divided by what was needed to be accomplished.

The efficiency of the Egyptians building skills allowed the scalable ability of work output to build larger and larger pyramids for their leaders.  The Great Pyramid’s construction took approximately fifty thousand men due to the increases distance and increased mass.  Although the exact number of laborers cannot be exactly determined the mathematics used lead to advancements in labor organization.  The division of labor created leadership roles spanning a chain of command of two thousand men and then divided again into two groups of one thousand and thus divided again into five groups equally distributed from that one thousand (Brier & Houdin, 2009).  The division of labor and command and control are still utilized today in military doctrine as well as the process for continual process improvement through statistical analysis and implementation systems such as lean management and six sigma practices.

During the building process of the pyramids each team of two hundred men were assigned to specific tasks in building, logistical actions such as transportation and supply.  There was also a second type of team system that involved the same technique for breaking down the teams into manageable workforces.

This included workers divided up into teams associated to points on a compass but only included the cardinal directions of North, South and West.  East was not referred to in any documentation and after further reading it was found out that east is associated to evil and was not assigned.  It was also noted that in the breakdown of labor associated to the directions on the compass workers were more specialized and skilled in the trade in which they were assigned.  This represents not only a separation of work duties but job fit and function were also taken into account when building the teams for the pyramid project.

It is without great coincidence that the Egyptians built mathematically sound pyramids based on their aforementioned mathematical prowess and distinguished use of record keeping, documentation and analysis.  The Egyptians used mathematics to record time, length, and areas of land, money and cooking.  Their mathematics revolved around certain computations and was closely related.

The Great Pyramid has an Egyptian based perimeter of 365.24 cubits which is also the amount of days in the year.  If you double the perimeter it is equivalent to one minute of one degree on at the equator.  The perimeter divided by two times the height is equal to 3.1416.  The measurements of the King’s chamber are 2x5x3 and 3x4x5 which are Pythagorean triangles ((Brier & Houdin, 2009).  All of these mathematical nuances were a way of life to the Egyptian culture and the use of specific measurements and calculations could be homeage to that of the earth and its relationship to other celestial occurrences such as the year passing, the Nile flooding and seasons changing.  Although why certain measurements were used it is for certain that the Egyptians utilized advanced mathematical calculations to conduct the project build of the pyramids.  Even with the advancements of today’s technology no country on earth has the available resources to replicate the build of the Egyptians.

The entire process for ramping up efforts, sustaining the work force and calculating the amount of effort, resources and timeline was exponentially larger than any other undertaking of the time.  Speaking from a purely project management standpoint the entire process must be funded with inputs of raw material such as stone, grain, tools, labor and time mixed together to form the perfect balance of inputs to outputs resulting in a larger than life pyramid.

During the process mathematics were just as critical as the rope pulling the stone up the face of the pyramid because without the mathematic calculations the stone would have never been cut, the pyramid face would not be the correct angle, the amount of labor would be incorrect, the amount of food and water would not reach the labor force, the area of the pyramid would be grossly misconstrued and the project would not have finished without catastrophic failure.

References

Brier, B., & Houdin, J. (2009). The secret of the great pyramid, how one man’s obsession led to the solution of ancient egypt’s greatest mystery. Harper Paperbacks.

Gillings, R. J. (1982). Mathematics in the time of the pharaohs. Dover Publications.

Imhausen, Annette (2006). Ancient Egyptian mathematics: new perspectives on old sources. The Mathematical Intelligencer, Vol 28, Nr 1, pg 19-27

Katz, V., Imhausen, A. (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton University.