In my article on the design of the Meydum
Pyramid in a recent issue of DE,
certain views which to judge from the response of Robins and Shute in
following issue, have not been clearly understood. My statement of
views was certainly very brief, and the time has come to clarify the
concerning which Robins and Shute appear to have had some difficulty.
comments have been very welcome, and have precipitated a significant
of the material presented in my earlier articles, as we will see.
Height and Base versus Seked
Robins and Shute
begin their article by dividing interpretations of the external form of
into two sorts, "according to whether the slope of the faces or the
of the apex is considered to be the most important determinant." Now it
happens that neither approach describes my own position, because
choice of slope may sometimes have been a priority, I believe that both
height and base were in general intimately connected with the slope.
I have referred to the casing-angle of a pyramid, I have at the same
deduced the height which resulted from that angle, knowing also the
of the base.
Robins and Shute, however, have taken a different
view, and have dealt with the slopes of pyramids almost exclusively.
to have orthodoxy on their side, and in an earlier article they tell
us:  "It
is clear from the pyramid problems in the Rhind Mathematical Papyrus,
56-59, that the slopes of pyramids were predetermined according to a
proportional measurement called the seked,
which was the
horizontal displacement in palms for a vertical drop of seven palms, or
royal cubit." Reference to the problems in question, however, will show
that in three out of five instances the seked
calculated from the dimensions first selected for the pyramid's height
base. So although it might be assumed that the slopes were
seked values, it would seem
more accurate to assert that they
were predetermined by the chosen height and base.
This is not how
Robins and Shute see the problem, however, for they have expressed the
that: "Taking the pyramids as a whole, it seems that the architects
not particularly concerned about the exact height, which emerged from
precisely selected seked and the
space available on the site for
the square base." To counter the objection that pyramidal heights are
specified in the Rhind mathematical papyrus, they say: "these were
intended as exercises, so that it would be wrong to infer from them
pyramid designers were particularly interested in the heights of the
I can only point out that constructing a pyramid was
not just an exercise, but something that would in some cases have been
on for at least twenty years; and I think we may assume that the
have been more than a little interested to know what the eventual
height of the
edifice would be. Certainly, the height was in a sense theoretical
could not be measured directly; but this in no way diminishes the
which would have been attached to the dimension by the builders.
must be said that the actual dimensions used in several of the pyramids
Fourth Dynasty are far from obvious in origin, and derived from
which were much more subtle than Robins and Shute had any reason to
expect. Even so, the casing-angles of the Meydum Pyramid and of the
Great Pyramid are
clearly explained by the dimensions of height and base, and one might
expected some recognition of this fact from Robins and Shute. But I can
mention in their articles of the actual dimensions of any pyramids,
the base of 150 cubits and the height of 100 cubits which were used in
pyramids from the end of the Vth Dynasty onwards.
In my previous
article for DE, I suggested that
the use of the seked
during the Old Kingdom was not proven, and Robins and Shute have indeed
unable to point to any seked-measuring
equipment or contemporary
texts which would show the use of the seked
at this time with
any certainty. The assumption of its use rests primarily upon the
of slope in the Rhind mathematical papyrus, which was copied from a
text of the
Middle Kingdom dated some 700 years after the Giza pyramids were
although the seked may date back
to the Old Kingdom, there is
no proof that it does; and it is unnecessary to assume its use in every
where a sloping surface had to be constructed. There was always a more
measurement of slope available to the builders, defined simply by the
between height and base; and the question is whether the slopes of
were originally understood in this form or as the more abstract seked
Since every seked value can be
an equivalent ratio between height and base, this question might appear
merely one of definition. Robins and Shute, however, have taken the
stage further, by asserting: "the general conclusion that the
architects determined slope consistently by one rule only, which
lateral displacement of palms and fingers for a drop of one royal
cubit." Now this requirement that sekeds
should be expressible in
numbers of palms and fingers did not apply in the Rhind mathematical
papyrus, and in one example the height and base selected for a pyramid
yielded a seked
of 5 1/25 palms. The impractical nature of this result shows the use
seked as a theoretical
concept divorced from the realities of
But let us now consider the seked
of five palms and one finger, which is supposed to have been used for
pyramids of the Old Kingdom. What was the significance of this value to
pyramid-builders? Robins and Shute suggest that it was used in one
the late Vth Dynasty and in four pyramids of the VIth Dynasty, to give
combination of a base of 150 royal cubits and a height of 100 royal
Could it not be, then, that the architects conceived the dimensions in
these simple terms? Since the ratio of height to base was in this case
2:3, the ratio between height and semi-base was 4:3, and the slope of
pyramid-casing represented the hypotenuse of a Pythagorean 3,4,5
as Robins and Shute have again pointed out, this result could have
"a convenient basis for set-squares used by the stonemasons."
since the casing could be constructed with 4 parts rise on 3 parts
reason had the architect to convert the slope into the seked
5 palms 1 finger? For this measurement had neither practical utility
numerical significance. The fact that the slopes of some pyramids can
expressed as seked values
involving palms and fingers, is no
proof that the slopes were conceived in those terms when the pyramids
By insisting upon seked values
in whole numbers
of palms and fingers, moreover, Robins and Shute needlessly exclude
slopes which have every reason to be considered for the sake of
simplicity, but which cannot be expressed as a seked
and fingers. One example of indeed questionable accuracy is the slope
of 5 rise
on 4 base which has been attributed by authorities to the Third Pyramid
at Giza, as Robins and Shute have noted, but requires a seked
of 5 3/5
To find support for their theory, Robins and Shute refer to
the lower slope of the Bent Pyramid of Dahshur, and state: "it is now
generally agreed that the value should be 54° 27' 44", to conform
with a seked of 5." They
base this conclusion on the
theoretical angle listed by Baines and Malek in their popular reference
work, but at the same time overlook the survey-data published more
century ago by Flinders Petrie, and the results of the survey
published by Josef Dorner in MDAIK.
They make no mention of
the article in GM last year, in
which I discussed the results of
these surveys in some detail. The two surveyors both concluded that
lower slope was 10 rise on 7 base, in close agreement with the observed
casing-angle of about 55°; but this is more than half a degree steeper
the slope required by the seked
But now Robins
and Shute have a problem, since this slope of 10 rise on 7 base cannot
expressed as a seked in palms
and fingers. A value can be found
by working with fifths of a palm instead of quarters, yet the builders
reason to seek out this result. For they could very easily have
slope by taking a vertical rise of 10 palms for each cubit measured
It so happens that a seked of 5,
slope of 7 rise on 5 base, can be ascribed to some upper parts of the
slope of the Bent Pyramid, though opinions differ as to the cause of
associated convexity of the faces; and Dorner discounts the upper parts
slope entirely. But if Dorner's view were to be accepted without
reservation, the seked of 5
would be half a degree in error.
now move on a few centuries, however, and assume the position of a
scribe of the
Middle Kingdom, who standing in awe and reverence in front of a mighty
of the Fourth Dynasty, wished to glean some knowledge of its structure.
now the seked became clearly the
most practical means by which
the scribe could measure and compare the slopes of already-existing
pyramids, to assist in the revival of pyramid-construction which took
place at this time. The cubit served suitably as the standard measure
of vertical height, against
which a horizontal offset to the slope of the pyramid could readily be
in palms and fingers - though fractions of a finger would sometimes
have to be
neglected. Taking a pyramid with a height-to-base ratio of 2 to 3,
however, the measurement could be exact; and indeed we can hear the
scribe calling out: "Lo! It is five palms and one finger!"
This result could
have been used to calculate the height of the pyramid, and would in
have shown that the builders had combined a base of 150 cubits with a
100 cubits. For the Second Pyramid of Giza, however, the design would
have been obvious, since the base was constrained by the requirements
Giza site plan to the value of 411 cubits. It was for this reason,
that the 7:11 height-to-base ratio of the Great Pyramid - as reflected
height of that pyramid of 280 cubits and the base of 440 cubits - gave
the ratio of 2:3; since for the base of 411 or 3 × 137 cubits, the
became just 2 × 137 or 274 cubits, and the dimensions were defined with
the greatest possible simplicity.
Derivation of Square Roots
With reference to my
findings regarding the Giza site
plan of three
pyramids, Robins and Shute continue their paper with a
discussion of square
roots - and in particular, my suggestion that numerical values for the
roots of 2 and 3, which are represented by the overall dimensions of
of 1000√2 and 1000√3
cubits, could have been calculated. It is quite possible, however,
these dimensions originated in the geometrical development of the site
it is by no means necessary to assume that the architect evolved square
abstract mathematical quantities. The square root of two could have
simply from the relationship between the side of a square and its
diagonal - a
significant possibility because, of course, every pyramid had to be
with a square base, and the lengths of the diagonals had to be equal.
In my paper describing the position of the Sphinx, I suggested that
this 1:√2 relationship
could have been found by direct
measurement, and this is in principle exactly the method now proposed
and Shute. Whilst they begin with a dimension of 1 cubit, however, I
initial dimension to be 10 cubits; and by thus making the divisions of
cubit more sensible to measurement, the result is considerably more
accurate. In a square of side 10 cubits or 70 palms, which I assume to
have been laid out
on an area of levelled pavement, the length of the diagonal is found to
almost exactly 99 palms; and reducing the result to unit fractions in
Egyptian manner, we obtain the following value for the square root of
99/70 = 1 + 1/5 + 1/7 + 1/14 =
It may be debated whether the
architect needed to carry out
this reduction, since the numbers 70 and 99 provided him with a method
obtaining the diagonal of any square, which was possibly all that he
required. Nonetheless, the above sum gives the value of the square root
of two with an
error of only one part in 19,600.
Since the side of the square of ten
cubits also measures 280 fingers, we can discern the origin of the
280 cubits which determined the height of the Great Pyramid, the
the enclosure of the Bent Pyramid, and several other dimensions which
prominently in the design of the Fourth-Dynasty pyramids. Any divisions
in this formative square, moreover, could have been translated into
dimensions by using a scale of one cubit for each finger. By laying off
semi-diagonal of the square of 2 × 99 or 198 fingers along one side,
example, we obtain the division in the height of the
Great Pyramid as
marked by the floor of the
King's Chamber, at the level of (280 - 198) or 82 cubits; while the
difference of (280 - 99) or 181 cubits defines the exact semi-base of
As I have previously shown, this last result can be
combined with the theoretical lower casing-angle of the
first proposed by Lauer - that is
√2 rise on 1 base -
to define the dimensions
and profile of this pyramid using the enclosure-width of 2 × 280
cubits, in a simple geometrical scheme. Whilst Lauer intended to invoke
a slope for the
pyramid corner-edges of just 1:1, the slope of the faces could also
controlled with the use of a set-square made simply by transferring the
of a square frame to one side. As we have now seen, an approximation to
slope is given by the ratio 99:70, which can be simplified with some
accuracy to 100:70 or 10:7, and 98:70 or 7:5.
Perhaps to facilitate
the positioning of the entrances to the Bent Pyramid in the north and
of the casing, the builders seem to have used the ratio of 10:7 for the
part of the lower slope, and to have corrected for the imbalance by
average slope of 7:5 for the remainder. This is suggested by Petrie's
measures for the lower and upper parts of the lower slope of 55° 1' and
31' respectively, giving an overall mean of 54° 46' which is very
close to the theoretical angle of 54° 44' for
√2 rise on 1 base.
At Dahshur, this
combination of slopes finds an interesting parallel in the Subsidiary
the Bent Pyramid, and in the Northern Stone Pyramid, both of which have
casing-angle of about 44° 30'. Flinders Petrie, observing that
Perring's result for the latter pyramid was about one degree in
error, ascribed to this angle the profile of 7 slope on 5 base; but
almost the same
theoretical angle of 44° 25' is in fact given by a profile of 10 slope
rise. In neither case can a seked
be obtained since the slant
height is involved; and the nearest values in palms and fingers are in
about half a degree in error.
If measurements involving the slant
height are now considered for the lower slope of the Bent Pyramid, we
once presented with the next problem posed by Robins and Shute, which
estimate a value for the square root of three. The lower slant height
Bent Pyramid represents in concrete form the hypotenuse of a right
√3 - the base being
1 and the vertical side
being √2. As I have
shown elsewhere, this
slant height is equal to the upper vertical height, so that the total
height of 200 cubits is divided in the ratio of √2:√3, or into parts of
110.1 cubits. In the
Giza site plan, the same proportion explains the major division of a
of 2000 cubits into parts of 899 and 1101 cubits at the south side of
Let us now, however, obtain an estimate for the square root
of three starting from an initial square of 10 cubits. By marking off
diagonal of this square along one side, we can construct a rectangle
measures 10 cubits by 10√2
diagonal length will be 10√3
measuring this diagonal to the nearest finger will yield a result of 17
palms 1 finger, or 485 fingers. Hence we have a value for the square
three of 485/280, or 97/56, which is correct to 1 part in 18,800.
this result to unit fractions we find:
97/56 = 1 + 1/2 + 1/7 + 1/14 +
1/56 = 1.73214...
It will be noted that any multiple
cubits can be expressed as a sum of cubits, palms and fingers, with
negligible error. In the Giza site plan, the dimension of 1000√3 cubits
would be expressed as
1732 cubits 1 palm,
which is not far removed from the theoretical number of 1732.05...
noted possible references to the approximation to the square root of
99/70, some indication of the above estimate for the square root of
also be expected; and indeed this is to be found in certain dimensions
Giza site plan which I determined long ago, but had not considered in
context. Numbers in the ratio 97:56 can be seen in the distance of (970
or 1530 cubits southwards from the north side of the Great Pyramid to
side of the Third Pyramid - this being divided at the north side of the
Pyramid into a northern part of (440 + 250) or 690 cubits, with a
840 cubits. The latter dimension is naturally divided into three parts
280 cubits, as I have previously noted, so that a division of the
of 1530 cubits into parts of 970 and 560 cubits occurs at a point 280
southwards from the north side of the Second Pyramid.
point does not appear to be marked by any known constructional feature,
indicate a geometrical development in which all the dimensions of the
fall out, as it were, at a stroke. This construction begins with a
side (440 + 560) or 1000 cubits, and yields the round-figure side of
Pyramid as the difference of (970 - 560) or 410 cubits, and the side of
Third Pyramid as about (1732 - 970 - 560) or 202 cubits.
the occurrence of the square roots of two and three in the Giza site
the minimum of mathematical elaboration, there remains a great question
how these numbers were really understood by the pyramid architects. Did
emerge merely as the result of geometrical experimentation, or did the
architects understand the properties of right-angled triangles, and
this knowledge through the values of the dimensions they employed? As
and Shute have observed, the contents of secular teaching texts such as
Rhind papyrus should not be assumed to represent the full extent of the
Egyptians' knowledge of mathematics; but equally, it would be wrong to
that the methods employed by perhaps only a handful of individuals
Fourth Dynasty, were common knowledge in the later periods of Egyptian
on the Meydum Pyramid
the dimensions of the Meydum Pyramid have been discussed by P. Testa in
article in DE, with
conclusions differing in some respects
from my own, it seems appropriate to explain here how these
Firstly, it must be emphasised that Testa never states the
actual measurements upon which he bases his conclusions, but only gives
theoretical dimensions in cubits according to his own interpretations.
turn depend upon an imagined variation in the length of the cubit used
different parts of the pyramid, of nearly three centimetres. In my own
the Fourth-Dynasty monuments, on the other hand, the variations I have
detected in the cubit amount to less than a millimetre, and I always
constant value to convert a given set of measurements into cubits.
to Testa, the sides of the Meydum Pyramid measure 280 cubits, this
approximate length suggested by Lauer. But as Maragioglio and Rinaldi
out, this dimension is "the length of the foundations protruding for a
short way from the base of the casing", and therefore not the
base of the pyramid at all. Following Petrie, the sides of the base
measure 275 cubits, for a length of cubit just 0.7 millimetre longer
'Giza' cubit of 0.52375 metres or 20.620 inches.
Since Testa has
taken the height of the Meydum Pyramid to be 175 cubits, which is
original height established in Petrie's survey, he asserts that the
the sides is 175 rise on 140 base, or 5 rise on 4 base. The theoretical
for this profile is not, however, 51° 34' 01" as Testa states, but
51.34019° = 51° 20' 25", or half a degree less than any measures
of the casing-angle have indicated.
1. J.A.R. Legon, DE 17 (1990),
2. G. Robins and C.C.D. Shute, DE
18 (1990), 43-53.
G. Robins and C.C.D. Shute, DE
16 (1990), 75-80; 75.
T.E. Peet, The Rhind Mathematical Papyrus
(Liverpool, 1923), 97-100. Problem no. 59 is in two parts.
5. G. Robins and C.C.D. Shute,
Historia Mathematica Vol.12 no.2 (May, 1985),
6. Ibid., 120.
7. G. Robins and C.C.D. Shute, The Rhind
Mathematical Papyrus (London, 1987), 11.
8. G. Robins and C.C.D. Shute,
GM 57 (1982), 49-54, 53.
See also op.cit. (n.5), 109.
Peet, (op.cit., 98), treated this value as a practical measure.
and Shute op.cit. (n.5), 112.
11. Ibid., 112; see also op.cit. (n.2), 49.
12. Robins and Shute op.cit. (n.2), 45.
13. J. Baines and J. Malek,
Atlas of Ancient Egypt (Oxford, 1980).
14. W.M.F. Petrie,
A Season in Egypt, 1887 (London, 1888), 27-32.
15. J. Dorner, MDAIK
42 (1986), 43-58.
16. J.A.R. Legon, GM 116 (1990),
17. J.A.R. Legon, DE 10 (1988),
34-40; 36, Table I.
19. J.A.R. Legon, DE 14 (1989),
20. J.A.R. Legon, DE 12 (1988),
43, fig.1; G.M. 108 (1989),
21. Legon, op.cit. (n.16), 69, figs.1, 2.
22. J-Ph. Lauer, Le myst�re
des pyramides (Paris, 1974), 306, 342.
23. Petrie, op.cit., 30.
Ibid., 27, 32; V. Maragioglio and C.A. Rinaldi, L'Architettura
Piramidi Menfite Vol. III (Rapallo, 1964), 76.
25. Perring's measure is
20 rise on 21 base. See H. Vyse, Appendix to Operations
carried on at the
Pyramids of Gizeh (1842), 65.
26. Legon, op.cit. (n.16), 72, fig. 3.
27. Legon, op.cit. (n.17), 37.
28. P. Testa, DE 18 (1990),
29. Legon, op.cit.
30. Maragioglio and Rinaldi, op.cit., 16.
31. See Legon,
op.cit. (n.1), 19-20.