The Seked was a term used by the ancient Egyptians to describe the slope of an inclined surface numerically The system was based on the Egyptian's linear measure known as the cubit. Rather than being divided into centimeters or inches, the cubit was subdivided into palms and digits. The inclination of measured slopes was therefore expressed as the number of palms and digits moved horizontally for each cubit rise.

One of the most widely known applications of the seked slope measurement system was in the construction of the pyramids of Egypt. Although there is no direct evidence of its application from the archaeology of the Old Kingdom, there are examples from mathematical papyrii dating to the Middle Kingdom that show the use of this system for defining the slopes of the sides of pyramids, based on their height and base dimensions. The most widely quoted example is perhaps problem 56 from the Rhind papyrus.

The most famous of all the pyramids of Egypt is the Great Pyramid of Giza built around 2,550 B.C.. Based on the surveys of this structure that have been carried out by Flinders Petrie and others, the slopes of the faces of this monument were a seked of 5 1/2, or 5 palms and 2 digits [See figure above], which equates to a slope of 51.84º from the horizontal, using the modern 360 degree system. This slope would probably have been accurately applied during construction by way of 'A frame' shaped wooden tools with plumb bobs, marked to the correct incline, so that slopes could be measured out and checked efficiently.

Furthermore, according to Flinders Petrie's survey data in "The Pyramids and Temples of Gizeh" the mean slope of the Great Pyramid's entrance passage is 26º 31' 23" ± 5". This is less than 1/20th of one degree in deviation from an ideal slope of 1 in 2, which is 26º 33' 54".

This equates to a seked of 14, and is generally considered to have been the intentional designed slope applied by the Old Kingdom builders for internal passages.

Pyramid Slopes

Many of the smaller pyramids in Egypt have varying slopes, however, like the Great Pyramid of Giza, the pyramid at Meidum is thought to have had sides that sloped by 51.842º or 51º 50' 35"

The Great Pyramid scholar Professor I.E.S Edwards considered this to have been the 'normal' or most typical slope choice for pyramids. Flinders Petrie also noted the similarity of the slope of this pyramid to that of the Great Pyramid at Giza, and both Egyptologists considered it to have been a deliberate choice, based on a desire to ensure that the circuit of the base of the pyramids precisely equalled the length of a circle that would be swept out if the pyramid's height were used as a radius . Petrie wrote "...these relations of areas and of circular ratio are so systematic that we should grant that they were in the builder's design".

The seked of a pyramid is described by Richard Gillings in his book 'Mathematics in the Time of the Pharaohs' as follows:

"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'."

See also

• Cubit
• Great Pyramid
• Rhind papyrus

References

1. Gillings: Mathematics in the Time of the Pharaohs 1982: pp 212
2. Petrie: The Pyramids and Temples of Gizeh 1893: pp58
3. Petrie: Medum 1892
4. Edwards. The Pyramids of Egypt 1969. pp269
5. Lightbody. Egyptian Tomb Architecture: The Archaeological Facts of Pharaonic Stone Masonry 2008: pp 22-27,
6. Petrie Wisdom of the Egyptians 1940: 30
7. Gillings: Mathematics in the Time of the Pharaohs 1982: pp 212

Further reading

• Verner, Miroslav, "The Pyramids - Their Archaeology and History", Atlantic Books, 2001, ISBN 1-84354-171-8
• Arnold, Dieter. "Building In Egypt: Pharaonic Stone Masory", 1991. Oxford: Oxford University Press
• Jackson, K & J. Stamp. "Pyramid : Beyond Imagination. Inside the Great Pyramid of Giza"BBC Worldwide Ltd, 2002, ISBN 978-0563488033