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 the
following issue, have not been clearly understood. My statement of those
views was certainly very brief, and the time has come to clarify the points
concerning which Robins and Shute appear to have had some difficulty. Their
comments have been very welcome, and have precipitated a significant enhancement
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 pyramids
into two sorts, "according to whether the slope of the faces or the height
of the apex is considered to be the most important determinant." Now it so
happens that neither approach describes my own position, because although the
choice of slope may sometimes have been a priority, I believe that both the
height and base were in general intimately connected with the slope. Whenever
I have referred to the casing-angle of a pyramid, I have at the same time
deduced the height which resulted from that angle, knowing also the dimensions
of the base.
Robins and Shute, however, have taken a different
view, and have dealt with the slopes of pyramids almost exclusively. They claim
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, nos.
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 one
royal cubit." Reference to the problems in question, however, will show
that in three out of five instances the seked values were
calculated from the dimensions first selected for the pyramid's height and
base. So although it might be assumed that the slopes were controlled by
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 view
that: "Taking the pyramids as a whole, it seems that the architects were
not particularly concerned about the exact height, which emerged from the very
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 that the
pyramid designers were particularly interested in the heights of the buildings
I can only point out that constructing a pyramid was
not just an exercise, but something that would in some cases have been carried
on for at least twenty years; and I think we may assume that the builders would
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 since it
could not be measured directly; but this in no way diminishes the importance
which would have been attached to the dimension by the builders.
must be said that the actual dimensions used in several of the pyramids of the
Fourth Dynasty are far from obvious in origin, and derived from considerations
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 have
expected some recognition of this fact from Robins and Shute. But I can find no
mention in their articles of the actual dimensions of any pyramids, except for
the base of 150 cubits and the height of 100 cubits which were used in some
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 been
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 calculations
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 built.
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 instance
where a sloping surface had to be constructed. There was always a more direct
measurement of slope available to the builders, defined simply by the ratio
between height and base; and the question is whether the slopes of pyramids
were originally understood in this form or as the more abstract seked
Since every seked value can be expressed as
an equivalent ratio between height and base, this question might appear to be
merely one of definition. Robins and Shute, however, have taken the problem a
stage further, by asserting: "the general conclusion that the pyramid
architects determined slope consistently by one rule only, which involved a
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 of the
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 several
pyramids of the Old Kingdom. What was the significance of this value to the
pyramid-builders? Robins and Shute suggest that it was used in one pyramid of
the late Vth Dynasty and in four pyramids of the VIth Dynasty, to give the "neat
combination of a base of 150 royal cubits and a height of 100 royal cubits."
Could it not be, then, that the architects conceived the dimensions in precisely
these simple terms? Since the ratio of height to base was in this case just
2:3, the ratio between height and semi-base was 4:3, and the slope of the
pyramid-casing represented the hypotenuse of a Pythagorean 3,4,5 triangle; and
as Robins and Shute have again pointed out, this result could have provided
"a convenient basis for set-squares used by the stonemasons."
since the casing could be constructed with 4 parts rise on 3 parts base, what
reason had the architect to convert the slope into the seked of
5 palms 1 finger? For this measurement had neither practical utility nor
numerical significance. The fact that the slopes of some pyramids can today be
expressed as seked values involving palms and fingers, is no
proof that the slopes were conceived in those terms when the pyramids were
By insisting upon seked values in whole numbers
of palms and fingers, moreover, Robins and Shute needlessly exclude some
slopes which have every reason to be considered for the sake of accuracy or
simplicity, but which cannot be expressed as a seked in palms
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 than a
century ago by Flinders Petrie, and the results of the survey recently
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 the
lower slope was 10 rise on 7 base, in close agreement with the observed lower
casing-angle of about 55°; but this is more than half a degree steeper than
the slope required by the seked of 5.
But now Robins
and Shute have a problem, since this slope of 10 rise on 7 base cannot be
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 had no
reason to seek out this result. For they could very easily have controlled the
slope by taking a vertical rise of 10 palms for each cubit measured
It so happens that a seked of 5, or
slope of 7 rise on 5 base, can be ascribed to some upper parts of the lower
slope of the Bent Pyramid, though opinions differ as to the cause of the
associated convexity of the faces; and Dorner discounts the upper parts of the
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 pyramid
of the Fourth Dynasty, wished to glean some knowledge of its structure. Because
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 obtained
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 some cases
have shown that the builders had combined a base of 150 cubits with a height of
100 cubits. For the Second Pyramid of Giza, however, the design would not
have been obvious, since the base was constrained by the requirements of the
Giza site plan to the value of 411 cubits. It was for this reason, I think,
that the 7:11 height-to-base ratio of the Great Pyramid - as reflected in the
height of that pyramid of 280 cubits and the base of 440 cubits - gave way to
the ratio of 2:3; since for the base of 411 or 3 × 137 cubits, the height
became just 2 × 137 or 274 cubits, and the dimensions were defined with
the greatest possible simplicity.
The 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 square
roots of 2 and 3, which are represented by the overall dimensions of the plan
of 1000Ö2 and 1000Ö3
cubits, could have been calculated. It is quite possible, however, that
these dimensions originated in the geometrical development of the site plan, and
it is by no means necessary to assume that the architect evolved square roots as
abstract mathematical quantities. The square root of two could have emerged
simply from the relationship between the side of a square and its diagonal - a
significant possibility because, of course, every pyramid had to be laid out
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:Ö2 relationship could have been found by direct
measurement, and this is in principle exactly the method now proposed by Robins
and Shute. Whilst they begin with a dimension of 1 cubit, however, I take the
initial dimension to be 10 cubits; and by thus making the divisions of the
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 be
almost exactly 99 palms; and reducing the result to unit fractions in the usual
Egyptian manner, we obtain the following value for the square root of two:
99/70 = 1 + 1/5 + 1/7 + 1/14 = 1.414285...
It may be debated whether the architect needed to carry out
this reduction, since the numbers 70 and 99 provided him with a method of
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 dimension of
280 cubits which determined the height of the Great Pyramid, the semi-width of
the enclosure of the Bent Pyramid, and several other dimensions which figure
prominently in the design of the Fourth-Dynasty pyramids. Any divisions evolved
in this formative square, moreover, could have been translated into actual
dimensions by using a scale of one cubit for each finger. By laying off the
semi-diagonal of the square of 2 × 99 or 198 fingers along one side, for
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 the Bent
As I have previously shown, this last result can be
combined with the theoretical lower casing-angle of the
Bent Pyramid first proposed by Lauer - that is
Ö2 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 have been
controlled with the use of a set-square made simply by transferring the diagonal
of a square frame to one side. As we have now seen, an approximation to this
slope is given by the ratio 99:70, which can be simplified with some loss of
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 west sides
of the casing, the builders seem to have used the ratio of 10:7 for the lower
part of the lower slope, and to have corrected for the imbalance by using an
average slope of 7:5 for the remainder. This is suggested by Petrie's mean
measures for the lower and upper parts of the lower slope of 55° 1' and 54°
31' respectively, giving an overall mean of 54° 46' which is very
close to the theoretical angle of 54° 44' for
Ö2 rise on 1 base.
At Dahshur, this
combination of slopes finds an interesting parallel in the Subsidiary Pyramid of
the Bent Pyramid, and in the Northern Stone Pyramid, both of which have a
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 in 7
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 any case
about half a degree in error.
If measurements involving the slant
height are now considered for the lower slope of the Bent Pyramid, we are at
once presented with the next problem posed by Robins and Shute, which is to
estimate a value for the square root of three. The lower slant height of the
Bent Pyramid represents in concrete form the hypotenuse of a right triangle with
Ö3 - the base being 1 and the vertical side
being Ö2. As I have shown elsewhere, this
slant height is equal to the upper vertical height, so that the total vertical
height of 200 cubits is divided in the ratio of Ö2:Ö3, or into parts of 89.9 and 110.1 cubits. In the
Giza site plan, the same proportion explains the major division of a dimension
of 2000 cubits into parts of 899 and 1101 cubits at the south side of the Second
Let us now, however, obtain an estimate for the square root
of three starting from an initial square of 10 cubits. By marking off the
diagonal of this square along one side, we can construct a rectangle which
measures 10 cubits by 10Ö2 cubits. The
diagonal length will be 10Ö3 cubits, and
measuring this diagonal to the nearest finger will yield a result of 17 cubits 2
palms 1 finger, or 485 fingers. Hence we have a value for the square root of
three of 485/280, or 97/56, which is correct to 1 part in 18,800. Reducing
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 of 2Ö3
cubits can be expressed as a sum of cubits, palms and fingers, with almost
negligible error. In the Giza site plan, the dimension of 1000Ö3 cubits would be expressed as 1732 cubits 1 palm,
which is not far removed from the theoretical number of 1732.05... cubits.
noted possible references to the approximation to the square root of two of
99/70, some indication of the above estimate for the square root of three might
also be expected; and indeed this is to be found in certain dimensions of the
Giza site plan which I determined long ago, but had not considered in this
context. Numbers in the ratio 97:56 can be seen in the distance of (970 + 560)
or 1530 cubits southwards from the north side of the Great Pyramid to the north
side of the Third Pyramid - this being divided at the north side of the Second
Pyramid into a northern part of (440 + 250) or 690 cubits, with a remainder of
840 cubits. The latter dimension is naturally divided into three parts each of
280 cubits, as I have previously noted, so that a division of the distance
of 1530 cubits into parts of 970 and 560 cubits occurs at a point 280 cubits
southwards from the north side of the Second Pyramid.
point does not appear to be marked by any known constructional feature, it does
indicate a geometrical development in which all the dimensions of the site plan
fall out, as it were, at a stroke. This construction begins with a square of
side (440 + 560) or 1000 cubits, and yields the round-figure side of the Second
Pyramid as the difference of (970 - 560) or 410 cubits, and the side of the
Third Pyramid as about (1732 - 970 - 560) or 202 cubits.
the occurrence of the square roots of two and three in the Giza site plan with
the minimum of mathematical elaboration, there remains a great question as to
how these numbers were really understood by the pyramid architects. Did they
emerge merely as the result of geometrical experimentation, or did the
architects understand the properties of right-angled triangles, and express
this knowledge through the values of the dimensions they employed? As Robins
and Shute have observed, the contents of secular teaching texts such as the
Rhind papyrus should not be assumed to represent the full extent of the ancient
Egyptians' knowledge of mathematics; but equally, it would be wrong to assume
that the methods employed by perhaps only a handful of individuals during the
Fourth Dynasty, were common knowledge in the later periods of Egyptian history.
A Note on the Meydum Pyramid
the dimensions of the Meydum Pyramid have been discussed by P. Testa in a recent
article in DE, with conclusions differing in some respects
from my own, it seems appropriate to explain here how these differences
Firstly, it must be emphasised that Testa never states the
actual measurements upon which he bases his conclusions, but only gives the
theoretical dimensions in cubits according to his own interpretations. These in
turn depend upon an imagined variation in the length of the cubit used in
different parts of the pyramid, of nearly three centimetres. In my own work on
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 employ a
constant value to convert a given set of measurements into cubits.
to Testa, the sides of the Meydum Pyramid measure 280 cubits, this being the
approximate length suggested by Lauer. But as Maragioglio and Rinaldi point
out, this dimension is "the length of the foundations protruding for a
short way from the base of the casing", and therefore not the actual
base of the pyramid at all. Following Petrie, the sides of the base actually
measure 275 cubits, for a length of cubit just 0.7 millimetre longer than the
'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 indeed the
original height established in Petrie's survey, he asserts that the profile of
the sides is 175 rise on 140 base, or 5 rise on 4 base. The theoretical angle
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.
John A.R. Legon
1. J.A.R. Legon, DE 17 (1990), 15-22.
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), 107-122, 112.
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), 65-72.
17. J.A.R. Legon, DE 10 (1988), 34-40; 36, Table I.
19. J.A.R. Legon, DE 14 (1989), 53-60; 59.
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 mystère
des pyramides (Paris, 1974), 306, 342.
23. Petrie, op.cit., 30.
Ibid., 27, 32; V. Maragioglio and C.A. Rinaldi, L'Architettura delle
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), 55-69.
29. Legon, op.cit.
30. Maragioglio and Rinaldi, op.cit., 16.
31. See Legon,
op.cit. (n.1), 19-20.