Hi All,

There is no direct evidence that proving the Ancient Egyptians had a working knowledge of pi, discounting the Pyramids of course. I was never inclined to agree with this line of thinking until I found the formula for calculating the area of a circle from a square using seked ratio demonstrated below. But, the question remains, if the Ancient Egyptians did not use pi to calculate the area of a circle, how did they do it? Does RMP 50 actually demonstrate the method of the Ancient Egyptians referencing a square to find the area of a circle? Could there be some other yet undiscovered method? The RMP tells us the Ancient Egyptians were aware of and used the formula we know as seked, the equivalent of our angle cotangent.

Wiki: "The seked of a right pyramid is the inclination of any one of the four triangular faces to the horizontal plane of its base, and is measured as so many horizontal units per one vertical unit rise. It is thus a measure equivalent to our modern cotangent of the angle of slope. In general, the seked of a pyramid is a kind of fraction, given as so many palms horizontally for each cubit of vertically, where 7 palm equal one cubit. The Egyptian word 'seked' is thus related to our modern word 'gradient'."

While the above statement is basically true, it falls far short of describing the abilities of the 14/11-seked ratio, which in fact it provides many characteristics of Ancient Egyptian system of mathematics until now unknown. Among the aspects of this particular seked, it is the ratio by which the Ancient Egyptians squared the circle. It seems the same formula that creates angles will also calculate the area of a circle, the circumference, diameter and radius from the square’s dimensions all accomplished without the use of pi.
Their method of calculating the area of a circle is demonstrated by G1 which is the answer to RMP 50 utilizing the seked ratio to illustrate the exchange and interface of shapes without the use of pi as practiced by our current system of mathematics.

It is as follows: Square’s area with a perimeter of (n) * 14/11 = circle’s area with a circumference of (n). Therefore, (x^2 14)/11 derives the same answer as provided by (22 r^2)/7)

(x^2 14)/11 when written as (x^2 56)/44 produces the values of the side length 44 units, diameter 56 units demonstrated by G1. The area of a circle is calculated from the area of a square by multiplying by the seked of G1. Using G1 dimensions in cubits divided by 10 we find the following values:

Perimeter or circumference = 176x units
Diameter = 56x units
Height or radius = 28x units
Side length of the square = 44x units.
Diameter x square side length = area of circle i.e. 56 x 44= 2464 area

((x^2 14)/11) as a geometric figure creates a parabola: the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

Rhind Mathematical Papyrus Problem #50 states: A circular field has diameter 9 khet. What is its area?
The written solution says, subtract 1/9 of the diameter, which leaves 8 khet. The area is 8 multiplied by 8, or 64 setat. Something is missing unless the ancient’s method of calculating the area of a circle by referencing the area of a square is understood, using the more accurate (x^2 14)/11 which should be a given in all applications of this kind. In light of this information it should become clear that RMP 50 is a shorthand method of obtaining an estimate figure for the area of a circle and/or square that is close enough for most practical real world applications but not accurate enough for civil applications.

To accurately calculate RMP 50 to civil engineering standards would be as follows:

(9 diameter /14 x 11 = 7 + 1/14 side length of square)

7 + 1/14 ^2 = 50 + 1/196 area of square

50 + 1/196 x 14/11 = 63 9/14 area of circle

(63 + 1/2 + 1/8 + 1/56 area of circle diameter of 9)

63 9/14: 64 ratio providing a margin of error of 0.005611%


Proofing for Comparison:
The (22 r^2)/7 part of the equation (below) is only to demonstrate equality of process and is not an essential part of the AE process.
(x^2 14)/11 = = (22 r^2)/7
121 r^2 = 49 x^2
+X = +(11r)/7
-X = -(11r)/7

I believed it has been proven many times by many people you cannot square a circle using pi. The ancient method circumnavigates this problem by not using pi at all and could account for the Ancient Egyptians seeming use of pi in their calculations?
Could it stem from a misconception that the concepts, principles and elements of the Ancient Egyptians were compatible with ours?


Regards,
Jacob