Seked (or seqed) is an ancient Egyptian term describing the inclination of the triangular faces of a right pyramid. The system was based on the Egyptians' length measure known as the royal cubit. It was subdivided into seven ''palms'', each of which was sub-divided into four ''digits''. The inclination of measured slopes was therefore expressed as the number of horizontal palms and digits relative to each royal cubit rise. The seked is proportional to the reciprocal of our modern measure of

slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...

or gradient, and to the cotangent of the angle of elevation.Gillings: Mathematics in the Time of the Pharaohs 1982: pp 212 Specifically, if ''s'' is the seked, ''m'' the slope (rise over run), and

\phi

the angle of elevation from horizontal, then: :

s = \frac = 7\cot(\phi).

The most famous example of a seked slope is of the

Great Pyramid of Giza The Great Pyramid of Giza is the biggest Egyptian pyramid and the tomb of Fourth Dynasty pharaoh Khufu. Built in the early 26th century BC during a period of around 27 years, the pyramid is the oldest of the Seven Wonders of the Ancient Worl ...

in Egypt built around 2550 BC. Based on modern surveys, the faces of this monument had a seked of 5½, or 5 palms and 2 digits, in modern terms equivalent to a slope of 1.27, a gradient of 127%, and an elevation of 51.84° from the horizontal (in our 360° system).

Overview

Information on the use of the seked in the design of pyramids has been obtained from two mathematical papyri: the

Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scotland, Scottish antiquarian, who ...

in the British Museum and the Moscow Mathematical Papyrus in the Museum of Fine Arts. Although there is no direct evidence of its application from the archaeology of the Old Kingdom, there are a number of examples from the two mathematical papyri, which date 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

. The most famous of all the pyramids of Egypt is the

Great Pyramid The Great Pyramid of Giza is the biggest Egyptian pyramid and the tomb of Fourth Dynasty pharaoh Khufu. Built in the early 26th century BC during a period of around 27 years, the pyramid is the oldest of the Seven Wonders of the Ancient World ...

of Giza built around 2550 BC. Based on the surveys of this structure that have been carried out by

Flinders Petrie Sir William Matthew Flinders Petrie ( – ), commonly known as simply Flinders Petrie, was a British Egyptologist and a pioneer of systematic methodology in archaeology and the preservation of artefacts. He held the first chair of Egyp ...

and others, the slopes of the faces of this monument were a seked of 5½, or 5 palms and 2 digits ee figure abovewhich equates to a slope of 51.84° from the horizontal, using the modern 360° 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 Petrie's survey data in "The Pyramids and Temples of Gizeh" the mean slope of the Great Pyramid's entrance passage is 26° 31' 23" ± 5". This is less than 1/20 of one degree in deviation from an ideal slope of 1 in 2, which is 26° 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

KhufuPyramidCasingStone-BritishMuseum-August19-08

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 palms = 1 cubit. The Egyptian word 'seked' is thus related '' n meaning, not origin/nowiki>'' to our modern word 'gradient'.

Many of the smaller pyramids in Egypt have varying slopes; however, like the Great Pyramid of Giza, the pyramid at

Meidum Meidum, Maydum or Maidum ( ar, ميدوم, , ) is an archaeological site in Lower Egypt. It contains a large pyramid and several mudbrick mastabas. The pyramid was Egypt's first straight-sided one, but it partially collapsed in ancient times. The ...

is thought to have had sides that sloped by 51.842° or 51° 50' 35", which is a seked of 5½ palms. 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 circumference 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".Petrie Wisdom of the Egyptians 1940: 30

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Overview

Pyramid slopes

See also

References

Further reading