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Messages from Michael L. Morton

Trigonometric Ratio

Gordon Tibbles wrote:

One problem we run into when attempting to resolve mathematical intent between cultures and ages is that not all factors are constant.

"Problem 56 of the Rhind Papyrus is of special interest in that it contains rudiments of trigonometry and a theory of similar triangles. In the construction of the pyramids it had been essential to maintain a uniform slope for the faces, and it may have been this concern that led the Egyptians to introduce a concept equivalent to the cotangent of an angle. For instance, in modern technology it is customary to measure the steepness of a straight line through the ratio of the "rise" to the "run". In Egypt it was customary to use the reciprocal of this ratio.

There the word "seqt" meant the horizontal departure of an oblique line from the vertical axis for every unit change in the height. The seqt thus corresponded, except for the units of measurement, to the batter used today by architects to describe inward slope of a masonry wall or pier. The vertical unit of length was the cubit; but in measuring the horizontal distance, the unit used was the "hand," of which there were seven in cubit. Hence, the seqt of the face of a pyramid was the ratio of run to rise, the former measured in hands, the latter in cubits.

In Problem 56 one is asked to find the seqt of a pyramid that is 250 ells or cubits high and has a square base 360 ells on a side. The scribe first divided 360 by 2 and then divided the result by 250, obtaining 1/2+1/5+1/50 in ells. Multiplying the result by 76, he gave the seqt as 5-1/25 in hands per ell. In other pyramid problems in the Ahmes Papyrus the seqt turns out to be 5-1/4, agreeing somewhat better with that of the great Cheops Pyramid, 440 ells wide and 280 high, the seqt being 5-1/2 hands per ell.

There are many stories about presumed geometric relationships among dimensions in the Great Pyramid, some of which are patently false. For instance, the story that the perimeter of the base was intended to be precisely equal to the circumference of a circle of which the radius is the height of the pyramid is not in agreement with the work of Ahmes. The ratio of perimeter to height is indeed very close to 44/7, which is just twice the value of 22/7 often used today for pi; but we must recall that the Ahmes value for pi is about 3-1/6, not 3-1/7. That Ahmes' value was used also by other Egyptians is confirmed in a papyrus roll from the twelfth dynasty (the Kahun Papyrus, now in London) in which the volume of a cylinder is found by multiplying the height by the area of the base, the base being determined according to Ahmes' rule."

A History of Mathematics by Carl B. Boyer (Wiley 1991 2nd Edition)

Michael responded

Munck's work has proven that the original perimeter of The Great Pyramid DID, indeed, give an exact ratio of 2Pi ... as in ... 6.283185307 ... in comparison to the original height 'including the capstone'. And the archaeomatrix as a whole supports this conclusion. In Feet ... 3018.110298 / 6.283185307 = 480.3471728 ... there it is. Those same figures show up all over the place in the archaeomatrix. -- Michael