It can be seen from the seked's used that
two ratios were adopted on more than one occasion. These are
the sekeds 5.5 [5 palms and 2 digits] and 5.25 [5 palms and 1
digit]. These two sekeds can be shown to be based on simple ratios;
the first being a height to base of 7:11 and the second a height
to base of 2:3 (4:6). This latter ratio is the derived from the
3 : 4 : 5 triangle.
We might question why the Ancient Egyptians
would have wished to choose these sekeds when, for practical
construction purposes, more simple seked ratios of 5 or 6 palms
would have been more preferable. The first produces an angle
of slope of 54.46° and the second of 49.4°. There is
no technical constructional reasons why either of these angles
could not have been adopted yet the only surviving pyramid that
uses either of these ratios, is the lower portion of the Bent
pyramid. We are therefore forced to ask what was so special about
the sekeds of 5.5 and 5.25?
Seked 5.25 (ratio
It has been claimed by a number of authorities
that the Ancient Egyptians did not know the 3:4:5 Pythagorean
ratio. For example T. L. Heath states in his book The Thirteen
Books of Euclid's Elements, Vol 1:
"There seems to be no evidence
that they (Egyptians) knew that triangle (3:4:5) is right-angled;
indeed according to the latest authority (T. Eric Peet, The Rhind
Mathematical Papyrus, 1923), nothing in Egyptian mathematics
suggests that the Egyptians were acquainted with this or any
special cases of the Pythagorean theorem."
This proposition is supported by Richard
Gillings based on the known textual information. Set against
this view there is considerable constructional and geometrical
evidence to indicate that the Egyptians were well aware of the
3 : 4 : 5 ratio.
Firstly this triangle was used, or the
ratios derived from it, in at least three pyramids including
that of Khafre on the Giza plateau. Khafre's pyramid was measured
by Petrie who gave the angle of slope as 53°-10' +/- 4'.
We can therefore be fairly certain that the seked of 5.25, which
produces an angle of 53°-7'-48", was intended in the
It is clear from the cubit measures that
have been recovered, that the Ancient Egyptians were quite capable
of measuring to an accuracy of 1/16 of a digit. It is just not
credible that they would never, whilst building or setting out
their triangles, have bothered to measure the length of the hypotenuse
of slope of the triangles based on their sekeds. Once a seked
of 5.25 had been adopted, sooner or later someone would have
measured the hypotenuse and discovered its relationship to the
other two sides.
From other problems in the RMP it is clear
that the Egyptians were quite capable of division. They would
certainly have discovered that a triangle with sides of 6:8:10
(RMP 59) could be reduced to a 3:4:5 ratio. Indeed it could be
argued that a whole number ratio of cubits, for the slope, base
and height of the pyramid, was precisely why they chose to adopt
a seked of 5.25. For this would considerably ease the technical
problems of ensuring that the correct angle of slope was always
maintained, in the final finishing of the casing stones. In support
of this argument problems 57, 58 and 59 in the RMP are based
on a seked of 5.25, which demonstrates its importance in pyramid
In addition there are two further items
of corroborative evidence. The first stems from Pythagoras's
exposure to Egyptian ideas during the ten years of his life that
he spent in Egypt as part of the priesthood. Whilst it is very
likely that he was the first individual to 'prove' the relationship
between the sides of a right angled triangle, based on the square
of its sides, we could also infer that he obtained a knowledge
of the 3 : 4 : 5 triangle from his time in Egypt.
Secondly , whilst the problems in the RMP
relate solely to the base and perpendicular sides of a right
angled triangle we know that the hypotenuse was important in
other calculations in relationship to areas. The unit of measure
of a Remen, or more correctly double Remen is the diagonal of
a square whose side is one cubit7. From this it is clear that
the measure of the hypotenuse was an aspect of Ancient Egyptian
mathematics and geometry which had practical application in the
surveying of land.
Omitting all other evidence, that the seked
of 5.25 occurs with such regularity both in the mathematic texts
as well as in the practical construction of at least three pyramids
lends substantial weight to the evidence that the 3 : 4 : 5 ratio
was known to the Ancient Egyptians. Whether out of curiosity
of intent the hypotenuse of a pyramid or triangle with this seked
would have been accurately measured. This in turn would inevitably
lead to a working knowledge of the 3 : 4 : 5 ratio.
Seked 5.5 (ratio
7:11 - height to base) - The Great Pyramid Ratio
We now need to question why a seked of
5.5 might have been used? This is not so obvious but an explanation
could lie with a further understanding of Egyptian measures,
particularly the relationship of the Royal cubit to the palm
and the digit.
In modern times we are used to working
with the metric system with its standard ten base ratios. Prior
to this, in Britain, Imperial measures were used which incorporated
a range of different ratios, 12 inches in one foot, 3 feet in
one yard and so on. All ancient measures incorporate practical
relationships to assist in the computation of lengths, areas
and volumes. The relationship of the primary Ancient Egyptian
measures has already been given. As a distance the cubit is generally
reckoned to be 20.6 inches or 523 millimetres. For comparison
the Sumerian cubit of 495 millimetres was divided into 30 digits
as opposed to 28 digits in Egyptian measures. The Greeks also
used a cubit of about 489 millimetres being divided into 24 digits.
Both the Greeks and the Egyptians used 4 digits to equal a palm;
giving 6 palms to one cubit in the case of Greek measures and
7 palms to one cubit in Egyptian measures.
In all ancient measures the division of
the cubit into seven parts is, to say the least, very curious.
As a measure it has no divisors being a prime number. It was
probably for this reason that a short cubit was introduced of
6 palms, which could then be divided into halves and thirds;
but not so the Royal cubit. It might be argued that the sevenfold
division held some magical or numerological significance, such
as a relationship to the 70 days between the rising and setting
of the Dog Star, and that such a relationship has now been lost.
There is however one practical reason why the Ancient Egyptians
might have chosen to divide their cubit into seven parts.
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