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October 18, 2019, 11:24 am UTC    
April 21, 2017 03:14PM
One of the very first questions that one should ask is what base did AE count in during the pyramid age.

In my monograph on the Grand Gallery of the Great Pyramid is a picture of the statue of Khasekhem, and at the base of the statue is the number of northern enemies that were killed.
The symbols are based on a base 10 counting system even before the Great Pyramid was built.

It seems likely that people counted on the fingers and thumbs of their hands as soon as they had the sense to count.

The relationship between 360 and the pi approximation 22/7 is given in my monograph in the context of squaring the circle.

The square of the cotangent to square the circle is 360/88 which can be pictured as two squares on the rise and run of a right angled triangle.

The square on the rise has an area of 88 square units and the square on the run has an area of 360 square units.

A circle with an area equal to 88 square units has an equal area circle with a radius equal to the square root of 28.

The hypotenuse of the triangle is equal to 4 times the radius of the circle.

This relationship is expressed in the geometric design of the Grand Gallery and Ascending Passage.

The slope of the triangle is a rise of sr28 for a run of sr360.

This slope equates to a rise of sr55 for a run of 15, and the hypotenuse of the triangle is sr280, as the square root of the height of the pyramid.

This triangle defines the height of the Grand Gallery near the north end wall which is sr280 cubits as the vertical height of the gallery and 15 cubits as the perpendicular height of the gallery (perpendicular to slope of floor).

The angle can be calculated as 26 degrees 18 minutes 30 seconds to the nearest arc second. Four times the radius of any circle can be projected onto the slope, and the vertical rise is equal to the side-length of its equal area square, exactly so for the pi approximation 22/7.

It seems likely that the pi approximation 22/7 was regarded as exact. A circle with a diameter of 1 cubit has a circumference of precisely 88 digits, as calculated from the pi approximation 22/7, but we can calculate that the circumference is, more precisely, 87.96.. digits, which is a difference of approximately 1/38 inch on a circle with a circumference of approximately 64.8 inches.

Therefore a circle with a diameter of 1 cubit of 7 palms or 28 digits has a circumference very close indeed to 22 palms or 88 digits, so any attempt to measure a circle would have yielded the ratio 22/7 which remains the same for any and every diameter, but with no particular need to ascribe a symbol to what is obvious when the numbers suffice well enough.

Knowledge of the circumference of a circle would have had little or no practical value in AE.

The area of a circle has value in that one can calculate the volume of a cylinder such as a granary. It is known that AE applied complex fractions to problems such as the approximation that the side length of a square is equal to 8/9 diameter of its equal area circle.

In reality the side length of the square equal in area to a circle with a diameter of 9 palms (26.51.. inches) is only 7.976 palms (23.49..inches) not 8 palms (23.56.. inches), but the difference of 0.024 palms (0.07 inches) isn't a big deal when it comes to taking stock of grain.

This doesn't mean that the 8/9 ready reckoner was the only AE method of squaring a circle, but it is the only one in the RMP.

Mark



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