**ON PYRAMID DIMENSIONS AND PROPORTIONS**

John A.R. Legon

Reproduced from *Discussions in Egyptology* 20 (1991),
25-34

In my article on the design of the Meydum
Pyramid in a recent issue of I expressed certain views which to judge from the response of Robins and Shute in the following issue,[2] 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 SekedRobins 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: [3] "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 , 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.[4] 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.sekedThis is not how Robins and Shute see the problem, however, for they have expressed the view that:[5] "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 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:[6] "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
as such."sekedI 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. It 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 , I suggested that the use of the DE
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.[7] sekedNow although the 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
values. sekedSince every 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:[8] "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 seked 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 sekeds
of 5 1/25 palms.[9] The impractical nature of this result shows the use of the
seked as a theoretical concept divorced from the realities of
actual measurement.sekedBut let us now consider the
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."[10]
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,[11] this result could have provided
"a convenient basis for set-squares used by the stonemasons."sekedBut 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 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
built.sekedBy insisting upon 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
palms.sekedTo 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 of 5."[12] They base this conclusion on the
theoretical angle listed by Baines and Malek in their popular reference
work,[13] but at the same time overlook the survey-data published more than a
century ago by Flinders Petrie,[14] and the results of the survey recently
published by Josef Dorner in seked.[15] They make no mention of
the article in MDAIK last year, in which I discussed the results of
these surveys in some detail.[16] 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 GM of 5.sekedBut now Robins and Shute have a problem, since this slope of 10 rise on 7 base cannot be expressed as a 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
horizontally.sekedIt so happens that a 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.sekedLet us 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 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!"sekedThis 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.[17] 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 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. 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 a length Ö3 - the base being 1 and the vertical side being Ö2. As I have shown elsewhere,[26] 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 Pyramid. 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. A Note on the Meydum Pyramid 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. According 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",[30] and therefore not the actual base of the pyramid at all. Following Petrie,[31] 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.
2. G. Robins and C.C.D. Shute, 18 (1990), 43-53. DE3. G. Robins and C.C.D. Shute, 16 (1990), 75-80; 75. DE4. 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, 57 (1982), 49-54, 53. See also op.cit. (n.5), 109.GM9. Peet, (op.cit., 98), treated this value as a practical measure. 10. Robins 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,
42 (1986), 43-58. MDAIK16. J.A.R. Legon, 116 (1990), 65-72.
GM17. J.A.R. Legon, 10 (1988), 34-40; 36, Table I. DE18. Ibid., 37. 19. J.A.R. Legon, 14 (1989), 53-60; 59.
DE20. J.A.R. Legon, 12 (1988), 43, fig.1; G.M. 108 (1989),
59.
DE21. 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. 24. 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, 18 (1990), 55-69. DE29. Legon, op.cit. (n.1). 30. Maragioglio and Rinaldi, op.cit., 16. 31. See Legon, op.cit. (n.1), 19-20. |